strong radiative shocks relevant for stellar environments
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Strong radiative shocks relevant for stellarenvironments experimental study and numerical
approachRaj Laxmi Singh
To cite this versionRaj Laxmi Singh Strong radiative shocks relevant for stellar environments experimental study andnumerical approach Physics [physics] Universiteacute Pierre et Marie Curie - Paris VI 2017 EnglishNNT 2017PA066092 tel-01722700
THESE DE DOCTORAT
SORBONNE UNIVERSITE - PIERRE ET MARIE CURIE
Eacute13ole do13torale Physique en Ile-de-Fran13e
Preacutepareacutee
au Laboratoire de Physique des Plasmas
et au Laboratoire dEacutetude du Rayonnement et de la Matiegravere en
Astrophysique et Atmosphegraveres
Preacutesenteacutee par
Raj Laxmi SINGH
Sujet de la thegravese
Strong radiative sho13ks relevant for stellar environments
experimental study and numeri13al approa13h
Soutenan13e le 02 Mars 2017
agrave Universiteacute Pierre et Marie Curie devant un jury 13omposeacute de
Mme FALK Katerina Senior S13ientist Rapporteur
ELI Beamlines Cze13h Republi13
Mme TURCK-CHIEZE Sylvaine Dire13tri13e de Re13her13hes Rapporteur
honoraire du CEA
M GONZALEZ Matthias Maicirctre de 13onfeacuteren13es Examinateur
U Denis Diderot
Mme RICONDA Caterina Professeur UPMC Examinateur
M LAROUR Jean Chargeacute de Re13her13he CNRS Dire13teur de thegravese
Mme STEHLEacute Chantal Dire13tri13e de Re13her13he CNRS (Inviteacutee) Co-dire13tri13e de thegravese
Ex13ept where otherwise noted this work is li13ensed under
http13reative13ommonsorgli13ensesby-n13-nd30
ACKNOWLEDGEMENTS
First and foremost I would like pay my sin13ere gratitude to Dr Jean La-
rour and Dr Chantal Stehleacute my thesis supervisors who have introdu13ed
me to the marvelous world of Laboratory Astrophysi13s I would express my
thanks for their 13ontinuous support during my PhD journey for en13ouraging
me to learn 13hallenging things Working under their guidan13e was a unique
experien13e and a great joy I am parti13ularly grateful for their serenity and
patien13e that helped me in developing my spa13e and style in resear13h Co-
ming all the way from India to a non-English speaking 13ommunity had been
presented by them to me like a se13ond home through their benign love and
ae13tion
I have imparted a signi13ant time during my PhD for 13ondu13ting expe-
riments in PALS laser fa13ility at Prague through LASERLAB a1313ess Being
newbie in su13h a large s13ale experimental fa13ility has never been a problem
for whi13h I am thankful to all the team members for their help and guidan13e
from Paris PALS Prague and Imperial College London Spe13ial thanks to
Dr M Kozlova Dr M Krus Dr J Nejdl Dr J Dostal and all other
sta members at the PALS laser fa13ility and the target fabri13ation to Mr
P Barrasso and team at the Observatoire de Paris without whom none of
these experiments 13ould have been possible Spe13ial a13knowledgement to
Dr F Suzuki Vidal Mr T Clayson P Barroso and Dr U Chaulagain for
sharing their insight and explanations during the experiments
I espe13ially appre13iate the 13onstru13tive help and 13omments of Dr F
Suzuki Vidal Dr M Kozlova Dr M Cotelo Dr R Rodriacuteguez and Dr U
Chaulagain in the data analysis arti13le and thesis writing
I am grateful to all the jury members of my PhD 13ommittee Dr Sylvaine
Tur13k-Chieze Dr Katedegina Falk Dr Caterina Ri13onda Dr Matthias Gon-
zalez Dr Chantal Stehleacute and Dr Jean Larour for taking time to attend
my PhD defense Thanking very mu13h to Dr Tur13k-Chieze and Dr Falk for
reviewing my thesis and providing valuable 13omments and feedba13k whi13h
have surely improved my thesis
I would like to express sin13ere thanks to the dire13tors of LPP Dr P
Chabert and of LERMA Dr D Lis for allowing me to work on the thesis
in su13h a fabulous university ambien13e It would be unjust if I do not men-
tion the administrative help re13eived from Dr D Zahorski former Proje13t
13hief of PlasPar as well as Dr X Fresquet the present proje13t 13hief of
PlasPar I am grateful to PlasPar Labex for funding my PhD studies in
Fran13e
Next I would like to thank Dr U Chaulagain for helping me in Paris
during my stay from the very rst day I am also grateful to Dr L de Sagrave
Dr L Ibgui Dr M Drouin Dr A Stan Mr B Khiar Mr L Ni13olas
Dr J Freundli13h for their help and support during my stay in Fran13e
My thanks is due to my friends for 13reating a homely atmosphere during
my thesis in Fran13e In parti13ular I would like to thank Dr R Mourya
and Mr A Ranjan for their 13ontinuous support in the administrative for-
malities and others whi13h helped me a lot to 13on13entrate and speed up my
thesis writing I would also like to mention the names of my friends Miss T
Bhowmik Miss V Shaw and Dr A Gupta for their support
Good 13ompany plays a very important role in the life In this I am par-
ti13ularly thankful to Dr S C Tripathi Dr J Pal Dr R Kumar Dr M
Dave Dr A Bharadwaj Mr K Singh Miss S Mishra and Miss S Pandey
for helping me in keeping pa13e with resear13h and daily life developments I
am indebted to my brother Dr A Awasthi for his sin13ere guidan13e en13oura-
gement and 13ooperation during my entire do13toral study Also I would like
to take this opportunity to thank my 13hildhood friend Miss R Yadav for her
love and en13ouragement during my thesis
Finally last but not the least I express my whole hearted gratitude to the
support I re13eived in the form of love and ae13tion from my grandparents
parents sisters brothers and my other family members This is something
beyond a13knowledgements They were always with me and the en13ourage-
ment I re13eived from them will never fade away
Raj Laxmi Singh
Strong radiative sho13ks relevant for stellar
environments experimental study and numeri13al
approa13h
ABSTRACT
Strong sho13ks are present in various astrophysi13al phenomena Su13h
sho13ks are strongly inuen13ed by the radiation through its 13oupling with
hydrodynami13s Thus their topology and dynami13s are quite 13omplex Ge-
nerating su13h hypersoni13 sho13ks in the laboratory with 13ontrolled 13ondi-
tions is thus an adequate tool to study the inuen13e of radiation and to
13ompare them with numeri13al simulations Su13h sho13ks 13an be generated
by intense lasers and ele13tromagneti13 devi13es The rst part of this dis-
sertation 13on13erns the numeri13al and experimental study of the intera13tion
of two 13ounter propagating laser-driven sho13ks The experiments perfor-
med at the kJ PALS laser fa13ility allowed to generate sho13ks with dierent
speeds (sim 30-55 kms and 10-25 kms) in noble gases and low pressure
(less than 1 bar) Several diagnosti13s were implemented visible interfero-
metry time- and spa13e-resolved visible spe13tros13opy and time integrated
XUV spe13tros13opy Our experiment shows a strong intera13tion of one radi-
ative pre13ursor onto the se13ond one The physi13al parameters of the plasma
were dedu13ed from the diagnosti13s and 13ompared with 1-D simulation re-
sults The se13ond part is devoted to the design of an experiment where the
sho13k is generated ele13tromagneti13ally The optimization of this generator is
presented and also the full experimental set up whi13h allows studying sho13k
sim 30 kms in noble gas at sim 1 mbar
Keywords Radiative sho13k hydrodynami13s laser-plasmas visible
and XUV spe13tros13opy
numeri13al simulation laboratory astrophysi13s high energy density
physi13s
Cho13s forts et radiatifs dinteacuterecirct pour les
environnements stellaires eacutetude expeacuterimentale et
appro13he numeacuterique
RESUME
Les 13ho13s forts sont preacutesents dans des pheacutenomegravenes astrophysiques varieacutes
De tels 13ho13s sont fortement inuen13eacutes par le rayonnement par son 13ouplage
ave13 lhydrodynamique Par suite leur topologie et leur dynamique sont
assez 13omplexes Geacuteneacuterer de tels 13ho13s hypersoniques en laboratoire dans
des 13onditions 13ontrleacutees est ainsi un outil pertinent pour eacutetudier linuen13e
du rayonnement et pour 13omparer aux reacutesultats des simulations numeacuteriques
Ces 13ho13s sont geacuteneacutereacutes par des lasers intenses et par des moyens eacutele13tromag-
neacutetiques La premiegravere partie du texte est 13onsa13reacute agrave leacutetude numeacuterique et
expeacuterimentale de lintera13tion de deux 13ho13s induits par laser se propageant
en sens 13ontraires Les expeacuterien13es ont eacuteteacute meneacutees sur linstallation laser kJ
PALS qui permet de former deux 13ho13s ave13 des vitesses propres dieacuteren-
tes (sim 30-55 et 10-25 kms respe13tivement) dans des gaz rares agrave pression
faible (moins de 1 bar) Des diagnosti13s ont eacuteteacute installeacutes interfeacuteromeacutetrie
visible spe13tros13opie visible agrave reacutesolution spatiale et temporelle spe13tros13o-
pie XUV inteacutegreacutee en temps Nos expeacuterien13es montrent une forte intera13tion
entre les deux preacute13urseurs radiatifs Les paramegravetres physiques du plasma
ont eacuteteacute deacuteduits de 13es diagnosti13s et 13ompareacutes aux reacutesultats de simulations
monodimensionnelles La se13onde partie est 13onsa13reacutee agrave la 13on13eption dune
expeacuterien13e ougrave le 13ho13 est geacuteneacutereacute de faccedilon eacutele13tromagneacutetique Loptimisation
de 13e geacuteneacuterateur est preacutesenteacutee ainsi que lenvironnement expeacuterimental per-
mettant deacutetudier des 13ho13s jusquagrave 30 kms dans des gaz rares peu denses
(1 mbar)
Mots 13leacutes 13ho13s radiatifs plasmas laser hydrodynamique spe13tros13opie
visible et XUV simulation numeacuterique astrophysique de laboratoire
physique agrave haute densiteacute deacutenergie
Reacutesumeacute long
En introdu13tion (13hapitre 1) les 13ho13s radiatifs (CR) sont preacutesenteacutes
dans de nombreux environnements astrophysiques en parti13ulier dans les
eacutetoiles Ce sont des 13ho13s forts ave13 un nombre de Ma13h eacuteleveacute (M1) et une
tregraves haute tempeacuterature 13e qui induit un rayonnement intense Alors que
lobservation de 13ette signature ave13 une reacutesolution spatiale est tregraves di13ile
en astrophysique une appro13he innovante 13elle dite des plasmas astrophy-
siques de laboratoire fournit un puissant moyen deacutetude des CR sur Terre
Depuis plus dune deacute13ennie 13es 13ho13s sont eacutetudieacutes en laboratoire prin13ipale-
ment sur de grandes installations laser dans les gaz rares et sous dieacuterentes
geacuteomeacutetries Lintera13tion entre un laser et une feuille min13e produit une
forte ablation et par eet fuseacutee a1313eacutelegravere la feuille qui se propage dans le
gaz agrave plusieurs dizaines de kms Pour des irradiations laser entre 10
14and
10
15W13m
2 on enregistre des vitesses de 13ho13s entre 40 et 150 kms En
parallegravele une des13ription ne est permise par des simulations numeacuteriques
Au deacutebut de 13e travail les expeacuterien13es eacutetaient 13on13entreacutees sur des CR
simples alors que la situation astrophysique est 13elle dun CR interagissant
ave13 un milieu plus dense donnant des 13ho13s reacuteeacute13his et transmis Des
exemples repreacutesentatifs de 13es pheacutenomegravenes sont la propagation de restes de
supernovae dans des nuages moleacute13ulaires denses les 13ho13s da1313reacutetion sur
la photosphegravere des eacutetoiles T-Tauri ou en13ore les 13ho13s deacutetrave agrave lavant
des jets stellaires La 13ollision (ou lintera13tion) de deux CR est bien sucircr un
eacutevegravenement rare en astrophysique mais un exemple est fourni par lintera13tion
de deacutebris de la supernova DEM L316 (13f Fig 1 de Williams et al (1997))
mecircme si lhypothegravese est 13ontesteacutee (Velarde et al (2006))
Il y a don13 une forte motivation pour reacutealiser une expeacuterien13e de labora-
toire pour eacutetudier la propagation et lintera13tion de deux 13ho13s fa13e agrave fa13e
et interpreacuteter les signatures observeacutees Cette appro13he 13onstitue une grande
part de mon travail En 13ompleacutement aux expeacuterien13es laser jai eacutetudieacute un
geacuteneacuterateur 13ompa13t de puissan13e pulseacutee alimentant un 13anon agrave plasma 13oax-
ial pour 13reacuteer des 13ho13s dinteacuterecirct astrophysique dans les gaz agrave basse pression
Les 13ho13s ainsi 13reacuteeacutes sont de plus grande taille et peuvent ecirctre eacutetudieacutes plus
fa13ilement (Kondo et al 2008) Pour 13ette thegravese je me suis atta13heacutee agrave preacute-
senter les 13ho13s radiatifs dans le 13ontexte de lastrophysique de laboratoire
en suivant 13es deux appro13hes
Le deuxiegraveme 13hapitre traite de la physique des CR Ils 13omposent
une 13lasse de 13ho13s supersoniques qui sont 13haueacutes agrave haute tempeacuterature
et sont par 13onseacutequent sour13es dun rayonnement intense En retour 13e
rayonnement modie la dynamique et la stru13ture du 13ho13 13e qui 13omplexie
la des13ription Les CR se ren13ontrent dans des situations astrophysiques
13omme les 13ho13s da1313reacutetion lors de la formation des protoeacutetoiles (Stahler
Palla et Salpeter 1986) lexplosion de supernovae et lintera13tion de leur
restes ave13 le milieu interstellaire dense (Chevalier 1977) ou en13ore les 13ho13s
deacutetrave en tecircte des jets stellaires (Hartigan et al 2001) Les CR sont
maintenant eacutetudieacutes en laboratoire 13e qui permet de 13omparer les reacutesultats
aux modegraveles de la litteacuterature et de veacuterier la possibiliteacute de les retrouver ave13
les simulations numeacuteriques disponibles (Bouquet et al (2004) Leygna13 et
al (2006)) Leacutetude expeacuterimentale des CR est 13ru13iale notamment pour
13omprendre les pro13essus eacutenergeacutetiques au sein des plasmas astrophysiques
Dans le 13as des gaz parfaits le saut au niveau dun 13ho13 est deacute13rit par les
eacutequations de Rankine-Hugoniot qui relient les quantiteacutes thermodynamiques
de part et dautre de la dis13ontinuiteacute La solution monodimensionnelle dun
13ho13 se propageant agrave la vitesse us dans un gaz au repos est usuellement
deacute13rite dans le repegravere lieacute au 13ho13 et on distingue la reacutegion amont (ou preacute-
13ho13) agrave la vitesse u1= - us et en arriegravere du 13ho13 (reacutegion post-13ho13 ou aval)
la vitesse est u2 selon la Fig 21 Pour les 13ho13s forts ougrave le nombre de Ma13h
M est tregraves grand devant 1 les sauts des quantiteacutes thermodynamiques dun
gaz parfait sont donneacutes par les eacuteq 21 agrave 24 Pour un gaz monoatomique
le rapport de 13ompression est de 4 et la tempeacuterature de la zone post-13ho13
est proportionnelle au 13arreacute de la vitesse du 13ho13 et agrave la masse atomique
La situation est plus 13omplexe pour un gaz reacuteel 13ar une part de leacutenergie
13ineacutetique sert agrave ex13iter et agrave ioniser le gaz du post 13ho13 La tempeacuterature
attendue est don13 infeacuterieure agrave 13elle du 13as ideacuteal en outre la pression est
modieacutee par lionisation (Mi13haut et al 2004) Un nouveau jeu deacutequations
in13orpore la 13harge ee13tive Z des ions (eacuteq 25 agrave 28) Ces modegraveles ignorent
le 13hauage et le refroidissement radiatifs alors que le rayonnement eacutemis
ae13te la stru13ture dun 13ho13 fort Cette des13ription demande dajouter aux
eacutequations de 13onservation de la masse quantiteacute de mouvement et eacutenergie les
13ontributions radiatives (ux eacutenergie et pression Le 13as est plus 13omplexe
et les eacutequations dhydrodynamique radiative ont eacuteteacute exprimeacutees dans le repegravere
du 13ho13 (Mihalas and Mihalas 1999 Drake 2006) selon les eacuteq 29 agrave 211
Les 13ontributions de la pression radiative et du 13hamp de rayonnement
sont importantes agrave tregraves haute tempeacuterature (vitesse) mais deacutejagrave pour nos 13ho13s
le ux radiatif est dominant Selon lopa13iteacute le rayonnement du 13ho13 peut
ecirctre absorbeacute par la region preacute-13ho13 induisant un 13hauage loin en avant de
la dis13ontinuiteacute Egalement la zone 13hoqueacutee pro13he de la dis13ontinuiteacute est
ae13teacutee par la 13ompeacutetition entre leacutemission (refroidissement) et labsorption
(13hauage) Ainsi leacutepaisseur optique lo13ale (eacuteq 211) devient le paramegravetre
pertinent Comme on raisonne sur le ux radiatif moyenneacute en freacutequen13e
13est lopa13iteacute de Rosseland qui est utiliseacutee et on dieacuteren13ie les reacutegimes op-
tiquement eacutepais et optiquement min13e 13e qui permet de 13lasser les 13ho13s
radiatifs
Notre inteacuterecirct sest 13on13entreacute sur les 13ho13s preacutesentant un preacute13urseur radi-
atif En laboratoire une intensiteacute laser au dessus de 10
14W13m
2le permet
et les installations sont rappeleacutees dans le tableau 21 Il est aussi possible de
lan13er des 13ho13s moins rapides don13 peu radiatifs (sim 10-30 kms) ave13 un
piston eacutele13tromagneacutetique
Le 13hapitre 3 est 13onsa13reacute agrave lanalyse numeacuterique des CR en deacute13rivant
les pro13essus physiques jouant un rle dans 13es systegravemes physiques De mecircme
que les observations les expeacuterien13es en laboratoire ne sont pas dire13tement
interpreacutetables et la simulation numeacuterique devient un outil preacute13ieux Diverses
appro13hes numeacuteriques sont utiliseacutees pour simuler les plasmas de laboratoire
On 13onsidegravere soit des grilles xes soit des grilles adaptables (Adaptative
Mesh Renement) pour maintenir une reacutesolution susante sur 13haque zone
du plasma Une autre appro13he utilise des grilles qui suivent le uide dans
son mouvement (modegravele Lagrangien) don13 sans passage de matiegravere dune
13ellule agrave la voisine Les eacutequations sont alors dieacuterentes (Orban et al 2013)
Pour deacute13rire les 13ollisions de 13ho13s jai utiliseacute le 13ode 13ommer13ial HE-
LIOS un 13ode Lagrangien mono-dimensionnel pouvant geacuterer le rayonnement
et lhydrodynamique (Ma13Farlane Golovkin et Woodru 2006) Assez fa13ile
agrave prendre en main il a le gros avantage de pouvoir simuler notre 13ongura-
tion deux 13ho13s de vitesses opposeacutees lan13eacutes par deux lasers HELIOS deacute13rit
les plasmas hors deacutequilibre thermodynamique (hors ETL) gracirc13e agrave un mo-
degravele 13ollisionnel-radiatif (HELIOS-CR) de 13al13ul des populations atomiques
agrave 13haque pas de simulation hydrodynamique De fait jai utiliseacute HELIOS
dans une approximation ETL qui 13orrespond aux 13onditions expeacuterimentales
(Rodriguez et al 2011) et 13ela est dis13uteacute en deacutetails
Le 13ode reacutesout les eacutequations uides en introduisant les pressions des eacutele13-
trons des ions et du rayonnement Ele13trons et ions sont deacute13rits par deux
uides en intera13tion de tempeacuteratures respe13tives Te et Ti La 13ondu13tion
thermique est geacutereacutee par 13elle des eacutele13trons agrave partir de la 13ondu13tiviteacute de
Spitzer et le deacutept deacutenergie laser par un modegravele de Bremsstrahlung inverse
Leacutemission et labsorption du rayonnement sont introduites dans leacutequation
deacutenergie des eacutele13trons et dans 13elles du transport radiatif Une meacutethode
utilise un modegravele de diusion du rayonnement multi-groupes et agrave ux li-
miteacute ougrave le ux radiatif est proportionnel au gradient de leacutenergie radiative
et inversement proportionnel agrave lopa13iteacute de Rosseland ave13 une pondeacuteration
pour obtenir une bonne des13ription du 13as optiquement min13e selon Olson
Auer et Hall (2000) Une se13onde meacutethode utilise un s13heacutema deacutependant du
temps agrave 13ourte distan13e et multi-angles Dans notre 13as ougrave rayonnement
et hydrodynamique sont fortement 13oupleacutes et aussi en variation rapide on
utilise le modegravele de diusion et les opa13iteacutes ETL multi-groupes de Plan13k et
Rosseland
En outre notre version dHELIOS utilise PROPACEOS (Ma13Farlane
Golovkin et Woodru 2006) une table deacutequation deacutetat et dopa13iteacute multi-
groupes geacuteneacutereacutee par le 13ode sans possibiliteacute dintroduire une autre table La
13onnaissan13e des 13onditions thermodynamiques est neacute13essaire pour la simu-
lation du plasma dans nos 13as masse volumique entre 10
minus4et 10
minus1g13m
3
tempeacuterature entre 0 et 50 eV Une eacutevaluation preacute13ise par Rodriguez et al
(13f Fig 2 de Rodriguez et al 2011) pour le Xeacutenon indique que les 13onditi-
ons thermodynamiques de nos CR 13orrespondent au reacutegime ETL Leacutequation
deacutetat relie pression ionisation et eacutenergie interne agrave la masse volumique et agrave
la tempeacuterature Notre version dHELIOS pour plasma ETL utilise PROPA-
CEOS qui est peu do13umenteacutee elle utilise le modegravele de More et al (1988)
en reacutegime de 13ouplage fort (haute densiteacute et basse tempeacuterature) et un modegravele
datome isoleacute pour un 13ouplage faible Des interpolations sont in13luses pour
passer au 13ouplage fort (httpwwwprism-13s13omSoftwarePROPACEOS)
Jai reacutealiseacute des simulations HELIOS 1D pour divers 13as de CR 13ho13 seul
ou 13ho13s en 13ollision qui sont deacute13rites dans 13e 13hapitre 3 Les CR se propa-
geant dans le Xe agrave 01 bar et agrave 50 kms sont 13ara13teacuteriseacutes par un preacute13urseur
radiatif eacutetendu Une 13ompression eacuteleveacutee (38) dans le post-13ho13 est attribueacutee
agrave lionisation du gaz et au refroidissement radiatif Les tempeacuteratures du
post-13ho13 et du preacute-13ho13 de part et dautre du pi13 sont identiques signe
dun 13ho13 super13ritique Le rle spe13ta13ulaire de lopa13iteacute a eacuteteacute remarqueacute
mais agrave 13onsideacuterer lin13ertitude sur lopa13iteacute du Xeacutenon et le traitement 1D
jai deacute13ideacute de ne pas 13her13her agrave raner la simulation en termes de nombre
de groupes
Leacutetude numeacuterique permet de retrouver les 13ara13teacuteristiques de lintera13tion
de deux 13ho13s opposeacutes ave13 des vitesses respe13tives 50-50 kms et 50-20 kms
Dans tous les 13as la prin13ipale signature de lintera13tion (don13 avant la 13ol-
lision) est le re13ouvrement des preacute13urseurs agrave t0+8 ns pour 50-50 kms et
agrave t0+15 ns pour 50-20 kms Ensuite on assiste agrave une remonteacutee reacuteguliegravere
de la densiteacute et de la tempeacuterature eacutele13troniques Linstant de la 13ollision
voit une augmentation soudaine de la densiteacute eacutele13tronique (par un ordre de
grandeur) qui atteint 66 times 10
21et 3 times 10
2113m
minus3respe13tivement alors que
la tempeacuterature monte agrave 39 et 28 eV
Dans le quatriegraveme 13hapitre je preacutesente linstallation laser PALS puis
la 13ellule dintera13tion son implantation et les diagnosti13s asso13ieacutes Rappe-
lons le prin13ipe expeacuterimental Quand une impulsion bregraveve de lumiegravere laser
est fo13aliseacutee sur une feuille min13e une part importante deacutenergie est trans-
feacutereacutee et le 13hauage du solide 13onduit agrave son ablation Un plasma 13oronal
13haud et peu dense est eacuteje13teacute vers larriegravere et un 13ho13 vers lavant se forme
dans la feuille par eet fuseacutee Comme 13ette feuille limite le tube le 13ho13
se transmet au gaz la feuille agissant 13omme un piston Des expeacuterien13es
ont eacuteteacute meneacutees sur le Prague Asterix Laser System (PALS) au printemps
2015 (5 semaines 20 avril - 22 mai) ave13 pour obje13tif la premiegravere eacutetude
de lintera13tion puis de la 13ollision de deux CR sous la13tion de deux lasers
au niveau de 10
14W13m
2 Le Prague Asterix Laser System (PALS) repose
sur un laser agrave iode infrarouge (Asterix IV Jungwirth et al 2001) Apregraves
ampli13ation il deacutelivre jusquagrave 1 kJ en 03 ns agrave la longueur donde fonda-
mentale de 1315 nm Des fais13eaux auxiliaires sont disponibles agrave freacutequen13e
doubleacutee (λ = 657 nm) ou tripleacutee (438nm) PALS deacutelivre deux tirs par heure
agrave haute eacutenergie et dans de bonnes 13onditions dhomogeacuteneacuteiteacute du fais13eau
Pour nos expeacuterien13es nous utilisons deux fais13eaux Le fais13eau fondamental
est diviseacute apregraves le 4egraveme ampli13ateur en deux fais13eaux deacutenergie 60 et
40 (voir Fig 42) Le plus puissant est inje13teacute dans le 5
meampli13ateur et
sa freacutequen13e est tripleacutee Ce fais13eau agrave 438 nm est nommeacute MAIN Le se13ond
fais13eau est utiliseacute sans modi13ation don13 agrave 1315 nm et est nommeacute AUX
Le s13heacutema de distribution est en Fig 42 et les proprieacuteteacutes de MAIN and
AUX laser sont rappeleacutees Tdans le ableau 41
PALS propose deux 13hambres agrave vide spheacuterique et 13ylindrique respe13ti-
vement nous avons utiliseacute la 13hambre spheacuterique (Fig 43) Dun diamegravetre
de 100 13m elle est a1313essible aux fais13eaux MAIN et AUX par deux hublots
de 80 13m et 50 13m Des portes sont ameacutenageacutees ainsi que de nombreux ports
A linteacuterieur la 13ellule et des diagnosti13s sont monteacutes sur une table optique
deacute13oupleacutee meacute13aniquement de la 13hambre Les 13ibles (Fig 45) 13omprennent
un petit tube long de 4 mm fermeacute agrave ses deux extreacutemiteacutes par des feuilles min-
13es de parylegravene-N doreacute de 11 microm sur lesquelles les deux lasers sont fo13aliseacutes
au niveau de 10
14W13m
minus2 Le tube est rempli du gaz dans lequel le 13ho13
se propagera agrave une vitesse attendue entre 30 et 60 kms Les feuilles min13es
assurent la 13onversion de leacutenergie laser en eacutenergie 13ineacutetique via lablation et
la geacuteneacuteration de 13ho13 Des 13ellules speacute13iques en dural massif permettent
lalignement et la mesure du diamegravetre du fais13eau Toutes les 13ellules sont
reacutealiseacutees par le Ple instrumental de lObservatoire de Paris
Les 13ellules agrave gaz sont remplies in situ agrave une fra13tion de bar ave13 Xe
Ar Xe90He10 et He permettant ainsi de bien 13onnaicirctre le milieu ougrave se
propagent les 13ho13s Le systegraveme de remplissage (Fig 410) permet de limiter
leacute13art de pression sur les fenecirctres de la 13ellule et permet de suivre la pression
jusquau moment du tir Cest 13ritique pour ne pas 13asser la fenecirctre ultra
ne de Si3N4 pour la spe13tros13opie XUV et pour geacuterer une fuite eacuteventuelle
Le reacuteglage de la position et de la fo13alisation permet dobtenir des ta13hes
laser 13entreacutees de diamegravetres 450 - 500 microm et 250 - 300 microm pour MAIN et
AUX respe13tivement Ce 13ontrle est reacutepeacuteteacute 13haque jour Les impa13ts sont
toujours suivis par des 13ameacuteras X au keV mecircme si limpa13t sur la feuille de
parylegravene-N donne un signal X faible 13ompareacute agrave 13elui du tir sur Al
Les diagnosti13s du plasma sont prin13ipalement linterfeacuteromeacutetrie visible et
la spe13tros13opie XUV et visible Lalignement est suivi par des 13ameacuteras dans
le visible Pour linterfeacuteromeacutetrie visible preacute13iseacutement on doit prendre une
image de reacutefeacuteren13e (sans franges) de 13haque 13ible (Fig 417a) Le fais13eau de
reacutefeacuteren13e de linterfeacuteromegravetre Ma13h-Zehnder est bloqueacute et le fais13eau sonde est
utiliseacute seul La fente est ensuite reacuteduite agrave 200 microm pour ne sonder que la zone
13entrale du tube mais seacutetendant le long de laxe jusquaux pistons pour avoir
une reacutefeacuteren13e des positions initiales Ensuite en mode interfeacuterogramme on
13reacutee un systegraveme de franges perpendi13ulaires agrave la fente don13 perpendi13ulaires
agrave la dire13tion de propagation du 13ho13 (13f Fig 417b et Fig 413) Le tube
de 13ho13 est imageacute sur la 13ameacutera ave13 un grandissement susant (54 mm) et
il est possible dassurer un balayage eacutele13tronique de la fente sur le deacutete13teur
jusquagrave 200 ns Un interfeacuterogramme dit streak permet ainsi de suivre limpa13t
des 13ho13s sur les franges en fon13tion du temps
Leacutemission XUV du plasma est enregistreacutee ave13 une inteacutegration spatiale
et temporelle sur un spe13trographe agrave 13hamp plan muni dun reacuteseau 13on13ave
Il est installeacute dans la partie haute de la 13hambre (13f Fig 418) et enregistre
leacutemission XUV traversant la fenecirctre de Si3N4 Une spe13tros13opie visible
reacutesolue en temps et en position longitudinale a eacuteteacute installeacutee (Fig 419a)
mais na pas en13ore pu ecirctre exploiteacutee 13omplegravetement
Le 13hapitre 5 preacutesente les reacutesultats des deux diagnosti13s surtout linterfeacute-
romeacutetrie qui a eacuteteacute 13omplegravetement analyseacutee Jai deacuteveloppeacute une pro13eacutedure
de deacutepouillement ave13 des outils numeacuteriques ad ho13 de 13al13ul et danalyse
dimage Certes linterfeacuteromeacutetrie ne permet pas de sonder des plasmas plus
denses que la densiteacute 13ritique imposeacutee par le laser (4 times 10
2113m
minus3agrave 627 nm)
et on 13onstate que le post-13ho13 est opaque Au 13ontraire le rayonnement
XUV est preacutesent dans toute la stru13ture et la spe13tros13opie XUV inteacutegreacutee
en temps et en espa13e permet dexplorer agrave la fois le post-13ho13 et le preacute13ur-
seur Lanalyse des interfeacuterogrammes streak en visible part dune eacutevaluation
de leet de lindi13e de reacutefra13tion des eacutele13trons deacutetailleacutee dans lappendi13e A
mais qui se heurte assez vite agrave une perte de 13ontraste et agrave une disparition
des franges bien en dessous de la valeur de la densiteacute 13ritique deacutejagrave menti-
onneacutee Les deacutephasages a1313essibles ave13 preacute13ision ne deacutepassent guegravere 2 ou 3
fois 2π La re13her13he de maxima des franges est semi automatiseacutee et permet
darriver agrave une densiteacute eacutele13tronique moyenneacutee sur les 600 microm de leacutepaisseur
du tube de 13ho13 Le prol non plan du 13ho13 13onduit agrave penser que la moyenne
sous-estime dun fa13teur 2 voire plus la densiteacute dans le preacute13urseur La dis-
parition des franges est attribueacutee agrave la preacutesen13e du front de 13ho13 Par suite
les interfeacuterogrammes permettent de mesurer des vitesses de 13ho13s de 30-55
et 10-30 kms pour MAIN et AUX respe13tivement ainsi que des densiteacutes
eacutele13tronique dans les preacute13urseurs entre 10
17et 10
1913m
minus3
Nous avons ainsi deacutemontreacute lintera13tion entre deux preacute13urseurs radiatifs
dans Xe at 01 bar pour deux 13ho13s agrave 54 et 23 kms Lintera13tion est
13lairement 13ara13teacuteriseacutee par laugmentation de londe dionisation puis par la
superposition des deux preacute13urseurs agrave t0+20ns La 13ollision est enregistreacutee agrave
t0+47 ns un reacutesultat retrouveacute par la simulation
Nous avons reacutepeacuteteacute mais agrave plus haute pression 02 bar et trouveacute des
vitesses de sim 41 kms pour MAIN et sim 18 kms pour AUX Cependant
nous navons pas de signature du preacute13urseur radiatif pour AUX et le temps de
13ollision na pas eacuteteacute a1313essible agrave lenregistrement Le preacute13urseur 13teacute MAIN
nest pas inuen13eacute par AUX jusquagrave la limite de t0+48 ns (Fig 59(b) et (13))
La simulation preacutedit un tregraves faible preacute13urseur 13teacute AUX et une intera13tion
des preacute13urseurs agrave t0+49 ns
Linterfeacuteromeacutetrie transverse agrave 02 bar ave13 des vitesses sim 40 et 20 kms
indique que le preacute13urseur 13teacute MAIN a une extension lateacuterale de sim 600 microm13ontre 300 microm pour AUX Le preacute13urseur de MAIN est leacutegegraverement ae13teacute
pregraves des parois alors que 13elui de AUX est fortement 13ourbeacute Les eets 2D
sont don13 pronon13eacutes pour AUX et faibles pour MAIN En outre pour le
Xeacutenon nous disposons de la tempeacuterature et de la 13harge moyenne par le
spe13tre XUV inteacutegreacute (tir48143 Xeacutenon 06 bar) On peut en 13on13lure que
la 13harge moyenne est au moins eacutegale agrave 6 et quon a atteint une tempeacuterature
de 15 eV
Nos simulations donnent une des13ription qualitative des CR en inte-
ra13tion agrave la reacuteserve pregraves que nous introduisions dans HELIOS une uen13e
laser ajusteacutee pour donner la bonne vitesse Cependant il est maintenant
admis que les simulations 2D (ave13 les opa13iteacutes 13orre13tes) deacute13rivent bien les
expeacuterien13es (Gonzaacutelez Audit et Stehleacute 2009 Leygna13 et al 2006 Stehleacute
et al 2010) A la mecircme eacutenergie laser le 13al13ul 2D 13onduit agrave diminuer la
vitesse du 13ho13 13ompare au 13al13ul 1D et aussi agrave une baisse de la densiteacute
eacutele13tronique Par exemple pour un 13ho13 lan13eacute agrave PALS par un laser agrave 1315
nm dans le Xeacutenon agrave 03 bar ave13 une uen13e de 85000 J13m
2 ARWEN 2D
donne une vitesse de 44 kms en a1313ord ave13 lexpeacuterien13e (Cotelo et al
2015) La simulation 1D demanderait 30000 J13m
2pour obtenir la mecircme
vitesse
De mecircme la spe13tros13opie XUV inteacutegreacutee agrave 06 bar pour des vitesses reacuteel-
les de sim 39 et 18 kms indique que la tempeacuterature a atteint 15 eV et que
la 13harge moyenne a atteint 6 ou 7 alors que la simulation 1D preacutedit 10-30
eV et 5-10 respe13tivement (Fig 518) On peut en 13on13lure quune eacutetude
deacutetailleacutee baseacutee sur des simulations 2D et un post-traitement du transfert de
rayonnement sont neacute13essaires pour raner lanalyse Pour les autres gaz
rares (Ar Kr) on a observeacute que agrave eacutenergie laser donneacutee la vitesse de CR est
une fon13tion deacute13roissante de la masse volumique Ce13i 13onrme que pour
une densiteacute et une vitesse les eets radiatifs augmentent ave13 le numeacutero
atomique Pour le Krypton on a observeacute un petit preacute13urseur mais au13une
intera13tion Celle-13i ne serait possible quen augmentant nettement la vi-
tesse don13 leacutenergie du laser Ce13i a eacuteteacute rendu possible sur linstallation
laser Orion (AWE Aldermaston G-B) ougrave la 13ollision de deux CR deacutegale
vitesse a eacuteteacute obtenue agrave sim 80 kms pour une uen13e laser sim 6 times 10
14W13m
2
(Clayson et al 2016 Suzuki-Vidal et al 2016) pour une large gamme de
gaz rares et des pressions entre 01 et 1 bar Pour eacutetudier agrave la fois les 13ho13s
et les preacute13urseurs radiatifs de nombreux diagnosti13s eacutetaient installeacutes radi-
ographie X imagerie optique en 13ameacutera agrave balayage de fente interfeacuteromeacutetrie
multi-vues et agrave balayage de fente Bien que je naie pas pu parti13iper aux
13ampagnes jai fait des simulations 1D pour interpreacuteter les reacutesultats Cet
aspe13t nest pas deacute13rit dans 13ette thegravese mais a eacuteteacute publieacute tregraves reacute13emment
(Clayson et al 2017)
Apregraves des 13hapitres 13onsa13reacutes aux 13ho13s radiatifs geacuteneacutereacutes par laser le
13hapitre 6 deacute13rit un moyen alternatif passant par la voie eacutele13trique agrave haute
puissan13e Les 13hires de la voie laser sont tregraves eacuteleveacutes en uen13e plus de
10
14W13m
2 mais repreacutesentent des eacutenergies modeacutereacutees dans limpulsion (120
et 60 J pour MAIN et AUX au PALS) Or un systegraveme eacutele13trique devant
deacutelivrer 100 J reste modeste pour sa partie sto13kage Les vitesses de 50
kms seront probablement di13iles agrave atteindre mais on peut tabler sur un
reacutegime 13ompleacutementaire des 13ho13s radiatifs ave13 plus de 10 kms si le ren-
dement de 13onversion en eacutenergie 13ineacutetique reste 13orre13t Dougrave un neacute13essaire
travail doptimisation Des vitesses jusquagrave 100 kms avaient eacuteteacute mesureacutees
tregraves tt dans des 13anons agrave plasma dhydrogegravene agrave basse pression (Lee 1969)
mais 13e13i a eacuteteacute exploiteacute ensuite pour dautres naliteacutes La litteacuterature ne
mentionne quun 13as de 13anon eacutele13tromagneacutetique 13oaxial deacutedieacute aux 13ho13s
dinteacuterecirct astrophysique (Kondo et al 2006) Il est signaleacute que le lan13eur
eacutele13tromagneacutetique produit des 13ho13s plans de grande dimension transverse
(1 ordre de grandeur par rapport agrave la voie laser) et don13 plus fa13iles agrave in-
vestiguer (Kondo et al 2008) Le prin13ipe de fon13tionnement est le suivant
i) 13reacuteation dun 13laquage haute tension initial sur une surfa13e dieacutele13trique
fa13e agrave du gaz ii) 13hauage ohmique du plasma par le 13ourant en 13roissan13e
rapide iii) a1313eacuteleacuteration de la 13ou13he de plasma sous la pression magneacuteti-
que auto-geacuteneacutereacutee par le 13ourant (dizaines de kA) iv) maintien (1 micros) de la
pression magneacutetique pour 13ommuniquer une impulsion importante Des tra-
vaux sur le Z-pin13h ou le plasma fo13us permettent de postuler des geacuteomeacutetries
13onvenables pour a1313eacuteleacuterer une masse de lordre de 10
minus510
minus4g
Apregraves avoir eacutetudieacute le fon13tionnement dun 13ir13uit eacutele13trique rapide jai
listeacute les points forts et les points faibles en fon13tion des obje13tifs et jai pro-
poseacute une geacuteomeacutetrie 13oaxiale 13ompa13te Loptimisation du 13anon agrave plasma a
eacuteteacute faite en terme de vitesse maximale agrave la sortie ave13 13omme seules 13ontrain-
tes une eacutenergie sto13keacutee de 1 kJ une dimension transverse de 4 mm et une
dimension axiale dun ordre plus grande Lobje13tif eacutetait le 13ho13 fortement
supersonique 10-30 kms agrave quelques mbar dAr ou de Xe Un modegravele simple
(0D) a eacuteteacute 13ompareacute agrave un modegravele 3D MHD par le 13ode GORGON qui deacute13rit
bien les expeacuterien13es de plasmas astrophysique par puissan13e pulseacutee (Ciardi
et al 2007)
La pression magneacutetique sexprimant par B
22micro0 un 13ourant supeacuterieur agrave
100 kA apparaicirct neacute13essaire et atteignable sur un geacuteneacuterateur 1 kJ - 1 micros 13om-
pa13t de type R-L-C Ensuite il a fallu monter un modegravele de 13ir13uit deacuteformable
13omportant une se13tion formeacutee de la lame de plasma en mouvement et qui
prenne en 13ompte la1313reacutetion du gaz dans un modegravele dit snowplow (Potter
1971) Des paramegravetres geacuteomeacutetriques 13omme la forme et les dimensions du
13anon ont pu ecirctre optimiseacutes pour donner une grande vitesse de sortie du
plasma pour Ar et Xe et dans la gamme 01 - 10 mbar Un 13anon a eacuteteacute 13on-
struit et 13onne13teacute agrave un geacuteneacuterateur eacutele13trique pulseacute existant Des observations
du mouvement du plasma par la dynamique du rayonnement radial donnent
des vitesses 13oheacuterentes ave13 le modegravele ave13 la simulation 3D MHD et des
nombres de Ma13h eacuteleveacutes de 20 agrave 60 Ces travaux ont soutenu la 13on13eption
dun autre geacuteneacuterateur pour obtenir des performan13es plus eacuteleveacutees en vitesse
de 13ho13 et en taux de reacutepeacutetition
Contents
List of Figures
List of Tables
1 Introdu13tion 1
11 General Context 1
12 Outline of Chapters 2
13 My 13ontribution 3
2 Radiative sho13k waves 5
21 Rankine-Hugoniot(R-H) relations 6
211 Jump relations for a real gas 9
212 Ee13t of radiation on sho13k waves 11
213 Radiative hydrodynami13s equations 13
22 Radiative sho13k regimes 15
221 Typi13al radiative sho13k waves with a pre13ursor 16
23 Radiative sho13k waves experiments 16
231 Laser-driven sho13k experiments 18
232 Ele13tromagneti13ally laun13hed sho13k 21
3 1D Simulations 23
31 HELIOS 23
32 LTE approximation 24
321 Mean opa13ity 25
322 Equation of State 27
33 Single radiative sho13k waves 27
34 Intera13ting radiative sho13k waves 40
35 Summary 44
4 Experimental Setup 47
41 Prague Asterix Laser System (PALS) fa13ility 48
42 Targets 49
421 Massive Targets 51
422 Gaseous Targets 51
423 Target holder 55
CONTENTS
424 Target lling 56
43 Laser Fo13using 57
431 Fo13al Lenses and Phase Zone Plates 57
432 Fo13using of the MAIN and AUX laser beams 58
44 Diagnosti13s 59
441 Visible interferometry 61
442 XUV spe13tros13opy 65
443 Visible spe13tros13opy 67
45 Summary 68
5 Results and Interpretation 71
51 Visible interferometry 71
511 Longitudinal interferometry sho13k speed and ele13tron
density 74
512 Transverse interferometry lateral extension of the sho13k 91
52 XUV Spe13tros13opy 92
53 Simulations based on experimental results 93
54 Summary 97
6 Ele13tromagneti13ally laun13hed sho13k 101
61 Prin13iples of operation of a high 13urrent generator 102
62 Prin13iples of the run-down phase in a PFD 103
63 Proposed design for the plasma gun 104
64 Dynami13 13ir13uit modelling 106
65 3-D MHD simulations using GORGON 13ode 111
66 Measurements 116
67 Summary 118
7 Con13lusion 121
71 Con13lusions 121
72 Perspe13tives 123
8 Thesis summary 127
Appendi13es 141
A Visible Interferometry 143
A01 Refra13tive index of a plasma 143
A02 Absorption of the laser beam 144
A03 Prin13iple of interferometry 145
A04 Ma13h Zehnder Interferometer 146
B Opa13ities and mean 13harge 149
Referen13es 152
List of Figures
21 S13hemati13 diagram for a 1-D sho13k moving in the x dire13tion
The speed of the sho13k front is us In the frame of referen13e of
the sho13k front the pre-sho13k region has a relative speed u1=
-us density ρ1 total pressure P1 temperature T1 whereas these
quantities are respe13tively u2 ρ2 P2 and T2 for the post-sho13k
region 6
22 Proles of temperature (Te = Ti = T ) and mass density (ρ) at 10ns for a sho13k propagating in Xenon at 01 bar (ρ1= 57 times 10
minus4
g13m
minus3) with a speed of sim 45 kms It is to note that the given
pressure value are at 298 K The simulation is performed with the
HELIOS 13ode ex13luding the radiation ee13t and for a polytropi13
gas (γ=53) The initial temperature of Xenon is set to 1 eV
and with a negligible (10
minus5WmK) thermal 13ondu13tivity The
Ma13h number is then equal to 40 For this simulation the sho13k
is laun13hed by a 100 microm thi13k Aluminium piston moving at the
velo13ity of 33 kms The position x=0 13orresponds to the position
of the XeAl interfa13e at time t=0 8
23 Ele13tron temperature (Te) ion temperature (Ti) and mass density
(ρ) proles at 10 ns derived from 1-D simulation for the 13ase of
ionisation (no radiation) The sho13k propagates in Xe gas at 01
bar (ρ1= 57 times 10
minus4g13m
minus3by sim 45 kms) It is to note that the
given pressure value is at 298 K The simulation is performed with
the HELIOS (Te 6= Ti 13ase) using the PROPACEOS equation
of state without any radiation and with a negligible thermal
13ondu13tivity (10
minus05WmK) The initial temperature in Xenon
is set to 1 eV and thus the Ma13h number is equal to 40 For this
simulation the sho13k is laun13hed by a 100 microm thi13k Aluminium
piston moving at the velo13ity of 38 kms The position x=0
13orresponds to the position of the XeAl interfa13e at time t=0 10
LIST OF FIGURES
24 Figure 14 amp 17 of Mi13haut et al (2004) (a) Compression ρ2ρ1(full 13ir13les thin line) kT2 in eV (inverted full triangles thin line)
in Argon (a) Xenon (b) for initial 13onditions kT1( = 10 eV for
Ar and = 01 eV for Xe) ρ1 = 5 times 10minus4 gcm3 versus sho13k
speed in kms with (full markers full lines) and without (empty
markers dashed lines) radiation The ionization stage lt z gt2 is
plotted with the diamond symbol 12
25 Simulated ele13tron temperature (Te) ion temperature (Ti) and
mass density (ρ) proles at 10 ns of a radiative sho13k propagatingwith a 15 kms (a sub-13riti13al) and 48 kms (b super13riti13al)
The sho13k is propagating in Xenon at 01 bar (ρ = 54 times 10
minus4
g13m
minus3 at 298 K) The initial temperature in Xenon is set to
1eV The two Ma13h numbers are respe13tively equal to 13 and 43
The radiation and ionisation ee13ts are in13luded in the simulation
whi13h is performed for two dierent temperatures (Te 6= Ti) and a
negligible thermal 13ondu13tivity (10
minus05WmK) For this simula-
tion the sho13k is laun13hed by a 100 microm thi13k Aluminium piston
moving at the speed of 12 kms and 45 kms for 13ase (a) and (b)
respe13tively The position x=0 13orresponds to the position of the
XeAl interfa13e at time t=0 17
31 PROPACEOS Mono13hromati13 opa13ity versus the photon energy
in eV of Xenon at 10 eV and at two ele13tron densities respe13tively
equal to 1018 and 1020 13m
minus3 26
32 Mass density and temperature (a) ele13tron density and mean
13harge (b) at 10 ns for a radiative sho13k of speed sim 47 kms
in Xenon at 01 bar The verti13al dotted bla13k lines show the
position of the interfa13e between piston and ba13king Xenon gas
Zero at x-axis is the interfa13e of piston and Xenon at time zero
Spitzer thermal 13ondu13tivity has been used in the simulation 29
33 Mean Rosseland opa13ity (in 13m
2g
minus1) and opa13ity (13m
minus1) at 10
ns for a radiative sho13k of speed sim 47 kms in Xenon at 01
bar The verti13al dotted bla13k lines show the position of the
interfa13e between piston and ba13king Xenon gas Zero at x-axis
is the interfa13e of piston and Xenon at time zero Spitzer thermal
13ondu13tivity has been used in the simulation 30
34 (a) Mass density and temperature (b) ele13tron density and
mean 13harge at 10 ns for a radiative sho13k of speed sim 47 kms
in Xenon at 01 bar The verti13al dotted bla13k lines show the
position of the interfa13e between piston and ba13king Xenon gas
Zero at x-axis is the interfa13e of piston and Xenon at time zero
A negligible 13onstant thermal 13ondu13tivity (10
minus12WmK) have
been used in the simulation for Xenon 31
LIST OF FIGURES
35 (a) Ele13tron density and ele13tron temperature (b) Mass density
and mean 13harge at 10 ns for a radiative sho13k of speed sim 48
kms in Xenon at 01 bar for the 13ase when the ele13tron and ion
temperature are dierent The verti13al dotted bla13k lines show
the position of the interfa13e between piston and ba13king Xenon
gas Zero at x-axis 13orresponds to the position of the interfa13e
between the piston and Xenon at time zero The Spitzer thermal
13ondu13tivity has been used in this simulation 32
36 (a) Mass density and ele13tron temperature (b) ele13tron density
and mean 13harge at 10 ns for a radiative sho13k of speed sim 45
kms in Xenon at 01 bar for the 13ase when the ele13tron and ion
temperature are dierent The verti13al dotted bla13k lines show
the position of the interfa13e between piston and ba13king Xenon
gas Zero at x-axis 13orresponds to the position of the interfa13e
between the piston and Xenon at time zero In this simulation
value of thermal 13ondu13tivity (10
minus12WmK) is negligible for
Xenon The peak value of ion temperature is found to be 405 eV
whereas theoreti13ally this value is expe13ted to be sim 600 eV It is
possible to a13hieve the expe13ted value by in13reasing the resolution
of the simulation 34
37 Ele13tron temperature proles for various numbers of frequen13y
groups N = 1 20 50 60 70 90 and 100 35
38 (a) Ele13tron temperature (a) mean 13harge (b) and mass density
(13) for four (1 10 30 and 40) multipliers of the Xenon opa13ity at
10 ns 36
39 Plan13k and Rosseland mono13hromati13 opa13ity proles Te for ρ=15 times 10
minus3from PROPACEOS (I) BIGBART (II) and Rodri-
guez et al (III) 38
310 Mono13hromati13 Xenon opa13ity versus the photon energy in eV
at T = 15 eV and for ρ = 15 times 10
minus3g13m
minus3for the PROPA-
CEOS (a) and Rodriacuteguez et al (2015) (Fig 16 of Rodriacuteguez et al
(2015)) (b) models 39
311 Ele13tron density Ne (a) and ele13tron temperature Te (b) versus
axial position (along a 04 13m long sho13k tube) at 3 10 20 30
and 38 ns from HELIOS simulations (with opa13ity times 20) for the
13ases of single sho13k of sim 50 kms (dotted line) and two identi-
13al 13ounter-propagating sho13ks of sim 50 kms (solid lines) The
verti13al dotted lines show the position of the interfa13e between
piston and ba13king Xenon gas 41
LIST OF FIGURES
312 Variations ofNe (a) and Te (b) versus axial position for the 13ase of
two identi13al 13ounter-propagating sho13ks (of speeds sim 50 kms)
at 3 10 20 30 35 and 40 ns as derived from HELIOS simulations
For these simulations we have negle13ted the ee13t of radiation
by keeping the Xe opa13ity equal to zero The verti13al dotted lines
show the position of the interfa13e between the piston and ba13king
Xenon gas 43
313 Variations of Ne (a) and Te (b) with axial position for the 13ase
of two non-identi13al 13ounter-propagating sho13ks (of speeds sim 50
amp 20 kms) and two single sho13ks (dotted lines) of speeds sim 50
amp 20 kms respe13tively at 3 10 30 35 and 46 ns as derived
from HELIOS simulations The verti13al dotted lines show the
position of the interfa13e between piston and ba13king Xenon gas
(with opa13ity times 20) 45
41 a Spatial prole of the PALS Laser beam b Intensity prole of
laser pulse I(t) with time 48
42 Energy distribution s13heme for MAIN and AUX laser beams 49
43 Snapshot of the spheri13al 13hamber showing the horizontal bred-
board and on the right the fo13using MAIN lens 50
44 Dimension drawing (a) and snapshot (b) of Aluminum massive
(mo13k) target 50
45 Values on the gures are in mm (a) Verti13al 13ross se13tion view
of the gaseous target (b) Horizontal 13ross se13tion view 52
46 Pi13ture of a the gaseous target on its base 53
47 S13hemati13 of the gaseous target 53
48 Transmission of 100 nm thi13k Si3N4 membrane (CXRO database) 54
49 Snapshot of a target holder with one V-shield regarding the AUX
laser whi13h will pass through the hole in this shield marked in
green in the Figure The target whi13h fa13es the two diagnosti13s
(tubes) and is lo13ated behind the V-shield is thus not visible 54
410 In-situ system for target lling and pressure monitoring 56
411 S13hemati13 of the keV 13amera 59
412 (a) Sign 13onvention for the respe13tive positions of the lens and
the target (b) keV image of a MAIN impa13t on a massive target
(lens position +1500 microm) The spot size is 475plusmn25 microm Pixel
size for MAIN keV 13amera is 56 microm (13) keV image of an AUX
impa13t (lens position -1500 microm) The spot size is sim 280plusmn20 micromon target One pixel = 66 microm on AUXs keV 13amera 60
413 First interferometri13 setup All the opti13al elements are 1 in13h
For simpli13ity the plasma slab is not reported in the probe beam 61
414 Se13ond interferometri13 setup 63
415 S13hemati13 of the Streak 13amera (from Hamamatsu noti13e) 64
LIST OF FIGURES
416 Imaging setup of the Ma13h Zehnder Interferometer setup opera-
ting in the longitudinal mode The 13hannel of the target is fully
illuminated by the probe beam In this mode the lens images
the axis of symmetry of the two sho13k waves along the slit of the
streak 13amera 65
417 (a) Referen13e 2D image of a target before the shot re13orded on
the Streak 13amera The positions of the two pistons 13losing the 4
mm long target are lo13ated at 800 and 4800 microm The dark zones
along them (between 800 to 1000 and 4600-4800 microm) 13orresponds
to the glue This glue is then visible through verti13al bla13k strips
in the interferometri13 re13ord (b) Corresponding interferometri13
time-spa13e re13ord 66
418 Typi13al ray tra13ing in a grazing in13iden13e XUV spe13trograph
The dete13tor (GMCP or image plate) is installed tangentially to
the Rowland 13ir13le 67
419 (a) Setup of the XUV spe13trometer in the spheri13al 13hamber (b)
Typi13al re13ord between sim 10 and 40 nm is shown in the bottom
panel 68
420 (a) S13hemati13 of setup employed to re13ord the time and spa13e
visible plasma emission (b) A typi13al spe13trogram 69
51 Original re13ord of shot 48055 (left) FFT of the original image
(13enter) and the frequen13y ltered image (right) 73
52 A 13ropped se13tion of the re13ord from shot the 48055 The
rst ve positions have been sele13ted manually `+ signs (in red
13olor) on ea13h fringe On this re13ord the distan13e between two
unperturbed fringes is 13orrespond to 15 pixels (ie 159 microm) 73
53 (a) Representative points of the fringes as derived from the spline
tting of the 5 manually sele13ted points (Fig 52 (II)) on ea13h
fringe (b) Positions of the fringes maxima along Y-axis for ea13h
fringe derived by lo13ating the points of maximum intensity in
X-dire13tion of the previous points obtained by spline t 75
54 Fringe maxima on the full image 76
55 Imaging setup of the Ma13h Zehnder Interferometer operating in
the longitudinal mode The lens images the axis of symmetry of
the two sho13k waves along the slit of the streak 13amera 76
56 Interferometri13 image re13orded for the shot 48055 in Xe at 01
bar The sho13k speeds for the sho13ks driven by MAIN (from left
side) land AUX (from right side) lasers are respe13tively equal to
sim 54 and 23 kms The time of laser arrival on the piston is at
146 ns The positions of the Au-Xe interfa13e on the re13ord are
respe13tively 950plusmn50 and 4950plusmn50 mi13rons 77
LIST OF FIGURES
57 (a) MAIN sho13k speed (13al13ulated by the last fringe method)
versus the MAIN energy for Xenon or XeHe mixture at dierent
pressures with the error bars (b) AUX sho13k speed versus AUX
energy also for Xenon or XeHe mixture at dierent pressures 79
58 MAIN sho13k speed (13al13ulated by the last fringe method) versus
the MAIN laser energy for Xe (at 01 bar only) Ar He and Kr
at dierent pressures with the error bars 82
59 Left panel interferometri13 re13ords 48055 in Xe at 01 bar (a)
48132 in Xe+He at 02 bar (b) and 48138 in Xe+He at 02 bar
(13) Right panel ele13tron density at 10 20 30 and 40 ns versus
distan13e for these re13ords The positions of maxima have been
identied on the re13ords in the left panel The time t = 0 13orre-
sponds to the time of laser arrival on the target and the position
x = 0 13orresponds to the interfa13e between the piston (Au layer)
and the gas Its determination is pre13ise within 100 mi13rons The
distan13es between two unperturbed fringes for re13ords 48055
48132 and 48138 are 159 244 and 244 microm respe13tively The
lt Ne gt un13ertainty (plusmn 2 pixels) is indi13ated by the error bar in
the right panels It 13orresponds respe13tively to plusmn 9 times 10
17 plusmn 6
times 10
17and plusmn 6 times 10
1713m
minus3for the gures (a) (b) and (13) The
limit of dete13tion (2 pixels) is presented by a dotted line on ea13h
gure 85
510 (a) Interferometri13 re13ord in Ar at 03 bar (48079) The esti-
mated speeds for MAIN and AUX are 49plusmn5 and 23plusmn3 kms (b)
Same re13ord where the fringes maxima are marked by points
The bins denition is as fellows bin 0 Ne le 11 times 10
1813m
minus3
(white) bin 1 11 - 18 times 10
1813m
minus3(yellow) and bin 2 39 -
57 times 10
1813m
minus3(red) The time t = 0 13orresponds to the time
of laser arrival on the target and the position x = 0 13orresponds
to the interfa13e between the piston (Au layer) and the gas Its
determination is pre13ise within 100 mi13rons 87
511 Results for temperature (a) and Rosseland opa13ity (b) obtained
from HELIOS simulation at 10 ns for two 13ounter-propagating
sho13ks at sim 50 and 18 kms for Xe (with opa13ity multiplier 1
and 20) Ar and Kr at 54 times 10
minus4g13m
minus3initial mass density
A 13omparison Ar Kr and Xe PROPACEOS opa13ity shown in
APPENDIX B 88
LIST OF FIGURES
512 (a) Raw interferometri13 re13ord in Kr at 02 bar (shot 48146)
The estimated speed for MAIN sho13k is 53plusmn2 kms The time
t = 0 13orresponds to the time of laser arrival on the target(b)
Same re13ord shown in (a) maxima is marked by 13olored points
The bins denition is as follows bin 0 Ne le 11 times 10
1813m
minus3
(white) bin 1 11 - 18 times 10
1813m
minus3(yellow) and bin 2 39 -
57 times 10
1813m
minus3(red) The time t = 0 13orresponds to the time
of laser arrival on the target and the position x = 0 13orresponds
to the interfa13e between the piston (Au layer) and the gas Its
determination is pre13ise within 100 mi13rons 90
513 S13hemati13s of the Ma13h Zehnder Interferometer setup to re13ord
transverse interferometri13 images The lens allows to make on
the slit of the 13amera the image of a se13tion perpendi13ular to
dire13tion of sho13k propagation 91
514 Transverse interferometri13 images for (a) shot48111 (MAIN
sho13k only) (b) shot48130 (AUX sho13k only) The time is
measured after an oset equal to 14 and 23 ns respe13tively after
the time of the laser arrival on the target The position zero on
the x-axis of ea13h image 13orresponds to the base of the target 92
515 Interferometri13 image for the shot48143 The time t = 0 13orre-
sponds to the time of laser arrival on the target and the position
x = 0 13orresponds to the interfa13e between the piston (Au layer)
and the gas Its determination is pre13ise within 100 mi13rons 94
516 Raw (a) and 13orre13ted XUV spe13trum (b) for the shot 48143 95
517 Re13orded ele13tron density (shot 48055) together with the HE-
LIOS results (with Xenon opa13ity times 20) at dierent times in
Xenon at 01 bar 97
518 Time evolution of the mass density (a) ele13tron temperature (b)
and mean 13harge (13) at 56 57 58 60 64 and 65 ns within the
sho13k tube derived from the HELIOS simulations (with Xenon
opa13ity multiplier = 20) for two 13ounter streaming sho13ks of sim39 and 18 kms in Xenon at 06 bar 98
61 (a) Sket13h of the sho13k generator showing the pulsed ele13tri13al
13ir13uit the set of 13oaxial 13oni13al ele13trodes with a 13onstant radial
gap and the plasti13 insulator featured in grey on whi13h a planar
surfa13e dis13harge is initiated The installation of three opti13al
bers allows looking radially at the plasma moving in the sho13k
tube (b) S13hemati13s of the plasma dynami13s inside the 13oaxial
gun in fast-pulse mode the ele13tri13al 13urrent ows in the super-
13ial layers of the two 13oaxial 13oni13al ele13trodes and through an
annular plasma layer The magneti13 pressure Pmag pushes the
dis13harge axially 105
62 Exploded view of the plasma gun 106
LIST OF FIGURES
63 Time dependen13e of the main parameters for Ar gas at 1 mbar
a1313ording to the 13ir13uit model (a) 13urrent (kA) and voltage
(V) (b) a1313reted mass (kg) and rate of a1313retion (kgs) (13)
speed (ms) and kineti13 energy (J) 108
64 Plot of the dierent output parameters (normalized with respe13t
to the peak values) obtained at the top of the inner ele13trode
For ea13h group upper panel represents the variation for Ar gas
whereas lower panel is for Xe gas Proles of output parameters
are given (a) with 13one angle (b) with a1313retion fa13tor (13) with
gas pressure and (d) with damping resistan13e 109
65 Example of mapping of the ele13tron density from a 3-D MHD
simulation (ba13kground gas Argon at 1 mbar) when the plasma
sheath is (left) at the exit of the plasma gun (right) in free
ight 13onditions (log s13ales for the false 13olors) 113
66 Time history of plasma merging and early free ight through
the mapping of the ele13tron density ele13tron temperature and
average ioni13 13harge as given by a 3-D MHD simulation (ba13k-
ground gas Argon at 1 mbar) (s13ales for the false 13olors) 114
67 Axial proles values taken in the sho13k tube along a line slightly
o-axis at a distan13e of 1 mm for mass density ion temperature
ion density average ion 13harge magneti13 eld ele13tron tempera-
ture ele13tron density and average speed at 1500 ns (ba13kground
gas Argon at 1 mbar) A maping of |B| is given with a dashed
line indi13ating sho13k front position as well 115
68 Experimental results (a) time history of the passive opti13al
re13ords 13ompared to the main 13urrent for Argon gas 13lose to the
referen13e pressure (b) time-integrated signal vs Ar pressure for
the dierent bers (13) averaged speed vs Ar ba13king pressure 117
71 Interferometri13 re13ord obtained at 546 ns for a single radiative
sho13k of speed sim 100 kms propagating in Xe+He gas at 06 bar
driven by laser at 348 nm with energy 170 J The dierent 13olors
in (b) 13orrespond to yellow le Ne le 35 times 10
1713m
minus3 13yan 37
- 74 times 10
1713m
minus3 blue74 - 15 times 10
1813m
minus3 green 15 - 18 times
10
1813m
minus3 magenta 18 - 22 times 10
1813m
minus3 red 22 - 26 times 10
18
13m
minus3 orange26 times 10
1813m
minus3- 18 times 10
1913m
minus3 124
A1 S13hemati13s of Ma13h Zehnder interferometer with two mirrors
two 50 beam splitters and the dete13tor A plasma slab is pla13ed
in one of the two arms of the interferometer 147
B1 Rosseland and Plan13k opa13ity for three gases Xe Kr and Ar at
mass densities 51 times 10
minus4g13m
3(a) and 16 times 10
minus3g13m
3(b) 150
LIST OF FIGURES
B2 Mean 13harge for three gases Xe Kr and Ar at mass densities 16
times 10
minus3g13m
3(a) and 51 times 10
minus4g13m
3(b) 151
List of Tables
21 Summary of the radiative sho13ks experiments and out13omes in
dierent laser installations All the experiments are performed in
Xe ex13ept Keiter et al (2002) and Vis13o et al (2012) In these
two experiments the rst uses SiO
lowast
2 and the se13ond uses Ar
lowastlowastas
a sho13ked medium 20
41 Nominal 13hara13teristi13s of MAIN and AUX laser beams 49
42 Spe13i13ations of the MAIN and AUX fo13al lenses and the f- num-
ber (fo13al lengthbeam diameter) is given (see table 41 for the
beam diameters) 57
43 Spe13i13ations of the two Phase Zone Plates 58
44 Spe13i13ations of the two uEye keV 13ameras (1stVsion 13ompany) 59
45 Spe13i13ations of Andor CCD 13amera 67
46 Spe13i13ations of the XUV grating 67
51 Sho13k speeds estimated from the `last fringe method `NA re-
presents the entries whi13h 13ould not be dedu13ed from the re13ord
Further entries in the bold font are dis13ussed in detail in this
13hapter 81
52 Atomi13 data and density at 01 bar (at room temperature) for
He Ar Kr and Xe 82
61 Input parameters their standard values and range of variation 110
62 Laun13hing performan13e ArXe 111
63 Values of variables in various 13onditions at rest (with seed ele13-
trons) inside the sho13k and inside post-sho13k region A star lowastdenotes the insulator surfa13e 112
Chapter 1
Introdu13tion
11 General Context
Radiative sho13ks (RS) are present in various astrophysi13al environments es-
pe13ially in stars They are then present in all the supersoni13 a1313retioneje13tion
pro13esses during the stellar evolution from the early phase of the proto-
stellar 13ollapse up to the magnetosphere a1313retion on to the already formed
T- Tauri stars up to the late stage of the supernovae sho13ks
Radiative sho13ks are strong sho13ks (ie Ma13h number Mgtgt1) whi13hrea13h high temperatures and thus are the sour13e of intense radiation Alt-
hough the observation of (ie spatially resolved ) signatures of su13h sho13ks
in the astrophysi13al environments is very di13ult an alternate approa13h na-
med laboratory plasma astrophysi13s provides a powerful tool to study them
on the Earth
These waves have been experimentally studied sin13e more than a de13ade
mostly on large-s13ale laser fa13ilities in noble gases and with dierent targets
geometries The ablation pro13ess generated by the intera13tion between the
laser and a foil indu13es a sho13k wave in the foil whi13h then propagates
in the gas at a speed of several tens of kms Numeri13al simulations 13an
des13ribe these sho13ks with an improving pre13ision With laser intensities on
target 13omprised between 1014 and 1015 W13m
2 these experiments allowed
to re13ord sho13k speeds ranging between 40 and 150 kms
All previous experimental studies have been fo13used on the 13ase of single
radiative sho13ks However in astrophysi13al 13onditions the radiative sho13k
in general intera13ts with a denser medium leading to the development of
ree13ted and transmitted sho13ks A few representative examples of su13h
phenomena are the intera13tion of supernovae remnants with dense mole13ular
13louds the a1313retion sho13ks on the photosphere of T-Tauri stars and the bow
sho13ks at the head of fast stellar jets The 13ollision (or the intera13tion) of
two radiative sho13k waves is obviously a rare astrophysi13al event and the
template 13ase of the supernova remnants DEM L316 (see Fig 1 of Williams
et al (1997)) is still the subje13t of debates (Velarde et al 2006) In this
regard the development of dedi13ated laboratory experiments to the study of
propagation and intera13tion of 13ounter-propagating sho13k waves is important
as a tool to 13hara13terize su13h events through their spe13i13 signatures
Complementary to laser experiments 13ompa13t pulsed power generators
may drive an ele13tromagneti13 13oaxial plasma gun to 13reate astrophysi13al
1
2 CHAPTER 1 INTRODUCTION
relevant sho13ks in lower pressure noble gases with a high availability and a
rather modest 13apital 13ost The ele13tromagneti13ally driven sho13k waves may
have larger s13ales than those by laser Thus they 13an be analyzed rather
easily with the adequate diagnosti13s (Kondo et al 2008)
In this thesis I am presenting radiative sho13ks in the 13ontext of labo-
ratory astrophysi13s using two experimental approa13hes laser-driven sho13ks
and ele13tromagneti13ally driven sho13ks
12 Outline of Chapters
In the rst 13hapter the physi13s of radiative sho13ks wave will be summarized
I will dis13uss the Rankine -Hugoniot (RH) for the perfe13t gas and genera-
lized RH equations for ionized gas Further the ee13ts of radiation on the
sho13k equations will be introdu13ed and I will review the dierent regimes of
the radiative sho13ks presenting a radiative pre13ursor Then I will present
a short histori13al review of laser and ele13tromagneti13ally laboratory experi-
ments on this topi13 Finally I will highlight the obje13tives of my work in
this framework
The 13hapter two is dedi13ated to the numeri13al analysis of radiative sho13k
waves and of the main physi13al pro13esses whi13h play a role in these 13omplex
pro13esses The simulations will be performed in 1D using the 13ommer13ial ra-
diation hydrodynami13 HELIOS 13ode (Ma13Farlane Golovkin and Woodru
2006) After a des13ription of the 13ode and his possibilities several simu-
lations will be performed to analyze the ee13ts of the two uids (ele13trons
and ions) approa13h 13ompared to the one uid 13ase (identi13al temperature
for both) of the thermal 13ondu13tivity and the multi-groups opa13ity des13rip-
tion Then a brief overview of single and two 13ounter-propagating radiative
sho13ks will be presented
Chapter three will start with a short overview of the Prague Asterix Laser
System fa13ility (PALS) It will be followed by the experimental setup in13lu-
ding the two high energy beams and targets des13ription These paragraphs
will be followed by the detail of the diagnosti13s
The fourth 13hapter will be dedi13ated to the data analysis the results
and the interpretation of the experimental re13ords I will rst des13ribe the
method I followed for the analysis of interferometri13 re13ords Subsequently
the results obtained from a few representative re13ords for Xe Ar and Kr
will be dis13ussed in details Complementary 1D simulations performed with
initial 13onditions similar to that in the experiment will be shown and help to
interpret the experimental results Finally the results of XUV spe13tros13opy
diagnosti13s will be presented
In the fth 13hapter I will present the 13alibration of a table top ele13tro-
magneti13 plasma generator whi13h is able to laun13h supersoni13 sho13ks up to
sim 40 kms speed at stati13 pressures of few mbar The results of this 13ali-
13 MY CONTRIBUTION 3
bration will also be presented and a 13omparison with 3-D MHD simulations
will be performed The diagnosti13s whi13h have been implemented will be
presented to illustrate the model as well as preliminary re13ords of the plasma
speed
The nal 13hapter of the thesis is dedi13ated to the summary of this work
together with perspe13tives for the follow-up of these experiments
13 My 13ontribution
I joined PhD on November 12 2013 under the supervision of Dr Jean
Larour and Dr Chantal Stehleacute My PhD thesis is 13omprised of two parts
the rst part of the work is to study the laser-driven sho13ks The se13ond
part of my thesis is to study the ele13tromagneti13ally laun13hed strong sho13ks
Con13erning the laser-driven sho13ks I have started to work on the analysis
and interpretation of XUV images re13orded previously during an experimen-
tal 13ampaign 13ondu13ted at the Prague Asterix Laser System (PALS) fa13ility
in 2011 This enabled me to estimate the lo13al opa13ity by employing Abel
inversion Although this work not presented in this thesis it shed light on
the basi13 understanding of radiative sho13k as seen in the experiments as well
as the knowledge of experimental setups Later in April - May 2015 I have
parti13ipated in a ve weeks experimental 13ampaign on PALS laser fa13ility
in Prague whi13h was dedi13ated to the study of the spe13tros13opi13 and in-
terferometri13 signatures of laser-produ13ed sho13ks Espe13ially the experiment
was designed to study the physi13s during the 13ollision of 13ounter-propagating
sho13ks In the preparation of this 13ampaign I be13ame a13quainted with the
instruments and the targets to be used as well as prepared a master re-
port 13ontaining all the 13ru13ial aspe13ts of the experiment to be referred by
all the parti13ipants during the experiments During the 13ampaign we have
performed around 55 shots 13orresponding to dierent gases (Xenon Argon
Helium and Krypton) at dierent pressure and used three types of diagnos-
ti13s ie visible interferometry visible spe13tros13opy and XUV spe13tros13opy
In this experiment parti13ularly I was in 13harge of the target sele13tion its
installation in the experimental 13hamber gas lling preparing logbook re-
13ording of the visible spe13tros13opy data from the 13ontrol room qui13k data
analysis for the spot size sho13k speed from interferometry data et13 In
addition I have worked on spe13tros13opi13 and interferometri13 data analysis
re13orded during the experiment Next in order to interpret the experimental
results I have performed several 1D hydrodynami13 simulations with varying
input 13onditions In parallel I have also performed simulations for another
experiment performed by my 13ollaborators at ORION laser fa13ility in the
United Kingdom However I have not presented this work in this thesis
Based on the results obtained from my analysis of re13ords obtained from
the PALS experiments in 2015 grabbing the opportunity we have planned
4 CHAPTER 1 INTRODUCTION
and parti13ipated to a 13omplementary experimental 13ampaign at PALS in
September 2016 The analysis of data re13orded during experiments is still
under progress and forms the future s13ope of the thesis
The se13ond part of my thesis has been 13omprised of the study of sho13k
laun13hed through the ele13tromagneti13 generator setup In this I have wor-
ked on the optimization of the design of the ele13tri13 generator in view of the
required sho13k parameters Moreover I have derived various sho13k 13hara13-
teristi13s viz radial dis13harge layer namely speed mass a1313eleration et13
at the dierent positions of its propagation In present a upgraded ele13tri13
generator with new diagnosti13s is ready to perform the new experiments
Chapter 2
Radiative sho13k waves
Contents
21 Rankine-Hugoniot(R-H) relations 6
211 Jump relations for a real gas 9
212 Ee13t of radiation on sho13k waves 11
213 Radiative hydrodynami13s equations 13
22 Radiative sho13k regimes 15
221 Typi13al radiative sho13k waves with a pre13ursor 16
23 Radiative sho13k waves experiments 16
231 Laser-driven sho13k experiments 18
232 Ele13tromagneti13ally laun13hed sho13k 21
Small-amplitude disturban13es in a medium often take the form of linear
waves On the 13ontrary strong disturban13es generally produ13ed by pro13esses
su13h as explosions lightening supersoni13 movements of bodies powerful
ele13tri13 dis13harges et13 where the uid properties 13hange rapidly generate
nonlinear waves like sho13k waves As a 13onsequen13e of sho13k waves the ow
moves faster than the sound speed A sho13k is 13hara13terized by a sudden
in13rease in the density and velo13ity of the medium The Ma13h number (M)
is 13ommonly used to 13hara13terize the strength of the sho13k This number
dened as the ratio of the sho13k speed to the speed of sound in the upstream
medium in whi13h the sho13k propagates
Radiative sho13k waves are hypersoni13 sho13k waves whi13h are heated to
high temperature and as a 13onsequen13e be13ome the sour13e of intense radi-
ation This radiation in turn modies the dynami13s and stru13ture of the
sho13k itself whi13h makes its stru13ture more 13ompli13ated Radiative sho13k
waves o1313ur in several astrophysi13al 13ir13umstan13es su13h as in the a1313retion
sho13k of protostellar formation (Stahler Palla and Salpeter 1986) the su-
pernova explosion and the intera13tion of their remnants with the dense in-
terstellar medium (Chevalier 1977) the bow sho13ks at the head of stellar
jets (Hartigan et al 2001)
Radiative sho13k waves 13an now be studied in the laboratory whi13h allows
to 13ompare the data with models existing in the literature and to 13he13k the
ability of the numeri13al 13odes to reprodu13e them (see for instan13e Bouquet
et al (2004) Leygna13 et al (2006)) The experimental study of radiative
5
6 CHAPTER 2 RADIATIVE SHOCK WAVES
sho13k 13hara13teristi13s is 13ru13ial in understanding these various energeti13 pro-
13esses o1313urring in the astrophysi13al plasmas In this 13hapter I will rst
present a short summary of the jump 13onditions (Rankine-Hugoniot rela-
tions) for the simple 13ase of an ideal gas whi13h rely on the values of the
thermodynami13al quantities on both sides of the dis13ontinuity Next I will
present the role played by the ionisation and ex13itation of the gas and then
the role played by the radiation energy and pressure The des13ription of the
dierent regimes of radiative sho13ks whi13h depend on the opa13ity of medium
and 13an be 13hara13terized by the presen13e or absen13e of a radiative pre13ursor
will follow Further I will give a brief overview of the literature fo13using on
the generation of radiative sho13ks in the laboratory and I will present the
motivations and obje13tives of my work
21 Rankine-Hugoniot(R-H) relations
Let us 13onsider a one-dimensional sho13k propagating in a gas at rest with
the speed us In the frame asso13iated with the sho13k front the pre-sho13k
(upstream) uid velo13ity is then u1 = - us while behind the sho13k (post-
sho13k or downstream region) this velo13ity is u2 as shown in the Fig 21
Figure 21 S13hemati13 diagram for a 1-D sho13k moving in the x dire13tion The
speed of the sho13k front is us In the frame of referen13e of the sho13k front the pre-
sho13k region has a relative speed u1= -us density ρ1 total pressure P1 temperature
T1 whereas these quantities are respe13tively u2 ρ2 P2 and T2 for the post-sho13k
region
Assuming that a stationary state is rea13hed the 13onditions whi13h deter-
mine the post-sho13k thermodynami13al quantities are the well known Rankine-
21 RANKINE-HUGONIOT(R-H) RELATIONS 7
Hugoniot equations (Ma13quorn Rankine 1870 Hugoniot 1887 1889) De-
du13ed from the Euler equations these 13onservation equations for the mass
momentum and energy are given by
ρ2u2 = ρ1u1 (21)
ρ2u22 + P2 = ρ1u
21 + P1 (22)
u2(ε2 +P2
ρ2) +
1
2u32 = u1(ε1 +
P1
ρ1) +
1
2u31 (23)
where ε1 and ε2 are the spe13i13 internal energies (energy per unit mass)
in the two (upstream and downstream) regions
For an ideal gas of atomi13 mass m =mPA (mP proton mass A atomi13
number) and having an adiabati13 index γ the speed of sound is given by
Cs =
radic
γP
ρ=
radic
γkBT
m(24)
and the Ma13h number (M) of the sho13k is then
M =u1Cs1
(25)
Inserting these the aforementioned form of M and Cs in the equations
21 22 and 23 we 13an derive the 13ompression pressure and temperature
ratios for the system as follows
ρ2ρ1
=u1u2
=M2(γ + 1)
2 +M2(γ minus 1)(26)
P2
P1=
2M2γ minus (γ minus 1)
(γ + 1)(27)
T2
T1=
P2
P1
ρ1ρ2
=(2M2γ minus (γ minus 1))(2 +M2(γ minus 1)
M2(γ + 1)2) (28)
Considering the 13ase of strong sho13ks where M may be 13onsidered to be
≫1 these equations be13ome
ρ2ρ1
=(γ + 1)
(γ minus 1)(29)
T2
T1=
2M2γ(γ minus 1)
(γ + 1)2(210)
kBT2 =2(γ minus 1)
(γ + 1)2mus
2(211)
8 CHAPTER 2 RADIATIVE SHOCK WAVES
For mono-atomi13 gases γ = 53 the 13ompression ratio is estimated to
be 4 and the post-sho13k temperature is
kBT2 =3
16mu1
2 =3
16mPAu1
2(212)
Temperature is thus proportional to the square of the sho13k velo13ity and
to the atomi13 mass Therefore to a13hieve a high temperature in the radiative
sho13k experiment we often use high atomi13 number and mass gases For Xe-
non (A=13129) the post-sho13k temperature is then equal to T2 = 0257u21(eV kms) whi13h for instan13e gives sim 520 eV for a sho13k propagating at
45 kms (see Fig 22)
Figure 22 Proles of temperature (Te = Ti = T ) and mass density (ρ) at 10ns for a sho13k propagating in Xenon at 01 bar (ρ1= 57 times 10
minus4g13m
minus3) with
a speed of sim 45 kms It is to note that the given pressure value are at 298 K
The simulation is performed with the HELIOS 13ode ex13luding the radiation ee13t
and for a polytropi13 gas (γ=53) The initial temperature of Xenon is set to 1
eV and with a negligible (10
minus5WmK) thermal 13ondu13tivity The Ma13h number
is then equal to 40 For this simulation the sho13k is laun13hed by a 100 microm thi13k
Aluminium piston moving at the velo13ity of 33 kms The position x=0 13orresponds
to the position of the XeAl interfa13e at time t=0
21 RANKINE-HUGONIOT(R-H) RELATIONS 9
211 Jump relations for a real gas
In 13ontrast to the dis13ussion on ideal gas presented in the previous se13tion
the 13ase of a real gas is a little more 13ompli13ated It is to note that in this
13ase a part of the kineti13 energy is used to ex13ite and ionize the post-sho13k
gas As a result its temperature is lower than that for the 13ase of ideal gas
In addition the pressure is also modied due to ionisation (Mi13haut et al
2004) as dis13ussed below
Let us now dene j as the ionization stage of the atom (varying between
0 and z) and i to be the atomi13 state in this ionisation stage We shall
denote by Pj amp Pji the fra13tional ionization of the ion and the population
of the atomi13 state i in the jth ionization state respe13tively Then the mean
ionization stage (average 13harge) per atom 13an be written as
z =
zsum
j=0
jPj (213)
and the ex13itation energy (in13luding ionization) per atom and per unit of
mass is
ǫexc =
sumzj=0
sum
i Pji Eji
mpA(214)
where Eji is the energy of the ionisation stage j in the atomi13 stage iThus the total pressure of gas 13an be written as
P = Pi + Pe (215)
where Pi and Pe are the ioni13 and ele13troni13 pressures 13ontributions
Using the 13harge neutrality 13ondition Ne = z Ni this gives
P = Ni(1 + z) kBT = ρkBT
m(1 + z) (216)
The thermal pressure and enthalpy of the real gas are given by
Pth = ρ(1 + z)
mkBT (217)
h =5
2
(1 + z)
mkBT + ǫexc (218)
Next the speed of sound in13luding the ee13t of ionization with the
average 13harge z in the medium is now modied by ionisation
Cs ≃radic
5
3
γ(z + 1)kBT
m(219)
Thus the 13ontinuity relations in the frame moving with the sho13k front
be13ome
ρ2u2 = ρ1u1 (220)
10 CHAPTER 2 RADIATIVE SHOCK WAVES
ρ2u22 + ρ2
kBT2
m(1 + z2) = ρ1u
21 + ρ1
kBT1
m(1 + z1) (221)
ρ2u2
[5
2
(1 + z2)
mkBT2 + ǫexc2 +
u222
]
= ρ1u1
[5
2
(1 + z1)
mkBT1 + ǫexc1 +
u212
]
(222)
Figure 23 Ele13tron temperature (Te) ion temperature (Ti) and mass density
(ρ) proles at 10 ns derived from 1-D simulation for the 13ase of ionisation (no
radiation) The sho13k propagates in Xe gas at 01 bar (ρ1= 57 times 10
minus4g13m
minus3by
sim 45 kms) It is to note that the given pressure value is at 298 K The simulation
is performed with the HELIOS (Te 6= Ti 13ase) using the PROPACEOS equation
of state without any radiation and with a negligible thermal 13ondu13tivity (10
minus05
WmK) The initial temperature in Xenon is set to 1 eV and thus the Ma13h
number is equal to 40 For this simulation the sho13k is laun13hed by a 100 microm thi13k
Aluminium piston moving at the velo13ity of 38 kms The position x=0 13orresponds
to the position of the XeAl interfa13e at time t=0
Similar to the Fig 22 however for the 13ase of a real gas the Fig
23 shows the variation the ele13tron and ion temperatures as well as the
mass density with the distan13e for a sho13k propagating in Xenon at sim 45
kms Now the post-sho13k ele13tron temperature peaks at 22 eV whi13h is
mu13h smaller than the maximum temperature of the ions (510 eV) The
13ompression is now equal to 10 instead of 4 as in the previous 13ase The ions
21 RANKINE-HUGONIOT(R-H) RELATIONS 11
are heated rst in the sho13k through ion-ion 13ollisions This ion temperature
de13reases and the ele13tron temperature in13reases as a 13onsequen13e of the
ele13tron-ion 13ollisions The two temperatures thus be13ome equal Sin13e the
momentum 13hange of the ions require only a few number of kineti13 13ollisions
the thi13kness of the abrupt transition between the pre-sho13k and the post-
sho13k is of the order of a mean free path of the ions
It should be noted that for sho13ks propagating with very high speed
the post-sho13k temperature is so high that the 13ompressed gas is fully ioni-
zed Therefore the ee13ts of ionisation and ex13itation be13ome negligible and
hen13e the 13ompression ratio rea13hes 4 as in the 13ase of the adiabati13 sho13k
212 Ee13t of radiation on sho13k waves
In the previous se13tion the ee13ts of radiation heating and 13ooling have
been negle13ted However the radiation emitted in a strong sho13k will ae13t
its stru13ture These ee13ts 13an be determined using mass momentum and
energy 13onservation equations whi13h now have to in13lude the 13ontributi-
ons of the radiative ux energy and pressure To this purpose and with
some simpli13ations we 13an use the same approa13h than in se13tion 21 still
assuming that the sho13k is stationary
Jump relations with radiation
Let us 13onsider a stationary sho13k propagating in an atomi13 gas far away from
the dis13ontinuity The medium is then opti13ally thi13k and we 13an negle13t
the 13ontribution of the radiative ux Let us also assume that the medium
is at LTE and that the radiation pressure and energy may be des13ribed as
follows
Prad =1
3Erad =
4
3cσT 4 =
1
3aradT
4(223)
where arad is the radiation density 13onstant and T is the sho13k radiation
temperature whi13h we shall suppose to be equal to the ele13tron temperature
Following Mi13haut et al (2004) the new jump 13onditions a13ross the sho13k
dis13ontinuity be13ome
ρ2u2 = ρ1u1 (224)
ρ2u22+ρ2
kBT2
m(1+z2)+
1
3aradT
42 = ρ1u
21+ρ1
kBT1
m(1+z1)+
1
3aradT
41 (225)
ρ2u2
(
5
2
(1 + z2)
mkBT2 + ǫexc2 +
u222
)
+4
3aradT
42 u2 =
ρ1u1
(
5
2
(1 + z1)
mkBT1 + ǫexc1 +
u212
)
+4
3aradT
41 u1 (226)
12 CHAPTER 2 RADIATIVE SHOCK WAVES
(a)
(b)
Figure 24 Figure 14 amp 17 of Mi13haut et al (2004) (a) Compression ρ2ρ1 (full13ir13les thin line) kT2 in eV (inverted full triangles thin line) in Argon (a) Xenon
(b) for initial 13onditions kT1( = 10 eV for Ar and = 01 eV for Xe) ρ1 = 5 times10minus4 gcm3
versus sho13k speed in kms with (full markers full lines) and without
(empty markers dashed lines) radiation The ionization stage lt z gt2 is plotted
with the diamond symbol
21 RANKINE-HUGONIOT(R-H) RELATIONS 13
In the radiative regime the 13ompression ratio ρ2ρ1 approa13hes to the
value of 7 at very high velo13ities (several hundred of kms) whereas it is equal
to 4 when the ee13ts of radiation energy and pressure have been negle13ted
This limit is the same for the 13ases of adiabati13 (Bouquet Teyssier and
Chieze 2000) as well as the real gas (Mi13haut et al 2004) This is due to
the fa13t that at very high velo13ities the medium be13omes fully ionized and
behaves as a perfe13t gas On the 13ontrary in the 13omparatively low velo13ity
regime the ee13t of ionisation and ex13itation dominates the radiation ee13ts
This is illustrated in Fig 24 taken from Mi13haut et al (2004) whi13h reports
the variations of the Ar and Xe 13ompression ratio versus the sho13k speed for
an initial mass density of 5 times 10
minus4g13m
3 At 50 kms the 13ompression
ratio and the temperature are respe13tively equal to (5 32eV) and (11 30
eV) for Argon and Xenon
The 13ompression ratio shows bumps for high velo13ities These bumps
may be attributed to the fa13t that as the temperature of the post-sho13k
in13reases with the velo13ity the atom ionises more and more Thus the ex-
ternal shells in its ioni13 atomi13 stru13ture are su1313essively opened indu13ing
a strong variation in the ex13itation energy
The post-sho13k temperature is very high for the very high-speed sho13ks
In su13h 13ases the 13ompressed gas is fully ionized and the ee13ts of ionisation
and ex13itation remain negligible like in the adiabati13 13ase It should be noted
that below 100 - 200 kms the ee13ts of radiation pressure remain negligible
Thus very energeti13 lasers like LMJ or NIF are required to rea13h a domain
where this ee13t 13an be observed
Thus whereas for the range of velo13ities of 20 - 50 kms 13overed by my
study the ee13ts of radiative energy and pressure 13an be negle13ted this is
not the 13ase for the radiative ux whi13h has been negle13ted in the previous
se13tions and whi13h may be13ome 13omparable to the material energy ux Its
ee13t on the system must be taken into a1313ount as will be shown below
213 Radiative hydrodynami13s equations
To in13lude the 13ontributions of radiation it is 13ompulsory to introdu13e the
relevant terms in the uid equations The propagation of the radiation in the
matter is governed by the pro13esses of absorption and emission of photons
whi13h respe13tively heats and 13ools the medium As the mean free path of
photons is always higher than the mean free path of parti13les the sho13k
stru13ture is more inuen13ed by the radiative transfer than the vis13osity of
the uid
The propagation of radiation in the matter is des13ribed by the radiative
transfer equation
(
1
c
part
partt+
part
parts
)
I(rn ν t) = η(r ν)minus χ(r ν) I(rn ν t) (227)
14 CHAPTER 2 RADIATIVE SHOCK WAVES
where χ(ρ ν T ) and η(ρ ν T ) are respe13tively the mono13hromati13 opa-
13ity and emissivity of the gas
The spe13i13 intensity I(rn ν t) (erg cmminus2 sminus1Hzminus1 srminus1) is the energy
radiated per units of surfa13e area time frequen13y and solid angle in the
dire13tion n
The radiative energy density Erad ux Frad and pressure Prad are re-
spe13tively dened as the zero rst and se13ond moments of the spe13i13 in-
tensity versus the angle after integration over the photon frequen13y ν
Erad =1
c
int ∮
I(rn ν t)dΩdν (228)
Frad =
int ∮
I (rn ν t) cos θ dΩdν (229)
Prad =1
c
int ∮
I (rn ν t) cos2 θ dΩdν (230)
(231)
Taking into a1313ount radiative ee13ts the equations of radiative hydro-
dynami13s in the 13o-moving frame of the sho13k (Mihalas and Mihalas 1999
Drake 2006) 13an be written as
partρ
partt= minusnabla(ρu) (232)
ρ
(
partu
partt+ unablau
)
= minusnabla(P + Prad) (233)
part
partt
(
ρu2
2+ ρǫ+ Erad
)
+nabla
[
ρu
(
ǫ+u2
2+
P
ρ
)
+ (Erad + Prad)u
]
= minusnablaFrad
(234)
As mentioned previously the 13ontributions linked to Prad and Erad are
ee13tive only at very high temperature (ie velo13ity) However for the
radiative sho13ks of interest in this work the 13ontribution of the radiative
ux Frad is non-negligible 13ompared to ρu3 These sho13ks are thus in the
radiation ux dominated regime
Most of the experimental sho13k waves are in this regime These hyd-
rodynami13 equations are strongly non-linear In the present work the one-
dimensional radiative-hydrodynami13 13ode `HELIOS (Ma13Farlane Golovkin
and Woodru 2006) has been employed to derive the physi13al parameters
of the sho13k in various 13onditions as presented in the following se13tion HE-
LIOS will be dis13ussed in detail in the next 13hapter
22 RADIATIVE SHOCK REGIMES 15
22 Radiative sho13k regimes
Radiative sho13ks are strong sho13ks (Ma13h number Mgtgt1) whi13h attain
high temperatures and thus are the sour13e of intense radiation (Mihalas
and Mihalas 1984 Zeldovi13h Zeldovi13h and Raizer 2002 Drake 2006)
Depending on the opa13ity the radiation emitted from the sho13k may be ab-
sorbed by the pre-sho13k region indu13ing its pre-heating Far away from the
dis13ontinuity the stru13ture of the upstream medium is determined by the
absorption On its side the stru13ture of the transition layer of the down-
stream medium 13lose to the dis13ontinuity depends on the balan13e between
the emission (13ooling) and the absorption (heating) Thus the full stru13ture
of the sho13k mainly depends on the variation of the opa13ity χ (13m
2g) The
opti13al depth (τ(s)) measured from the position of the jump is then the
relevant parameter for the absorption It is given by
τ(s) =
int s
sjump
χ(sprime)ρ(sprime) dsprime (235)
where sprime 13hara13terizes the path of the radiation As we are interested in
the mean frequen13y averaged radiative ux the relevant opa13ity here will be
the averaged Rosseland opa13ity whi13h will be dened in due 13ourse
If τ gtgt 1 the regime is referred as opti13ally thi13k while in the opposite
13ase (τ lt 1) it is opti13ally thin Flux dominated radiative sho13ks have
been 13lassied depending on the opti13al depth of upstream and downstream
regimes
An attempt of radiative sho13k 13lassi13ation has been performed by Drake
(Drake (2006)) a1313ording to the opti13al the thi13kness of the upstream and
downstream regimes resulting in the denition of four dierent regimes
namely a thin-thi13k thin-thin thi13k-thi13k thi13k-thin radiative sho13ks
Following this author a thin-thi13k type of radiative sho13k probably exists
only in theory In the 13ase of a thin-thin radiative sho13ks both upstream
and downstream regimes are opti13ally thin and the sho13k radiates away
its energy in both dire13tions This regime of radiative sho13k is present in
various astrophysi13al 13ontexts for instan13e in the internal sho13ks of stellar
jets (Hartigan 2003) It is possible to generate these type of sho13ks in the
laboratory in very low-density gases In thi13k-thi13k radiative sho13ks both
upstream and downstream zones are opti13ally thi13k This regime o1313urs
in the stellar interiors The radiation 13oming from the hot downstream
region heats the 13ooler upstream material and forms a pre13ursor The fourth
regime is the thi13k-thin radiative sho13k whi13h is 13hara13terized by an opti13ally
thi13k downstream and an opti13ally thin upstream region The radiation
from the heated downstream region passes ahead of the sho13k and 13reates a
13ooling layer in the downstream before rea13hing the nal state There is no
absorption of the radiation in the upstream region and therefore 13ontrary to
16 CHAPTER 2 RADIATIVE SHOCK WAVES
the thi13k-thi13k 13ase there is no radiative pre13ursor Sho13ks generated in the
laser experiments are usually between the thi13k-thi13k and thin-thin regimes
221 Typi13al radiative sho13k waves with a pre13ursor
The radiation from the post-sho13ked region just behind the sho13k passes
ahead of the sho13k and generates the pre13ursor However it also leads to a
radiative 13ooling layer downstream that extension depends on the opa13ity
A typi13al stru13ture of su13h sho13ks is presented in M13Clarren et al (2010)
In these 13ases the 13ompression in the 13ooling layer behind the sho13k is
mu13h higher than that in the hydrodynami13 13ase This kind of sho13k is
most 13ommonly generated in the laboratory experiments (Bozier et al 1986
Keiter et al 2002 Reighard et al 2006 Fleury et al 2002) In astrophysi13s
these sho13ks are present for instan13e in supernovae when the blast wave
emerges from the stellar interior (Ensman and Burrows 1992)
A detailed 13lassi13ation of the sho13ks presenting a radiative pre13ursor
is reported in several referen13es (Zeldovi13h Zeldovi13h and Raizer 2002
Ensman 1994 Vaytet et al 2013)
In the 13ase of sub-13riti13al sho13k the temperature just ahead of the sho13k
front is smaller than the post-sho13k temperature Su13h sho13ks are relatively
weak and hen13e the radiative ux transfer from the post-sho13k to the pre-
sho13k is small A piston moving with a 13onstant speed of 12 kms in Xenon
gas at 01 bar is able to generate su13h a sub-13riti13al radiative sho13k The
resulting mass density and ele13tron temperature proles have been presented
at 10 ns in the Fig 25a
The other regime of radiative sho13k is known as super13riti13al in whi13h
the pre-sho13k and the nal post-sho13k temperatures are the same The mass
density and ele13tron temperature proles of super13riti13al sho13k driven by a
13onstant moving piston with speed 45 kms in Xenon at 01 bar are presented
in Fig 25b
The speeds a13hieved in the PALS experiment range between sim 15 - 55
kms for pressure 13omprised between 01- 06 Our simulations indi13ate
that depending on the speeds the regimes 13over either the sub13riti13al or
super13riti13al 13ases
23 Radiative sho13k waves experiments
Radiative sho13k waves have been studied experimentally for more than a
de13ade on high-energy lasers fast magneti13 pin13h ma13hines and pulsed po-
wer fa13ilities Su13h fa13ilities are able to a13hieve strong sho13k 13onditions
(high-temperature high-pressure) over a very small volume and time Va-
rious diagnosti13s are able to re13ord the plasma 13onditions whi13h are then
13ompared with the results of numeri13al simulations The results of these
experiments are also employed to validate numeri13al 13odes some of them
23 RADIATIVE SHOCK WAVES EXPERIMENTS 17
(a)
(b)
Figure 25 Simulated ele13tron temperature (Te) ion temperature (Ti) and mass
density (ρ) proles at 10 ns of a radiative sho13k propagating with a 15 kms (a sub-
13riti13al) and 48 kms (b super13riti13al) The sho13k is propagating in Xenon at 01
bar (ρ = 54 times 10
minus4g13m
minus3 at 298 K) The initial temperature in Xenon is set to
1eV The two Ma13h numbers are respe13tively equal to 13 and 43 The radiation and
ionisation ee13ts are in13luded in the simulation whi13h is performed for two dierent
temperatures (Te 6= Ti) and a negligible thermal 13ondu13tivity (10
minus05WmK) For
this simulation the sho13k is laun13hed by a 100 microm thi13k Aluminium piston moving
at the speed of 12 kms and 45 kms for 13ase (a) and (b) respe13tively The position
x=0 13orresponds to the position of the XeAl interfa13e at time t=0
18 CHAPTER 2 RADIATIVE SHOCK WAVES
being relevant to the astrophysi13s like FLASH (Fryxell et al 2000) and
HERACLES (Gonzaacutelez Audit and Huynh 2007)
231 Laser-driven sho13k experiments
Most of laboratory experiments on radiative sho13ks have been performed
on large-s13ale laser fa13ilities (Bouquet et al 2004 Chaulagain et al 2015
Gonzaacutelez et al 2006 Reighard et al 2006 Stehleacute et al 2010 Doss et al
2009 Drake et al 2011 Diziegravere et al 2011 Stehleacute et al 2012) in noble
gases at pressures between 01 and 1 bar With laser intensities on the
target 13omprised between 1014 and 1015 W13m
2 these experiments allowed
to re13ord sho13k speeds ranging between 40 and 150 kms
Bozier et al (1986) for the rst time experimentally eviden13ed the ge-
neration of a radiative wave propagating ahead of a super13riti13al sho13k in
xenon (pressure up to 6 bars) at the CEAs Limeil laser fa13ilities (irradian13e
of 5times 10
14W13m
minus2) In this experiment the laser beam (wavelength of 106
microm duration 1 ns) of 40 J energy was fo13used on an aluminum foil a13ting
as a piston A sho13k of speed sim 50 kms was produ13ed in the gas A visible
streak 13amera was used to re13ord the pre13ursor emissivity over 10 ns
Later Keiter et al (2002) performed a similar experiment at the OMEGA
laser fa13ility (25 kJ 1ns fo13al spot diameter 600 microm) still in a planar
geometry The sho13k propagated here in a low-density SiO2 aerogel foam
(ρ sim 96 mg13m
3) In this experiment X-ray absorption spe13tros13opi13 data
were re13orded to qualitatively predi13t the temperature of the pre13ursor
In the year 2000 a new radiative sho13k experiment was performed in
Xenon at 02 bar at the LULI laser fa13ility (sim 100 J λ = 053 microm 4-
6times 10
13W13m
minus2) (Fleury et al (2002) Bouquet et al (2004)) This expe-
riment used a three-layered piston 13omposed of a thin polystyrene a13ting
as ablator (2 microm thi13k) a titanium foil (3microm) a13ting as x-ray shield and a
layer of a polyethylene foam (25 microm) as an a1313elerator Further a VISAR
(velo13ity interferometer system for any ree13tor) and a Ma13h - Zehnder in-
terferometer were used as a diagnosti13s to measure the front sho13k speed and
the pre13ursor ele13tron density Two streak 13ameras re13orded the sho13k pro-
pagation in longitudinal and transverse dire13tions This was the rst attempt
to estimate ele13tron density within the pre13ursor Their measurements indi-
13ate an ele13tron density ranging between 1018minus1020cm3 The experimental
results were in qualitative agreements with 1D numeri13al MULTI and FCI
simulations However the speed of the pre13ursor in these 1D simulations
was too large This was then explained later when multidimensional ee13ts
were in13luded in the simulations (Leygna13 et al 2006 Gonzaacutelez Audit and
Stehleacute 2009)
Complementary experiments were then performed by Gonzaacutelez et al
(2006) at the PALS laser fa13ility (60 J 035 ns 438 nm spot diameter 600microm
sim 7times 10
13W13m
minus2) in Xe gas The goal was to study the sho13ks over longer
23 RADIATIVE SHOCK WAVES EXPERIMENTS 19
times and to inspe13t the ee13ts of the walls albedo A doubled layered pis-
ton made of 10microm polystyrene and 05 microm gold was used The pre13ursor
edge was re13orded by shadowgraphy using a visible streak 13amera The ex-
perimental results were 13ompared with 2D numeri13al simulations performed
with HERACLES (Gonzaacutelez Audit and Huynh 2007) This 13omparison
allowed quantifying the albedo of the walls It also showed that the radia-
tive losses at the walls lead to a small 13urvature of the ionization front and
to a redu13tion of its longitudinal extension (Leygna13 et al 2006 Gonzaacutelez
Audit and Stehleacute 2009)
In parallel another experiment was performed at the OMEGA laser fa-
13ility (035microm spot sim 700 - 800microm sim 2times 10
15W13m
minus2) by Reighardt et
al (Reighard et al (2006) Reighard (2007)) The sho13k was imaged by an
X-ray ba13k-lighter at dierent times whi13h enabled to inspe13t for the rst
time the shape of the sho13k front At higher speeds (sim 100 kms) X-ray
radiography pointed out a 13ollapse of the post-sho13k (Reighard et al 2006)
whi13h was attributed to the radiation losses trough the sho13k front For
these high-speed 13onditions the wall heating leads to the development of se-
13ondary wall sho13ks intera13ting with the primary sho13k (Doss et al 2009)
Su13h wall sho13ks have not yet been observed at lower speeds
Subsequently an experiment by Stehleacute et al (2010) was performed at
PALS laser fa13ility with the similar laser parameters as taken in Gonzalez
et al (2006) In this experiment a Ma13h-Zehnder interferometer and a
streak 13amera were used to re13ord the sho13k propagation over 50 ns Time-
integrated XUV spe13tros13opi13 analysis between 16 and 22 nm of sho13k was
performed The experiment showed that of a quasi-stationary regime was
rea13hed after 20 ns
The rst experimental study in Argon gas was reported in 2012 by Vis13o
et al (2012) at OMEGA fa13ility (035microm spotsim 600 microm sim 7times 10
14W13m
minus2)
The ele13tron temperatures in the radiative pre13ursor and sho13k-front were
measured to be 34 eV and 60 eV respe13tively using X-ray Thomson s13atte-
ring diagnosti13
In addition a new experiment at PALS (Jungwirth et al 2001) was
presented by Chaulagain et al (2015) using for the rst time a Zn x-ray
laser for XUV imaging at 21 nm This diagnosti13 allowed imaging both the
pre13ursor and post-sho13k stru13ture of radiative sho13ks in Xe gas at 03 bar
In this experiment where the sho13k wave did not ll the tube 2D ee13ts
be13ame pronoun13ed The experimental results were su1313essfully 13ompared
with the results of 2D ARWEN simulations (Cotelo et al 2015)
A summary of these dierent experiments and their prin13ipal diagnosti13s
is presented in table 21
20
CHAPTER2RADIATIVESHOCKWAVES
Experiment Laser fa13ility I (times 1013) λ amp τ ρ0 u Prin13ipal diagnosti13s
W13m
minus2 microm ns mg 13m
minus3
km s
minus1
Bozier et al (1986) CEAs Limeil 20-50 0351 1 10 -30 50-80 Visible streak Camera
Fleury et al (2002) LULI 5 053 05 1 60-65 Visible streak Camera
Visible interferometry
VISAR
Keiter et al (2002) OMEGA 35- 85 0351 1 5 -15
lowast
100 x-ray spe13tros13opy
Gonzaacutelez et al (2006) PALS 10-15 0438 035 1 65 Visible streak 13amera
Visible interferometry
Reighard (2007) OMEGA 70-100 035 1 6 100-140 x-ray ba13k-lighting
VISAR
Stehleacute et al (2010) PALS 10-20 1315 035 1 60 Visible Interferometry
XUV spe13trometer
Stehleacute et al (2012) PALS 10-20 1315 035 15 50-55 XUV imaging
XUV fast Si diodes
Vis13o et al (2012) OMEGA 70 035 1 196
lowastlowast
100-150 x-ray Thomson S13atte-
ring
Chaulagain et al (2015) PALS 10-20 1315 035 15 50 XUV fast Si diodes
Table 21 Summary of the radiative sho13ks experiments and out13omes in dierent laser installations All the experiments are performed
in Xe ex13ept Keiter et al (2002) and Vis13o et al (2012) In these two experiments the rst uses SiO
lowast
2 and the se13ond uses Ar
lowastlowast
as a
sho13ked medium
23 RADIATIVE SHOCK WAVES EXPERIMENTS 21
Laser driven radiative sho13ks Goal of my thesis
All these experimental studies have been fo13used on the 13ase of single ra-
diative sho13ks propagating in a tube However in astrophysi13al 13onditions
radiative sho13ks often intera13t with a denser medium leading to the deve-
lopment of ree13ted and transmitted sho13ks A few representative examples
of su13h phenomena are the intera13tion of supernovae remnants with dense
mole13ular 13louds (Gonzaacutelez Audit and Stehleacute 2009) the a1313retion sho13ks
on the photosphere of T-Tauri stars (Orlando et al 2013) and the bow
sho13ks at the head of stellar jets (Hartigan 1989 Raga et al 1999) The
13ollision (or the intera13tion) of two radiative sho13k waves is obviously a rare
astrophysi13al event and the template 13ase of supernova remnant DEM L316
(see Fig 1 of Williams et al (1997)) is still the subje13t of debates (Williams
et al 2005 Toledo-Roy et al 2009 Velarde et al 2006) as the observation
of these two dierent sho13ks 13an be also interpreted as the superposition of
two blast waves in the eld of view of the teles13ope In this regard the de-
velopment of dedi13ated laboratory experiments to the study of propagation
and intera13tion of 13ounter-propagating sho13k waves is important as a tool to
13hara13terize su13h events through their spe13i13 signatures
In this thesis I will present the results of experiments performed at the
Prague Asterix Laser System (PALS) fa13ility (Jungwirth et al (2001)) on
the study of the intera13tion of two radiative sho13k waves The experimental
obje13tives were
To generate two sho13ks of dierent speeds propagating in opposite
dire13tions in the tube lled with Xenon gas at low pressure (lt 1 bar)
with speeds 13omprised between 12 and 55 kms
To re13ord the ele13tron density by using Ma13h-Zehnder Interferome-
ter over a duration 13omprised between 50 and 200 ns This allowed
studying the ee13t of pre13ursor intera13tion on to the other one
To determine the temperature and the ion 13harge of gas using a time
and spa13e integrated XUV emission spe13tros13opy
The experimental results were interpreted with the help of one-dimensional
simulations using the Lagrangian radiation hydrodynami13 13ode HELIOS
232 Ele13tromagneti13ally laun13hed sho13k by a 13ompa13t pul-
sed power devi13e
As dis13ussed in the previous se13tion most of the plasma laboratory astrop-
hysi13s studies are mainly performed on large-s13ale laser fa13ilities addressing
pure hydrodynami13 radiative sho13ks at very high speed (50 - 150 kms)
and moderate pressure (01 - 1 bar) Complementary to laser experiments
22 CHAPTER 2 RADIATIVE SHOCK WAVES
dis13harge produ13ed plasma has also been a well-established method for ge-
nerating the strong sho13ks (Kondo et al 2008 2009ab Larour et al 2015)
Su13h devi13e presents a high exibility and a larger repetition rate whi13h al-
lows a deeper understanding of sho13k physi13s in 13onditions of astrophysi13al
interest for instan13e in the 13ontext of stellar jets
In prin13iple a 13oaxial ele13trode pair lled with gas followed by an intense
ele13tri13al surfa13e dis13harge generates a thin layer of plasma (plasma sheath)
at the bottom of the ele13trode Due to the high rising 13urrent in the plasma
layer a magneti13 pressure is produ13ed whi13h a1313elerates the plasma layer
also termed as plasma fo13us (Gonzalez Brollo and Clausse 2009) During
this motion the plasma sheath a1313retes a noti13eable part of the ba13kground
gas Experiments performed by Serban and Lee (1995) in the light gasses
H2 D2 have shown fast axial sho13k waves (100 kms) but the plasma sheath
was annular
Later on 13ompa13t pulsed power ele13tri13 generators were able to laun13h
astrophysi13ally relevant strong sho13ks in low-pressure noble gases (Kondo
et al 2006) These authors reported sho13k of speed 45 kms using 13oni13al
13oaxial ele13trodes and Xenon gas (006 - 11 mbar) The ion - ele13tron energy
transfer and ionization relaxation pro13ess as also the radiation ee13t were
investigated by Kondo et al (2008 2009ab)
My work on ele13tromagneti13ally laun13hed sho13k experiment
My work fo13using on the ele13tromagneti13ally laun13hed strong sho13ks rele-
vant for astrophysi13s aimed to design and to optimize an ele13tromagneti13
generator to be able to produ13e strong sho13ks in noble gasses at low pres-
sures of few mbar The ele13tromagneti13 generator is a powerful a1313elerating
devi13e whi13h eje13ts a quasi-planar plasma sheath out of a set of 13oaxial 13o-
ni13al ele13trodes where a pulsed 100-kA 13urrent is passing
Due to the ele13tri13al dis13harge in the gas a hot and fast moving plasma
is generated whi13h travels along the ele13trodes The high-intensity 13urrents
produ13e a magneti13 eld of several Teslas whi13h a13ts as a piston a1313elerating
an annular plasma sheath and driving a strong sho13k towards the top of the
13one I have employed a simple model to optimize the operation parame-
ters With su13h optimization our ele13tromagneti13 generator should produ13e
strong sho13ks over few nanose13onds
Our preliminary experiments show that the generator is 13apable of laun-
13hing supersoni13 sho13ks in Argon in the form of a thin plasma layer with
the speed of sim 1 - 30 kms Three-dimensional MHD simulation performed
by A Ciardi is 13onsistent with the rst observations This is presented in
detail in the 13hapter 6 of the thesis
Chapter 3
1D Simulations
Contents
31 HELIOS 23
32 LTE approximation 24
321 Mean opa13ity 25
322 Equation of State 27
33 Single radiative sho13k waves 27
34 Intera13ting radiative sho13k waves 40
35 Summary 44
Radiative sho13ks are present in various astrophysi13al pro13esses implying
extreme 13onditions Laboratory experiments then allow investigating the un-
derlying physi13al pro13esses whi13h take pla13e in these 13onditions and whi13h
are not observable from the Earth due to a la13k of spatial resolution of the te-
les13opes instruments However experiments are not always straightforward
to interpret and the help of numeri13al simulations be13omes pre13ious
In this 13hapter I will present with the help of 1D radiative hydrodynami13
simulations a brief overview of the physi13s of isolated radiative sho13ks and
of two intera13ting radiative 13ounter-propagating sho13k waves
31 HELIOS
Dierent numeri13al approa13hes are used for the simulation of laboratory
plasmas Some of them use xed grids Other introdu13e the possibility of
renement (on the y) in some meshes (Adaptative Mesh Renement) to
maintain a good resolution in the interesting parts of the plasma Another
approa13h is based on meshes following the uid (Lagrangian des13ription)
whi13h means that the meshes move with the uid and no matter may go
through a mesh to another The form of the equations dier in this 13ase (see
for instan13e Orban et al (2013))
HELIOS is su13h a one-dimensional Lagrangian radiation hydrodynami13
13ommer13ial 13ode (Ma13Farlane Golovkin and Woodru 2006) and I have
used it to simulate our experiment of intera13ting sho13k waves This 13ode
is easy to handle and has the presently interesting 13apability to be able to
simulate the 13ase of two sho13k waves propagating in opposite dire13tions and
laun13hed by two dierent lasers HELIOS may des13ribe non-LTE plasmas as
23
24 CHAPTER 3 1D SIMULATIONS
it in13ludes an in-line 13ollisional radiative (HELIOS-CR) model for 13omputing
non-LTE atomi13 level populations at ea13h time step of the hydrodynami13s
simulation However I used HELIOS in the LTE approximation whi13h is
appropriate to our experimental 13onditions (Rodriguez et al 2011) as will
be dis13ussed in the next se13tion
The 13ode solves the uid equations of motion using the pressure 13ontri-
butions of ele13trons ions and radiation It may des13ribe the ele13trons and
ions as two intera13ting uids at respe13tive temperatures Te and Ti Ther-
mal 13ondu13tion is des13ribed within a ux-limited ele13tron 13ondu13tion model
based on the Spitzer 13ondu13tivity (Burhop and Spitzer 1957) and the laser
energy deposition is 13omputed with an inverse Bremsstrahlung model ()
The radiation emission and absorption terms are introdu13ed in the equa-
tion of energy for the ele13trons and in the radiation transport equations
One of the methods whi13h are proposed is the ux-limited multi-group
radiation diusion model (Gonzaacutelez et al 2015) where the radiative ux is
proportional to the gradient of the radiative energy and is inversely propor-
tional to the Rosseland opa13ity (Dobbs-Dixon Cumming and Lin 2010)
The expression is pondered by a ux-limited diusion 13oe13ient in order
to obtain the good opti13ally thin limit This 13oe13ient follows the Larsen
expression des13ribed in Olson Auer and Hall (2000) The se13ond method is
a (time independent) multi-angle short-13hara13teristi13s s13heme In our 13ase
where radiation and hydrodynami13s are strongly 13oupled and thus the ra-
diation eld varies rapidly we used the diusion model together with LTE
Plan13k and Rosseland multi-groups opa13ities
In addition our version of HELIOS uses the PROPACEOS (Ma13Farlane
Golovkin and Woodru 2006) Equation of State (EOS) and multi-group
opa13ity tables whi13h are generated by the 13ode
1
and it is not possible to
in13lude any other opa13ity or EOS table
32 Lo13al Thermodynami13 Equilibrium
The knowledge of the thermodynami13al 13onditions is required for an ade-
quate simulation of the plasma des13ription
The Lo13al Thermodynami13 Equilibrium (LTE) regime is valid at large
ele13tron densities In that 13ase the 13ollisions between the ele13trons and the
ions and atoms are so frequent that they maintain at ea13h time a steady-
state populations of ions Moreover within a given ioni13 stage the popu-
lations of the dierent energy levels follow the Boltzmann equilibrium and
the populations of the dierent ions the Saha-Boltzmann equation (Fadeyev
and Gillet 2001)
On the 13ontrary in the 13ase of Non Lo13al Thermodynami13 Equilibrium
(NLTE) regime one has to take into a1313ount the ex13itationdeex13itation
1
see http wwwprismminus cscomSoftwarePROPACEOS
32 LTE APPROXIMATION 25
indu13ed by the ele13tron 13ollisions and various radiative pro13esses This leads
to a set of dierential time dependent equations (rate equations) whi13h has
to be solved through and adequate Collisional Radiative model (CR) (Kunze
2009)
Therefore it is 13ru13ial to determine if the regime is LTE or NLTE A
13riterium is given by Griem (2005)
In our experiment the mass density ranges between sim 10
minus4and 10
minus1
g13m
3 while the temperature values are 13omprised between andsim 0 to 50 eV
The pre13ise 13al13ulations performed by Rodriguez et al (Fig2 of Rodriguez
et al (2011)) for Xenon indi13ate that the thermodynami13al 13onditions of
our radiative sho13k experiments 13orrespond to the LTE regime
321 Mean opa13ity
The opa13ity is a fundamental ingredient of the radiative transfer equations
whi13h provide the spe13i13 intensity and its moments (radiative ux energy
and pressure) These last ones enter in the uid equations (see se13tion 213
in 13hapter 2) The 13oupling of the uid equations with the radiative transfer
equation is demanding in terms of 13omputational resour13es and most of the
13odes use simplied radiation transport equations whi13h require the know-
ledge of adequate frequen13y averaged opa13ity 13oe13ients The form of the
average of the opa13ity is not obvious as the average has to be done on the
radiative ux or energy whi13h are unknown before the 13omputation Then
the 13ommonly used opa13ities are the Plan13k and Rosseland mean opa13ities
(Mihalas 1978 Seaton 1987)
The Rosseland mean opa13ity (κR in 13m
minus1and χR = κRρ in 13m
2g) isderived by 13onsidering harmoni13 weighting over the temperature derivative
of the Plan13k fun13tion It gives the 13orre13t radiative ux in the opti13ally
thi13k diusion limit The average is performed over the total mono13hromati13
opa13ity κtotν = κabsν +κscatν (in 13m
minus1) where in the present 13ase the s13attering
13ontribution 13omes from the ele13tron Thomson s13attering
1
κR=
int
infin
01
κtotν
dBν
dT dνint
infin
0dBν
dT dν(31)
The Plan13k mean opa13ity (κP in 13m
minus1and χP = κP ρ in 13m
2g) employs
the normalized Plan13k bla13kbody energy density distribution as a weighting
fa13tor It will give the 13orre13t value for the integrated thermal LTE emission
of an opti13ally thin plasma and is expressed as
κP =
int
infin
0 κabsν Bνdνint
infin
0 Bνdν(32)
In the multi-groups des13ription the radiation transport takes into a13-
13ount the propagation of radiation in N dierent wavelength bands (νk minusνk+1 k = 1 N) Hen13e for instan13e the medium may be transparent for
26 CHAPTER 3 1D SIMULATIONS
some wavelengths and opaque for other ones The radiation transport equa-
tions split into N 13ontributions ea13h of them being asso13iated to adequate
opa13ity 13oe13ients The expression of the 13oe13ients is similar to the pre-
vious grey 13ase (N =1) ex13ept that the integrals are performed between
the boundary frequen13ies of the group
In HELIOS the mono13hromati13 opa13ities κν are 13omputed for the die-
rent radiation groups using the ATBASE atomi13 suite of 13odes Energy levels
and other atomi13 data relevant for bound-bound and bound-free transitions
are 13omputed within a 13onguration intera13tion model using Hartree-Fo13k
wave-fun13tions (Ma13Farlane Golovkin and Woodru 2006)
Dividing the opa13ity by the mass density redu13es the variation of this
quantity with ρ The literature then often uses the mass absorption 13oe-
13ient (also termed as opa13ity) whi13h is dened as
χν = κνρ (33)
Figure 31 PROPACEOS Mono13hromati13 opa13ity versus the photon energy in
eV of Xenon at 10 eV and at two ele13tron densities respe13tively equal to 1018 and
1020 13mminus3
An example of the variations of the PROPACEOS opa13ity χν versus the
frequen13y in eV is given in Fig 310a for Xenon at 10 eV and two ele13tron
densities respe13tively equal to 1018 and 1020 13m
minus3 whi13h do 13orrespond to
typi13al 13onditions of the pre13ursor and post-sho13k of our laser generated
radiative sho13ks
33 SINGLE RADIATIVE SHOCK WAVES 27
322 Equation of State
The equation of state (EOS) denes the dependen13e of the pressure ionisa-
tion internal energy with the mass density and temperature Several models
do exist in the literature
Our version of HELIOS for LTE plasmas may handle polytropi13 or PRO-
PACEOS equation of state Unfortunately there are not many details about
this last method The authors of the 13ode mention that it uses a QEOS-type
model (More et al 1988) in the strong 13oupling regime (ie at relatively
high densities and low temperatures) and an isolated atom model whi13h
uses a detailed atomi13 stru13ture modeling in the weak 13oupling region The
properties between the weak and strong 13oupling regimes are said to be obtai-
ned by interpolating in a thermodynami13ally 13onsistent manner This EOS
in13ludes the 13ontributions from the translation of ions and ele13trons ele13-
tron degenera13y atom ionization and ex13itation and Coulomb intera13tions
(Debye - Hu13kel 13orre13tion)
Comparisons on spe13i13 13ases have been performed by the authors of the
13ode with Los Alamos SESAME data for CH Al and Au showing a good
agreement when using these two equations of state
In this 13hapter HELIOS 13ode will then be employed to simulate the
detailed 13hara13teristi13s of single and 13ounter-propagating radiative sho13ks
driven by one or two lasers in the 13onditions of our experiments (Chapter 5)
The target 13ell (4-mm length) is lled with Xenon gas at 01 bar This valueof pressure is representative and 13orrespond to standard ambient tempera-
ture ie 298 K and not the initial temperature 13onsidered in the simulations
This terminology is maintained throughout the thesis Two lasers are inte-
ra13ting at both ends with two gold-13oated `CH foils 11microm CH and 06 micromAu) ea13h of them resulting in a piston of total thi13kness 116 microm 13losing the
13ell
33 Single radiative sho13k waves
Typi13al simulations of the radiative sho13ks espe13ially multidimensional ones
whi13h are time-13onsuming use a frequen13y independent (grey or one group)
opa13ities However then the opa13ity of the gas under investigation presents
strong variations with the frequen13y a multi-group approa13h may be suitable
(Vaytet et al 2011 2013) When the number N of groups tends to innity
ea13h group opa13ity κkR and κkP tends to the lo13al mono13hromati13 opa13ity κν whi13h solves the question of the adequate opa13ity average
In the following I will investigate the inuen13e of the opa13ity and the
number of groups on the stru13ture of a sho13k wave propagating at sim 50 kms
in Xenon at 01 bar To this purpose I have performed HELIOS simulations
for a laser of uen13e 28000 Jcm2whi13h is able to drive a sho13k with the
relevant velo13ity In addition I have also analyzed the ee13ts of taking into
28 CHAPTER 3 1D SIMULATIONS
a1313ount the dierent ion and ele13tron temperatures
Referen13e simulation (1 group Te = Ti = T )
A result of the simulation is reported at 10 ns in the Fig 32 whi13h reports
the variations of ele13tron density (Ne) temperature (T ) mean 13harge (Z)and mass density (ρ) The initial 13onditions are 45 times 10
minus4g13m
3and T =
1 eV
The sho13k front is lo13ated at sim 0047 13m (speed of 47 kms) At this
position the temperature peaks at 22 eV The bla13k verti13al dotted line shows
the position of the interfa13e between the piston and the ba13king Xenon gas
This interfa13e is lo13ated at sim 20 microm from to sho13k front The post-sho13k gas
mass density pressure ele13tron density and ion 13harge are respe13tively equal
to 002 g13m
3 11times 10
4bars 52times 10
2013m
3and 56 The high 13ompression
is due to ionisation and radiation 13ooling (13hapter 2) Its temperature of 12
eV is the same than in the pre13ursor where it is 13onstant over the 2 mm of the
simulation The pre13ursor ele13tron density and mean 13harge are respe13tively
equal to 18 times 10
1913m
minus3and 7 The pre13ursor is not 13ompressed
The variations of the Rosseland opa13ities (13m
2g
minus1and 13m
minus1) with the
distan13e are reported in the Fig 33 The Rosseland opa13ity is low in the
pre13ursor Hen13e 600 13m
2g
minus113orresponds to 05 13m
minus1 whi13h means that
an opti13al depth of 1 is rea13hed at 2 13m from the sho13k front This explains
the at prole of the pre13ursor With a typi13al value in the post-sho13k of
2577 in 13m
2g
minus1 this opa13ity is now 53 13m
minus1 and an opti13al depth of 1 is
here rea13hed at 200 microm from the sho13k front (whi13h is mu13h smaller than
the post sho13k extension)
This simulation was performed using the Spitzer thermal 13ondu13tivity
To 13he13k the 13ondu13tivity ee13t I performed another simulation with a
negligible thermal 13ondu13tivity (ie 10
minus12WmK) for the Xenon layers
only The results are reported in the Fig 34 The interfa13e between the
piston and the ba13king Xenon gas is still 13lose to the sho13k front Ex13ept
for the peak of temperature whi13h is higher (sim 29 eV instead of 22 eV) and
thiner the sho13k stru13ture is similar to the previous 13ase
Two uids with dierent temperatures Te and Ti (1 group)
In the previous se13tion HELIOS simulations have been 13arried out for the
13ase of equal ion and ele13tron temperatures I will now investigate the ee13t
of su13h approximation on the simulated results employing 1) Spitzer and 2)
negligible 13onstant thermal 13ondu13tivity in the Xenon layers It is important
to note that this two uids approximation 13an not be restri13ted to the Xenon
layers only and thus it is valid from the piston (CHAu) up to the Xenon
gas Thus the dynami13s and 13onditions of the piston are also modied whi13h
may slightly modify the sho13k velo13ity
33 SINGLE RADIATIVE SHOCK WAVES 29
(a)
(b)
Figure 32 Mass density and temperature (a) ele13tron density and mean 13harge
(b) at 10 ns for a radiative sho13k of speed sim 47 kms in Xenon at 01 bar The
verti13al dotted bla13k lines show the position of the interfa13e between piston and
ba13king Xenon gas Zero at x-axis is the interfa13e of piston and Xenon at time zero
Spitzer thermal 13ondu13tivity has been used in the simulation
30 CHAPTER 3 1D SIMULATIONS
Figure 33 Mean Rosseland opa13ity (in 13m
2g
minus1) and opa13ity (13m
minus1) at 10 ns
for a radiative sho13k of speed sim 47 kms in Xenon at 01 bar The verti13al dotted
bla13k lines show the position of the interfa13e between piston and ba13king Xenon gas
Zero at x-axis is the interfa13e of piston and Xenon at time zero Spitzer thermal
13ondu13tivity has been used in the simulation
33 SINGLE RADIATIVE SHOCK WAVES 31
(a)
(b)
Figure 34 (a) Mass density and temperature (b) ele13tron density and mean
13harge at 10 ns for a radiative sho13k of speed sim 47 kms in Xenon at 01 bar The
verti13al dotted bla13k lines show the position of the interfa13e between piston and
ba13king Xenon gas Zero at x-axis is the interfa13e of piston and Xenon at time zero
A negligible 13onstant thermal 13ondu13tivity (10
minus12WmK) have been used in the
simulation for Xenon
32 CHAPTER 3 1D SIMULATIONS
(a)
(b)
Figure 35 (a) Ele13tron density and ele13tron temperature (b) Mass density and
mean 13harge at 10 ns for a radiative sho13k of speed sim 48 kms in Xenon at 01
bar for the 13ase when the ele13tron and ion temperature are dierent The verti13al
dotted bla13k lines show the position of the interfa13e between piston and ba13king
Xenon gas Zero at x-axis 13orresponds to the position of the interfa13e between the
piston and Xenon at time zero The Spitzer thermal 13ondu13tivity has been used in
this simulation
33 SINGLE RADIATIVE SHOCK WAVES 33
The Fig 35a reports the sho13k stru13ture at 10 ns for the 13ase of Spitzer
thermal 13ondu13tivity The sho13k is lo13ated at 0048 13m The ion temperature
peaks at 53 eV and the ele13tron temperature peaks at the same value of 20
eV than previously Ex13ept for the ion temperature the sho13k stru13ture is
not signi13antly 13hanged 13ompared to the 13ase of equal temperatures
Let us now inspe13t the role played by the thermal 13ondu13tivity (see Fig
36) In the 13ase of a negligible thermal 13ondu13tivity of (10
minus12WmK) for
the Xenon layers only the ele13tron temperature (and as a 13onsequen13e the
mean 13harge) peaks at higher value 25 eV (instead of 20 eV with the Spitzer
13ondu13tivity) and the ion temperature rea13hes to 405 eV (instead of 53 eV)
The rest of the sho13k is not ae13ted by this 13hange in the 13ondu13tivity This
13onrms that the width and height of the ion peak temperature are a dire13t
13onsequen13e of the thermal 13ondu13tivity
Several groups for the opa13ity one temperature
As mentioned earlier the number of frequen13y groups inuen13es the dyn-
ami13s and morphology of radiative sho13ks A set of simulations has been
performed 13onsidering (only for Xenon) dierent numbers (N) of frequen13y
group varying between 1 and 100 for Te = Ti The distribution in energy
of the groups is logarithmi13 and the grid is set up with approximately 85
of the groups having photon energies between 01 eV and 3 KeV while the
remaining 15 lie between 3 KeV and 1 MeV
The Figures 37 (a) and (b) show the variations of the ele13tron tempe-
rature with the number of groups (from 1 to 100) for the pre13ursor (a) and
the front (b) In13reasing the number of groups ae13ts mostly the pre13ursor
A similar behavior is found in Vaytet et al (2013)
It should be noted that this multigroup opa13ity treatment is applied
to all the layers (ex13ept gold only one group opa13ity is provided by the
13ompany) in the simulation This 13ould ae13t slightly the dynami13s of the
piston Con13erning the pre13ursor at this time of the simulation and taking
an a1313ountable variation in temperature of 10 the 13onvergen13e is a13hieved
after 20 groups
Variation with the opa13ity
In order to estimate the ee13t of the opa13ity I have performed several simu-
lations by in13reasing the Plan13k and Rosseland opa13ities in Xenon (Stehleacute
et al 2014b) using a 13ommon multiplier ranging between 1 to 40 Su13h
pro13edure is often used to redu13e the gap between the available and more
sophisti13ated opa13ities To ex13lude the impa13ts of the radiation 13oming from
the post-sho13k to the piston layers the opa13ity of the piston (CH and Au)
layers have been set to zero
I present in Fig 38 the results (proles of temperature mass density and
34 CHAPTER 3 1D SIMULATIONS
(a)
(b)
Figure 36 (a) Mass density and ele13tron temperature (b) ele13tron density and
mean 13harge at 10 ns for a radiative sho13k of speed sim 45 kms in Xenon at 01 bar
for the 13ase when the ele13tron and ion temperature are dierent The verti13al dotted
bla13k lines show the position of the interfa13e between piston and ba13king Xenon
gas Zero at x-axis 13orresponds to the position of the interfa13e between the piston
and Xenon at time zero In this simulation value of thermal 13ondu13tivity (10
minus12
WmK) is negligible for Xenon The peak value of ion temperature is found to be
405 eV whereas theoreti13ally this value is expe13ted to be sim 600 eV It is possible
to a13hieve the expe13ted value by in13reasing the resolution of the simulation
33 SINGLE RADIATIVE SHOCK WAVES 35
Figure 37 Ele13tron temperature proles for various numbers of frequen13y groups
N = 1 20 50 60 70 90 and 100
mean 13harge) of four 13ases 13orresponding to an opa13ity multiplier equal to
1 10 30 and 40 The ele13tron and ion temperatures are taken to be equal
In13reasing the opa13ity in13reases the absorption in the pre13ursor and thus
de13reases its length as the photons emitted by the sho13k are more rapidly
absorbed by the 13old pre-sho13k Taking as a referen13e the previous value of
the Rosseland opa13ity of 600 13m
2g
minus1and a multipli13ative fa13tor of 40 the
opti13al depth of 1 will now be rea13hed at 005 13m (instead of 2 13m without
any multiplying fa13tor)
The temperature peak value at the sho13k front also de13reases I do believe
that this 13an be attributed to the in13rease of the radiative 13ooling of the peak
of temperature with the multiplying fa13tor
I adopted here an oversimplied method in the sele13tion of the groups
boundaries A more realisti13 one would be to take into a1313ount the variation
of the opa13ity in the dierent layers with the wavelength In the 13ase of
xenon for instan13e it would be then more adapted to rene the des13ription
of the opa13ity between 5 and 100 eV (see Figure 310a)
Opa13ities 13omparison
The previous study illustrates the role played by the opa13ity for the radi-
ative sho13ks waves with a pre13ursor It seams then logi13al to 13ompare the
36 CHAPTER 3 1D SIMULATIONS
(a)
(b)
(13)
Figure 38 (a) Ele13tron temperature (a) mean 13harge (b) and mass density (13)
for four (1 10 30 and 40) multipliers of the Xenon opa13ity at 10 ns
33 SINGLE RADIATIVE SHOCK WAVES 37
PROPACEOS opa13ity whi13h is used by HELIOS with more sophisti13ated
13al13ulations
The Figures 39a and 39b report the variations of the Plan13k and Ros-
seland grey opa13ities 13omputed for the density of the pre13ursor ρ = 15 times10
minus3g13m
3by two dierent 13odes PROPACEOS (Ma13Farlane Golovkin
and Woodru 2006) and BiGBART (de la Varga et al 2011 Ogando and
Velarde 2001) PROPACEOS opa13ities and EOS used in HELIOS simula-
tions 13omputes frequen13y-dependent opa13ities along with spe13i13 internal
energies and pressures The atomi13 13ode BiGBART able to 13al13ulate two
type of opa13ities dependent on atomi13 stru13ture and frequen13y dependent
The opa13ities 13omputed by Rodriacuteguez et al (2015) are also reported for 13om-
parison The two last 13al13ulations use the FAC (Flexible Atomi13 Code) (Gu
2008) for the 13al13ulation of the atomi13 data
We see that for a temperature equal to 10 eV the Rosseland opa13ity is
equal respe13tively to 800 15000 and 8000 g13m
2 whereas the Plan13k opa13ity
is equal to 18000 65000 370000 g13m
2 This means that the PROPACEOS
opa13ity whi13h is used in HELIOS is smaller by more than one order of
magnitude than the two last opa13ities As it is not possible to 13hange the
opa13ity model within HELIOS we then de13ided to use the PROPACEOS
opa13ity with a multipli13ative fa13tor whi13h we have set equal to 20
The dis13repan13ies in the average opa13ity are a 13onsequen13e of important
dieren13es in the mono13hromati13 opa13ities This is illustrated in the Fig
310 whi13h 13ompares the mono13hromati13 opa13ities given by PROPACEOS
and Rodriacuteguez et al (2015) (see Fig 16 of Rodriacuteguez et al (2015)) at 15
eV and 15 times 10
minus3g13m
minus3) This gure indi13ates a noti13eable dieren13e
between 5 and 150 eV with more bb transitions for the last method This
explains the dieren13es observed in the 13orresponding averages
Synthesis
These dierent simulations for a sho13k propagating at sim 50 kms in Xenon
at 01 bar show that the simulations performed with only one temperature
are adequate for the des13ription of the sho13k
The opa13ity strongly ae13ts the pre13ursor extension whi13h de13reases from
more than 4 mm for 1 group to 3 mm then the number of groups is larger
than 20 The 13onvergen13e versus the number of groups is rea13hed after 20
groups
To t with more sophisti13ated opa13ities and as it was impossible to
in13lude them in HELIOS I de13ided to multiply the opa13ities of Xenon by a
fa13tor of 20 This redu13es strongly the pre13ursor extension up to sim 1 mm
at 10 ns Taking into a1313ount the present huge un13ertainty in the opa13ity
I 13onsidered that it was not ne13essary to use multi-group opa13ities This
also means that our 1D simulations will be used - only - for a qualitative
interpretation of the experimental results A quantitative 13omparison should
38 CHAPTER 3 1D SIMULATIONS
(a)
(b)
Figure 39 Plan13k and Rosseland mono13hromati13 opa13ity proles Te for ρ =15
times 10
minus3from PROPACEOS (I) BIGBART (II) and Rodriguez et al (III)
33 SINGLE RADIATIVE SHOCK WAVES 39
(a)
(b)
Figure 310 Mono13hromati13 Xenon opa13ity versus the photon energy in eV at T
= 15 eV and for ρ = 15 times 10
minus3g13m
minus3for the PROPACEOS (a) and Rodriacuteguez
et al (2015) (Fig 16 of Rodriacuteguez et al (2015)) (b) models
40 CHAPTER 3 1D SIMULATIONS
be in the future based on adequate opa13ities and multi-groups modeling
34 Intera13ting radiative sho13k waves
In order to pre13ise what is expe13ted from the experimental study I will now
investigate the 13hara13teristi13 parameters of two 13ounter-streaming sho13ks
propagating in Xenon at 01 bar (ρ = 54 times 10
minus4g13m
minus3) with equal and
dierent velo13ities To highlight the ee13t of the intera13tion I will 13ompare
the results with those obtained with 1D simulations All the simulations will
be performed with HELIOS For the Xenon opa13ity a multiplier times 20 will
be used The number of groups will be set to be 1 and the ele13tron and ion
temperatures will be supposed to be equal
Three representative sets will be 13onsidered
Case(I) two identi13al radiative sho13ks (2RS) at sim 50 kms propaga-
ting in opposite dire13tions (ie starting from the left and right end
respe13tively)
Case(II) same 13onditions but without any 13oupling with radiation
Case(III) two 13ounter-propagating radiative sho13ks propagating with
dierent speeds sim 50 kms from the left side of the 13ell and sim 20
kms from the right side
To a13hieve the aforementioned speeds on the left and right sides the
uen13es of the two laser beams have been adjusted The pulse duration is
set to 03 ns (peak at 015 ns) to reprodu13e the experimental 13onditions
detailed later in the manus13ript
Case(I) 13ounter propagating sho13ks with the same speed
This a13ademi13al 13ase is fully symmetri13al and it is equivalent to the 13ase of
one RS with a fully ree13tive boundary (for hydrodynami13s and radiation)
in the middle of the tube The Fig 311 shows the variations of the ele13tron
density (Ne) and temperature (Te) in the Xenon layers The two sho13ks
appear in Xenon at sim 2 ns and the 13ollision o1313urs at sim 38 ns At 3 ns the
pre13ursor extension is sim 008 13m whereas the post-sho13k ele13tron density
and ele13tron temperature are 78 times 10
2013m
minus3and 16 eV respe13tively The
length of pre13ursor in13reases rapidly with time and the two pre13ursors merge
suddenly at sim 8 ns After this time the merging ee13t in13reases signi13antly
It is 13hara13terized by a at 13ommon pre13ursor those ele13tron density and
temperature are in13reasing with time At the time of the 13ollision (sim 38
ns) the post-sho13k mass and ele13tron density in13rease from 0011 to 014
g13m
minus3and 67 times 10
20to 66 times 10
2113m
minus3 whereas the temperature rises
up to 39 eV The 13ollision leads to the development of two reverse sho13k
34 INTERACTING RADIATIVE SHOCK WAVES 41
(a)
(b)
Figure 311 Ele13tron density Ne (a) and ele13tron temperature Te (b) versus axial
position (along a 04 13m long sho13k tube) at 3 10 20 30 and 38 ns from HELIOS
simulations (with opa13ity times 20) for the 13ases of single sho13k of sim 50 kms (dotted
line) and two identi13al 13ounter-propagating sho13ks of sim 50 kms (solid lines) The
verti13al dotted lines show the position of the interfa13e between piston and ba13king
Xenon gas
42 CHAPTER 3 1D SIMULATIONS
waves propagating ba13k with a speed of sim 15 kms rst in Xenon and later
on in the dierent piston layers (not shown in the Figure) These reverse
sho13ks lead to a dense plasma (Ne gt 10
2113m
minus3) whi13h is not a1313essible to
the experimental diagnosti13s and will not be detailed here
To investigate the ee13ts of the intera13tion I have reported in Fig 311
the 13ase of 1RS moving from the left to the right dire13tion in the 13ell (dotted
lines) The wave propagates identi13ally to 2RS until 10 ns After this time
the proles of the temperature and ele13tron density dier strongly from the
previous 13ase and their values are lower than for the 2RS The post-sho13k
extension is slightly smaller than for the 2RS and the sho13k speed is thus
slightly smaller too This last ee13t is due to the fa13t that for the two inte-
ra13ting waves the pre13ursor is at a higher temperature than for the isolated
sho13k and thus the sho13k propagates in a warmer medium then modifying
the sho13k 13onditions (sound speed opa13ity et13 ) A similar ee13t in the post
- sho13k extension may be seen for an isolated sho13k wave propagating in a
warmer pre-sho13k gas
Case (II) Identi13al 13ounter-propagating sho13ks for an ideal gas
In order to highlight the ee13t of the radiation another simulation has been
performed with the same set of parameters as in the 13ase I however putting
the Xenon opa13ity equal to zero I still use here the PROPACEOS equation
of state The result of the simulation is presented in Fig312 The 13ollision
time is now 40 ns instead of 38 ns (thus the sho13k speed is slightly smaller)
The post-sho13k is no more 13ompressed by radiation 13ooling Its 13ompression
at 10 ns is 10 instead of 35 This 13ompression is due to the vis13ous sho13k
(fa13tor 4) followed by the ionisationex13itation of the gasThere is obviously
no radiative pre13ursor Moreover there are no dieren13es in the Ne and Tproles of the single sho13k and that of the two 13ounter-propagating sho13ks
before the 13ollision time
Case (III) Two sho13ks at dierent speeds
The spatial and temporal variations of Ne and Te are plotted at times 3
10 20 30 38 and 49 ns in Fig 313a and 313b The left and the right
sho13ks appear in Xenon at sim 2 and 3 ns respe13tively Later at 10 ns
the two pre13ursor extensions are respe13tively equal to 018 and 0034 13m
The merging of the two pre13ursors starts at sim 15 ns As expe13ted from the
values of sho13k speeds the 13ollision time o1313urs at 49 ns whi13h is delayed
in 13omparison to the 13ase (I)
In 2RS the two radiative pre13ursors merging results in a plateau for the
ele13tron density and the temperature The temperature at 13ollision time is
now 28 eV instead of 39 eV in the 13ase I and the ele13tron density rea13hes
up to 31 times 10
2113m
minus3instead of 66 times 10
2113m
minus3
34 INTERACTING RADIATIVE SHOCK WAVES 43
(a)
(b)
Figure 312 Variations of Ne (a) and Te (b) versus axial position for the 13ase
of two identi13al 13ounter-propagating sho13ks (of speeds sim 50 kms) at 3 10 20
30 35 and 40 ns as derived from HELIOS simulations For these simulations we
have negle13ted the ee13t of radiation by keeping the Xe opa13ity equal to zero
The verti13al dotted lines show the position of the interfa13e between the piston and
ba13king Xenon gas
44 CHAPTER 3 1D SIMULATIONS
It may be noted that up to this 13ollision time the post-sho13k density
13onditions and speed are identi13al for the 13ase of an isolated RS propagating
from the left (1RSL) at 50 kms and this present left sho13k This reveals
that in the 13ase of 2RS there is no noti13eable ee13t of the right sho13k with
speed 20 kms on the left post-sho13k of 50 kms
On the 13ontrary we note a dieren13e in extension of the right post sho13k
from the 2RS 13ompared with the 13ase of an isolated sho13k (1RSR) propa-
gating at 20 kms from the right whi13h indi13ates that the left sho13k of the
2RS has an inuen13e on the right post-sho13k (Fig 313a)
35 Summary
Radiative sho13k waves propagating in Xenon at 01 bar with a velo13ity of 50
kms are 13hara13terized by the development of an extended radiative pre13ur-
sor The huge 13ompression of 38 in the post-sho13k is a 13onsequen13e of both
the sho13k and the gas ionisation (fa13tor of 10) as also from the radiative
13ooling The post-sho13k and pre-sho13k temperature on both side of the peak
are identi13al indi13ating that the sho13k is super13riti13al
The spe13ta13ular role of the opa13ity has been highlighted Due to the
un13ertainty in the opa13ity used for Xenon in our simulations I de13ided that
at this stage it was not ne13essary to rene the simulation in terms of group
numbers within the 1D des13ription of the sho13k wave
Our numeri13al study gives the main 13hara13teristi13s of the intera13tion of
two 13ounter-propagating sho13k waves with speeds equal to 50-50 kms and
50-20 kms The 13ase of identi13al speeds is simpler due to the symmetry of
the problem However whatever the speeds the most important signature
of the intera13tion is the merging of the pre13ursor at 8 ns for 50-50 kms
and at 15 ns for 50-20 kms This merging is followed by a regular in13rease
with time of the ele13tron density and the temperature The 13ollision time
is 13hara13terized by a sudden in13rease of the ele13tron density by an order of
magnitude rea13hing 66 times 10
21and 3 times 10
2113m
minus3respe13tively whereas the
temperature in13reases up to 39 and 28 eV
35 SUMMARY 45
(a)
(b)
Figure 313 Variations of Ne (a) and Te (b) with axial position for the 13ase of
two non-identi13al 13ounter-propagating sho13ks (of speeds sim 50 amp 20 kms) and two
single sho13ks (dotted lines) of speeds sim 50 amp 20 kms respe13tively at 3 10 30 35
and 46 ns as derived from HELIOS simulations The verti13al dotted lines show the
position of the interfa13e between piston and ba13king Xenon gas (with opa13ity times20)
46 CHAPTER 3 1D SIMULATIONS
Chapter 4
Laser-driven radiative sho13k
Experimental Setup
Contents
41 Prague Asterix Laser System (PALS) fa13ility 48
42 Targets 49
421 Massive Targets 51
422 Gaseous Targets 51
423 Target holder 55
424 Target lling 56
43 Laser Fo13using 57
431 Fo13al Lenses and Phase Zone Plates 57
432 Fo13using of the MAIN and AUX laser beams 58
44 Diagnosti13s 59
441 Visible interferometry 61
442 XUV spe13tros13opy 65
443 Visible spe13tros13opy 67
45 Summary 68
When a short pulse of a high-power laser beam is fo13ussed on a thin foil
it transfers a huge amount of energy to the foil material Almost instantane-
ously the laser absorption heats it up to the ablation A hot and low density
13orona is generated ba13kwards by this pro13ess Beyond the ablation surfa13e
a sho13k is formed in the foil as a 13onsequen13e of the momentum 13onservation
(ro13ket ee13t) The sho13k moves inward in the foil Our 13ase of the experi-
mental setup as the foil 13loses a tube lled with gas the sho13k propagates
then in the gas where it is studied through various kinds of diagnosti13s The
foil then a13ts as a piston
We performed the experiments at the Prague Asterix Laser System (PALS)
in the Spring 2015 (ve weeks 20th April - 22th May) The obje13tive of the
experimental 13ampaign was to study the evolution of two 13ounter streaming
radiative sho13ks driven by two 13ounter-propagating lasers with an irradian13e
in the range of 10
14W13m
minus2 and therefore to 13hara13terize their intera13tion
and 13ollision In this 13hapter I will rst present a brief des13ription of the
PALS laser fa13ility and of the laser beams used in our experiment This will
47
48 CHAPTER 4 EXPERIMENTAL SETUP
be followed by a presentation of the targets design and an overview of the
general setup and of our main diagnosti13s
41 Prague Asterix Laser System (PALS) fa13ility
The Prague Asterix Laser System (PALS) is a laser fa13ility based on an
Infrared high-power iodine laser system (Asterix IV) (Jungwirth et al 2001)
Using dierent amplifying stages the laser fa13ility is able to deliver energy
up to 1 kJ in 03 ns at the fundamental wavelength 1315 nm The output
laser beam 13an be further subdivided in few auxiliary beams All auxiliary
beams may be frequen13y doubled (λ = 657 nm) or tripled (438 nm) The
PALS laser fa13ility is 13apable of ring up to two high energy laser shots per
hour Compared with solid state lasers this gas laser is known to deliver a
quite homogenous beam intensity without hot spots A spatial prole of the
PALS laser beam is shown in the Fig 41a (re13orded by PALS team during
the experimental 13ampaign) while its pulse 13hara13teristi13s are plotted with
time in the Fig 41b
(a)
(b)
Figure 41 a Spatial prole of the PALS Laser beam b Intensity prole of
laser pulse I(t) with time
For our experiments we used two independent laser beams to drive the
two 13ounter streaming sho13ks in the gaseous target For this purpose the
fundamental PALS laser beam has been subdivided after the fourth amplier
(V4) into two beams with respe13tive energy fra13tions of 60 and 40 (see
Fig 42) The most energeti13 of these laser beams is then inje13ted through
the fth (V5) amplier and its frequen13y are then tripled This beam at 438
nm will be hereafter termed termed as the MAIN laser beam
The se13ond of the two previous laser beams propagates then without any
modi13ation It will be hereafter 13alled the AUX laser beam (1315 nm)
The s13heme of the energy distribution is shown in the Fig 42 and the
13hara13teristi13s of MAIN and AUX laser are presented in Table 41
For the experiments PALS oers two va13uum 13hambers with a spheri13al
42 TARGETS 49
Figure 42 Energy distribution s13heme for MAIN and AUX laser beams
LASER MAIN AUX
Typi13al laser energy (J) 120 60
Beam diameter (mm) 290 148
Wavelength (nm) 1315 438
Pulse duration (ps) 350 350
Table 41 Nominal 13hara13teristi13s of MAIN and AUX laser beams
and 13ylindri13al geometry respe13tively The spheri13al 13hamber assembly used
in our experiments is shown in Fig 43 This 13hamber having a diameter
of 100 13m is 13losed by two entran13e 13ir13ular windows of respe13tive diameters
80 13m and 50 13m to allow the MAIN and AUX beams to penetrate in the
13hamber and to rea13h the target Ea13h of these windows is outtted with a
hinged door Several 13ir13ular ports of various diameters also allow for the
dierent diagnosti13s The target holder and some diagnosti13s are mounted
on an opti13al ben13h whi13h is de13oupled from the 13hamber walls
42 Targets
Our targets s13hemati13ally 13onsist in small tubes of 4 mm length 13losed on
both sides by two spe13i13 foils of thi13kness 11 microm on whi13h the laser beams
are fo13used (one laser per foil) with an irradian13e of about 10
1413m
minus3 The
tube is lled with gas in whi13h the radiative sho13k propagates with a velo-
13ity of 30 - 60 kms The two foils 13losing the target insure the 13onversion
through ablation and sho13k generation of the laser energy into me13hani13al
energy
In our experiments we used spe13i13 targets for alignment and 13hara13te-
rization of the beam size These targets were manufa13tured at the Pole
instrumental of Observatoire de Paris
50 CHAPTER 4 EXPERIMENTAL SETUP
Figure 43 Snapshot of the spheri13al 13hamber showing the horizontal bredboard
and on the right the fo13using MAIN lens
(a) (b)
Figure 44 Dimension drawing (a) and snapshot (b) of Aluminum massive
(mo13k) target
42 TARGETS 51
421 Massive Targets
For the laser alignment on the target we have used spe13ial targets also
termed as mo13k targets A mo13k target (Fig 44) is an aluminum blo13k
with a base Three 13uts made on it help to fo13us the laser beam at the
desired position The 13enter of ea13h of the two laser spots should be lo13ated
at the interse13tion of the verti13al and horizontal marks and the pre13ision of
this alignment is less than 20 microm
Beside alignment purpose su13h targets are used to 13hara13terize the MAIN
and AUX spot diameters in the planes whi13h 13orrespond to the lo13ation of
the two Parylene foils Hen13e the laser beam fo13ussed on these targets
generates a 13rater (Chaulagain 2015) The detailed analysis of the shape
and size of these 13raters provides a quantitative information about the ho-
mogeneity of the in13ident beam and the size of the fo13al spot A detailed
dis13ussion about the fo13alisation of the laser with a massive target will be
presented in the se13tion 432
422 Gaseous Targets
The 13ore of the gaseous targets (Stehleacute et al 2014a) 13onsists of a 13hannel of
a parallel pipe shape having the dimension of 09 times 06 times 4 mm pla13ed at
the top of an aluminum stru13ture The pipe in whi13h the sho13ks propagate
is dened by two 500 microm thi13k verti13al lateral sides made with 13oated fused
sili13a (SiO2) windows (09 times 4 mm) as shown in the Fig 45 Su13h windows
are suitable for the visible interferometry diagnosti13 whi13h uses an auxiliary
Q-swit13hed NdYLF green Evolution laser (see Se13tion 441)
The 13hannel is 13losed at the top with a window made of a 100 nm thi13k
Si3N4 membrane whi13h is itself supported is by a 200 microm thi13k sili13on frame
(26 times 4 mm) made by SILSON 13ompany Su13h membrane is transparent
to XUV radiation whi13h is ne13essary for the XUV spe13tros13opi13 diagnosti13
used in the experiment The transmission of the aforementioned material is
obtained from the 13enter for X-ray opti13s (CXRO) database
1
and is plotted
between 0 and 40 nm in Fig 48
The pistons 13losing the tube from opposite sides and whi13h will laun13h
the two sho13ks are made of Parylene-N (11 microm) 13oated by Au (06 microm)
(SCITECH 13ompany) They are glued on 01 mm thi13k Ni13kel disks with
external diameter 5 mm and internal diameter 1 mm This disk helps in
assembling the targets It also 13ontributes together with a larger V-shield
(se13tion 49) to prevent hard X-ray emitted at the laser impa13t on the foil
to rea13h the gas in the tube The Parylene layer is fa13ing the laser and plays
the role of an ablator The gold layer whi13h is 13oated on the Parylene and
is lo13ated between the Parylene and the gas aims at blo13king the X-rays
generated by the laser plasma intera13tion to rea13h and preheat the gas in
1
httphenkelblgovopti13al_13onstants
52 CHAPTER 4 EXPERIMENTAL SETUP
(a)
(b)
Figure 45 Values on the gures are in mm (a) Verti13al 13ross se13tion view of
the gaseous target (b) Horizontal 13ross se13tion view
42 TARGETS 53
Figure 46 Pi13ture of a the gaseous target on its base
Figure 47 S13hemati13 of the gaseous target
54 CHAPTER 4 EXPERIMENTAL SETUP
Figure 48 Transmission of 100 nm thi13k Si3N4 membrane (CXRO database)
Figure 49 Snapshot of a target holder with one V-shield regarding the AUX
laser whi13h will pass through the hole in this shield marked in green in the Figure
The target whi13h fa13es the two diagnosti13s (tubes) and is lo13ated behind the
V-shield is thus not visible
42 TARGETS 55
the 13ell These various elements are manually glued on the base of the target
to built the sho13k 13hannel Ea13h target has been tested against leaks before
the experiment and before ea13h shot The axis of symmetry of the sho13k
waves is required to be lo13ated at 03 mm from the base and at 03 mm from
the two verti13al SiO2 windows S13hemati13s of verti13al and horizontal 13ross
se13tion views of the target are presented in Fig 45b The axis of symmetry
is visible in the verti13al view while the position of the laser fo13us point (`+
sign) in the horizontal view This is a13hieved during the alignment pro13edure
employing mo13k targets as dis13ussed in the se13tion 432 The base of the
target 13onsists of a mi13ro-ma13hined aluminum stru13ture A gas pipe at the
bottom of the stru13ture allows for in situ gas lling at a pressure of a fra13tion
of a bar
A total of 55 targets were manufa13tured for the experiment Among
them 31 gave valuable results
423 Target holder
We x the targets on a stainless holder and then pla13e the system into the
va13uum 13hamber The target holder has been designed and manufa13tured
at Observatoire de Paris It allows a very reprodu13ible positioning of the
target on the va13uum 13hamber breadboard The main element of the target
holder is a base where ea13h target is rst xed by s13rews and removed after
ea13h shot to be repla13ed by a new one It also in13ludes a diagnosti13s support
(Fig 46) whi13h is atta13hed on it from the top with s13rews One of these
diagnosti13s 13onsists in a fast XUV Si diode with a pinhole The diode is
atta13hed to the bla13k tube whi13h is visible on the left part the target holder
in Fig 47 This last diagnosti13 aimed for sho13k timing as the pinhole-diode
system allows to re13ord the sho13k passing at the imaged position of the tube
(Stehleacute et al 2012 Chaulagain 2015) Unfortunately this diagnosti13 did
not work well and will not be mentioned further in the thesis
The se13ond diagnosti13 is a visible time-and-spa13e-resolved visible spe13tro-
meter On the target holder this diagnosti13 13ontains a lens whi13h is lo13ated
inside the bla13k tube on the right part of Fig 49 The lens allows imaging
the sho13k on a linear bundle of 21 bers 13onne13ted to the visible spe13trometer
through a feedthrough (se13tion 443)
A snapshot of the target holder is shown in the gure 49 The red arrows
in the Fig 49 shows the distribution of the various diagnosti13s
In addition we used two 3 mm thi13k stainless steel V-shape shields on
both sides of the target whi13h were xed to the holder These V-shields have
a hole (sim25 mm) shaped entran13e to allow laser in13iden13e on the target (in
green on the Figure 49) The role of the V-shields is to prevent the target
from the extra laser or another kind of hard X-ray plasma self-emission linked
to the ablation pro13ess
56 CHAPTER 4 EXPERIMENTAL SETUP
424 Target lling
The gaseous targets were lled in situ at a fra13tion of 1 bar with a gas (viz
Xe Ar a mixture of Xe and He) whi13h thus provide the medium for the
sho13k propagation We 13hose an in-situ lling system (Fig 410) to redu13e
the dieren13e of pressures inside and outside of the tube This is 13riti13al as
a large pressure dieren13e between the target 13ell and the va13uum 13hamber
13an result in the breaking of the ultra-thin Si3N4 windows of the target
To 13ontrol the pressure we used a piezo-resistive pressure transdu13er
(Swagelok PTI-S-AC6-32) whi13h has 10 kΩ bridge resistan13e and works
with 14-30 V biasing voltage A sensitive Bourdan manometer and a gauge
allowed monitoring the pressure inside the target 13ell To read the pressure
remotely (from the 13ontrol room) until the last se13ond before the shot we
13onne13ted the system with a long 13able
The transdu13er has been operated with a +17 V DC biasing voltage The
pressure is read using an industrial pressure transdu13er from Swagelok (0 to
10 Volts -10 bar to 06 bar 05 pre13ision) and reading is made with a
voltmeter at better than 05 This system generally took approximately
one minute to stabilize A s13hemati13 of the aforementioned gas lling system
is presented in the Figure 410
The pro13edure of the target lling is presented as follows
Figure 410 In-situ system for target lling and pressure monitoring
The target is pumped during the 13hamber eva13uation
The target is isolated at the end of the pumping
The target is lled to the desired pressure
43 LASER FOCUSING 57
The target is isolated from the gas bottle to avoid any leakage in
13hamber
Then the pressure 13an be read from the 13ontrol room till the last se13ond
before the shot
The gases are taken from high grade bottles 99995 purity The Xe+He
mixture is 1007plusmn050 He molar fra13tion The lling pro13edure is pre-
venting the presen13e of impurities like air When 13losing the 13hamber and
during the primary pumping the lling 13ir13uit 13ommuni13ates with the 13ham-
ber thus the whole tubing and the target are eva13uated down to 001 mbar
After the turbo pump is a13tivated the 13onne13tion to 13hamber is 13losed and
the gas in inje13ted When the target pressure is rea13hed the 13ell is isolated
from the pressure regulator In 13ase there is an una1313eptable overpressure
it is possible to re13onne13t to 13hamber to eva13uate and to restart the lling
The pressure reading is possible in the 13ontrol room still at pre13ision better
than 1 and the value kept for the re13ords is the one just before the shot
whi13h happens normally 1-2 minutes after 13losing
43 Laser Fo13using
431 Fo13al Lenses and Phase Zone Plates
Two fo13al lenses are used to fo13us the laser beams on the target piston foil for
MAIN and AUX respe13tively The MAIN lens is lo13ated inside the va13uum
13hamber whereas the AUX lens is outside this 13hamber The spe13i13ations
of these lenses are given in the Table 42
laser Diameter (mm) fo13al length (mm) f-number
MAIN 340 564 at 3ω 19
AUX 150 1022 at ω 69
Table 42 Spe13i13ations of the MAIN and AUX fo13al lenses and the f- number
(fo13al lengthbeam diameter) is given (see table 41 for the beam diameters)
Whereas these lenses provide a smooth intensity at the fo13us this is not
su13ient to laun13h a planar sho13k wave To this purpose phase zone plates
(PZP) have been used between the output laser beam and the lens
A PZP 13omprises several phase zone elements whi13h divide the input
beam into several beamlets whi13h 13ombined with the main fo13al lens provi-
ding a uniform intensity distribution on the target with a spe13ied diameter
(Ross Pepler and Danson 1995 Bett et al 1995) The PZP are 13ommonly
used in EOS laser experiments whi13h require a uniform intensity distribution
on the target (Koenig et al 1994 Batani et al 2003)
Two PZP as dis13ussed below were used in the experiment
58 CHAPTER 4 EXPERIMENTAL SETUP
The rst one made by COLSICOAT was already used at PALS for
radiative sho13k experiments (Stehleacute et al 2010) This PZP used for
MAIN (438 nm) is made on a 13oated BK7 substrate of a diameter
of 310 mm and se13tion of 8 mm It is designed to provide a uniform
13ir13ular fo13al spot over a diameter of 05 mm
The se13ond plate made by SCITECH has been used for AUX (1315
nm) The plate is made on a 63 mm thi13k 13oated Viosil substrate
with a square 153 x 153 mm shape It is designed to provide a uniform
13ir13ular fo13al spot over a diameter of 0250 mm
Laser Spe13ied spot diameter (mm)
MAIN 05
AUX 025
Table 43 Spe13i13ations of the two Phase Zone Plates
In terms of energy distribution sim 80 of the laser energy is in the laser
spot (C Spindloe private 13omm) The energy of the laser whi13h is given
by the PALS installation should be 13orre13ted by this fa13tor of 80 For
AUX another 13orre13tion of sim 10 has to be added whi13h is linked to the
transmission of the entran13e window of the va13uum 13hamber As a result
for nominal energies of the MAIN and AUX lasers equal to 120 and 60 J (see
table 41) the uen13es on the target are equal to sim 48800 and 59500 J13m
2
respe13tively Taking into a1313ount the 03 ns laser duration of the lasers this
results in an intensity of sim 1-2 times 10
14W13m
2
432 Fo13using of the MAIN and AUX laser beams
The determinations of the fo13i of the two lenses were performed with a
Hartmann setup using a photographi13 paper and without any PZP plate
The determination of the size of the beam at the impa13t is important both to
know the initial 13onditions for the sho13k waves (laser irradian13e whi13h should
be few 10
14W13m
minus2and se13tion of the sho13k wave) The sele13tion of the
best spots size on the piston was done with mo13k targets and by translating
the two laser lenses The in13iden13e of a laser beam on a solid planar target
generates a 13rater in the foil and an ablated plume propagating ba13kward
at high temperature (Mahmood and Ur-Rehman 2009 Singh and Thakur
2007) The 13rater is the result of boiling and ablation of the material The
ablated plasma whi13h expands towards the in13ident laser beam emits in the
keV range
A standard pro13edure whi13h works well at low energy (sim 10 J) 13onsists
of the estimation of the spot size by measuring the 13rater on the bulk target
using a standard mi13ros13ope At higher energy the boiling ee13t makes
44 DIAGNOSTICS 59
this estimation impre13ise and we preferred to use a keV imaging diagnosti13s
(Chaulagain et al 2012) whi13h is based on a pinhole (25 microm for MAIN keV
and 20 microm for AUX) and a keV 13amera These two keV 13ameras have been
used during ea13h laser shot to image the impa13ts of the MAIN and AUX
laser beams on the two respe13tive pistons of the target Ea13h keV 13amera
is a CMOS dete13tor without any glass It is prote13ted by an Aluminium
lter (200 microm for MAIN and 42 microm for AUX) to blo13k the visible light and
to prote13t the ship from the debris The spe13i13ations of the 13ameras are
presented in the Table 44 and the keV 13amera system is shown in the Fig
411 The two PALS laser lenses were translated up to a13hieve the suitable
diameters on the target whi13h were nally set to 450 - 500 microm and 250 -
300 microm for the MAIN and AUX beams
The size of the impa13t was 13ontrolled ea13h day on mo13k targets before
using the laser beams for real shots on gaseous targets It is worth to pre13ise
that the keV diagnosti13 was still in pla13e for the shots on the gaseous targets
In that 13ase the beams impa13t Parylene-N instead of Aluminium The keV
signal is then weaker than for the mo13k aluminum targets
Figure 411 S13hemati13 of the keV 13amera
KeV 13amera KeV 13amera
MAIN laser AUX laser
Model UI-122xLE UI-164xLE
Resolution 752(H) x 480(V) 1280(H) x 1024(V)
Pixel size (microm x microm) 60 x 60 36 x 36
A13tive Area (mm x mm) 451 (H) x 288 (V) 46 (H) x 37 (V)
Magni13ation 107 068
Pinhole Material Ta Pt
Table 44 Spe13i13ations of the two uEye keV 13ameras (1stVsion 13ompany)
44 Diagnosti13s
Various kinds of diagnosti13s were implemented namely visible interfero-
metry XUV and visible spe13tros13opy For the same two keV 13ameras are
indu13ted to monitor the impa13t and two visible 13ameras for target align-
60 CHAPTER 4 EXPERIMENTAL SETUP
(a)
(b)
(13)
Figure 412 (a) Sign 13onvention for the respe13tive positions of the lens and the
target (b) keV image of a MAIN impa13t on a massive target (lens position +1500
microm) The spot size is 475plusmn25 microm Pixel size for MAIN keV 13amera is 56 microm (13)
keV image of an AUX impa13t (lens position -1500 microm) The spot size is sim 280plusmn20microm on target One pixel = 66 microm on AUXs keV 13amera
44 DIAGNOSTICS 61
ment In this se13tion I will des13ribe the setups of the visible interferometry
XUV spe13tros13opy and visible spe13tros13opy The analysis of the re13ords of
visible interferometry and XUV spe13tros13opy will be dis13ussed in subsequent
13hapters
441 Visible Interferometry
A Ma13h-Zehnder Interferometer (MZI) has been employed to perform visible
interferometry The re13orded interferometri13 images have been used to study
the sho13k speed and the ele13tron density during the sho13ks propagation The
experimental setup of this interferometer is presented below
Ma13h Zehnder Interferometer
We pla13ed the target in whi13h sho13k is produ13ed in one of the two arms
of the interferometer The probe laser wavelength (527 nm) is supposed to
be far away of any atomi13 absorption resonan13es in the plasma and thus the
13ontributions to the refra13tive index 13ome from the plasma free ele13trons and
not from bound ele13trons (Harilal and Tilla13k 2004)
Figure 413 First interferometri13 setup All the opti13al elements are 1 in13h For
simpli13ity the plasma slab is not reported in the probe beam
The rst experimental setup shown in Fig 413 was used until 04052015
62 CHAPTER 4 EXPERIMENTAL SETUP
In this setup
the 300 ns long probe laser of wavelength 527nm (beam diameter asymp 1
in13h energy on the target few tens of mi13rojoules) rst passes through
an alignment pinhole PH1
the non-polarizing beam splitter BS1 transmits 50 of the intensity
of the in13ident beam and the remaining 50 of intensity in the per-
pendi13ular dire13tion At this stage the two arms of the interferometer
namely the referen13e and probe beams are produ13ed
two half wave plates HWP1 and HWP2 are pla13ed respe13tively in
the path of the probe and referen13e beams They allow to play with
the polarization in order to optimize the fringes 13ontrast as it will be
explained below
the verti13ally polarized probe beam passes through the target (plasma
slab) Due to its polarization it is fully ree13ted by the polarizing 13ube
beam splitter PBS1 towards the polarizing 13ube beam splitter PBS2
the referen13e beam is ree13ted by the mirror M4 on to this 13ube splitter
PBS2
At the output of PBS2 the two beams overlap but they have ortho-
gonal polarizations and 13ant interfere They then travel through a
polarizing beam splitter PBS3 those axis makes an angle of 45
with
the 13ommon dire13tion of in13iden13e of the two beams This rotates
the polarizations of the referen13e (horizontally polarized) and probe
(verti13ally polarized) beams along the same axis
At this stage the beams share the same polarization and interfere
A last non-polarizing beamsplitter plate (90-10 ) BS2 is pla13ed after
PBS3 to enable the propagation of the two beams on the same axis as
the laser beam at the entran13e of the interferometer
The fringes are re13orded on a HAMAMATSU C7700 VIS Streak Camera
lo13ated after the last alignment pinhole PH4 To this purpose an imaging
setup makes the image of the target longitudinal axis (ie along the dire13tion
of the sho13k propagation) on the streak slit This is done with an a13hromati13
spheri13al doublet of fo13al length equal to 1000 mm The magni13ation is
measured using an AGAR grid to 226 whi13h means that 106 microm on target
are re13orded by one pixel (size of the pixel 24 microm) on the streak on both
dire13tions (Fig 413)
44 DIAGNOSTICS 63
Figure 414 Se13ond interferometri13 setup
Role of the polarizing devi13es
Let us suppose that the polarization state of the in13oming beam is unknown
The polarization of the probe beam is then xed to be verti13al by the pola-
rizer HWP1 Ree13ted by the polarizing 13ube PBS2 it remains verti13al at
the entran13e of PBS3 On this side the polarization of the referen13e beam
is set to be horizontal after the 13ube BS1 using the polariser HWP2 and is
transmitted as horizontal by PBS2 The rotation of PBS3 allows to proje13t
part of the two beams on a 13ommon polarization axis and to interfere
This setup presents the following interest the intensity of the laser beam
in the two arms of the interferometer 13an be modied (and thus the 13ontrast
of the fringes) 13an be optimized using the two polarizing plates HPW1 and
HPW2 However as the multimode laser is not fully 13oherent the dark
fringe (zero net intensity) has never been obtained
Moreover the setup is designed to also allow alignment and adjustment
of the imaging setup with a green HeNe laser whi13h shares the same path
as the probe beam up to PBS1
A more optimized setup was used after 05052015 (ie shot number
48033) where the mirror M1 in the referen13e beam was repla13ed by a
polarizing beam splitter PBS4 ree13ting the probe beam like the mirror M1
(Fig 414) This was done in order to redu13e the dieren13e between the
64 CHAPTER 4 EXPERIMENTAL SETUP
two paths l1 and l2 and then to optimize the fringe 13ontrast This was more
suitable due to the short 13oheren13e length (1- 2 13m) of the probe laser
Streak 13amera
Figure 415 S13hemati13 of the Streak 13amera (from Hamamatsu noti13e)
A streak 13amera is a setup whi13h allows measuring the temporal varia-
tion of a fast signal whi13h is re13orded on its slit As indi13ated in the noti13e)
of the Hamamatsu C770 13amera the light pulse rst passes through the slit
then it is fo13used on the photo13athode (size 7times17 mm) of the streak by an
input opti13s The visible photons are 13onverted into a number of ele13trons
whi13h is proportional to the intensity of the in13ident light These ele13trons
are a1313elerated and 13ondu13ted towards the phosphor s13reen when a high-
speed voltage whi13h is syn13hronized to the in13ident light is applied As a
result the ele13trons are swept at high speed from the top to the bottom
after whi13h they are bombarded against the phosphor s13reen of the streak
tube and 13onverted through an output opti13s to an opti13al image on a dual
mode 13ooled CCD 13amera with 512 times 512 pixels (Fig 415) In the PALS
experiment the sweeping times were kept either 50 or 200 ns The CCD
pixel size is 24 x 24 microm
The streak 13amera 13an be operated with time swap mode with the slit
of the 13amera almost 13losed The re13ord 13onsisted then in images (position
along the slit versus time) as dis13ussed above or in a stati13 mode (no time
swap) whatever the slit aperture In that 13ase the re13ord is a usual 2D
spatial image of the obje13t
During the experiment we have taken referen13e images (without any
fringe) for every target as shown in the Fig 417a for one target example
In order to re13ord su13h images we blo13ked the referen13e beam and let the
probe beam only to illuminate the target in a stati13 mode and with the slit
open Then we redu13ed the slit width to its nominal value (200 microm) This
enabled us to have the visualization of the portion of the target whi13h was
44 DIAGNOSTICS 65
Figure 416 Imaging setup of the Ma13h Zehnder Interferometer setup operating
in the longitudinal mode The 13hannel of the target is fully illuminated by the
probe beam In this mode the lens images the axis of symmetry of the two sho13k
waves along the slit of the streak 13amera
imaged in the slit on the target and also to lo13ate the position of the pistons
before the shot This image is termed as the referen13e target image Then
the system was pla13ed in the interferometri13 mode and the fringe pattern
was optimized to be perpendi13ular to the slit and thus in the longitudinal
mode perpendi13ular to the dire13tion of the sho13k propagation
The Fig 417b reports su13h a typi13al interferogram re13orded during our
experiment (13f Fig 413) The horizontal axis 13orresponds to the dire13tion
of the sho13k propagation (with a total s13ale of 54 mm on the target) and
the verti13al axis 13orresponds to time (s13aling 200 ns)
442 XUV Spe13tros13opy
The time and spa13e integrated XUV plasma emission are re13orded with a
Flat Field XUV spe13trometer (grazing in13iden13e XUV spe13trometer) using
a 13on13ave grating Typi13al ray tra13ing a grazing in13iden13e XUV spe13tro-
meter is shown in Fig 418 In pra13ti13al the entran13e slit was removed
as the plasma extension was small In the Fig 419 the drawing of the
XUV spe13trometer setup s13heme (red 13olor) is presented together with the
spheri13al 13hamber and the target The XUV spe13trometer is installed on
the top of the spheri13al 13hamber whi13h allows the XUV emission passing
through the Si3N4 membrane on the top of the target (see se13tion 422) to
be re13orded
A 13ooled Andor DX440 CCD (without glass prote13tion) is atta13hed to
the spe13trometer to re13ord the spe13trum of the XUV radiation between 12
66 CHAPTER 4 EXPERIMENTAL SETUP
(a)
(b)
Figure 417 (a) Referen13e 2D image of a target before the shot re13orded on the
Streak 13amera The positions of the two pistons 13losing the 4 mm long target are
lo13ated at 800 and 4800 microm The dark zones along them (between 800 to 1000 and
4600-4800 microm) 13orresponds to the glue This glue is then visible through verti13al
bla13k strips in the interferometri13 re13ord (b) Corresponding interferometri13 time-
spa13e re13ord
44 DIAGNOSTICS 67
Figure 418 Typi13al ray tra13ing in a grazing in13iden13e XUV spe13trograph The
dete13tor (GMCP or image plate) is installed tangentially to the Rowland 13ir13le
and 40 nm (31- 100 eV) Its 13hara13teristi13s are given in the Table 45 Two Al
lters of thi13kness 08 and 16 microm prote13t the grating and the CCD 13amera
from debris and visible light The XUV images re13orded on the CCD are
time (300 micros) and spa13e (sim 3 mm) integrated
Andor 13amera (DX440)
Pixel Size 135 x 135 micromArea 276 (H) x 69 (V) mm
A13tive pixels 2048 x 512
Table 45 Spe13i13ations of Andor CCD 13amera
The 13urved grating (Table 46) used to fo13us the XUV emission on to
the CCD 13hip has a variable growes spa13ing as shown in the bottom panel
of the Fig 419 A typi13al output re13ord is also presented for illustration
Grating
Type Dira13tion
Growes per mm 1200
Dimensions 30 x 50 x 10 mmRadius of 13urvature 5649 mm
Blazing angle 37 degree
Blazing wavelength 16 nm
Table 46 Spe13i13ations of the XUV grating
443 Visible spe13tros13opy
In addition to the interferometry and XUV spe13tros13opy we have also im-
plemented a time- and spa13e-resolved visible spe13tros13opi13 plasma emission
diagnosti13
68 CHAPTER 4 EXPERIMENTAL SETUP
Figure 419 (a) Setup of the XUV spe13trometer in the spheri13al 13hamber (b)
Typi13al re13ord between sim 10 and 40 nm is shown in the bottom panel
The 13ore of the diagnosti13s is a Czerny-Turner 300 mm Spe13trometer
2
lo13ated outside the va13uum 13hamber The light of the plasma is inje13ted on
the slit of this spe13trometer through a bers bundle This bundle 13onsists of
three sets of an aligned seven individual bers looking at dierent se13tions
of the target 13hannel through an imaging lens lo13ated in the bla13k tube of
Fig 49 The time resolution is performed through an iCCD 13amera lo13ated
at the exit port of the spe13trometer (Fig 420a)
A typi13al re13ord is reported in Fig 420b The results of this diagnosti13s
show several signatures whi13h dier along the tube Most of them are in
absorption and 13ould be linked to the heating of the target sili13a window
Their interpretation is not yet a13hieved and will not be presented in this
study
45 Summary
In this 13hapter I have presented the details of our experimental setup and
diagnosti13s The next 13hapter will be dedi13ated to the presentation of our
re13ords results and their analysis
2
see http wwwandorcomspectrographshamrock minus spectrographminus series
45 SUMMARY 69
(a)
(b)
Figure 420 (a) S13hemati13 of setup employed to re13ord the time and spa13e
visible plasma emission (b) A typi13al spe13trogram
70 CHAPTER 4 EXPERIMENTAL SETUP
Chapter 5
Results and interpretation
Contents
51 Visible interferometry 71
511 Longitudinal interferometry sho13k speed and
ele13tron density 74
512 Transverse interferometry lateral extension of
the sho13k 91
52 XUV Spe13tros13opy 92
53 Simulations based on experimental results 93
54 Summary 97
In the previous 13hapter I presented a brief des13ription of our experimen-
tal setup This setup in13ludes two main diagnosti13s namely visible interfe-
rometry and XUV spe13tros13opy The visible interferometry is an adequate
tool to probe the ele13tron density of the radiative pre13ursor up to the 13riti-
13al density (4 times 10
2113m
minus3at the wavelength 527 nm of the probing laser)
whereas the denser post-sho13k region is opaque to the visible light XUV
radiation is present in the whole sho13k stru13ture and thus the time- and
spa13e-integrated XUV spe13tros13opy may allow exploring the self-emission
13oming from the post-sho13k and pre13ursor regions
In this 13hapter
1
I present the results of the both diagnosti13s with a
parti13ular fo13us on the interferometri13 data whi13h I have extensively studied
with the help of a spe13i13 data analysis pro13edure that I have developed I
will then present the main results of the XUV spe13tros13opi13 diagnosti13s The
results will then be interpreted with the help of numeri13al simulations
51 Visible interferometry
In the visible interferometry the interferen13es between the main and refe-
ren13e beams overlapping on the dete13tor produ13e a pattern of fringes whi13h
follow the relative phase variations between the probe (whi13h passes through
the plasma) and the referen13e beams (see se13tion 441 of the previous 13hapter
for the experimental setup) The phase variation between two 13onse13utive
fringes of the unperturbed beams (ie without any plasma) is equal to 2π
1
Most of results presented in this 13hapter has been published in Singh et al (2017)
71
72 CHAPTER 5 RESULTS AND INTERPRETATION
Then in the presen13e of the plasma in the probe beam these fringes be-
13ome shifted due to the modi13ation of the refra13tive index More details
about the prin13iple of the interferometry and the Ma13h-Zehnder interfero-
metri13 setup may be found in the APPENDIX A and we re13all here only
the expression of the phase shift
∆φ asymp minus πd
λNclt Ne gt (51)
where λ = 527 nm is the wavelength of the probing laser Nc = 4 times 10
21
13m
minus3the 13riti13al density at this wavelength and lt Ne gt is the ele13tron
density averaged over the laser path d in the 13ell and is dened as
lt Ne gt=
int d
0
Ne(z t)dy
d(52)
In order to dedu13e the ele13tron density it is ne13essary to follow the
evolution of the positions of the maxima (respe13tively minima) of the fringes
versus time To this purpose I have developed a spe13i13 data treatment
pipeline in the intera13tive data language (IDL)
2
To improve the visibility of
the fringes obtained in the interferograms I pro13essed the images with the
Fast Fourier Transform (FFT) (Proakis 2001) This treatment transforms
the image from the spatial domain (ie our re13ords) into the frequen13y
domain It then allows to dene spe13i13 bands of frequen13y whi13h represent
noise for the image and to 13lean them The 13omplete pro13edure is des13ribed
below
First the FFT of the (512 times 512) interferometri13 image is derived In
the next step the low and high-frequen13y lterings within the FFT of the
image are done using low-pass Fminus and high-pass F+ Butterworth frequen13y
lters (Proakis 2001) as follows
Fminus(u) = 1[1 + C(uu0)2n] with C = 1 n = 1
F+(u) = 1[1 + C(u0u)2n] with C = 1 n = 1
(53)
where u represents the position in the FFT image and u0 is the nominal
lter 13uto frequen13y (represented as the width of the region in pixels) For
the low-pass Butterworth frequen13y lter u0 is 13onsidered to be 55 whereas
for high-pass Butterworth frequen13y lter it is taken to be 45 The values of
u0 are dened manually by a trial method with the aim to result in better
noise redu13tion
After this step the inverse FFT of the ltered FFT image is performed
whi13h results in a 13lean image in the spatial domain In Fig 51 the original
re13ord for the shot number 48055 (left) the FFT of this image (13enter)
and the nal frequen13y ltered image (right) are presented showing how the
2
httpwwwast13ama13uk~vasilyidlidl_introdu13tionpdf
51 VISIBLE INTERFEROMETRY 73
frequen13y ltering improves the fringes 13ontrast and thus fa13ilitates their
analysis
Figure 51 Original re13ord of shot 48055 (left) FFT of the original image
(13enter) and the frequen13y ltered image (right)
Next the 13ru13ial task is to identify and to follow the fringes with the
help of a pre13ise determination of the intensity (pixel 13ounts) maxima In
order to lo13ate these maxima I have adopted the following steps
Figure 52 A 13ropped se13tion of the re13ord from shot the 48055 The rst ve
positions have been sele13ted manually `+ signs (in red 13olor) on ea13h fringe On
this re13ord the distan13e between two unperturbed fringes is 13orrespond to 15 pixels
(ie 159 microm)
Firstly I have manually 13hosen a se13tion of a typi13al re13ord with the
aim to sele13t only the area asso13iated with the sho13k dynami13s
74 CHAPTER 5 RESULTS AND INTERPRETATION
Next ve representative points are sele13ted visually (`+ marks in red
13olor) on ea13h fringe 13overing the important lo13ations on the fringe
(13f Fig 52) They serve as the input to the spline interpolation for
estimating all intermediate pixels lo13ations along the fringe (see Fig
53a)
In order to obtain the lo13ations representing the fringe maxima the
spline interpolated lo13ation points are further rened by sele13ting the
pixel of maximum intensity within plusmn5 pixels of the respe13tive spline
lo13ations in the X-dire13tion (see Fig 53b)
The position of fringe maxima for the 13ropped image obtained in an
aforesaid manner is shifted so as to represent the same points however
on the full-image (see Fig 54)
The re13ords obtained from the experiments have been pro13essed in this
way to estimate the sho13k se13tion speed ele13tron temperature and density
511 Longitudinal interferometry sho13k speed and ele13tron
density
In the longitudinal interferometry we perform on the slit of the streak 13a-
mera the image of the axis of symmetry of the sho13k propagation along the
tube to analyse the sho13k propagation in this dire13tion
The interferometri13 images have been pro13essed as explained previously
to enhan13e the fringes 13ontrast The lo13ations of the maximum intensity in
ea13h fringe are then used to derive the sho13k speed and the average ele13tron
density as will be presented below
Sho13k speed dedu13ed from the last fringe method
An estimation of the sho13k speed 13an be obtained through the interferometri13
image In this se13tion I will present the method to derive sho13k speed
based on the last visible end points of the fringes also 13alled last fringe
method This determination provides rapidly a qualitative estimation of the
sho13k speed whi13h is based on the absorption behavior of the plasma (see
Equation A11 in appendix A) and not on the real position of the front
dis13ontinuity The front dis13ontinuity is not seen in the re13ords due to the
strong absorption
At the positions of the last visible end points of the fringes the ele13tron
density rea13hes the maximum value a1313essible to the diagnosti13 The frin-
ges are strongly bent and the absorption of the visible light be13omes also
important Therefore it is believed that the sho13k front is 13lose from this
lo13ation
The lo13ations of the fringe maxima have been already dedu13ed following
the analysis presented in previous se13tions In this regard the best visible end
51 VISIBLE INTERFEROMETRY 75
(a)
(b)
Figure 53 (a) Representative points of the fringes as derived from the spline
tting of the 5 manually sele13ted points (Fig 52 (II)) on ea13h fringe (b) Positions
of the fringes maxima along Y-axis for ea13h fringe derived by lo13ating the points
of maximum intensity in X-dire13tion of the previous points obtained by spline t
76 CHAPTER 5 RESULTS AND INTERPRETATION
Figure 54 Fringe maxima on the full image
Figure 55 Imaging setup of the Ma13h Zehnder Interferometer operating in the
longitudinal mode The lens images the axis of symmetry of the two sho13k waves
along the slit of the streak 13amera
51 VISIBLE INTERFEROMETRY 77
point of ea13h respe13tive fringe is 13onsidered for the average speed estimation
for the sho13k
At very initial time the in13rease in sho13k speed is non-linear This non-
linear trend may result in non-physi13al speed estimations Therefore I dis-
13arded the end points whi13h appear to be within the ve nanose13onds after
the sho13k laun13hing time (eg t = 145 ns on Fig 56)
However the above pro13edure may result in un13ertainties as the sele13ted
end points may have dierent intensity (13ounts) values In this regard I
have evaluated the un13ertainty in the determination of the average speeds
To this purpose I rst sele13ted the best visible end-point of a random fringe
Su13h an end point gives a referen13e intensity whi13h was then used to obtain
on the rest of fringes the end points having 13lose intensity This pro13edure
was performed thri13e in order to dedu13e three sets of su13h end-points (see
Fig 56) Ea13h of these three sets of points was then used to estimate the
speed through a linear-t method (lines in white red and green) It may be
noted that these sets of points and tted lines are very 13lose to ea13h other
Therefore it is tough to distinguish them on the Fig 56 These three speed
determinations enable us to derive an average value with an un13ertainty
whi13h is equal to the standard deviation of these three values
Figure 56 Interferometri13 image re13orded for the shot 48055 in Xe at 01 bar
The sho13k speeds for the sho13ks driven by MAIN (from left side) land AUX (from
right side) lasers are respe13tively equal to sim 54 and 23 kms The time of laser
arrival on the piston is at 146 ns The positions of the Au-Xe interfa13e on the
re13ord are respe13tively 950plusmn50 and 4950plusmn50 mi13rons
This estimation of the sho13k average speed is satisfa13tory for the re13ords
in whi13h the sho13k speed is almost 13onstant throughout its propagation Ho-
wever the variation with the time of the speed may be substantial for some
re13ords In su13h 13ases the un13ertainty in the average speed will be in prin-
13iple higher than the previous estimations Therefore I introdu13ed another
78 CHAPTER 5 RESULTS AND INTERPRETATION
method to determine the average velo13ity this velo13ity is then estimated
during the initial and in nal durations of the sho13k propagation by deriving
the slope of points on the rst middle and last few fringes respe13tively
The standard deviation estimated by this method is then termed as the
un13ertainty in the average speed of respe13tive re13ord
Using these two methods I have obtained two values of un13ertainties for
all the MAIN and AUX sho13ks observed in the experimental re13ords Finally
the largest un13ertainty value of the two is 13onsidered as the nal un13ertainty
in the estimation of the respe13tive speed and is shown by error bars in Fig
57a and 57b 13orresponding to the MAIN and AUX sho13ks respe13tively
The table 51 reports for dierent noble gases the values of the velo13ities
re13orded in several shots for the 13ounter-propagating sho13k waves In addi-
tion one re13ord (48131) 13orresponds to the 13ase of an isolated sho13k wave
laun13hed by the MAIN laser The values of the MAIN and AUX laser ener-
gies are also reported In some 13ases the glue on the window prevented the
probe laser to pass through the 13hannel Then it be13ame impossible to derive
any value for the velo13ity In su13h 13ases I put `NA for the 13orresponding
sho13k velo13ity
It may be noted that the sho13k speeds generated by the MAIN laser vary
in the range of 30-55 kms while for the sho13k originated from AUX laser
the speeds vary between 10 and 30 kms
Variations with laser energy and pressure the 13ase of Xenon
When the number of re13ords is su13ient to make a statisti13al analysis we
may analyse the variations of the sho13k speeds with the laser energy of
the MAIN and AUX lasers This is the 13ase for Xe or XeHe (90-10)
mixture and for gas pressures ranging between 01 and 02 bar At 01 bar
all the measurements 13on13ern Xenon (ρ= 54 times 10
minus4g13m
minus3) At 02 bar
the re13ords 13on13ern a mixture XeHe (90-10 ρ=10 times 10
minus3g13m
minus3)
ex13ept for one point at 121 J whi13h 13orrespond to the 13ase of pure Xe (ρ =
108 times 10
minus3g13m
minus3) We do believe that taking into a1313ount the pre13ision
of our re13ord the introdu13tion of tra13es of Helium do not ae13t mu13h the
dynami13s of the sho13k wave as the 13orresponding variation of mass density
is negligible
As expe13ted the sho13k speed in13reases with the laser energy and de13rea-
ses with the pressure (Fig 57a) A linear t gives the following dependen13e
of the speed in kms with the MAIN laser energy (in J) at 01 and 02 bar
respe13tively
3
v01bar = 1423 + 030E (54)
3
Two outlier points have been dis13arded from the set 13orresponding to 01 bar They
are shown for information on the Figure
51 VISIBLE INTERFEROMETRY 79
(a)
(b)
Figure 57 (a) MAIN sho13k speed (13al13ulated by the last fringe method) versus
the MAIN energy for Xenon or XeHe mixture at dierent pressures with the error
bars (b) AUX sho13k speed versus AUX energy also for Xenon or XeHe mixture
at dierent pressures
80 CHAPTER 5 RESULTS AND INTERPRETATION
v02bar = minus1405 + 047E (55)
The speeds generated by the AUX laser are lower than the previous ones
Although the re13ords are more sparse we note that the speeds at 01 bar
also in13rease with the laser energy However at 02 bar I 13ould not nd
similar trend due to the la13k of re13ords
Variations with the gas
As 13an be seen from the Table 51 we have also performed several shots in
Ar at 01 02 03 and 08 bar one shot in He at 05 and one in Kr at 02
bar
To 13ompare the velo13ities obtained for the dierent gases one 13an either
keep the pressure 13onstant (ie the number of atoms) or the mass density
For a given mass density and laser energy the velo13ity should be the same
in the adiabati13 limit However due to the radiation 13ooling and ionisation
this 13an not be the 13ase Nonetheless if we 13ompare the two re13ords of Ar
at 03 bar (ρ = 493 times 10
minus4g13m
minus3 E = 121 J) with Xe at 01 bar (54
times 10
minus4g13m
minus3 E = 133 J) we measure similar speeds (49 and 54 kms)
These two re13ords are thus interesting to 13ompare This will be done later
on in this 13hapter
More generally we note that the speed of Helium at 05 bar is higher by
sim 40 than for Ar at the same energy (see Fig 58) This is not surprising
for this lighter element (Table 52) However it is important to note that
the la13k of shots prevents us to make any pre13ise 13on13lusion
Ex13ept for this 13ase and for the unique re13ord of Ar at 02 bar whi13h
gives also a higher velo13ity the velo13ities of the other re13ords follow more or
less the variation with the energy than Xenon
To 13ompare the radiative ee13ts it seems justied to perform the 13om-
parison at a given sho13k speed independent of the laser energy and for mass
densities whi13h are 13lose to ea13h other In the following we shall thus 13om-
pare the results of Xe at 01 bar (54 kms 54times 10
minus4g13m
minus3) Ar at 03
bar (49 kms 49times 10
minus4g13m
minus3) and Kr at 02 bar (55 kms 68 times 10
minus4
g13m
minus3)
Line averaged ele13tron density lt Ne gt
I have estimated the ele13tron density lt Ne gt of the plasma averaged along
the path of the probe laser beam by analyzing the interferogram re13ords
assuming the same se13tion d of the plasma layer of 600 microm for both MAIN
and AUX side sho13ks
51 VISIBLE INTERFEROMETRY 81
Shot Gas Pressure E3ω MAIN sho13k Eω AUX sho13k
(bar) (J) speed (kms) (J) speed (kms)
at 298 K
48033 Air 03 124 54plusmn1 66 20plusmn248034 Air 03 131 52plusmn1 74 27plusmn248076 Ar 05 100 41plusmn1 66 18plusmn148077 Ar 01 115 42plusmn1 67 18plusmn148078 Ar 03 112 46plusmn3 65 25plusmn248079 Ar 03 121 49plusmn5 67 23plusmn348080 Ar 08 103 38plusmn2 62 21plusmn248081 Ar 08 113 38plusmn1 68 NA
48082 Ar 08 107 36plusmn1 65 NA
48141 Ar 02 111 63plusmn1 57 NA
48083 He 05 106 57plusmn3 63 NA
48146 Kr 02 125 55plusmn2 53 NA
48043 Xe 03 138 57plusmn1 72 25plusmn148051 Xe 01 123 58plusmn1 67 30plusmn148055 Xe 01 133 54plusmn3 68 22plusmn348057 Xe 01 127 53plusmn1 68 23plusmn148058 Xe 01 115 48plusmn3 63 18plusmn148059 Xe 01 116 50plusmn1 67 21plusmn148061 Xe 01 127 53plusmn1 67 17plusmn248065 Xe 01 122 52plusmn4 68 23plusmn348066 Xe 01 114 50plusmn3 67 15plusmn248067 Xe 05 115 39plusmn2 65 NA
48068 Xe 05 109 36plusmn2 65 12plusmn248070 Xe 05 109 33plusmn3 65 NA
48138 Xe 02 121 45plusmn5 0 0
48131 Xe +He 02 112 38plusmn1 0 0
48132 Xe +He 02 118 41plusmn4 56 18plusmn248133 Xe +He 02 112 41plusmn3 56 NA
48134 Xe +He 02 111 38plusmn1 60 NA
48136 Xe +He 02 115 39plusmn3 59 14plusmn148143 Xe +He 06 123 39plusmn4 63 18plusmn548144 Xe+He 02 133 45plusmn3 66 NA
Table 51 Sho13k speeds estimated from the `last fringe method `NA represents
the entries whi13h 13ould not be dedu13ed from the re13ord Further entries in the
bold font are dis13ussed in detail in this 13hapter
82 CHAPTER 5 RESULTS AND INTERPRETATION
Helium Argon Krypton Xenon
Atomi13 Number 2 18 36 54
Atomi13 Mass 4 3995 8380 13129
First Ionization Energy (eV) 246 157 14 12
Density (10
minus4g13m
minus3) at 01 bar 016 164 344 539
Table 52 Atomi13 data and density at 01 bar (at room temperature) for He Ar
Kr and Xe
Figure 58 MAIN sho13k speed (13al13ulated by the last fringe method) versus the
MAIN laser energy for Xe (at 01 bar only) Ar He and Kr at dierent pressures
with the error bars
51 VISIBLE INTERFEROMETRY 83
It should be noted that the beam se13tions dMAIN and dAUX of MAIN
and AUX lasers are approximately 600 and 300 microm respe13tively on the two
pistons and that the transverse horizontal se13tion of the sho13k tube is equal
to dtube = 600 microm Thus the value of lt Ne gt 13omputed for the sho13k wave
laun13hed by MAIN should be 13lose to the physi13al lo13al value Ne supposing
that the plasma is homogeneous along the transverse se13tion However
the value obtained for AUX is 13ertainly larger by a fa13tor whi13h 13an be
estimated at a rst step as the fra13tion of dtubedAUX giving a fa13tor of
about 2
As explained in the previous se13tion the pixels representing the positions
of the fringe maxima have been already determined with the best possible
pre13ision Therefore it is possible to 13al13ulate the relative phase shift (with
respe13t to zero time) variation along y-axis (time axis) at ea13h fringe maxima
Zero time is time of MAIN and AUX laser rival on the target With these
phase shifts estimated at ea13h pixel of ea13h fringe I then derived lt Ne gtusing Equation A22
Let us take the 13ase of three shots in Xenon for illustrating the method
whi13h will be followed (see the left panel of the Figure 59) The maximum
density is estimated to be sim11 times 1019cmminus3(13orresponding to phase shift sim
16) The number of subdivisions is taken to 5 bins with the following phase
(∆φ) ranges (in the units of 2π) and average ele13tron densities (ltNegt)
bin 1 ∆φ le 06 lt Ne gtle 39 10
1813m
minus3(white)
bin 2 06 - 08 39 - 57 10
1813m
minus3(red)
bin 3 08 - 11 57 - 75 10
1813m
minus3(blue)
bin 4 11 - 13 75 - 93 10
1813m
minus3(green)
bin 5 gt 13 gt 93 10
1813m
minus3(magenta)
All the re13ords shown in the Table 51 have been pro13essed using this
method The limit of dete13tion for the phase shift 13orresponds to 2 pixels
giving a threshold for the Ne measurement lt Nemin gt This threshold
diers from one re13ord to another due to the variation in the number of
fringes (and thus the distan13e between two unperturbed fringes)
In order to make a 13omparative investigation of the sho13ks propagation
and intera13tion in dierent gases as already mentioned previously I sele13ted
three 13ases (shot48055 shot48132 and shot48138) relative to Xe (or Xe-
He mixture) one 13ase relative to Ar (shot48079) and Kr (shot48146)
The 13orresponding experimental 13onditions are noted in bold in the Table
51
84 CHAPTER 5 RESULTS AND INTERPRETATION
lt Ne gt in Xenon
The Fig 59 reports the interferometri13 re13ords for the three Xenon 13ases
The top and the middle panels 13orrespond to the propagation of two 13ounter-
streaming radiative sho13k waves at 01 (shot48055) and 02 bar (48132)
For 13omparison one re13ord (48138 bottom panel) is dedi13ated to the pro-
pagation of single sho13k (MAIN) at 02 bar The Xe-He mixture (90 - 10
in numbers of atoms) is used for the investigation in the 13ase of 02 bar
pressure while it is only Xe in the 13ase of 01 bar As mentioned previously
at the pre13ision of our re13ords we 13onsider that this impurity 13on13entration
has a negligible ee13t on the sho13k speed and the pre13ursor ele13tron density
The limit of dete13tion of lt Ne gt over the se13tion of the tube (06 mm) is
13orresponds respe13tively to 9 times 10
17 6 times 10
17and 6 times 10
1713m
minus3for the
Figures 59(a) (b) and (13)
The variations of lt Ne gt with the distan13e along the sho13k tube (ie
parallel to the dire13tion of the sho13ks) are reported in the right panel of Fig
59 at 10 ns (in red) 20 ns (in blue) 30 ns (in green) and 40 ns (in magenta)
The intera13tion between the two pre13ursors is 13learly visible at 01 bar
(Fig 59(a)) at 10 ns the intera13tion of the 13ounter-propagating sho13ks
has either not yet started or is below the sensitivity of this diagnosti13 The
intera13tion o1313urs at later times with a typi13al signature whi13h is as follows
the slope of lt Ne gt is de13reasing from the left (MAIN pre13ursor) passes
through a minimum and in13reases at the right (AUX) The minimum itself
in13reases with time up to 7 times 10
1813m
minus3at 40 ns
At 02 bar we have not been able to re13ord the 13ollision in the temporal
range (50 ns) of the streak However the two re13ords (with MAIN only and
with the two sho13k waves) indi13ate a pre13ursor for MAIN The two gures
(Fig 59(b)) and (Fig 59(13)) show that the pre13ursor of the MAIN sho13k
wave is not inuen13ed by the presen13e of AUX sho13k wave up to 45 ns
There is no obvious indi13ation about a pre13ursor for AUX in the 13ase of
two 13ounter-propagating sho13k waves (Fig 59(b)) At this pressure and
13ompared with the previous 13ase at 01 bar the absen13e of pre13ursor for
AUX may be attributed to i) a low sho13k speed (18 kms) 13ombined with a
larger pressure (hen13e for a given gas the pre13ursor extension in13reases with
the speed and de13reases with pressure) ii) a too small longitudinal extension
of the eventual pre13ursor (see Fig 59(b)) 13ompared with the resolution of
20 mi13rons (2 pixels) Our 1D numeri13al simulations with Xenon opa13ity
multiplier times 20 (not presented here) indi13ate a small pre13ursor for AUX
sho13k At 15 ns its extension is 50 microm (900 microm for MAIN sho13k) with
a typi13al ele13tron density sim 35 times 10
1913m
minus3(23 times 10
1913m
minus3for MAIN
sho13k) whi13h does not agree with the re13ord At 42 ns the pre13ursor of MAIN
rea13hes the AUX sho13k front and the prole is similar to the prole at 20 ns
shown in Fig 313a at 01 bar with a plateau of almost 13onstant ele13tron
density between the two fronts This might be 13ompatible with small bending
51 VISIBLE INTERFEROMETRY 85
Figure 59 Left panel interferometri13 re13ords 48055 in Xe at 01 bar (a)
48132 in Xe+He at 02 bar (b) and 48138 in Xe+He at 02 bar (13) Right
panel ele13tron density at 10 20 30 and 40 ns versus distan13e for these re13ords
The positions of maxima have been identied on the re13ords in the left panel
The time t = 0 13orresponds to the time of laser arrival on the target and the
position x = 0 13orresponds to the interfa13e between the piston (Au layer) and the
gas Its determination is pre13ise within 100 mi13rons The distan13es between two
unperturbed fringes for re13ords 48055 48132 and 48138 are 159 244 and 244
microm respe13tively The lt Ne gt un13ertainty (plusmn 2 pixels) is indi13ated by the error bar
in the right panels It 13orresponds respe13tively to plusmn 9 times 10
17 plusmn 6 times 10
17and plusmn
6 times 10
1713m
minus3for the gures (a) (b) and (13) The limit of dete13tion (2 pixels) is
presented by a dotted line on ea13h gure
86 CHAPTER 5 RESULTS AND INTERPRETATION
of the 4
thfringe (from the right) between 45 and 50 ns As 1D simulations
are known to overestimate the pre13ursor ele13tron density 2D simulations are
ne13essary for a more pre13ise interpretation of the experimental result
lt Ne gt in Argon and in Krypton
As seen in the Fig 58 the variation with the laser energy of the sho13k speed
in Argon at 03 bar (in the red dashed 13ir13le) is in good agreement with the
13orresponding one of Xenon at 01 bar (blue linet) As these two 13ases have
a 13omparable mass density of sim 5 times 10
minus4g13m
3 this qualitative agreement
is expe13ted
Taking into a1313ount the larger pressure (and thus density) the slope of
the variation of the slower sho13k speeds in Ar at 08 bar (in a red dash-
dot 13ir13le) with the laser intensity is 13ompatible with the previous trends
However any 13on13lusion 13an not be derived for Argon at 02 and 05 bar
owing to the s13ar13ity of the re13ords
In Fig 510 the interferometri13 re13ord for Ar at 03 bar (shot 48079
49 times 10
minus4g13m
minus3) is presented The estimated speeds of MAIN and AUX
sho13ks dedu13ed from the last fringe method are respe13tively equal to 49 and
23 kms whi13h are 13lose to the speeds of 54 and 23 kms re13orded in Xe
at 01 bar (shot 48055 539 times 10
minus4g13m
minus3 Fig 59(a)) Contrarily to
the Xenon 13ase we do not see any signi13ant bending in the fringes The
maximum fringe shift is reported in red on the fringe 1 The 13orresponding
ele13tron density is equal to 38 times 10
1813m
minus3
Thus obviously the radiation ee13ts are less important for Argon than
for Xenon at the same density and speed Part of this dieren13e 13ould be
attributed to the huge dieren13e in the atomi13 numbers (40 and 131) of the
two gases whi13h leads to dierent peak temperatures for the ions as expe13ted
from equation 84 of 13hapter 2 However the post-sho13k temperatures dedu-
13ed from the jump relations for a real gas (Chapter 2) without radiation are
very 13lose (sim 25 eV) and thus this 13ould be not the relevant explanation
To 13larify this I performed HELIOSPROPACEOS simulations for two
13ounter-propagating sho13ks at 53 and 23 kms respe13tively in Ar and Xe
at the same initial mass density (54 times 10
minus4g13m
minus3) without any opa13ity
multiplier I have also reported for information the 13ase of Xenon with
an opa13ity multiplier equal to 20 The simulated temperature proles are
shown in the Fig 511 One notes immediately that the pre13ursor extension
is smaller for Ar than for Xe The peak temperatures of the MAIN sho13k
are respe13tively sim 21 eV (16 eV for opa13ity multiplier=20) for Xe and 21
eV for Ar In all the 13ases the post-sho13k temperatures are equal to sim 11
eV This indi13ates that the temperature is not the main explanation of the
dieren13es in the pre13ursor length
More interesting are the dieren13es in the Rosseland opa13ity (here per
unit of length) whi13h is reported in the Fig 511b We see that 13lose to the
51 VISIBLE INTERFEROMETRY 87
(a)
(b)
Figure 510 (a) Interferometri13 re13ord in Ar at 03 bar (48079) The estimated
speeds for MAIN and AUX are 49plusmn5 and 23plusmn3 kms (b) Same re13ord where the
fringes maxima are marked by points The bins denition is as fellows bin 0 Ne le11 times 10
1813m
minus3(white) bin 1 11 - 18 times 10
1813m
minus3(yellow) and bin 2 39 - 57
times 10
1813m
minus3(red) The time t = 0 13orresponds to the time of laser arrival on the
target and the position x = 0 13orresponds to the interfa13e between the piston (Au
layer) and the gas Its determination is pre13ise within 100 mi13rons
88 CHAPTER 5 RESULTS AND INTERPRETATION
(a)
(b)
Figure 511 Results for temperature (a) and Rosseland opa13ity (b) obtained from
HELIOS simulation at 10 ns for two 13ounter-propagating sho13ks at sim 50 and 18
kms for Xe (with opa13ity multiplier 1 and 20) Ar and Kr at 54 times 10
minus4g13m
minus3
initial mass density A 13omparison Ar Kr and Xe PROPACEOS opa13ity shown in
APPENDIX B
51 VISIBLE INTERFEROMETRY 89
front the Argon opa13ity is 13lose to 4 13m
minus1 whi13h means that the radiation
from the sho13k is absorbed within 25 mm This has to be 13ompared with the
Xenon 13ase (no multiplier) where it is equal to sim 03 13m
minus1 whi13h means
that the hot pre13ursor is almost transparent over the length of the tube
Thus the dierent behaviors of the opa13ity in the upstream gas explain the
quantitative dieren13e in the development of the pre13ursor The sharp peak
of the Xenon opa13ity with the multiplier of times 20 is due to the strong rise
of the opa13ity when the temperature de13reases below 5 eV whi13h marks the
end of the pre13ursor
I will present now the 13ase of two 13ounter-propagating sho13ks in Kr
(48146 in Fig 512) at 02 bar (69 times 10
minus4g13m
minus3) In this 13ase unfortu-
nately the AUX sho13k 13ould not be imaged 13ompletely due to the presen13e
of glue at the right edge of the target 13ell but the 13ollision was re13orded
at sim 40 ns The speed of the MAIN sho13k is estimated to be 55 kms
Thus despite a slightly higher initial mass density this sho13k wave may be
13ompared with the shot (48055) in Xe at 01 bar (54 times 10
minus4g13m
minus3 54
kms)
Despite the poor quality of the re13ord and the inadequate temporal range
(150 ns) sele13ted on the streak 13amera we note some bending in all the
fringes indi13ating the presen13e of pre13ursor The maximum fringe deviation
(063 of the distan13e between two fringes) is noted at the end of fringe 2
whi13h 13orresponds to the lt Ne gt value 45 times 10
1813m
minus3 The pre13ursor
seams to be more developed here than for the previous Ar 13ase
The result of the simulation in Xe and Kr with equal initial mass density
54 times 10
minus4g13m
minus3 and for the two respe13tive sho13k speeds of 50 and 20
kms is reported in the Figure 511 The peak temperatures of the MAIN
sho13k are respe13tively sim 21 eV (16 eV for opa13ity multiplier = 20) for Xe
and 22 eV for Kr The post-sho13k temperatures (sim 10-11 eV) are 13lose from
ea13h other
The pre13ursor extension in Krypton is smaller than in Xenon whi13h is lo-
gi13al if we 13onsider the opa13ity values (Fig 511b) However if we take into
a1313ount the multiplying fa13tor for the Xenon opa13ity the Krypton pre13ursor
is now more extended than for xenon whi13h is in 13ontradi13tion with the ex-
periment This probably means that either the Kr opa13ity is under estimated
by PROPACEOS or that our multiplier for Xenon is overestimated
Synthesis
The previous experimental investigation 13onrms that the sho13k velo13ity in-
13reases with the laser energy and de13reases with the mass density Moreover
for our given sho13k velo13ity and mass density the extension of the radiative
pre13ursor in the experiment de13reases with the atomi13 number Among Ar
Kr and Xe this last one appears to be the most adapted for the investiga-
90 CHAPTER 5 RESULTS AND INTERPRETATION
(a)
(b)
Figure 512 (a) Raw interferometri13 re13ord in Kr at 02 bar (shot 48146) The
estimated speed for MAIN sho13k is 53plusmn2 kms The time t = 0 13orresponds to the
time of laser arrival on the target(b) Same re13ord shown in (a) maxima is marked
by 13olored points The bins denition is as follows bin 0 Ne le 11 times 10
1813m
minus3
(white) bin 1 11 - 18 times 10
1813m
minus3(yellow) and bin 2 39 - 57 times 10
1813m
minus3
(red) The time t = 0 13orresponds to the time of laser arrival on the target and
the position x = 0 13orresponds to the interfa13e between the piston (Au layer) and
the gas Its determination is pre13ise within 100 mi13rons
51 VISIBLE INTERFEROMETRY 91
tion of the radiative pre13ursor A similar behaviour is found also at ORION
with higher speeds (Clayson et al 2017)
512 Transverse interferometry lateral extension of the sho13k
The transverse interferometry is a dierent geometri13al imaging setup of the
Ma13h-Zehnder interferometer whi13h provides qualitative information about
the 13urvature lo13alisation and transverse extension of the radiative pre13ur-
sor and thus of the sho13k
In the transverse interferometry we perform the image of a transverse
se13tion (on the slit of the streak 13amera) of the tube whi13h is lo13ated at a
distan13e dslit equal to 3 mm from the initial position of the MAIN piston
As the setup is originally adapted for the longitudinal interferometry we use
a Dove prism between the va13uum 13hamber and the streak 13amera to rotate
the image by 90 degrees
Figure 513 S13hemati13s of the Ma13h Zehnder Interferometer setup to re13ord
transverse interferometri13 images The lens allows to make on the slit of the
13amera the image of a se13tion perpendi13ular to dire13tion of sho13k propagation
A transverse interferometri13 re13ord for the MAIN sho13k alone in Xe at
02 bar is reported in Fig514(a) The streak 13amera rst re13ords the unper-
turbed plasma on the se13tion dslit in the tube Then it re13ords su1313essively
the pre13ursor and the post-sho13k (whi13h is opaque to visible radiation and
appears as a dark zone in the re13ords) Taking into a1313ount the oset of 14
ns the time of sho13k arrival is re13orded at 72 ns after the time t0 of laser
arrival on the target and the sho13k speed is estimated to be sim 35 kms Due
to glue on one lateral window (on the right part of the gure) only 6 fringes
92 CHAPTER 5 RESULTS AND INTERPRETATION
Figure 514 Transverse interferometri13 images for (a) shot48111 (MAIN sho13k
only) (b) shot48130 (AUX sho13k only) The time is measured after an oset
equal to 14 and 23 ns respe13tively after the time of the laser arrival on the target
The position zero on the x-axis of ea13h image 13orresponds to the base of the target
are visible The lateral extension of the sho13k stru13ture at this time is deri-
ved to be sim 570plusmn30 microm in qualitative agreement with the spe13i13ations of
the MAIN phase plate and the shape of the pre13ursor is relatively at The
axis of symmetry of the sho13k is determined to be at sim 350 mi13rons from
the base of the target (ie 50 microm above the nominal value of 300 microm)
A re13ord for the AUX sho13k alone is shown in Fig 514(b) whi13h 13orre-
sponds to a gate opening of 50 ns The start time of the image has an oset
of +23 ns from t0 and the distan13e dslit is set to 700 mi13rons from the initial
position of the AUX piston The AUX sho13k duration extends from 30 ns
to at least 34 ns after t0 The sho13k speed is then estimated to be ranging
between 23 and 20 kms The shape of the pre13ursor is strongly bent and
we note a tiny shift of the se13ond and third fringes (starting from the left)
at sim 8 ns It may further be noted that the lateral spread of the opaque
sho13k is ranging between 275plusmn25 microm (whi13h is also in agreement with the
spe13i13ations of the AUX phase plate) and that the axis of symmetry of the
sho13k system is also lo13ated at about 350 microm from the bottom of the 13ell
52 XUV Spe13tros13opy
XUV emission originates from the 13omparatively hot portions of the plasma
In the present 13ase it 13orresponds to the post-sho13k and the pre13ursor re-
53 SIMULATIONS BASED ON EXPERIMENTAL RESULTS 93
gion whi13h is 13lose to the sho13k front The spe13trum re13orded by the XUV
spe13trometer enables us to analyze the spe13tros13opi13 signatures of the radi-
ative sho13k One of the aims of this diagnosti13s was to identify the sho13ks
13ollision whi13h leads to higher temperatures through 13omparing the spe13tra
obtained for the 13ases of single and 13ounter-propagating sho13k waves Howe-
ver only a few re13ords were possible to be obtained during the experiment
and unfortunately the 13omparison with the 13ase of an isolated sho13k wave
was not performed
Among the shots re13orded the XUV spe13trum of the shot 48143 is
presented herewith in detail This shot was performed for [Xe (90)+ He
(10) mixture at 06 bar with laser energies of 123 J for MAIN and 63 J
for AUX The interferometri13 re13ord of this shot is shown in Fig 515 The
MAIN sho13k speed has been estimated to be sim 39 plusmn 4 kms The estimated
AUX sho13k speed (18 plusmn 5 kms) is not pre13ise due to the presen13e of glue
on the right se13tion of the re13ord (note Fig 515) In this interferometri13
re13ord we have not been able to retrieve the 13ollision time However an
extrapolation of the slope 13orresponding to the speeds sim 39 kms (MAIN)
and sim 18 kms (AUX) enables us to approximately determine the 13ollision
time to be between 60 to 65 ns
The raw spe13trum (shown in Fig 516a) re13orded for the wavelength
range of 15-35 nm (35-82 eV) shows the `L edge of Aluminum at 17 nm
(34 nm in se13ond order) in the rst and se13ond (34 nm)orders These two
wavelengths will be used for the wavelength 13alibration The net spe13trum
13orre13tion for the transmission (Henke Gullikson and Davis 1993) of the
100 nm thi13k Si3N4 window (3 mm times 04 mm) is introdu13ed in Fig 516b
A remarkable feature is a strong absorption dip between 19 and 22 nm (56-65
eV) This absorption probably 13omes from the 13older layers (thi13kness 300
microm) between the sho13k heated plasma and the Si3N4 window Few lines
of Xe VII-VIII are identied through NIST database
4
as also Oxygen IV
and V lines Lyman lines of He II (from 1-2 to 1-7) are also present in the
spe13trum This information will be useful for the estimation of the ele13tron
temperature
53 Simulations based on experimental results
In this se13tion I will 13ompare the experimental sho13k 13hara13teristi13s with the
results of HELIOS simulations using the PROPACEOS equation of state and
opa13ity (limited to 1 group) As indi13ated in the 13hapter 3 this opa13ity has
been multiplied by 20 for the Xenon only As our interest is to understand the
sho13k stru13ture in Xenon and not the laser matter intera13tion on the piston
we performed several simulations with Xe gas for various sets of uen13es for
4
[httpphysi13snistgovPhysRefDataASDlines_formhtml
94 CHAPTER 5 RESULTS AND INTERPRETATION
Figure 515 Interferometri13 image for the shot48143 The time t = 0 13orre-
sponds to the time of laser arrival on the target and the position x = 0 13orresponds
to the interfa13e between the piston (Au layer) and the gas Its determination is
pre13ise within 100 mi13rons
53 SIMULATIONS BASED ON EXPERIMENTAL RESULTS 95
(a)
(b)
Figure 516 Raw (a) and 13orre13ted XUV spe13trum (b) for the shot 48143
96 CHAPTER 5 RESULTS AND INTERPRETATION
the MAIN and AUX beams in order to obtain the best agreement between
the simulated and measured sho13k speeds
To analyse the results from the shot 48055 (Fig 59(a)) we set the u-
en13es to 32000 amp 7500 J13m
2 This allows produ13ing the experimental sho13k
speeds 54 and 23 kms in Xenon at 01 bar for the MAIN and AUX beams
respe13tively The two sho13ks appear in Xenon at 2 and 3 ns respe13tively for
MAIN and AUX The merging of the two pre13ursors starts at sim 15 ns and
the sho13k 13ollision time o1313urs at 47 ns In Fig 517 we present the ele13tron
density proles from the simulation (dotted lines) and the experiment (solid
lines) at 10 20 30 and 40 ns
At 10 ns the two simulated pre13ursor extensions are 0165 and 0022 13m
for MAIN and AUX respe13tively The ele13tron density is larger by a fa13tor
of 4 than in the experiment The shapes of the pre13ursors are also very dif-
ferent However this 1D pi13ture supposes the plasma to be uniform within
the tube In reality in the transverse dire13tion Ne is de13reasing from the
13enter to the walls whi13h results in a lower estimation of the average value
(lt Ne gt) 13ompared to the value at the target 13enter and in a smoother pro-
le near 02 amp 035 13m at 10 ns It is also important to note that for AUX
sho13k the average lt Ne gt value underestimates the lo13al one by a fa13tor
of about 2 (as it is averaged over 06 mm instead of 03 mm) Moreover
our 1D simulation suers from an inexa13t opa13ity and 2D ee13ts are pro-
bably important espe13ially for AUX Thus we have here only a qualitative
interpretation of the experimental results
The intera13tion between the two HELIOS radiative pre13ursors starts be-
tween 10 and 20 ns like in the experiment However the shape as well as
absolute values of the simulated ele13tron density 13urves are not in agreement
with the experimental results and the intera13tion is stronger in the simulation
than in the experiment
In order to interpret the spe13tros13opi13 data presented in se13tion 52 we
performed another 1D simulation in Xenon at 06 bar and adapted the
uen13es to generate two 13ounter-propagating sho13ks with the speeds 36 and
18 kms 13lose to the experiment The time evolutions of the ele13tron density
mean 13harge and ele13tron temperature at 56 57 58 60 64 and 65 ns are
presented in Fig 518 The two sho13ks appear in Xenon at 2 and 3 ns
respe13tively for MAIN and AUX Con13erning AUX the 13ombination of a
small speed and a relative high pressure does not allow to develop a radiative
pre13ursor in agreement with the experimental results (Fig 515) whereas
the MAIN sho13k has a pre13ursor and its length is in13reasing with time
The post sho13k temperature of the MAIN is sim 21 eV and the ion 13harge
sim 9 At 57 ns the pre13ursor of MAIN rea13hes the AUX sho13k front This
time is out of our re13ord (see Fig 515) whi13h means that the intera13tion
ee13t is either absent or o1313urs at later times The stru13ture of the AUX
post sho13k is modied by the intera13tion with the MAIN pre13ursor (Fig
518b) The sho13k 13ollision o1313urs at 65 ns (Fig 518a) resulting in the
54 SUMMARY 97
Figure 517 Re13orded ele13tron density (shot 48055) together with the HELIOS
results (with Xenon opa13ity times 20) at dierent times in Xenon at 01 bar
development of two reserve sho13k waves At the 13ollision time the ele13tron
density mass density ele13tron temperature and ion 13harge rea13h respe13tively
sim 14 times 10
2113m
minus3 0034 g13m
minus3 26 eV and 10 Mean 13harge is varying
between 5-10 whi13h 13ompatible with the presen13e of lines of Xe VII-VIII in
the experimental re13ord shown in Fig 516b
Moreover in order to interpret XUV spe13tros13opi13 results shown in se13tion
52 Rodriguez performed qualitative preliminary 13omputations (as des13ribed
in Rodriacuteguez et al (2015)) of the XUV spe13tra emerging from a 600 microm thi13k
plasma with two representative values of the mass density ρ = 32 times 10
minus2
and 33 times 10
minus3g13m
3 They show that the lines of HeII 13an only be ob-
served at a temperature of sim 15 eV and for the lowest density ie in the
radiative pre13ursor
54 Summary
In this 13hapter I have presented an extensive data analysis of few representa-
tive interferometri13 and spe13tros13opi13 re13ords The average sho13k speed and
ele13tron density have been estimated from the interferograms The sho13k
speeds of the MAIN and AUX radiative sho13k waves vary between sim 30-55
and 10-30 kms respe13tively and the averaged pre13ursor ele13tron density
ranges between 10
17and 10
1913m
minus3during the sho13ks propagation
We have demonstrated the intera13tion ee13t between the two radiative
pre13ursors in the 13ase of Xe at 01 bar at 54 and 23 kms The intera13tion is
13learly 13hara13terized in the experiment by the enhan13ement of the ionisation
wave followed by the merging of the two radiative pre13ursors at 20 ns The
13ollision time is re13orded at 47 ns Su13h behavior is reprodu13ed by the
98 CHAPTER 5 RESULTS AND INTERPRETATION
(a)
(b)
(13)
Figure 518 Time evolution of the mass density (a) ele13tron temperature (b) and
mean 13harge (13) at 56 57 58 60 64 and 65 ns within the sho13k tube derived from
the HELIOS simulations (with Xenon opa13ity multiplier = 20) for two 13ounter
streaming sho13ks of sim 39 and 18 kms in Xenon at 06 bar
54 SUMMARY 99
simulation
We have investigated this intera13tion at a larger pressure 02 bar with
the following speeds sim 41 kms for the MAIN and sim 18 kms for the AUX
sho13k waves We do not re13ord any experimental signature of the radiative
pre13ursor for AUX Further we have not been able to 13at13h experimentally
the 13ollision time The re13orded pre13ursor of MAIN is not inuen13ed by AUX
wave up to 48 ns (Fig 59(b) and (13) in data analysis 13hapter) whi13h is the
limit of the re13ord On its side the 1D simulation predi13ts a tiny pre13ursor
for AUX and that both pre13ursors start to intera13t at 49 ns This plausible
sho13ks intera13tion o1313urring at times whi13h are outside of the re13ord 13an
not be 13onrmed by our experiment
The results of the transverse interferometry at 02 bar with speeds of sim40 and 20 kms indi13ate that the MAIN pre13ursor has a lateral extension of
sim 600 microm whereas it is 300 microm for AUX The pre13ursor of MAIN is almost
at with a probable small bending at the edges of the tube whereas the
AUX pre13ursor is more 13urved This means that the 2D ee13ts are more
important for AUX than for MAIN Still in the 13ase of Xenon we have
obtained information about the temperature and the mean 13harge of gas
through our time integrated XUV spe13tra (shot48143) in Xenon at 06
bar From these results we may 13on13lude that the mean ion 13harge is at
least equal to 6 and that the temperature has rea13hed 15 eV
Our simulations give a qualitative des13ription of the sho13ks intera13tion
when the laser uen13e is adjusted to give the 13orre13t sho13k velo13ities Howe-
ver it is now well known that 2D simulations (together with state of the art
opa13ities) t better with experiments (Gonzaacutelez Audit and Stehleacute 2009
Leygna13 et al 2006 Stehleacute et al 2010) For the same laser energy the 2D
simulations lead to a diminution of the sho13k speed 13ompared to 1D as also
to a diminution of the ele13tron density For instan13e in the 13ase of a sho13k
wave laun13hed by a laser beam at 1315 nm in Xenon at 03 bar at PALS and
with a laser uen13e of 85000 J13m
2 ARWEN 2D simulations give a sho13k
speed of 44 kms in agreement with the experimental one (Cotelo et al
2015) 1D simulation would require in this 13ase a uen13e of 30000 J13m
2to
a13hieve the same velo13ity
The spa13e and time integrated XUV re13ords at 06 bar for respe13tive
speeds whi13h are equal to sim 39 and 18 kms indi13ate that the temperature
of the sho13k has rea13hed values up to 15 eV and that the Xenon mean ion
13harge has also rea13hed values of 6 - 7 whereas 1D simulations predi13t ele13tron
temperature 10-30 eV and ion 13harge 5-10 (Fig 51813) A more detailed
study based on 2D simulation and radiative transfer post-pro13essing will be
ne13essary to rene the analysis
We have investigated the 13ase of other noble gases (Ar Kr) and we have
observed that for a given laser energy the sho13k velo13ity de13reases with the
mass density This study has 13onrmed that for a given density and sho13k
velo13ity the radiative ee13ts in13rease with the atomi13 number For Kr we
100 CHAPTER 5 RESULTS AND INTERPRETATION
noti13ed a tiny pre13ursor without any intera13tion To get su13h ee13t it will
be ne13essary to in13rease the sho13k velo13ity and thus the laser energy
This has been done at the ORION laser fa13ility in UK where the 13ol-
lision of two 13ounter-propagating sho13ks at equal speed sim 80 kms with
laser uen13e sim 6 times 10
14W13m
2has been performed (Clayson et al 2017
Suzuki-Vidal et al 2016) for dierent noble gases and for pressures 13ompri-
sed between 01 and 1 bar A number of diagnosti13s setup X-ray ba13klig-
hting opti13al self-emission streak imaging and interferometry (multi-frame
and streak imaging) were used to study both the post-sho13k and the radia-
tive pre13ursor Although I have not parti13ipated in the experiments I have
performed 1D simulations to interpret ORION experimental results This
work is not presented in this thesis
Chapter 6
Optimization of an
ele13tromagneti13 generator for
strong sho13ks in low pressure
gas
Contents
61 Prin13iples of operation of a high 13urrent generator 102
62 Prin13iples of the run-down phase in a PFD 103
63 Proposed design for the plasma gun 104
64 Dynami13 13ir13uit modelling 106
65 3-D MHD simulations using GORGON 13ode 111
66 Measurements 116
67 Summary 118
The previous 13hapters were dedi13ated to the study of laser generated
radiative sho13ks With irradian13e more than 10
14W13m
2 we were able to
study sho13k waves propagating at velo13ity up to 50 kms in noble gases with
an initial mass density 13omprised between 5 times 10
minus4- 3 times 10
minus3g13m
minus3
Complementary to laser experiments 13ompa13t pulsed power generators may
drive an ele13tromagneti13 13oaxial plasma gun to 13reate astrophysi13al relevant
sho13ks in lower pressure noble gases (Kondo et al 2006) with a high avai-
lability and a rather modest 13apital 13ost The ele13tromagneti13ally driven
sho13k waves may have larger s13ales than those by laser thus they 13an be
observed rather easily (Kondo et al 2008)
Su13h ele13tri13al pulsed power devi13es may then i) 13reate a high-voltage
breakdown through a gas or more easily at the surfa13e of a diele13tri13 ii)
produ13e relatively hot plasma by ohmi13 heating when the rising 13urrent in the
devi13e is passing through a portion of gas iii) a1313elerate plasma layers under
the magneti13 pressure asso13iated with the self-generated magneti13 eld when
high ele13tri13al 13urrent (say 10s of kA) is maintained for a short but su13ient
time (say 1 micros) Su13h ionization and a1313eleration are present in Z-pin13h
plasmas and in plasma fo13us devi13es (PFD) providing a 13lever geometry is
employed and the mass under 13onsideration (say 1 mg) is 13ompatible with
the stored energy (Martin Williams and Kristiansen 1996)
101
102 CHAPTER 6 ELECTROMAGNETICALLY LAUNCHED SHOCK
In this 13hapter we will see rst how a fast ele13tri13al 13ir13uit works then
we will dis13uss the prin13iples of a 13oaxial plasma gun and of PFD We will
see what is in favor of our obje13tive and what has to be avoided A geometry
13apable to rea13h our obje13tive will be proposed
The optimization of the 13oaxial gun (in term of plasma speed) will be
performed and some typi13al gures will be given in the 13ase of a generator
with a stored energy around 1 kJ and a plasma slab of 4-mm diameter ie
quite 10 times the transverse dimension of a laser driven RS The operating
13onditions are supersoni13 sho13ks up to 10-30 kms speed at stati13 pressures
of few mbar in heavy rare gases (Ar Xe) The results of this 0-D model will
be then 13ompared to those obtained with 3-D MHD simulations performed
with the 13ode GORGON (by Andrea Ciardi at LERMA) whi13h has been
used su1313essfully to des13ribe other pulse-power driven plasma experiments
(Chittenden et al 2004) as well as laboratory plasma astrophysi13s experi-
ments (Ciardi et al 2007) The diagnosti13s whi13h have been implemented
will be presented to illustrate the model as well as preliminary re13ords of the
plasma speed
61 Prin13iples of operation of a high 13urrent gene-
rator
Our aim being to a1313elerate a plasma slab using the magneti13 pressure it
is obviously needed to drive a high intensity 13urrent be13ause the magneti13
pressure is expressed by B
22micro0 in the region where exists an indu13tion B
The pressure in bars is simply 4B
2 with B in Teslas and in a 13ylindri13al
geometry 1 Tesla is the eld around a 50-kA 13urrent at a radius of 1 13m
Thus we should 13ount on roughly 100 kA delivered by the generator but
su13h a high 13urrent is delivered only in a pulsed mode by a laboratory s13ale
devi13e
There are numbers of te13hni13al solutions to a13hieve su13h ele13tri13al pa-
rameters one may use rather slow generators at moderate high voltage like
13apa13itors bank with the advantage of well know te13hniques but with limited
adjustments A1313ording to an abundant literature
1
a 13onvenient devi13e is a
medium-energy 13apa13itor bank feeding a low-indu13tan13e 13ir13uit An R-L-C
13ir13uit is a well-known 13ombination delivering a high peak 13urrent in the
pseudo-periodi13 mode For a 13apa13itor C initially 13harged under U0 key
performan13es are as follows
Current intensity I(t) is a damped sinusoid
Pseudo pulsation ω =radic[1(LC)minusR2(4L2)]
1
Institute for Plasma Fo13us Studies resour13e website httpplasmafo13usnet last
13onne13tion in 2016
62 PRINCIPLES OF THE RUN-DOWN PHASE IN A PFD 103
First 13urrent peak Ipeak = U0[radic(LC) + 08R]
Time of 13urrent peaking T4 = π(2ω)
Equivalent impedan13e Z =radic(LC)
For a safe handling in air a voltage not higher than 30 kV is re13ommen-
ded Taking a total 13apa13itan13e of 6 microF 2700 J are stored under 30 kV For
an indu13tan13e of 240 nH the expe13ted peak 13urrent might be 150 kA at 19
micros This set of values will be a guideline for the development presented in
further se13tions
62 Prin13iples of the run-down phase in a PFD
Our aim is thus to 13reate a fast moving plasma sheath with quite a one-
dimension geometry The issues are to initiate this plasma at the best then
to maintain even improve its stru13ture during the a1313eleration nally to
laun13h it
Instead of 13reating a gas breakdown in volume whi13h would lead ra-
pidly to 13on13entrate the 13urrent in an ar13 the idea for PFD operation is to
start from a surfa13e dis13harge (Lee 1969 Bernard 2002) In the 13lassi13al
13oaxial geometry of the Mather-type PFD the 13entral ele13trode (usually the
anode) is a 13ut metal 13ylinder and the outer one is a se13ond metal 13ylin-
der of the same length or better a squirrel 13age allowing many viewpoints
to the plasma Both ele13trodes are atta13hed to a diele13tri13 bottom plate
made of polymer or 13erami13 When the 13apa13itor bank is swit13hed on the
high voltage is applied to this 13m-size gap and a radial surfa13e dis13harge is
initiated at the interfa13e with the gas forming a quite uniform ring-shaped
layer of thi13kness around 1 mm An insulating sleeve is adjusted around
the 13entral ele13trode to for13e the plasma to ow rapidly upwards under the
magneti13 pressure in this region The result is an elongation the dis13harge
path preventing an ar13ing in the plasma layer
At later times as the 13urrent is growing the plasma layer is strongly
inuen13ed it is pushed upwards by the magneti13 pressure jtimesB it is heatedby the ohmi13 ee13t then its ionization degree in13reases it be13omes denser
as a per13entage (10-40) of the heavy parti13les from the swept volume are
a1313reted (Potter 1971) This ele13tromagneti13ally driven motion is 13alled the
rundown phase Due to the a1313retion it is des13ribed by a so-13alled snowplow
model whi13h will be dis13ussed later in detail In quite all the designs even
if very high speeds are reported (Lee 1969 Serban 1995) the plasma is bell
shaped whi13h does not fulll our requirements
At the end of the rundown phase whi13h mat13hes roughly with the 13urrent
peak the plasma stays 13onne13ting the upper ends of the ele13trodes and the
radial 13omponent of the magneti13 for13es grows rapidly Then the plasma is
104 CHAPTER 6 ELECTROMAGNETICALLY LAUNCHED SHOCK
strongly pushed to the axis giving the so-13alled Z-pin13h This stage made
the PFD popular be13ause the pin13hed plasma is a sour13e of fast ele13trons
fast ions hard radiation and possibly neutron beams it must be avoided
in our 13ase It was also quoted (Lee and Serban 1996) that optimal PFD
are mat13hing a universal fa13tor 13omprising anode radius peak 13urrent and
gas density meaning a robustness of the design when a geometry has been
13hosen
63 Proposed design for the plasma gun
The previous des13ription has shown the positive inuen13e of a surfa13e dis-
13harge the modi13ation of the shape by the sleeve and the issue of the
pin13hing ee13t A1313ordingly the following design is proposed following a
work by Kondo et al (Kondo et al 2006 2008)
To ensure a rapid dis13harge with the values given in the se13tion 61 a
highly 13oaxial 13ir13uit is 13hosen for all the large parts the energy bank the
13losing swit13h and all the 13onne13tions The initial phase is kept with two
13oaxial ele13trodes pressed on a at insulator Homogeneity of the plasma
sheath is expe13ted by 13hoosing a rather small radial gap of 25 mm The ee13t
of magneti13 for13es during the rundown phase must be enhan13ed espe13ially
due to the in13rease of the mass of the plasma as des13ribed above As the
13urrent 13urve is the rising part of a sinusoid a tri13k is used to in13rease
the lo13al magneti13 eld B around a 13ondu13tor varies as the inverse of the
distan13e to axis so the proposed shape for the anode is a 13oneHowever a
nal divergen13e must be avoided and that is obtained by a rounded tip In
order to keep 13onstant the plasma length along the 13urrent path the radial
gap is kept 13onstant so the outer ele13trode is a hollow 13one with the same
angle Above the 13one the a1313elerator is 13onne13ted to a 13ondu13ting tube
where the plasma 13an propagate freely in the ba13kground gas The distan13e
of the plasma sheath to the 13one tip is in13reasing rapidly and the main
13urrent whi13h is still high will pass preferably through the diuse plasma
remaining between the ele13trodes Thus there will be no magneti13 pressure
anymore a13ting on the sheath whi13h 13an propagate freely The resulting
devi13e is des13ribed in Fig 61a and the a13ting magneti13 for13es shown in Fig
61b
Te13hni13ally su13h a 13onguration is 13onvenient even at 15 kV be13ause the
sharp edge of the anode tou13hing the insulator forms a so-13alled triple point
where the ele13tri13 eld is enhan13ed at the surfa13e of the insulator whi13h
triggers e13iently a dis13harge in presen13e of gas at low pressure (01 10
mbar) as proposed by Kondo et al (Kondo et al 2006) Polya13etal is a
13onvenient material for diele13tri13 and me13hani13al performan13es
The dire13tion and the orientation of the magneti13 for13e are other points to
be dis13ussed The high-intensity 13urrent (up to sim 150 kA) generates a strong
63 PROPOSED DESIGN FOR THE PLASMA GUN 105
(a)
(b)
Figure 61 (a) Sket13h of the sho13k generator showing the pulsed ele13tri13al 13ir13uit
the set of 13oaxial 13oni13al ele13trodes with a 13onstant radial gap and the plasti13
insulator featured in grey on whi13h a planar surfa13e dis13harge is initiated The
installation of three opti13al bers allows looking radially at the plasma moving in
the sho13k tube (b) S13hemati13s of the plasma dynami13s inside the 13oaxial gun
in fast-pulse mode the ele13tri13al 13urrent ows in the super13ial layers of the two
13oaxial 13oni13al ele13trodes and through an annular plasma layer The magneti13
pressure Pmag pushes the dis13harge axially
106 CHAPTER 6 ELECTROMAGNETICALLY LAUNCHED SHOCK
Figure 62 Exploded view of the plasma gun
azimuthal magneti13 eld between the ele13trodes thus the magneti13 pressure
jtimesB whi13h a1313elerates the annular plasma sheath stays perpendi13ular to the
sheath lo13ally (Fig 61b) The more planar is the initial sheath the more
axial will be the magneti13 pressure Finally if the roles of the ele13trodes are
ex13hanged the dire13tion of the 13urrent will be inverted and the same for B
so the pressure will stay oriented in order to expel the plasma Compared
with the Mather-type plasma fo13us (Potter 1971 Zambra et al 2009) the
plasma sheath 13onsidered here is quite planar 13lose to the insulating surfa13e
and later is expe13ted to stay planar and perpendi13ular to the axis As in a
PFD the thi13kness of the plasma sheath in13reases gradually (Zambra et al
2009) but it is assumed to stay in the mm-range We employ 13oaxial 13oni13al
brass ele13trodes ea13h 42-mm high with a 13onstant gap of 25 mm At the
bottom level internal and external radii of ele13trodes are 125 mm and 15
mm respe13tively This small-size dis13harge 13ell on top of the generator will
be modelled in the following
64 Dynami13 13ir13uit modelling
We optimize the ele13tri13 generator for various gases namely Ar and Xe
with the motivation to produ13e plasma sho13ks with speeds sim 1 - 30 kms
ie Ma13h numbers up to 200 For that we design the ele13trodes and set
64 DYNAMIC CIRCUIT MODELLING 107
the ele13tri13al 13ir13uit parameters to produ13e 1-micros pulses in the gas 13hamber
The geometry of the ele13trodes is as mentioned above The half angle of the
13oni13al ele13trode is 13hosen to be α =15deg and thus the height of the 13one is h
sim 42 mm
Eleven (N = 11) 13apa13itors ea13h with 13apa13itan13e C0 of 06 microF are13onne13ted in parallel giving an equivalent 13apa13itan13e of the bank C = 66
microF 13harging voltage U0 of the bank is set to 15 kV Thus the available
ele13tri13al energy is 750 J The 13ir13uit resistan13e is used to be 1 m Ω and
damping resistan13e is set to zero whi13h yields a total resistan13e (R = Rcir+
Rdamp N ) of 1 mΩ and a pseudo-periodi13 regime is expe13ted
Based on the above geometri13al and ele13tri13al parameters we have esti-
mated the time evolution of various parameters of the radial dis13harge layer
namely speed mass a1313eleration et13 at a dierent position along the axis
from z = 0 to h For this 13al13ulation we need to solve a set of three equa-
tions whi13h give the main 13urrent (- dqdt) supposed to ow only through
the plasma layer the mass a1313retion rate (dMdt) and the velo13ity v of
the plasma These equations together with initial 13onditions are explained
below The ele13tri13al equation is written 13lassi13ally
Ld2qdt2 +Rdqdt+ qC = 0 (61)
When the R L and C parameters are not time-dependent the 13ir13uit
equation 61 has an analyti13al solution a damped sinusoid for the 13urrent I
= -dqdt and that was used as a test of the iterative solving pro13ess and to
13onrm the parameters of the external 13ir13uit with a short 13ir13uit load The
indu13tan13e of the external 13ir13uit L0 is 13omputed from the ringing frequen13y
in short 13ir13uit (66 nH) whi13h results in a 13ir13uit impedan13e Z0 =
radic
L0Cto be 01 Ω In the plasma gun mode the sheath 13arrying the 13urrent (see
Fig 61b) is a1313elerated by the magneti13 pressure and it behaves as the de-
formable part of the 13ir13uit then L is time-dependent and a numeri13al solver
is ne13essary for equation 61 We used a lumped-parameter model whi13h has
proven to des13ribe similar 13ir13uits su1313essfully (Potter 1971 Zambra et al
2009) Gonzalez (Gonzalez et al 2004) showed that the mass and momen-
tum equations for a 13urrent sheath in the shape of an annular piston moving
forward in the axial dire13tion 13an be given by
dMdt = ερ0π(R2e minusR2
i)v (62)
where ε is a1313retion fa13tor and that the transformation of magneti13 to
kineti13 energy is expressed by
d(Mv)dt = lI22 (63)
where l is the linear indu13tan13e of the gun and 13an be expressed as l =
(micro04π) ln (ReRi) the radii being estimated where the plasma sheath is
13urrently lo13ated The equations 62 and 63 have been introdu13ed in the
108 CHAPTER 6 ELECTROMAGNETICALLY LAUNCHED SHOCK
(a)
(b)
(13)
Figure 63 Time dependen13e of the main parameters for Ar gas at 1 mbar a1313or-
ding to the 13ir13uit model (a) 13urrent (kA) and voltage (V) (b) a1313reted mass
(kg) and rate of a1313retion (kgs) (13) speed (ms) and kineti13 energy (J)
64 DYNAMIC CIRCUIT MODELLING 109
Figure 64 Plot of the dierent output parameters (normalized with respe13t to
the peak values) obtained at the top of the inner ele13trode For ea13h group upper
panel represents the variation for Ar gas whereas lower panel is for Xe gas Proles
of output parameters are given (a) with 13one angle (b) with a1313retion fa13tor (13)
with gas pressure and (d) with damping resistan13e
110 CHAPTER 6 ELECTROMAGNETICALLY LAUNCHED SHOCK
time-dependent iterative solving of equation 61 Therefore from equations
62 and 63 we get
dvdt =[ lI
2
2 minus ερ0π(R2e minusR2
i)v2]
M(64)
From equation 64 we may note that initial value of mass M(t0) 13annot
be set to zero as it would lead to a nonphysi13al divergen13e of the a1313eleration
In this regard we have 13onsidered an initial thi13kness of the 13urrent sheath
of 1 mm and integrated the gas density at rest leading to M(t0) equal to
35 times 10
minus10kg for Ar and 13 times 10
minus9kg for Xe A higher limit has to be
put too M(t0) must be small 13ompared with the mass at the end of the
laun13hing phase that is the mass 13ontained in the gap times the a1313retion
fa13tor The above value for M(t0) follows that 13riterion and within these
limits the 13hoi13e of initial mass is not inuen13ing the results
Parameter Referen13e Value Variation
Half angle of the 13one 15
13
14
15
16
middot middot middot 24A1313retion fa13tor ǫ 04 01 02 03 middot middot middot 09
Pressure P 100 Pa 10 20 50 100 200 500 1000
Damping resistan13e Rdamp 0 ohm 0 01 02
Table 61 Input parameters their standard values and range of variation
As the motion is mainly along the z-axis the speed v is the derivative
of the plasma position v = dzdt We solved these equations by a time-
dependent method where the initial 13ondition is that at the onset time t0
the 13harge q is set to CU0 Further the 13urrent i = -dqdt is taken to be zero
at t0 Thus using equation 61 the initial rate of 13hange of the 13urrent is
d
2qdt
2= - qL0C Then all the rst time-derivatives are updated at the rst
time step t0+∆t with ∆t=043 ns using equations 61-64 The quantities
like q M v z are updated at the next step then the iteration is pursued
Initially position a1313eleration and velo13ity of the sheath are set to be zero
as well as the rate of 13hange of the mass
As we initiate the ele13tri13al 13urrent into the gas inside the 13hamber the
plasma sheath is a1313elerated with respe13t to its initial mass as seen above
At later times the mass a1313retion is masking the inuen13e of this arbitrary
value We have 13al13ulated the plasma evolution in the dis13harge for dierent
sets of initial 13onditions whi13h are reported in Table 61
Fig 63 shows the behavior for the standard values reported in Table 61
in the 13ase of Argon at 1 mbar The 13orresponding values of 13harge mass
and velo13ity will be hereafter 13alled the referen13e values In Fig 63a the
variation of 13urrent and voltage in the rst half-period is 13lassi13al showing
that the pulse generator departs slightly from a pure RLC 13ir13uit Fig
63b represents the a1313reted mass whi13h in13reases in time as well whereas
65 3-D MHD SIMULATIONS USING GORGON CODE 111
the rate of mass a1313retion 13ulminates with a 13ompetition of speed growth
and de13rease of the swept volume per time step be13ause it depends on the
squared radii In Fig 6313 plotted are the speed and kineti13 energy of the
plasma sheath First the speed of the plasma is in13reasing slowly with time
quite linearly as the mass and the kineti13 energy is roughly a third power
of the time
That leads to 13hoose a design where the a1313eleration is imposed for a
su13iently long time However the estimated speed of plasma is sim 140 kms
whi13h is a very optimisti13 value The reason may 13ome from two reasons
(i) a part of the 13urrent ows through the large sho13ked volume redu13ing
the magneti13 pressure a13ting on the plasma sheath (ii) energy transfer from
magneti13 to thermal energy and ionization is redu13ing the kineti13 one The
observation of a very bright emission from the plasma will be a 13lear eviden13e
of this redistribution of the energy
Further we made a 13omparative study of the parameters indi13ated in
Table 61 and applied to two gases namely Ar and Xe In order to 13he13k the
ee13ts of the initial 13onditions (13one angle a1313retion fa13tor pressure and
damping resistan13e) we have performed a systemati13 study by varying only
one parameter and keeping the others 13onstant and equal to the referen13e
values given by the se13ond 13olumn of Table 61 The tested range is listed
in the third 13olumn of Table 61 Similarly for ea13h set the variation of the
plasma 13onditions at the top of the muzzle is normalized to the referen13e
value Fig 64 summarizes the variation of the sele13ted dierent output
parameters the time tm to rea13h the muzzle (z = h) the ele13tri13al 13urrent
and the axial speed at muzzle the laun13hed mass M the kineti13 energy and
the average speed at the top of the ele13trode 13al13ulated from the initial
point v=h( tm - t0)
Table 62 summarizes the respe13tive ratios for Ar and Xe at the top of
the 13oaxial ele13trodes As the values of interest are obtained at 13onstant
stored energy in the 13apa13itors there is a benet to work with a lighter gas
like Argon However the results obtained using this 13ir13uit model show that
the referen13e values are ensuring a satisfa13tory behavior for both gases
Speed Kineti13 energy Time to laun13h Mass
222 151 66 31
Table 62 Laun13hing performan13e ArXe
65 3-D MHD simulations using GORGON 13ode
Our simplied model (13f se13tion 64) is 13onvenient to perform qui13kly some
parametri13 tests in the a1313eleration phase and to help designing the devi13e
but it is not 13apable to treat the sho13k dynami13s in detail nor to give valuable
112 CHAPTER 6 ELECTROMAGNETICALLY LAUNCHED SHOCK
information on the plasma parameters in the sho13k region So rened nume-
ri13al simulations of the experiments were performed in 13ollaboration with A
Ciardi (Ciardi 2014) using the GORGON 13ode (Chittenden et al 2004 Ci-
ardi et al 2007 Suzuki-Vidal et al 2014) GORGON is an expli13it parallel
13ode designed to solve the resistive MHD equations on a three-dimensional
(3-D) Cartesian grid employing a Van Leer type algorithm The 13ode treats
the plasma as a single uid but solves separately the energy equations for
ions and ele13trons allowing dierent temperatures for the two spe13ies Both
thermal 13ondu13tion and resistive diusion are treated using Braginskii-like
transport 13oe13ients
Variables At rest Sho13k Post-sho13k
Ar density ρ (g13m
3) 18E-6 25E-5 10E-5
Ele13tron density Ne (13m3) 10 E+15 20E+18 15E+18
Ele13tron Temperature Te (eV) 003 (1lowast) 10 15-18
Ion density ni (13m3) 30E+16 35E+17 2-3E+7
Ion Temperature Ti 0 50 15-20
Average Speed ltVgt kms 0 50 20-40
ltzgt 02 6 7-9
Table 63 Values of variables in various 13onditions at rest (with seed ele13trons)
inside the sho13k and inside post-sho13k region A star lowast denotes the insulator surfa13e
This 13ode des13ribes the 13reation of the surfa13e dis13harge then its a1313ele-
ration by the magneti13 pressure It provides the mapping of all the plasma
parameters and lo13al B-eld in the laun13hing 13one as well as in the free ight
region The 3-D simulation was performed for Argon gas at 1 mbar only the
dis13harge 13urrent law being taken from experiment The Table 63 presents
the ranges of all the variables in the regions of interest
As an example Fig 65 shows an enlarged mapping of log (ne) at two
times in the laun13hing phase then when the sho13k has deeply penetrated
in the sho13k tube Taking the zero time as a referen13e the averaged speed
zt is 40 kms in this simulation When the two snapshots are 13onsidered
the estimate of the instantaneous speed zt is higher at 56 kms The
time history of plasma merging and early free ight into the sho13k tube
is presented in Fig 66 During the a1313eleration (inside the double 13one)
the plasma is well lo13alized on a planar annular sheath whi13h is 13arrying
the 13urrent as assumed in the lumped parameter model We see that the
merging is well a13hieved at the muzzle shortly before 1250 ns When the
sho13k has penetrated inside the tube (1500 ns) the 13omputed B-eld is
negligible at this lo13ation and the propagation is quite free as expe13ted for
the 13hosen design of the devi13e However a high 13urrent is still passing
through the gas at that time a su13ient ele13tron density remaining in the
13oni13al gap for that
65 3-D MHD SIMULATIONS USING GORGON CODE 113
Figure 65 Example of mapping of the ele13tron density from a 3-D MHD simu-
lation (ba13kground gas Argon at 1 mbar) when the plasma sheath is (left) at
the exit of the plasma gun (right) in free ight 13onditions (log s13ales for the false
13olors)
114 CHAPTER 6 ELECTROMAGNETICALLY LAUNCHED SHOCK
Figure 66 Time history of plasma merging and early free ight through the
mapping of the ele13tron density ele13tron temperature and average ioni13 13harge as
given by a 3-D MHD simulation (ba13kground gas Argon at 1 mbar) (s13ales for the
false 13olors)
65 3-D MHD SIMULATIONS USING GORGON CODE 115
Figure 67 Axial proles values taken in the sho13k tube along a line slightly o-
axis at a distan13e of 1 mm for mass density ion temperature ion density average
ion 13harge magneti13 eld ele13tron temperature ele13tron density and average speed
at 1500 ns (ba13kground gas Argon at 1 mbar) A maping of |B| is given with a
dashed line indi13ating sho13k front position as well
116 CHAPTER 6 ELECTROMAGNETICALLY LAUNCHED SHOCK
To 13onrm the steep stru13ture of the sho13k Fig 67 presents the axial
proles of ele13tron temperature Te number density Ne mass density ρ and
average ion 13harge ltZgt at 1500 ns inside the tube with an enlarged axial
position s13ale Steep fronts are observed for mass density ele13tron density
and temperature as in the earliest simulations (Potter 1971 Kondo et al
2008) The sho13k region is at 10 eV and sim2times1018 ele13trons13m
minus3 the post-
sho13k region is hotter Argon ionization stages of 5 (sho13k) to 9 (post-sho13k)
are obtained fore13asting an emission spe13trum in the UV or harder A wider
view of the plasma is given by other quantities at the sho13k front and behind
ion temperature average speed and lo13al magneti13 eld in Fig 67 We get
the 13onrmation of the extin13tion of B after the rst 5 mm inside the sho13k
tube as well as the sho13k speed at 50 kms 13oherent with the 13onstant speed
of 56 kms mentioned just above At present these parameters are guiding
the 13hoi13e of diagnosti13s like ultra-fast interferometry or UV-spe13tros13opy
66 Measurements
The aim of these tests was to show the 13onsisten13e of the above model by me-
asuring the plasma speed 13lose to the plasma gun muzzle The parameters of
the devi13e are the referen13e ones as dened in Table 61 A rather simple and
noise-free method for diagnosing a plasma gun during the rundown phase
is to re13ord the light emitted by the plasma sheath with a spatial resolution
13ompatible with the plasma stru13ture If we 13onsider a line of sight whi13h
is radial or tangential ie perpendi13ular to the dire13tion of propagation
one expe13ts to see a sudden rise when the plasma enters the dete13tion vo-
lume possibly a plateau when the plasma travels inside the volume then a
slow de13rease when the hotter plasma leaves the volume but when the sho13k
remnants are still present So the expe13ted signal is a triangular asymme-
tri13 pulse (Serban and Lee 1995) or some kind of double exponential pulse
(Stehleacute et al 2012) A peak indi13ates the time of ight for the given posi-
tion allowing to 13ompute a mean speed providing the laun13hing time (t=0)
and all positions are known An instant speed is a1313essible providing two
13lose dete13tors are 13onsidered That has been proposed and tested su1313ess-
fully in the 13ase of radiative sho13ks driven by one PALS laser beam (Stehleacute
et al 2012) the dete13tors being 13ollimated and ltered sili13on diodes For
13ooler plasmas the visible emission is more relevant Inside a squirrel 13age
plasma fo13us Serban has installed a set of opti13al ber looking tangentially
at the drift zone (Serban and Lee 1995) Despite a 13oni13al eld of view the
results were 13on13lusive at speeds as high as 100 kms in D2 An improved
version with 13ollimated opti13al bers (a1313eptan13e angle 2deg) was developed
too (Serban and Lee 1995) A non intrusive method was developed (Veloso
et al 2012ba) by imaging the plasma on a set of opti13al bers allowing to
play with the magni13ation
66 MEASUREMENTS 117
(a)
(b)
Figure 68 Experimental results (a) time history of the passive opti13al re13ords
13ompared to the main 13urrent for Argon gas 13lose to the referen13e pressure (b)
time-integrated signal vs Ar pressure for the dierent bers (13) averaged speed vs
Ar ba13king pressure
118 CHAPTER 6 ELECTROMAGNETICALLY LAUNCHED SHOCK
In our 13ase (see Fig 61a) three 1-mm PMMA (Poly(methyl metha13ry-
late)) opti13al bers are installed radially along a transparent se13tion of the
sho13k tube at z sim 625 mm 70 mm and 775 mm and fast DET10A dete13tors
re13ord the transient presen13e of the plasma sheath in the respe13tive lines of
sight The re13orded spe13tral range is limited to the whole visible in a preli-
minary stage a band-pass ltering might give information on the presen13e of
spe13i13 ioni13 spe13ies 13hara13teristi13 of a plasma temperature level however
the amount of light was not su13ient to do so Fig 68a presents the time
evolution of the ele13tri13al 13urrent with a period of 4 micros and the delayed
side-on observational data of the moving plasma inside the sho13k tube The
behavior of the ele13tri13al 13ir13uit is following well the expe13ted ringing of a
weakly damped RLC 13ir13uit the zero-time being xed at the onset of the
13urrent On opti13al 13hannels a steep rise and a peak are the signature of a
mm-thi13k fast-sho13k stru13ture (Serban and Lee 1995) 13onsistent with early
numeri13al simulations (Potter 1971 Tou Lee and Kwek 1989 Kwek Tou
and Lee 1990 Veloso et al 2012b) On ea13h ber signal se13ondary peaks
namely the one delayed by 4-5 micros are attributed to 13urrent restrikes at the
pseudoperiod To avoid the mixing of sho13ks further experiments should be
13ondu13ted in an aperiodi13 regime with a heavy-duty damping resistor added
in series to ea13h 13apa13itor Rdamp sim NZ
On Fig 68b and 13 are presented the exploitation of opti13al re13ords with
the Ar ba13kground pressure this parameter varying in the widest range (b)
the integrated emission whi13h presents a maximum between low pressure side
when the temperature is high but the number of emitters is low and the high
pressure side when the temperature is mu13h lower (13) the averaged speed
ziti for the rst peak on ea13h ber The relative intensities re13orded in the
three lo13ation s are not signi13ant be13ause of a strong 13opper deposition
13loser to the muzzle whi13h ae13ted the dete13tion The long-life noise on
the opti13al signals was making the lo13al speed 13al13ulated as (zi-zj)(ti-tj) a
non-reliable output One 13an note Ma13h numbers in the range 20 60 with
a pressure dependen13e similar to previous work (Kondo et al 2006) The
reported observations are 13onsistent with a high-speed millimetri13 planar
plasma travelling in the sho13k tube The speed is lower than dedu13ed from
the simplied model possibly due to the la13k of dissipation terms whi13h
is overestimating the transfer to kineti13 energy At the 13ontrary a realisti13
simulation as the one performed using GORGON is in better agreement
67 Summary
For this part of the work the obje13tive was to show that in 13omplement of
laser experiments a 13ompa13t pulsed power generator might drive astrophy-
si13ally relevant sho13ks in low pressure noble gases with a higher availability
and at a rather modest 13apital 13ost The longitudinal a1313eleration of a
67 SUMMARY 119
plasma sheath in a low ba13king gas pressure has been demonstrated in 13o-
axial plasma guns as early as in the 70s even leading to speeds of 100 kms
in light gas as H2 whi13h were measured and modeled A 13hoi13e was made
to revisit this type of devi13e At present the ele13tromagneti13 13oaxial plasma
guns are quite only 13onsidered as the initiators of a strong radial plasma
implosion plasma fo13us devi13es (PFD) due to a good ability to produ13e a
pin13h plasma and emission of energeti13 parti13les and hard X-ray For that
reason it was ne13essary to adapt the prin13iple of a PFD with two major
13onstraints in13rease the 13onversion of stored energy into the kineti13 energy
of a plasma sheath prevent totally any pin13hing
That obje13tive was rea13hed by dening the ele13tri13al and geometri13al
parameters of a low-indu13tan13e and 13ompa13t pulsed power generator An
optimization pro13ess has been established to mat13h a13hievable ele13tri13al pa-
rameters of the 13ir13uit (13apa13itan13e indu13tan13e peaking 13urrent intensity)
with a plasma motion in the a1313elerating tube over few 13m leading to a nal
speed of 10-30 kms A model was built up to des13ribe the dynami13s of an
RLC 13ir13uit with a varying element 13omposed of the moving plasma sheath
a1313elerated by the self generated magneti13 pressure The 13hange in plasma
position along the axis is in13reasing the indu13tan13e and the mass a1313retion
is in13reasing the plasma mass It was shown that a 13hange in the ele13trode
geometry namely a 30deg 13oni13al shape was in13reasing the energy 13onversion
and preventing any fo13using The further step in 13ontrast with PFD is the
merging of the plasma ring into a rather planar one whi13h is inje13ted in a
drift tube
Considering Ar or Xe at gas pressure in the range 01 10 mbar and
a highly 13oaxial 1-kJ devi13e the simple model was su13ient to predi13t
performan13es agreeing with the only work quoted in the literature Basi13
observations of the moving plasma using side-on dete13tion of the visible
emission give features 13oherent with the model Ma13h numbers from 20
to 60 are obtained tting with the obje13tive of strong sho13k formation A
rened 3-D MHD numeri13al simulation indi13ates very promising features of
the sho13k in view of exploring mm-size sho13ks in a regime 13omplementary
to laser driven ones and on a table-top devi13e allowing a higher repetition
rate
120 CHAPTER 6 ELECTROMAGNETICALLY LAUNCHED SHOCK
Chapter 7
Con13lusions and perspe13tives
71 Con13lusions
Although ubiquitous in the astrophysi13al environments radiative sho13ks are
13omplex phenomena whi13h still require an important eort to understand
them Beside observations whi13h suer from a la13k of angular resolution
laboratory experiments provide today an interesting approa13h to improve
our present knowledge In this 13ontext my thesis work is dedi13ated to the
experimental and numeri13al study of the 13hara13teristi13s of two 13ounter pro-
pagating radiative sho13ks propagating at dierent velo13ities (20-50 kms)
Most of the work presented here is the out13ome of the rst-of-its-kind ex-
perimental 13ampaign held in year 2015 at Prague and Laser Asterix system
(PALS) laser fa13ility
In this manus13ript I presented the setup for the aforesaid sho13k expe-
riment Following this the data analysis results interpretation as well as
relevant numeri13al simulations 13on13erning various diagnosti13s viz visible in-
terferometry and XUV spe13tros13opy have been presented Although a large
fra13tion of the experimental re13ords were performed for Xenon at various
pressures I have also studied the sho13k 13hara13teristi13s of dierent gases like
Ar Kr and He Our results are 13omplementary to those obtained at ORION
laser fa13ility also in 2015 relative to the 13ollision of identi13al radiative sho13k
waves at higher velo13ities (80 kms) In parallel in this thesis I have also
worked on the optimization of a setup dedi13ated to ele13tromagneti13ally ge-
nerated strong sho13ks at lower velo13ities In the following I briey present
a brief summary of the main results of my thesis work
In the third 13hapter I have presented a series of 1D hydrodynami13 simu-
lations (realized with the 13ommer13ial 13ode HELIOS) on isolated and 13ounter-
propagating sho13k waves at equal and dierent velo13ities The simulations
13onrm that a single radiative sho13k propagating in Xenon gas at 01 bar
is 13hara13terized by an extended pre13ursor and a large 13ompression of 30 in
the post sho13k At 50 kms the temperature evolution with the distan13e
shows identi13al post-sho13k and pre-sho13k values This indi13ates that the
numeri13al sho13k is of super13riti13al nature However I nd that there is an
important un13ertainty in the Xenon opa13ity whi13h makes the renement of
the simulation unne13essary espe13ially in terms of group numbers Next I
have investigated the intera13tion two 13ounter propagating sho13ks for iden-
ti13al (50-50 kms) and non identi13al (50-20 kms) sho13k speeds For the
121
122 CHAPTER 7 CONCLUSION
13ase of identi13al speeds the pre13ursors merge together at around 10 ns The
intera13tion is then 13hara13terized by a regular in13rease of the ele13tron density
and the temperature with the time The sho13ks 13ollision at 38 ns leads to a
jump in the ele13tron density (6 times 10
2113m
minus3) and temperature (39 eV) On
the other hand for the 13ase of non-identi13al sho13k speeds (13ase representing
our experiments) the pre13ursors intera13tion starts later than in the former
13ase The sho13ks 13ollision o1313urs at 49 ns and it is 13hara13terized by a sudden
in13rease of the ele13tron density also by an order of magnitude (4 times 10
21
13m
minus3) whereas the temperature in13reases up to 29 eV
Although the data analysis of all the experimental re13ords obtained du-
ring the PALS experimental 13ampaign has been 13arried out only a few repre-
sentative interferometri13 and spe13tros13opi13 re13ords for Xe gas are dis13ussed
in details in this thesis The interferometri13 re13ords allowed me to estimate
the average sho13k speed and time variation of the ele13tron density during the
sho13k propagation The sho13k speeds of the MAIN and AUX radiative sho13k
waves are found to be ranging between sim 30-55 and 12-25 kms respe13tively
whereas the averaged pre13ursor ele13tron density is varying between 10
17and
10
1913m
minus3during the sho13ks propagation
From the interferometri13 re13ords I have investigated the ee13t of the
intera13tion between the two radiative pre13ursors for the sho13ks propagating
in Xenon at 01 bar with respe13tive speeds of 50plusmn3 and 23plusmn3 kms The
intera13tion starts at 20 ns and is followed by the merging of the two ra-
diative pre13ursors This pre13ursor intera13tion is 13hara13terized through the
enhan13ement of the ionisation wave The sho13ks 13ollision is re13orded at 50
ns On the 13ontrary the same intera13tion behavior is not seen at 02 bar
(sho13k speeds sim 38plusmn4 and 18plusmn2 kms for the MAIN and AUX sho13k waves
respe13tively) Moreover in this 13ase there is no signature of a radiative pre-
13ursor for the AUX sho13k The pre13ursors intera13tion if any should then
13ould o1313ur at times whi13h are outside of the re13ord and 13an not be 13on-
rmed by the experiment This indi13ates that for a given gas and sho13k
speed the radiative ee13ts de13reases with the initial mass density
The investigation of the lateral extension of the sho13k has been made
through the analysis of transverse interferograms At 02 bar the MAIN
sho13k with a speed sim 40 kms has a lateral pre13ursor extension of sim570plusmn30microm whereas it is 275plusmn25 microm for AUX sho13k whi13h is propagating
with a speed of 20 kms The MAIN sho13k pre13ursor is almost of a at
spatial prole whereas the AUX pre13ursor is more 13urved suggesting that
the 2D ee13ts are mu13h more important for AUX than for MAIN One of
the explanations is that the spot size of AUX laser on the target is smaller
than the target width whereas the MAIN laser has a spot size whi13h is equal
to the target width
In order to interpret the experimental results with appropriate 1D simu-
lations I have optimized the lasers uen13es to numeri13ally obtain a sho13k
speed equal to that re13orded in the experiment In the 13orresponding simu-
72 PERSPECTIVES 123
lations for Xenon at 01 bar I note the intera13tion of the two pre13ursors
However I do not nd a good quantitative agreement for the ele13tron den-
sity Beside the question of the impre13ise opa13ities this disagreement might
also be attributed to 2D ee13ts (Gonzaacutelez Audit and Stehleacute 2009 Leygna13
et al 2006 Stehleacute et al 2010)(Cotelo et al 2015)
The analysis of spa13e- and time-integrated XUV re13ord at 06 bar is
presented for two 13ounter propagating sho13ks of speeds sim 36plusmn4 and 18plusmn5kms for MAIN and AUX respe13tively The presen13e of HeII Balmer lines
and Xenon lines tends to indi13ate a temperature of the sho13k of about 15
eV and a Xenon mean ion 13harge around 6 - 7 The 1D simulations predi13t
ele13tron temperature in the range of 10-30 eV while ion 13harge to be ranging
between 5 and 10
In addition to this study whi13h was fo13used on the Xe 13ase I have
investigated the sho13k 13hara13teristi13s for Ar Kr at 13lose mass density (sim 6
times 10
minus4g13m3) whi13h 13orrespond to 03 bar and Kr gas 02 bar Almost no
pre13ursor is found to be present in 13ase of Ar whereas a tiny pre13ursor have
been observed for Kr This indi13ates that for a given density and sho13k
speed the radiative ee13ts in13rease with the atomi13 number
In parallel I have worked on the optimization of the design of an experi-
mental setup where the sho13k is generated ele13tromagneti13ally This setup
allows studying sho13k sim 30 kms in noble gas at sim1 mbar The parameters
of a low-indu13tan13e and 13ompa13t pulsed power devi13e have been optimized
with a simple model in view of building-up and a1313elerating a plasma sheath
in Ar and Xe at gas pressure in the range 01-10 mbar Simple observations
of the moving plasma using side-on dete13tion of the visible emission give
features 13oherent with the model Sho13ks 13orresponding to ma13h numbers
ranging from 20 to 60 are obtained
72 Perspe13tives
The experimental results presented in this thesis 13orrespond to the 13ase of
two 13ounter-propagating radiative sho13ks We have been able to study the
time variations of the pre13ursor intera13tion and the ee13t of gas pressure
with 1D time dependent interferometri13 re13ords obtained with a streak 13a-
mera Together with the help of XUV spe13tros13opy we 13ould estimate the
sho13k speed ele13tron density and sho13k temperature However no informa-
tion is provided about the lateral extension of the pre13ursor and we were
not able to re13ord XUV data for all the shots Therefore in order to make
these results more 13omprehensive we have designed and parti13ipated in a
follow-up experimental 13ampaign in September 2016 at PALS This time we
fo13ussed on single sho13ks in 13lose experimental 13onditions but with a wider
range in terms of speeds (45-100 kms) We repla13ed the time dependent
streak 13amera interferometri13 study by an instantaneous 2D interferometri13
124 CHAPTER 7 CONCLUSION
image (GOI) obtained with a short pulse sub pi13ose13ond laser and 13ameras
XUV spe13tra were also re13orded The re13ords were performed at three dif-
ferent times of the sho13k propagation allowing to dedu13e the sho13k speed
and the time evolution of the pre13ursor morphology This follow up experi-
ment will allow to 13omplete the analysis of the laser generated sho13k waves
presented in this thesis
The preliminary analysis I have performed on these new re13ords is very
promising An example of these interferometri13 re13ords is given in the Figure
71a for the shot 30364 in Xe+He at 06 bar The high speed of the sho13k is
related to the energy of the laser whi13h is here 170 J whereas it was less than
120 J in the previous experiment The 13omparison with the interferometri13
re13ord (not shown) before the shot allows to dedu13e the ele13tron density
whi13h rea13hes 18 times 10
1913m
3 The high quality re13ord shows 13learly 2D
ee13ts for this sho13k wave
(a)
(b)
Figure 71 Interferometri13 re13ord obtained at 546 ns for a single radiative sho13k
of speed sim 100 kms propagating in Xe+He gas at 06 bar driven by laser at 348
nm with energy 170 J The dierent 13olors in (b) 13orrespond to yellow le Ne le35 times 10
1713m
minus3 13yan 37 - 74 times 10
1713m
minus3 blue74 - 15 times 10
1813m
minus3 green
15 - 18 times 10
1813m
minus3 magenta 18 - 22 times 10
1813m
minus3 red 22 - 26 times 10
1813m
minus3
orange26 times 10
1813m
minus3- 18 times 10
1913m
minus3
This analysis is now ongoing and the results of this experimental 13am-
72 PERSPECTIVES 125
paign will be 13ompared with that obtained from the analysis of re13ords from
2015 presented in this manus13ript This instantaneous imaging interferome-
tri13 diagnosti13 does not follow the sho13k 13ontinuously with the time but it
gives a pre13ise map of the ele13tron density whi13h 13an be inverted through
Abel inversion to obtain the lo13al estimation of the ele13tron density
In parallel in the ele13tromagneti13ally laun13hed sho13ks we have upgra-
ded our table top setup to a13hieve higher speeds For this we have used 12
13apa13itors ea13h of 1 microF at the pla13e of the previous 11 13apa13itors ea13h of
06 microF We have implemented new diagnosti13s namely a visible interferome-
ter working in the Gated Opti13al Imaging mode and time and spa13e visible
spe13tros13opy similar to the one des13ribed in the 13hapter 4 For the visible in-
terferometry we have Ma13h Zehnder Interferometer setup and we will image
the sho13k at dierent times with the help of a SL300 pi13ose13ond high energy
NdYAG laser (λ = 532 nm and energy = 024 J) The experiment will soon
be performed on this new setup and the analysis of su13h re13ords will also be
undertaken in future allowing a 13omparison of ele13tromagneti13ally laun13hed
sho13ks with the laser-driven sho13k whi13h should make the bridge between
thin and opti13ally thi13ker sho13ks
126 CHAPTER 7 CONCLUSION
Chapter 8
Thesis summary
Radiative sho13ks (RS) are present in various astrophysi13al environments es-
pe13ially in stars They are then present in all the supersoni13 a1313retioneje13tion
pro13esses during the stellar evolution from the early phase of the proto-
stellar 13ollapse up to the magnetosphere a1313retion on to the already formed
T- Tauri stars up to the late stage of the supernovae sho13ks
Radiative sho13ks are strong sho13ks (ie Ma13h number Mgtgt1) whi13hrea13h high temperatures and thus are the sour13e of intense radiation Alt-
hough the observation of (ie spatially resolved ) signatures of su13h sho13ks
in the astrophysi13al environments is very di13ult an alternate approa13h na-
med laboratory plasma astrophysi13s provides a powerful tool to study them
on the Earth
These waves have been experimentally studied sin13e more than a de13ade
mostly on large-s13ale laser fa13ilities in noble gases and with dierent targets
geometries The ablation pro13ess generated by the intera13tion between the
laser and a foil indu13es a sho13k wave in the foil whi13h then propagates
in the gas at a speed of several tens of kms Numeri13al simulations 13an
des13ribe these sho13ks with an improving pre13ision With laser intensities on
target 13omprised between 1014 and 1015 W13m
2 these experiments allowed
to re13ord sho13k speeds ranging between 40 and 150 kms
All previous experimental studies have been fo13used on the 13ase of single
radiative sho13ks However in astrophysi13al 13onditions the radiative sho13k
in general intera13ts with a denser medium leading to the development of
ree13ted and transmitted sho13ks A few representative examples of su13h
phenomena are the intera13tion of supernovae remnants with dense mole13ular
13louds the a1313retion sho13ks on the photosphere of T-Tauri stars and the bow
sho13ks at the head of fast stellar jets The 13ollision (or the intera13tion) of
two radiative sho13k waves is obviously a rare astrophysi13al event and the
template 13ase of the supernova remnants DEM L316 (see Fig 1 of Williams
et al (1997)) is still the subje13t of debates Williams et al (2005) Toledo-
Roy et al (2009) Velarde et al (2006) In this regard the development of
dedi13ated laboratory experiments to the study of propagation and intera13tion
of 13ounter-propagating sho13k waves is important as a tool to 13hara13terize
su13h events through their spe13i13 signatures
Complementary to laser experiments 13ompa13t pulsed power generators
may drive an ele13tromagneti13 13oaxial plasma gun to 13reate astrophysi13al
relevant sho13ks in lower pressure noble gases with a high availability and a
127
128 CHAPTER 8 THESIS SUMMARY
rather modest 13apital 13ost The ele13tromagneti13ally driven sho13k waves may
have larger s13ales than those by laser Thus they 13an be analyzed rather
easily with the adequate diagnosti13s (Kondo et al 2008)
In this thesis I am presenting radiative sho13ks in the 13ontext of labo-
ratory astrophysi13s using two experimental approa13hes laser-driven sho13ks
and ele13tromagneti13ally driven sho13ks
First 13hapter of my thesis is Radiative sho13k waves in whi13h the physi13s
of radiative sho13ks have been studded Radiative sho13k waves are hypersoni13
sho13k waves whi13h are heated to high temperature and as a 13onsequen13e
be13ome the sour13e of intense radiation This radiation in turn modies
the dynami13s and stru13ture of the sho13k itself whi13h makes its stru13ture
more 13ompli13ated Radiative sho13k waves o1313ur in several astrophysi13al 13ir-
13umstan13es su13h as in the a1313retion sho13k of protostellar formation (Stahler
Palla and Salpeter 1986) the supernova explosion and the intera13tion of
their remnants with the dense interstellar medium (Chevalier 1977) the bow
sho13ks at the head of stellar jets (Hartigan et al 2001)
Radiative sho13k waves 13an now be studied in the laboratory whi13h allows
to 13ompare the data with models existing in the literature and to 13he13k the
ability of the numeri13al 13odes to reprodu13e them (see for instan13e Bouquet
et al (2004) Leygna13 et al (2006)) The experimental study of radiative
sho13k 13hara13teristi13s is 13ru13ial in understanding these various energeti13 pro-
13esses o1313urring in the astrophysi13al plasmas
The jump 13onditions (Rankine-Hugoniot relations) for the simple 13ase
of an ideal gas whi13h rely on the values of the thermodynami13al quantities
on both sides of the dis13ontinuity Let us 13onsider a one-dimensional sho13k
propagating in a gas at rest with the speed us In the frame asso13iated with
the sho13k front the pre-sho13k (upstream) uid velo13ity is then u1 = - us
while behind the sho13k (post-sho13k or downstream region) this velo13ity is
u2 as shown in the Fig 21 Considering the 13ase of strong sho13ks where
M may be 13onsidered to be ≫1 jump 13onditions for ideal gas are
ρ2ρ1
=(γ + 1)
(γ minus 1)(81)
T2
T1=
2M2γ(γ minus 1)
(γ + 1)2(82)
kBT2 =2(γ minus 1)
(γ + 1)2mus
2(83)
For mono-atomi13 gases γ = 53 the 13ompression ratio is estimated to
be 4 and the post-sho13k temperature is
kBT2 =3
16mu1
2 =3
16mPAu1
2(84)
129
This temperature is thus proportional to the square of the sho13k velo13ity
and to the atomi13 mass In 13ontrast to the dis13ussion on ideal gas in the
13ase of a real gas is a little more 13ompli13ate It is to note that in this 13ase
a part of the kineti13 energy is used to ex13ite and ionize the post-sho13k gas
As a result its temperature is lower than that for the 13ase of ideal gas
In addition the pressure is also modied due to ionisation (Mi13haut et al
2004) In13luding the ee13t of ionization with the average 13harge z in the
medium for the 13ase real gas jump relations for real gas are
ρ2u2 = ρ1u1 (85)
ρ2u22 + ρ2
kBT2
m(1 + z2) = ρ1u
21 + ρ1
kBT1
m(1 + z1) (86)
ρ2u2
[5
2
(1 + z2)
mkBT2 + ǫexc2 +
u222
]
= ρ1u1
[5
2
(1 + z1)
mkBT1 + ǫexc1 +
u212
]
(87)
In the previous both 13ases the ee13ts of radiation heating and 13ooling
have been negle13ted However the radiation emitted in a strong sho13k will
ae13t its stru13ture These ee13ts 13an be determined using mass momentum
and energy 13onservation equations whi13h now have to in13lude the 13ontribu-
tions of the radiative ux energy and pressure This 13ase is more 13omplex
now Taking into a1313ount radiative ee13ts the equations of radiative hydro-
dynami13s in the 13o-moving frame of the sho13k (Mihalas and Mihalas 1999
Drake 2006) 13an be written as
partρ
partt= minusnabla(ρu) (88)
ρ
(
partu
partt+ unablau
)
= minusnabla(P + Prad) (89)
part
partt
(
ρu2
2+ ρǫ+ Erad
)
+nabla
[
ρu
(
ǫ+u2
2+
P
ρ
)
+ (Erad + Prad)u
]
= minusnablaFrad
(810)
The 13ontributions linked to Prad and Erad are ee13tive only at very high
temperature (ie velo13ity) However for the radiative sho13ks of interest in
our experiments the 13ontribution of the radiative ux Frad is non-negligible
13ompared to ρu3 These sho13ks are thus in the radiation ux dominated
regime Most of the experimental sho13k waves are in this regime These
hydrodynami13 equations are strongly non-linear
Depending on the opa13ity the radiation emitted from the sho13k may be
absorbed by the pre-sho13k region indu13ing its pre-heating Far away from
the dis13ontinuity the stru13ture of the upstream medium is determined by
the absorption On its side the stru13ture of the transition layer of the down-
stream medium 13lose to the dis13ontinuity depends on the balan13e between
130 CHAPTER 8 THESIS SUMMARY
the emission (13ooling) and the absorption (heating) Thus the full stru13ture
of the sho13k mainly depends on the variation of the opa13ity χ (13m
2g) The
opti13al depth (τ(s)) measured from the position of the jump is then the
relevant parameter for the absorption It is given by
τ(s) =
int s
sjump
χ(sprime)ρ(sprime) dsprime (811)
where sprime 13hara13terizes the path of the radiation As we are interested in
the mean frequen13y averaged radiative ux the relevant opa13ity here will be
the averaged Rosseland opa13ity whi13h will be dened in due 13ourse
If τ gtgt 1 the regime is referred as opti13ally thi13k while in the opposite
13ase (τ lt 1) it is opti13ally thin Flux dominated radiative sho13ks have
been 13lassied depending on the opti13al depth of upstream and downstream
regimes
We are more interested in the kind of sho13k have a radiative pre13ursor It
is possible to produ13e a radiative sho13k at the high energy laser fa13ilities with
laser intensity of sim 10
14W13m
2 A short summary of number of experiment
performed at high power laser fa13ilities have been listed in table 21 Other
than laser driven radiative sho13k it is possible to laun13h 13omparatively slower
radiative sho13ks (sim 10-30 kms) using ele13tromagneti13 generators
The 13hapter two is dedi13ated to the numeri13al analysis of radiative sho13k
waves and of the main physi13al pro13esses whi13h play a role in these 13omplex
pro13esses Radiative sho13ks are present in various astrophysi13al pro13esses
implying extreme 13onditions Laboratory experiments then allow investiga-
ting the underlying physi13al pro13esses whi13h take pla13e in these 13onditions
and whi13h are not observable from the Earth due to a la13k of spatial resolu-
tion However experiments are not always straightforward to interpret and
the help of numeri13al simulations be13omes pre13ious
Dierent numeri13al approa13hes are used for the simulation of laboratory
plasmas Some of them use xed grids Other introdu13e the possibility of
renement (on the y) in some meshes (Adaptative Mesh Renement) to
maintain a good resolution in the interesting parts of the plasma Another
approa13h is based on meshes following the uid (Lagrangian des13ription)
whi13h means that the meshes move with the uid and no matter may go
through a mesh to another The form of the equations dier in this 13ase (see
for instan13e Orban et al (2013))
HELIOS is su13h a one-dimensional Lagrangian radiation hydrodynami13
13ommer13ial 13ode (Ma13Farlane Golovkin and Woodru 2006) and I have
used it to simulate our experiment of intera13ting sho13k waves This 13ode
is easy to handle and has the presently interesting 13apability to be able to
simulate the 13ase of two sho13k waves propagating in opposite dire13tions and
laun13hed by two dierent lasers HELIOS may des13ribe non-LTE plasmas as
it in13ludes an in-line 13ollisional radiative (HELIOS-CR) model for 13omputing
131
non-LTE atomi13 level populations at ea13h time step of the hydrodynami13s
simulation However I used HELIOS in the LTE approximation whi13h is
appropriate to our experimental 13onditions (Rodriguez et al 2011) as will
be dis13ussed in the next se13tion
The 13ode solves the uid equations of motion using the pressure 13ontri-
butions of ele13trons ions and radiation It may des13ribe the ele13trons and
ions as two intera13ting uids at respe13tive temperatures Te and Ti Thermal
13ondu13tion is des13ribed within a ux-limited ele13tron 13ondu13tion model ba-
sed on the Spitzer 13ondu13tivity and the laser energy deposition is 13omputed
with an inverse Bremsstrahlung model
The radiation emission and absorption terms are introdu13ed in the equa-
tion of energy for the ele13trons and in the radiation transport equations One
of the methods whi13h are proposed is the ux-limited multi-group radia-
tion diusion model where the radiative ux is proportional to the gradient
of the radiative energy and is inversely proportional to the Rosseland opa-
13ity The expression is pondered by a ux-limited diusion 13oe13ient in
order to obtain the good opti13ally thin limit This 13oe13ient follows the
Larsen expression des13ribed in Olson Auer and Hall (2000) The se13ond
method is a (time independent) multi-angle short-13hara13teristi13s s13heme In
our 13ase where radiation and hydrodynami13s are strongly 13oupled and thus
the radiation eld varies rapidly we used the diusion model together with
LTE Plan13k and Rosseland multi-groups opa13ities
In addition our version of HELIOS uses the PROPACEOS (Ma13Farlane
Golovkin and Woodru 2006) Equation of State (EOS) and multi-group
opa13ity tables whi13h are generated by the 13ode
1
and it is not possible
to in13lude any other opa13ity or EOS table The knowledge of the thermo-
dynami13al 13onditions is required for an adequate simulation of the plasma
des13ription In our experiment the mass density ranges between sim 10
minus4and
10
minus1g13m
3 while the temperature values are 13omprised between and sim 0 to
50 eV The pre13ise 13al13ulations performed by Rodriguez et al (Fig2 of Rodri-
guez et al (2011)) for Xenon indi13ate that the thermodynami13al 13onditions
of our radiative sho13k experiments 13orrespond to the LTE regime
The equation of state (EOS) denes the dependen13e of the pressure
ionisation internal energy with the mass density and temperature Several
models do exist in the literature
Our version of HELIOS for LTE plasmas may handle polytropi13 or PRO-
PACEOS equation of state Unfortunately there are not many details about
this last method The authors of the 13ode mention that it uses a QEOS-type
model (More et al 1988) in the strong 13oupling regime (ie at relatively
high densities and low temperatures) and an isolated atom model whi13h
uses a detailed atomi13 stru13ture modeling in the weak 13oupling region The
properties between the weak and strong 13oupling regimes are said to be obtai-
1
see http wwwprism minus cscomSoftwarePROPACEOS
132 CHAPTER 8 THESIS SUMMARY
ned by interpolating in a thermodynami13ally 13onsistent manner This EOS
in13ludes the 13ontributions from the translation of ions and ele13trons ele13-
tron degenera13y atom ionization and ex13itation and Coulomb intera13tions
(Debye - Hu13kel 13orre13tion)
I have performed HELIOS 1D simulations for dierent 13ases for single
and 13ounter propagating sho13ks in 13hapter 2 of my thesis Radiative sho13k
waves propagating in Xenon at 01 bar with a velo13ity of 50 kms are 13ha-
ra13terized by the development of an extended radiative pre13ursor The huge
13ompression of 38 in the post-sho13k is a 13onsequen13e of both the sho13k and
the gas ionisation (fa13tor of 10) as also from the radiative 13ooling The post-
sho13k and pre-sho13k temperature on both side of the peak are identi13al
indi13ating that the sho13k is super13riti13al
The spe13ta13ular role of the opa13ity has been highlighted Due to the
un13ertainty in the opa13ity used for Xenon in our simulations I de13ided that
at this stage it was not ne13essary to rene the simulation in terms of group
numbers within the 1D des13ription of the sho13k wave
Our numeri13al study gives the main 13hara13teristi13s of the intera13tion of
two 13ounter-propagating sho13k waves with speeds equal to 50-50 kms and
50-20 kms The 13ase of identi13al speeds is simpler due to the symmetry of
the problem However whatever the speeds the most important signature
of the intera13tion is the merging of the pre13ursor at 8 ns for 50-50 kms
and at 15 ns for 50-20 kms This merging is followed by a regular in13rease
with time of the ele13tron density and the temperature The 13ollision time
is 13hara13terized by a sudden in13rease of the ele13tron density by an order of
magnitude rea13hing 66 times 10
21and 3 times 10
2113m
minus3respe13tively whereas the
temperature in13reases up to 39 and 28 eV
When a short pulse of a high-power laser beam is fo13ussed on a thin foil
it transfers a huge amount of energy to the foil material Almost instantane-
ously the laser absorption heats it up to the ablation A hot and low density
13orona is generated ba13kwards by this pro13ess Beyond the ablation surfa13e
a sho13k is formed in the foil as a 13onsequen13e of the momentum 13onservation
(ro13ket ee13t) The sho13k moves inward in the foil Our 13ase of the experi-
mental setup as the foil 13loses a tube lled with gas the sho13k propagates
then in the gas where it is studied through various kinds of diagnosti13s The
foil then a13ts as a piston
We performed the experiments at Prague Asterix Laser System (PALS)
in the Spring 2015 (ve weeks 20th April - 22th May) The obje13tive of the
experimental 13ampaign was to study the evolution of two 13ounter streaming
radiative sho13ks driven by two 13ounter-propagating lasers with an irradian13e
in the range of 10
14W13m
minus2 and therefore to 13hara13terize their intera13tion
and 13ollision In the 13hapter third 13hapter I have rst presented a brief
des13ription of the PALS laser fa13ility and of the laser beams used in our
experiment This followed by a presentation of the targets design and an
overview of the general setup and of our main diagnosti13s
133
The Prague Asterix Laser System (PALS) is a laser fa13ility based on
an Infrared high-power iodine laser system (Asterix IV) (Jungwirth et al
2001) Using dierent amplifying stages the laser fa13ility is able to deliver
energy up to 1 kJ in 03 ns at the fundamental wavelength 1315 nm The
output laser beam 13an be further subdivided in few auxiliary beams All
auxiliary beams may be frequen13y doubled (λ = 657 nm) or tripled (438
nm) The PALS laser fa13ility is 13apable of ring up to two high energy laser
shots per hour Compared with solid states lasers this gas laser is known to
deliver a quite homogenous beam intensity without hot spots
For our experiments we used two independent laser beams to drive the
two 13ounter streaming sho13ks in the gaseous target For this purpose the
fundamental PALS laser beam has been subdivided after the fourth amplier
(V4) into two beams with respe13tive energy fra13tions of 60 and 40 (see
Fig 42) The most energeti13 of these laser beams is then inje13ted through
the fth (V5) amplier and its frequen13y are then tripled This beam at 438
nm will be hereafter termed termed as the MAIN laser beam
The se13ond of the two previous laser beams propagates then without any
modi13ation It will be hereafter 13alled the AUX laser beam (1315 nm)
The s13heme of the energy distribution is shown in the Fig 42 and the
13hara13teristi13s of MAIN and AUX laser are presented in Table 41
For the experiments PALS oers two va13uum 13hambers with a spheri13al
and 13ylindri13al geometry respe13tively The spheri13al 13hamber assembly used
in our experiments is shown in Fig 43 This 13hamber having a diameter
of 100 13m is 13losed by two entran13e 13ir13ular windows of respe13tive diameters
80 13m and 50 13m to allow the MAIN and AUX beams to penetrate in the
13hamber and to rea13h the target Ea13h of these windows is outtted with a
hinged door Several 13ir13ular ports of various diameters also allow for the
dierent diagnosti13s The target holder and some diagnosti13s are mounted
on an opti13al ben13h whi13h is de13oupled from the 13hamber walls
Our targets (see Fig 45) s13hemati13ally 13onsist in small tubes of 4 mm
length 13losed on both sides by two spe13i13 foils of thi13kness 11 microm on whi13h
the laser beams are fo13used (one laser per foil) with an irradian13e of about
10
1413m
minus3 The tube is lled with gas in whi13h the radiative sho13k propa-
gates with a velo13ity of 30 - 60 kms The two foils 13losing the target insure
the 13onversion through ablation and sho13k generation of the laser energy
into me13hani13al energy
In our experiments we used spe13i13 targets for alignment and 13hara13te-
rization of the beam size These targets were manufa13tured at the Pole
instrumental of Observatoire de Paris
The gaseous targets were lled in situ at a fra13tion of 1 bar with a gas
(viz Xe Ar a mixture of Xe and He) whi13h thus provide the medium for the
sho13k propagation We 13hose an in-situ lling system (Fig 410) to redu13e
the dieren13e of pressures inside and outside of the tube This is 13riti13al as
a large pressure dieren13e between the target 13ell and the va13uum 13hamber
134 CHAPTER 8 THESIS SUMMARY
13an result in the breaking of the ultra-thin SiN windows of the target
First of all we performed fo13alization of MAIN and AUX laser beam
on target by using Al-massive targets The two PALS laser lenses were
translated up to a13hieve the suitable diameters on the target whi13h were
nally set to 450 - 500 microm and 250 - 300 microm for the MAIN and AUX beams
The size of the impa13t was 13ontrolled ea13h day on mo13k targets before using
the laser beams for real shots on gaseous targets It is worth to pre13ise that
the keV diagnosti13 was still in pla13e for the shots on the gaseous targets
In that 13ase the beams impa13t Parylene-N instead of Aluminium The keV
signal is then weaker than for the mo13k aluminum targets
Various kinds of diagnosti13s were implemented namely visible interfero-
metry XUV and visible spe13tros13opy For the same two keV 13ameras are
indu13ted to monitor the impa13t and two visible 13ameras for target alignment
For visible interferometory we have taken referen13e images (without any
fringe) for every target as shown in the Fig 417a for one target example
In order to re13ord su13h images we blo13ked the referen13e beam and let the
probe beam only to illuminate the target in a stati13 mode and with the slit
open Then we redu13ed the slit width to its nominal value (200 microm) This
enabled us to have the visualization of the portion of the target whi13h was
imaged in the slit on the target and also to lo13ate the position of the pistons
before the shot This image is termed as the referen13e target image Then
the system was pla13ed in the interferometri13 mode and the fringe pattern
was optimized to be perpendi13ular to the slit and thus in the longitudinal
mode perpendi13ular to the dire13tion of the sho13k propagation The Fig
417b reports su13h a typi13al interferogram re13orded during our experiment
(13f Fig 413) The horizontal axis 13orresponds to the dire13tion of the sho13k
propagation (with a total s13ale of 54 mm on the target) and the verti13al
axis 13orresponds to time (s13aling 200 ns)
The time and spa13e integrated XUV plasma emission are re13orded with
a Flat Field XUV spe13trometer using a 13on13ave grating In the Fig 419
the drawing of the XUV spe13trometer setup s13heme (red 13olor) is presented
together with the spheri13al 13hamber and the target The XUV spe13trometer
is installed on the top of the spheri13al 13hamber whi13h allows the XUV
emission passing through the Si3N4 membrane on the top of the target (see
se13tion 422) to be re13orded
In addition to the interferometry and XUV spe13tros13opy we have also im-
plemented a time- and spa13e-resolved visible spe13tros13opi13 (see Fig 420a)plasma
emission diagnosti13
In 13hapter 5 I present the results of the two diagnosti13s with a parti13u-
lar fo13us on the interferometri13 data whi13h I have extensively studied with
the help of a spe13i13 data analysis pro13edure that I have developed The
visible interferometry is an adequate tool to probe the ele13tron density of
the radiative pre13ursor up to the 13riti13al density (4 times 10
2113m
minus3at the wa-
velength 527 nm of the probing laser) whereas the denser post-sho13k region
135
is opaque to the visible light XUV radiation is present in the whole sho13k
stru13ture and thus the time- and spa13e-integrated XUV spe13tros13opy may
allow exploring the self-emission 13oming from the post-sho13k and pre13ursor
regions
In the visible interferometry the interferen13es between the main and
referen13e beams overlapping on the dete13tor produ13e a pattern of fringes
whi13h follow the relative phase variations between the probe (whi13h passes
through the plasma) and the referen13e beams (see se13tion 441 of the previ-
ous 13hapter for the experimental setup) The phase variation between two
13onse13utive fringes of the unperturbed beams (ie without any plasma) is
equal to 2π Then in the presen13e of the plasma in the probe beam these
fringes be13ome shifted due to the modi13ation of the refra13tive index More
details about the prin13iple of the interferometry and the Ma13h-Zehnder in-
terferometri13 setup may be found in the APPENDIX A and we re13all here
only the expression of the phase shift
∆φ asymp minus πd
λNclt Ne gt (812)
where λ = 527 nm is the wavelength of the probing laser Nc = 4 times 10
21
13m
minus3the 13riti13al density at this wavelength and lt Ne gt is the ele13tron
density averaged over the laser path d in the 13ell and is dened as
lt Ne gt=
int d
0
Ne(z t)dy
d(813)
In this 13hapter I have presented an extensive data analysis of few re-
presentative interferometri13 and spe13tros13opi13 re13ords I am presenting here
summary of results obtained bellow
The average sho13k speed and ele13tron density have been estimated from
the interferograms The sho13k speeds of the MAIN and AUX radiative sho13k
waves vary between sim 30-55 and 10-30 kms respe13tively and the averaged
pre13ursor ele13tron density ranges between 10
17and 10
1913m
minus3during the
sho13ks propagation
We have demonstrated the intera13tion ee13t between the two radiative
pre13ursors in the 13ase of Xe at 01 bar at 54 and 23 kms The intera13tion is
13learly 13hara13terized in the experiment by the enhan13ement of the ionisation
wave followed by the merging of the two radiative pre13ursors at 20 ns The
13ollision time is re13orded at 47 ns Su13h behavior is reprodu13ed by the
simulation
We have investigated this intera13tion at a larger pressure 02 bar with
the following speeds sim 41 kms for the MAIN and sim 18 kms for the AUX
sho13k waves We do not re13ord any experimental signature of the radiative
pre13ursor for AUX Further we have not been able to 13at13h experimentally
the 13ollision time The re13orded pre13ursor of MAIN is not inuen13ed by AUX
wave up to 48 ns (Fig 59(b) and (13) in data analysis 13hapter) whi13h is the
136 CHAPTER 8 THESIS SUMMARY
limit of the re13ord On its side the 1D simulation predi13ts a tiny pre13ursor
for AUX and that both pre13ursors start to intera13t at 49 ns This plausible
sho13ks intera13tion o1313urring at times whi13h are outside of the re13ord 13an
not be 13onrmed by our experiment
The results of the transverse interferometry at 02 bar with speeds of sim40 and 20 kms indi13ate that the MAIN pre13ursor has a lateral extension of
sim 600 microm whereas it is 300 microm for AUX The pre13ursor of MAIN is almost
at with a probable small bending at the edges of the tube whereas the
AUX pre13ursor is more 13urved This means that the 2D ee13ts are more
important for AUX than for MAIN Still in the 13ase of Xenon we have
obtained information about the temperature and the mean 13harge of gas
through our time integrated XUV spe13tra (shot48143) in Xenon at 06
bar From these results we may 13on13lude that the mean ion 13harge is at
least equal to 6 and that the temperature has rea13hed 15 eV
Our simulations give a qualitative des13ription of the sho13ks intera13tion
when the laser uen13e is adjusted to give the 13orre13t sho13k velo13ities Howe-
ver it is now well known that 2D simulations (together with state of the art
opa13ities) t better with experiments (Gonzaacutelez Audit and Stehleacute 2009
Leygna13 et al 2006 Stehleacute et al 2010) For the same laser energy the 2D
simulations lead to a diminution of the sho13k speed 13ompared to 1D as also
to a diminution of the ele13tron density For instan13e in the 13ase of a sho13k
wave laun13hed by a laser beam at 1315 nm in Xenon at 03 bar at PALS and
with a laser uen13e of 85000 J13m
2 ARWEN 2D simulations give a sho13k
speed of 44 kms in agreement with the experimental one (Cotelo et al
2015) 1D simulation would require in this 13ase a uen13e of 30000 J13m
2to
a13hieve the same velo13ity
The spa13e and time integrated XUV re13ords at 06 bar for respe13tive
speeds whi13h are equal to sim 39 and 18 kms indi13ate that the temperature
of the sho13k has rea13hed values up to 15 eV and that the Xenon mean ion
13harge has also rea13hed values of 6 - 7 whereas 1D simulations predi13t ele13tron
temperature 10-30 eV and ion 13harge 5-10 (Fig 51813) A more detailed
study based on 2D simulation and radiative transfer post-pro13essing will be
ne13essary to rene the analysis
We have investigated the 13ase of other noble gases (Ar Kr) and we have
observed that for a given laser energy the sho13k velo13ity de13reases with the
mass density This study has 13onrmed that for a given density and sho13k
velo13ity the radiative ee13ts in13rease with the atomi13 number For Kr we
noti13ed a tiny pre13ursor without any intera13tion To get su13h ee13t it will
be ne13essary to in13rease the sho13k velo13ity and thus the laser energy
This has been done at the ORION laser fa13ility in UK where the 13ol-
lision of two 13ounter-propagating sho13ks at equal speed sim 80 kms with
laser uen13e sim 6 times 10
14W13m
2has been performed (Clayson et al 2016
Suzuki-Vidal et al 2016) for dierent noble gases and for pressures 13ompri-
sed between 01 and 1 bar A number of diagnosti13s setup X-ray ba13klig-
137
hting opti13al self-emission streak imaging and interferometry (multi-frame
and streak imaging) were used to study both the post-sho13k and the radia-
tive pre13ursor Although I have not parti13ipated in the experiments I have
performed 1D simulations to interpret ORION experimental results This
work is not presented in this thesis
The previous 13hapters were dedi13ated to the study of laser generated
radiative sho13ks With irradian13e more than 10
14W13m
2 we were able to
study sho13k waves propagating at velo13ity up to 50 kms in noble gases with
an initial mass density 13omprised between 5 times 10
minus4- 3 times 10
minus3g13m
minus3
Complementary to laser experiments 13ompa13t pulsed power generators may
drive an ele13tromagneti13 13oaxial plasma gun to 13reate astrophysi13al relevant
sho13ks in lower pressure noble gases (Kondo et al 2006) with a high avai-
lability and a rather modest 13apital 13ost The ele13tromagneti13ally driven
sho13k waves may have larger s13ales than those by laser thus they 13an be
observed rather easily (Kondo et al 2008)
Su13h ele13tri13al pulsed power devi13es may then i) 13reate a high-voltage
breakdown through a gas or more easily at the surfa13e of a diele13tri13 ii)
produ13e relatively hot plasma by ohmi13 heating when the rising 13urrent in the
devi13e is passing through a portion of gas iii) a1313elerate plasma layers under
the magneti13 pressure asso13iated with the self-generated magneti13 eld when
high ele13tri13al 13urrent (say 10s of kA) is maintained for a short but su13ient
time (say 1 micros) Su13h ionization and a1313eleration are present in Z-pin13h
plasmas and in plasma fo13us devi13es (PFD) providing a 13lever geometry is
employed and the mass under 13onsideration (say 1 mg) is 13ompatible with
the stored energy (Martin Williams and Kristiansen 1996)
In this 13hapter we will see rst how a fast ele13tri13al 13ir13uit works then
we will dis13uss the prin13iples of a 13oaxial plasma gun and of PFD We will
see what is in favor of our obje13tive and what has to be avoided A geometry
13apable to rea13h our obje13tive will be proposed
The optimization of the 13oaxial gun (in term of plasma speed) will be
performed and some typi13al gures will be given in the 13ase of a generator
with a stored energy around 1 kJ and a plasma slab of 4-mm diameter ie
quite 10 times the transverse dimension of a laser driven RS The operating
13onditions are supersoni13 sho13ks up to 10-30 kms speed at stati13 pressures
of few mbar in heavy rare gases (Ar Xe) The results of this 0-D model will
be then 13ompared to those obtained with 3-D MHD simulations performed
with the 13ode GORGON (by Andrea Ciardi at LERMA) whi13h has been
used su1313essfully to des13ribe other pulse-power driven plasma experiments
(Chittenden et al 2004) as well as laboratory plasma astrophysi13s experi-
ments (Ciardi et al 2007) The diagnosti13s whi13h have been implemented
will be presented to illustrate the model as well as preliminary re13ords of the
plasma speed
Our aim being to a1313elerate a plasma slab using the magneti13 pressure
it is obviously needed to drive a high intensity 13urrent be13ause the magneti13
138 CHAPTER 8 THESIS SUMMARY
pressure is expressed by B
22micro0 in the region where exists an indu13tion B
The pressure in bars is simply 4B
2 with B in Teslas and in a 13ylindri13al
geometry 1 Tesla is the eld around a 50-kA 13urrent at a radius of 1 13m
Thus we should 13ount on roughly 100 kA delivered by the generator but
su13h a high 13urrent is delivered only in a pulsed mode by a laboratory s13ale
devi13e
There are numbers of te13hni13al solutions to a13hieve su13h ele13tri13al pa-
rameters one may use rather slow generators at moderate high voltage like
13apa13itors bank with the advantage of well know te13hniques but with limited
adjustments A1313ording to an abundant literature
1
a 13onvenient devi13e is a
medium-energy 13apa13itor bank feeding a low-indu13tan13e 13ir13uit An R-L-C
13ir13uit is a well-known 13ombination delivering a high peak 13urrent in the
pseudo-periodi13 mode For a 13apa13itor C initially 13harged under U0 key
performan13es are as follows
Current intensity I(t) is a damped sinusoid
Pseudo pulsation ω =radic[1(LC)minusR2(4L2)]
First 13urrent peak Ipeak = U0[radic(LC) + 08R]
Time of 13urrent peaking T4 = π(2ω)
Equivalent impedan13e Z =radic(LC)
For a safe handling in air a voltage not higher than 30 kV is re13ommen-
ded Taking a total 13apa13itan13e of 6 microF 2700 J are stored under 30 kV For
an indu13tan13e of 240 nH the expe13ted peak 13urrent might be 150 kA at 19
micros This set of values will be a guideline for the development presented in
further se13tions
For this part of the work the obje13tive was to show that in 13omple-
ment of laser experiments a 13ompa13t pulsed power generator might drive
astrophysi13ally relevant sho13ks in low pressure noble gases with a higher
availability and at a rather modest 13apital 13ost The longitudinal a1313elera-
tion of a plasma sheath in a low ba13king gas pressure has been demonstrated
in 13oaxial plasma guns as early as in the 70s even leading to speeds of 100
kms in light gas as H2 whi13h were measured and modeled A 13hoi13e was
made to revisit this type of devi13e At present the ele13tromagneti13 13oaxial
plasma guns are quite only 13onsidered as the initiators of a strong radial
plasma implosion plasma fo13us devi13es (PFD) due to a good ability to pro-
du13e a pin13h plasma and emission of energeti13 parti13les and hard X-ray For
that reason it was ne13essary to adapt the prin13iple of a PFD with two major
13onstraints in13rease the 13onversion of stored energy into the kineti13 energy
of a plasma sheath prevent totally any pin13hing
1
Institute for Plasma Fo13us Studies resour13e website httpplasmafo13usnet last
13onne13tion in 2016
139
That obje13tive was rea13hed by dening the ele13tri13al and geometri13al
parameters of a low-indu13tan13e and 13ompa13t pulsed power generator An
optimization pro13ess has been established to mat13h a13hievable ele13tri13al pa-
rameters of the 13ir13uit (13apa13itan13e indu13tan13e peaking 13urrent intensity)
with a plasma motion in the a1313elerating tube over few 13m leading to a nal
speed of 10-30 kms A model was built up to des13ribe the dynami13s of an
RLC 13ir13uit with a varying element 13omposed of the moving plasma sheath
a1313elerated by the self generated magneti13 pressure The 13hange in plasma
position along the axis is in13reasing the indu13tan13e and the mass a1313retion
is in13reasing the plasma mass It was shown that a 13hange in the ele13trode
geometry namely a 30deg 13oni13al shape was in13reasing the energy 13onversion
and preventing any fo13using The further step in 13ontrast with PFD is the
merging of the plasma ring into a rather planar one whi13h is inje13ted in a
drift tube
Considering Ar or Xe at gas pressure in the range 01 10 mbar and
a highly 13oaxial 1-kJ devi13e the simple model was su13ient to predi13t
performan13es agreeing with the only work quoted in the literature Basi13
observations of the moving plasma using side-on dete13tion of the visible
emission give features 13oherent with the model Ma13h numbers from 20
to 60 are obtained tting with the obje13tive of strong sho13k formation A
rened 3-D MHD numeri13al simulation indi13ates very promising features of
the sho13k in view of exploring mm-size sho13ks in a regime 13omplementary
to laser driven ones and on a table-top devi13e allowing a higher repetition
rate
140 CHAPTER 8 THESIS SUMMARY
Appendi13es
141
Appendix A
Visible Interferometry
Laser interferometry is used in plasmas to dedu13e the ele13tron density through
the variation of the refra13tive index 13aused by the ionisation of the matter
In this appendix rstly the experimental setup of Ma13h Zehnder interfero-
metry as well as the underlying physi13al prin13iples are presented
A01 Refra13tive index of a plasma
In order to derive the refra13tive index of a gas one starts with the set of
Maxwells equations for a mono13hromati13 plane wave propagating in the
plasma (see for instan13e equation 411 of referen13e (Hut13hinson 2002))
Free ele13trons and ions are supposed to be distributed uniformly and the net
total 13harge density is equal to zero
For harmoni13 waves propagating in the dire13tion of +x and with the
pulsation ω one obvious solution is given by
E(x t) = E0ei(kxminusωt)
(A1)
where E0 and k are 13omplex quantities
The relation between k and ω is given by the dispersion relation
k2 =εmicroω2
c2(1 + i
4πσ
ωε) (A2)
where micro is the ele13tri13 permitivity and ε is permeability
The free ele13trons are a1313elerated by the ele13tri13 eld and therefore equa-
tion their motion is given as
medv
dt= minuseE0e
minusiωt(A3)
The obvious solution for the ele13tron velo13ity v is
v = minusie
meωE (A4)
and thus the ele13trons 13arry a 13urrent with a density J
J = Neev = iNee
2
meωE (A5)
where Ne is the ele13tron density Due to the larger mass of the ions the
13orresponding 13urrent is negligible
143
144 APPENDIX A VISIBLE INTERFEROMETRY
The previous equation allows to derive the plasma 13ondu13tivity σ whi13his equal to iNee
2ωme Putting this value in the equation (A2) and assu-
ming a thin medium where ε asymp 1 and micro = 1 one obtains nally
k2 =ω2
c2(1minus
ω2p
ω2) (A6)
where ωp is the plasma frequen13y dened as
ω2p =
4πNee2
me(A7)
Equation A6 shows that a ele13tromagneti13 wave with a pulsation larger than
ωp 13ant propagate in the plasma This allows to derive a 13riti13al ele13tron
density Nc in 13m
minus3as
Nc =meω
2
4πe2=
4πmec2
e2λ2(A8)
For instan13e for a radiation of 1 microm wavelength this upper limit is equal
to Nc = 446times 1021cmminus3
Therefore the index of refra13tion is given by the expression
n =
radic
1minusω2p
ω2=
radic
1minus Ne
Nc(A9)
Finally one obtains
E(x t) = E0ei(kxminusωt) with k = n
ω
c(A10)
As a 13onsequen13e of the variation of the refra13tive index in the dierent
plasma layers the light is refra13ted Indeed the Snell laws stipulate that if a
beam is in13ident on a plane interfa13e between layers (of refra13tive index n1
and n2) it undergoes a bending from the angle of in13iden13e The in13ident
beam the refra13ted beam and the normal to the interfa13e lie in the same
plane and one has n1 sinθ1 = n2 sinθ2 where θ1 is the angle of in13iden13e
and θ2 is the angle of refra13tion Thus a mono13hromati13 beam propagating
in a plasma with in13reasing density be13omes more and more dee13ted
A02 Absorption of the laser beam
Beside refra13tion the light is absorbed by the plasma and the absorption
also in13reases rapidly near the 13riti13al density In the absen13e of any ato-
mi13 bound-bound or bound-free transition this absorption is due to the
inverse Bremsstrahlung pro13ess (ion-ele13tron 13ollision (Pfalzner 2006)) by
145
the plasma free ele13trons The absorption 13oe13ient κ (in 13m
minus1) is then
given by (NRL formulary)
κ(Ne Tev) = 317 10minus7LnΛ lt Z gt
(
Ne
ω
)2 1
T32ev (1minusNeNc)12
(A11)
where Tev is the temperature in eV Ne is the ele13tron density in 13m
minus3
and LnΛ is the Coulomb Logarithm dened as Λ = Max (2 Min(XY))
with (NRL formulary)
X = 23minus log
(
lt Z gtN
12e
T32ev
)
and Y = 24minus log
(
N12e
Tev
)
(A12)
The transmission T (d) through a layer of thi13kness d is given by
T (d) = exp(minusint d
0κ(l)dl) (A13)
For instan13e for λ = 527 nm taking Ne= 101913mminus3 ltZgt= 10 T = 10 eV
whi13h are typi13al values for the post-sho13k in our experiments in Xenon and
supposing a homogenous plasma thi13kness d = 600 microm the transmission is
about 09 whereas at 5times 1019cmminus3 it falls to 01
A03 Prin13iple of interferometry
In the opti13al interferometry te13hnique (Ovsyannikov and Zhukov 2000) the
interferen13e of two plane waves is performed by addition of the two 13orre-
sponding 13oherent ele13tri13 elds The analyzed medium (here the plasma)
in whi13h one of the beam is propagating indu13es a deshaping between the
beams whi13h ae13ts the resulting intensity depending if the elds interfere
in phase or out of phase ie 13onstru13tively or destru13tively
Two te13hniques are used the wave front and the amplitude division
The wave front division is obtained by using two portions (Fresnel mir-
rors Youngs double slit Lloyds mirror prisms et13) of the original wave
front whi13h are then superimposed to produ13e interferen13e
In the amplitude division the two beams are separated by division of
the amplitude over the same se13tion of the wave front Mi13helson and Ma13h
Zehnder interferometers et13 are the typi13al examples of this te13hnique of
amplitude division
The prin13iple of interferen13e of two 13oherent mono13hromati13 traveling
waves 1 and 2 with ele13tri13 elds E1 and E2 propagating in the same
dire13tion and polarized in the same plane 13an be understood as follows for
a given time t at the point r where these waves superpose the amplitude of
the ele13tri13 eld is the ve13tor addition of the two 13orresponding elementary
amplitudes ie
146 APPENDIX A VISIBLE INTERFEROMETRY
E(r t) = E1(r t) + E2(r t)
= a1exp(ikr + ωt+ iφ1) + a2exp(ikr + ωt+ iφ2)
where φ1 and φ2 are the phases of the two waves at r The total intensity
I(r t) = |E1 + E2|2 (A14)
One gets
I = I1 + I2 + 2radic
(I1I2)cos∆φ (A15)
where I1 = a12and I2 = a2
2are the intensities at the point of interferen13e
due to the two waves a13ting independently and ∆φ = φ1minusφ2 represents the
phase dieren13e between the two waves In the 13ase of equal intensities I1one gets
I = 4I1cos2(∆φ2) (A16)
and the interferometri13 pattern 13onsists of dark (∆φ = π + 2nπ) and
bright patterns (∆φ = 2nπ)In general the two beams are only partially 13oherent and the minimum
of the fringe intensity is not equal to zero The fringe 13ontrast
FC =Imax minus Imin
Imax + Imin(A17)
is then a measure of the interferen13e quality FC is maximum and equal to
1 in the 13ase of fully 13oherent beams of the same intensity I1 = I2
A04 Ma13h Zehnder Interferometer
The Ma13h Zehnder Interferometer is a devi13e 13ommonly and e13iently used
to estimate the 13hange in refra13tive index by the plasma Su13h 13hange is in
most the 13ases related to variations in the ele13tron density We employed
this interferometri13 setup in our experiment
The prin13iple of this interferometer is based on the division of the in13ident
light beam into two beams with the help of a beam-splitter whi13h 13an be
a 13oated glass plate or a 13ube After this division one of the beams (probe
beam) is allowed to pass through the medium of interest (a plasma in our
13ase) and then made to interfere with the other unae13ted part of the beam
(referen13e beam) with the help of se13ond beam-splitter as may be noted
from Fig A1
In order to understand the prin13iple of interferen13e and its use in esti-
mating plasma properties let us 13onsider l1 and l2 whi13h are the total path
lengths (in13luding the path in the opti13al elements) for the light travelling
from the sour13e to the dete13tor for the upper and lower paths (13f Figure A1)
respe13tively Suppose for the moment that the sample is removed from beam
147
Figure A1 S13hemati13s of Ma13h Zehnder interferometer with two mirrors two
50 beam splitters and the dete13tor A plasma slab is pla13ed in one of the two
arms of the interferometer
1 (see Figure A1) the two opti13al paths of the probe and referen13e beams
dier in terms of phases Simplify to supposing that the beams propagate
in air (n=1) these phases are then equal to the 2πl1λ and 2πl2λThus a1313ounting for the phase dieren13e δ between the two beams tra-
velled up to dete13tor A 13an be given as
2πl1 minus l2
λ= δ (A18)
If the two beams are perfe13tly parallel at the position of the dete13tor the
interferometri13 gure presents a at intensity whi13h passes through minima
and maxima when one length of the 2 arms is slightly 13hanged Usually one
introdu13es a small tilt between the two dire13tions of propagations whi13h
leads to the apparition of parallel linear dark and bright fringes
In order to obtain interferen13e patterns it is important to make the paths
l1 and l2 as 13lose as possible 13ompared to the 13oheren13e length of the laser
For the multimode Evolution 15 laser used in the PALS experiment this
length is approximately 1 13m (value 13ommuni13ated by the 13onstru13tor)
If we introdu13e the sample in the probe beam 1 this sample will introdu13e
an additional phase denoted by δsample and the net phase shift is then δ +δsample
The prin13iple of the dedu13tion of the ele13tron density with the Ma13h
Zehnder interferometer is to measure the refra13tive index of the test plasma
slab whi13h is pla13ed in the probe beam We assume for the moment that
148 APPENDIX A VISIBLE INTERFEROMETRY
this plasma slab is a phase obje13t for the in13oming beam whi13h means that
it does not signi13antly ae13t the intensity of the probe beam while it only
13hanges the phase of the wave [5]The additive phase dieren13e in the probe arm introdu13ed by the plasma
13an be dened as
φ =
int d
0kplasmadl =
int d
0nω
cdl (A19)
where n is the lo13al refra13tive index at position l of the plasma slab and
the integral is performed other the slab thi13kness d Therefore the phase
dieren13e introdu13ed by the plasma relative to the propagation into va13uum
of the referen13e beam is equal to
∆φ =
int d
0(kplasma minus k0)dl (A20)
where k0 = ωc = 2πλ is the wave ve13tor of the beam propagating in
va13uum
As long as Ne lt Nc the beam propagates in the plasma the plasma
refra13tive index is given by equation A9 and one has
∆φ =ω
c
int d
0
[
(
(1minus Ne
Nc
)12
minus 1]
dl (A21)
At very small ele13tron density (Ne ≪ Nc) the phase shift 13an be written as
∆φ asymp minus ω
2cNc
int d
0Nedl = minus πd
λNclt Ne gt (A22)
where
lt Ne gt=
int d
0
Ne(z t)dy
d(A23)
is the ele13tron density averaged over the path dIn pra13ti13e a referen13e re13ord is rst taken without plasma To dedu13e
the density one has to analyze how the fringes depart from their referen13e
positions
The spa13e between two fringes (inter-fringe) 13orresponds to a phase va-
riation of 2 π A shift of the fringe by 2 π from its referen13e position (where
Ne = 0) 13orresponds to an ele13tron density variation equal to lt Ne gt = 2
λNcd For λ = 527 nm and supposing an homogeneous plasma of thi13kness
600 microm this gives lt Ne gt= 7 times 10
1813m
minus3
Appendix B
Opa13ities and mean 13harge
The variations in logarithmi13 s13ale with the temperature of the PROPA-
CEOS Rosseland and Plan13k opa13ities (se13tion of 13hapter 5 see for intense
Fig 511)as also of the mean 13harge of the plasma are reported for two
densities 16 times 10
minus3g13m
3and 51 times 10
minus4g13m
3for three noble gases Ar
Kr and Xe in Fig B1 and B2
ρ = 51 times 10
minus4g13m
313orrespond to typi13al density of the pre13ursor
For this density Xenon and Krypton have quite similar variations for the
Rosseland opa13ity with a bump of 7000 13m
2g at 3 eV for Xe and 12000
13m
2g at 4 eV for Krypton At 10 eV the two opa13ities de13rease respe13tively
to 600 and 900 13m
2g In 13omparison with these two 13ases the opa13ity of
Argon is broader and more regular Its opa13ity peaks at 6 eV to 13000 13m
2g
At 10 eV it have de13reased to 7000 13m
2g whi13h is higher than for the two
previous gases
The mean 13harge variations in13rease regularly with the temperature At
10 eV these mean 13harges are respe13tively equal to 65 55 and 5 for Xe Kr
and Ar
149
150 APPENDIX B OPACITIES AND MEAN CHARGE
(a)
(b)
Figure B1 Rosseland and Plan13k opa13ity for three gases Xe Kr and Ar at mass
densities 51 times 10
minus4g13m
3(a) and 16 times 10
minus3g13m
3(b)
151
(a)
(b)
Figure B2 Mean 13harge for three gases Xe Kr and Ar at mass densities 16 times10
minus3g13m
3(a) and 51 times 10
minus4g13m
3(b)
152 APPENDIX B OPACITIES AND MEAN CHARGE
Referen13es
Batani D Strati F Telaro B Loumlwer T Hall T Benuzzi-Mounaix A
Koenig M 2003 Produ13tion of high quality sho13ks for equation of state
experiments European Physi13al Journal D 23 99 DOI ADS
Bernard A 2002 Plasma fo13us et appli13ations Le Vide 57(306)
Bett TH Jinks P Stevenson RM Danson CN Pepler DA Ross
IN 1995 Binary phase zone-plate arrays for laser-beam spatial-intensity
distribution 13onversion Applied Opti13s 34 4025 DOI ADS
Bouquet S Teyssier R Chieze JP 2000 Analyti13al study and stru13-
ture of a stationary radiative sho13k The Astrophysi13al Journal Supplement
Series 127(2) 245
Bouquet S Stehleacute C Koenig M Chiegraveze J Benuzzi-Mounaix A Ba-
tani D Leygna13 S Fleury X Merdji H Mi13haut C et al 2004
Observation of laser driven super13riti13al radiative sho13k pre13ursors Phy-
si13al review letters 92 225001
Bozier J Thiell G Le Breton J Azra S De13roisette M S13hirmann
D 1986 Experimental observation of a radiative wave generated in xenon
by a laser-driven super13riti13al sho13k Physi13al review letters 57(11) 1304
Burhop E Spitzer L 1957 Physi13s of fully ionized gases inters13ien13e
tra13ts on physi13s and astronomy no 3 JSTOR
Chaulagain U 2015 Radiative sho13ks expriments modellin and links to
Astrophysi13s PhD thesis Pierre and Marie Curie University
Chaulagain U Stehleacute C de Saacute L Larour J Auvray P Kozlova M
Krus M Dostal J Propupek J Suzuki-Vidal F et al 2012 Labo-
ratory experiments of radiative sho13ks in the 13ontext of stellar a1313retion
In SF2A-2012 Pro13eedings of the Annual meeting of the Fren13h So13iety
of Astronomy and Astrophysi13s 1 305
Chaulagain U Stehleacute C Larour J Kozlovaacute M Suzuki-Vidal F Bar-
roso P Cotelo M Velarde P Rodriguez R Gil JM Ciardi A
A13ef O Nejdl J de Saacute L Singh RL Ibgui L Champion N 2015
Stru13ture of a laser-driven radiative sho13k High Energy Density Physi13s
17 106 DOI ADS
Chevalier RA 1977 The intera13tion of supernovae with the interstellar
medium Annual review of astronomy and astrophysi13s 15 175 DOI ADS
153
154 REFERENCES
Chittenden JP Lebedev SV Jennings CA Bland SN Ciardi A
2004 X-ray generation me13hanisms in three-dimensional simulations of
wire array Z-pin13hesthe evolution of magneti13 tower jets in the laboratory
Plasma Physi13s and Controlled Fusion 46 B457 DOI ADS
Ciardi A 2014 private 13ommuni13ation
Ciardi A Lebedev SV Frank A Bla13kman EG Chittenden JP
Jennings CJ Ampleford DJ Bland SN Bott SC Rapley J Hall
GN Suzuki-Vidal FA Maro1313hino A Lery T Stehle C 2007 The
evolution of magneti13 tower jets in the laboratory Physi13s of Plasmas
14(5) 056501 DOI ADS
Clayson T Suzuki-Vidal F Lebedev SV Swadling GF Burdiak GC
Patankar S Smith RA Foster J Skidmore J Gumbrell E Gra-
ham P Danson C Stehleacute C Singh RL Chaulagain U Larour
J Kozlova M Spindloe C 2016 Counter-propagating radiative sho13k
experiments on the Orion laser fa13ility In APS Meeting Abstra13ts DOI
ADS
Clayson T Suzuki-Vidal F Lebedev S Swadling G Stehleacute C Bur-
diak G Foster J Skidmore J Graham P Gumbrell E et al 2017
Counter-propagating radiative sho13k experiments on the orion laser and
the formation of radiative pre13ursors High Energy Density Physi13s 23
60
Cotelo M Velarde P de la Varga AG Portillo D Stehleacute C Chaul-
again U Kozlova M Larour J Suzuki-Vidal F 2015 Simulation
of radiative sho13k waves in Xe of last PALS experiments High Energy
Density Physi13s 17 68 DOI ADS
de la Varga AG Velarde P Cotelo M de Gaufridy F
Zeitoun P 2011 Radiative properties for warm and hot
dense matter High Energy Density Physi13s 7(3) 163 DOI
httpwwws13ien13edire13t13oms13ien13earti13lepiiS1574181811000498
Diziegravere A Mi13haut C Koenig M Gregory C Ravasio A Sakawa
Y Kuramitsu Y Morita T Ide T Tanji H et al 2011 Highly
radiative sho13k experiments driven by gekko xii Astrophysi13s and Spa13e
S13ien13e 336(1) 213
Dobbs-Dixon I Cumming A Lin DNC 2010 Radiative Hydrodyna-
mi13 Simulations of HD209458b Temporal Variability The Astrophysi13al
Journal 710 1395 DOI ADS
Doss FW Robey HF Drake RP Kuranz CC 2009 Wall sho13ks in
high-energy-density sho13k tube experiments Physi13s of Plasmas 16(11)
112705
REFERENCES 155
Drake RP 2006 High-energy-density physi13s Fundamentals inertial fu-
sion and experimental astrophysi13s Sho13k wave and high pressure phe-
nomena Springer Berlin Heidelberg New York ISBN 3-540-29314-0
httpopa13inriafrre13ord=b1128433
Drake R Doss F M13Clarren R Adams M Amato N Bingham D
Chou C DiStefano C Fidkowski K Fryxell B et al 2011 Radiative
ee13ts in radiative sho13ks in sho13k tubes High Energy Density Physi13s
7(3) 130
Ensman L 1994 Test problems for radiation and radiation-hydrodynami13s
13odes The Astrophysi13al Journal 424 275 DOI ADS
Ensman L Burrows A 1992 Sho13k breakout in sn 1987a The Astrophy-
si13al Journal 393 742
Fadeyev YA Gillet D 2001 The stru13ture of radiative sho13k waves
III The model grid for partially ionized hydrogen gas Astronomy and
Astrophysi13s 368 901 DOI ADS
Fleury X Bouquet S Stehleacute C Koenig M Batani D Benuzzi-
Mounaix A Chieze J-P Grandjouan N Grenier J Hall T et al
2002 A laser experiment for studying radiative sho13ks in astrophysi13s
Laser and Parti13le Beams 20(02) 263
Fryxell B Olson K Ri13ker P Timmes FX Zingale M Lamb DQ
Ma13Nei13e P Rosner R Truran JW Tufo H 2000 FLASH An
Adaptive Mesh Hydrodynami13s Code for Modeling Astrophysi13al Ther-
monu13lear Flasheshera13les a three-dimensional radiation hydrodynami13s
13ode Astrophysi13al Journal Supplement 131 273 DOI ADS
Gonzalez JH Brollo FR Clausse A 2009 Modeling of the Dynami13
Plasma Pin13h in Plasma Fo13us Dis13harges Based in Von Karman Approx-
imations IEEE Transa13tions on Plasma S13ien13e 37 2178 DOI ADS
Gonzalez JH Clausse A Bruzzone H Florido PCMZ 2004 A Lum-
ped Parameter Model of Plasma Fo13us IEEE Transa13tions on Plasma
S13ien13e 32 1383 DOI ADS
Gonzaacutelez M Audit E Huynh P 2007 HERACLES a three-dimensional
radiation hydrodynami13s 13ode Astronomy and Astrophysi13s 464 429
DOI ADS
Gonzaacutelez M Audit E Huynh P 2007 Hera13les a three-dimensional
radiation hydrodynami13s 13ode Astronomy and Astrophysi13s 464(2) 429
Gonzaacutelez M Audit E Stehleacute C 2009 2D numeri13al study of the ra-
diation inuen13e on sho13k stru13ture relevant to laboratory astrophysi13s
Astronomy and Astrophysi13s 497 27 DOI ADS
156 REFERENCES
Gonzaacutelez M Stehleacute C Rus B Busquet M Thais F A13ef O Barroso
P Bar-Shalom A Bauduin D Kozlova M Lery T Madouri A To-
mas M Polan J 2006 Astrophysi13al radiative sho13ks From modeling
to laboratory experiments Laser and Parti13le Beams 24 535540 DOI
ADS
Gonzalez M StehlE C Audit E Busquet M Rus B Thais F A13ef
O Barroso P Bar-Shalom A Bauduin D et al 2006 Astrophysi13al
radiative sho13ks From modeling to laboratory experiments Laser and
Parti13le Beams-Pulse Power and High Energy Densities 24(4) 535
Gonzaacutelez M Vaytet N Commerccedilon B Masson J 2015 Multigroup
radiation hydrodynami13s with ux-limited diusion and adaptive mesh
renement Astronomy and Astrophysi13s 578 A12 DOI ADS
Griem HR 2005 Prin13iples of Plasma Spe13tros13opy 386 ADS
Gu MF 2008 The exible atomi13 13ode Canadian Journal of Physi13s
86(5) 675
Harilal SS Tilla13k MS 2004 Laser plasma
density measurements using interferometry
httplinkapsorgdoi101103PhysRevE64066412
Hartigan P 1989 The visibility of the Ma13h disk and the bow sho13k of a
stellar jet Astrophysi13al Journal 339 987 DOI ADS
Hartigan P 2003 Sho13k Waves in Outows from Young Stars Astrophysi13s
and Spa13e S13ien13e 287 111 DOI ADS
Hartigan P Morse JA Reipurth B Heath13ote S Bally J 2001 Pro-
per Motions of the HH 111 Jet Observed with the Hubble Spa13e Teles13ope
Astrophysi13al Journal Letters 559 L157 DOI ADS
Henke BL Gullikson E Davis JC 1993 X-ray intera13tions Photoab-
sorption s13attering transmission and ree13tion atlt igt eltigt= 50-30000
evlt igt zltigt= 1-92 Atomi13 data and nu13lear data tables 54(2) 181
Hugoniot PH 1887 Meacutemoire sur la propagation du mouvement dans les
13orps et ples speacute13ialement dans les gaz parfaits Journal de matheacutematiques
pures et appliqueacutees 3 477
Hugoniot PH 1889 Meacutemoire sur la propagation des mouvements dans les
13orps et speacute13ialement dans les gaz parfaits (deuxiegraveme partie) Journal de
lEacute13ole Polyte13hnique 58 1
Hut13hinson IH 2002 Prin13iples of Plasma Diagnosti13s 458 ADS
REFERENCES 157
Jungwirth K Cejnarova A Juha L Kralikova B Krasa J Krousky
E Krupi13kova P Laska L Masek K Mo13ek T et al 2001 The
prague asterix laser system Physi13s of Plasmas 8 2495
Keiter P Drake R Perry T Robey H Remington B Iglesias C
Walla13e R Knauer J 2002 Observation of a hydrodynami13ally driven
radiative-pre13ursor sho13k Physi13al review letters 89(16) 165003
Koenig M Faral B Boudenne JM Batani D Benuzzi A Bossi S
1994 Opti13al smoothing te13hniques for sho13k wave generation in laser-
produ13ed plasmas Physi13al Review E 50 R3314 DOI ADS
Kondo K Nakajima M Kawamura T Horioka K 2006 Compa13t pulse
power devi13e for generation of one-dimensional strong sho13k waves Review
of S13ienti13 Instruments 77(3) 036104 DOI ADS
Kondo K Nakajima M Kawamura T Horioka K 2008 Relaxation
layer in ele13tro-magneti13ally driven strong sho13ks Journal of Physi13s Con-
feren13e Series 112(4) 042028 DOI ADS
Kondo K Nakajima M Kawamura T Horioka K 2009a Atomi13 pro-
13ess in ele13tro-magneti13ally driven strong sho13k wave Nu13lear Instruments
and Methods in Physi13s Resear13h A 606 223 DOI ADS
Kondo K Nakajima M Kawamura T Horioka K 2009b Radiation ef-
fe13t and relaxation layer in ele13tro-magneti13ally driven strong sho13k waves
In APS Meeting Abstra13ts ADS
Kunze H-J 2009 Springer series on atomi13 opti13al and
plasma physi13s 125 Springer Berlin Heidelberg New York
httpwwwspringer13omseries411
Kwek K Tou T Lee S 1990 Current sheath stru13ture of the plasma
fo13us in the run-down phase IEEE Transa13tions on Plasma S13ien13e 18(5)
826
Larour J Singh RL Stehleacute C Ciardi A Chaulagain U Suzuki-Vidal
F 2015 Optimization of an ele13tromagneti13 generator for strong sho13ks
in low pressure gas High Energy Density Physi13s 17 129 DOI ADS
Lee S 1969 Transverse ionizing sho13k waves in a planar ele13tromagneti13
sho13k tube PhD thesis Australian National University
Lee S Serban A 1996 Dimensions and lifetime of the plasma fo13us pin13h
IEEE Transa13tions on plasma s13ien13e 24(3) 1101
Leygna13 S Boireau L Mi13haut C Lanz T Stehleacute C Clique C
Bouquet S 2006 Modeling multidimensional ee13ts in the propagation
of radiative sho13ks Physi13s of Plasmas (1994-present) 13(11) 113301
158 REFERENCES
Ma13Farlane JJ Golovkin IE Woodru PR 2006 HELIOS-CR A 1-
D radiation-magnetohydrodynami13s 13ode with inline atomi13 kineti13s mo-
deling Journal of Quantitative Spe13tros13opy and Radiative Transfer 99
381 DOI ADS
Ma13quorn Rankine WJ 1870 On the Thermodynami13 Theory of Waves of
Finite Longitudinal Disturban13e Philosophi13al Transa13tions of the Royal
So13iety of London Series I 160 277 ADS
Mahmood S Ur-Rehman H 2009 Ele13trostati13 solitons in unmagnetized
hot ele13tron-positron-ion plasmas Physi13s Letters A 373 2255 DOI ADS
Martin TH Williams M Kristiansen M 1996 J13 martin on pulsed
power 3 Springer S13ien13e amp Business Media
M13Clarren RG Drake RP Morel JE Holloway JP 2010 Theory of
radiative sho13ks in the mixed opti13ally thi13k-thin 13ase Physi13s of Plasmas
17(9) 093301
Mi13haut C Stehleacute C Leygna13 S Lanz T Boireau L 2004 Jump
13onditions in hypersoni13 sho13ks Quantitative ee13ts of ioni13 ex13itation
and radiation European Physi13al Journal D 28 381 DOI ADS
Mihalas D 1978 Stellar atmospheres 2nd edition ADS
Mihalas D Mihalas BW 1984 Foundations of radiation hydrodynami13s
ADS
Mihalas D Mihalas BW 1999 Foundations of radiation hydrodynami13s
More R Warren K Young D Zimmerman G 1988 A new quotidian
equation of state (qeos) for hot dense matter Physi13s of Fluids (1958-
1988) 31(10) 3059
Ogando F Velarde P 2001 Development of a radiation transport uid
dynami13 13ode under amr s13heme Journal of Quantitative Spe13tros13opy and
Radiative Transfer 71(2) 541
Olson GL Auer LH Hall ML 2000 Diusion P1 and other approx-
imate forms of radiation transport Journal of Quantitative Spe13tros13opy
amp Radiative Transfer 64 619 DOI ADS
Orban C Fatenejad M Chawla S Wilks SC Lamb DQ 2013 A
Radiation-Hydrodynami13s Code Comparison for Laser-Produ13ed Plasmas
FLASH versus HYDRA and the Results of Validation Experiments ArXiv
e-prints ADS
REFERENCES 159
Orlando S Bonito R Argiro C Reale F Peres G Mi13eli M Mat-
sakos T Stehleacute C Ibgui L de Saacute L Chiegraveze J-P Lanz T 2013
Radiative a1313retion sho13ks along nonuniform stellar magneti13 elds in 13las-
si13al T Tauri stars Astronomy amp Astrophysi13s
Ovsyannikov AA Zhukov MF 2000 Plasma diagnos-
ti13s Cambridge International S13ien13e Publishing Cambridge
httppubli13eblib13om13hoi13epubli13fullre13ordaspxp=258189
Pfalzner S 2006 An introdu13tion to inertial 13onnement fusion 1st
edn Series in Plasma Physi13s Taylor amp Fran13is Taylor amp Fran13is
httpgenlibruse13bookindexphpmd5=EACF38292FD79113F78BB532AB5301B8
Potter DE 1971 Numeri13al Studies of the Plasma Fo13us Physi13s of Fluids
14 1911 DOI ADS
Proakis JG 2001 Digital signal pro13essins Wiley Online Library
Raga A Mellema G Arthur S Binette L Ferruit P Steen W 1999
3d transfer of the diuse ionizing radiation in ism ows and the preioni-
zation of a herbig-haro working surfa13e Revista Mexi13ana de Astronomia
y Astrosi13a 35 123
Reighard AB 2007 Collapsing radiative sho13k experiments on the omega
laser PhD thesis The University of Mi13higan
Reighard A Drake R Dannenberg K Kremer D Grosskopf M Har-
ding E Leibrandt D Glendinning S Perry T Remington B et al
2006 Observation of 13ollapsing radiative sho13ks in laboratory experiments
Physi13s of Plasmas (1994-present) 13(8) 082901
Rodriguez R Gil JM Florido R Rubiano JG Mendoza MA Mar-
tel P Minguez E Symes DR Hohenberger M Smith RA 2011
Determination of the average ionization and thermodynami13 regimes of
xenon plasmas with an appli13ation to the 13hara13terization of blast waves
laun13hed in xenon 13lusters High Energy Density Physi13s 7(2) 71 DOI
httpwwws13ien13edire13t13oms13ien13earti13lepiiS157418181100022X
Rodriacuteguez R Espinosa G Gil JM Stehleacute C Suzuki-Vidal F Rubi-
ano JG Martel P Miacutenguez E 2015 Mi13ros13opi13 properties of xenon
plasmas for density and temperature regimes of laboratory astrophysi13s
experiments on radiative sho13ks Physi13al Review E 91(5) 053106 DOI
ADS
Ross IN Pepler DA Danson CN 1995 Binary phase zone plate de-
signs using 13al13ulations of far-eld distributions Opti13s Communi13ations
116 55 DOI ADS
160 REFERENCES
Seaton MJ 1987 Atomi13 data for opa13ity 13al13ulations I General des13rip-
tion Journal of Physi13s B Atomi13 and Mole13ular Physi13s 20(23) 6363
Serban A 1995 Anode geometry and fo13us 13hara13teristi13s PhD thesis
Nanyang Te13hnology University httphdlhandlenet104971568
Serban A Lee S 1995 A simple opti13al ber axial speed dete13tor Review
of S13ienti13 Instruments 66 4958 DOI ADS
Serban A Lee S 1995 A simple opti13al ber axial speed dete13tor Review
of s13ienti13 instruments 66(10) 4958
Singh JP Thakur SN (eds) 2007 Laser-indu13ed breakdown
spe13tros13opy Elsevier s13ien13edire13t
Singh R Stehleacute C Suzuki-Vidal F Kozlova M Larour J Chaulagain
U Clayson T Rodriguez R Gil J Nejdl J et al 2017 Experimental
study of the intera13tion of two laser-driven radiative sho13ks at the pals
laser High Energy Density Physi13s 23 20
Stahler SW Palla F Salpeter EE 1986 Primordial stellar evolution -
The protostar phase The Astrophysi13al Journal 302 590 DOI ADS
Stehleacute C Gonzaacutelez M Kozlova M Rus B Mo13ek T A13ef O Colom-
bier JP Lanz T Champion N Jakub13zak K et al 2010 Experi-
mental study of radiative sho13ks at pals fa13ility Laser and parti13le beams
28(02) 253
Stehleacute C Kozlovaacute M Larour J Nejdl J Champion N Barroso P
Suzuki-Vidal F A13ef O Delattre P-A Dostaacutel J et al 2012 New
probing te13hniques of radiative sho13ks Opti13s Communi13ations 285(1)
64
Stehleacute C Lefegravevre R Chaulagain U Champion N Barroso P Reix
F Jagourel P Larour J Melt13hakov E Mer13ier R et al 2014a
Criti13al 13omponents for xuv probing of laser driven sho13ks In X-Ray
Lasers 2012 Springer 239
Stehleacute C Kozlovaacute M Larour J Nejdl J Suzuki-Vidal F Cohen M
Chaulagain UP Champion N Barroso P A13ef O et al 2014b
Reply on the 13omment of the paper new probing te13hniques of radiative
sho13ks Opti13s Communi13ations 318 226
Suzuki-Vidal F Lebedev S Pi13kworth L Swadling G Burdiak G
Skidmore J Hall G Bennett M Bland S Chittenden J et al
2014 Sho13k formation in 13ounter-streaming jets on the magpie pulsed-
power generator In APS Meeting Abstra13ts 1 5005
REFERENCES 161
Suzuki-Vidal F Clayson T Swadling GF Patankar S Burdiak GC
Lebedev SV Smith RA Stehle C Chaulagain U Singh RL La-
rour J Kozlova M Spindloe C Foster J Skidmore J Gumbrell E
Graham P Danson C 2016 Counter-streaming radiative sho13k experi-
ments on the Orion laser In APS Meeting Abstra13ts ADS
Toledo-Roy JC Velaacutezquez PF de Colle F Gonzaacutelez RF Reynoso
EM Kurtz SE Reyes-Iturbide J 2009 Numeri13al model for the SNR
DEM L316 simulated X-ray emission Monthly Noti13es of the RAS 395
351 DOI ADS
Tou T Lee S Kwek K 1989 Nonperturbing plasma-fo13us measurements
in the run-down phase IEEE Transa13tions on Plasma S13ien13e 17(2) 311
Vaytet NMH Audit E Dubro13a B Delahaye F 2011 A numeri13al
model for multigroup radiation hydrodynami13s Journal of Quantitiative
Spe13tros13opy and Radiative Transfer 112 1323 DOI ADS
Vaytet N Chabrier G Audit E Commerccedilon B Masson J Ferguson
J Delahaye F 2013 Simulations of protostellar 13ollapse using mul-
tigroup radiation hydrodynami13s ii the se13ond 13ollapse arXiv preprint
arXiv13071010
Vaytet N Gonzaacutelez M Audit E Chabrier G 2013 The inuen13e
of frequen13y-dependent radiative transfer on the stru13tures of radiative
sho13ks Journal of Quantitiative Spe13tros13opy and Radiative Transfer 125
105 DOI ADS
Velarde P Gar13iacutea-Senz D Bravo E Ogando F Relantildeo A Gar13iacutea C
Oliva E 2006 Intera13tion of supernova remnants From the 13ir13umstellar
medium to the terrestrial laboratory Physi13s of Plasmas 13(9) 092901
DOI ADS
Veloso F Moreno J Tarifentildeo-Saldivia A Pavez C Zambra M Soto
L 2012a Non-intrusive plasma diagnosti13s for measuring sheath kine-
mati13s in plasma fo13us dis13harges Measurement S13ien13e and Te13hnology
23(8) 087002
Veloso F Tarifentildeo-Saldivia A Pavez C Moreno J Zambra M Soto
L 2012b Plasma sheath kinemati13s and some impli13ations on the mo-
deling of very low energy plasma fo13us devi13es Plasma Physi13s and Con-
trolled Fusion 54(9) 095007
Vis13o A Drake R Glenzer S Doumlppner T Gregori G Froula D
Grosskopf M 2012 Measurement of radiative sho13k properties by x-ray
thomson s13attering Physi13al review letters 108(14) 145001
162 REFERENCES
Williams RM Chu Y-H Di13kel JR Beyer R Petre R Smith RC
Milne DK 1997 Supernova Remnants in the Magellani13 Clouds I The
Colliding Remnants DEM L316 Astrophysi13al Journal 480 618 ADS
Williams RM Chu Y-H Di13kel JR Gruendl RA Seward FD
Guerrero MA Hobbs G 2005 Supernova Remnants in the Magella-
ni13 Clouds V The Complex Interior Stru13ture of the N206 Supernova
Remnant Astrophysi13al Journal 628 704 DOI ADS
Zambra M Kalise D Fernandez J Hernandez E Pas-
ten D Munoz V 2009 Current sheet thi13kness in the
plasma fo13us snowplow model J Plasma Fusion 8(879-882)
httpspeopleri13amoeawa13atdkalisefilesplasmapdf
Zeldovi13h IB Zeldovi13h YB Raizer YP 2002 Physi13s of sho13k waves
and high-temperature hydrodynami13 phenomena Courier Dover Publi13ati-
ons London New York Sydney
REFERENCES 163
THESE DE DOCTORAT
SORBONNE UNIVERSITE - PIERRE ET MARIE CURIE
Eacute13ole do13torale Physique en Ile-de-Fran13e
Preacutepareacutee
au Laboratoire de Physique des Plasmas
et au Laboratoire dEacutetude du Rayonnement et de la Matiegravere en
Astrophysique et Atmosphegraveres
Preacutesenteacutee par
Raj Laxmi SINGH
Sujet de la thegravese
Strong radiative sho13ks relevant for stellar environments
experimental study and numeri13al approa13h
Soutenan13e le 02 Mars 2017
agrave Universiteacute Pierre et Marie Curie devant un jury 13omposeacute de
Mme FALK Katerina Senior S13ientist Rapporteur
ELI Beamlines Cze13h Republi13
Mme TURCK-CHIEZE Sylvaine Dire13tri13e de Re13her13hes Rapporteur
honoraire du CEA
M GONZALEZ Matthias Maicirctre de 13onfeacuteren13es Examinateur
U Denis Diderot
Mme RICONDA Caterina Professeur UPMC Examinateur
M LAROUR Jean Chargeacute de Re13her13he CNRS Dire13teur de thegravese
Mme STEHLEacute Chantal Dire13tri13e de Re13her13he CNRS (Inviteacutee) Co-dire13tri13e de thegravese
Ex13ept where otherwise noted this work is li13ensed under
http13reative13ommonsorgli13ensesby-n13-nd30
ACKNOWLEDGEMENTS
First and foremost I would like pay my sin13ere gratitude to Dr Jean La-
rour and Dr Chantal Stehleacute my thesis supervisors who have introdu13ed
me to the marvelous world of Laboratory Astrophysi13s I would express my
thanks for their 13ontinuous support during my PhD journey for en13ouraging
me to learn 13hallenging things Working under their guidan13e was a unique
experien13e and a great joy I am parti13ularly grateful for their serenity and
patien13e that helped me in developing my spa13e and style in resear13h Co-
ming all the way from India to a non-English speaking 13ommunity had been
presented by them to me like a se13ond home through their benign love and
ae13tion
I have imparted a signi13ant time during my PhD for 13ondu13ting expe-
riments in PALS laser fa13ility at Prague through LASERLAB a1313ess Being
newbie in su13h a large s13ale experimental fa13ility has never been a problem
for whi13h I am thankful to all the team members for their help and guidan13e
from Paris PALS Prague and Imperial College London Spe13ial thanks to
Dr M Kozlova Dr M Krus Dr J Nejdl Dr J Dostal and all other
sta members at the PALS laser fa13ility and the target fabri13ation to Mr
P Barrasso and team at the Observatoire de Paris without whom none of
these experiments 13ould have been possible Spe13ial a13knowledgement to
Dr F Suzuki Vidal Mr T Clayson P Barroso and Dr U Chaulagain for
sharing their insight and explanations during the experiments
I espe13ially appre13iate the 13onstru13tive help and 13omments of Dr F
Suzuki Vidal Dr M Kozlova Dr M Cotelo Dr R Rodriacuteguez and Dr U
Chaulagain in the data analysis arti13le and thesis writing
I am grateful to all the jury members of my PhD 13ommittee Dr Sylvaine
Tur13k-Chieze Dr Katedegina Falk Dr Caterina Ri13onda Dr Matthias Gon-
zalez Dr Chantal Stehleacute and Dr Jean Larour for taking time to attend
my PhD defense Thanking very mu13h to Dr Tur13k-Chieze and Dr Falk for
reviewing my thesis and providing valuable 13omments and feedba13k whi13h
have surely improved my thesis
I would like to express sin13ere thanks to the dire13tors of LPP Dr P
Chabert and of LERMA Dr D Lis for allowing me to work on the thesis
in su13h a fabulous university ambien13e It would be unjust if I do not men-
tion the administrative help re13eived from Dr D Zahorski former Proje13t
13hief of PlasPar as well as Dr X Fresquet the present proje13t 13hief of
PlasPar I am grateful to PlasPar Labex for funding my PhD studies in
Fran13e
Next I would like to thank Dr U Chaulagain for helping me in Paris
during my stay from the very rst day I am also grateful to Dr L de Sagrave
Dr L Ibgui Dr M Drouin Dr A Stan Mr B Khiar Mr L Ni13olas
Dr J Freundli13h for their help and support during my stay in Fran13e
My thanks is due to my friends for 13reating a homely atmosphere during
my thesis in Fran13e In parti13ular I would like to thank Dr R Mourya
and Mr A Ranjan for their 13ontinuous support in the administrative for-
malities and others whi13h helped me a lot to 13on13entrate and speed up my
thesis writing I would also like to mention the names of my friends Miss T
Bhowmik Miss V Shaw and Dr A Gupta for their support
Good 13ompany plays a very important role in the life In this I am par-
ti13ularly thankful to Dr S C Tripathi Dr J Pal Dr R Kumar Dr M
Dave Dr A Bharadwaj Mr K Singh Miss S Mishra and Miss S Pandey
for helping me in keeping pa13e with resear13h and daily life developments I
am indebted to my brother Dr A Awasthi for his sin13ere guidan13e en13oura-
gement and 13ooperation during my entire do13toral study Also I would like
to take this opportunity to thank my 13hildhood friend Miss R Yadav for her
love and en13ouragement during my thesis
Finally last but not the least I express my whole hearted gratitude to the
support I re13eived in the form of love and ae13tion from my grandparents
parents sisters brothers and my other family members This is something
beyond a13knowledgements They were always with me and the en13ourage-
ment I re13eived from them will never fade away
Raj Laxmi Singh
Strong radiative sho13ks relevant for stellar
environments experimental study and numeri13al
approa13h
ABSTRACT
Strong sho13ks are present in various astrophysi13al phenomena Su13h
sho13ks are strongly inuen13ed by the radiation through its 13oupling with
hydrodynami13s Thus their topology and dynami13s are quite 13omplex Ge-
nerating su13h hypersoni13 sho13ks in the laboratory with 13ontrolled 13ondi-
tions is thus an adequate tool to study the inuen13e of radiation and to
13ompare them with numeri13al simulations Su13h sho13ks 13an be generated
by intense lasers and ele13tromagneti13 devi13es The rst part of this dis-
sertation 13on13erns the numeri13al and experimental study of the intera13tion
of two 13ounter propagating laser-driven sho13ks The experiments perfor-
med at the kJ PALS laser fa13ility allowed to generate sho13ks with dierent
speeds (sim 30-55 kms and 10-25 kms) in noble gases and low pressure
(less than 1 bar) Several diagnosti13s were implemented visible interfero-
metry time- and spa13e-resolved visible spe13tros13opy and time integrated
XUV spe13tros13opy Our experiment shows a strong intera13tion of one radi-
ative pre13ursor onto the se13ond one The physi13al parameters of the plasma
were dedu13ed from the diagnosti13s and 13ompared with 1-D simulation re-
sults The se13ond part is devoted to the design of an experiment where the
sho13k is generated ele13tromagneti13ally The optimization of this generator is
presented and also the full experimental set up whi13h allows studying sho13k
sim 30 kms in noble gas at sim 1 mbar
Keywords Radiative sho13k hydrodynami13s laser-plasmas visible
and XUV spe13tros13opy
numeri13al simulation laboratory astrophysi13s high energy density
physi13s
Cho13s forts et radiatifs dinteacuterecirct pour les
environnements stellaires eacutetude expeacuterimentale et
appro13he numeacuterique
RESUME
Les 13ho13s forts sont preacutesents dans des pheacutenomegravenes astrophysiques varieacutes
De tels 13ho13s sont fortement inuen13eacutes par le rayonnement par son 13ouplage
ave13 lhydrodynamique Par suite leur topologie et leur dynamique sont
assez 13omplexes Geacuteneacuterer de tels 13ho13s hypersoniques en laboratoire dans
des 13onditions 13ontrleacutees est ainsi un outil pertinent pour eacutetudier linuen13e
du rayonnement et pour 13omparer aux reacutesultats des simulations numeacuteriques
Ces 13ho13s sont geacuteneacutereacutes par des lasers intenses et par des moyens eacutele13tromag-
neacutetiques La premiegravere partie du texte est 13onsa13reacute agrave leacutetude numeacuterique et
expeacuterimentale de lintera13tion de deux 13ho13s induits par laser se propageant
en sens 13ontraires Les expeacuterien13es ont eacuteteacute meneacutees sur linstallation laser kJ
PALS qui permet de former deux 13ho13s ave13 des vitesses propres dieacuteren-
tes (sim 30-55 et 10-25 kms respe13tivement) dans des gaz rares agrave pression
faible (moins de 1 bar) Des diagnosti13s ont eacuteteacute installeacutes interfeacuteromeacutetrie
visible spe13tros13opie visible agrave reacutesolution spatiale et temporelle spe13tros13o-
pie XUV inteacutegreacutee en temps Nos expeacuterien13es montrent une forte intera13tion
entre les deux preacute13urseurs radiatifs Les paramegravetres physiques du plasma
ont eacuteteacute deacuteduits de 13es diagnosti13s et 13ompareacutes aux reacutesultats de simulations
monodimensionnelles La se13onde partie est 13onsa13reacutee agrave la 13on13eption dune
expeacuterien13e ougrave le 13ho13 est geacuteneacutereacute de faccedilon eacutele13tromagneacutetique Loptimisation
de 13e geacuteneacuterateur est preacutesenteacutee ainsi que lenvironnement expeacuterimental per-
mettant deacutetudier des 13ho13s jusquagrave 30 kms dans des gaz rares peu denses
(1 mbar)
Mots 13leacutes 13ho13s radiatifs plasmas laser hydrodynamique spe13tros13opie
visible et XUV simulation numeacuterique astrophysique de laboratoire
physique agrave haute densiteacute deacutenergie
Reacutesumeacute long
En introdu13tion (13hapitre 1) les 13ho13s radiatifs (CR) sont preacutesenteacutes
dans de nombreux environnements astrophysiques en parti13ulier dans les
eacutetoiles Ce sont des 13ho13s forts ave13 un nombre de Ma13h eacuteleveacute (M1) et une
tregraves haute tempeacuterature 13e qui induit un rayonnement intense Alors que
lobservation de 13ette signature ave13 une reacutesolution spatiale est tregraves di13ile
en astrophysique une appro13he innovante 13elle dite des plasmas astrophy-
siques de laboratoire fournit un puissant moyen deacutetude des CR sur Terre
Depuis plus dune deacute13ennie 13es 13ho13s sont eacutetudieacutes en laboratoire prin13ipale-
ment sur de grandes installations laser dans les gaz rares et sous dieacuterentes
geacuteomeacutetries Lintera13tion entre un laser et une feuille min13e produit une
forte ablation et par eet fuseacutee a1313eacutelegravere la feuille qui se propage dans le
gaz agrave plusieurs dizaines de kms Pour des irradiations laser entre 10
14and
10
15W13m
2 on enregistre des vitesses de 13ho13s entre 40 et 150 kms En
parallegravele une des13ription ne est permise par des simulations numeacuteriques
Au deacutebut de 13e travail les expeacuterien13es eacutetaient 13on13entreacutees sur des CR
simples alors que la situation astrophysique est 13elle dun CR interagissant
ave13 un milieu plus dense donnant des 13ho13s reacuteeacute13his et transmis Des
exemples repreacutesentatifs de 13es pheacutenomegravenes sont la propagation de restes de
supernovae dans des nuages moleacute13ulaires denses les 13ho13s da1313reacutetion sur
la photosphegravere des eacutetoiles T-Tauri ou en13ore les 13ho13s deacutetrave agrave lavant
des jets stellaires La 13ollision (ou lintera13tion) de deux CR est bien sucircr un
eacutevegravenement rare en astrophysique mais un exemple est fourni par lintera13tion
de deacutebris de la supernova DEM L316 (13f Fig 1 de Williams et al (1997))
mecircme si lhypothegravese est 13ontesteacutee (Velarde et al (2006))
Il y a don13 une forte motivation pour reacutealiser une expeacuterien13e de labora-
toire pour eacutetudier la propagation et lintera13tion de deux 13ho13s fa13e agrave fa13e
et interpreacuteter les signatures observeacutees Cette appro13he 13onstitue une grande
part de mon travail En 13ompleacutement aux expeacuterien13es laser jai eacutetudieacute un
geacuteneacuterateur 13ompa13t de puissan13e pulseacutee alimentant un 13anon agrave plasma 13oax-
ial pour 13reacuteer des 13ho13s dinteacuterecirct astrophysique dans les gaz agrave basse pression
Les 13ho13s ainsi 13reacuteeacutes sont de plus grande taille et peuvent ecirctre eacutetudieacutes plus
fa13ilement (Kondo et al 2008) Pour 13ette thegravese je me suis atta13heacutee agrave preacute-
senter les 13ho13s radiatifs dans le 13ontexte de lastrophysique de laboratoire
en suivant 13es deux appro13hes
Le deuxiegraveme 13hapitre traite de la physique des CR Ils 13omposent
une 13lasse de 13ho13s supersoniques qui sont 13haueacutes agrave haute tempeacuterature
et sont par 13onseacutequent sour13es dun rayonnement intense En retour 13e
rayonnement modie la dynamique et la stru13ture du 13ho13 13e qui 13omplexie
la des13ription Les CR se ren13ontrent dans des situations astrophysiques
13omme les 13ho13s da1313reacutetion lors de la formation des protoeacutetoiles (Stahler
Palla et Salpeter 1986) lexplosion de supernovae et lintera13tion de leur
restes ave13 le milieu interstellaire dense (Chevalier 1977) ou en13ore les 13ho13s
deacutetrave en tecircte des jets stellaires (Hartigan et al 2001) Les CR sont
maintenant eacutetudieacutes en laboratoire 13e qui permet de 13omparer les reacutesultats
aux modegraveles de la litteacuterature et de veacuterier la possibiliteacute de les retrouver ave13
les simulations numeacuteriques disponibles (Bouquet et al (2004) Leygna13 et
al (2006)) Leacutetude expeacuterimentale des CR est 13ru13iale notamment pour
13omprendre les pro13essus eacutenergeacutetiques au sein des plasmas astrophysiques
Dans le 13as des gaz parfaits le saut au niveau dun 13ho13 est deacute13rit par les
eacutequations de Rankine-Hugoniot qui relient les quantiteacutes thermodynamiques
de part et dautre de la dis13ontinuiteacute La solution monodimensionnelle dun
13ho13 se propageant agrave la vitesse us dans un gaz au repos est usuellement
deacute13rite dans le repegravere lieacute au 13ho13 et on distingue la reacutegion amont (ou preacute-
13ho13) agrave la vitesse u1= - us et en arriegravere du 13ho13 (reacutegion post-13ho13 ou aval)
la vitesse est u2 selon la Fig 21 Pour les 13ho13s forts ougrave le nombre de Ma13h
M est tregraves grand devant 1 les sauts des quantiteacutes thermodynamiques dun
gaz parfait sont donneacutes par les eacuteq 21 agrave 24 Pour un gaz monoatomique
le rapport de 13ompression est de 4 et la tempeacuterature de la zone post-13ho13
est proportionnelle au 13arreacute de la vitesse du 13ho13 et agrave la masse atomique
La situation est plus 13omplexe pour un gaz reacuteel 13ar une part de leacutenergie
13ineacutetique sert agrave ex13iter et agrave ioniser le gaz du post 13ho13 La tempeacuterature
attendue est don13 infeacuterieure agrave 13elle du 13as ideacuteal en outre la pression est
modieacutee par lionisation (Mi13haut et al 2004) Un nouveau jeu deacutequations
in13orpore la 13harge ee13tive Z des ions (eacuteq 25 agrave 28) Ces modegraveles ignorent
le 13hauage et le refroidissement radiatifs alors que le rayonnement eacutemis
ae13te la stru13ture dun 13ho13 fort Cette des13ription demande dajouter aux
eacutequations de 13onservation de la masse quantiteacute de mouvement et eacutenergie les
13ontributions radiatives (ux eacutenergie et pression Le 13as est plus 13omplexe
et les eacutequations dhydrodynamique radiative ont eacuteteacute exprimeacutees dans le repegravere
du 13ho13 (Mihalas and Mihalas 1999 Drake 2006) selon les eacuteq 29 agrave 211
Les 13ontributions de la pression radiative et du 13hamp de rayonnement
sont importantes agrave tregraves haute tempeacuterature (vitesse) mais deacutejagrave pour nos 13ho13s
le ux radiatif est dominant Selon lopa13iteacute le rayonnement du 13ho13 peut
ecirctre absorbeacute par la region preacute-13ho13 induisant un 13hauage loin en avant de
la dis13ontinuiteacute Egalement la zone 13hoqueacutee pro13he de la dis13ontinuiteacute est
ae13teacutee par la 13ompeacutetition entre leacutemission (refroidissement) et labsorption
(13hauage) Ainsi leacutepaisseur optique lo13ale (eacuteq 211) devient le paramegravetre
pertinent Comme on raisonne sur le ux radiatif moyenneacute en freacutequen13e
13est lopa13iteacute de Rosseland qui est utiliseacutee et on dieacuteren13ie les reacutegimes op-
tiquement eacutepais et optiquement min13e 13e qui permet de 13lasser les 13ho13s
radiatifs
Notre inteacuterecirct sest 13on13entreacute sur les 13ho13s preacutesentant un preacute13urseur radi-
atif En laboratoire une intensiteacute laser au dessus de 10
14W13m
2le permet
et les installations sont rappeleacutees dans le tableau 21 Il est aussi possible de
lan13er des 13ho13s moins rapides don13 peu radiatifs (sim 10-30 kms) ave13 un
piston eacutele13tromagneacutetique
Le 13hapitre 3 est 13onsa13reacute agrave lanalyse numeacuterique des CR en deacute13rivant
les pro13essus physiques jouant un rle dans 13es systegravemes physiques De mecircme
que les observations les expeacuterien13es en laboratoire ne sont pas dire13tement
interpreacutetables et la simulation numeacuterique devient un outil preacute13ieux Diverses
appro13hes numeacuteriques sont utiliseacutees pour simuler les plasmas de laboratoire
On 13onsidegravere soit des grilles xes soit des grilles adaptables (Adaptative
Mesh Renement) pour maintenir une reacutesolution susante sur 13haque zone
du plasma Une autre appro13he utilise des grilles qui suivent le uide dans
son mouvement (modegravele Lagrangien) don13 sans passage de matiegravere dune
13ellule agrave la voisine Les eacutequations sont alors dieacuterentes (Orban et al 2013)
Pour deacute13rire les 13ollisions de 13ho13s jai utiliseacute le 13ode 13ommer13ial HE-
LIOS un 13ode Lagrangien mono-dimensionnel pouvant geacuterer le rayonnement
et lhydrodynamique (Ma13Farlane Golovkin et Woodru 2006) Assez fa13ile
agrave prendre en main il a le gros avantage de pouvoir simuler notre 13ongura-
tion deux 13ho13s de vitesses opposeacutees lan13eacutes par deux lasers HELIOS deacute13rit
les plasmas hors deacutequilibre thermodynamique (hors ETL) gracirc13e agrave un mo-
degravele 13ollisionnel-radiatif (HELIOS-CR) de 13al13ul des populations atomiques
agrave 13haque pas de simulation hydrodynamique De fait jai utiliseacute HELIOS
dans une approximation ETL qui 13orrespond aux 13onditions expeacuterimentales
(Rodriguez et al 2011) et 13ela est dis13uteacute en deacutetails
Le 13ode reacutesout les eacutequations uides en introduisant les pressions des eacutele13-
trons des ions et du rayonnement Ele13trons et ions sont deacute13rits par deux
uides en intera13tion de tempeacuteratures respe13tives Te et Ti La 13ondu13tion
thermique est geacutereacutee par 13elle des eacutele13trons agrave partir de la 13ondu13tiviteacute de
Spitzer et le deacutept deacutenergie laser par un modegravele de Bremsstrahlung inverse
Leacutemission et labsorption du rayonnement sont introduites dans leacutequation
deacutenergie des eacutele13trons et dans 13elles du transport radiatif Une meacutethode
utilise un modegravele de diusion du rayonnement multi-groupes et agrave ux li-
miteacute ougrave le ux radiatif est proportionnel au gradient de leacutenergie radiative
et inversement proportionnel agrave lopa13iteacute de Rosseland ave13 une pondeacuteration
pour obtenir une bonne des13ription du 13as optiquement min13e selon Olson
Auer et Hall (2000) Une se13onde meacutethode utilise un s13heacutema deacutependant du
temps agrave 13ourte distan13e et multi-angles Dans notre 13as ougrave rayonnement
et hydrodynamique sont fortement 13oupleacutes et aussi en variation rapide on
utilise le modegravele de diusion et les opa13iteacutes ETL multi-groupes de Plan13k et
Rosseland
En outre notre version dHELIOS utilise PROPACEOS (Ma13Farlane
Golovkin et Woodru 2006) une table deacutequation deacutetat et dopa13iteacute multi-
groupes geacuteneacutereacutee par le 13ode sans possibiliteacute dintroduire une autre table La
13onnaissan13e des 13onditions thermodynamiques est neacute13essaire pour la simu-
lation du plasma dans nos 13as masse volumique entre 10
minus4et 10
minus1g13m
3
tempeacuterature entre 0 et 50 eV Une eacutevaluation preacute13ise par Rodriguez et al
(13f Fig 2 de Rodriguez et al 2011) pour le Xeacutenon indique que les 13onditi-
ons thermodynamiques de nos CR 13orrespondent au reacutegime ETL Leacutequation
deacutetat relie pression ionisation et eacutenergie interne agrave la masse volumique et agrave
la tempeacuterature Notre version dHELIOS pour plasma ETL utilise PROPA-
CEOS qui est peu do13umenteacutee elle utilise le modegravele de More et al (1988)
en reacutegime de 13ouplage fort (haute densiteacute et basse tempeacuterature) et un modegravele
datome isoleacute pour un 13ouplage faible Des interpolations sont in13luses pour
passer au 13ouplage fort (httpwwwprism-13s13omSoftwarePROPACEOS)
Jai reacutealiseacute des simulations HELIOS 1D pour divers 13as de CR 13ho13 seul
ou 13ho13s en 13ollision qui sont deacute13rites dans 13e 13hapitre 3 Les CR se propa-
geant dans le Xe agrave 01 bar et agrave 50 kms sont 13ara13teacuteriseacutes par un preacute13urseur
radiatif eacutetendu Une 13ompression eacuteleveacutee (38) dans le post-13ho13 est attribueacutee
agrave lionisation du gaz et au refroidissement radiatif Les tempeacuteratures du
post-13ho13 et du preacute-13ho13 de part et dautre du pi13 sont identiques signe
dun 13ho13 super13ritique Le rle spe13ta13ulaire de lopa13iteacute a eacuteteacute remarqueacute
mais agrave 13onsideacuterer lin13ertitude sur lopa13iteacute du Xeacutenon et le traitement 1D
jai deacute13ideacute de ne pas 13her13her agrave raner la simulation en termes de nombre
de groupes
Leacutetude numeacuterique permet de retrouver les 13ara13teacuteristiques de lintera13tion
de deux 13ho13s opposeacutes ave13 des vitesses respe13tives 50-50 kms et 50-20 kms
Dans tous les 13as la prin13ipale signature de lintera13tion (don13 avant la 13ol-
lision) est le re13ouvrement des preacute13urseurs agrave t0+8 ns pour 50-50 kms et
agrave t0+15 ns pour 50-20 kms Ensuite on assiste agrave une remonteacutee reacuteguliegravere
de la densiteacute et de la tempeacuterature eacutele13troniques Linstant de la 13ollision
voit une augmentation soudaine de la densiteacute eacutele13tronique (par un ordre de
grandeur) qui atteint 66 times 10
21et 3 times 10
2113m
minus3respe13tivement alors que
la tempeacuterature monte agrave 39 et 28 eV
Dans le quatriegraveme 13hapitre je preacutesente linstallation laser PALS puis
la 13ellule dintera13tion son implantation et les diagnosti13s asso13ieacutes Rappe-
lons le prin13ipe expeacuterimental Quand une impulsion bregraveve de lumiegravere laser
est fo13aliseacutee sur une feuille min13e une part importante deacutenergie est trans-
feacutereacutee et le 13hauage du solide 13onduit agrave son ablation Un plasma 13oronal
13haud et peu dense est eacuteje13teacute vers larriegravere et un 13ho13 vers lavant se forme
dans la feuille par eet fuseacutee Comme 13ette feuille limite le tube le 13ho13
se transmet au gaz la feuille agissant 13omme un piston Des expeacuterien13es
ont eacuteteacute meneacutees sur le Prague Asterix Laser System (PALS) au printemps
2015 (5 semaines 20 avril - 22 mai) ave13 pour obje13tif la premiegravere eacutetude
de lintera13tion puis de la 13ollision de deux CR sous la13tion de deux lasers
au niveau de 10
14W13m
2 Le Prague Asterix Laser System (PALS) repose
sur un laser agrave iode infrarouge (Asterix IV Jungwirth et al 2001) Apregraves
ampli13ation il deacutelivre jusquagrave 1 kJ en 03 ns agrave la longueur donde fonda-
mentale de 1315 nm Des fais13eaux auxiliaires sont disponibles agrave freacutequen13e
doubleacutee (λ = 657 nm) ou tripleacutee (438nm) PALS deacutelivre deux tirs par heure
agrave haute eacutenergie et dans de bonnes 13onditions dhomogeacuteneacuteiteacute du fais13eau
Pour nos expeacuterien13es nous utilisons deux fais13eaux Le fais13eau fondamental
est diviseacute apregraves le 4egraveme ampli13ateur en deux fais13eaux deacutenergie 60 et
40 (voir Fig 42) Le plus puissant est inje13teacute dans le 5
meampli13ateur et
sa freacutequen13e est tripleacutee Ce fais13eau agrave 438 nm est nommeacute MAIN Le se13ond
fais13eau est utiliseacute sans modi13ation don13 agrave 1315 nm et est nommeacute AUX
Le s13heacutema de distribution est en Fig 42 et les proprieacuteteacutes de MAIN and
AUX laser sont rappeleacutees Tdans le ableau 41
PALS propose deux 13hambres agrave vide spheacuterique et 13ylindrique respe13ti-
vement nous avons utiliseacute la 13hambre spheacuterique (Fig 43) Dun diamegravetre
de 100 13m elle est a1313essible aux fais13eaux MAIN et AUX par deux hublots
de 80 13m et 50 13m Des portes sont ameacutenageacutees ainsi que de nombreux ports
A linteacuterieur la 13ellule et des diagnosti13s sont monteacutes sur une table optique
deacute13oupleacutee meacute13aniquement de la 13hambre Les 13ibles (Fig 45) 13omprennent
un petit tube long de 4 mm fermeacute agrave ses deux extreacutemiteacutes par des feuilles min-
13es de parylegravene-N doreacute de 11 microm sur lesquelles les deux lasers sont fo13aliseacutes
au niveau de 10
14W13m
minus2 Le tube est rempli du gaz dans lequel le 13ho13
se propagera agrave une vitesse attendue entre 30 et 60 kms Les feuilles min13es
assurent la 13onversion de leacutenergie laser en eacutenergie 13ineacutetique via lablation et
la geacuteneacuteration de 13ho13 Des 13ellules speacute13iques en dural massif permettent
lalignement et la mesure du diamegravetre du fais13eau Toutes les 13ellules sont
reacutealiseacutees par le Ple instrumental de lObservatoire de Paris
Les 13ellules agrave gaz sont remplies in situ agrave une fra13tion de bar ave13 Xe
Ar Xe90He10 et He permettant ainsi de bien 13onnaicirctre le milieu ougrave se
propagent les 13ho13s Le systegraveme de remplissage (Fig 410) permet de limiter
leacute13art de pression sur les fenecirctres de la 13ellule et permet de suivre la pression
jusquau moment du tir Cest 13ritique pour ne pas 13asser la fenecirctre ultra
ne de Si3N4 pour la spe13tros13opie XUV et pour geacuterer une fuite eacuteventuelle
Le reacuteglage de la position et de la fo13alisation permet dobtenir des ta13hes
laser 13entreacutees de diamegravetres 450 - 500 microm et 250 - 300 microm pour MAIN et
AUX respe13tivement Ce 13ontrle est reacutepeacuteteacute 13haque jour Les impa13ts sont
toujours suivis par des 13ameacuteras X au keV mecircme si limpa13t sur la feuille de
parylegravene-N donne un signal X faible 13ompareacute agrave 13elui du tir sur Al
Les diagnosti13s du plasma sont prin13ipalement linterfeacuteromeacutetrie visible et
la spe13tros13opie XUV et visible Lalignement est suivi par des 13ameacuteras dans
le visible Pour linterfeacuteromeacutetrie visible preacute13iseacutement on doit prendre une
image de reacutefeacuteren13e (sans franges) de 13haque 13ible (Fig 417a) Le fais13eau de
reacutefeacuteren13e de linterfeacuteromegravetre Ma13h-Zehnder est bloqueacute et le fais13eau sonde est
utiliseacute seul La fente est ensuite reacuteduite agrave 200 microm pour ne sonder que la zone
13entrale du tube mais seacutetendant le long de laxe jusquaux pistons pour avoir
une reacutefeacuteren13e des positions initiales Ensuite en mode interfeacuterogramme on
13reacutee un systegraveme de franges perpendi13ulaires agrave la fente don13 perpendi13ulaires
agrave la dire13tion de propagation du 13ho13 (13f Fig 417b et Fig 413) Le tube
de 13ho13 est imageacute sur la 13ameacutera ave13 un grandissement susant (54 mm) et
il est possible dassurer un balayage eacutele13tronique de la fente sur le deacutete13teur
jusquagrave 200 ns Un interfeacuterogramme dit streak permet ainsi de suivre limpa13t
des 13ho13s sur les franges en fon13tion du temps
Leacutemission XUV du plasma est enregistreacutee ave13 une inteacutegration spatiale
et temporelle sur un spe13trographe agrave 13hamp plan muni dun reacuteseau 13on13ave
Il est installeacute dans la partie haute de la 13hambre (13f Fig 418) et enregistre
leacutemission XUV traversant la fenecirctre de Si3N4 Une spe13tros13opie visible
reacutesolue en temps et en position longitudinale a eacuteteacute installeacutee (Fig 419a)
mais na pas en13ore pu ecirctre exploiteacutee 13omplegravetement
Le 13hapitre 5 preacutesente les reacutesultats des deux diagnosti13s surtout linterfeacute-
romeacutetrie qui a eacuteteacute 13omplegravetement analyseacutee Jai deacuteveloppeacute une pro13eacutedure
de deacutepouillement ave13 des outils numeacuteriques ad ho13 de 13al13ul et danalyse
dimage Certes linterfeacuteromeacutetrie ne permet pas de sonder des plasmas plus
denses que la densiteacute 13ritique imposeacutee par le laser (4 times 10
2113m
minus3agrave 627 nm)
et on 13onstate que le post-13ho13 est opaque Au 13ontraire le rayonnement
XUV est preacutesent dans toute la stru13ture et la spe13tros13opie XUV inteacutegreacutee
en temps et en espa13e permet dexplorer agrave la fois le post-13ho13 et le preacute13ur-
seur Lanalyse des interfeacuterogrammes streak en visible part dune eacutevaluation
de leet de lindi13e de reacutefra13tion des eacutele13trons deacutetailleacutee dans lappendi13e A
mais qui se heurte assez vite agrave une perte de 13ontraste et agrave une disparition
des franges bien en dessous de la valeur de la densiteacute 13ritique deacutejagrave menti-
onneacutee Les deacutephasages a1313essibles ave13 preacute13ision ne deacutepassent guegravere 2 ou 3
fois 2π La re13her13he de maxima des franges est semi automatiseacutee et permet
darriver agrave une densiteacute eacutele13tronique moyenneacutee sur les 600 microm de leacutepaisseur
du tube de 13ho13 Le prol non plan du 13ho13 13onduit agrave penser que la moyenne
sous-estime dun fa13teur 2 voire plus la densiteacute dans le preacute13urseur La dis-
parition des franges est attribueacutee agrave la preacutesen13e du front de 13ho13 Par suite
les interfeacuterogrammes permettent de mesurer des vitesses de 13ho13s de 30-55
et 10-30 kms pour MAIN et AUX respe13tivement ainsi que des densiteacutes
eacutele13tronique dans les preacute13urseurs entre 10
17et 10
1913m
minus3
Nous avons ainsi deacutemontreacute lintera13tion entre deux preacute13urseurs radiatifs
dans Xe at 01 bar pour deux 13ho13s agrave 54 et 23 kms Lintera13tion est
13lairement 13ara13teacuteriseacutee par laugmentation de londe dionisation puis par la
superposition des deux preacute13urseurs agrave t0+20ns La 13ollision est enregistreacutee agrave
t0+47 ns un reacutesultat retrouveacute par la simulation
Nous avons reacutepeacuteteacute mais agrave plus haute pression 02 bar et trouveacute des
vitesses de sim 41 kms pour MAIN et sim 18 kms pour AUX Cependant
nous navons pas de signature du preacute13urseur radiatif pour AUX et le temps de
13ollision na pas eacuteteacute a1313essible agrave lenregistrement Le preacute13urseur 13teacute MAIN
nest pas inuen13eacute par AUX jusquagrave la limite de t0+48 ns (Fig 59(b) et (13))
La simulation preacutedit un tregraves faible preacute13urseur 13teacute AUX et une intera13tion
des preacute13urseurs agrave t0+49 ns
Linterfeacuteromeacutetrie transverse agrave 02 bar ave13 des vitesses sim 40 et 20 kms
indique que le preacute13urseur 13teacute MAIN a une extension lateacuterale de sim 600 microm13ontre 300 microm pour AUX Le preacute13urseur de MAIN est leacutegegraverement ae13teacute
pregraves des parois alors que 13elui de AUX est fortement 13ourbeacute Les eets 2D
sont don13 pronon13eacutes pour AUX et faibles pour MAIN En outre pour le
Xeacutenon nous disposons de la tempeacuterature et de la 13harge moyenne par le
spe13tre XUV inteacutegreacute (tir48143 Xeacutenon 06 bar) On peut en 13on13lure que
la 13harge moyenne est au moins eacutegale agrave 6 et quon a atteint une tempeacuterature
de 15 eV
Nos simulations donnent une des13ription qualitative des CR en inte-
ra13tion agrave la reacuteserve pregraves que nous introduisions dans HELIOS une uen13e
laser ajusteacutee pour donner la bonne vitesse Cependant il est maintenant
admis que les simulations 2D (ave13 les opa13iteacutes 13orre13tes) deacute13rivent bien les
expeacuterien13es (Gonzaacutelez Audit et Stehleacute 2009 Leygna13 et al 2006 Stehleacute
et al 2010) A la mecircme eacutenergie laser le 13al13ul 2D 13onduit agrave diminuer la
vitesse du 13ho13 13ompare au 13al13ul 1D et aussi agrave une baisse de la densiteacute
eacutele13tronique Par exemple pour un 13ho13 lan13eacute agrave PALS par un laser agrave 1315
nm dans le Xeacutenon agrave 03 bar ave13 une uen13e de 85000 J13m
2 ARWEN 2D
donne une vitesse de 44 kms en a1313ord ave13 lexpeacuterien13e (Cotelo et al
2015) La simulation 1D demanderait 30000 J13m
2pour obtenir la mecircme
vitesse
De mecircme la spe13tros13opie XUV inteacutegreacutee agrave 06 bar pour des vitesses reacuteel-
les de sim 39 et 18 kms indique que la tempeacuterature a atteint 15 eV et que
la 13harge moyenne a atteint 6 ou 7 alors que la simulation 1D preacutedit 10-30
eV et 5-10 respe13tivement (Fig 518) On peut en 13on13lure quune eacutetude
deacutetailleacutee baseacutee sur des simulations 2D et un post-traitement du transfert de
rayonnement sont neacute13essaires pour raner lanalyse Pour les autres gaz
rares (Ar Kr) on a observeacute que agrave eacutenergie laser donneacutee la vitesse de CR est
une fon13tion deacute13roissante de la masse volumique Ce13i 13onrme que pour
une densiteacute et une vitesse les eets radiatifs augmentent ave13 le numeacutero
atomique Pour le Krypton on a observeacute un petit preacute13urseur mais au13une
intera13tion Celle-13i ne serait possible quen augmentant nettement la vi-
tesse don13 leacutenergie du laser Ce13i a eacuteteacute rendu possible sur linstallation
laser Orion (AWE Aldermaston G-B) ougrave la 13ollision de deux CR deacutegale
vitesse a eacuteteacute obtenue agrave sim 80 kms pour une uen13e laser sim 6 times 10
14W13m
2
(Clayson et al 2016 Suzuki-Vidal et al 2016) pour une large gamme de
gaz rares et des pressions entre 01 et 1 bar Pour eacutetudier agrave la fois les 13ho13s
et les preacute13urseurs radiatifs de nombreux diagnosti13s eacutetaient installeacutes radi-
ographie X imagerie optique en 13ameacutera agrave balayage de fente interfeacuteromeacutetrie
multi-vues et agrave balayage de fente Bien que je naie pas pu parti13iper aux
13ampagnes jai fait des simulations 1D pour interpreacuteter les reacutesultats Cet
aspe13t nest pas deacute13rit dans 13ette thegravese mais a eacuteteacute publieacute tregraves reacute13emment
(Clayson et al 2017)
Apregraves des 13hapitres 13onsa13reacutes aux 13ho13s radiatifs geacuteneacutereacutes par laser le
13hapitre 6 deacute13rit un moyen alternatif passant par la voie eacutele13trique agrave haute
puissan13e Les 13hires de la voie laser sont tregraves eacuteleveacutes en uen13e plus de
10
14W13m
2 mais repreacutesentent des eacutenergies modeacutereacutees dans limpulsion (120
et 60 J pour MAIN et AUX au PALS) Or un systegraveme eacutele13trique devant
deacutelivrer 100 J reste modeste pour sa partie sto13kage Les vitesses de 50
kms seront probablement di13iles agrave atteindre mais on peut tabler sur un
reacutegime 13ompleacutementaire des 13ho13s radiatifs ave13 plus de 10 kms si le ren-
dement de 13onversion en eacutenergie 13ineacutetique reste 13orre13t Dougrave un neacute13essaire
travail doptimisation Des vitesses jusquagrave 100 kms avaient eacuteteacute mesureacutees
tregraves tt dans des 13anons agrave plasma dhydrogegravene agrave basse pression (Lee 1969)
mais 13e13i a eacuteteacute exploiteacute ensuite pour dautres naliteacutes La litteacuterature ne
mentionne quun 13as de 13anon eacutele13tromagneacutetique 13oaxial deacutedieacute aux 13ho13s
dinteacuterecirct astrophysique (Kondo et al 2006) Il est signaleacute que le lan13eur
eacutele13tromagneacutetique produit des 13ho13s plans de grande dimension transverse
(1 ordre de grandeur par rapport agrave la voie laser) et don13 plus fa13iles agrave in-
vestiguer (Kondo et al 2008) Le prin13ipe de fon13tionnement est le suivant
i) 13reacuteation dun 13laquage haute tension initial sur une surfa13e dieacutele13trique
fa13e agrave du gaz ii) 13hauage ohmique du plasma par le 13ourant en 13roissan13e
rapide iii) a1313eacuteleacuteration de la 13ou13he de plasma sous la pression magneacuteti-
que auto-geacuteneacutereacutee par le 13ourant (dizaines de kA) iv) maintien (1 micros) de la
pression magneacutetique pour 13ommuniquer une impulsion importante Des tra-
vaux sur le Z-pin13h ou le plasma fo13us permettent de postuler des geacuteomeacutetries
13onvenables pour a1313eacuteleacuterer une masse de lordre de 10
minus510
minus4g
Apregraves avoir eacutetudieacute le fon13tionnement dun 13ir13uit eacutele13trique rapide jai
listeacute les points forts et les points faibles en fon13tion des obje13tifs et jai pro-
poseacute une geacuteomeacutetrie 13oaxiale 13ompa13te Loptimisation du 13anon agrave plasma a
eacuteteacute faite en terme de vitesse maximale agrave la sortie ave13 13omme seules 13ontrain-
tes une eacutenergie sto13keacutee de 1 kJ une dimension transverse de 4 mm et une
dimension axiale dun ordre plus grande Lobje13tif eacutetait le 13ho13 fortement
supersonique 10-30 kms agrave quelques mbar dAr ou de Xe Un modegravele simple
(0D) a eacuteteacute 13ompareacute agrave un modegravele 3D MHD par le 13ode GORGON qui deacute13rit
bien les expeacuterien13es de plasmas astrophysique par puissan13e pulseacutee (Ciardi
et al 2007)
La pression magneacutetique sexprimant par B
22micro0 un 13ourant supeacuterieur agrave
100 kA apparaicirct neacute13essaire et atteignable sur un geacuteneacuterateur 1 kJ - 1 micros 13om-
pa13t de type R-L-C Ensuite il a fallu monter un modegravele de 13ir13uit deacuteformable
13omportant une se13tion formeacutee de la lame de plasma en mouvement et qui
prenne en 13ompte la1313reacutetion du gaz dans un modegravele dit snowplow (Potter
1971) Des paramegravetres geacuteomeacutetriques 13omme la forme et les dimensions du
13anon ont pu ecirctre optimiseacutes pour donner une grande vitesse de sortie du
plasma pour Ar et Xe et dans la gamme 01 - 10 mbar Un 13anon a eacuteteacute 13on-
struit et 13onne13teacute agrave un geacuteneacuterateur eacutele13trique pulseacute existant Des observations
du mouvement du plasma par la dynamique du rayonnement radial donnent
des vitesses 13oheacuterentes ave13 le modegravele ave13 la simulation 3D MHD et des
nombres de Ma13h eacuteleveacutes de 20 agrave 60 Ces travaux ont soutenu la 13on13eption
dun autre geacuteneacuterateur pour obtenir des performan13es plus eacuteleveacutees en vitesse
de 13ho13 et en taux de reacutepeacutetition
Contents
List of Figures
List of Tables
1 Introdu13tion 1
11 General Context 1
12 Outline of Chapters 2
13 My 13ontribution 3
2 Radiative sho13k waves 5
21 Rankine-Hugoniot(R-H) relations 6
211 Jump relations for a real gas 9
212 Ee13t of radiation on sho13k waves 11
213 Radiative hydrodynami13s equations 13
22 Radiative sho13k regimes 15
221 Typi13al radiative sho13k waves with a pre13ursor 16
23 Radiative sho13k waves experiments 16
231 Laser-driven sho13k experiments 18
232 Ele13tromagneti13ally laun13hed sho13k 21
3 1D Simulations 23
31 HELIOS 23
32 LTE approximation 24
321 Mean opa13ity 25
322 Equation of State 27
33 Single radiative sho13k waves 27
34 Intera13ting radiative sho13k waves 40
35 Summary 44
4 Experimental Setup 47
41 Prague Asterix Laser System (PALS) fa13ility 48
42 Targets 49
421 Massive Targets 51
422 Gaseous Targets 51
423 Target holder 55
CONTENTS
424 Target lling 56
43 Laser Fo13using 57
431 Fo13al Lenses and Phase Zone Plates 57
432 Fo13using of the MAIN and AUX laser beams 58
44 Diagnosti13s 59
441 Visible interferometry 61
442 XUV spe13tros13opy 65
443 Visible spe13tros13opy 67
45 Summary 68
5 Results and Interpretation 71
51 Visible interferometry 71
511 Longitudinal interferometry sho13k speed and ele13tron
density 74
512 Transverse interferometry lateral extension of the sho13k 91
52 XUV Spe13tros13opy 92
53 Simulations based on experimental results 93
54 Summary 97
6 Ele13tromagneti13ally laun13hed sho13k 101
61 Prin13iples of operation of a high 13urrent generator 102
62 Prin13iples of the run-down phase in a PFD 103
63 Proposed design for the plasma gun 104
64 Dynami13 13ir13uit modelling 106
65 3-D MHD simulations using GORGON 13ode 111
66 Measurements 116
67 Summary 118
7 Con13lusion 121
71 Con13lusions 121
72 Perspe13tives 123
8 Thesis summary 127
Appendi13es 141
A Visible Interferometry 143
A01 Refra13tive index of a plasma 143
A02 Absorption of the laser beam 144
A03 Prin13iple of interferometry 145
A04 Ma13h Zehnder Interferometer 146
B Opa13ities and mean 13harge 149
Referen13es 152
List of Figures
21 S13hemati13 diagram for a 1-D sho13k moving in the x dire13tion
The speed of the sho13k front is us In the frame of referen13e of
the sho13k front the pre-sho13k region has a relative speed u1=
-us density ρ1 total pressure P1 temperature T1 whereas these
quantities are respe13tively u2 ρ2 P2 and T2 for the post-sho13k
region 6
22 Proles of temperature (Te = Ti = T ) and mass density (ρ) at 10ns for a sho13k propagating in Xenon at 01 bar (ρ1= 57 times 10
minus4
g13m
minus3) with a speed of sim 45 kms It is to note that the given
pressure value are at 298 K The simulation is performed with the
HELIOS 13ode ex13luding the radiation ee13t and for a polytropi13
gas (γ=53) The initial temperature of Xenon is set to 1 eV
and with a negligible (10
minus5WmK) thermal 13ondu13tivity The
Ma13h number is then equal to 40 For this simulation the sho13k
is laun13hed by a 100 microm thi13k Aluminium piston moving at the
velo13ity of 33 kms The position x=0 13orresponds to the position
of the XeAl interfa13e at time t=0 8
23 Ele13tron temperature (Te) ion temperature (Ti) and mass density
(ρ) proles at 10 ns derived from 1-D simulation for the 13ase of
ionisation (no radiation) The sho13k propagates in Xe gas at 01
bar (ρ1= 57 times 10
minus4g13m
minus3by sim 45 kms) It is to note that the
given pressure value is at 298 K The simulation is performed with
the HELIOS (Te 6= Ti 13ase) using the PROPACEOS equation
of state without any radiation and with a negligible thermal
13ondu13tivity (10
minus05WmK) The initial temperature in Xenon
is set to 1 eV and thus the Ma13h number is equal to 40 For this
simulation the sho13k is laun13hed by a 100 microm thi13k Aluminium
piston moving at the velo13ity of 38 kms The position x=0
13orresponds to the position of the XeAl interfa13e at time t=0 10
LIST OF FIGURES
24 Figure 14 amp 17 of Mi13haut et al (2004) (a) Compression ρ2ρ1(full 13ir13les thin line) kT2 in eV (inverted full triangles thin line)
in Argon (a) Xenon (b) for initial 13onditions kT1( = 10 eV for
Ar and = 01 eV for Xe) ρ1 = 5 times 10minus4 gcm3 versus sho13k
speed in kms with (full markers full lines) and without (empty
markers dashed lines) radiation The ionization stage lt z gt2 is
plotted with the diamond symbol 12
25 Simulated ele13tron temperature (Te) ion temperature (Ti) and
mass density (ρ) proles at 10 ns of a radiative sho13k propagatingwith a 15 kms (a sub-13riti13al) and 48 kms (b super13riti13al)
The sho13k is propagating in Xenon at 01 bar (ρ = 54 times 10
minus4
g13m
minus3 at 298 K) The initial temperature in Xenon is set to
1eV The two Ma13h numbers are respe13tively equal to 13 and 43
The radiation and ionisation ee13ts are in13luded in the simulation
whi13h is performed for two dierent temperatures (Te 6= Ti) and a
negligible thermal 13ondu13tivity (10
minus05WmK) For this simula-
tion the sho13k is laun13hed by a 100 microm thi13k Aluminium piston
moving at the speed of 12 kms and 45 kms for 13ase (a) and (b)
respe13tively The position x=0 13orresponds to the position of the
XeAl interfa13e at time t=0 17
31 PROPACEOS Mono13hromati13 opa13ity versus the photon energy
in eV of Xenon at 10 eV and at two ele13tron densities respe13tively
equal to 1018 and 1020 13m
minus3 26
32 Mass density and temperature (a) ele13tron density and mean
13harge (b) at 10 ns for a radiative sho13k of speed sim 47 kms
in Xenon at 01 bar The verti13al dotted bla13k lines show the
position of the interfa13e between piston and ba13king Xenon gas
Zero at x-axis is the interfa13e of piston and Xenon at time zero
Spitzer thermal 13ondu13tivity has been used in the simulation 29
33 Mean Rosseland opa13ity (in 13m
2g
minus1) and opa13ity (13m
minus1) at 10
ns for a radiative sho13k of speed sim 47 kms in Xenon at 01
bar The verti13al dotted bla13k lines show the position of the
interfa13e between piston and ba13king Xenon gas Zero at x-axis
is the interfa13e of piston and Xenon at time zero Spitzer thermal
13ondu13tivity has been used in the simulation 30
34 (a) Mass density and temperature (b) ele13tron density and
mean 13harge at 10 ns for a radiative sho13k of speed sim 47 kms
in Xenon at 01 bar The verti13al dotted bla13k lines show the
position of the interfa13e between piston and ba13king Xenon gas
Zero at x-axis is the interfa13e of piston and Xenon at time zero
A negligible 13onstant thermal 13ondu13tivity (10
minus12WmK) have
been used in the simulation for Xenon 31
LIST OF FIGURES
35 (a) Ele13tron density and ele13tron temperature (b) Mass density
and mean 13harge at 10 ns for a radiative sho13k of speed sim 48
kms in Xenon at 01 bar for the 13ase when the ele13tron and ion
temperature are dierent The verti13al dotted bla13k lines show
the position of the interfa13e between piston and ba13king Xenon
gas Zero at x-axis 13orresponds to the position of the interfa13e
between the piston and Xenon at time zero The Spitzer thermal
13ondu13tivity has been used in this simulation 32
36 (a) Mass density and ele13tron temperature (b) ele13tron density
and mean 13harge at 10 ns for a radiative sho13k of speed sim 45
kms in Xenon at 01 bar for the 13ase when the ele13tron and ion
temperature are dierent The verti13al dotted bla13k lines show
the position of the interfa13e between piston and ba13king Xenon
gas Zero at x-axis 13orresponds to the position of the interfa13e
between the piston and Xenon at time zero In this simulation
value of thermal 13ondu13tivity (10
minus12WmK) is negligible for
Xenon The peak value of ion temperature is found to be 405 eV
whereas theoreti13ally this value is expe13ted to be sim 600 eV It is
possible to a13hieve the expe13ted value by in13reasing the resolution
of the simulation 34
37 Ele13tron temperature proles for various numbers of frequen13y
groups N = 1 20 50 60 70 90 and 100 35
38 (a) Ele13tron temperature (a) mean 13harge (b) and mass density
(13) for four (1 10 30 and 40) multipliers of the Xenon opa13ity at
10 ns 36
39 Plan13k and Rosseland mono13hromati13 opa13ity proles Te for ρ=15 times 10
minus3from PROPACEOS (I) BIGBART (II) and Rodri-
guez et al (III) 38
310 Mono13hromati13 Xenon opa13ity versus the photon energy in eV
at T = 15 eV and for ρ = 15 times 10
minus3g13m
minus3for the PROPA-
CEOS (a) and Rodriacuteguez et al (2015) (Fig 16 of Rodriacuteguez et al
(2015)) (b) models 39
311 Ele13tron density Ne (a) and ele13tron temperature Te (b) versus
axial position (along a 04 13m long sho13k tube) at 3 10 20 30
and 38 ns from HELIOS simulations (with opa13ity times 20) for the
13ases of single sho13k of sim 50 kms (dotted line) and two identi-
13al 13ounter-propagating sho13ks of sim 50 kms (solid lines) The
verti13al dotted lines show the position of the interfa13e between
piston and ba13king Xenon gas 41
LIST OF FIGURES
312 Variations ofNe (a) and Te (b) versus axial position for the 13ase of
two identi13al 13ounter-propagating sho13ks (of speeds sim 50 kms)
at 3 10 20 30 35 and 40 ns as derived from HELIOS simulations
For these simulations we have negle13ted the ee13t of radiation
by keeping the Xe opa13ity equal to zero The verti13al dotted lines
show the position of the interfa13e between the piston and ba13king
Xenon gas 43
313 Variations of Ne (a) and Te (b) with axial position for the 13ase
of two non-identi13al 13ounter-propagating sho13ks (of speeds sim 50
amp 20 kms) and two single sho13ks (dotted lines) of speeds sim 50
amp 20 kms respe13tively at 3 10 30 35 and 46 ns as derived
from HELIOS simulations The verti13al dotted lines show the
position of the interfa13e between piston and ba13king Xenon gas
(with opa13ity times 20) 45
41 a Spatial prole of the PALS Laser beam b Intensity prole of
laser pulse I(t) with time 48
42 Energy distribution s13heme for MAIN and AUX laser beams 49
43 Snapshot of the spheri13al 13hamber showing the horizontal bred-
board and on the right the fo13using MAIN lens 50
44 Dimension drawing (a) and snapshot (b) of Aluminum massive
(mo13k) target 50
45 Values on the gures are in mm (a) Verti13al 13ross se13tion view
of the gaseous target (b) Horizontal 13ross se13tion view 52
46 Pi13ture of a the gaseous target on its base 53
47 S13hemati13 of the gaseous target 53
48 Transmission of 100 nm thi13k Si3N4 membrane (CXRO database) 54
49 Snapshot of a target holder with one V-shield regarding the AUX
laser whi13h will pass through the hole in this shield marked in
green in the Figure The target whi13h fa13es the two diagnosti13s
(tubes) and is lo13ated behind the V-shield is thus not visible 54
410 In-situ system for target lling and pressure monitoring 56
411 S13hemati13 of the keV 13amera 59
412 (a) Sign 13onvention for the respe13tive positions of the lens and
the target (b) keV image of a MAIN impa13t on a massive target
(lens position +1500 microm) The spot size is 475plusmn25 microm Pixel
size for MAIN keV 13amera is 56 microm (13) keV image of an AUX
impa13t (lens position -1500 microm) The spot size is sim 280plusmn20 micromon target One pixel = 66 microm on AUXs keV 13amera 60
413 First interferometri13 setup All the opti13al elements are 1 in13h
For simpli13ity the plasma slab is not reported in the probe beam 61
414 Se13ond interferometri13 setup 63
415 S13hemati13 of the Streak 13amera (from Hamamatsu noti13e) 64
LIST OF FIGURES
416 Imaging setup of the Ma13h Zehnder Interferometer setup opera-
ting in the longitudinal mode The 13hannel of the target is fully
illuminated by the probe beam In this mode the lens images
the axis of symmetry of the two sho13k waves along the slit of the
streak 13amera 65
417 (a) Referen13e 2D image of a target before the shot re13orded on
the Streak 13amera The positions of the two pistons 13losing the 4
mm long target are lo13ated at 800 and 4800 microm The dark zones
along them (between 800 to 1000 and 4600-4800 microm) 13orresponds
to the glue This glue is then visible through verti13al bla13k strips
in the interferometri13 re13ord (b) Corresponding interferometri13
time-spa13e re13ord 66
418 Typi13al ray tra13ing in a grazing in13iden13e XUV spe13trograph
The dete13tor (GMCP or image plate) is installed tangentially to
the Rowland 13ir13le 67
419 (a) Setup of the XUV spe13trometer in the spheri13al 13hamber (b)
Typi13al re13ord between sim 10 and 40 nm is shown in the bottom
panel 68
420 (a) S13hemati13 of setup employed to re13ord the time and spa13e
visible plasma emission (b) A typi13al spe13trogram 69
51 Original re13ord of shot 48055 (left) FFT of the original image
(13enter) and the frequen13y ltered image (right) 73
52 A 13ropped se13tion of the re13ord from shot the 48055 The
rst ve positions have been sele13ted manually `+ signs (in red
13olor) on ea13h fringe On this re13ord the distan13e between two
unperturbed fringes is 13orrespond to 15 pixels (ie 159 microm) 73
53 (a) Representative points of the fringes as derived from the spline
tting of the 5 manually sele13ted points (Fig 52 (II)) on ea13h
fringe (b) Positions of the fringes maxima along Y-axis for ea13h
fringe derived by lo13ating the points of maximum intensity in
X-dire13tion of the previous points obtained by spline t 75
54 Fringe maxima on the full image 76
55 Imaging setup of the Ma13h Zehnder Interferometer operating in
the longitudinal mode The lens images the axis of symmetry of
the two sho13k waves along the slit of the streak 13amera 76
56 Interferometri13 image re13orded for the shot 48055 in Xe at 01
bar The sho13k speeds for the sho13ks driven by MAIN (from left
side) land AUX (from right side) lasers are respe13tively equal to
sim 54 and 23 kms The time of laser arrival on the piston is at
146 ns The positions of the Au-Xe interfa13e on the re13ord are
respe13tively 950plusmn50 and 4950plusmn50 mi13rons 77
LIST OF FIGURES
57 (a) MAIN sho13k speed (13al13ulated by the last fringe method)
versus the MAIN energy for Xenon or XeHe mixture at dierent
pressures with the error bars (b) AUX sho13k speed versus AUX
energy also for Xenon or XeHe mixture at dierent pressures 79
58 MAIN sho13k speed (13al13ulated by the last fringe method) versus
the MAIN laser energy for Xe (at 01 bar only) Ar He and Kr
at dierent pressures with the error bars 82
59 Left panel interferometri13 re13ords 48055 in Xe at 01 bar (a)
48132 in Xe+He at 02 bar (b) and 48138 in Xe+He at 02 bar
(13) Right panel ele13tron density at 10 20 30 and 40 ns versus
distan13e for these re13ords The positions of maxima have been
identied on the re13ords in the left panel The time t = 0 13orre-
sponds to the time of laser arrival on the target and the position
x = 0 13orresponds to the interfa13e between the piston (Au layer)
and the gas Its determination is pre13ise within 100 mi13rons The
distan13es between two unperturbed fringes for re13ords 48055
48132 and 48138 are 159 244 and 244 microm respe13tively The
lt Ne gt un13ertainty (plusmn 2 pixels) is indi13ated by the error bar in
the right panels It 13orresponds respe13tively to plusmn 9 times 10
17 plusmn 6
times 10
17and plusmn 6 times 10
1713m
minus3for the gures (a) (b) and (13) The
limit of dete13tion (2 pixels) is presented by a dotted line on ea13h
gure 85
510 (a) Interferometri13 re13ord in Ar at 03 bar (48079) The esti-
mated speeds for MAIN and AUX are 49plusmn5 and 23plusmn3 kms (b)
Same re13ord where the fringes maxima are marked by points
The bins denition is as fellows bin 0 Ne le 11 times 10
1813m
minus3
(white) bin 1 11 - 18 times 10
1813m
minus3(yellow) and bin 2 39 -
57 times 10
1813m
minus3(red) The time t = 0 13orresponds to the time
of laser arrival on the target and the position x = 0 13orresponds
to the interfa13e between the piston (Au layer) and the gas Its
determination is pre13ise within 100 mi13rons 87
511 Results for temperature (a) and Rosseland opa13ity (b) obtained
from HELIOS simulation at 10 ns for two 13ounter-propagating
sho13ks at sim 50 and 18 kms for Xe (with opa13ity multiplier 1
and 20) Ar and Kr at 54 times 10
minus4g13m
minus3initial mass density
A 13omparison Ar Kr and Xe PROPACEOS opa13ity shown in
APPENDIX B 88
LIST OF FIGURES
512 (a) Raw interferometri13 re13ord in Kr at 02 bar (shot 48146)
The estimated speed for MAIN sho13k is 53plusmn2 kms The time
t = 0 13orresponds to the time of laser arrival on the target(b)
Same re13ord shown in (a) maxima is marked by 13olored points
The bins denition is as follows bin 0 Ne le 11 times 10
1813m
minus3
(white) bin 1 11 - 18 times 10
1813m
minus3(yellow) and bin 2 39 -
57 times 10
1813m
minus3(red) The time t = 0 13orresponds to the time
of laser arrival on the target and the position x = 0 13orresponds
to the interfa13e between the piston (Au layer) and the gas Its
determination is pre13ise within 100 mi13rons 90
513 S13hemati13s of the Ma13h Zehnder Interferometer setup to re13ord
transverse interferometri13 images The lens allows to make on
the slit of the 13amera the image of a se13tion perpendi13ular to
dire13tion of sho13k propagation 91
514 Transverse interferometri13 images for (a) shot48111 (MAIN
sho13k only) (b) shot48130 (AUX sho13k only) The time is
measured after an oset equal to 14 and 23 ns respe13tively after
the time of the laser arrival on the target The position zero on
the x-axis of ea13h image 13orresponds to the base of the target 92
515 Interferometri13 image for the shot48143 The time t = 0 13orre-
sponds to the time of laser arrival on the target and the position
x = 0 13orresponds to the interfa13e between the piston (Au layer)
and the gas Its determination is pre13ise within 100 mi13rons 94
516 Raw (a) and 13orre13ted XUV spe13trum (b) for the shot 48143 95
517 Re13orded ele13tron density (shot 48055) together with the HE-
LIOS results (with Xenon opa13ity times 20) at dierent times in
Xenon at 01 bar 97
518 Time evolution of the mass density (a) ele13tron temperature (b)
and mean 13harge (13) at 56 57 58 60 64 and 65 ns within the
sho13k tube derived from the HELIOS simulations (with Xenon
opa13ity multiplier = 20) for two 13ounter streaming sho13ks of sim39 and 18 kms in Xenon at 06 bar 98
61 (a) Sket13h of the sho13k generator showing the pulsed ele13tri13al
13ir13uit the set of 13oaxial 13oni13al ele13trodes with a 13onstant radial
gap and the plasti13 insulator featured in grey on whi13h a planar
surfa13e dis13harge is initiated The installation of three opti13al
bers allows looking radially at the plasma moving in the sho13k
tube (b) S13hemati13s of the plasma dynami13s inside the 13oaxial
gun in fast-pulse mode the ele13tri13al 13urrent ows in the super-
13ial layers of the two 13oaxial 13oni13al ele13trodes and through an
annular plasma layer The magneti13 pressure Pmag pushes the
dis13harge axially 105
62 Exploded view of the plasma gun 106
LIST OF FIGURES
63 Time dependen13e of the main parameters for Ar gas at 1 mbar
a1313ording to the 13ir13uit model (a) 13urrent (kA) and voltage
(V) (b) a1313reted mass (kg) and rate of a1313retion (kgs) (13)
speed (ms) and kineti13 energy (J) 108
64 Plot of the dierent output parameters (normalized with respe13t
to the peak values) obtained at the top of the inner ele13trode
For ea13h group upper panel represents the variation for Ar gas
whereas lower panel is for Xe gas Proles of output parameters
are given (a) with 13one angle (b) with a1313retion fa13tor (13) with
gas pressure and (d) with damping resistan13e 109
65 Example of mapping of the ele13tron density from a 3-D MHD
simulation (ba13kground gas Argon at 1 mbar) when the plasma
sheath is (left) at the exit of the plasma gun (right) in free
ight 13onditions (log s13ales for the false 13olors) 113
66 Time history of plasma merging and early free ight through
the mapping of the ele13tron density ele13tron temperature and
average ioni13 13harge as given by a 3-D MHD simulation (ba13k-
ground gas Argon at 1 mbar) (s13ales for the false 13olors) 114
67 Axial proles values taken in the sho13k tube along a line slightly
o-axis at a distan13e of 1 mm for mass density ion temperature
ion density average ion 13harge magneti13 eld ele13tron tempera-
ture ele13tron density and average speed at 1500 ns (ba13kground
gas Argon at 1 mbar) A maping of |B| is given with a dashed
line indi13ating sho13k front position as well 115
68 Experimental results (a) time history of the passive opti13al
re13ords 13ompared to the main 13urrent for Argon gas 13lose to the
referen13e pressure (b) time-integrated signal vs Ar pressure for
the dierent bers (13) averaged speed vs Ar ba13king pressure 117
71 Interferometri13 re13ord obtained at 546 ns for a single radiative
sho13k of speed sim 100 kms propagating in Xe+He gas at 06 bar
driven by laser at 348 nm with energy 170 J The dierent 13olors
in (b) 13orrespond to yellow le Ne le 35 times 10
1713m
minus3 13yan 37
- 74 times 10
1713m
minus3 blue74 - 15 times 10
1813m
minus3 green 15 - 18 times
10
1813m
minus3 magenta 18 - 22 times 10
1813m
minus3 red 22 - 26 times 10
18
13m
minus3 orange26 times 10
1813m
minus3- 18 times 10
1913m
minus3 124
A1 S13hemati13s of Ma13h Zehnder interferometer with two mirrors
two 50 beam splitters and the dete13tor A plasma slab is pla13ed
in one of the two arms of the interferometer 147
B1 Rosseland and Plan13k opa13ity for three gases Xe Kr and Ar at
mass densities 51 times 10
minus4g13m
3(a) and 16 times 10
minus3g13m
3(b) 150
LIST OF FIGURES
B2 Mean 13harge for three gases Xe Kr and Ar at mass densities 16
times 10
minus3g13m
3(a) and 51 times 10
minus4g13m
3(b) 151
List of Tables
21 Summary of the radiative sho13ks experiments and out13omes in
dierent laser installations All the experiments are performed in
Xe ex13ept Keiter et al (2002) and Vis13o et al (2012) In these
two experiments the rst uses SiO
lowast
2 and the se13ond uses Ar
lowastlowastas
a sho13ked medium 20
41 Nominal 13hara13teristi13s of MAIN and AUX laser beams 49
42 Spe13i13ations of the MAIN and AUX fo13al lenses and the f- num-
ber (fo13al lengthbeam diameter) is given (see table 41 for the
beam diameters) 57
43 Spe13i13ations of the two Phase Zone Plates 58
44 Spe13i13ations of the two uEye keV 13ameras (1stVsion 13ompany) 59
45 Spe13i13ations of Andor CCD 13amera 67
46 Spe13i13ations of the XUV grating 67
51 Sho13k speeds estimated from the `last fringe method `NA re-
presents the entries whi13h 13ould not be dedu13ed from the re13ord
Further entries in the bold font are dis13ussed in detail in this
13hapter 81
52 Atomi13 data and density at 01 bar (at room temperature) for
He Ar Kr and Xe 82
61 Input parameters their standard values and range of variation 110
62 Laun13hing performan13e ArXe 111
63 Values of variables in various 13onditions at rest (with seed ele13-
trons) inside the sho13k and inside post-sho13k region A star lowastdenotes the insulator surfa13e 112
Chapter 1
Introdu13tion
11 General Context
Radiative sho13ks (RS) are present in various astrophysi13al environments es-
pe13ially in stars They are then present in all the supersoni13 a1313retioneje13tion
pro13esses during the stellar evolution from the early phase of the proto-
stellar 13ollapse up to the magnetosphere a1313retion on to the already formed
T- Tauri stars up to the late stage of the supernovae sho13ks
Radiative sho13ks are strong sho13ks (ie Ma13h number Mgtgt1) whi13hrea13h high temperatures and thus are the sour13e of intense radiation Alt-
hough the observation of (ie spatially resolved ) signatures of su13h sho13ks
in the astrophysi13al environments is very di13ult an alternate approa13h na-
med laboratory plasma astrophysi13s provides a powerful tool to study them
on the Earth
These waves have been experimentally studied sin13e more than a de13ade
mostly on large-s13ale laser fa13ilities in noble gases and with dierent targets
geometries The ablation pro13ess generated by the intera13tion between the
laser and a foil indu13es a sho13k wave in the foil whi13h then propagates
in the gas at a speed of several tens of kms Numeri13al simulations 13an
des13ribe these sho13ks with an improving pre13ision With laser intensities on
target 13omprised between 1014 and 1015 W13m
2 these experiments allowed
to re13ord sho13k speeds ranging between 40 and 150 kms
All previous experimental studies have been fo13used on the 13ase of single
radiative sho13ks However in astrophysi13al 13onditions the radiative sho13k
in general intera13ts with a denser medium leading to the development of
ree13ted and transmitted sho13ks A few representative examples of su13h
phenomena are the intera13tion of supernovae remnants with dense mole13ular
13louds the a1313retion sho13ks on the photosphere of T-Tauri stars and the bow
sho13ks at the head of fast stellar jets The 13ollision (or the intera13tion) of
two radiative sho13k waves is obviously a rare astrophysi13al event and the
template 13ase of the supernova remnants DEM L316 (see Fig 1 of Williams
et al (1997)) is still the subje13t of debates (Velarde et al 2006) In this
regard the development of dedi13ated laboratory experiments to the study of
propagation and intera13tion of 13ounter-propagating sho13k waves is important
as a tool to 13hara13terize su13h events through their spe13i13 signatures
Complementary to laser experiments 13ompa13t pulsed power generators
may drive an ele13tromagneti13 13oaxial plasma gun to 13reate astrophysi13al
1
2 CHAPTER 1 INTRODUCTION
relevant sho13ks in lower pressure noble gases with a high availability and a
rather modest 13apital 13ost The ele13tromagneti13ally driven sho13k waves may
have larger s13ales than those by laser Thus they 13an be analyzed rather
easily with the adequate diagnosti13s (Kondo et al 2008)
In this thesis I am presenting radiative sho13ks in the 13ontext of labo-
ratory astrophysi13s using two experimental approa13hes laser-driven sho13ks
and ele13tromagneti13ally driven sho13ks
12 Outline of Chapters
In the rst 13hapter the physi13s of radiative sho13ks wave will be summarized
I will dis13uss the Rankine -Hugoniot (RH) for the perfe13t gas and genera-
lized RH equations for ionized gas Further the ee13ts of radiation on the
sho13k equations will be introdu13ed and I will review the dierent regimes of
the radiative sho13ks presenting a radiative pre13ursor Then I will present
a short histori13al review of laser and ele13tromagneti13ally laboratory experi-
ments on this topi13 Finally I will highlight the obje13tives of my work in
this framework
The 13hapter two is dedi13ated to the numeri13al analysis of radiative sho13k
waves and of the main physi13al pro13esses whi13h play a role in these 13omplex
pro13esses The simulations will be performed in 1D using the 13ommer13ial ra-
diation hydrodynami13 HELIOS 13ode (Ma13Farlane Golovkin and Woodru
2006) After a des13ription of the 13ode and his possibilities several simu-
lations will be performed to analyze the ee13ts of the two uids (ele13trons
and ions) approa13h 13ompared to the one uid 13ase (identi13al temperature
for both) of the thermal 13ondu13tivity and the multi-groups opa13ity des13rip-
tion Then a brief overview of single and two 13ounter-propagating radiative
sho13ks will be presented
Chapter three will start with a short overview of the Prague Asterix Laser
System fa13ility (PALS) It will be followed by the experimental setup in13lu-
ding the two high energy beams and targets des13ription These paragraphs
will be followed by the detail of the diagnosti13s
The fourth 13hapter will be dedi13ated to the data analysis the results
and the interpretation of the experimental re13ords I will rst des13ribe the
method I followed for the analysis of interferometri13 re13ords Subsequently
the results obtained from a few representative re13ords for Xe Ar and Kr
will be dis13ussed in details Complementary 1D simulations performed with
initial 13onditions similar to that in the experiment will be shown and help to
interpret the experimental results Finally the results of XUV spe13tros13opy
diagnosti13s will be presented
In the fth 13hapter I will present the 13alibration of a table top ele13tro-
magneti13 plasma generator whi13h is able to laun13h supersoni13 sho13ks up to
sim 40 kms speed at stati13 pressures of few mbar The results of this 13ali-
13 MY CONTRIBUTION 3
bration will also be presented and a 13omparison with 3-D MHD simulations
will be performed The diagnosti13s whi13h have been implemented will be
presented to illustrate the model as well as preliminary re13ords of the plasma
speed
The nal 13hapter of the thesis is dedi13ated to the summary of this work
together with perspe13tives for the follow-up of these experiments
13 My 13ontribution
I joined PhD on November 12 2013 under the supervision of Dr Jean
Larour and Dr Chantal Stehleacute My PhD thesis is 13omprised of two parts
the rst part of the work is to study the laser-driven sho13ks The se13ond
part of my thesis is to study the ele13tromagneti13ally laun13hed strong sho13ks
Con13erning the laser-driven sho13ks I have started to work on the analysis
and interpretation of XUV images re13orded previously during an experimen-
tal 13ampaign 13ondu13ted at the Prague Asterix Laser System (PALS) fa13ility
in 2011 This enabled me to estimate the lo13al opa13ity by employing Abel
inversion Although this work not presented in this thesis it shed light on
the basi13 understanding of radiative sho13k as seen in the experiments as well
as the knowledge of experimental setups Later in April - May 2015 I have
parti13ipated in a ve weeks experimental 13ampaign on PALS laser fa13ility
in Prague whi13h was dedi13ated to the study of the spe13tros13opi13 and in-
terferometri13 signatures of laser-produ13ed sho13ks Espe13ially the experiment
was designed to study the physi13s during the 13ollision of 13ounter-propagating
sho13ks In the preparation of this 13ampaign I be13ame a13quainted with the
instruments and the targets to be used as well as prepared a master re-
port 13ontaining all the 13ru13ial aspe13ts of the experiment to be referred by
all the parti13ipants during the experiments During the 13ampaign we have
performed around 55 shots 13orresponding to dierent gases (Xenon Argon
Helium and Krypton) at dierent pressure and used three types of diagnos-
ti13s ie visible interferometry visible spe13tros13opy and XUV spe13tros13opy
In this experiment parti13ularly I was in 13harge of the target sele13tion its
installation in the experimental 13hamber gas lling preparing logbook re-
13ording of the visible spe13tros13opy data from the 13ontrol room qui13k data
analysis for the spot size sho13k speed from interferometry data et13 In
addition I have worked on spe13tros13opi13 and interferometri13 data analysis
re13orded during the experiment Next in order to interpret the experimental
results I have performed several 1D hydrodynami13 simulations with varying
input 13onditions In parallel I have also performed simulations for another
experiment performed by my 13ollaborators at ORION laser fa13ility in the
United Kingdom However I have not presented this work in this thesis
Based on the results obtained from my analysis of re13ords obtained from
the PALS experiments in 2015 grabbing the opportunity we have planned
4 CHAPTER 1 INTRODUCTION
and parti13ipated to a 13omplementary experimental 13ampaign at PALS in
September 2016 The analysis of data re13orded during experiments is still
under progress and forms the future s13ope of the thesis
The se13ond part of my thesis has been 13omprised of the study of sho13k
laun13hed through the ele13tromagneti13 generator setup In this I have wor-
ked on the optimization of the design of the ele13tri13 generator in view of the
required sho13k parameters Moreover I have derived various sho13k 13hara13-
teristi13s viz radial dis13harge layer namely speed mass a1313eleration et13
at the dierent positions of its propagation In present a upgraded ele13tri13
generator with new diagnosti13s is ready to perform the new experiments
Chapter 2
Radiative sho13k waves
Contents
21 Rankine-Hugoniot(R-H) relations 6
211 Jump relations for a real gas 9
212 Ee13t of radiation on sho13k waves 11
213 Radiative hydrodynami13s equations 13
22 Radiative sho13k regimes 15
221 Typi13al radiative sho13k waves with a pre13ursor 16
23 Radiative sho13k waves experiments 16
231 Laser-driven sho13k experiments 18
232 Ele13tromagneti13ally laun13hed sho13k 21
Small-amplitude disturban13es in a medium often take the form of linear
waves On the 13ontrary strong disturban13es generally produ13ed by pro13esses
su13h as explosions lightening supersoni13 movements of bodies powerful
ele13tri13 dis13harges et13 where the uid properties 13hange rapidly generate
nonlinear waves like sho13k waves As a 13onsequen13e of sho13k waves the ow
moves faster than the sound speed A sho13k is 13hara13terized by a sudden
in13rease in the density and velo13ity of the medium The Ma13h number (M)
is 13ommonly used to 13hara13terize the strength of the sho13k This number
dened as the ratio of the sho13k speed to the speed of sound in the upstream
medium in whi13h the sho13k propagates
Radiative sho13k waves are hypersoni13 sho13k waves whi13h are heated to
high temperature and as a 13onsequen13e be13ome the sour13e of intense radi-
ation This radiation in turn modies the dynami13s and stru13ture of the
sho13k itself whi13h makes its stru13ture more 13ompli13ated Radiative sho13k
waves o1313ur in several astrophysi13al 13ir13umstan13es su13h as in the a1313retion
sho13k of protostellar formation (Stahler Palla and Salpeter 1986) the su-
pernova explosion and the intera13tion of their remnants with the dense in-
terstellar medium (Chevalier 1977) the bow sho13ks at the head of stellar
jets (Hartigan et al 2001)
Radiative sho13k waves 13an now be studied in the laboratory whi13h allows
to 13ompare the data with models existing in the literature and to 13he13k the
ability of the numeri13al 13odes to reprodu13e them (see for instan13e Bouquet
et al (2004) Leygna13 et al (2006)) The experimental study of radiative
5
6 CHAPTER 2 RADIATIVE SHOCK WAVES
sho13k 13hara13teristi13s is 13ru13ial in understanding these various energeti13 pro-
13esses o1313urring in the astrophysi13al plasmas In this 13hapter I will rst
present a short summary of the jump 13onditions (Rankine-Hugoniot rela-
tions) for the simple 13ase of an ideal gas whi13h rely on the values of the
thermodynami13al quantities on both sides of the dis13ontinuity Next I will
present the role played by the ionisation and ex13itation of the gas and then
the role played by the radiation energy and pressure The des13ription of the
dierent regimes of radiative sho13ks whi13h depend on the opa13ity of medium
and 13an be 13hara13terized by the presen13e or absen13e of a radiative pre13ursor
will follow Further I will give a brief overview of the literature fo13using on
the generation of radiative sho13ks in the laboratory and I will present the
motivations and obje13tives of my work
21 Rankine-Hugoniot(R-H) relations
Let us 13onsider a one-dimensional sho13k propagating in a gas at rest with
the speed us In the frame asso13iated with the sho13k front the pre-sho13k
(upstream) uid velo13ity is then u1 = - us while behind the sho13k (post-
sho13k or downstream region) this velo13ity is u2 as shown in the Fig 21
Figure 21 S13hemati13 diagram for a 1-D sho13k moving in the x dire13tion The
speed of the sho13k front is us In the frame of referen13e of the sho13k front the pre-
sho13k region has a relative speed u1= -us density ρ1 total pressure P1 temperature
T1 whereas these quantities are respe13tively u2 ρ2 P2 and T2 for the post-sho13k
region
Assuming that a stationary state is rea13hed the 13onditions whi13h deter-
mine the post-sho13k thermodynami13al quantities are the well known Rankine-
21 RANKINE-HUGONIOT(R-H) RELATIONS 7
Hugoniot equations (Ma13quorn Rankine 1870 Hugoniot 1887 1889) De-
du13ed from the Euler equations these 13onservation equations for the mass
momentum and energy are given by
ρ2u2 = ρ1u1 (21)
ρ2u22 + P2 = ρ1u
21 + P1 (22)
u2(ε2 +P2
ρ2) +
1
2u32 = u1(ε1 +
P1
ρ1) +
1
2u31 (23)
where ε1 and ε2 are the spe13i13 internal energies (energy per unit mass)
in the two (upstream and downstream) regions
For an ideal gas of atomi13 mass m =mPA (mP proton mass A atomi13
number) and having an adiabati13 index γ the speed of sound is given by
Cs =
radic
γP
ρ=
radic
γkBT
m(24)
and the Ma13h number (M) of the sho13k is then
M =u1Cs1
(25)
Inserting these the aforementioned form of M and Cs in the equations
21 22 and 23 we 13an derive the 13ompression pressure and temperature
ratios for the system as follows
ρ2ρ1
=u1u2
=M2(γ + 1)
2 +M2(γ minus 1)(26)
P2
P1=
2M2γ minus (γ minus 1)
(γ + 1)(27)
T2
T1=
P2
P1
ρ1ρ2
=(2M2γ minus (γ minus 1))(2 +M2(γ minus 1)
M2(γ + 1)2) (28)
Considering the 13ase of strong sho13ks where M may be 13onsidered to be
≫1 these equations be13ome
ρ2ρ1
=(γ + 1)
(γ minus 1)(29)
T2
T1=
2M2γ(γ minus 1)
(γ + 1)2(210)
kBT2 =2(γ minus 1)
(γ + 1)2mus
2(211)
8 CHAPTER 2 RADIATIVE SHOCK WAVES
For mono-atomi13 gases γ = 53 the 13ompression ratio is estimated to
be 4 and the post-sho13k temperature is
kBT2 =3
16mu1
2 =3
16mPAu1
2(212)
Temperature is thus proportional to the square of the sho13k velo13ity and
to the atomi13 mass Therefore to a13hieve a high temperature in the radiative
sho13k experiment we often use high atomi13 number and mass gases For Xe-
non (A=13129) the post-sho13k temperature is then equal to T2 = 0257u21(eV kms) whi13h for instan13e gives sim 520 eV for a sho13k propagating at
45 kms (see Fig 22)
Figure 22 Proles of temperature (Te = Ti = T ) and mass density (ρ) at 10ns for a sho13k propagating in Xenon at 01 bar (ρ1= 57 times 10
minus4g13m
minus3) with
a speed of sim 45 kms It is to note that the given pressure value are at 298 K
The simulation is performed with the HELIOS 13ode ex13luding the radiation ee13t
and for a polytropi13 gas (γ=53) The initial temperature of Xenon is set to 1
eV and with a negligible (10
minus5WmK) thermal 13ondu13tivity The Ma13h number
is then equal to 40 For this simulation the sho13k is laun13hed by a 100 microm thi13k
Aluminium piston moving at the velo13ity of 33 kms The position x=0 13orresponds
to the position of the XeAl interfa13e at time t=0
21 RANKINE-HUGONIOT(R-H) RELATIONS 9
211 Jump relations for a real gas
In 13ontrast to the dis13ussion on ideal gas presented in the previous se13tion
the 13ase of a real gas is a little more 13ompli13ated It is to note that in this
13ase a part of the kineti13 energy is used to ex13ite and ionize the post-sho13k
gas As a result its temperature is lower than that for the 13ase of ideal gas
In addition the pressure is also modied due to ionisation (Mi13haut et al
2004) as dis13ussed below
Let us now dene j as the ionization stage of the atom (varying between
0 and z) and i to be the atomi13 state in this ionisation stage We shall
denote by Pj amp Pji the fra13tional ionization of the ion and the population
of the atomi13 state i in the jth ionization state respe13tively Then the mean
ionization stage (average 13harge) per atom 13an be written as
z =
zsum
j=0
jPj (213)
and the ex13itation energy (in13luding ionization) per atom and per unit of
mass is
ǫexc =
sumzj=0
sum
i Pji Eji
mpA(214)
where Eji is the energy of the ionisation stage j in the atomi13 stage iThus the total pressure of gas 13an be written as
P = Pi + Pe (215)
where Pi and Pe are the ioni13 and ele13troni13 pressures 13ontributions
Using the 13harge neutrality 13ondition Ne = z Ni this gives
P = Ni(1 + z) kBT = ρkBT
m(1 + z) (216)
The thermal pressure and enthalpy of the real gas are given by
Pth = ρ(1 + z)
mkBT (217)
h =5
2
(1 + z)
mkBT + ǫexc (218)
Next the speed of sound in13luding the ee13t of ionization with the
average 13harge z in the medium is now modied by ionisation
Cs ≃radic
5
3
γ(z + 1)kBT
m(219)
Thus the 13ontinuity relations in the frame moving with the sho13k front
be13ome
ρ2u2 = ρ1u1 (220)
10 CHAPTER 2 RADIATIVE SHOCK WAVES
ρ2u22 + ρ2
kBT2
m(1 + z2) = ρ1u
21 + ρ1
kBT1
m(1 + z1) (221)
ρ2u2
[5
2
(1 + z2)
mkBT2 + ǫexc2 +
u222
]
= ρ1u1
[5
2
(1 + z1)
mkBT1 + ǫexc1 +
u212
]
(222)
Figure 23 Ele13tron temperature (Te) ion temperature (Ti) and mass density
(ρ) proles at 10 ns derived from 1-D simulation for the 13ase of ionisation (no
radiation) The sho13k propagates in Xe gas at 01 bar (ρ1= 57 times 10
minus4g13m
minus3by
sim 45 kms) It is to note that the given pressure value is at 298 K The simulation
is performed with the HELIOS (Te 6= Ti 13ase) using the PROPACEOS equation
of state without any radiation and with a negligible thermal 13ondu13tivity (10
minus05
WmK) The initial temperature in Xenon is set to 1 eV and thus the Ma13h
number is equal to 40 For this simulation the sho13k is laun13hed by a 100 microm thi13k
Aluminium piston moving at the velo13ity of 38 kms The position x=0 13orresponds
to the position of the XeAl interfa13e at time t=0
Similar to the Fig 22 however for the 13ase of a real gas the Fig
23 shows the variation the ele13tron and ion temperatures as well as the
mass density with the distan13e for a sho13k propagating in Xenon at sim 45
kms Now the post-sho13k ele13tron temperature peaks at 22 eV whi13h is
mu13h smaller than the maximum temperature of the ions (510 eV) The
13ompression is now equal to 10 instead of 4 as in the previous 13ase The ions
21 RANKINE-HUGONIOT(R-H) RELATIONS 11
are heated rst in the sho13k through ion-ion 13ollisions This ion temperature
de13reases and the ele13tron temperature in13reases as a 13onsequen13e of the
ele13tron-ion 13ollisions The two temperatures thus be13ome equal Sin13e the
momentum 13hange of the ions require only a few number of kineti13 13ollisions
the thi13kness of the abrupt transition between the pre-sho13k and the post-
sho13k is of the order of a mean free path of the ions
It should be noted that for sho13ks propagating with very high speed
the post-sho13k temperature is so high that the 13ompressed gas is fully ioni-
zed Therefore the ee13ts of ionisation and ex13itation be13ome negligible and
hen13e the 13ompression ratio rea13hes 4 as in the 13ase of the adiabati13 sho13k
212 Ee13t of radiation on sho13k waves
In the previous se13tion the ee13ts of radiation heating and 13ooling have
been negle13ted However the radiation emitted in a strong sho13k will ae13t
its stru13ture These ee13ts 13an be determined using mass momentum and
energy 13onservation equations whi13h now have to in13lude the 13ontributi-
ons of the radiative ux energy and pressure To this purpose and with
some simpli13ations we 13an use the same approa13h than in se13tion 21 still
assuming that the sho13k is stationary
Jump relations with radiation
Let us 13onsider a stationary sho13k propagating in an atomi13 gas far away from
the dis13ontinuity The medium is then opti13ally thi13k and we 13an negle13t
the 13ontribution of the radiative ux Let us also assume that the medium
is at LTE and that the radiation pressure and energy may be des13ribed as
follows
Prad =1
3Erad =
4
3cσT 4 =
1
3aradT
4(223)
where arad is the radiation density 13onstant and T is the sho13k radiation
temperature whi13h we shall suppose to be equal to the ele13tron temperature
Following Mi13haut et al (2004) the new jump 13onditions a13ross the sho13k
dis13ontinuity be13ome
ρ2u2 = ρ1u1 (224)
ρ2u22+ρ2
kBT2
m(1+z2)+
1
3aradT
42 = ρ1u
21+ρ1
kBT1
m(1+z1)+
1
3aradT
41 (225)
ρ2u2
(
5
2
(1 + z2)
mkBT2 + ǫexc2 +
u222
)
+4
3aradT
42 u2 =
ρ1u1
(
5
2
(1 + z1)
mkBT1 + ǫexc1 +
u212
)
+4
3aradT
41 u1 (226)
12 CHAPTER 2 RADIATIVE SHOCK WAVES
(a)
(b)
Figure 24 Figure 14 amp 17 of Mi13haut et al (2004) (a) Compression ρ2ρ1 (full13ir13les thin line) kT2 in eV (inverted full triangles thin line) in Argon (a) Xenon
(b) for initial 13onditions kT1( = 10 eV for Ar and = 01 eV for Xe) ρ1 = 5 times10minus4 gcm3
versus sho13k speed in kms with (full markers full lines) and without
(empty markers dashed lines) radiation The ionization stage lt z gt2 is plotted
with the diamond symbol
21 RANKINE-HUGONIOT(R-H) RELATIONS 13
In the radiative regime the 13ompression ratio ρ2ρ1 approa13hes to the
value of 7 at very high velo13ities (several hundred of kms) whereas it is equal
to 4 when the ee13ts of radiation energy and pressure have been negle13ted
This limit is the same for the 13ases of adiabati13 (Bouquet Teyssier and
Chieze 2000) as well as the real gas (Mi13haut et al 2004) This is due to
the fa13t that at very high velo13ities the medium be13omes fully ionized and
behaves as a perfe13t gas On the 13ontrary in the 13omparatively low velo13ity
regime the ee13t of ionisation and ex13itation dominates the radiation ee13ts
This is illustrated in Fig 24 taken from Mi13haut et al (2004) whi13h reports
the variations of the Ar and Xe 13ompression ratio versus the sho13k speed for
an initial mass density of 5 times 10
minus4g13m
3 At 50 kms the 13ompression
ratio and the temperature are respe13tively equal to (5 32eV) and (11 30
eV) for Argon and Xenon
The 13ompression ratio shows bumps for high velo13ities These bumps
may be attributed to the fa13t that as the temperature of the post-sho13k
in13reases with the velo13ity the atom ionises more and more Thus the ex-
ternal shells in its ioni13 atomi13 stru13ture are su1313essively opened indu13ing
a strong variation in the ex13itation energy
The post-sho13k temperature is very high for the very high-speed sho13ks
In su13h 13ases the 13ompressed gas is fully ionized and the ee13ts of ionisation
and ex13itation remain negligible like in the adiabati13 13ase It should be noted
that below 100 - 200 kms the ee13ts of radiation pressure remain negligible
Thus very energeti13 lasers like LMJ or NIF are required to rea13h a domain
where this ee13t 13an be observed
Thus whereas for the range of velo13ities of 20 - 50 kms 13overed by my
study the ee13ts of radiative energy and pressure 13an be negle13ted this is
not the 13ase for the radiative ux whi13h has been negle13ted in the previous
se13tions and whi13h may be13ome 13omparable to the material energy ux Its
ee13t on the system must be taken into a1313ount as will be shown below
213 Radiative hydrodynami13s equations
To in13lude the 13ontributions of radiation it is 13ompulsory to introdu13e the
relevant terms in the uid equations The propagation of the radiation in the
matter is governed by the pro13esses of absorption and emission of photons
whi13h respe13tively heats and 13ools the medium As the mean free path of
photons is always higher than the mean free path of parti13les the sho13k
stru13ture is more inuen13ed by the radiative transfer than the vis13osity of
the uid
The propagation of radiation in the matter is des13ribed by the radiative
transfer equation
(
1
c
part
partt+
part
parts
)
I(rn ν t) = η(r ν)minus χ(r ν) I(rn ν t) (227)
14 CHAPTER 2 RADIATIVE SHOCK WAVES
where χ(ρ ν T ) and η(ρ ν T ) are respe13tively the mono13hromati13 opa-
13ity and emissivity of the gas
The spe13i13 intensity I(rn ν t) (erg cmminus2 sminus1Hzminus1 srminus1) is the energy
radiated per units of surfa13e area time frequen13y and solid angle in the
dire13tion n
The radiative energy density Erad ux Frad and pressure Prad are re-
spe13tively dened as the zero rst and se13ond moments of the spe13i13 in-
tensity versus the angle after integration over the photon frequen13y ν
Erad =1
c
int ∮
I(rn ν t)dΩdν (228)
Frad =
int ∮
I (rn ν t) cos θ dΩdν (229)
Prad =1
c
int ∮
I (rn ν t) cos2 θ dΩdν (230)
(231)
Taking into a1313ount radiative ee13ts the equations of radiative hydro-
dynami13s in the 13o-moving frame of the sho13k (Mihalas and Mihalas 1999
Drake 2006) 13an be written as
partρ
partt= minusnabla(ρu) (232)
ρ
(
partu
partt+ unablau
)
= minusnabla(P + Prad) (233)
part
partt
(
ρu2
2+ ρǫ+ Erad
)
+nabla
[
ρu
(
ǫ+u2
2+
P
ρ
)
+ (Erad + Prad)u
]
= minusnablaFrad
(234)
As mentioned previously the 13ontributions linked to Prad and Erad are
ee13tive only at very high temperature (ie velo13ity) However for the
radiative sho13ks of interest in this work the 13ontribution of the radiative
ux Frad is non-negligible 13ompared to ρu3 These sho13ks are thus in the
radiation ux dominated regime
Most of the experimental sho13k waves are in this regime These hyd-
rodynami13 equations are strongly non-linear In the present work the one-
dimensional radiative-hydrodynami13 13ode `HELIOS (Ma13Farlane Golovkin
and Woodru 2006) has been employed to derive the physi13al parameters
of the sho13k in various 13onditions as presented in the following se13tion HE-
LIOS will be dis13ussed in detail in the next 13hapter
22 RADIATIVE SHOCK REGIMES 15
22 Radiative sho13k regimes
Radiative sho13ks are strong sho13ks (Ma13h number Mgtgt1) whi13h attain
high temperatures and thus are the sour13e of intense radiation (Mihalas
and Mihalas 1984 Zeldovi13h Zeldovi13h and Raizer 2002 Drake 2006)
Depending on the opa13ity the radiation emitted from the sho13k may be ab-
sorbed by the pre-sho13k region indu13ing its pre-heating Far away from the
dis13ontinuity the stru13ture of the upstream medium is determined by the
absorption On its side the stru13ture of the transition layer of the down-
stream medium 13lose to the dis13ontinuity depends on the balan13e between
the emission (13ooling) and the absorption (heating) Thus the full stru13ture
of the sho13k mainly depends on the variation of the opa13ity χ (13m
2g) The
opti13al depth (τ(s)) measured from the position of the jump is then the
relevant parameter for the absorption It is given by
τ(s) =
int s
sjump
χ(sprime)ρ(sprime) dsprime (235)
where sprime 13hara13terizes the path of the radiation As we are interested in
the mean frequen13y averaged radiative ux the relevant opa13ity here will be
the averaged Rosseland opa13ity whi13h will be dened in due 13ourse
If τ gtgt 1 the regime is referred as opti13ally thi13k while in the opposite
13ase (τ lt 1) it is opti13ally thin Flux dominated radiative sho13ks have
been 13lassied depending on the opti13al depth of upstream and downstream
regimes
An attempt of radiative sho13k 13lassi13ation has been performed by Drake
(Drake (2006)) a1313ording to the opti13al the thi13kness of the upstream and
downstream regimes resulting in the denition of four dierent regimes
namely a thin-thi13k thin-thin thi13k-thi13k thi13k-thin radiative sho13ks
Following this author a thin-thi13k type of radiative sho13k probably exists
only in theory In the 13ase of a thin-thin radiative sho13ks both upstream
and downstream regimes are opti13ally thin and the sho13k radiates away
its energy in both dire13tions This regime of radiative sho13k is present in
various astrophysi13al 13ontexts for instan13e in the internal sho13ks of stellar
jets (Hartigan 2003) It is possible to generate these type of sho13ks in the
laboratory in very low-density gases In thi13k-thi13k radiative sho13ks both
upstream and downstream zones are opti13ally thi13k This regime o1313urs
in the stellar interiors The radiation 13oming from the hot downstream
region heats the 13ooler upstream material and forms a pre13ursor The fourth
regime is the thi13k-thin radiative sho13k whi13h is 13hara13terized by an opti13ally
thi13k downstream and an opti13ally thin upstream region The radiation
from the heated downstream region passes ahead of the sho13k and 13reates a
13ooling layer in the downstream before rea13hing the nal state There is no
absorption of the radiation in the upstream region and therefore 13ontrary to
16 CHAPTER 2 RADIATIVE SHOCK WAVES
the thi13k-thi13k 13ase there is no radiative pre13ursor Sho13ks generated in the
laser experiments are usually between the thi13k-thi13k and thin-thin regimes
221 Typi13al radiative sho13k waves with a pre13ursor
The radiation from the post-sho13ked region just behind the sho13k passes
ahead of the sho13k and generates the pre13ursor However it also leads to a
radiative 13ooling layer downstream that extension depends on the opa13ity
A typi13al stru13ture of su13h sho13ks is presented in M13Clarren et al (2010)
In these 13ases the 13ompression in the 13ooling layer behind the sho13k is
mu13h higher than that in the hydrodynami13 13ase This kind of sho13k is
most 13ommonly generated in the laboratory experiments (Bozier et al 1986
Keiter et al 2002 Reighard et al 2006 Fleury et al 2002) In astrophysi13s
these sho13ks are present for instan13e in supernovae when the blast wave
emerges from the stellar interior (Ensman and Burrows 1992)
A detailed 13lassi13ation of the sho13ks presenting a radiative pre13ursor
is reported in several referen13es (Zeldovi13h Zeldovi13h and Raizer 2002
Ensman 1994 Vaytet et al 2013)
In the 13ase of sub-13riti13al sho13k the temperature just ahead of the sho13k
front is smaller than the post-sho13k temperature Su13h sho13ks are relatively
weak and hen13e the radiative ux transfer from the post-sho13k to the pre-
sho13k is small A piston moving with a 13onstant speed of 12 kms in Xenon
gas at 01 bar is able to generate su13h a sub-13riti13al radiative sho13k The
resulting mass density and ele13tron temperature proles have been presented
at 10 ns in the Fig 25a
The other regime of radiative sho13k is known as super13riti13al in whi13h
the pre-sho13k and the nal post-sho13k temperatures are the same The mass
density and ele13tron temperature proles of super13riti13al sho13k driven by a
13onstant moving piston with speed 45 kms in Xenon at 01 bar are presented
in Fig 25b
The speeds a13hieved in the PALS experiment range between sim 15 - 55
kms for pressure 13omprised between 01- 06 Our simulations indi13ate
that depending on the speeds the regimes 13over either the sub13riti13al or
super13riti13al 13ases
23 Radiative sho13k waves experiments
Radiative sho13k waves have been studied experimentally for more than a
de13ade on high-energy lasers fast magneti13 pin13h ma13hines and pulsed po-
wer fa13ilities Su13h fa13ilities are able to a13hieve strong sho13k 13onditions
(high-temperature high-pressure) over a very small volume and time Va-
rious diagnosti13s are able to re13ord the plasma 13onditions whi13h are then
13ompared with the results of numeri13al simulations The results of these
experiments are also employed to validate numeri13al 13odes some of them
23 RADIATIVE SHOCK WAVES EXPERIMENTS 17
(a)
(b)
Figure 25 Simulated ele13tron temperature (Te) ion temperature (Ti) and mass
density (ρ) proles at 10 ns of a radiative sho13k propagating with a 15 kms (a sub-
13riti13al) and 48 kms (b super13riti13al) The sho13k is propagating in Xenon at 01
bar (ρ = 54 times 10
minus4g13m
minus3 at 298 K) The initial temperature in Xenon is set to
1eV The two Ma13h numbers are respe13tively equal to 13 and 43 The radiation and
ionisation ee13ts are in13luded in the simulation whi13h is performed for two dierent
temperatures (Te 6= Ti) and a negligible thermal 13ondu13tivity (10
minus05WmK) For
this simulation the sho13k is laun13hed by a 100 microm thi13k Aluminium piston moving
at the speed of 12 kms and 45 kms for 13ase (a) and (b) respe13tively The position
x=0 13orresponds to the position of the XeAl interfa13e at time t=0
18 CHAPTER 2 RADIATIVE SHOCK WAVES
being relevant to the astrophysi13s like FLASH (Fryxell et al 2000) and
HERACLES (Gonzaacutelez Audit and Huynh 2007)
231 Laser-driven sho13k experiments
Most of laboratory experiments on radiative sho13ks have been performed
on large-s13ale laser fa13ilities (Bouquet et al 2004 Chaulagain et al 2015
Gonzaacutelez et al 2006 Reighard et al 2006 Stehleacute et al 2010 Doss et al
2009 Drake et al 2011 Diziegravere et al 2011 Stehleacute et al 2012) in noble
gases at pressures between 01 and 1 bar With laser intensities on the
target 13omprised between 1014 and 1015 W13m
2 these experiments allowed
to re13ord sho13k speeds ranging between 40 and 150 kms
Bozier et al (1986) for the rst time experimentally eviden13ed the ge-
neration of a radiative wave propagating ahead of a super13riti13al sho13k in
xenon (pressure up to 6 bars) at the CEAs Limeil laser fa13ilities (irradian13e
of 5times 10
14W13m
minus2) In this experiment the laser beam (wavelength of 106
microm duration 1 ns) of 40 J energy was fo13used on an aluminum foil a13ting
as a piston A sho13k of speed sim 50 kms was produ13ed in the gas A visible
streak 13amera was used to re13ord the pre13ursor emissivity over 10 ns
Later Keiter et al (2002) performed a similar experiment at the OMEGA
laser fa13ility (25 kJ 1ns fo13al spot diameter 600 microm) still in a planar
geometry The sho13k propagated here in a low-density SiO2 aerogel foam
(ρ sim 96 mg13m
3) In this experiment X-ray absorption spe13tros13opi13 data
were re13orded to qualitatively predi13t the temperature of the pre13ursor
In the year 2000 a new radiative sho13k experiment was performed in
Xenon at 02 bar at the LULI laser fa13ility (sim 100 J λ = 053 microm 4-
6times 10
13W13m
minus2) (Fleury et al (2002) Bouquet et al (2004)) This expe-
riment used a three-layered piston 13omposed of a thin polystyrene a13ting
as ablator (2 microm thi13k) a titanium foil (3microm) a13ting as x-ray shield and a
layer of a polyethylene foam (25 microm) as an a1313elerator Further a VISAR
(velo13ity interferometer system for any ree13tor) and a Ma13h - Zehnder in-
terferometer were used as a diagnosti13s to measure the front sho13k speed and
the pre13ursor ele13tron density Two streak 13ameras re13orded the sho13k pro-
pagation in longitudinal and transverse dire13tions This was the rst attempt
to estimate ele13tron density within the pre13ursor Their measurements indi-
13ate an ele13tron density ranging between 1018minus1020cm3 The experimental
results were in qualitative agreements with 1D numeri13al MULTI and FCI
simulations However the speed of the pre13ursor in these 1D simulations
was too large This was then explained later when multidimensional ee13ts
were in13luded in the simulations (Leygna13 et al 2006 Gonzaacutelez Audit and
Stehleacute 2009)
Complementary experiments were then performed by Gonzaacutelez et al
(2006) at the PALS laser fa13ility (60 J 035 ns 438 nm spot diameter 600microm
sim 7times 10
13W13m
minus2) in Xe gas The goal was to study the sho13ks over longer
23 RADIATIVE SHOCK WAVES EXPERIMENTS 19
times and to inspe13t the ee13ts of the walls albedo A doubled layered pis-
ton made of 10microm polystyrene and 05 microm gold was used The pre13ursor
edge was re13orded by shadowgraphy using a visible streak 13amera The ex-
perimental results were 13ompared with 2D numeri13al simulations performed
with HERACLES (Gonzaacutelez Audit and Huynh 2007) This 13omparison
allowed quantifying the albedo of the walls It also showed that the radia-
tive losses at the walls lead to a small 13urvature of the ionization front and
to a redu13tion of its longitudinal extension (Leygna13 et al 2006 Gonzaacutelez
Audit and Stehleacute 2009)
In parallel another experiment was performed at the OMEGA laser fa-
13ility (035microm spot sim 700 - 800microm sim 2times 10
15W13m
minus2) by Reighardt et
al (Reighard et al (2006) Reighard (2007)) The sho13k was imaged by an
X-ray ba13k-lighter at dierent times whi13h enabled to inspe13t for the rst
time the shape of the sho13k front At higher speeds (sim 100 kms) X-ray
radiography pointed out a 13ollapse of the post-sho13k (Reighard et al 2006)
whi13h was attributed to the radiation losses trough the sho13k front For
these high-speed 13onditions the wall heating leads to the development of se-
13ondary wall sho13ks intera13ting with the primary sho13k (Doss et al 2009)
Su13h wall sho13ks have not yet been observed at lower speeds
Subsequently an experiment by Stehleacute et al (2010) was performed at
PALS laser fa13ility with the similar laser parameters as taken in Gonzalez
et al (2006) In this experiment a Ma13h-Zehnder interferometer and a
streak 13amera were used to re13ord the sho13k propagation over 50 ns Time-
integrated XUV spe13tros13opi13 analysis between 16 and 22 nm of sho13k was
performed The experiment showed that of a quasi-stationary regime was
rea13hed after 20 ns
The rst experimental study in Argon gas was reported in 2012 by Vis13o
et al (2012) at OMEGA fa13ility (035microm spotsim 600 microm sim 7times 10
14W13m
minus2)
The ele13tron temperatures in the radiative pre13ursor and sho13k-front were
measured to be 34 eV and 60 eV respe13tively using X-ray Thomson s13atte-
ring diagnosti13
In addition a new experiment at PALS (Jungwirth et al 2001) was
presented by Chaulagain et al (2015) using for the rst time a Zn x-ray
laser for XUV imaging at 21 nm This diagnosti13 allowed imaging both the
pre13ursor and post-sho13k stru13ture of radiative sho13ks in Xe gas at 03 bar
In this experiment where the sho13k wave did not ll the tube 2D ee13ts
be13ame pronoun13ed The experimental results were su1313essfully 13ompared
with the results of 2D ARWEN simulations (Cotelo et al 2015)
A summary of these dierent experiments and their prin13ipal diagnosti13s
is presented in table 21
20
CHAPTER2RADIATIVESHOCKWAVES
Experiment Laser fa13ility I (times 1013) λ amp τ ρ0 u Prin13ipal diagnosti13s
W13m
minus2 microm ns mg 13m
minus3
km s
minus1
Bozier et al (1986) CEAs Limeil 20-50 0351 1 10 -30 50-80 Visible streak Camera
Fleury et al (2002) LULI 5 053 05 1 60-65 Visible streak Camera
Visible interferometry
VISAR
Keiter et al (2002) OMEGA 35- 85 0351 1 5 -15
lowast
100 x-ray spe13tros13opy
Gonzaacutelez et al (2006) PALS 10-15 0438 035 1 65 Visible streak 13amera
Visible interferometry
Reighard (2007) OMEGA 70-100 035 1 6 100-140 x-ray ba13k-lighting
VISAR
Stehleacute et al (2010) PALS 10-20 1315 035 1 60 Visible Interferometry
XUV spe13trometer
Stehleacute et al (2012) PALS 10-20 1315 035 15 50-55 XUV imaging
XUV fast Si diodes
Vis13o et al (2012) OMEGA 70 035 1 196
lowastlowast
100-150 x-ray Thomson S13atte-
ring
Chaulagain et al (2015) PALS 10-20 1315 035 15 50 XUV fast Si diodes
Table 21 Summary of the radiative sho13ks experiments and out13omes in dierent laser installations All the experiments are performed
in Xe ex13ept Keiter et al (2002) and Vis13o et al (2012) In these two experiments the rst uses SiO
lowast
2 and the se13ond uses Ar
lowastlowast
as a
sho13ked medium
23 RADIATIVE SHOCK WAVES EXPERIMENTS 21
Laser driven radiative sho13ks Goal of my thesis
All these experimental studies have been fo13used on the 13ase of single ra-
diative sho13ks propagating in a tube However in astrophysi13al 13onditions
radiative sho13ks often intera13t with a denser medium leading to the deve-
lopment of ree13ted and transmitted sho13ks A few representative examples
of su13h phenomena are the intera13tion of supernovae remnants with dense
mole13ular 13louds (Gonzaacutelez Audit and Stehleacute 2009) the a1313retion sho13ks
on the photosphere of T-Tauri stars (Orlando et al 2013) and the bow
sho13ks at the head of stellar jets (Hartigan 1989 Raga et al 1999) The
13ollision (or the intera13tion) of two radiative sho13k waves is obviously a rare
astrophysi13al event and the template 13ase of supernova remnant DEM L316
(see Fig 1 of Williams et al (1997)) is still the subje13t of debates (Williams
et al 2005 Toledo-Roy et al 2009 Velarde et al 2006) as the observation
of these two dierent sho13ks 13an be also interpreted as the superposition of
two blast waves in the eld of view of the teles13ope In this regard the de-
velopment of dedi13ated laboratory experiments to the study of propagation
and intera13tion of 13ounter-propagating sho13k waves is important as a tool to
13hara13terize su13h events through their spe13i13 signatures
In this thesis I will present the results of experiments performed at the
Prague Asterix Laser System (PALS) fa13ility (Jungwirth et al (2001)) on
the study of the intera13tion of two radiative sho13k waves The experimental
obje13tives were
To generate two sho13ks of dierent speeds propagating in opposite
dire13tions in the tube lled with Xenon gas at low pressure (lt 1 bar)
with speeds 13omprised between 12 and 55 kms
To re13ord the ele13tron density by using Ma13h-Zehnder Interferome-
ter over a duration 13omprised between 50 and 200 ns This allowed
studying the ee13t of pre13ursor intera13tion on to the other one
To determine the temperature and the ion 13harge of gas using a time
and spa13e integrated XUV emission spe13tros13opy
The experimental results were interpreted with the help of one-dimensional
simulations using the Lagrangian radiation hydrodynami13 13ode HELIOS
232 Ele13tromagneti13ally laun13hed sho13k by a 13ompa13t pul-
sed power devi13e
As dis13ussed in the previous se13tion most of the plasma laboratory astrop-
hysi13s studies are mainly performed on large-s13ale laser fa13ilities addressing
pure hydrodynami13 radiative sho13ks at very high speed (50 - 150 kms)
and moderate pressure (01 - 1 bar) Complementary to laser experiments
22 CHAPTER 2 RADIATIVE SHOCK WAVES
dis13harge produ13ed plasma has also been a well-established method for ge-
nerating the strong sho13ks (Kondo et al 2008 2009ab Larour et al 2015)
Su13h devi13e presents a high exibility and a larger repetition rate whi13h al-
lows a deeper understanding of sho13k physi13s in 13onditions of astrophysi13al
interest for instan13e in the 13ontext of stellar jets
In prin13iple a 13oaxial ele13trode pair lled with gas followed by an intense
ele13tri13al surfa13e dis13harge generates a thin layer of plasma (plasma sheath)
at the bottom of the ele13trode Due to the high rising 13urrent in the plasma
layer a magneti13 pressure is produ13ed whi13h a1313elerates the plasma layer
also termed as plasma fo13us (Gonzalez Brollo and Clausse 2009) During
this motion the plasma sheath a1313retes a noti13eable part of the ba13kground
gas Experiments performed by Serban and Lee (1995) in the light gasses
H2 D2 have shown fast axial sho13k waves (100 kms) but the plasma sheath
was annular
Later on 13ompa13t pulsed power ele13tri13 generators were able to laun13h
astrophysi13ally relevant strong sho13ks in low-pressure noble gases (Kondo
et al 2006) These authors reported sho13k of speed 45 kms using 13oni13al
13oaxial ele13trodes and Xenon gas (006 - 11 mbar) The ion - ele13tron energy
transfer and ionization relaxation pro13ess as also the radiation ee13t were
investigated by Kondo et al (2008 2009ab)
My work on ele13tromagneti13ally laun13hed sho13k experiment
My work fo13using on the ele13tromagneti13ally laun13hed strong sho13ks rele-
vant for astrophysi13s aimed to design and to optimize an ele13tromagneti13
generator to be able to produ13e strong sho13ks in noble gasses at low pres-
sures of few mbar The ele13tromagneti13 generator is a powerful a1313elerating
devi13e whi13h eje13ts a quasi-planar plasma sheath out of a set of 13oaxial 13o-
ni13al ele13trodes where a pulsed 100-kA 13urrent is passing
Due to the ele13tri13al dis13harge in the gas a hot and fast moving plasma
is generated whi13h travels along the ele13trodes The high-intensity 13urrents
produ13e a magneti13 eld of several Teslas whi13h a13ts as a piston a1313elerating
an annular plasma sheath and driving a strong sho13k towards the top of the
13one I have employed a simple model to optimize the operation parame-
ters With su13h optimization our ele13tromagneti13 generator should produ13e
strong sho13ks over few nanose13onds
Our preliminary experiments show that the generator is 13apable of laun-
13hing supersoni13 sho13ks in Argon in the form of a thin plasma layer with
the speed of sim 1 - 30 kms Three-dimensional MHD simulation performed
by A Ciardi is 13onsistent with the rst observations This is presented in
detail in the 13hapter 6 of the thesis
Chapter 3
1D Simulations
Contents
31 HELIOS 23
32 LTE approximation 24
321 Mean opa13ity 25
322 Equation of State 27
33 Single radiative sho13k waves 27
34 Intera13ting radiative sho13k waves 40
35 Summary 44
Radiative sho13ks are present in various astrophysi13al pro13esses implying
extreme 13onditions Laboratory experiments then allow investigating the un-
derlying physi13al pro13esses whi13h take pla13e in these 13onditions and whi13h
are not observable from the Earth due to a la13k of spatial resolution of the te-
les13opes instruments However experiments are not always straightforward
to interpret and the help of numeri13al simulations be13omes pre13ious
In this 13hapter I will present with the help of 1D radiative hydrodynami13
simulations a brief overview of the physi13s of isolated radiative sho13ks and
of two intera13ting radiative 13ounter-propagating sho13k waves
31 HELIOS
Dierent numeri13al approa13hes are used for the simulation of laboratory
plasmas Some of them use xed grids Other introdu13e the possibility of
renement (on the y) in some meshes (Adaptative Mesh Renement) to
maintain a good resolution in the interesting parts of the plasma Another
approa13h is based on meshes following the uid (Lagrangian des13ription)
whi13h means that the meshes move with the uid and no matter may go
through a mesh to another The form of the equations dier in this 13ase (see
for instan13e Orban et al (2013))
HELIOS is su13h a one-dimensional Lagrangian radiation hydrodynami13
13ommer13ial 13ode (Ma13Farlane Golovkin and Woodru 2006) and I have
used it to simulate our experiment of intera13ting sho13k waves This 13ode
is easy to handle and has the presently interesting 13apability to be able to
simulate the 13ase of two sho13k waves propagating in opposite dire13tions and
laun13hed by two dierent lasers HELIOS may des13ribe non-LTE plasmas as
23
24 CHAPTER 3 1D SIMULATIONS
it in13ludes an in-line 13ollisional radiative (HELIOS-CR) model for 13omputing
non-LTE atomi13 level populations at ea13h time step of the hydrodynami13s
simulation However I used HELIOS in the LTE approximation whi13h is
appropriate to our experimental 13onditions (Rodriguez et al 2011) as will
be dis13ussed in the next se13tion
The 13ode solves the uid equations of motion using the pressure 13ontri-
butions of ele13trons ions and radiation It may des13ribe the ele13trons and
ions as two intera13ting uids at respe13tive temperatures Te and Ti Ther-
mal 13ondu13tion is des13ribed within a ux-limited ele13tron 13ondu13tion model
based on the Spitzer 13ondu13tivity (Burhop and Spitzer 1957) and the laser
energy deposition is 13omputed with an inverse Bremsstrahlung model ()
The radiation emission and absorption terms are introdu13ed in the equa-
tion of energy for the ele13trons and in the radiation transport equations
One of the methods whi13h are proposed is the ux-limited multi-group
radiation diusion model (Gonzaacutelez et al 2015) where the radiative ux is
proportional to the gradient of the radiative energy and is inversely propor-
tional to the Rosseland opa13ity (Dobbs-Dixon Cumming and Lin 2010)
The expression is pondered by a ux-limited diusion 13oe13ient in order
to obtain the good opti13ally thin limit This 13oe13ient follows the Larsen
expression des13ribed in Olson Auer and Hall (2000) The se13ond method is
a (time independent) multi-angle short-13hara13teristi13s s13heme In our 13ase
where radiation and hydrodynami13s are strongly 13oupled and thus the ra-
diation eld varies rapidly we used the diusion model together with LTE
Plan13k and Rosseland multi-groups opa13ities
In addition our version of HELIOS uses the PROPACEOS (Ma13Farlane
Golovkin and Woodru 2006) Equation of State (EOS) and multi-group
opa13ity tables whi13h are generated by the 13ode
1
and it is not possible to
in13lude any other opa13ity or EOS table
32 Lo13al Thermodynami13 Equilibrium
The knowledge of the thermodynami13al 13onditions is required for an ade-
quate simulation of the plasma des13ription
The Lo13al Thermodynami13 Equilibrium (LTE) regime is valid at large
ele13tron densities In that 13ase the 13ollisions between the ele13trons and the
ions and atoms are so frequent that they maintain at ea13h time a steady-
state populations of ions Moreover within a given ioni13 stage the popu-
lations of the dierent energy levels follow the Boltzmann equilibrium and
the populations of the dierent ions the Saha-Boltzmann equation (Fadeyev
and Gillet 2001)
On the 13ontrary in the 13ase of Non Lo13al Thermodynami13 Equilibrium
(NLTE) regime one has to take into a1313ount the ex13itationdeex13itation
1
see http wwwprismminus cscomSoftwarePROPACEOS
32 LTE APPROXIMATION 25
indu13ed by the ele13tron 13ollisions and various radiative pro13esses This leads
to a set of dierential time dependent equations (rate equations) whi13h has
to be solved through and adequate Collisional Radiative model (CR) (Kunze
2009)
Therefore it is 13ru13ial to determine if the regime is LTE or NLTE A
13riterium is given by Griem (2005)
In our experiment the mass density ranges between sim 10
minus4and 10
minus1
g13m
3 while the temperature values are 13omprised between andsim 0 to 50 eV
The pre13ise 13al13ulations performed by Rodriguez et al (Fig2 of Rodriguez
et al (2011)) for Xenon indi13ate that the thermodynami13al 13onditions of
our radiative sho13k experiments 13orrespond to the LTE regime
321 Mean opa13ity
The opa13ity is a fundamental ingredient of the radiative transfer equations
whi13h provide the spe13i13 intensity and its moments (radiative ux energy
and pressure) These last ones enter in the uid equations (see se13tion 213
in 13hapter 2) The 13oupling of the uid equations with the radiative transfer
equation is demanding in terms of 13omputational resour13es and most of the
13odes use simplied radiation transport equations whi13h require the know-
ledge of adequate frequen13y averaged opa13ity 13oe13ients The form of the
average of the opa13ity is not obvious as the average has to be done on the
radiative ux or energy whi13h are unknown before the 13omputation Then
the 13ommonly used opa13ities are the Plan13k and Rosseland mean opa13ities
(Mihalas 1978 Seaton 1987)
The Rosseland mean opa13ity (κR in 13m
minus1and χR = κRρ in 13m
2g) isderived by 13onsidering harmoni13 weighting over the temperature derivative
of the Plan13k fun13tion It gives the 13orre13t radiative ux in the opti13ally
thi13k diusion limit The average is performed over the total mono13hromati13
opa13ity κtotν = κabsν +κscatν (in 13m
minus1) where in the present 13ase the s13attering
13ontribution 13omes from the ele13tron Thomson s13attering
1
κR=
int
infin
01
κtotν
dBν
dT dνint
infin
0dBν
dT dν(31)
The Plan13k mean opa13ity (κP in 13m
minus1and χP = κP ρ in 13m
2g) employs
the normalized Plan13k bla13kbody energy density distribution as a weighting
fa13tor It will give the 13orre13t value for the integrated thermal LTE emission
of an opti13ally thin plasma and is expressed as
κP =
int
infin
0 κabsν Bνdνint
infin
0 Bνdν(32)
In the multi-groups des13ription the radiation transport takes into a13-
13ount the propagation of radiation in N dierent wavelength bands (νk minusνk+1 k = 1 N) Hen13e for instan13e the medium may be transparent for
26 CHAPTER 3 1D SIMULATIONS
some wavelengths and opaque for other ones The radiation transport equa-
tions split into N 13ontributions ea13h of them being asso13iated to adequate
opa13ity 13oe13ients The expression of the 13oe13ients is similar to the pre-
vious grey 13ase (N =1) ex13ept that the integrals are performed between
the boundary frequen13ies of the group
In HELIOS the mono13hromati13 opa13ities κν are 13omputed for the die-
rent radiation groups using the ATBASE atomi13 suite of 13odes Energy levels
and other atomi13 data relevant for bound-bound and bound-free transitions
are 13omputed within a 13onguration intera13tion model using Hartree-Fo13k
wave-fun13tions (Ma13Farlane Golovkin and Woodru 2006)
Dividing the opa13ity by the mass density redu13es the variation of this
quantity with ρ The literature then often uses the mass absorption 13oe-
13ient (also termed as opa13ity) whi13h is dened as
χν = κνρ (33)
Figure 31 PROPACEOS Mono13hromati13 opa13ity versus the photon energy in
eV of Xenon at 10 eV and at two ele13tron densities respe13tively equal to 1018 and
1020 13mminus3
An example of the variations of the PROPACEOS opa13ity χν versus the
frequen13y in eV is given in Fig 310a for Xenon at 10 eV and two ele13tron
densities respe13tively equal to 1018 and 1020 13m
minus3 whi13h do 13orrespond to
typi13al 13onditions of the pre13ursor and post-sho13k of our laser generated
radiative sho13ks
33 SINGLE RADIATIVE SHOCK WAVES 27
322 Equation of State
The equation of state (EOS) denes the dependen13e of the pressure ionisa-
tion internal energy with the mass density and temperature Several models
do exist in the literature
Our version of HELIOS for LTE plasmas may handle polytropi13 or PRO-
PACEOS equation of state Unfortunately there are not many details about
this last method The authors of the 13ode mention that it uses a QEOS-type
model (More et al 1988) in the strong 13oupling regime (ie at relatively
high densities and low temperatures) and an isolated atom model whi13h
uses a detailed atomi13 stru13ture modeling in the weak 13oupling region The
properties between the weak and strong 13oupling regimes are said to be obtai-
ned by interpolating in a thermodynami13ally 13onsistent manner This EOS
in13ludes the 13ontributions from the translation of ions and ele13trons ele13-
tron degenera13y atom ionization and ex13itation and Coulomb intera13tions
(Debye - Hu13kel 13orre13tion)
Comparisons on spe13i13 13ases have been performed by the authors of the
13ode with Los Alamos SESAME data for CH Al and Au showing a good
agreement when using these two equations of state
In this 13hapter HELIOS 13ode will then be employed to simulate the
detailed 13hara13teristi13s of single and 13ounter-propagating radiative sho13ks
driven by one or two lasers in the 13onditions of our experiments (Chapter 5)
The target 13ell (4-mm length) is lled with Xenon gas at 01 bar This valueof pressure is representative and 13orrespond to standard ambient tempera-
ture ie 298 K and not the initial temperature 13onsidered in the simulations
This terminology is maintained throughout the thesis Two lasers are inte-
ra13ting at both ends with two gold-13oated `CH foils 11microm CH and 06 micromAu) ea13h of them resulting in a piston of total thi13kness 116 microm 13losing the
13ell
33 Single radiative sho13k waves
Typi13al simulations of the radiative sho13ks espe13ially multidimensional ones
whi13h are time-13onsuming use a frequen13y independent (grey or one group)
opa13ities However then the opa13ity of the gas under investigation presents
strong variations with the frequen13y a multi-group approa13h may be suitable
(Vaytet et al 2011 2013) When the number N of groups tends to innity
ea13h group opa13ity κkR and κkP tends to the lo13al mono13hromati13 opa13ity κν whi13h solves the question of the adequate opa13ity average
In the following I will investigate the inuen13e of the opa13ity and the
number of groups on the stru13ture of a sho13k wave propagating at sim 50 kms
in Xenon at 01 bar To this purpose I have performed HELIOS simulations
for a laser of uen13e 28000 Jcm2whi13h is able to drive a sho13k with the
relevant velo13ity In addition I have also analyzed the ee13ts of taking into
28 CHAPTER 3 1D SIMULATIONS
a1313ount the dierent ion and ele13tron temperatures
Referen13e simulation (1 group Te = Ti = T )
A result of the simulation is reported at 10 ns in the Fig 32 whi13h reports
the variations of ele13tron density (Ne) temperature (T ) mean 13harge (Z)and mass density (ρ) The initial 13onditions are 45 times 10
minus4g13m
3and T =
1 eV
The sho13k front is lo13ated at sim 0047 13m (speed of 47 kms) At this
position the temperature peaks at 22 eV The bla13k verti13al dotted line shows
the position of the interfa13e between the piston and the ba13king Xenon gas
This interfa13e is lo13ated at sim 20 microm from to sho13k front The post-sho13k gas
mass density pressure ele13tron density and ion 13harge are respe13tively equal
to 002 g13m
3 11times 10
4bars 52times 10
2013m
3and 56 The high 13ompression
is due to ionisation and radiation 13ooling (13hapter 2) Its temperature of 12
eV is the same than in the pre13ursor where it is 13onstant over the 2 mm of the
simulation The pre13ursor ele13tron density and mean 13harge are respe13tively
equal to 18 times 10
1913m
minus3and 7 The pre13ursor is not 13ompressed
The variations of the Rosseland opa13ities (13m
2g
minus1and 13m
minus1) with the
distan13e are reported in the Fig 33 The Rosseland opa13ity is low in the
pre13ursor Hen13e 600 13m
2g
minus113orresponds to 05 13m
minus1 whi13h means that
an opti13al depth of 1 is rea13hed at 2 13m from the sho13k front This explains
the at prole of the pre13ursor With a typi13al value in the post-sho13k of
2577 in 13m
2g
minus1 this opa13ity is now 53 13m
minus1 and an opti13al depth of 1 is
here rea13hed at 200 microm from the sho13k front (whi13h is mu13h smaller than
the post sho13k extension)
This simulation was performed using the Spitzer thermal 13ondu13tivity
To 13he13k the 13ondu13tivity ee13t I performed another simulation with a
negligible thermal 13ondu13tivity (ie 10
minus12WmK) for the Xenon layers
only The results are reported in the Fig 34 The interfa13e between the
piston and the ba13king Xenon gas is still 13lose to the sho13k front Ex13ept
for the peak of temperature whi13h is higher (sim 29 eV instead of 22 eV) and
thiner the sho13k stru13ture is similar to the previous 13ase
Two uids with dierent temperatures Te and Ti (1 group)
In the previous se13tion HELIOS simulations have been 13arried out for the
13ase of equal ion and ele13tron temperatures I will now investigate the ee13t
of su13h approximation on the simulated results employing 1) Spitzer and 2)
negligible 13onstant thermal 13ondu13tivity in the Xenon layers It is important
to note that this two uids approximation 13an not be restri13ted to the Xenon
layers only and thus it is valid from the piston (CHAu) up to the Xenon
gas Thus the dynami13s and 13onditions of the piston are also modied whi13h
may slightly modify the sho13k velo13ity
33 SINGLE RADIATIVE SHOCK WAVES 29
(a)
(b)
Figure 32 Mass density and temperature (a) ele13tron density and mean 13harge
(b) at 10 ns for a radiative sho13k of speed sim 47 kms in Xenon at 01 bar The
verti13al dotted bla13k lines show the position of the interfa13e between piston and
ba13king Xenon gas Zero at x-axis is the interfa13e of piston and Xenon at time zero
Spitzer thermal 13ondu13tivity has been used in the simulation
30 CHAPTER 3 1D SIMULATIONS
Figure 33 Mean Rosseland opa13ity (in 13m
2g
minus1) and opa13ity (13m
minus1) at 10 ns
for a radiative sho13k of speed sim 47 kms in Xenon at 01 bar The verti13al dotted
bla13k lines show the position of the interfa13e between piston and ba13king Xenon gas
Zero at x-axis is the interfa13e of piston and Xenon at time zero Spitzer thermal
13ondu13tivity has been used in the simulation
33 SINGLE RADIATIVE SHOCK WAVES 31
(a)
(b)
Figure 34 (a) Mass density and temperature (b) ele13tron density and mean
13harge at 10 ns for a radiative sho13k of speed sim 47 kms in Xenon at 01 bar The
verti13al dotted bla13k lines show the position of the interfa13e between piston and
ba13king Xenon gas Zero at x-axis is the interfa13e of piston and Xenon at time zero
A negligible 13onstant thermal 13ondu13tivity (10
minus12WmK) have been used in the
simulation for Xenon
32 CHAPTER 3 1D SIMULATIONS
(a)
(b)
Figure 35 (a) Ele13tron density and ele13tron temperature (b) Mass density and
mean 13harge at 10 ns for a radiative sho13k of speed sim 48 kms in Xenon at 01
bar for the 13ase when the ele13tron and ion temperature are dierent The verti13al
dotted bla13k lines show the position of the interfa13e between piston and ba13king
Xenon gas Zero at x-axis 13orresponds to the position of the interfa13e between the
piston and Xenon at time zero The Spitzer thermal 13ondu13tivity has been used in
this simulation
33 SINGLE RADIATIVE SHOCK WAVES 33
The Fig 35a reports the sho13k stru13ture at 10 ns for the 13ase of Spitzer
thermal 13ondu13tivity The sho13k is lo13ated at 0048 13m The ion temperature
peaks at 53 eV and the ele13tron temperature peaks at the same value of 20
eV than previously Ex13ept for the ion temperature the sho13k stru13ture is
not signi13antly 13hanged 13ompared to the 13ase of equal temperatures
Let us now inspe13t the role played by the thermal 13ondu13tivity (see Fig
36) In the 13ase of a negligible thermal 13ondu13tivity of (10
minus12WmK) for
the Xenon layers only the ele13tron temperature (and as a 13onsequen13e the
mean 13harge) peaks at higher value 25 eV (instead of 20 eV with the Spitzer
13ondu13tivity) and the ion temperature rea13hes to 405 eV (instead of 53 eV)
The rest of the sho13k is not ae13ted by this 13hange in the 13ondu13tivity This
13onrms that the width and height of the ion peak temperature are a dire13t
13onsequen13e of the thermal 13ondu13tivity
Several groups for the opa13ity one temperature
As mentioned earlier the number of frequen13y groups inuen13es the dyn-
ami13s and morphology of radiative sho13ks A set of simulations has been
performed 13onsidering (only for Xenon) dierent numbers (N) of frequen13y
group varying between 1 and 100 for Te = Ti The distribution in energy
of the groups is logarithmi13 and the grid is set up with approximately 85
of the groups having photon energies between 01 eV and 3 KeV while the
remaining 15 lie between 3 KeV and 1 MeV
The Figures 37 (a) and (b) show the variations of the ele13tron tempe-
rature with the number of groups (from 1 to 100) for the pre13ursor (a) and
the front (b) In13reasing the number of groups ae13ts mostly the pre13ursor
A similar behavior is found in Vaytet et al (2013)
It should be noted that this multigroup opa13ity treatment is applied
to all the layers (ex13ept gold only one group opa13ity is provided by the
13ompany) in the simulation This 13ould ae13t slightly the dynami13s of the
piston Con13erning the pre13ursor at this time of the simulation and taking
an a1313ountable variation in temperature of 10 the 13onvergen13e is a13hieved
after 20 groups
Variation with the opa13ity
In order to estimate the ee13t of the opa13ity I have performed several simu-
lations by in13reasing the Plan13k and Rosseland opa13ities in Xenon (Stehleacute
et al 2014b) using a 13ommon multiplier ranging between 1 to 40 Su13h
pro13edure is often used to redu13e the gap between the available and more
sophisti13ated opa13ities To ex13lude the impa13ts of the radiation 13oming from
the post-sho13k to the piston layers the opa13ity of the piston (CH and Au)
layers have been set to zero
I present in Fig 38 the results (proles of temperature mass density and
34 CHAPTER 3 1D SIMULATIONS
(a)
(b)
Figure 36 (a) Mass density and ele13tron temperature (b) ele13tron density and
mean 13harge at 10 ns for a radiative sho13k of speed sim 45 kms in Xenon at 01 bar
for the 13ase when the ele13tron and ion temperature are dierent The verti13al dotted
bla13k lines show the position of the interfa13e between piston and ba13king Xenon
gas Zero at x-axis 13orresponds to the position of the interfa13e between the piston
and Xenon at time zero In this simulation value of thermal 13ondu13tivity (10
minus12
WmK) is negligible for Xenon The peak value of ion temperature is found to be
405 eV whereas theoreti13ally this value is expe13ted to be sim 600 eV It is possible
to a13hieve the expe13ted value by in13reasing the resolution of the simulation
33 SINGLE RADIATIVE SHOCK WAVES 35
Figure 37 Ele13tron temperature proles for various numbers of frequen13y groups
N = 1 20 50 60 70 90 and 100
mean 13harge) of four 13ases 13orresponding to an opa13ity multiplier equal to
1 10 30 and 40 The ele13tron and ion temperatures are taken to be equal
In13reasing the opa13ity in13reases the absorption in the pre13ursor and thus
de13reases its length as the photons emitted by the sho13k are more rapidly
absorbed by the 13old pre-sho13k Taking as a referen13e the previous value of
the Rosseland opa13ity of 600 13m
2g
minus1and a multipli13ative fa13tor of 40 the
opti13al depth of 1 will now be rea13hed at 005 13m (instead of 2 13m without
any multiplying fa13tor)
The temperature peak value at the sho13k front also de13reases I do believe
that this 13an be attributed to the in13rease of the radiative 13ooling of the peak
of temperature with the multiplying fa13tor
I adopted here an oversimplied method in the sele13tion of the groups
boundaries A more realisti13 one would be to take into a1313ount the variation
of the opa13ity in the dierent layers with the wavelength In the 13ase of
xenon for instan13e it would be then more adapted to rene the des13ription
of the opa13ity between 5 and 100 eV (see Figure 310a)
Opa13ities 13omparison
The previous study illustrates the role played by the opa13ity for the radi-
ative sho13ks waves with a pre13ursor It seams then logi13al to 13ompare the
36 CHAPTER 3 1D SIMULATIONS
(a)
(b)
(13)
Figure 38 (a) Ele13tron temperature (a) mean 13harge (b) and mass density (13)
for four (1 10 30 and 40) multipliers of the Xenon opa13ity at 10 ns
33 SINGLE RADIATIVE SHOCK WAVES 37
PROPACEOS opa13ity whi13h is used by HELIOS with more sophisti13ated
13al13ulations
The Figures 39a and 39b report the variations of the Plan13k and Ros-
seland grey opa13ities 13omputed for the density of the pre13ursor ρ = 15 times10
minus3g13m
3by two dierent 13odes PROPACEOS (Ma13Farlane Golovkin
and Woodru 2006) and BiGBART (de la Varga et al 2011 Ogando and
Velarde 2001) PROPACEOS opa13ities and EOS used in HELIOS simula-
tions 13omputes frequen13y-dependent opa13ities along with spe13i13 internal
energies and pressures The atomi13 13ode BiGBART able to 13al13ulate two
type of opa13ities dependent on atomi13 stru13ture and frequen13y dependent
The opa13ities 13omputed by Rodriacuteguez et al (2015) are also reported for 13om-
parison The two last 13al13ulations use the FAC (Flexible Atomi13 Code) (Gu
2008) for the 13al13ulation of the atomi13 data
We see that for a temperature equal to 10 eV the Rosseland opa13ity is
equal respe13tively to 800 15000 and 8000 g13m
2 whereas the Plan13k opa13ity
is equal to 18000 65000 370000 g13m
2 This means that the PROPACEOS
opa13ity whi13h is used in HELIOS is smaller by more than one order of
magnitude than the two last opa13ities As it is not possible to 13hange the
opa13ity model within HELIOS we then de13ided to use the PROPACEOS
opa13ity with a multipli13ative fa13tor whi13h we have set equal to 20
The dis13repan13ies in the average opa13ity are a 13onsequen13e of important
dieren13es in the mono13hromati13 opa13ities This is illustrated in the Fig
310 whi13h 13ompares the mono13hromati13 opa13ities given by PROPACEOS
and Rodriacuteguez et al (2015) (see Fig 16 of Rodriacuteguez et al (2015)) at 15
eV and 15 times 10
minus3g13m
minus3) This gure indi13ates a noti13eable dieren13e
between 5 and 150 eV with more bb transitions for the last method This
explains the dieren13es observed in the 13orresponding averages
Synthesis
These dierent simulations for a sho13k propagating at sim 50 kms in Xenon
at 01 bar show that the simulations performed with only one temperature
are adequate for the des13ription of the sho13k
The opa13ity strongly ae13ts the pre13ursor extension whi13h de13reases from
more than 4 mm for 1 group to 3 mm then the number of groups is larger
than 20 The 13onvergen13e versus the number of groups is rea13hed after 20
groups
To t with more sophisti13ated opa13ities and as it was impossible to
in13lude them in HELIOS I de13ided to multiply the opa13ities of Xenon by a
fa13tor of 20 This redu13es strongly the pre13ursor extension up to sim 1 mm
at 10 ns Taking into a1313ount the present huge un13ertainty in the opa13ity
I 13onsidered that it was not ne13essary to use multi-group opa13ities This
also means that our 1D simulations will be used - only - for a qualitative
interpretation of the experimental results A quantitative 13omparison should
38 CHAPTER 3 1D SIMULATIONS
(a)
(b)
Figure 39 Plan13k and Rosseland mono13hromati13 opa13ity proles Te for ρ =15
times 10
minus3from PROPACEOS (I) BIGBART (II) and Rodriguez et al (III)
33 SINGLE RADIATIVE SHOCK WAVES 39
(a)
(b)
Figure 310 Mono13hromati13 Xenon opa13ity versus the photon energy in eV at T
= 15 eV and for ρ = 15 times 10
minus3g13m
minus3for the PROPACEOS (a) and Rodriacuteguez
et al (2015) (Fig 16 of Rodriacuteguez et al (2015)) (b) models
40 CHAPTER 3 1D SIMULATIONS
be in the future based on adequate opa13ities and multi-groups modeling
34 Intera13ting radiative sho13k waves
In order to pre13ise what is expe13ted from the experimental study I will now
investigate the 13hara13teristi13 parameters of two 13ounter-streaming sho13ks
propagating in Xenon at 01 bar (ρ = 54 times 10
minus4g13m
minus3) with equal and
dierent velo13ities To highlight the ee13t of the intera13tion I will 13ompare
the results with those obtained with 1D simulations All the simulations will
be performed with HELIOS For the Xenon opa13ity a multiplier times 20 will
be used The number of groups will be set to be 1 and the ele13tron and ion
temperatures will be supposed to be equal
Three representative sets will be 13onsidered
Case(I) two identi13al radiative sho13ks (2RS) at sim 50 kms propaga-
ting in opposite dire13tions (ie starting from the left and right end
respe13tively)
Case(II) same 13onditions but without any 13oupling with radiation
Case(III) two 13ounter-propagating radiative sho13ks propagating with
dierent speeds sim 50 kms from the left side of the 13ell and sim 20
kms from the right side
To a13hieve the aforementioned speeds on the left and right sides the
uen13es of the two laser beams have been adjusted The pulse duration is
set to 03 ns (peak at 015 ns) to reprodu13e the experimental 13onditions
detailed later in the manus13ript
Case(I) 13ounter propagating sho13ks with the same speed
This a13ademi13al 13ase is fully symmetri13al and it is equivalent to the 13ase of
one RS with a fully ree13tive boundary (for hydrodynami13s and radiation)
in the middle of the tube The Fig 311 shows the variations of the ele13tron
density (Ne) and temperature (Te) in the Xenon layers The two sho13ks
appear in Xenon at sim 2 ns and the 13ollision o1313urs at sim 38 ns At 3 ns the
pre13ursor extension is sim 008 13m whereas the post-sho13k ele13tron density
and ele13tron temperature are 78 times 10
2013m
minus3and 16 eV respe13tively The
length of pre13ursor in13reases rapidly with time and the two pre13ursors merge
suddenly at sim 8 ns After this time the merging ee13t in13reases signi13antly
It is 13hara13terized by a at 13ommon pre13ursor those ele13tron density and
temperature are in13reasing with time At the time of the 13ollision (sim 38
ns) the post-sho13k mass and ele13tron density in13rease from 0011 to 014
g13m
minus3and 67 times 10
20to 66 times 10
2113m
minus3 whereas the temperature rises
up to 39 eV The 13ollision leads to the development of two reverse sho13k
34 INTERACTING RADIATIVE SHOCK WAVES 41
(a)
(b)
Figure 311 Ele13tron density Ne (a) and ele13tron temperature Te (b) versus axial
position (along a 04 13m long sho13k tube) at 3 10 20 30 and 38 ns from HELIOS
simulations (with opa13ity times 20) for the 13ases of single sho13k of sim 50 kms (dotted
line) and two identi13al 13ounter-propagating sho13ks of sim 50 kms (solid lines) The
verti13al dotted lines show the position of the interfa13e between piston and ba13king
Xenon gas
42 CHAPTER 3 1D SIMULATIONS
waves propagating ba13k with a speed of sim 15 kms rst in Xenon and later
on in the dierent piston layers (not shown in the Figure) These reverse
sho13ks lead to a dense plasma (Ne gt 10
2113m
minus3) whi13h is not a1313essible to
the experimental diagnosti13s and will not be detailed here
To investigate the ee13ts of the intera13tion I have reported in Fig 311
the 13ase of 1RS moving from the left to the right dire13tion in the 13ell (dotted
lines) The wave propagates identi13ally to 2RS until 10 ns After this time
the proles of the temperature and ele13tron density dier strongly from the
previous 13ase and their values are lower than for the 2RS The post-sho13k
extension is slightly smaller than for the 2RS and the sho13k speed is thus
slightly smaller too This last ee13t is due to the fa13t that for the two inte-
ra13ting waves the pre13ursor is at a higher temperature than for the isolated
sho13k and thus the sho13k propagates in a warmer medium then modifying
the sho13k 13onditions (sound speed opa13ity et13 ) A similar ee13t in the post
- sho13k extension may be seen for an isolated sho13k wave propagating in a
warmer pre-sho13k gas
Case (II) Identi13al 13ounter-propagating sho13ks for an ideal gas
In order to highlight the ee13t of the radiation another simulation has been
performed with the same set of parameters as in the 13ase I however putting
the Xenon opa13ity equal to zero I still use here the PROPACEOS equation
of state The result of the simulation is presented in Fig312 The 13ollision
time is now 40 ns instead of 38 ns (thus the sho13k speed is slightly smaller)
The post-sho13k is no more 13ompressed by radiation 13ooling Its 13ompression
at 10 ns is 10 instead of 35 This 13ompression is due to the vis13ous sho13k
(fa13tor 4) followed by the ionisationex13itation of the gasThere is obviously
no radiative pre13ursor Moreover there are no dieren13es in the Ne and Tproles of the single sho13k and that of the two 13ounter-propagating sho13ks
before the 13ollision time
Case (III) Two sho13ks at dierent speeds
The spatial and temporal variations of Ne and Te are plotted at times 3
10 20 30 38 and 49 ns in Fig 313a and 313b The left and the right
sho13ks appear in Xenon at sim 2 and 3 ns respe13tively Later at 10 ns
the two pre13ursor extensions are respe13tively equal to 018 and 0034 13m
The merging of the two pre13ursors starts at sim 15 ns As expe13ted from the
values of sho13k speeds the 13ollision time o1313urs at 49 ns whi13h is delayed
in 13omparison to the 13ase (I)
In 2RS the two radiative pre13ursors merging results in a plateau for the
ele13tron density and the temperature The temperature at 13ollision time is
now 28 eV instead of 39 eV in the 13ase I and the ele13tron density rea13hes
up to 31 times 10
2113m
minus3instead of 66 times 10
2113m
minus3
34 INTERACTING RADIATIVE SHOCK WAVES 43
(a)
(b)
Figure 312 Variations of Ne (a) and Te (b) versus axial position for the 13ase
of two identi13al 13ounter-propagating sho13ks (of speeds sim 50 kms) at 3 10 20
30 35 and 40 ns as derived from HELIOS simulations For these simulations we
have negle13ted the ee13t of radiation by keeping the Xe opa13ity equal to zero
The verti13al dotted lines show the position of the interfa13e between the piston and
ba13king Xenon gas
44 CHAPTER 3 1D SIMULATIONS
It may be noted that up to this 13ollision time the post-sho13k density
13onditions and speed are identi13al for the 13ase of an isolated RS propagating
from the left (1RSL) at 50 kms and this present left sho13k This reveals
that in the 13ase of 2RS there is no noti13eable ee13t of the right sho13k with
speed 20 kms on the left post-sho13k of 50 kms
On the 13ontrary we note a dieren13e in extension of the right post sho13k
from the 2RS 13ompared with the 13ase of an isolated sho13k (1RSR) propa-
gating at 20 kms from the right whi13h indi13ates that the left sho13k of the
2RS has an inuen13e on the right post-sho13k (Fig 313a)
35 Summary
Radiative sho13k waves propagating in Xenon at 01 bar with a velo13ity of 50
kms are 13hara13terized by the development of an extended radiative pre13ur-
sor The huge 13ompression of 38 in the post-sho13k is a 13onsequen13e of both
the sho13k and the gas ionisation (fa13tor of 10) as also from the radiative
13ooling The post-sho13k and pre-sho13k temperature on both side of the peak
are identi13al indi13ating that the sho13k is super13riti13al
The spe13ta13ular role of the opa13ity has been highlighted Due to the
un13ertainty in the opa13ity used for Xenon in our simulations I de13ided that
at this stage it was not ne13essary to rene the simulation in terms of group
numbers within the 1D des13ription of the sho13k wave
Our numeri13al study gives the main 13hara13teristi13s of the intera13tion of
two 13ounter-propagating sho13k waves with speeds equal to 50-50 kms and
50-20 kms The 13ase of identi13al speeds is simpler due to the symmetry of
the problem However whatever the speeds the most important signature
of the intera13tion is the merging of the pre13ursor at 8 ns for 50-50 kms
and at 15 ns for 50-20 kms This merging is followed by a regular in13rease
with time of the ele13tron density and the temperature The 13ollision time
is 13hara13terized by a sudden in13rease of the ele13tron density by an order of
magnitude rea13hing 66 times 10
21and 3 times 10
2113m
minus3respe13tively whereas the
temperature in13reases up to 39 and 28 eV
35 SUMMARY 45
(a)
(b)
Figure 313 Variations of Ne (a) and Te (b) with axial position for the 13ase of
two non-identi13al 13ounter-propagating sho13ks (of speeds sim 50 amp 20 kms) and two
single sho13ks (dotted lines) of speeds sim 50 amp 20 kms respe13tively at 3 10 30 35
and 46 ns as derived from HELIOS simulations The verti13al dotted lines show the
position of the interfa13e between piston and ba13king Xenon gas (with opa13ity times20)
46 CHAPTER 3 1D SIMULATIONS
Chapter 4
Laser-driven radiative sho13k
Experimental Setup
Contents
41 Prague Asterix Laser System (PALS) fa13ility 48
42 Targets 49
421 Massive Targets 51
422 Gaseous Targets 51
423 Target holder 55
424 Target lling 56
43 Laser Fo13using 57
431 Fo13al Lenses and Phase Zone Plates 57
432 Fo13using of the MAIN and AUX laser beams 58
44 Diagnosti13s 59
441 Visible interferometry 61
442 XUV spe13tros13opy 65
443 Visible spe13tros13opy 67
45 Summary 68
When a short pulse of a high-power laser beam is fo13ussed on a thin foil
it transfers a huge amount of energy to the foil material Almost instantane-
ously the laser absorption heats it up to the ablation A hot and low density
13orona is generated ba13kwards by this pro13ess Beyond the ablation surfa13e
a sho13k is formed in the foil as a 13onsequen13e of the momentum 13onservation
(ro13ket ee13t) The sho13k moves inward in the foil Our 13ase of the experi-
mental setup as the foil 13loses a tube lled with gas the sho13k propagates
then in the gas where it is studied through various kinds of diagnosti13s The
foil then a13ts as a piston
We performed the experiments at the Prague Asterix Laser System (PALS)
in the Spring 2015 (ve weeks 20th April - 22th May) The obje13tive of the
experimental 13ampaign was to study the evolution of two 13ounter streaming
radiative sho13ks driven by two 13ounter-propagating lasers with an irradian13e
in the range of 10
14W13m
minus2 and therefore to 13hara13terize their intera13tion
and 13ollision In this 13hapter I will rst present a brief des13ription of the
PALS laser fa13ility and of the laser beams used in our experiment This will
47
48 CHAPTER 4 EXPERIMENTAL SETUP
be followed by a presentation of the targets design and an overview of the
general setup and of our main diagnosti13s
41 Prague Asterix Laser System (PALS) fa13ility
The Prague Asterix Laser System (PALS) is a laser fa13ility based on an
Infrared high-power iodine laser system (Asterix IV) (Jungwirth et al 2001)
Using dierent amplifying stages the laser fa13ility is able to deliver energy
up to 1 kJ in 03 ns at the fundamental wavelength 1315 nm The output
laser beam 13an be further subdivided in few auxiliary beams All auxiliary
beams may be frequen13y doubled (λ = 657 nm) or tripled (438 nm) The
PALS laser fa13ility is 13apable of ring up to two high energy laser shots per
hour Compared with solid state lasers this gas laser is known to deliver a
quite homogenous beam intensity without hot spots A spatial prole of the
PALS laser beam is shown in the Fig 41a (re13orded by PALS team during
the experimental 13ampaign) while its pulse 13hara13teristi13s are plotted with
time in the Fig 41b
(a)
(b)
Figure 41 a Spatial prole of the PALS Laser beam b Intensity prole of
laser pulse I(t) with time
For our experiments we used two independent laser beams to drive the
two 13ounter streaming sho13ks in the gaseous target For this purpose the
fundamental PALS laser beam has been subdivided after the fourth amplier
(V4) into two beams with respe13tive energy fra13tions of 60 and 40 (see
Fig 42) The most energeti13 of these laser beams is then inje13ted through
the fth (V5) amplier and its frequen13y are then tripled This beam at 438
nm will be hereafter termed termed as the MAIN laser beam
The se13ond of the two previous laser beams propagates then without any
modi13ation It will be hereafter 13alled the AUX laser beam (1315 nm)
The s13heme of the energy distribution is shown in the Fig 42 and the
13hara13teristi13s of MAIN and AUX laser are presented in Table 41
For the experiments PALS oers two va13uum 13hambers with a spheri13al
42 TARGETS 49
Figure 42 Energy distribution s13heme for MAIN and AUX laser beams
LASER MAIN AUX
Typi13al laser energy (J) 120 60
Beam diameter (mm) 290 148
Wavelength (nm) 1315 438
Pulse duration (ps) 350 350
Table 41 Nominal 13hara13teristi13s of MAIN and AUX laser beams
and 13ylindri13al geometry respe13tively The spheri13al 13hamber assembly used
in our experiments is shown in Fig 43 This 13hamber having a diameter
of 100 13m is 13losed by two entran13e 13ir13ular windows of respe13tive diameters
80 13m and 50 13m to allow the MAIN and AUX beams to penetrate in the
13hamber and to rea13h the target Ea13h of these windows is outtted with a
hinged door Several 13ir13ular ports of various diameters also allow for the
dierent diagnosti13s The target holder and some diagnosti13s are mounted
on an opti13al ben13h whi13h is de13oupled from the 13hamber walls
42 Targets
Our targets s13hemati13ally 13onsist in small tubes of 4 mm length 13losed on
both sides by two spe13i13 foils of thi13kness 11 microm on whi13h the laser beams
are fo13used (one laser per foil) with an irradian13e of about 10
1413m
minus3 The
tube is lled with gas in whi13h the radiative sho13k propagates with a velo-
13ity of 30 - 60 kms The two foils 13losing the target insure the 13onversion
through ablation and sho13k generation of the laser energy into me13hani13al
energy
In our experiments we used spe13i13 targets for alignment and 13hara13te-
rization of the beam size These targets were manufa13tured at the Pole
instrumental of Observatoire de Paris
50 CHAPTER 4 EXPERIMENTAL SETUP
Figure 43 Snapshot of the spheri13al 13hamber showing the horizontal bredboard
and on the right the fo13using MAIN lens
(a) (b)
Figure 44 Dimension drawing (a) and snapshot (b) of Aluminum massive
(mo13k) target
42 TARGETS 51
421 Massive Targets
For the laser alignment on the target we have used spe13ial targets also
termed as mo13k targets A mo13k target (Fig 44) is an aluminum blo13k
with a base Three 13uts made on it help to fo13us the laser beam at the
desired position The 13enter of ea13h of the two laser spots should be lo13ated
at the interse13tion of the verti13al and horizontal marks and the pre13ision of
this alignment is less than 20 microm
Beside alignment purpose su13h targets are used to 13hara13terize the MAIN
and AUX spot diameters in the planes whi13h 13orrespond to the lo13ation of
the two Parylene foils Hen13e the laser beam fo13ussed on these targets
generates a 13rater (Chaulagain 2015) The detailed analysis of the shape
and size of these 13raters provides a quantitative information about the ho-
mogeneity of the in13ident beam and the size of the fo13al spot A detailed
dis13ussion about the fo13alisation of the laser with a massive target will be
presented in the se13tion 432
422 Gaseous Targets
The 13ore of the gaseous targets (Stehleacute et al 2014a) 13onsists of a 13hannel of
a parallel pipe shape having the dimension of 09 times 06 times 4 mm pla13ed at
the top of an aluminum stru13ture The pipe in whi13h the sho13ks propagate
is dened by two 500 microm thi13k verti13al lateral sides made with 13oated fused
sili13a (SiO2) windows (09 times 4 mm) as shown in the Fig 45 Su13h windows
are suitable for the visible interferometry diagnosti13 whi13h uses an auxiliary
Q-swit13hed NdYLF green Evolution laser (see Se13tion 441)
The 13hannel is 13losed at the top with a window made of a 100 nm thi13k
Si3N4 membrane whi13h is itself supported is by a 200 microm thi13k sili13on frame
(26 times 4 mm) made by SILSON 13ompany Su13h membrane is transparent
to XUV radiation whi13h is ne13essary for the XUV spe13tros13opi13 diagnosti13
used in the experiment The transmission of the aforementioned material is
obtained from the 13enter for X-ray opti13s (CXRO) database
1
and is plotted
between 0 and 40 nm in Fig 48
The pistons 13losing the tube from opposite sides and whi13h will laun13h
the two sho13ks are made of Parylene-N (11 microm) 13oated by Au (06 microm)
(SCITECH 13ompany) They are glued on 01 mm thi13k Ni13kel disks with
external diameter 5 mm and internal diameter 1 mm This disk helps in
assembling the targets It also 13ontributes together with a larger V-shield
(se13tion 49) to prevent hard X-ray emitted at the laser impa13t on the foil
to rea13h the gas in the tube The Parylene layer is fa13ing the laser and plays
the role of an ablator The gold layer whi13h is 13oated on the Parylene and
is lo13ated between the Parylene and the gas aims at blo13king the X-rays
generated by the laser plasma intera13tion to rea13h and preheat the gas in
1
httphenkelblgovopti13al_13onstants
52 CHAPTER 4 EXPERIMENTAL SETUP
(a)
(b)
Figure 45 Values on the gures are in mm (a) Verti13al 13ross se13tion view of
the gaseous target (b) Horizontal 13ross se13tion view
42 TARGETS 53
Figure 46 Pi13ture of a the gaseous target on its base
Figure 47 S13hemati13 of the gaseous target
54 CHAPTER 4 EXPERIMENTAL SETUP
Figure 48 Transmission of 100 nm thi13k Si3N4 membrane (CXRO database)
Figure 49 Snapshot of a target holder with one V-shield regarding the AUX
laser whi13h will pass through the hole in this shield marked in green in the Figure
The target whi13h fa13es the two diagnosti13s (tubes) and is lo13ated behind the
V-shield is thus not visible
42 TARGETS 55
the 13ell These various elements are manually glued on the base of the target
to built the sho13k 13hannel Ea13h target has been tested against leaks before
the experiment and before ea13h shot The axis of symmetry of the sho13k
waves is required to be lo13ated at 03 mm from the base and at 03 mm from
the two verti13al SiO2 windows S13hemati13s of verti13al and horizontal 13ross
se13tion views of the target are presented in Fig 45b The axis of symmetry
is visible in the verti13al view while the position of the laser fo13us point (`+
sign) in the horizontal view This is a13hieved during the alignment pro13edure
employing mo13k targets as dis13ussed in the se13tion 432 The base of the
target 13onsists of a mi13ro-ma13hined aluminum stru13ture A gas pipe at the
bottom of the stru13ture allows for in situ gas lling at a pressure of a fra13tion
of a bar
A total of 55 targets were manufa13tured for the experiment Among
them 31 gave valuable results
423 Target holder
We x the targets on a stainless holder and then pla13e the system into the
va13uum 13hamber The target holder has been designed and manufa13tured
at Observatoire de Paris It allows a very reprodu13ible positioning of the
target on the va13uum 13hamber breadboard The main element of the target
holder is a base where ea13h target is rst xed by s13rews and removed after
ea13h shot to be repla13ed by a new one It also in13ludes a diagnosti13s support
(Fig 46) whi13h is atta13hed on it from the top with s13rews One of these
diagnosti13s 13onsists in a fast XUV Si diode with a pinhole The diode is
atta13hed to the bla13k tube whi13h is visible on the left part the target holder
in Fig 47 This last diagnosti13 aimed for sho13k timing as the pinhole-diode
system allows to re13ord the sho13k passing at the imaged position of the tube
(Stehleacute et al 2012 Chaulagain 2015) Unfortunately this diagnosti13 did
not work well and will not be mentioned further in the thesis
The se13ond diagnosti13 is a visible time-and-spa13e-resolved visible spe13tro-
meter On the target holder this diagnosti13 13ontains a lens whi13h is lo13ated
inside the bla13k tube on the right part of Fig 49 The lens allows imaging
the sho13k on a linear bundle of 21 bers 13onne13ted to the visible spe13trometer
through a feedthrough (se13tion 443)
A snapshot of the target holder is shown in the gure 49 The red arrows
in the Fig 49 shows the distribution of the various diagnosti13s
In addition we used two 3 mm thi13k stainless steel V-shape shields on
both sides of the target whi13h were xed to the holder These V-shields have
a hole (sim25 mm) shaped entran13e to allow laser in13iden13e on the target (in
green on the Figure 49) The role of the V-shields is to prevent the target
from the extra laser or another kind of hard X-ray plasma self-emission linked
to the ablation pro13ess
56 CHAPTER 4 EXPERIMENTAL SETUP
424 Target lling
The gaseous targets were lled in situ at a fra13tion of 1 bar with a gas (viz
Xe Ar a mixture of Xe and He) whi13h thus provide the medium for the
sho13k propagation We 13hose an in-situ lling system (Fig 410) to redu13e
the dieren13e of pressures inside and outside of the tube This is 13riti13al as
a large pressure dieren13e between the target 13ell and the va13uum 13hamber
13an result in the breaking of the ultra-thin Si3N4 windows of the target
To 13ontrol the pressure we used a piezo-resistive pressure transdu13er
(Swagelok PTI-S-AC6-32) whi13h has 10 kΩ bridge resistan13e and works
with 14-30 V biasing voltage A sensitive Bourdan manometer and a gauge
allowed monitoring the pressure inside the target 13ell To read the pressure
remotely (from the 13ontrol room) until the last se13ond before the shot we
13onne13ted the system with a long 13able
The transdu13er has been operated with a +17 V DC biasing voltage The
pressure is read using an industrial pressure transdu13er from Swagelok (0 to
10 Volts -10 bar to 06 bar 05 pre13ision) and reading is made with a
voltmeter at better than 05 This system generally took approximately
one minute to stabilize A s13hemati13 of the aforementioned gas lling system
is presented in the Figure 410
The pro13edure of the target lling is presented as follows
Figure 410 In-situ system for target lling and pressure monitoring
The target is pumped during the 13hamber eva13uation
The target is isolated at the end of the pumping
The target is lled to the desired pressure
43 LASER FOCUSING 57
The target is isolated from the gas bottle to avoid any leakage in
13hamber
Then the pressure 13an be read from the 13ontrol room till the last se13ond
before the shot
The gases are taken from high grade bottles 99995 purity The Xe+He
mixture is 1007plusmn050 He molar fra13tion The lling pro13edure is pre-
venting the presen13e of impurities like air When 13losing the 13hamber and
during the primary pumping the lling 13ir13uit 13ommuni13ates with the 13ham-
ber thus the whole tubing and the target are eva13uated down to 001 mbar
After the turbo pump is a13tivated the 13onne13tion to 13hamber is 13losed and
the gas in inje13ted When the target pressure is rea13hed the 13ell is isolated
from the pressure regulator In 13ase there is an una1313eptable overpressure
it is possible to re13onne13t to 13hamber to eva13uate and to restart the lling
The pressure reading is possible in the 13ontrol room still at pre13ision better
than 1 and the value kept for the re13ords is the one just before the shot
whi13h happens normally 1-2 minutes after 13losing
43 Laser Fo13using
431 Fo13al Lenses and Phase Zone Plates
Two fo13al lenses are used to fo13us the laser beams on the target piston foil for
MAIN and AUX respe13tively The MAIN lens is lo13ated inside the va13uum
13hamber whereas the AUX lens is outside this 13hamber The spe13i13ations
of these lenses are given in the Table 42
laser Diameter (mm) fo13al length (mm) f-number
MAIN 340 564 at 3ω 19
AUX 150 1022 at ω 69
Table 42 Spe13i13ations of the MAIN and AUX fo13al lenses and the f- number
(fo13al lengthbeam diameter) is given (see table 41 for the beam diameters)
Whereas these lenses provide a smooth intensity at the fo13us this is not
su13ient to laun13h a planar sho13k wave To this purpose phase zone plates
(PZP) have been used between the output laser beam and the lens
A PZP 13omprises several phase zone elements whi13h divide the input
beam into several beamlets whi13h 13ombined with the main fo13al lens provi-
ding a uniform intensity distribution on the target with a spe13ied diameter
(Ross Pepler and Danson 1995 Bett et al 1995) The PZP are 13ommonly
used in EOS laser experiments whi13h require a uniform intensity distribution
on the target (Koenig et al 1994 Batani et al 2003)
Two PZP as dis13ussed below were used in the experiment
58 CHAPTER 4 EXPERIMENTAL SETUP
The rst one made by COLSICOAT was already used at PALS for
radiative sho13k experiments (Stehleacute et al 2010) This PZP used for
MAIN (438 nm) is made on a 13oated BK7 substrate of a diameter
of 310 mm and se13tion of 8 mm It is designed to provide a uniform
13ir13ular fo13al spot over a diameter of 05 mm
The se13ond plate made by SCITECH has been used for AUX (1315
nm) The plate is made on a 63 mm thi13k 13oated Viosil substrate
with a square 153 x 153 mm shape It is designed to provide a uniform
13ir13ular fo13al spot over a diameter of 0250 mm
Laser Spe13ied spot diameter (mm)
MAIN 05
AUX 025
Table 43 Spe13i13ations of the two Phase Zone Plates
In terms of energy distribution sim 80 of the laser energy is in the laser
spot (C Spindloe private 13omm) The energy of the laser whi13h is given
by the PALS installation should be 13orre13ted by this fa13tor of 80 For
AUX another 13orre13tion of sim 10 has to be added whi13h is linked to the
transmission of the entran13e window of the va13uum 13hamber As a result
for nominal energies of the MAIN and AUX lasers equal to 120 and 60 J (see
table 41) the uen13es on the target are equal to sim 48800 and 59500 J13m
2
respe13tively Taking into a1313ount the 03 ns laser duration of the lasers this
results in an intensity of sim 1-2 times 10
14W13m
2
432 Fo13using of the MAIN and AUX laser beams
The determinations of the fo13i of the two lenses were performed with a
Hartmann setup using a photographi13 paper and without any PZP plate
The determination of the size of the beam at the impa13t is important both to
know the initial 13onditions for the sho13k waves (laser irradian13e whi13h should
be few 10
14W13m
minus2and se13tion of the sho13k wave) The sele13tion of the
best spots size on the piston was done with mo13k targets and by translating
the two laser lenses The in13iden13e of a laser beam on a solid planar target
generates a 13rater in the foil and an ablated plume propagating ba13kward
at high temperature (Mahmood and Ur-Rehman 2009 Singh and Thakur
2007) The 13rater is the result of boiling and ablation of the material The
ablated plasma whi13h expands towards the in13ident laser beam emits in the
keV range
A standard pro13edure whi13h works well at low energy (sim 10 J) 13onsists
of the estimation of the spot size by measuring the 13rater on the bulk target
using a standard mi13ros13ope At higher energy the boiling ee13t makes
44 DIAGNOSTICS 59
this estimation impre13ise and we preferred to use a keV imaging diagnosti13s
(Chaulagain et al 2012) whi13h is based on a pinhole (25 microm for MAIN keV
and 20 microm for AUX) and a keV 13amera These two keV 13ameras have been
used during ea13h laser shot to image the impa13ts of the MAIN and AUX
laser beams on the two respe13tive pistons of the target Ea13h keV 13amera
is a CMOS dete13tor without any glass It is prote13ted by an Aluminium
lter (200 microm for MAIN and 42 microm for AUX) to blo13k the visible light and
to prote13t the ship from the debris The spe13i13ations of the 13ameras are
presented in the Table 44 and the keV 13amera system is shown in the Fig
411 The two PALS laser lenses were translated up to a13hieve the suitable
diameters on the target whi13h were nally set to 450 - 500 microm and 250 -
300 microm for the MAIN and AUX beams
The size of the impa13t was 13ontrolled ea13h day on mo13k targets before
using the laser beams for real shots on gaseous targets It is worth to pre13ise
that the keV diagnosti13 was still in pla13e for the shots on the gaseous targets
In that 13ase the beams impa13t Parylene-N instead of Aluminium The keV
signal is then weaker than for the mo13k aluminum targets
Figure 411 S13hemati13 of the keV 13amera
KeV 13amera KeV 13amera
MAIN laser AUX laser
Model UI-122xLE UI-164xLE
Resolution 752(H) x 480(V) 1280(H) x 1024(V)
Pixel size (microm x microm) 60 x 60 36 x 36
A13tive Area (mm x mm) 451 (H) x 288 (V) 46 (H) x 37 (V)
Magni13ation 107 068
Pinhole Material Ta Pt
Table 44 Spe13i13ations of the two uEye keV 13ameras (1stVsion 13ompany)
44 Diagnosti13s
Various kinds of diagnosti13s were implemented namely visible interfero-
metry XUV and visible spe13tros13opy For the same two keV 13ameras are
indu13ted to monitor the impa13t and two visible 13ameras for target align-
60 CHAPTER 4 EXPERIMENTAL SETUP
(a)
(b)
(13)
Figure 412 (a) Sign 13onvention for the respe13tive positions of the lens and the
target (b) keV image of a MAIN impa13t on a massive target (lens position +1500
microm) The spot size is 475plusmn25 microm Pixel size for MAIN keV 13amera is 56 microm (13)
keV image of an AUX impa13t (lens position -1500 microm) The spot size is sim 280plusmn20microm on target One pixel = 66 microm on AUXs keV 13amera
44 DIAGNOSTICS 61
ment In this se13tion I will des13ribe the setups of the visible interferometry
XUV spe13tros13opy and visible spe13tros13opy The analysis of the re13ords of
visible interferometry and XUV spe13tros13opy will be dis13ussed in subsequent
13hapters
441 Visible Interferometry
A Ma13h-Zehnder Interferometer (MZI) has been employed to perform visible
interferometry The re13orded interferometri13 images have been used to study
the sho13k speed and the ele13tron density during the sho13ks propagation The
experimental setup of this interferometer is presented below
Ma13h Zehnder Interferometer
We pla13ed the target in whi13h sho13k is produ13ed in one of the two arms
of the interferometer The probe laser wavelength (527 nm) is supposed to
be far away of any atomi13 absorption resonan13es in the plasma and thus the
13ontributions to the refra13tive index 13ome from the plasma free ele13trons and
not from bound ele13trons (Harilal and Tilla13k 2004)
Figure 413 First interferometri13 setup All the opti13al elements are 1 in13h For
simpli13ity the plasma slab is not reported in the probe beam
The rst experimental setup shown in Fig 413 was used until 04052015
62 CHAPTER 4 EXPERIMENTAL SETUP
In this setup
the 300 ns long probe laser of wavelength 527nm (beam diameter asymp 1
in13h energy on the target few tens of mi13rojoules) rst passes through
an alignment pinhole PH1
the non-polarizing beam splitter BS1 transmits 50 of the intensity
of the in13ident beam and the remaining 50 of intensity in the per-
pendi13ular dire13tion At this stage the two arms of the interferometer
namely the referen13e and probe beams are produ13ed
two half wave plates HWP1 and HWP2 are pla13ed respe13tively in
the path of the probe and referen13e beams They allow to play with
the polarization in order to optimize the fringes 13ontrast as it will be
explained below
the verti13ally polarized probe beam passes through the target (plasma
slab) Due to its polarization it is fully ree13ted by the polarizing 13ube
beam splitter PBS1 towards the polarizing 13ube beam splitter PBS2
the referen13e beam is ree13ted by the mirror M4 on to this 13ube splitter
PBS2
At the output of PBS2 the two beams overlap but they have ortho-
gonal polarizations and 13ant interfere They then travel through a
polarizing beam splitter PBS3 those axis makes an angle of 45
with
the 13ommon dire13tion of in13iden13e of the two beams This rotates
the polarizations of the referen13e (horizontally polarized) and probe
(verti13ally polarized) beams along the same axis
At this stage the beams share the same polarization and interfere
A last non-polarizing beamsplitter plate (90-10 ) BS2 is pla13ed after
PBS3 to enable the propagation of the two beams on the same axis as
the laser beam at the entran13e of the interferometer
The fringes are re13orded on a HAMAMATSU C7700 VIS Streak Camera
lo13ated after the last alignment pinhole PH4 To this purpose an imaging
setup makes the image of the target longitudinal axis (ie along the dire13tion
of the sho13k propagation) on the streak slit This is done with an a13hromati13
spheri13al doublet of fo13al length equal to 1000 mm The magni13ation is
measured using an AGAR grid to 226 whi13h means that 106 microm on target
are re13orded by one pixel (size of the pixel 24 microm) on the streak on both
dire13tions (Fig 413)
44 DIAGNOSTICS 63
Figure 414 Se13ond interferometri13 setup
Role of the polarizing devi13es
Let us suppose that the polarization state of the in13oming beam is unknown
The polarization of the probe beam is then xed to be verti13al by the pola-
rizer HWP1 Ree13ted by the polarizing 13ube PBS2 it remains verti13al at
the entran13e of PBS3 On this side the polarization of the referen13e beam
is set to be horizontal after the 13ube BS1 using the polariser HWP2 and is
transmitted as horizontal by PBS2 The rotation of PBS3 allows to proje13t
part of the two beams on a 13ommon polarization axis and to interfere
This setup presents the following interest the intensity of the laser beam
in the two arms of the interferometer 13an be modied (and thus the 13ontrast
of the fringes) 13an be optimized using the two polarizing plates HPW1 and
HPW2 However as the multimode laser is not fully 13oherent the dark
fringe (zero net intensity) has never been obtained
Moreover the setup is designed to also allow alignment and adjustment
of the imaging setup with a green HeNe laser whi13h shares the same path
as the probe beam up to PBS1
A more optimized setup was used after 05052015 (ie shot number
48033) where the mirror M1 in the referen13e beam was repla13ed by a
polarizing beam splitter PBS4 ree13ting the probe beam like the mirror M1
(Fig 414) This was done in order to redu13e the dieren13e between the
64 CHAPTER 4 EXPERIMENTAL SETUP
two paths l1 and l2 and then to optimize the fringe 13ontrast This was more
suitable due to the short 13oheren13e length (1- 2 13m) of the probe laser
Streak 13amera
Figure 415 S13hemati13 of the Streak 13amera (from Hamamatsu noti13e)
A streak 13amera is a setup whi13h allows measuring the temporal varia-
tion of a fast signal whi13h is re13orded on its slit As indi13ated in the noti13e)
of the Hamamatsu C770 13amera the light pulse rst passes through the slit
then it is fo13used on the photo13athode (size 7times17 mm) of the streak by an
input opti13s The visible photons are 13onverted into a number of ele13trons
whi13h is proportional to the intensity of the in13ident light These ele13trons
are a1313elerated and 13ondu13ted towards the phosphor s13reen when a high-
speed voltage whi13h is syn13hronized to the in13ident light is applied As a
result the ele13trons are swept at high speed from the top to the bottom
after whi13h they are bombarded against the phosphor s13reen of the streak
tube and 13onverted through an output opti13s to an opti13al image on a dual
mode 13ooled CCD 13amera with 512 times 512 pixels (Fig 415) In the PALS
experiment the sweeping times were kept either 50 or 200 ns The CCD
pixel size is 24 x 24 microm
The streak 13amera 13an be operated with time swap mode with the slit
of the 13amera almost 13losed The re13ord 13onsisted then in images (position
along the slit versus time) as dis13ussed above or in a stati13 mode (no time
swap) whatever the slit aperture In that 13ase the re13ord is a usual 2D
spatial image of the obje13t
During the experiment we have taken referen13e images (without any
fringe) for every target as shown in the Fig 417a for one target example
In order to re13ord su13h images we blo13ked the referen13e beam and let the
probe beam only to illuminate the target in a stati13 mode and with the slit
open Then we redu13ed the slit width to its nominal value (200 microm) This
enabled us to have the visualization of the portion of the target whi13h was
44 DIAGNOSTICS 65
Figure 416 Imaging setup of the Ma13h Zehnder Interferometer setup operating
in the longitudinal mode The 13hannel of the target is fully illuminated by the
probe beam In this mode the lens images the axis of symmetry of the two sho13k
waves along the slit of the streak 13amera
imaged in the slit on the target and also to lo13ate the position of the pistons
before the shot This image is termed as the referen13e target image Then
the system was pla13ed in the interferometri13 mode and the fringe pattern
was optimized to be perpendi13ular to the slit and thus in the longitudinal
mode perpendi13ular to the dire13tion of the sho13k propagation
The Fig 417b reports su13h a typi13al interferogram re13orded during our
experiment (13f Fig 413) The horizontal axis 13orresponds to the dire13tion
of the sho13k propagation (with a total s13ale of 54 mm on the target) and
the verti13al axis 13orresponds to time (s13aling 200 ns)
442 XUV Spe13tros13opy
The time and spa13e integrated XUV plasma emission are re13orded with a
Flat Field XUV spe13trometer (grazing in13iden13e XUV spe13trometer) using
a 13on13ave grating Typi13al ray tra13ing a grazing in13iden13e XUV spe13tro-
meter is shown in Fig 418 In pra13ti13al the entran13e slit was removed
as the plasma extension was small In the Fig 419 the drawing of the
XUV spe13trometer setup s13heme (red 13olor) is presented together with the
spheri13al 13hamber and the target The XUV spe13trometer is installed on
the top of the spheri13al 13hamber whi13h allows the XUV emission passing
through the Si3N4 membrane on the top of the target (see se13tion 422) to
be re13orded
A 13ooled Andor DX440 CCD (without glass prote13tion) is atta13hed to
the spe13trometer to re13ord the spe13trum of the XUV radiation between 12
66 CHAPTER 4 EXPERIMENTAL SETUP
(a)
(b)
Figure 417 (a) Referen13e 2D image of a target before the shot re13orded on the
Streak 13amera The positions of the two pistons 13losing the 4 mm long target are
lo13ated at 800 and 4800 microm The dark zones along them (between 800 to 1000 and
4600-4800 microm) 13orresponds to the glue This glue is then visible through verti13al
bla13k strips in the interferometri13 re13ord (b) Corresponding interferometri13 time-
spa13e re13ord
44 DIAGNOSTICS 67
Figure 418 Typi13al ray tra13ing in a grazing in13iden13e XUV spe13trograph The
dete13tor (GMCP or image plate) is installed tangentially to the Rowland 13ir13le
and 40 nm (31- 100 eV) Its 13hara13teristi13s are given in the Table 45 Two Al
lters of thi13kness 08 and 16 microm prote13t the grating and the CCD 13amera
from debris and visible light The XUV images re13orded on the CCD are
time (300 micros) and spa13e (sim 3 mm) integrated
Andor 13amera (DX440)
Pixel Size 135 x 135 micromArea 276 (H) x 69 (V) mm
A13tive pixels 2048 x 512
Table 45 Spe13i13ations of Andor CCD 13amera
The 13urved grating (Table 46) used to fo13us the XUV emission on to
the CCD 13hip has a variable growes spa13ing as shown in the bottom panel
of the Fig 419 A typi13al output re13ord is also presented for illustration
Grating
Type Dira13tion
Growes per mm 1200
Dimensions 30 x 50 x 10 mmRadius of 13urvature 5649 mm
Blazing angle 37 degree
Blazing wavelength 16 nm
Table 46 Spe13i13ations of the XUV grating
443 Visible spe13tros13opy
In addition to the interferometry and XUV spe13tros13opy we have also im-
plemented a time- and spa13e-resolved visible spe13tros13opi13 plasma emission
diagnosti13
68 CHAPTER 4 EXPERIMENTAL SETUP
Figure 419 (a) Setup of the XUV spe13trometer in the spheri13al 13hamber (b)
Typi13al re13ord between sim 10 and 40 nm is shown in the bottom panel
The 13ore of the diagnosti13s is a Czerny-Turner 300 mm Spe13trometer
2
lo13ated outside the va13uum 13hamber The light of the plasma is inje13ted on
the slit of this spe13trometer through a bers bundle This bundle 13onsists of
three sets of an aligned seven individual bers looking at dierent se13tions
of the target 13hannel through an imaging lens lo13ated in the bla13k tube of
Fig 49 The time resolution is performed through an iCCD 13amera lo13ated
at the exit port of the spe13trometer (Fig 420a)
A typi13al re13ord is reported in Fig 420b The results of this diagnosti13s
show several signatures whi13h dier along the tube Most of them are in
absorption and 13ould be linked to the heating of the target sili13a window
Their interpretation is not yet a13hieved and will not be presented in this
study
45 Summary
In this 13hapter I have presented the details of our experimental setup and
diagnosti13s The next 13hapter will be dedi13ated to the presentation of our
re13ords results and their analysis
2
see http wwwandorcomspectrographshamrock minus spectrographminus series
45 SUMMARY 69
(a)
(b)
Figure 420 (a) S13hemati13 of setup employed to re13ord the time and spa13e
visible plasma emission (b) A typi13al spe13trogram
70 CHAPTER 4 EXPERIMENTAL SETUP
Chapter 5
Results and interpretation
Contents
51 Visible interferometry 71
511 Longitudinal interferometry sho13k speed and
ele13tron density 74
512 Transverse interferometry lateral extension of
the sho13k 91
52 XUV Spe13tros13opy 92
53 Simulations based on experimental results 93
54 Summary 97
In the previous 13hapter I presented a brief des13ription of our experimen-
tal setup This setup in13ludes two main diagnosti13s namely visible interfe-
rometry and XUV spe13tros13opy The visible interferometry is an adequate
tool to probe the ele13tron density of the radiative pre13ursor up to the 13riti-
13al density (4 times 10
2113m
minus3at the wavelength 527 nm of the probing laser)
whereas the denser post-sho13k region is opaque to the visible light XUV
radiation is present in the whole sho13k stru13ture and thus the time- and
spa13e-integrated XUV spe13tros13opy may allow exploring the self-emission
13oming from the post-sho13k and pre13ursor regions
In this 13hapter
1
I present the results of the both diagnosti13s with a
parti13ular fo13us on the interferometri13 data whi13h I have extensively studied
with the help of a spe13i13 data analysis pro13edure that I have developed I
will then present the main results of the XUV spe13tros13opi13 diagnosti13s The
results will then be interpreted with the help of numeri13al simulations
51 Visible interferometry
In the visible interferometry the interferen13es between the main and refe-
ren13e beams overlapping on the dete13tor produ13e a pattern of fringes whi13h
follow the relative phase variations between the probe (whi13h passes through
the plasma) and the referen13e beams (see se13tion 441 of the previous 13hapter
for the experimental setup) The phase variation between two 13onse13utive
fringes of the unperturbed beams (ie without any plasma) is equal to 2π
1
Most of results presented in this 13hapter has been published in Singh et al (2017)
71
72 CHAPTER 5 RESULTS AND INTERPRETATION
Then in the presen13e of the plasma in the probe beam these fringes be-
13ome shifted due to the modi13ation of the refra13tive index More details
about the prin13iple of the interferometry and the Ma13h-Zehnder interfero-
metri13 setup may be found in the APPENDIX A and we re13all here only
the expression of the phase shift
∆φ asymp minus πd
λNclt Ne gt (51)
where λ = 527 nm is the wavelength of the probing laser Nc = 4 times 10
21
13m
minus3the 13riti13al density at this wavelength and lt Ne gt is the ele13tron
density averaged over the laser path d in the 13ell and is dened as
lt Ne gt=
int d
0
Ne(z t)dy
d(52)
In order to dedu13e the ele13tron density it is ne13essary to follow the
evolution of the positions of the maxima (respe13tively minima) of the fringes
versus time To this purpose I have developed a spe13i13 data treatment
pipeline in the intera13tive data language (IDL)
2
To improve the visibility of
the fringes obtained in the interferograms I pro13essed the images with the
Fast Fourier Transform (FFT) (Proakis 2001) This treatment transforms
the image from the spatial domain (ie our re13ords) into the frequen13y
domain It then allows to dene spe13i13 bands of frequen13y whi13h represent
noise for the image and to 13lean them The 13omplete pro13edure is des13ribed
below
First the FFT of the (512 times 512) interferometri13 image is derived In
the next step the low and high-frequen13y lterings within the FFT of the
image are done using low-pass Fminus and high-pass F+ Butterworth frequen13y
lters (Proakis 2001) as follows
Fminus(u) = 1[1 + C(uu0)2n] with C = 1 n = 1
F+(u) = 1[1 + C(u0u)2n] with C = 1 n = 1
(53)
where u represents the position in the FFT image and u0 is the nominal
lter 13uto frequen13y (represented as the width of the region in pixels) For
the low-pass Butterworth frequen13y lter u0 is 13onsidered to be 55 whereas
for high-pass Butterworth frequen13y lter it is taken to be 45 The values of
u0 are dened manually by a trial method with the aim to result in better
noise redu13tion
After this step the inverse FFT of the ltered FFT image is performed
whi13h results in a 13lean image in the spatial domain In Fig 51 the original
re13ord for the shot number 48055 (left) the FFT of this image (13enter)
and the nal frequen13y ltered image (right) are presented showing how the
2
httpwwwast13ama13uk~vasilyidlidl_introdu13tionpdf
51 VISIBLE INTERFEROMETRY 73
frequen13y ltering improves the fringes 13ontrast and thus fa13ilitates their
analysis
Figure 51 Original re13ord of shot 48055 (left) FFT of the original image
(13enter) and the frequen13y ltered image (right)
Next the 13ru13ial task is to identify and to follow the fringes with the
help of a pre13ise determination of the intensity (pixel 13ounts) maxima In
order to lo13ate these maxima I have adopted the following steps
Figure 52 A 13ropped se13tion of the re13ord from shot the 48055 The rst ve
positions have been sele13ted manually `+ signs (in red 13olor) on ea13h fringe On
this re13ord the distan13e between two unperturbed fringes is 13orrespond to 15 pixels
(ie 159 microm)
Firstly I have manually 13hosen a se13tion of a typi13al re13ord with the
aim to sele13t only the area asso13iated with the sho13k dynami13s
74 CHAPTER 5 RESULTS AND INTERPRETATION
Next ve representative points are sele13ted visually (`+ marks in red
13olor) on ea13h fringe 13overing the important lo13ations on the fringe
(13f Fig 52) They serve as the input to the spline interpolation for
estimating all intermediate pixels lo13ations along the fringe (see Fig
53a)
In order to obtain the lo13ations representing the fringe maxima the
spline interpolated lo13ation points are further rened by sele13ting the
pixel of maximum intensity within plusmn5 pixels of the respe13tive spline
lo13ations in the X-dire13tion (see Fig 53b)
The position of fringe maxima for the 13ropped image obtained in an
aforesaid manner is shifted so as to represent the same points however
on the full-image (see Fig 54)
The re13ords obtained from the experiments have been pro13essed in this
way to estimate the sho13k se13tion speed ele13tron temperature and density
511 Longitudinal interferometry sho13k speed and ele13tron
density
In the longitudinal interferometry we perform on the slit of the streak 13a-
mera the image of the axis of symmetry of the sho13k propagation along the
tube to analyse the sho13k propagation in this dire13tion
The interferometri13 images have been pro13essed as explained previously
to enhan13e the fringes 13ontrast The lo13ations of the maximum intensity in
ea13h fringe are then used to derive the sho13k speed and the average ele13tron
density as will be presented below
Sho13k speed dedu13ed from the last fringe method
An estimation of the sho13k speed 13an be obtained through the interferometri13
image In this se13tion I will present the method to derive sho13k speed
based on the last visible end points of the fringes also 13alled last fringe
method This determination provides rapidly a qualitative estimation of the
sho13k speed whi13h is based on the absorption behavior of the plasma (see
Equation A11 in appendix A) and not on the real position of the front
dis13ontinuity The front dis13ontinuity is not seen in the re13ords due to the
strong absorption
At the positions of the last visible end points of the fringes the ele13tron
density rea13hes the maximum value a1313essible to the diagnosti13 The frin-
ges are strongly bent and the absorption of the visible light be13omes also
important Therefore it is believed that the sho13k front is 13lose from this
lo13ation
The lo13ations of the fringe maxima have been already dedu13ed following
the analysis presented in previous se13tions In this regard the best visible end
51 VISIBLE INTERFEROMETRY 75
(a)
(b)
Figure 53 (a) Representative points of the fringes as derived from the spline
tting of the 5 manually sele13ted points (Fig 52 (II)) on ea13h fringe (b) Positions
of the fringes maxima along Y-axis for ea13h fringe derived by lo13ating the points
of maximum intensity in X-dire13tion of the previous points obtained by spline t
76 CHAPTER 5 RESULTS AND INTERPRETATION
Figure 54 Fringe maxima on the full image
Figure 55 Imaging setup of the Ma13h Zehnder Interferometer operating in the
longitudinal mode The lens images the axis of symmetry of the two sho13k waves
along the slit of the streak 13amera
51 VISIBLE INTERFEROMETRY 77
point of ea13h respe13tive fringe is 13onsidered for the average speed estimation
for the sho13k
At very initial time the in13rease in sho13k speed is non-linear This non-
linear trend may result in non-physi13al speed estimations Therefore I dis-
13arded the end points whi13h appear to be within the ve nanose13onds after
the sho13k laun13hing time (eg t = 145 ns on Fig 56)
However the above pro13edure may result in un13ertainties as the sele13ted
end points may have dierent intensity (13ounts) values In this regard I
have evaluated the un13ertainty in the determination of the average speeds
To this purpose I rst sele13ted the best visible end-point of a random fringe
Su13h an end point gives a referen13e intensity whi13h was then used to obtain
on the rest of fringes the end points having 13lose intensity This pro13edure
was performed thri13e in order to dedu13e three sets of su13h end-points (see
Fig 56) Ea13h of these three sets of points was then used to estimate the
speed through a linear-t method (lines in white red and green) It may be
noted that these sets of points and tted lines are very 13lose to ea13h other
Therefore it is tough to distinguish them on the Fig 56 These three speed
determinations enable us to derive an average value with an un13ertainty
whi13h is equal to the standard deviation of these three values
Figure 56 Interferometri13 image re13orded for the shot 48055 in Xe at 01 bar
The sho13k speeds for the sho13ks driven by MAIN (from left side) land AUX (from
right side) lasers are respe13tively equal to sim 54 and 23 kms The time of laser
arrival on the piston is at 146 ns The positions of the Au-Xe interfa13e on the
re13ord are respe13tively 950plusmn50 and 4950plusmn50 mi13rons
This estimation of the sho13k average speed is satisfa13tory for the re13ords
in whi13h the sho13k speed is almost 13onstant throughout its propagation Ho-
wever the variation with the time of the speed may be substantial for some
re13ords In su13h 13ases the un13ertainty in the average speed will be in prin-
13iple higher than the previous estimations Therefore I introdu13ed another
78 CHAPTER 5 RESULTS AND INTERPRETATION
method to determine the average velo13ity this velo13ity is then estimated
during the initial and in nal durations of the sho13k propagation by deriving
the slope of points on the rst middle and last few fringes respe13tively
The standard deviation estimated by this method is then termed as the
un13ertainty in the average speed of respe13tive re13ord
Using these two methods I have obtained two values of un13ertainties for
all the MAIN and AUX sho13ks observed in the experimental re13ords Finally
the largest un13ertainty value of the two is 13onsidered as the nal un13ertainty
in the estimation of the respe13tive speed and is shown by error bars in Fig
57a and 57b 13orresponding to the MAIN and AUX sho13ks respe13tively
The table 51 reports for dierent noble gases the values of the velo13ities
re13orded in several shots for the 13ounter-propagating sho13k waves In addi-
tion one re13ord (48131) 13orresponds to the 13ase of an isolated sho13k wave
laun13hed by the MAIN laser The values of the MAIN and AUX laser ener-
gies are also reported In some 13ases the glue on the window prevented the
probe laser to pass through the 13hannel Then it be13ame impossible to derive
any value for the velo13ity In su13h 13ases I put `NA for the 13orresponding
sho13k velo13ity
It may be noted that the sho13k speeds generated by the MAIN laser vary
in the range of 30-55 kms while for the sho13k originated from AUX laser
the speeds vary between 10 and 30 kms
Variations with laser energy and pressure the 13ase of Xenon
When the number of re13ords is su13ient to make a statisti13al analysis we
may analyse the variations of the sho13k speeds with the laser energy of
the MAIN and AUX lasers This is the 13ase for Xe or XeHe (90-10)
mixture and for gas pressures ranging between 01 and 02 bar At 01 bar
all the measurements 13on13ern Xenon (ρ= 54 times 10
minus4g13m
minus3) At 02 bar
the re13ords 13on13ern a mixture XeHe (90-10 ρ=10 times 10
minus3g13m
minus3)
ex13ept for one point at 121 J whi13h 13orrespond to the 13ase of pure Xe (ρ =
108 times 10
minus3g13m
minus3) We do believe that taking into a1313ount the pre13ision
of our re13ord the introdu13tion of tra13es of Helium do not ae13t mu13h the
dynami13s of the sho13k wave as the 13orresponding variation of mass density
is negligible
As expe13ted the sho13k speed in13reases with the laser energy and de13rea-
ses with the pressure (Fig 57a) A linear t gives the following dependen13e
of the speed in kms with the MAIN laser energy (in J) at 01 and 02 bar
respe13tively
3
v01bar = 1423 + 030E (54)
3
Two outlier points have been dis13arded from the set 13orresponding to 01 bar They
are shown for information on the Figure
51 VISIBLE INTERFEROMETRY 79
(a)
(b)
Figure 57 (a) MAIN sho13k speed (13al13ulated by the last fringe method) versus
the MAIN energy for Xenon or XeHe mixture at dierent pressures with the error
bars (b) AUX sho13k speed versus AUX energy also for Xenon or XeHe mixture
at dierent pressures
80 CHAPTER 5 RESULTS AND INTERPRETATION
v02bar = minus1405 + 047E (55)
The speeds generated by the AUX laser are lower than the previous ones
Although the re13ords are more sparse we note that the speeds at 01 bar
also in13rease with the laser energy However at 02 bar I 13ould not nd
similar trend due to the la13k of re13ords
Variations with the gas
As 13an be seen from the Table 51 we have also performed several shots in
Ar at 01 02 03 and 08 bar one shot in He at 05 and one in Kr at 02
bar
To 13ompare the velo13ities obtained for the dierent gases one 13an either
keep the pressure 13onstant (ie the number of atoms) or the mass density
For a given mass density and laser energy the velo13ity should be the same
in the adiabati13 limit However due to the radiation 13ooling and ionisation
this 13an not be the 13ase Nonetheless if we 13ompare the two re13ords of Ar
at 03 bar (ρ = 493 times 10
minus4g13m
minus3 E = 121 J) with Xe at 01 bar (54
times 10
minus4g13m
minus3 E = 133 J) we measure similar speeds (49 and 54 kms)
These two re13ords are thus interesting to 13ompare This will be done later
on in this 13hapter
More generally we note that the speed of Helium at 05 bar is higher by
sim 40 than for Ar at the same energy (see Fig 58) This is not surprising
for this lighter element (Table 52) However it is important to note that
the la13k of shots prevents us to make any pre13ise 13on13lusion
Ex13ept for this 13ase and for the unique re13ord of Ar at 02 bar whi13h
gives also a higher velo13ity the velo13ities of the other re13ords follow more or
less the variation with the energy than Xenon
To 13ompare the radiative ee13ts it seems justied to perform the 13om-
parison at a given sho13k speed independent of the laser energy and for mass
densities whi13h are 13lose to ea13h other In the following we shall thus 13om-
pare the results of Xe at 01 bar (54 kms 54times 10
minus4g13m
minus3) Ar at 03
bar (49 kms 49times 10
minus4g13m
minus3) and Kr at 02 bar (55 kms 68 times 10
minus4
g13m
minus3)
Line averaged ele13tron density lt Ne gt
I have estimated the ele13tron density lt Ne gt of the plasma averaged along
the path of the probe laser beam by analyzing the interferogram re13ords
assuming the same se13tion d of the plasma layer of 600 microm for both MAIN
and AUX side sho13ks
51 VISIBLE INTERFEROMETRY 81
Shot Gas Pressure E3ω MAIN sho13k Eω AUX sho13k
(bar) (J) speed (kms) (J) speed (kms)
at 298 K
48033 Air 03 124 54plusmn1 66 20plusmn248034 Air 03 131 52plusmn1 74 27plusmn248076 Ar 05 100 41plusmn1 66 18plusmn148077 Ar 01 115 42plusmn1 67 18plusmn148078 Ar 03 112 46plusmn3 65 25plusmn248079 Ar 03 121 49plusmn5 67 23plusmn348080 Ar 08 103 38plusmn2 62 21plusmn248081 Ar 08 113 38plusmn1 68 NA
48082 Ar 08 107 36plusmn1 65 NA
48141 Ar 02 111 63plusmn1 57 NA
48083 He 05 106 57plusmn3 63 NA
48146 Kr 02 125 55plusmn2 53 NA
48043 Xe 03 138 57plusmn1 72 25plusmn148051 Xe 01 123 58plusmn1 67 30plusmn148055 Xe 01 133 54plusmn3 68 22plusmn348057 Xe 01 127 53plusmn1 68 23plusmn148058 Xe 01 115 48plusmn3 63 18plusmn148059 Xe 01 116 50plusmn1 67 21plusmn148061 Xe 01 127 53plusmn1 67 17plusmn248065 Xe 01 122 52plusmn4 68 23plusmn348066 Xe 01 114 50plusmn3 67 15plusmn248067 Xe 05 115 39plusmn2 65 NA
48068 Xe 05 109 36plusmn2 65 12plusmn248070 Xe 05 109 33plusmn3 65 NA
48138 Xe 02 121 45plusmn5 0 0
48131 Xe +He 02 112 38plusmn1 0 0
48132 Xe +He 02 118 41plusmn4 56 18plusmn248133 Xe +He 02 112 41plusmn3 56 NA
48134 Xe +He 02 111 38plusmn1 60 NA
48136 Xe +He 02 115 39plusmn3 59 14plusmn148143 Xe +He 06 123 39plusmn4 63 18plusmn548144 Xe+He 02 133 45plusmn3 66 NA
Table 51 Sho13k speeds estimated from the `last fringe method `NA represents
the entries whi13h 13ould not be dedu13ed from the re13ord Further entries in the
bold font are dis13ussed in detail in this 13hapter
82 CHAPTER 5 RESULTS AND INTERPRETATION
Helium Argon Krypton Xenon
Atomi13 Number 2 18 36 54
Atomi13 Mass 4 3995 8380 13129
First Ionization Energy (eV) 246 157 14 12
Density (10
minus4g13m
minus3) at 01 bar 016 164 344 539
Table 52 Atomi13 data and density at 01 bar (at room temperature) for He Ar
Kr and Xe
Figure 58 MAIN sho13k speed (13al13ulated by the last fringe method) versus the
MAIN laser energy for Xe (at 01 bar only) Ar He and Kr at dierent pressures
with the error bars
51 VISIBLE INTERFEROMETRY 83
It should be noted that the beam se13tions dMAIN and dAUX of MAIN
and AUX lasers are approximately 600 and 300 microm respe13tively on the two
pistons and that the transverse horizontal se13tion of the sho13k tube is equal
to dtube = 600 microm Thus the value of lt Ne gt 13omputed for the sho13k wave
laun13hed by MAIN should be 13lose to the physi13al lo13al value Ne supposing
that the plasma is homogeneous along the transverse se13tion However
the value obtained for AUX is 13ertainly larger by a fa13tor whi13h 13an be
estimated at a rst step as the fra13tion of dtubedAUX giving a fa13tor of
about 2
As explained in the previous se13tion the pixels representing the positions
of the fringe maxima have been already determined with the best possible
pre13ision Therefore it is possible to 13al13ulate the relative phase shift (with
respe13t to zero time) variation along y-axis (time axis) at ea13h fringe maxima
Zero time is time of MAIN and AUX laser rival on the target With these
phase shifts estimated at ea13h pixel of ea13h fringe I then derived lt Ne gtusing Equation A22
Let us take the 13ase of three shots in Xenon for illustrating the method
whi13h will be followed (see the left panel of the Figure 59) The maximum
density is estimated to be sim11 times 1019cmminus3(13orresponding to phase shift sim
16) The number of subdivisions is taken to 5 bins with the following phase
(∆φ) ranges (in the units of 2π) and average ele13tron densities (ltNegt)
bin 1 ∆φ le 06 lt Ne gtle 39 10
1813m
minus3(white)
bin 2 06 - 08 39 - 57 10
1813m
minus3(red)
bin 3 08 - 11 57 - 75 10
1813m
minus3(blue)
bin 4 11 - 13 75 - 93 10
1813m
minus3(green)
bin 5 gt 13 gt 93 10
1813m
minus3(magenta)
All the re13ords shown in the Table 51 have been pro13essed using this
method The limit of dete13tion for the phase shift 13orresponds to 2 pixels
giving a threshold for the Ne measurement lt Nemin gt This threshold
diers from one re13ord to another due to the variation in the number of
fringes (and thus the distan13e between two unperturbed fringes)
In order to make a 13omparative investigation of the sho13ks propagation
and intera13tion in dierent gases as already mentioned previously I sele13ted
three 13ases (shot48055 shot48132 and shot48138) relative to Xe (or Xe-
He mixture) one 13ase relative to Ar (shot48079) and Kr (shot48146)
The 13orresponding experimental 13onditions are noted in bold in the Table
51
84 CHAPTER 5 RESULTS AND INTERPRETATION
lt Ne gt in Xenon
The Fig 59 reports the interferometri13 re13ords for the three Xenon 13ases
The top and the middle panels 13orrespond to the propagation of two 13ounter-
streaming radiative sho13k waves at 01 (shot48055) and 02 bar (48132)
For 13omparison one re13ord (48138 bottom panel) is dedi13ated to the pro-
pagation of single sho13k (MAIN) at 02 bar The Xe-He mixture (90 - 10
in numbers of atoms) is used for the investigation in the 13ase of 02 bar
pressure while it is only Xe in the 13ase of 01 bar As mentioned previously
at the pre13ision of our re13ords we 13onsider that this impurity 13on13entration
has a negligible ee13t on the sho13k speed and the pre13ursor ele13tron density
The limit of dete13tion of lt Ne gt over the se13tion of the tube (06 mm) is
13orresponds respe13tively to 9 times 10
17 6 times 10
17and 6 times 10
1713m
minus3for the
Figures 59(a) (b) and (13)
The variations of lt Ne gt with the distan13e along the sho13k tube (ie
parallel to the dire13tion of the sho13ks) are reported in the right panel of Fig
59 at 10 ns (in red) 20 ns (in blue) 30 ns (in green) and 40 ns (in magenta)
The intera13tion between the two pre13ursors is 13learly visible at 01 bar
(Fig 59(a)) at 10 ns the intera13tion of the 13ounter-propagating sho13ks
has either not yet started or is below the sensitivity of this diagnosti13 The
intera13tion o1313urs at later times with a typi13al signature whi13h is as follows
the slope of lt Ne gt is de13reasing from the left (MAIN pre13ursor) passes
through a minimum and in13reases at the right (AUX) The minimum itself
in13reases with time up to 7 times 10
1813m
minus3at 40 ns
At 02 bar we have not been able to re13ord the 13ollision in the temporal
range (50 ns) of the streak However the two re13ords (with MAIN only and
with the two sho13k waves) indi13ate a pre13ursor for MAIN The two gures
(Fig 59(b)) and (Fig 59(13)) show that the pre13ursor of the MAIN sho13k
wave is not inuen13ed by the presen13e of AUX sho13k wave up to 45 ns
There is no obvious indi13ation about a pre13ursor for AUX in the 13ase of
two 13ounter-propagating sho13k waves (Fig 59(b)) At this pressure and
13ompared with the previous 13ase at 01 bar the absen13e of pre13ursor for
AUX may be attributed to i) a low sho13k speed (18 kms) 13ombined with a
larger pressure (hen13e for a given gas the pre13ursor extension in13reases with
the speed and de13reases with pressure) ii) a too small longitudinal extension
of the eventual pre13ursor (see Fig 59(b)) 13ompared with the resolution of
20 mi13rons (2 pixels) Our 1D numeri13al simulations with Xenon opa13ity
multiplier times 20 (not presented here) indi13ate a small pre13ursor for AUX
sho13k At 15 ns its extension is 50 microm (900 microm for MAIN sho13k) with
a typi13al ele13tron density sim 35 times 10
1913m
minus3(23 times 10
1913m
minus3for MAIN
sho13k) whi13h does not agree with the re13ord At 42 ns the pre13ursor of MAIN
rea13hes the AUX sho13k front and the prole is similar to the prole at 20 ns
shown in Fig 313a at 01 bar with a plateau of almost 13onstant ele13tron
density between the two fronts This might be 13ompatible with small bending
51 VISIBLE INTERFEROMETRY 85
Figure 59 Left panel interferometri13 re13ords 48055 in Xe at 01 bar (a)
48132 in Xe+He at 02 bar (b) and 48138 in Xe+He at 02 bar (13) Right
panel ele13tron density at 10 20 30 and 40 ns versus distan13e for these re13ords
The positions of maxima have been identied on the re13ords in the left panel
The time t = 0 13orresponds to the time of laser arrival on the target and the
position x = 0 13orresponds to the interfa13e between the piston (Au layer) and the
gas Its determination is pre13ise within 100 mi13rons The distan13es between two
unperturbed fringes for re13ords 48055 48132 and 48138 are 159 244 and 244
microm respe13tively The lt Ne gt un13ertainty (plusmn 2 pixels) is indi13ated by the error bar
in the right panels It 13orresponds respe13tively to plusmn 9 times 10
17 plusmn 6 times 10
17and plusmn
6 times 10
1713m
minus3for the gures (a) (b) and (13) The limit of dete13tion (2 pixels) is
presented by a dotted line on ea13h gure
86 CHAPTER 5 RESULTS AND INTERPRETATION
of the 4
thfringe (from the right) between 45 and 50 ns As 1D simulations
are known to overestimate the pre13ursor ele13tron density 2D simulations are
ne13essary for a more pre13ise interpretation of the experimental result
lt Ne gt in Argon and in Krypton
As seen in the Fig 58 the variation with the laser energy of the sho13k speed
in Argon at 03 bar (in the red dashed 13ir13le) is in good agreement with the
13orresponding one of Xenon at 01 bar (blue linet) As these two 13ases have
a 13omparable mass density of sim 5 times 10
minus4g13m
3 this qualitative agreement
is expe13ted
Taking into a1313ount the larger pressure (and thus density) the slope of
the variation of the slower sho13k speeds in Ar at 08 bar (in a red dash-
dot 13ir13le) with the laser intensity is 13ompatible with the previous trends
However any 13on13lusion 13an not be derived for Argon at 02 and 05 bar
owing to the s13ar13ity of the re13ords
In Fig 510 the interferometri13 re13ord for Ar at 03 bar (shot 48079
49 times 10
minus4g13m
minus3) is presented The estimated speeds of MAIN and AUX
sho13ks dedu13ed from the last fringe method are respe13tively equal to 49 and
23 kms whi13h are 13lose to the speeds of 54 and 23 kms re13orded in Xe
at 01 bar (shot 48055 539 times 10
minus4g13m
minus3 Fig 59(a)) Contrarily to
the Xenon 13ase we do not see any signi13ant bending in the fringes The
maximum fringe shift is reported in red on the fringe 1 The 13orresponding
ele13tron density is equal to 38 times 10
1813m
minus3
Thus obviously the radiation ee13ts are less important for Argon than
for Xenon at the same density and speed Part of this dieren13e 13ould be
attributed to the huge dieren13e in the atomi13 numbers (40 and 131) of the
two gases whi13h leads to dierent peak temperatures for the ions as expe13ted
from equation 84 of 13hapter 2 However the post-sho13k temperatures dedu-
13ed from the jump relations for a real gas (Chapter 2) without radiation are
very 13lose (sim 25 eV) and thus this 13ould be not the relevant explanation
To 13larify this I performed HELIOSPROPACEOS simulations for two
13ounter-propagating sho13ks at 53 and 23 kms respe13tively in Ar and Xe
at the same initial mass density (54 times 10
minus4g13m
minus3) without any opa13ity
multiplier I have also reported for information the 13ase of Xenon with
an opa13ity multiplier equal to 20 The simulated temperature proles are
shown in the Fig 511 One notes immediately that the pre13ursor extension
is smaller for Ar than for Xe The peak temperatures of the MAIN sho13k
are respe13tively sim 21 eV (16 eV for opa13ity multiplier=20) for Xe and 21
eV for Ar In all the 13ases the post-sho13k temperatures are equal to sim 11
eV This indi13ates that the temperature is not the main explanation of the
dieren13es in the pre13ursor length
More interesting are the dieren13es in the Rosseland opa13ity (here per
unit of length) whi13h is reported in the Fig 511b We see that 13lose to the
51 VISIBLE INTERFEROMETRY 87
(a)
(b)
Figure 510 (a) Interferometri13 re13ord in Ar at 03 bar (48079) The estimated
speeds for MAIN and AUX are 49plusmn5 and 23plusmn3 kms (b) Same re13ord where the
fringes maxima are marked by points The bins denition is as fellows bin 0 Ne le11 times 10
1813m
minus3(white) bin 1 11 - 18 times 10
1813m
minus3(yellow) and bin 2 39 - 57
times 10
1813m
minus3(red) The time t = 0 13orresponds to the time of laser arrival on the
target and the position x = 0 13orresponds to the interfa13e between the piston (Au
layer) and the gas Its determination is pre13ise within 100 mi13rons
88 CHAPTER 5 RESULTS AND INTERPRETATION
(a)
(b)
Figure 511 Results for temperature (a) and Rosseland opa13ity (b) obtained from
HELIOS simulation at 10 ns for two 13ounter-propagating sho13ks at sim 50 and 18
kms for Xe (with opa13ity multiplier 1 and 20) Ar and Kr at 54 times 10
minus4g13m
minus3
initial mass density A 13omparison Ar Kr and Xe PROPACEOS opa13ity shown in
APPENDIX B
51 VISIBLE INTERFEROMETRY 89
front the Argon opa13ity is 13lose to 4 13m
minus1 whi13h means that the radiation
from the sho13k is absorbed within 25 mm This has to be 13ompared with the
Xenon 13ase (no multiplier) where it is equal to sim 03 13m
minus1 whi13h means
that the hot pre13ursor is almost transparent over the length of the tube
Thus the dierent behaviors of the opa13ity in the upstream gas explain the
quantitative dieren13e in the development of the pre13ursor The sharp peak
of the Xenon opa13ity with the multiplier of times 20 is due to the strong rise
of the opa13ity when the temperature de13reases below 5 eV whi13h marks the
end of the pre13ursor
I will present now the 13ase of two 13ounter-propagating sho13ks in Kr
(48146 in Fig 512) at 02 bar (69 times 10
minus4g13m
minus3) In this 13ase unfortu-
nately the AUX sho13k 13ould not be imaged 13ompletely due to the presen13e
of glue at the right edge of the target 13ell but the 13ollision was re13orded
at sim 40 ns The speed of the MAIN sho13k is estimated to be 55 kms
Thus despite a slightly higher initial mass density this sho13k wave may be
13ompared with the shot (48055) in Xe at 01 bar (54 times 10
minus4g13m
minus3 54
kms)
Despite the poor quality of the re13ord and the inadequate temporal range
(150 ns) sele13ted on the streak 13amera we note some bending in all the
fringes indi13ating the presen13e of pre13ursor The maximum fringe deviation
(063 of the distan13e between two fringes) is noted at the end of fringe 2
whi13h 13orresponds to the lt Ne gt value 45 times 10
1813m
minus3 The pre13ursor
seams to be more developed here than for the previous Ar 13ase
The result of the simulation in Xe and Kr with equal initial mass density
54 times 10
minus4g13m
minus3 and for the two respe13tive sho13k speeds of 50 and 20
kms is reported in the Figure 511 The peak temperatures of the MAIN
sho13k are respe13tively sim 21 eV (16 eV for opa13ity multiplier = 20) for Xe
and 22 eV for Kr The post-sho13k temperatures (sim 10-11 eV) are 13lose from
ea13h other
The pre13ursor extension in Krypton is smaller than in Xenon whi13h is lo-
gi13al if we 13onsider the opa13ity values (Fig 511b) However if we take into
a1313ount the multiplying fa13tor for the Xenon opa13ity the Krypton pre13ursor
is now more extended than for xenon whi13h is in 13ontradi13tion with the ex-
periment This probably means that either the Kr opa13ity is under estimated
by PROPACEOS or that our multiplier for Xenon is overestimated
Synthesis
The previous experimental investigation 13onrms that the sho13k velo13ity in-
13reases with the laser energy and de13reases with the mass density Moreover
for our given sho13k velo13ity and mass density the extension of the radiative
pre13ursor in the experiment de13reases with the atomi13 number Among Ar
Kr and Xe this last one appears to be the most adapted for the investiga-
90 CHAPTER 5 RESULTS AND INTERPRETATION
(a)
(b)
Figure 512 (a) Raw interferometri13 re13ord in Kr at 02 bar (shot 48146) The
estimated speed for MAIN sho13k is 53plusmn2 kms The time t = 0 13orresponds to the
time of laser arrival on the target(b) Same re13ord shown in (a) maxima is marked
by 13olored points The bins denition is as follows bin 0 Ne le 11 times 10
1813m
minus3
(white) bin 1 11 - 18 times 10
1813m
minus3(yellow) and bin 2 39 - 57 times 10
1813m
minus3
(red) The time t = 0 13orresponds to the time of laser arrival on the target and
the position x = 0 13orresponds to the interfa13e between the piston (Au layer) and
the gas Its determination is pre13ise within 100 mi13rons
51 VISIBLE INTERFEROMETRY 91
tion of the radiative pre13ursor A similar behaviour is found also at ORION
with higher speeds (Clayson et al 2017)
512 Transverse interferometry lateral extension of the sho13k
The transverse interferometry is a dierent geometri13al imaging setup of the
Ma13h-Zehnder interferometer whi13h provides qualitative information about
the 13urvature lo13alisation and transverse extension of the radiative pre13ur-
sor and thus of the sho13k
In the transverse interferometry we perform the image of a transverse
se13tion (on the slit of the streak 13amera) of the tube whi13h is lo13ated at a
distan13e dslit equal to 3 mm from the initial position of the MAIN piston
As the setup is originally adapted for the longitudinal interferometry we use
a Dove prism between the va13uum 13hamber and the streak 13amera to rotate
the image by 90 degrees
Figure 513 S13hemati13s of the Ma13h Zehnder Interferometer setup to re13ord
transverse interferometri13 images The lens allows to make on the slit of the
13amera the image of a se13tion perpendi13ular to dire13tion of sho13k propagation
A transverse interferometri13 re13ord for the MAIN sho13k alone in Xe at
02 bar is reported in Fig514(a) The streak 13amera rst re13ords the unper-
turbed plasma on the se13tion dslit in the tube Then it re13ords su1313essively
the pre13ursor and the post-sho13k (whi13h is opaque to visible radiation and
appears as a dark zone in the re13ords) Taking into a1313ount the oset of 14
ns the time of sho13k arrival is re13orded at 72 ns after the time t0 of laser
arrival on the target and the sho13k speed is estimated to be sim 35 kms Due
to glue on one lateral window (on the right part of the gure) only 6 fringes
92 CHAPTER 5 RESULTS AND INTERPRETATION
Figure 514 Transverse interferometri13 images for (a) shot48111 (MAIN sho13k
only) (b) shot48130 (AUX sho13k only) The time is measured after an oset
equal to 14 and 23 ns respe13tively after the time of the laser arrival on the target
The position zero on the x-axis of ea13h image 13orresponds to the base of the target
are visible The lateral extension of the sho13k stru13ture at this time is deri-
ved to be sim 570plusmn30 microm in qualitative agreement with the spe13i13ations of
the MAIN phase plate and the shape of the pre13ursor is relatively at The
axis of symmetry of the sho13k is determined to be at sim 350 mi13rons from
the base of the target (ie 50 microm above the nominal value of 300 microm)
A re13ord for the AUX sho13k alone is shown in Fig 514(b) whi13h 13orre-
sponds to a gate opening of 50 ns The start time of the image has an oset
of +23 ns from t0 and the distan13e dslit is set to 700 mi13rons from the initial
position of the AUX piston The AUX sho13k duration extends from 30 ns
to at least 34 ns after t0 The sho13k speed is then estimated to be ranging
between 23 and 20 kms The shape of the pre13ursor is strongly bent and
we note a tiny shift of the se13ond and third fringes (starting from the left)
at sim 8 ns It may further be noted that the lateral spread of the opaque
sho13k is ranging between 275plusmn25 microm (whi13h is also in agreement with the
spe13i13ations of the AUX phase plate) and that the axis of symmetry of the
sho13k system is also lo13ated at about 350 microm from the bottom of the 13ell
52 XUV Spe13tros13opy
XUV emission originates from the 13omparatively hot portions of the plasma
In the present 13ase it 13orresponds to the post-sho13k and the pre13ursor re-
53 SIMULATIONS BASED ON EXPERIMENTAL RESULTS 93
gion whi13h is 13lose to the sho13k front The spe13trum re13orded by the XUV
spe13trometer enables us to analyze the spe13tros13opi13 signatures of the radi-
ative sho13k One of the aims of this diagnosti13s was to identify the sho13ks
13ollision whi13h leads to higher temperatures through 13omparing the spe13tra
obtained for the 13ases of single and 13ounter-propagating sho13k waves Howe-
ver only a few re13ords were possible to be obtained during the experiment
and unfortunately the 13omparison with the 13ase of an isolated sho13k wave
was not performed
Among the shots re13orded the XUV spe13trum of the shot 48143 is
presented herewith in detail This shot was performed for [Xe (90)+ He
(10) mixture at 06 bar with laser energies of 123 J for MAIN and 63 J
for AUX The interferometri13 re13ord of this shot is shown in Fig 515 The
MAIN sho13k speed has been estimated to be sim 39 plusmn 4 kms The estimated
AUX sho13k speed (18 plusmn 5 kms) is not pre13ise due to the presen13e of glue
on the right se13tion of the re13ord (note Fig 515) In this interferometri13
re13ord we have not been able to retrieve the 13ollision time However an
extrapolation of the slope 13orresponding to the speeds sim 39 kms (MAIN)
and sim 18 kms (AUX) enables us to approximately determine the 13ollision
time to be between 60 to 65 ns
The raw spe13trum (shown in Fig 516a) re13orded for the wavelength
range of 15-35 nm (35-82 eV) shows the `L edge of Aluminum at 17 nm
(34 nm in se13ond order) in the rst and se13ond (34 nm)orders These two
wavelengths will be used for the wavelength 13alibration The net spe13trum
13orre13tion for the transmission (Henke Gullikson and Davis 1993) of the
100 nm thi13k Si3N4 window (3 mm times 04 mm) is introdu13ed in Fig 516b
A remarkable feature is a strong absorption dip between 19 and 22 nm (56-65
eV) This absorption probably 13omes from the 13older layers (thi13kness 300
microm) between the sho13k heated plasma and the Si3N4 window Few lines
of Xe VII-VIII are identied through NIST database
4
as also Oxygen IV
and V lines Lyman lines of He II (from 1-2 to 1-7) are also present in the
spe13trum This information will be useful for the estimation of the ele13tron
temperature
53 Simulations based on experimental results
In this se13tion I will 13ompare the experimental sho13k 13hara13teristi13s with the
results of HELIOS simulations using the PROPACEOS equation of state and
opa13ity (limited to 1 group) As indi13ated in the 13hapter 3 this opa13ity has
been multiplied by 20 for the Xenon only As our interest is to understand the
sho13k stru13ture in Xenon and not the laser matter intera13tion on the piston
we performed several simulations with Xe gas for various sets of uen13es for
4
[httpphysi13snistgovPhysRefDataASDlines_formhtml
94 CHAPTER 5 RESULTS AND INTERPRETATION
Figure 515 Interferometri13 image for the shot48143 The time t = 0 13orre-
sponds to the time of laser arrival on the target and the position x = 0 13orresponds
to the interfa13e between the piston (Au layer) and the gas Its determination is
pre13ise within 100 mi13rons
53 SIMULATIONS BASED ON EXPERIMENTAL RESULTS 95
(a)
(b)
Figure 516 Raw (a) and 13orre13ted XUV spe13trum (b) for the shot 48143
96 CHAPTER 5 RESULTS AND INTERPRETATION
the MAIN and AUX beams in order to obtain the best agreement between
the simulated and measured sho13k speeds
To analyse the results from the shot 48055 (Fig 59(a)) we set the u-
en13es to 32000 amp 7500 J13m
2 This allows produ13ing the experimental sho13k
speeds 54 and 23 kms in Xenon at 01 bar for the MAIN and AUX beams
respe13tively The two sho13ks appear in Xenon at 2 and 3 ns respe13tively for
MAIN and AUX The merging of the two pre13ursors starts at sim 15 ns and
the sho13k 13ollision time o1313urs at 47 ns In Fig 517 we present the ele13tron
density proles from the simulation (dotted lines) and the experiment (solid
lines) at 10 20 30 and 40 ns
At 10 ns the two simulated pre13ursor extensions are 0165 and 0022 13m
for MAIN and AUX respe13tively The ele13tron density is larger by a fa13tor
of 4 than in the experiment The shapes of the pre13ursors are also very dif-
ferent However this 1D pi13ture supposes the plasma to be uniform within
the tube In reality in the transverse dire13tion Ne is de13reasing from the
13enter to the walls whi13h results in a lower estimation of the average value
(lt Ne gt) 13ompared to the value at the target 13enter and in a smoother pro-
le near 02 amp 035 13m at 10 ns It is also important to note that for AUX
sho13k the average lt Ne gt value underestimates the lo13al one by a fa13tor
of about 2 (as it is averaged over 06 mm instead of 03 mm) Moreover
our 1D simulation suers from an inexa13t opa13ity and 2D ee13ts are pro-
bably important espe13ially for AUX Thus we have here only a qualitative
interpretation of the experimental results
The intera13tion between the two HELIOS radiative pre13ursors starts be-
tween 10 and 20 ns like in the experiment However the shape as well as
absolute values of the simulated ele13tron density 13urves are not in agreement
with the experimental results and the intera13tion is stronger in the simulation
than in the experiment
In order to interpret the spe13tros13opi13 data presented in se13tion 52 we
performed another 1D simulation in Xenon at 06 bar and adapted the
uen13es to generate two 13ounter-propagating sho13ks with the speeds 36 and
18 kms 13lose to the experiment The time evolutions of the ele13tron density
mean 13harge and ele13tron temperature at 56 57 58 60 64 and 65 ns are
presented in Fig 518 The two sho13ks appear in Xenon at 2 and 3 ns
respe13tively for MAIN and AUX Con13erning AUX the 13ombination of a
small speed and a relative high pressure does not allow to develop a radiative
pre13ursor in agreement with the experimental results (Fig 515) whereas
the MAIN sho13k has a pre13ursor and its length is in13reasing with time
The post sho13k temperature of the MAIN is sim 21 eV and the ion 13harge
sim 9 At 57 ns the pre13ursor of MAIN rea13hes the AUX sho13k front This
time is out of our re13ord (see Fig 515) whi13h means that the intera13tion
ee13t is either absent or o1313urs at later times The stru13ture of the AUX
post sho13k is modied by the intera13tion with the MAIN pre13ursor (Fig
518b) The sho13k 13ollision o1313urs at 65 ns (Fig 518a) resulting in the
54 SUMMARY 97
Figure 517 Re13orded ele13tron density (shot 48055) together with the HELIOS
results (with Xenon opa13ity times 20) at dierent times in Xenon at 01 bar
development of two reserve sho13k waves At the 13ollision time the ele13tron
density mass density ele13tron temperature and ion 13harge rea13h respe13tively
sim 14 times 10
2113m
minus3 0034 g13m
minus3 26 eV and 10 Mean 13harge is varying
between 5-10 whi13h 13ompatible with the presen13e of lines of Xe VII-VIII in
the experimental re13ord shown in Fig 516b
Moreover in order to interpret XUV spe13tros13opi13 results shown in se13tion
52 Rodriguez performed qualitative preliminary 13omputations (as des13ribed
in Rodriacuteguez et al (2015)) of the XUV spe13tra emerging from a 600 microm thi13k
plasma with two representative values of the mass density ρ = 32 times 10
minus2
and 33 times 10
minus3g13m
3 They show that the lines of HeII 13an only be ob-
served at a temperature of sim 15 eV and for the lowest density ie in the
radiative pre13ursor
54 Summary
In this 13hapter I have presented an extensive data analysis of few representa-
tive interferometri13 and spe13tros13opi13 re13ords The average sho13k speed and
ele13tron density have been estimated from the interferograms The sho13k
speeds of the MAIN and AUX radiative sho13k waves vary between sim 30-55
and 10-30 kms respe13tively and the averaged pre13ursor ele13tron density
ranges between 10
17and 10
1913m
minus3during the sho13ks propagation
We have demonstrated the intera13tion ee13t between the two radiative
pre13ursors in the 13ase of Xe at 01 bar at 54 and 23 kms The intera13tion is
13learly 13hara13terized in the experiment by the enhan13ement of the ionisation
wave followed by the merging of the two radiative pre13ursors at 20 ns The
13ollision time is re13orded at 47 ns Su13h behavior is reprodu13ed by the
98 CHAPTER 5 RESULTS AND INTERPRETATION
(a)
(b)
(13)
Figure 518 Time evolution of the mass density (a) ele13tron temperature (b) and
mean 13harge (13) at 56 57 58 60 64 and 65 ns within the sho13k tube derived from
the HELIOS simulations (with Xenon opa13ity multiplier = 20) for two 13ounter
streaming sho13ks of sim 39 and 18 kms in Xenon at 06 bar
54 SUMMARY 99
simulation
We have investigated this intera13tion at a larger pressure 02 bar with
the following speeds sim 41 kms for the MAIN and sim 18 kms for the AUX
sho13k waves We do not re13ord any experimental signature of the radiative
pre13ursor for AUX Further we have not been able to 13at13h experimentally
the 13ollision time The re13orded pre13ursor of MAIN is not inuen13ed by AUX
wave up to 48 ns (Fig 59(b) and (13) in data analysis 13hapter) whi13h is the
limit of the re13ord On its side the 1D simulation predi13ts a tiny pre13ursor
for AUX and that both pre13ursors start to intera13t at 49 ns This plausible
sho13ks intera13tion o1313urring at times whi13h are outside of the re13ord 13an
not be 13onrmed by our experiment
The results of the transverse interferometry at 02 bar with speeds of sim40 and 20 kms indi13ate that the MAIN pre13ursor has a lateral extension of
sim 600 microm whereas it is 300 microm for AUX The pre13ursor of MAIN is almost
at with a probable small bending at the edges of the tube whereas the
AUX pre13ursor is more 13urved This means that the 2D ee13ts are more
important for AUX than for MAIN Still in the 13ase of Xenon we have
obtained information about the temperature and the mean 13harge of gas
through our time integrated XUV spe13tra (shot48143) in Xenon at 06
bar From these results we may 13on13lude that the mean ion 13harge is at
least equal to 6 and that the temperature has rea13hed 15 eV
Our simulations give a qualitative des13ription of the sho13ks intera13tion
when the laser uen13e is adjusted to give the 13orre13t sho13k velo13ities Howe-
ver it is now well known that 2D simulations (together with state of the art
opa13ities) t better with experiments (Gonzaacutelez Audit and Stehleacute 2009
Leygna13 et al 2006 Stehleacute et al 2010) For the same laser energy the 2D
simulations lead to a diminution of the sho13k speed 13ompared to 1D as also
to a diminution of the ele13tron density For instan13e in the 13ase of a sho13k
wave laun13hed by a laser beam at 1315 nm in Xenon at 03 bar at PALS and
with a laser uen13e of 85000 J13m
2 ARWEN 2D simulations give a sho13k
speed of 44 kms in agreement with the experimental one (Cotelo et al
2015) 1D simulation would require in this 13ase a uen13e of 30000 J13m
2to
a13hieve the same velo13ity
The spa13e and time integrated XUV re13ords at 06 bar for respe13tive
speeds whi13h are equal to sim 39 and 18 kms indi13ate that the temperature
of the sho13k has rea13hed values up to 15 eV and that the Xenon mean ion
13harge has also rea13hed values of 6 - 7 whereas 1D simulations predi13t ele13tron
temperature 10-30 eV and ion 13harge 5-10 (Fig 51813) A more detailed
study based on 2D simulation and radiative transfer post-pro13essing will be
ne13essary to rene the analysis
We have investigated the 13ase of other noble gases (Ar Kr) and we have
observed that for a given laser energy the sho13k velo13ity de13reases with the
mass density This study has 13onrmed that for a given density and sho13k
velo13ity the radiative ee13ts in13rease with the atomi13 number For Kr we
100 CHAPTER 5 RESULTS AND INTERPRETATION
noti13ed a tiny pre13ursor without any intera13tion To get su13h ee13t it will
be ne13essary to in13rease the sho13k velo13ity and thus the laser energy
This has been done at the ORION laser fa13ility in UK where the 13ol-
lision of two 13ounter-propagating sho13ks at equal speed sim 80 kms with
laser uen13e sim 6 times 10
14W13m
2has been performed (Clayson et al 2017
Suzuki-Vidal et al 2016) for dierent noble gases and for pressures 13ompri-
sed between 01 and 1 bar A number of diagnosti13s setup X-ray ba13klig-
hting opti13al self-emission streak imaging and interferometry (multi-frame
and streak imaging) were used to study both the post-sho13k and the radia-
tive pre13ursor Although I have not parti13ipated in the experiments I have
performed 1D simulations to interpret ORION experimental results This
work is not presented in this thesis
Chapter 6
Optimization of an
ele13tromagneti13 generator for
strong sho13ks in low pressure
gas
Contents
61 Prin13iples of operation of a high 13urrent generator 102
62 Prin13iples of the run-down phase in a PFD 103
63 Proposed design for the plasma gun 104
64 Dynami13 13ir13uit modelling 106
65 3-D MHD simulations using GORGON 13ode 111
66 Measurements 116
67 Summary 118
The previous 13hapters were dedi13ated to the study of laser generated
radiative sho13ks With irradian13e more than 10
14W13m
2 we were able to
study sho13k waves propagating at velo13ity up to 50 kms in noble gases with
an initial mass density 13omprised between 5 times 10
minus4- 3 times 10
minus3g13m
minus3
Complementary to laser experiments 13ompa13t pulsed power generators may
drive an ele13tromagneti13 13oaxial plasma gun to 13reate astrophysi13al relevant
sho13ks in lower pressure noble gases (Kondo et al 2006) with a high avai-
lability and a rather modest 13apital 13ost The ele13tromagneti13ally driven
sho13k waves may have larger s13ales than those by laser thus they 13an be
observed rather easily (Kondo et al 2008)
Su13h ele13tri13al pulsed power devi13es may then i) 13reate a high-voltage
breakdown through a gas or more easily at the surfa13e of a diele13tri13 ii)
produ13e relatively hot plasma by ohmi13 heating when the rising 13urrent in the
devi13e is passing through a portion of gas iii) a1313elerate plasma layers under
the magneti13 pressure asso13iated with the self-generated magneti13 eld when
high ele13tri13al 13urrent (say 10s of kA) is maintained for a short but su13ient
time (say 1 micros) Su13h ionization and a1313eleration are present in Z-pin13h
plasmas and in plasma fo13us devi13es (PFD) providing a 13lever geometry is
employed and the mass under 13onsideration (say 1 mg) is 13ompatible with
the stored energy (Martin Williams and Kristiansen 1996)
101
102 CHAPTER 6 ELECTROMAGNETICALLY LAUNCHED SHOCK
In this 13hapter we will see rst how a fast ele13tri13al 13ir13uit works then
we will dis13uss the prin13iples of a 13oaxial plasma gun and of PFD We will
see what is in favor of our obje13tive and what has to be avoided A geometry
13apable to rea13h our obje13tive will be proposed
The optimization of the 13oaxial gun (in term of plasma speed) will be
performed and some typi13al gures will be given in the 13ase of a generator
with a stored energy around 1 kJ and a plasma slab of 4-mm diameter ie
quite 10 times the transverse dimension of a laser driven RS The operating
13onditions are supersoni13 sho13ks up to 10-30 kms speed at stati13 pressures
of few mbar in heavy rare gases (Ar Xe) The results of this 0-D model will
be then 13ompared to those obtained with 3-D MHD simulations performed
with the 13ode GORGON (by Andrea Ciardi at LERMA) whi13h has been
used su1313essfully to des13ribe other pulse-power driven plasma experiments
(Chittenden et al 2004) as well as laboratory plasma astrophysi13s experi-
ments (Ciardi et al 2007) The diagnosti13s whi13h have been implemented
will be presented to illustrate the model as well as preliminary re13ords of the
plasma speed
61 Prin13iples of operation of a high 13urrent gene-
rator
Our aim being to a1313elerate a plasma slab using the magneti13 pressure it
is obviously needed to drive a high intensity 13urrent be13ause the magneti13
pressure is expressed by B
22micro0 in the region where exists an indu13tion B
The pressure in bars is simply 4B
2 with B in Teslas and in a 13ylindri13al
geometry 1 Tesla is the eld around a 50-kA 13urrent at a radius of 1 13m
Thus we should 13ount on roughly 100 kA delivered by the generator but
su13h a high 13urrent is delivered only in a pulsed mode by a laboratory s13ale
devi13e
There are numbers of te13hni13al solutions to a13hieve su13h ele13tri13al pa-
rameters one may use rather slow generators at moderate high voltage like
13apa13itors bank with the advantage of well know te13hniques but with limited
adjustments A1313ording to an abundant literature
1
a 13onvenient devi13e is a
medium-energy 13apa13itor bank feeding a low-indu13tan13e 13ir13uit An R-L-C
13ir13uit is a well-known 13ombination delivering a high peak 13urrent in the
pseudo-periodi13 mode For a 13apa13itor C initially 13harged under U0 key
performan13es are as follows
Current intensity I(t) is a damped sinusoid
Pseudo pulsation ω =radic[1(LC)minusR2(4L2)]
1
Institute for Plasma Fo13us Studies resour13e website httpplasmafo13usnet last
13onne13tion in 2016
62 PRINCIPLES OF THE RUN-DOWN PHASE IN A PFD 103
First 13urrent peak Ipeak = U0[radic(LC) + 08R]
Time of 13urrent peaking T4 = π(2ω)
Equivalent impedan13e Z =radic(LC)
For a safe handling in air a voltage not higher than 30 kV is re13ommen-
ded Taking a total 13apa13itan13e of 6 microF 2700 J are stored under 30 kV For
an indu13tan13e of 240 nH the expe13ted peak 13urrent might be 150 kA at 19
micros This set of values will be a guideline for the development presented in
further se13tions
62 Prin13iples of the run-down phase in a PFD
Our aim is thus to 13reate a fast moving plasma sheath with quite a one-
dimension geometry The issues are to initiate this plasma at the best then
to maintain even improve its stru13ture during the a1313eleration nally to
laun13h it
Instead of 13reating a gas breakdown in volume whi13h would lead ra-
pidly to 13on13entrate the 13urrent in an ar13 the idea for PFD operation is to
start from a surfa13e dis13harge (Lee 1969 Bernard 2002) In the 13lassi13al
13oaxial geometry of the Mather-type PFD the 13entral ele13trode (usually the
anode) is a 13ut metal 13ylinder and the outer one is a se13ond metal 13ylin-
der of the same length or better a squirrel 13age allowing many viewpoints
to the plasma Both ele13trodes are atta13hed to a diele13tri13 bottom plate
made of polymer or 13erami13 When the 13apa13itor bank is swit13hed on the
high voltage is applied to this 13m-size gap and a radial surfa13e dis13harge is
initiated at the interfa13e with the gas forming a quite uniform ring-shaped
layer of thi13kness around 1 mm An insulating sleeve is adjusted around
the 13entral ele13trode to for13e the plasma to ow rapidly upwards under the
magneti13 pressure in this region The result is an elongation the dis13harge
path preventing an ar13ing in the plasma layer
At later times as the 13urrent is growing the plasma layer is strongly
inuen13ed it is pushed upwards by the magneti13 pressure jtimesB it is heatedby the ohmi13 ee13t then its ionization degree in13reases it be13omes denser
as a per13entage (10-40) of the heavy parti13les from the swept volume are
a1313reted (Potter 1971) This ele13tromagneti13ally driven motion is 13alled the
rundown phase Due to the a1313retion it is des13ribed by a so-13alled snowplow
model whi13h will be dis13ussed later in detail In quite all the designs even
if very high speeds are reported (Lee 1969 Serban 1995) the plasma is bell
shaped whi13h does not fulll our requirements
At the end of the rundown phase whi13h mat13hes roughly with the 13urrent
peak the plasma stays 13onne13ting the upper ends of the ele13trodes and the
radial 13omponent of the magneti13 for13es grows rapidly Then the plasma is
104 CHAPTER 6 ELECTROMAGNETICALLY LAUNCHED SHOCK
strongly pushed to the axis giving the so-13alled Z-pin13h This stage made
the PFD popular be13ause the pin13hed plasma is a sour13e of fast ele13trons
fast ions hard radiation and possibly neutron beams it must be avoided
in our 13ase It was also quoted (Lee and Serban 1996) that optimal PFD
are mat13hing a universal fa13tor 13omprising anode radius peak 13urrent and
gas density meaning a robustness of the design when a geometry has been
13hosen
63 Proposed design for the plasma gun
The previous des13ription has shown the positive inuen13e of a surfa13e dis-
13harge the modi13ation of the shape by the sleeve and the issue of the
pin13hing ee13t A1313ordingly the following design is proposed following a
work by Kondo et al (Kondo et al 2006 2008)
To ensure a rapid dis13harge with the values given in the se13tion 61 a
highly 13oaxial 13ir13uit is 13hosen for all the large parts the energy bank the
13losing swit13h and all the 13onne13tions The initial phase is kept with two
13oaxial ele13trodes pressed on a at insulator Homogeneity of the plasma
sheath is expe13ted by 13hoosing a rather small radial gap of 25 mm The ee13t
of magneti13 for13es during the rundown phase must be enhan13ed espe13ially
due to the in13rease of the mass of the plasma as des13ribed above As the
13urrent 13urve is the rising part of a sinusoid a tri13k is used to in13rease
the lo13al magneti13 eld B around a 13ondu13tor varies as the inverse of the
distan13e to axis so the proposed shape for the anode is a 13oneHowever a
nal divergen13e must be avoided and that is obtained by a rounded tip In
order to keep 13onstant the plasma length along the 13urrent path the radial
gap is kept 13onstant so the outer ele13trode is a hollow 13one with the same
angle Above the 13one the a1313elerator is 13onne13ted to a 13ondu13ting tube
where the plasma 13an propagate freely in the ba13kground gas The distan13e
of the plasma sheath to the 13one tip is in13reasing rapidly and the main
13urrent whi13h is still high will pass preferably through the diuse plasma
remaining between the ele13trodes Thus there will be no magneti13 pressure
anymore a13ting on the sheath whi13h 13an propagate freely The resulting
devi13e is des13ribed in Fig 61a and the a13ting magneti13 for13es shown in Fig
61b
Te13hni13ally su13h a 13onguration is 13onvenient even at 15 kV be13ause the
sharp edge of the anode tou13hing the insulator forms a so-13alled triple point
where the ele13tri13 eld is enhan13ed at the surfa13e of the insulator whi13h
triggers e13iently a dis13harge in presen13e of gas at low pressure (01 10
mbar) as proposed by Kondo et al (Kondo et al 2006) Polya13etal is a
13onvenient material for diele13tri13 and me13hani13al performan13es
The dire13tion and the orientation of the magneti13 for13e are other points to
be dis13ussed The high-intensity 13urrent (up to sim 150 kA) generates a strong
63 PROPOSED DESIGN FOR THE PLASMA GUN 105
(a)
(b)
Figure 61 (a) Sket13h of the sho13k generator showing the pulsed ele13tri13al 13ir13uit
the set of 13oaxial 13oni13al ele13trodes with a 13onstant radial gap and the plasti13
insulator featured in grey on whi13h a planar surfa13e dis13harge is initiated The
installation of three opti13al bers allows looking radially at the plasma moving in
the sho13k tube (b) S13hemati13s of the plasma dynami13s inside the 13oaxial gun
in fast-pulse mode the ele13tri13al 13urrent ows in the super13ial layers of the two
13oaxial 13oni13al ele13trodes and through an annular plasma layer The magneti13
pressure Pmag pushes the dis13harge axially
106 CHAPTER 6 ELECTROMAGNETICALLY LAUNCHED SHOCK
Figure 62 Exploded view of the plasma gun
azimuthal magneti13 eld between the ele13trodes thus the magneti13 pressure
jtimesB whi13h a1313elerates the annular plasma sheath stays perpendi13ular to the
sheath lo13ally (Fig 61b) The more planar is the initial sheath the more
axial will be the magneti13 pressure Finally if the roles of the ele13trodes are
ex13hanged the dire13tion of the 13urrent will be inverted and the same for B
so the pressure will stay oriented in order to expel the plasma Compared
with the Mather-type plasma fo13us (Potter 1971 Zambra et al 2009) the
plasma sheath 13onsidered here is quite planar 13lose to the insulating surfa13e
and later is expe13ted to stay planar and perpendi13ular to the axis As in a
PFD the thi13kness of the plasma sheath in13reases gradually (Zambra et al
2009) but it is assumed to stay in the mm-range We employ 13oaxial 13oni13al
brass ele13trodes ea13h 42-mm high with a 13onstant gap of 25 mm At the
bottom level internal and external radii of ele13trodes are 125 mm and 15
mm respe13tively This small-size dis13harge 13ell on top of the generator will
be modelled in the following
64 Dynami13 13ir13uit modelling
We optimize the ele13tri13 generator for various gases namely Ar and Xe
with the motivation to produ13e plasma sho13ks with speeds sim 1 - 30 kms
ie Ma13h numbers up to 200 For that we design the ele13trodes and set
64 DYNAMIC CIRCUIT MODELLING 107
the ele13tri13al 13ir13uit parameters to produ13e 1-micros pulses in the gas 13hamber
The geometry of the ele13trodes is as mentioned above The half angle of the
13oni13al ele13trode is 13hosen to be α =15deg and thus the height of the 13one is h
sim 42 mm
Eleven (N = 11) 13apa13itors ea13h with 13apa13itan13e C0 of 06 microF are13onne13ted in parallel giving an equivalent 13apa13itan13e of the bank C = 66
microF 13harging voltage U0 of the bank is set to 15 kV Thus the available
ele13tri13al energy is 750 J The 13ir13uit resistan13e is used to be 1 m Ω and
damping resistan13e is set to zero whi13h yields a total resistan13e (R = Rcir+
Rdamp N ) of 1 mΩ and a pseudo-periodi13 regime is expe13ted
Based on the above geometri13al and ele13tri13al parameters we have esti-
mated the time evolution of various parameters of the radial dis13harge layer
namely speed mass a1313eleration et13 at a dierent position along the axis
from z = 0 to h For this 13al13ulation we need to solve a set of three equa-
tions whi13h give the main 13urrent (- dqdt) supposed to ow only through
the plasma layer the mass a1313retion rate (dMdt) and the velo13ity v of
the plasma These equations together with initial 13onditions are explained
below The ele13tri13al equation is written 13lassi13ally
Ld2qdt2 +Rdqdt+ qC = 0 (61)
When the R L and C parameters are not time-dependent the 13ir13uit
equation 61 has an analyti13al solution a damped sinusoid for the 13urrent I
= -dqdt and that was used as a test of the iterative solving pro13ess and to
13onrm the parameters of the external 13ir13uit with a short 13ir13uit load The
indu13tan13e of the external 13ir13uit L0 is 13omputed from the ringing frequen13y
in short 13ir13uit (66 nH) whi13h results in a 13ir13uit impedan13e Z0 =
radic
L0Cto be 01 Ω In the plasma gun mode the sheath 13arrying the 13urrent (see
Fig 61b) is a1313elerated by the magneti13 pressure and it behaves as the de-
formable part of the 13ir13uit then L is time-dependent and a numeri13al solver
is ne13essary for equation 61 We used a lumped-parameter model whi13h has
proven to des13ribe similar 13ir13uits su1313essfully (Potter 1971 Zambra et al
2009) Gonzalez (Gonzalez et al 2004) showed that the mass and momen-
tum equations for a 13urrent sheath in the shape of an annular piston moving
forward in the axial dire13tion 13an be given by
dMdt = ερ0π(R2e minusR2
i)v (62)
where ε is a1313retion fa13tor and that the transformation of magneti13 to
kineti13 energy is expressed by
d(Mv)dt = lI22 (63)
where l is the linear indu13tan13e of the gun and 13an be expressed as l =
(micro04π) ln (ReRi) the radii being estimated where the plasma sheath is
13urrently lo13ated The equations 62 and 63 have been introdu13ed in the
108 CHAPTER 6 ELECTROMAGNETICALLY LAUNCHED SHOCK
(a)
(b)
(13)
Figure 63 Time dependen13e of the main parameters for Ar gas at 1 mbar a1313or-
ding to the 13ir13uit model (a) 13urrent (kA) and voltage (V) (b) a1313reted mass
(kg) and rate of a1313retion (kgs) (13) speed (ms) and kineti13 energy (J)
64 DYNAMIC CIRCUIT MODELLING 109
Figure 64 Plot of the dierent output parameters (normalized with respe13t to
the peak values) obtained at the top of the inner ele13trode For ea13h group upper
panel represents the variation for Ar gas whereas lower panel is for Xe gas Proles
of output parameters are given (a) with 13one angle (b) with a1313retion fa13tor (13)
with gas pressure and (d) with damping resistan13e
110 CHAPTER 6 ELECTROMAGNETICALLY LAUNCHED SHOCK
time-dependent iterative solving of equation 61 Therefore from equations
62 and 63 we get
dvdt =[ lI
2
2 minus ερ0π(R2e minusR2
i)v2]
M(64)
From equation 64 we may note that initial value of mass M(t0) 13annot
be set to zero as it would lead to a nonphysi13al divergen13e of the a1313eleration
In this regard we have 13onsidered an initial thi13kness of the 13urrent sheath
of 1 mm and integrated the gas density at rest leading to M(t0) equal to
35 times 10
minus10kg for Ar and 13 times 10
minus9kg for Xe A higher limit has to be
put too M(t0) must be small 13ompared with the mass at the end of the
laun13hing phase that is the mass 13ontained in the gap times the a1313retion
fa13tor The above value for M(t0) follows that 13riterion and within these
limits the 13hoi13e of initial mass is not inuen13ing the results
Parameter Referen13e Value Variation
Half angle of the 13one 15
13
14
15
16
middot middot middot 24A1313retion fa13tor ǫ 04 01 02 03 middot middot middot 09
Pressure P 100 Pa 10 20 50 100 200 500 1000
Damping resistan13e Rdamp 0 ohm 0 01 02
Table 61 Input parameters their standard values and range of variation
As the motion is mainly along the z-axis the speed v is the derivative
of the plasma position v = dzdt We solved these equations by a time-
dependent method where the initial 13ondition is that at the onset time t0
the 13harge q is set to CU0 Further the 13urrent i = -dqdt is taken to be zero
at t0 Thus using equation 61 the initial rate of 13hange of the 13urrent is
d
2qdt
2= - qL0C Then all the rst time-derivatives are updated at the rst
time step t0+∆t with ∆t=043 ns using equations 61-64 The quantities
like q M v z are updated at the next step then the iteration is pursued
Initially position a1313eleration and velo13ity of the sheath are set to be zero
as well as the rate of 13hange of the mass
As we initiate the ele13tri13al 13urrent into the gas inside the 13hamber the
plasma sheath is a1313elerated with respe13t to its initial mass as seen above
At later times the mass a1313retion is masking the inuen13e of this arbitrary
value We have 13al13ulated the plasma evolution in the dis13harge for dierent
sets of initial 13onditions whi13h are reported in Table 61
Fig 63 shows the behavior for the standard values reported in Table 61
in the 13ase of Argon at 1 mbar The 13orresponding values of 13harge mass
and velo13ity will be hereafter 13alled the referen13e values In Fig 63a the
variation of 13urrent and voltage in the rst half-period is 13lassi13al showing
that the pulse generator departs slightly from a pure RLC 13ir13uit Fig
63b represents the a1313reted mass whi13h in13reases in time as well whereas
65 3-D MHD SIMULATIONS USING GORGON CODE 111
the rate of mass a1313retion 13ulminates with a 13ompetition of speed growth
and de13rease of the swept volume per time step be13ause it depends on the
squared radii In Fig 6313 plotted are the speed and kineti13 energy of the
plasma sheath First the speed of the plasma is in13reasing slowly with time
quite linearly as the mass and the kineti13 energy is roughly a third power
of the time
That leads to 13hoose a design where the a1313eleration is imposed for a
su13iently long time However the estimated speed of plasma is sim 140 kms
whi13h is a very optimisti13 value The reason may 13ome from two reasons
(i) a part of the 13urrent ows through the large sho13ked volume redu13ing
the magneti13 pressure a13ting on the plasma sheath (ii) energy transfer from
magneti13 to thermal energy and ionization is redu13ing the kineti13 one The
observation of a very bright emission from the plasma will be a 13lear eviden13e
of this redistribution of the energy
Further we made a 13omparative study of the parameters indi13ated in
Table 61 and applied to two gases namely Ar and Xe In order to 13he13k the
ee13ts of the initial 13onditions (13one angle a1313retion fa13tor pressure and
damping resistan13e) we have performed a systemati13 study by varying only
one parameter and keeping the others 13onstant and equal to the referen13e
values given by the se13ond 13olumn of Table 61 The tested range is listed
in the third 13olumn of Table 61 Similarly for ea13h set the variation of the
plasma 13onditions at the top of the muzzle is normalized to the referen13e
value Fig 64 summarizes the variation of the sele13ted dierent output
parameters the time tm to rea13h the muzzle (z = h) the ele13tri13al 13urrent
and the axial speed at muzzle the laun13hed mass M the kineti13 energy and
the average speed at the top of the ele13trode 13al13ulated from the initial
point v=h( tm - t0)
Table 62 summarizes the respe13tive ratios for Ar and Xe at the top of
the 13oaxial ele13trodes As the values of interest are obtained at 13onstant
stored energy in the 13apa13itors there is a benet to work with a lighter gas
like Argon However the results obtained using this 13ir13uit model show that
the referen13e values are ensuring a satisfa13tory behavior for both gases
Speed Kineti13 energy Time to laun13h Mass
222 151 66 31
Table 62 Laun13hing performan13e ArXe
65 3-D MHD simulations using GORGON 13ode
Our simplied model (13f se13tion 64) is 13onvenient to perform qui13kly some
parametri13 tests in the a1313eleration phase and to help designing the devi13e
but it is not 13apable to treat the sho13k dynami13s in detail nor to give valuable
112 CHAPTER 6 ELECTROMAGNETICALLY LAUNCHED SHOCK
information on the plasma parameters in the sho13k region So rened nume-
ri13al simulations of the experiments were performed in 13ollaboration with A
Ciardi (Ciardi 2014) using the GORGON 13ode (Chittenden et al 2004 Ci-
ardi et al 2007 Suzuki-Vidal et al 2014) GORGON is an expli13it parallel
13ode designed to solve the resistive MHD equations on a three-dimensional
(3-D) Cartesian grid employing a Van Leer type algorithm The 13ode treats
the plasma as a single uid but solves separately the energy equations for
ions and ele13trons allowing dierent temperatures for the two spe13ies Both
thermal 13ondu13tion and resistive diusion are treated using Braginskii-like
transport 13oe13ients
Variables At rest Sho13k Post-sho13k
Ar density ρ (g13m
3) 18E-6 25E-5 10E-5
Ele13tron density Ne (13m3) 10 E+15 20E+18 15E+18
Ele13tron Temperature Te (eV) 003 (1lowast) 10 15-18
Ion density ni (13m3) 30E+16 35E+17 2-3E+7
Ion Temperature Ti 0 50 15-20
Average Speed ltVgt kms 0 50 20-40
ltzgt 02 6 7-9
Table 63 Values of variables in various 13onditions at rest (with seed ele13trons)
inside the sho13k and inside post-sho13k region A star lowast denotes the insulator surfa13e
This 13ode des13ribes the 13reation of the surfa13e dis13harge then its a1313ele-
ration by the magneti13 pressure It provides the mapping of all the plasma
parameters and lo13al B-eld in the laun13hing 13one as well as in the free ight
region The 3-D simulation was performed for Argon gas at 1 mbar only the
dis13harge 13urrent law being taken from experiment The Table 63 presents
the ranges of all the variables in the regions of interest
As an example Fig 65 shows an enlarged mapping of log (ne) at two
times in the laun13hing phase then when the sho13k has deeply penetrated
in the sho13k tube Taking the zero time as a referen13e the averaged speed
zt is 40 kms in this simulation When the two snapshots are 13onsidered
the estimate of the instantaneous speed zt is higher at 56 kms The
time history of plasma merging and early free ight into the sho13k tube
is presented in Fig 66 During the a1313eleration (inside the double 13one)
the plasma is well lo13alized on a planar annular sheath whi13h is 13arrying
the 13urrent as assumed in the lumped parameter model We see that the
merging is well a13hieved at the muzzle shortly before 1250 ns When the
sho13k has penetrated inside the tube (1500 ns) the 13omputed B-eld is
negligible at this lo13ation and the propagation is quite free as expe13ted for
the 13hosen design of the devi13e However a high 13urrent is still passing
through the gas at that time a su13ient ele13tron density remaining in the
13oni13al gap for that
65 3-D MHD SIMULATIONS USING GORGON CODE 113
Figure 65 Example of mapping of the ele13tron density from a 3-D MHD simu-
lation (ba13kground gas Argon at 1 mbar) when the plasma sheath is (left) at
the exit of the plasma gun (right) in free ight 13onditions (log s13ales for the false
13olors)
114 CHAPTER 6 ELECTROMAGNETICALLY LAUNCHED SHOCK
Figure 66 Time history of plasma merging and early free ight through the
mapping of the ele13tron density ele13tron temperature and average ioni13 13harge as
given by a 3-D MHD simulation (ba13kground gas Argon at 1 mbar) (s13ales for the
false 13olors)
65 3-D MHD SIMULATIONS USING GORGON CODE 115
Figure 67 Axial proles values taken in the sho13k tube along a line slightly o-
axis at a distan13e of 1 mm for mass density ion temperature ion density average
ion 13harge magneti13 eld ele13tron temperature ele13tron density and average speed
at 1500 ns (ba13kground gas Argon at 1 mbar) A maping of |B| is given with a
dashed line indi13ating sho13k front position as well
116 CHAPTER 6 ELECTROMAGNETICALLY LAUNCHED SHOCK
To 13onrm the steep stru13ture of the sho13k Fig 67 presents the axial
proles of ele13tron temperature Te number density Ne mass density ρ and
average ion 13harge ltZgt at 1500 ns inside the tube with an enlarged axial
position s13ale Steep fronts are observed for mass density ele13tron density
and temperature as in the earliest simulations (Potter 1971 Kondo et al
2008) The sho13k region is at 10 eV and sim2times1018 ele13trons13m
minus3 the post-
sho13k region is hotter Argon ionization stages of 5 (sho13k) to 9 (post-sho13k)
are obtained fore13asting an emission spe13trum in the UV or harder A wider
view of the plasma is given by other quantities at the sho13k front and behind
ion temperature average speed and lo13al magneti13 eld in Fig 67 We get
the 13onrmation of the extin13tion of B after the rst 5 mm inside the sho13k
tube as well as the sho13k speed at 50 kms 13oherent with the 13onstant speed
of 56 kms mentioned just above At present these parameters are guiding
the 13hoi13e of diagnosti13s like ultra-fast interferometry or UV-spe13tros13opy
66 Measurements
The aim of these tests was to show the 13onsisten13e of the above model by me-
asuring the plasma speed 13lose to the plasma gun muzzle The parameters of
the devi13e are the referen13e ones as dened in Table 61 A rather simple and
noise-free method for diagnosing a plasma gun during the rundown phase
is to re13ord the light emitted by the plasma sheath with a spatial resolution
13ompatible with the plasma stru13ture If we 13onsider a line of sight whi13h
is radial or tangential ie perpendi13ular to the dire13tion of propagation
one expe13ts to see a sudden rise when the plasma enters the dete13tion vo-
lume possibly a plateau when the plasma travels inside the volume then a
slow de13rease when the hotter plasma leaves the volume but when the sho13k
remnants are still present So the expe13ted signal is a triangular asymme-
tri13 pulse (Serban and Lee 1995) or some kind of double exponential pulse
(Stehleacute et al 2012) A peak indi13ates the time of ight for the given posi-
tion allowing to 13ompute a mean speed providing the laun13hing time (t=0)
and all positions are known An instant speed is a1313essible providing two
13lose dete13tors are 13onsidered That has been proposed and tested su1313ess-
fully in the 13ase of radiative sho13ks driven by one PALS laser beam (Stehleacute
et al 2012) the dete13tors being 13ollimated and ltered sili13on diodes For
13ooler plasmas the visible emission is more relevant Inside a squirrel 13age
plasma fo13us Serban has installed a set of opti13al ber looking tangentially
at the drift zone (Serban and Lee 1995) Despite a 13oni13al eld of view the
results were 13on13lusive at speeds as high as 100 kms in D2 An improved
version with 13ollimated opti13al bers (a1313eptan13e angle 2deg) was developed
too (Serban and Lee 1995) A non intrusive method was developed (Veloso
et al 2012ba) by imaging the plasma on a set of opti13al bers allowing to
play with the magni13ation
66 MEASUREMENTS 117
(a)
(b)
Figure 68 Experimental results (a) time history of the passive opti13al re13ords
13ompared to the main 13urrent for Argon gas 13lose to the referen13e pressure (b)
time-integrated signal vs Ar pressure for the dierent bers (13) averaged speed vs
Ar ba13king pressure
118 CHAPTER 6 ELECTROMAGNETICALLY LAUNCHED SHOCK
In our 13ase (see Fig 61a) three 1-mm PMMA (Poly(methyl metha13ry-
late)) opti13al bers are installed radially along a transparent se13tion of the
sho13k tube at z sim 625 mm 70 mm and 775 mm and fast DET10A dete13tors
re13ord the transient presen13e of the plasma sheath in the respe13tive lines of
sight The re13orded spe13tral range is limited to the whole visible in a preli-
minary stage a band-pass ltering might give information on the presen13e of
spe13i13 ioni13 spe13ies 13hara13teristi13 of a plasma temperature level however
the amount of light was not su13ient to do so Fig 68a presents the time
evolution of the ele13tri13al 13urrent with a period of 4 micros and the delayed
side-on observational data of the moving plasma inside the sho13k tube The
behavior of the ele13tri13al 13ir13uit is following well the expe13ted ringing of a
weakly damped RLC 13ir13uit the zero-time being xed at the onset of the
13urrent On opti13al 13hannels a steep rise and a peak are the signature of a
mm-thi13k fast-sho13k stru13ture (Serban and Lee 1995) 13onsistent with early
numeri13al simulations (Potter 1971 Tou Lee and Kwek 1989 Kwek Tou
and Lee 1990 Veloso et al 2012b) On ea13h ber signal se13ondary peaks
namely the one delayed by 4-5 micros are attributed to 13urrent restrikes at the
pseudoperiod To avoid the mixing of sho13ks further experiments should be
13ondu13ted in an aperiodi13 regime with a heavy-duty damping resistor added
in series to ea13h 13apa13itor Rdamp sim NZ
On Fig 68b and 13 are presented the exploitation of opti13al re13ords with
the Ar ba13kground pressure this parameter varying in the widest range (b)
the integrated emission whi13h presents a maximum between low pressure side
when the temperature is high but the number of emitters is low and the high
pressure side when the temperature is mu13h lower (13) the averaged speed
ziti for the rst peak on ea13h ber The relative intensities re13orded in the
three lo13ation s are not signi13ant be13ause of a strong 13opper deposition
13loser to the muzzle whi13h ae13ted the dete13tion The long-life noise on
the opti13al signals was making the lo13al speed 13al13ulated as (zi-zj)(ti-tj) a
non-reliable output One 13an note Ma13h numbers in the range 20 60 with
a pressure dependen13e similar to previous work (Kondo et al 2006) The
reported observations are 13onsistent with a high-speed millimetri13 planar
plasma travelling in the sho13k tube The speed is lower than dedu13ed from
the simplied model possibly due to the la13k of dissipation terms whi13h
is overestimating the transfer to kineti13 energy At the 13ontrary a realisti13
simulation as the one performed using GORGON is in better agreement
67 Summary
For this part of the work the obje13tive was to show that in 13omplement of
laser experiments a 13ompa13t pulsed power generator might drive astrophy-
si13ally relevant sho13ks in low pressure noble gases with a higher availability
and at a rather modest 13apital 13ost The longitudinal a1313eleration of a
67 SUMMARY 119
plasma sheath in a low ba13king gas pressure has been demonstrated in 13o-
axial plasma guns as early as in the 70s even leading to speeds of 100 kms
in light gas as H2 whi13h were measured and modeled A 13hoi13e was made
to revisit this type of devi13e At present the ele13tromagneti13 13oaxial plasma
guns are quite only 13onsidered as the initiators of a strong radial plasma
implosion plasma fo13us devi13es (PFD) due to a good ability to produ13e a
pin13h plasma and emission of energeti13 parti13les and hard X-ray For that
reason it was ne13essary to adapt the prin13iple of a PFD with two major
13onstraints in13rease the 13onversion of stored energy into the kineti13 energy
of a plasma sheath prevent totally any pin13hing
That obje13tive was rea13hed by dening the ele13tri13al and geometri13al
parameters of a low-indu13tan13e and 13ompa13t pulsed power generator An
optimization pro13ess has been established to mat13h a13hievable ele13tri13al pa-
rameters of the 13ir13uit (13apa13itan13e indu13tan13e peaking 13urrent intensity)
with a plasma motion in the a1313elerating tube over few 13m leading to a nal
speed of 10-30 kms A model was built up to des13ribe the dynami13s of an
RLC 13ir13uit with a varying element 13omposed of the moving plasma sheath
a1313elerated by the self generated magneti13 pressure The 13hange in plasma
position along the axis is in13reasing the indu13tan13e and the mass a1313retion
is in13reasing the plasma mass It was shown that a 13hange in the ele13trode
geometry namely a 30deg 13oni13al shape was in13reasing the energy 13onversion
and preventing any fo13using The further step in 13ontrast with PFD is the
merging of the plasma ring into a rather planar one whi13h is inje13ted in a
drift tube
Considering Ar or Xe at gas pressure in the range 01 10 mbar and
a highly 13oaxial 1-kJ devi13e the simple model was su13ient to predi13t
performan13es agreeing with the only work quoted in the literature Basi13
observations of the moving plasma using side-on dete13tion of the visible
emission give features 13oherent with the model Ma13h numbers from 20
to 60 are obtained tting with the obje13tive of strong sho13k formation A
rened 3-D MHD numeri13al simulation indi13ates very promising features of
the sho13k in view of exploring mm-size sho13ks in a regime 13omplementary
to laser driven ones and on a table-top devi13e allowing a higher repetition
rate
120 CHAPTER 6 ELECTROMAGNETICALLY LAUNCHED SHOCK
Chapter 7
Con13lusions and perspe13tives
71 Con13lusions
Although ubiquitous in the astrophysi13al environments radiative sho13ks are
13omplex phenomena whi13h still require an important eort to understand
them Beside observations whi13h suer from a la13k of angular resolution
laboratory experiments provide today an interesting approa13h to improve
our present knowledge In this 13ontext my thesis work is dedi13ated to the
experimental and numeri13al study of the 13hara13teristi13s of two 13ounter pro-
pagating radiative sho13ks propagating at dierent velo13ities (20-50 kms)
Most of the work presented here is the out13ome of the rst-of-its-kind ex-
perimental 13ampaign held in year 2015 at Prague and Laser Asterix system
(PALS) laser fa13ility
In this manus13ript I presented the setup for the aforesaid sho13k expe-
riment Following this the data analysis results interpretation as well as
relevant numeri13al simulations 13on13erning various diagnosti13s viz visible in-
terferometry and XUV spe13tros13opy have been presented Although a large
fra13tion of the experimental re13ords were performed for Xenon at various
pressures I have also studied the sho13k 13hara13teristi13s of dierent gases like
Ar Kr and He Our results are 13omplementary to those obtained at ORION
laser fa13ility also in 2015 relative to the 13ollision of identi13al radiative sho13k
waves at higher velo13ities (80 kms) In parallel in this thesis I have also
worked on the optimization of a setup dedi13ated to ele13tromagneti13ally ge-
nerated strong sho13ks at lower velo13ities In the following I briey present
a brief summary of the main results of my thesis work
In the third 13hapter I have presented a series of 1D hydrodynami13 simu-
lations (realized with the 13ommer13ial 13ode HELIOS) on isolated and 13ounter-
propagating sho13k waves at equal and dierent velo13ities The simulations
13onrm that a single radiative sho13k propagating in Xenon gas at 01 bar
is 13hara13terized by an extended pre13ursor and a large 13ompression of 30 in
the post sho13k At 50 kms the temperature evolution with the distan13e
shows identi13al post-sho13k and pre-sho13k values This indi13ates that the
numeri13al sho13k is of super13riti13al nature However I nd that there is an
important un13ertainty in the Xenon opa13ity whi13h makes the renement of
the simulation unne13essary espe13ially in terms of group numbers Next I
have investigated the intera13tion two 13ounter propagating sho13ks for iden-
ti13al (50-50 kms) and non identi13al (50-20 kms) sho13k speeds For the
121
122 CHAPTER 7 CONCLUSION
13ase of identi13al speeds the pre13ursors merge together at around 10 ns The
intera13tion is then 13hara13terized by a regular in13rease of the ele13tron density
and the temperature with the time The sho13ks 13ollision at 38 ns leads to a
jump in the ele13tron density (6 times 10
2113m
minus3) and temperature (39 eV) On
the other hand for the 13ase of non-identi13al sho13k speeds (13ase representing
our experiments) the pre13ursors intera13tion starts later than in the former
13ase The sho13ks 13ollision o1313urs at 49 ns and it is 13hara13terized by a sudden
in13rease of the ele13tron density also by an order of magnitude (4 times 10
21
13m
minus3) whereas the temperature in13reases up to 29 eV
Although the data analysis of all the experimental re13ords obtained du-
ring the PALS experimental 13ampaign has been 13arried out only a few repre-
sentative interferometri13 and spe13tros13opi13 re13ords for Xe gas are dis13ussed
in details in this thesis The interferometri13 re13ords allowed me to estimate
the average sho13k speed and time variation of the ele13tron density during the
sho13k propagation The sho13k speeds of the MAIN and AUX radiative sho13k
waves are found to be ranging between sim 30-55 and 12-25 kms respe13tively
whereas the averaged pre13ursor ele13tron density is varying between 10
17and
10
1913m
minus3during the sho13ks propagation
From the interferometri13 re13ords I have investigated the ee13t of the
intera13tion between the two radiative pre13ursors for the sho13ks propagating
in Xenon at 01 bar with respe13tive speeds of 50plusmn3 and 23plusmn3 kms The
intera13tion starts at 20 ns and is followed by the merging of the two ra-
diative pre13ursors This pre13ursor intera13tion is 13hara13terized through the
enhan13ement of the ionisation wave The sho13ks 13ollision is re13orded at 50
ns On the 13ontrary the same intera13tion behavior is not seen at 02 bar
(sho13k speeds sim 38plusmn4 and 18plusmn2 kms for the MAIN and AUX sho13k waves
respe13tively) Moreover in this 13ase there is no signature of a radiative pre-
13ursor for the AUX sho13k The pre13ursors intera13tion if any should then
13ould o1313ur at times whi13h are outside of the re13ord and 13an not be 13on-
rmed by the experiment This indi13ates that for a given gas and sho13k
speed the radiative ee13ts de13reases with the initial mass density
The investigation of the lateral extension of the sho13k has been made
through the analysis of transverse interferograms At 02 bar the MAIN
sho13k with a speed sim 40 kms has a lateral pre13ursor extension of sim570plusmn30microm whereas it is 275plusmn25 microm for AUX sho13k whi13h is propagating
with a speed of 20 kms The MAIN sho13k pre13ursor is almost of a at
spatial prole whereas the AUX pre13ursor is more 13urved suggesting that
the 2D ee13ts are mu13h more important for AUX than for MAIN One of
the explanations is that the spot size of AUX laser on the target is smaller
than the target width whereas the MAIN laser has a spot size whi13h is equal
to the target width
In order to interpret the experimental results with appropriate 1D simu-
lations I have optimized the lasers uen13es to numeri13ally obtain a sho13k
speed equal to that re13orded in the experiment In the 13orresponding simu-
72 PERSPECTIVES 123
lations for Xenon at 01 bar I note the intera13tion of the two pre13ursors
However I do not nd a good quantitative agreement for the ele13tron den-
sity Beside the question of the impre13ise opa13ities this disagreement might
also be attributed to 2D ee13ts (Gonzaacutelez Audit and Stehleacute 2009 Leygna13
et al 2006 Stehleacute et al 2010)(Cotelo et al 2015)
The analysis of spa13e- and time-integrated XUV re13ord at 06 bar is
presented for two 13ounter propagating sho13ks of speeds sim 36plusmn4 and 18plusmn5kms for MAIN and AUX respe13tively The presen13e of HeII Balmer lines
and Xenon lines tends to indi13ate a temperature of the sho13k of about 15
eV and a Xenon mean ion 13harge around 6 - 7 The 1D simulations predi13t
ele13tron temperature in the range of 10-30 eV while ion 13harge to be ranging
between 5 and 10
In addition to this study whi13h was fo13used on the Xe 13ase I have
investigated the sho13k 13hara13teristi13s for Ar Kr at 13lose mass density (sim 6
times 10
minus4g13m3) whi13h 13orrespond to 03 bar and Kr gas 02 bar Almost no
pre13ursor is found to be present in 13ase of Ar whereas a tiny pre13ursor have
been observed for Kr This indi13ates that for a given density and sho13k
speed the radiative ee13ts in13rease with the atomi13 number
In parallel I have worked on the optimization of the design of an experi-
mental setup where the sho13k is generated ele13tromagneti13ally This setup
allows studying sho13k sim 30 kms in noble gas at sim1 mbar The parameters
of a low-indu13tan13e and 13ompa13t pulsed power devi13e have been optimized
with a simple model in view of building-up and a1313elerating a plasma sheath
in Ar and Xe at gas pressure in the range 01-10 mbar Simple observations
of the moving plasma using side-on dete13tion of the visible emission give
features 13oherent with the model Sho13ks 13orresponding to ma13h numbers
ranging from 20 to 60 are obtained
72 Perspe13tives
The experimental results presented in this thesis 13orrespond to the 13ase of
two 13ounter-propagating radiative sho13ks We have been able to study the
time variations of the pre13ursor intera13tion and the ee13t of gas pressure
with 1D time dependent interferometri13 re13ords obtained with a streak 13a-
mera Together with the help of XUV spe13tros13opy we 13ould estimate the
sho13k speed ele13tron density and sho13k temperature However no informa-
tion is provided about the lateral extension of the pre13ursor and we were
not able to re13ord XUV data for all the shots Therefore in order to make
these results more 13omprehensive we have designed and parti13ipated in a
follow-up experimental 13ampaign in September 2016 at PALS This time we
fo13ussed on single sho13ks in 13lose experimental 13onditions but with a wider
range in terms of speeds (45-100 kms) We repla13ed the time dependent
streak 13amera interferometri13 study by an instantaneous 2D interferometri13
124 CHAPTER 7 CONCLUSION
image (GOI) obtained with a short pulse sub pi13ose13ond laser and 13ameras
XUV spe13tra were also re13orded The re13ords were performed at three dif-
ferent times of the sho13k propagation allowing to dedu13e the sho13k speed
and the time evolution of the pre13ursor morphology This follow up experi-
ment will allow to 13omplete the analysis of the laser generated sho13k waves
presented in this thesis
The preliminary analysis I have performed on these new re13ords is very
promising An example of these interferometri13 re13ords is given in the Figure
71a for the shot 30364 in Xe+He at 06 bar The high speed of the sho13k is
related to the energy of the laser whi13h is here 170 J whereas it was less than
120 J in the previous experiment The 13omparison with the interferometri13
re13ord (not shown) before the shot allows to dedu13e the ele13tron density
whi13h rea13hes 18 times 10
1913m
3 The high quality re13ord shows 13learly 2D
ee13ts for this sho13k wave
(a)
(b)
Figure 71 Interferometri13 re13ord obtained at 546 ns for a single radiative sho13k
of speed sim 100 kms propagating in Xe+He gas at 06 bar driven by laser at 348
nm with energy 170 J The dierent 13olors in (b) 13orrespond to yellow le Ne le35 times 10
1713m
minus3 13yan 37 - 74 times 10
1713m
minus3 blue74 - 15 times 10
1813m
minus3 green
15 - 18 times 10
1813m
minus3 magenta 18 - 22 times 10
1813m
minus3 red 22 - 26 times 10
1813m
minus3
orange26 times 10
1813m
minus3- 18 times 10
1913m
minus3
This analysis is now ongoing and the results of this experimental 13am-
72 PERSPECTIVES 125
paign will be 13ompared with that obtained from the analysis of re13ords from
2015 presented in this manus13ript This instantaneous imaging interferome-
tri13 diagnosti13 does not follow the sho13k 13ontinuously with the time but it
gives a pre13ise map of the ele13tron density whi13h 13an be inverted through
Abel inversion to obtain the lo13al estimation of the ele13tron density
In parallel in the ele13tromagneti13ally laun13hed sho13ks we have upgra-
ded our table top setup to a13hieve higher speeds For this we have used 12
13apa13itors ea13h of 1 microF at the pla13e of the previous 11 13apa13itors ea13h of
06 microF We have implemented new diagnosti13s namely a visible interferome-
ter working in the Gated Opti13al Imaging mode and time and spa13e visible
spe13tros13opy similar to the one des13ribed in the 13hapter 4 For the visible in-
terferometry we have Ma13h Zehnder Interferometer setup and we will image
the sho13k at dierent times with the help of a SL300 pi13ose13ond high energy
NdYAG laser (λ = 532 nm and energy = 024 J) The experiment will soon
be performed on this new setup and the analysis of su13h re13ords will also be
undertaken in future allowing a 13omparison of ele13tromagneti13ally laun13hed
sho13ks with the laser-driven sho13k whi13h should make the bridge between
thin and opti13ally thi13ker sho13ks
126 CHAPTER 7 CONCLUSION
Chapter 8
Thesis summary
Radiative sho13ks (RS) are present in various astrophysi13al environments es-
pe13ially in stars They are then present in all the supersoni13 a1313retioneje13tion
pro13esses during the stellar evolution from the early phase of the proto-
stellar 13ollapse up to the magnetosphere a1313retion on to the already formed
T- Tauri stars up to the late stage of the supernovae sho13ks
Radiative sho13ks are strong sho13ks (ie Ma13h number Mgtgt1) whi13hrea13h high temperatures and thus are the sour13e of intense radiation Alt-
hough the observation of (ie spatially resolved ) signatures of su13h sho13ks
in the astrophysi13al environments is very di13ult an alternate approa13h na-
med laboratory plasma astrophysi13s provides a powerful tool to study them
on the Earth
These waves have been experimentally studied sin13e more than a de13ade
mostly on large-s13ale laser fa13ilities in noble gases and with dierent targets
geometries The ablation pro13ess generated by the intera13tion between the
laser and a foil indu13es a sho13k wave in the foil whi13h then propagates
in the gas at a speed of several tens of kms Numeri13al simulations 13an
des13ribe these sho13ks with an improving pre13ision With laser intensities on
target 13omprised between 1014 and 1015 W13m
2 these experiments allowed
to re13ord sho13k speeds ranging between 40 and 150 kms
All previous experimental studies have been fo13used on the 13ase of single
radiative sho13ks However in astrophysi13al 13onditions the radiative sho13k
in general intera13ts with a denser medium leading to the development of
ree13ted and transmitted sho13ks A few representative examples of su13h
phenomena are the intera13tion of supernovae remnants with dense mole13ular
13louds the a1313retion sho13ks on the photosphere of T-Tauri stars and the bow
sho13ks at the head of fast stellar jets The 13ollision (or the intera13tion) of
two radiative sho13k waves is obviously a rare astrophysi13al event and the
template 13ase of the supernova remnants DEM L316 (see Fig 1 of Williams
et al (1997)) is still the subje13t of debates Williams et al (2005) Toledo-
Roy et al (2009) Velarde et al (2006) In this regard the development of
dedi13ated laboratory experiments to the study of propagation and intera13tion
of 13ounter-propagating sho13k waves is important as a tool to 13hara13terize
su13h events through their spe13i13 signatures
Complementary to laser experiments 13ompa13t pulsed power generators
may drive an ele13tromagneti13 13oaxial plasma gun to 13reate astrophysi13al
relevant sho13ks in lower pressure noble gases with a high availability and a
127
128 CHAPTER 8 THESIS SUMMARY
rather modest 13apital 13ost The ele13tromagneti13ally driven sho13k waves may
have larger s13ales than those by laser Thus they 13an be analyzed rather
easily with the adequate diagnosti13s (Kondo et al 2008)
In this thesis I am presenting radiative sho13ks in the 13ontext of labo-
ratory astrophysi13s using two experimental approa13hes laser-driven sho13ks
and ele13tromagneti13ally driven sho13ks
First 13hapter of my thesis is Radiative sho13k waves in whi13h the physi13s
of radiative sho13ks have been studded Radiative sho13k waves are hypersoni13
sho13k waves whi13h are heated to high temperature and as a 13onsequen13e
be13ome the sour13e of intense radiation This radiation in turn modies
the dynami13s and stru13ture of the sho13k itself whi13h makes its stru13ture
more 13ompli13ated Radiative sho13k waves o1313ur in several astrophysi13al 13ir-
13umstan13es su13h as in the a1313retion sho13k of protostellar formation (Stahler
Palla and Salpeter 1986) the supernova explosion and the intera13tion of
their remnants with the dense interstellar medium (Chevalier 1977) the bow
sho13ks at the head of stellar jets (Hartigan et al 2001)
Radiative sho13k waves 13an now be studied in the laboratory whi13h allows
to 13ompare the data with models existing in the literature and to 13he13k the
ability of the numeri13al 13odes to reprodu13e them (see for instan13e Bouquet
et al (2004) Leygna13 et al (2006)) The experimental study of radiative
sho13k 13hara13teristi13s is 13ru13ial in understanding these various energeti13 pro-
13esses o1313urring in the astrophysi13al plasmas
The jump 13onditions (Rankine-Hugoniot relations) for the simple 13ase
of an ideal gas whi13h rely on the values of the thermodynami13al quantities
on both sides of the dis13ontinuity Let us 13onsider a one-dimensional sho13k
propagating in a gas at rest with the speed us In the frame asso13iated with
the sho13k front the pre-sho13k (upstream) uid velo13ity is then u1 = - us
while behind the sho13k (post-sho13k or downstream region) this velo13ity is
u2 as shown in the Fig 21 Considering the 13ase of strong sho13ks where
M may be 13onsidered to be ≫1 jump 13onditions for ideal gas are
ρ2ρ1
=(γ + 1)
(γ minus 1)(81)
T2
T1=
2M2γ(γ minus 1)
(γ + 1)2(82)
kBT2 =2(γ minus 1)
(γ + 1)2mus
2(83)
For mono-atomi13 gases γ = 53 the 13ompression ratio is estimated to
be 4 and the post-sho13k temperature is
kBT2 =3
16mu1
2 =3
16mPAu1
2(84)
129
This temperature is thus proportional to the square of the sho13k velo13ity
and to the atomi13 mass In 13ontrast to the dis13ussion on ideal gas in the
13ase of a real gas is a little more 13ompli13ate It is to note that in this 13ase
a part of the kineti13 energy is used to ex13ite and ionize the post-sho13k gas
As a result its temperature is lower than that for the 13ase of ideal gas
In addition the pressure is also modied due to ionisation (Mi13haut et al
2004) In13luding the ee13t of ionization with the average 13harge z in the
medium for the 13ase real gas jump relations for real gas are
ρ2u2 = ρ1u1 (85)
ρ2u22 + ρ2
kBT2
m(1 + z2) = ρ1u
21 + ρ1
kBT1
m(1 + z1) (86)
ρ2u2
[5
2
(1 + z2)
mkBT2 + ǫexc2 +
u222
]
= ρ1u1
[5
2
(1 + z1)
mkBT1 + ǫexc1 +
u212
]
(87)
In the previous both 13ases the ee13ts of radiation heating and 13ooling
have been negle13ted However the radiation emitted in a strong sho13k will
ae13t its stru13ture These ee13ts 13an be determined using mass momentum
and energy 13onservation equations whi13h now have to in13lude the 13ontribu-
tions of the radiative ux energy and pressure This 13ase is more 13omplex
now Taking into a1313ount radiative ee13ts the equations of radiative hydro-
dynami13s in the 13o-moving frame of the sho13k (Mihalas and Mihalas 1999
Drake 2006) 13an be written as
partρ
partt= minusnabla(ρu) (88)
ρ
(
partu
partt+ unablau
)
= minusnabla(P + Prad) (89)
part
partt
(
ρu2
2+ ρǫ+ Erad
)
+nabla
[
ρu
(
ǫ+u2
2+
P
ρ
)
+ (Erad + Prad)u
]
= minusnablaFrad
(810)
The 13ontributions linked to Prad and Erad are ee13tive only at very high
temperature (ie velo13ity) However for the radiative sho13ks of interest in
our experiments the 13ontribution of the radiative ux Frad is non-negligible
13ompared to ρu3 These sho13ks are thus in the radiation ux dominated
regime Most of the experimental sho13k waves are in this regime These
hydrodynami13 equations are strongly non-linear
Depending on the opa13ity the radiation emitted from the sho13k may be
absorbed by the pre-sho13k region indu13ing its pre-heating Far away from
the dis13ontinuity the stru13ture of the upstream medium is determined by
the absorption On its side the stru13ture of the transition layer of the down-
stream medium 13lose to the dis13ontinuity depends on the balan13e between
130 CHAPTER 8 THESIS SUMMARY
the emission (13ooling) and the absorption (heating) Thus the full stru13ture
of the sho13k mainly depends on the variation of the opa13ity χ (13m
2g) The
opti13al depth (τ(s)) measured from the position of the jump is then the
relevant parameter for the absorption It is given by
τ(s) =
int s
sjump
χ(sprime)ρ(sprime) dsprime (811)
where sprime 13hara13terizes the path of the radiation As we are interested in
the mean frequen13y averaged radiative ux the relevant opa13ity here will be
the averaged Rosseland opa13ity whi13h will be dened in due 13ourse
If τ gtgt 1 the regime is referred as opti13ally thi13k while in the opposite
13ase (τ lt 1) it is opti13ally thin Flux dominated radiative sho13ks have
been 13lassied depending on the opti13al depth of upstream and downstream
regimes
We are more interested in the kind of sho13k have a radiative pre13ursor It
is possible to produ13e a radiative sho13k at the high energy laser fa13ilities with
laser intensity of sim 10
14W13m
2 A short summary of number of experiment
performed at high power laser fa13ilities have been listed in table 21 Other
than laser driven radiative sho13k it is possible to laun13h 13omparatively slower
radiative sho13ks (sim 10-30 kms) using ele13tromagneti13 generators
The 13hapter two is dedi13ated to the numeri13al analysis of radiative sho13k
waves and of the main physi13al pro13esses whi13h play a role in these 13omplex
pro13esses Radiative sho13ks are present in various astrophysi13al pro13esses
implying extreme 13onditions Laboratory experiments then allow investiga-
ting the underlying physi13al pro13esses whi13h take pla13e in these 13onditions
and whi13h are not observable from the Earth due to a la13k of spatial resolu-
tion However experiments are not always straightforward to interpret and
the help of numeri13al simulations be13omes pre13ious
Dierent numeri13al approa13hes are used for the simulation of laboratory
plasmas Some of them use xed grids Other introdu13e the possibility of
renement (on the y) in some meshes (Adaptative Mesh Renement) to
maintain a good resolution in the interesting parts of the plasma Another
approa13h is based on meshes following the uid (Lagrangian des13ription)
whi13h means that the meshes move with the uid and no matter may go
through a mesh to another The form of the equations dier in this 13ase (see
for instan13e Orban et al (2013))
HELIOS is su13h a one-dimensional Lagrangian radiation hydrodynami13
13ommer13ial 13ode (Ma13Farlane Golovkin and Woodru 2006) and I have
used it to simulate our experiment of intera13ting sho13k waves This 13ode
is easy to handle and has the presently interesting 13apability to be able to
simulate the 13ase of two sho13k waves propagating in opposite dire13tions and
laun13hed by two dierent lasers HELIOS may des13ribe non-LTE plasmas as
it in13ludes an in-line 13ollisional radiative (HELIOS-CR) model for 13omputing
131
non-LTE atomi13 level populations at ea13h time step of the hydrodynami13s
simulation However I used HELIOS in the LTE approximation whi13h is
appropriate to our experimental 13onditions (Rodriguez et al 2011) as will
be dis13ussed in the next se13tion
The 13ode solves the uid equations of motion using the pressure 13ontri-
butions of ele13trons ions and radiation It may des13ribe the ele13trons and
ions as two intera13ting uids at respe13tive temperatures Te and Ti Thermal
13ondu13tion is des13ribed within a ux-limited ele13tron 13ondu13tion model ba-
sed on the Spitzer 13ondu13tivity and the laser energy deposition is 13omputed
with an inverse Bremsstrahlung model
The radiation emission and absorption terms are introdu13ed in the equa-
tion of energy for the ele13trons and in the radiation transport equations One
of the methods whi13h are proposed is the ux-limited multi-group radia-
tion diusion model where the radiative ux is proportional to the gradient
of the radiative energy and is inversely proportional to the Rosseland opa-
13ity The expression is pondered by a ux-limited diusion 13oe13ient in
order to obtain the good opti13ally thin limit This 13oe13ient follows the
Larsen expression des13ribed in Olson Auer and Hall (2000) The se13ond
method is a (time independent) multi-angle short-13hara13teristi13s s13heme In
our 13ase where radiation and hydrodynami13s are strongly 13oupled and thus
the radiation eld varies rapidly we used the diusion model together with
LTE Plan13k and Rosseland multi-groups opa13ities
In addition our version of HELIOS uses the PROPACEOS (Ma13Farlane
Golovkin and Woodru 2006) Equation of State (EOS) and multi-group
opa13ity tables whi13h are generated by the 13ode
1
and it is not possible
to in13lude any other opa13ity or EOS table The knowledge of the thermo-
dynami13al 13onditions is required for an adequate simulation of the plasma
des13ription In our experiment the mass density ranges between sim 10
minus4and
10
minus1g13m
3 while the temperature values are 13omprised between and sim 0 to
50 eV The pre13ise 13al13ulations performed by Rodriguez et al (Fig2 of Rodri-
guez et al (2011)) for Xenon indi13ate that the thermodynami13al 13onditions
of our radiative sho13k experiments 13orrespond to the LTE regime
The equation of state (EOS) denes the dependen13e of the pressure
ionisation internal energy with the mass density and temperature Several
models do exist in the literature
Our version of HELIOS for LTE plasmas may handle polytropi13 or PRO-
PACEOS equation of state Unfortunately there are not many details about
this last method The authors of the 13ode mention that it uses a QEOS-type
model (More et al 1988) in the strong 13oupling regime (ie at relatively
high densities and low temperatures) and an isolated atom model whi13h
uses a detailed atomi13 stru13ture modeling in the weak 13oupling region The
properties between the weak and strong 13oupling regimes are said to be obtai-
1
see http wwwprism minus cscomSoftwarePROPACEOS
132 CHAPTER 8 THESIS SUMMARY
ned by interpolating in a thermodynami13ally 13onsistent manner This EOS
in13ludes the 13ontributions from the translation of ions and ele13trons ele13-
tron degenera13y atom ionization and ex13itation and Coulomb intera13tions
(Debye - Hu13kel 13orre13tion)
I have performed HELIOS 1D simulations for dierent 13ases for single
and 13ounter propagating sho13ks in 13hapter 2 of my thesis Radiative sho13k
waves propagating in Xenon at 01 bar with a velo13ity of 50 kms are 13ha-
ra13terized by the development of an extended radiative pre13ursor The huge
13ompression of 38 in the post-sho13k is a 13onsequen13e of both the sho13k and
the gas ionisation (fa13tor of 10) as also from the radiative 13ooling The post-
sho13k and pre-sho13k temperature on both side of the peak are identi13al
indi13ating that the sho13k is super13riti13al
The spe13ta13ular role of the opa13ity has been highlighted Due to the
un13ertainty in the opa13ity used for Xenon in our simulations I de13ided that
at this stage it was not ne13essary to rene the simulation in terms of group
numbers within the 1D des13ription of the sho13k wave
Our numeri13al study gives the main 13hara13teristi13s of the intera13tion of
two 13ounter-propagating sho13k waves with speeds equal to 50-50 kms and
50-20 kms The 13ase of identi13al speeds is simpler due to the symmetry of
the problem However whatever the speeds the most important signature
of the intera13tion is the merging of the pre13ursor at 8 ns for 50-50 kms
and at 15 ns for 50-20 kms This merging is followed by a regular in13rease
with time of the ele13tron density and the temperature The 13ollision time
is 13hara13terized by a sudden in13rease of the ele13tron density by an order of
magnitude rea13hing 66 times 10
21and 3 times 10
2113m
minus3respe13tively whereas the
temperature in13reases up to 39 and 28 eV
When a short pulse of a high-power laser beam is fo13ussed on a thin foil
it transfers a huge amount of energy to the foil material Almost instantane-
ously the laser absorption heats it up to the ablation A hot and low density
13orona is generated ba13kwards by this pro13ess Beyond the ablation surfa13e
a sho13k is formed in the foil as a 13onsequen13e of the momentum 13onservation
(ro13ket ee13t) The sho13k moves inward in the foil Our 13ase of the experi-
mental setup as the foil 13loses a tube lled with gas the sho13k propagates
then in the gas where it is studied through various kinds of diagnosti13s The
foil then a13ts as a piston
We performed the experiments at Prague Asterix Laser System (PALS)
in the Spring 2015 (ve weeks 20th April - 22th May) The obje13tive of the
experimental 13ampaign was to study the evolution of two 13ounter streaming
radiative sho13ks driven by two 13ounter-propagating lasers with an irradian13e
in the range of 10
14W13m
minus2 and therefore to 13hara13terize their intera13tion
and 13ollision In the 13hapter third 13hapter I have rst presented a brief
des13ription of the PALS laser fa13ility and of the laser beams used in our
experiment This followed by a presentation of the targets design and an
overview of the general setup and of our main diagnosti13s
133
The Prague Asterix Laser System (PALS) is a laser fa13ility based on
an Infrared high-power iodine laser system (Asterix IV) (Jungwirth et al
2001) Using dierent amplifying stages the laser fa13ility is able to deliver
energy up to 1 kJ in 03 ns at the fundamental wavelength 1315 nm The
output laser beam 13an be further subdivided in few auxiliary beams All
auxiliary beams may be frequen13y doubled (λ = 657 nm) or tripled (438
nm) The PALS laser fa13ility is 13apable of ring up to two high energy laser
shots per hour Compared with solid states lasers this gas laser is known to
deliver a quite homogenous beam intensity without hot spots
For our experiments we used two independent laser beams to drive the
two 13ounter streaming sho13ks in the gaseous target For this purpose the
fundamental PALS laser beam has been subdivided after the fourth amplier
(V4) into two beams with respe13tive energy fra13tions of 60 and 40 (see
Fig 42) The most energeti13 of these laser beams is then inje13ted through
the fth (V5) amplier and its frequen13y are then tripled This beam at 438
nm will be hereafter termed termed as the MAIN laser beam
The se13ond of the two previous laser beams propagates then without any
modi13ation It will be hereafter 13alled the AUX laser beam (1315 nm)
The s13heme of the energy distribution is shown in the Fig 42 and the
13hara13teristi13s of MAIN and AUX laser are presented in Table 41
For the experiments PALS oers two va13uum 13hambers with a spheri13al
and 13ylindri13al geometry respe13tively The spheri13al 13hamber assembly used
in our experiments is shown in Fig 43 This 13hamber having a diameter
of 100 13m is 13losed by two entran13e 13ir13ular windows of respe13tive diameters
80 13m and 50 13m to allow the MAIN and AUX beams to penetrate in the
13hamber and to rea13h the target Ea13h of these windows is outtted with a
hinged door Several 13ir13ular ports of various diameters also allow for the
dierent diagnosti13s The target holder and some diagnosti13s are mounted
on an opti13al ben13h whi13h is de13oupled from the 13hamber walls
Our targets (see Fig 45) s13hemati13ally 13onsist in small tubes of 4 mm
length 13losed on both sides by two spe13i13 foils of thi13kness 11 microm on whi13h
the laser beams are fo13used (one laser per foil) with an irradian13e of about
10
1413m
minus3 The tube is lled with gas in whi13h the radiative sho13k propa-
gates with a velo13ity of 30 - 60 kms The two foils 13losing the target insure
the 13onversion through ablation and sho13k generation of the laser energy
into me13hani13al energy
In our experiments we used spe13i13 targets for alignment and 13hara13te-
rization of the beam size These targets were manufa13tured at the Pole
instrumental of Observatoire de Paris
The gaseous targets were lled in situ at a fra13tion of 1 bar with a gas
(viz Xe Ar a mixture of Xe and He) whi13h thus provide the medium for the
sho13k propagation We 13hose an in-situ lling system (Fig 410) to redu13e
the dieren13e of pressures inside and outside of the tube This is 13riti13al as
a large pressure dieren13e between the target 13ell and the va13uum 13hamber
134 CHAPTER 8 THESIS SUMMARY
13an result in the breaking of the ultra-thin SiN windows of the target
First of all we performed fo13alization of MAIN and AUX laser beam
on target by using Al-massive targets The two PALS laser lenses were
translated up to a13hieve the suitable diameters on the target whi13h were
nally set to 450 - 500 microm and 250 - 300 microm for the MAIN and AUX beams
The size of the impa13t was 13ontrolled ea13h day on mo13k targets before using
the laser beams for real shots on gaseous targets It is worth to pre13ise that
the keV diagnosti13 was still in pla13e for the shots on the gaseous targets
In that 13ase the beams impa13t Parylene-N instead of Aluminium The keV
signal is then weaker than for the mo13k aluminum targets
Various kinds of diagnosti13s were implemented namely visible interfero-
metry XUV and visible spe13tros13opy For the same two keV 13ameras are
indu13ted to monitor the impa13t and two visible 13ameras for target alignment
For visible interferometory we have taken referen13e images (without any
fringe) for every target as shown in the Fig 417a for one target example
In order to re13ord su13h images we blo13ked the referen13e beam and let the
probe beam only to illuminate the target in a stati13 mode and with the slit
open Then we redu13ed the slit width to its nominal value (200 microm) This
enabled us to have the visualization of the portion of the target whi13h was
imaged in the slit on the target and also to lo13ate the position of the pistons
before the shot This image is termed as the referen13e target image Then
the system was pla13ed in the interferometri13 mode and the fringe pattern
was optimized to be perpendi13ular to the slit and thus in the longitudinal
mode perpendi13ular to the dire13tion of the sho13k propagation The Fig
417b reports su13h a typi13al interferogram re13orded during our experiment
(13f Fig 413) The horizontal axis 13orresponds to the dire13tion of the sho13k
propagation (with a total s13ale of 54 mm on the target) and the verti13al
axis 13orresponds to time (s13aling 200 ns)
The time and spa13e integrated XUV plasma emission are re13orded with
a Flat Field XUV spe13trometer using a 13on13ave grating In the Fig 419
the drawing of the XUV spe13trometer setup s13heme (red 13olor) is presented
together with the spheri13al 13hamber and the target The XUV spe13trometer
is installed on the top of the spheri13al 13hamber whi13h allows the XUV
emission passing through the Si3N4 membrane on the top of the target (see
se13tion 422) to be re13orded
In addition to the interferometry and XUV spe13tros13opy we have also im-
plemented a time- and spa13e-resolved visible spe13tros13opi13 (see Fig 420a)plasma
emission diagnosti13
In 13hapter 5 I present the results of the two diagnosti13s with a parti13u-
lar fo13us on the interferometri13 data whi13h I have extensively studied with
the help of a spe13i13 data analysis pro13edure that I have developed The
visible interferometry is an adequate tool to probe the ele13tron density of
the radiative pre13ursor up to the 13riti13al density (4 times 10
2113m
minus3at the wa-
velength 527 nm of the probing laser) whereas the denser post-sho13k region
135
is opaque to the visible light XUV radiation is present in the whole sho13k
stru13ture and thus the time- and spa13e-integrated XUV spe13tros13opy may
allow exploring the self-emission 13oming from the post-sho13k and pre13ursor
regions
In the visible interferometry the interferen13es between the main and
referen13e beams overlapping on the dete13tor produ13e a pattern of fringes
whi13h follow the relative phase variations between the probe (whi13h passes
through the plasma) and the referen13e beams (see se13tion 441 of the previ-
ous 13hapter for the experimental setup) The phase variation between two
13onse13utive fringes of the unperturbed beams (ie without any plasma) is
equal to 2π Then in the presen13e of the plasma in the probe beam these
fringes be13ome shifted due to the modi13ation of the refra13tive index More
details about the prin13iple of the interferometry and the Ma13h-Zehnder in-
terferometri13 setup may be found in the APPENDIX A and we re13all here
only the expression of the phase shift
∆φ asymp minus πd
λNclt Ne gt (812)
where λ = 527 nm is the wavelength of the probing laser Nc = 4 times 10
21
13m
minus3the 13riti13al density at this wavelength and lt Ne gt is the ele13tron
density averaged over the laser path d in the 13ell and is dened as
lt Ne gt=
int d
0
Ne(z t)dy
d(813)
In this 13hapter I have presented an extensive data analysis of few re-
presentative interferometri13 and spe13tros13opi13 re13ords I am presenting here
summary of results obtained bellow
The average sho13k speed and ele13tron density have been estimated from
the interferograms The sho13k speeds of the MAIN and AUX radiative sho13k
waves vary between sim 30-55 and 10-30 kms respe13tively and the averaged
pre13ursor ele13tron density ranges between 10
17and 10
1913m
minus3during the
sho13ks propagation
We have demonstrated the intera13tion ee13t between the two radiative
pre13ursors in the 13ase of Xe at 01 bar at 54 and 23 kms The intera13tion is
13learly 13hara13terized in the experiment by the enhan13ement of the ionisation
wave followed by the merging of the two radiative pre13ursors at 20 ns The
13ollision time is re13orded at 47 ns Su13h behavior is reprodu13ed by the
simulation
We have investigated this intera13tion at a larger pressure 02 bar with
the following speeds sim 41 kms for the MAIN and sim 18 kms for the AUX
sho13k waves We do not re13ord any experimental signature of the radiative
pre13ursor for AUX Further we have not been able to 13at13h experimentally
the 13ollision time The re13orded pre13ursor of MAIN is not inuen13ed by AUX
wave up to 48 ns (Fig 59(b) and (13) in data analysis 13hapter) whi13h is the
136 CHAPTER 8 THESIS SUMMARY
limit of the re13ord On its side the 1D simulation predi13ts a tiny pre13ursor
for AUX and that both pre13ursors start to intera13t at 49 ns This plausible
sho13ks intera13tion o1313urring at times whi13h are outside of the re13ord 13an
not be 13onrmed by our experiment
The results of the transverse interferometry at 02 bar with speeds of sim40 and 20 kms indi13ate that the MAIN pre13ursor has a lateral extension of
sim 600 microm whereas it is 300 microm for AUX The pre13ursor of MAIN is almost
at with a probable small bending at the edges of the tube whereas the
AUX pre13ursor is more 13urved This means that the 2D ee13ts are more
important for AUX than for MAIN Still in the 13ase of Xenon we have
obtained information about the temperature and the mean 13harge of gas
through our time integrated XUV spe13tra (shot48143) in Xenon at 06
bar From these results we may 13on13lude that the mean ion 13harge is at
least equal to 6 and that the temperature has rea13hed 15 eV
Our simulations give a qualitative des13ription of the sho13ks intera13tion
when the laser uen13e is adjusted to give the 13orre13t sho13k velo13ities Howe-
ver it is now well known that 2D simulations (together with state of the art
opa13ities) t better with experiments (Gonzaacutelez Audit and Stehleacute 2009
Leygna13 et al 2006 Stehleacute et al 2010) For the same laser energy the 2D
simulations lead to a diminution of the sho13k speed 13ompared to 1D as also
to a diminution of the ele13tron density For instan13e in the 13ase of a sho13k
wave laun13hed by a laser beam at 1315 nm in Xenon at 03 bar at PALS and
with a laser uen13e of 85000 J13m
2 ARWEN 2D simulations give a sho13k
speed of 44 kms in agreement with the experimental one (Cotelo et al
2015) 1D simulation would require in this 13ase a uen13e of 30000 J13m
2to
a13hieve the same velo13ity
The spa13e and time integrated XUV re13ords at 06 bar for respe13tive
speeds whi13h are equal to sim 39 and 18 kms indi13ate that the temperature
of the sho13k has rea13hed values up to 15 eV and that the Xenon mean ion
13harge has also rea13hed values of 6 - 7 whereas 1D simulations predi13t ele13tron
temperature 10-30 eV and ion 13harge 5-10 (Fig 51813) A more detailed
study based on 2D simulation and radiative transfer post-pro13essing will be
ne13essary to rene the analysis
We have investigated the 13ase of other noble gases (Ar Kr) and we have
observed that for a given laser energy the sho13k velo13ity de13reases with the
mass density This study has 13onrmed that for a given density and sho13k
velo13ity the radiative ee13ts in13rease with the atomi13 number For Kr we
noti13ed a tiny pre13ursor without any intera13tion To get su13h ee13t it will
be ne13essary to in13rease the sho13k velo13ity and thus the laser energy
This has been done at the ORION laser fa13ility in UK where the 13ol-
lision of two 13ounter-propagating sho13ks at equal speed sim 80 kms with
laser uen13e sim 6 times 10
14W13m
2has been performed (Clayson et al 2016
Suzuki-Vidal et al 2016) for dierent noble gases and for pressures 13ompri-
sed between 01 and 1 bar A number of diagnosti13s setup X-ray ba13klig-
137
hting opti13al self-emission streak imaging and interferometry (multi-frame
and streak imaging) were used to study both the post-sho13k and the radia-
tive pre13ursor Although I have not parti13ipated in the experiments I have
performed 1D simulations to interpret ORION experimental results This
work is not presented in this thesis
The previous 13hapters were dedi13ated to the study of laser generated
radiative sho13ks With irradian13e more than 10
14W13m
2 we were able to
study sho13k waves propagating at velo13ity up to 50 kms in noble gases with
an initial mass density 13omprised between 5 times 10
minus4- 3 times 10
minus3g13m
minus3
Complementary to laser experiments 13ompa13t pulsed power generators may
drive an ele13tromagneti13 13oaxial plasma gun to 13reate astrophysi13al relevant
sho13ks in lower pressure noble gases (Kondo et al 2006) with a high avai-
lability and a rather modest 13apital 13ost The ele13tromagneti13ally driven
sho13k waves may have larger s13ales than those by laser thus they 13an be
observed rather easily (Kondo et al 2008)
Su13h ele13tri13al pulsed power devi13es may then i) 13reate a high-voltage
breakdown through a gas or more easily at the surfa13e of a diele13tri13 ii)
produ13e relatively hot plasma by ohmi13 heating when the rising 13urrent in the
devi13e is passing through a portion of gas iii) a1313elerate plasma layers under
the magneti13 pressure asso13iated with the self-generated magneti13 eld when
high ele13tri13al 13urrent (say 10s of kA) is maintained for a short but su13ient
time (say 1 micros) Su13h ionization and a1313eleration are present in Z-pin13h
plasmas and in plasma fo13us devi13es (PFD) providing a 13lever geometry is
employed and the mass under 13onsideration (say 1 mg) is 13ompatible with
the stored energy (Martin Williams and Kristiansen 1996)
In this 13hapter we will see rst how a fast ele13tri13al 13ir13uit works then
we will dis13uss the prin13iples of a 13oaxial plasma gun and of PFD We will
see what is in favor of our obje13tive and what has to be avoided A geometry
13apable to rea13h our obje13tive will be proposed
The optimization of the 13oaxial gun (in term of plasma speed) will be
performed and some typi13al gures will be given in the 13ase of a generator
with a stored energy around 1 kJ and a plasma slab of 4-mm diameter ie
quite 10 times the transverse dimension of a laser driven RS The operating
13onditions are supersoni13 sho13ks up to 10-30 kms speed at stati13 pressures
of few mbar in heavy rare gases (Ar Xe) The results of this 0-D model will
be then 13ompared to those obtained with 3-D MHD simulations performed
with the 13ode GORGON (by Andrea Ciardi at LERMA) whi13h has been
used su1313essfully to des13ribe other pulse-power driven plasma experiments
(Chittenden et al 2004) as well as laboratory plasma astrophysi13s experi-
ments (Ciardi et al 2007) The diagnosti13s whi13h have been implemented
will be presented to illustrate the model as well as preliminary re13ords of the
plasma speed
Our aim being to a1313elerate a plasma slab using the magneti13 pressure
it is obviously needed to drive a high intensity 13urrent be13ause the magneti13
138 CHAPTER 8 THESIS SUMMARY
pressure is expressed by B
22micro0 in the region where exists an indu13tion B
The pressure in bars is simply 4B
2 with B in Teslas and in a 13ylindri13al
geometry 1 Tesla is the eld around a 50-kA 13urrent at a radius of 1 13m
Thus we should 13ount on roughly 100 kA delivered by the generator but
su13h a high 13urrent is delivered only in a pulsed mode by a laboratory s13ale
devi13e
There are numbers of te13hni13al solutions to a13hieve su13h ele13tri13al pa-
rameters one may use rather slow generators at moderate high voltage like
13apa13itors bank with the advantage of well know te13hniques but with limited
adjustments A1313ording to an abundant literature
1
a 13onvenient devi13e is a
medium-energy 13apa13itor bank feeding a low-indu13tan13e 13ir13uit An R-L-C
13ir13uit is a well-known 13ombination delivering a high peak 13urrent in the
pseudo-periodi13 mode For a 13apa13itor C initially 13harged under U0 key
performan13es are as follows
Current intensity I(t) is a damped sinusoid
Pseudo pulsation ω =radic[1(LC)minusR2(4L2)]
First 13urrent peak Ipeak = U0[radic(LC) + 08R]
Time of 13urrent peaking T4 = π(2ω)
Equivalent impedan13e Z =radic(LC)
For a safe handling in air a voltage not higher than 30 kV is re13ommen-
ded Taking a total 13apa13itan13e of 6 microF 2700 J are stored under 30 kV For
an indu13tan13e of 240 nH the expe13ted peak 13urrent might be 150 kA at 19
micros This set of values will be a guideline for the development presented in
further se13tions
For this part of the work the obje13tive was to show that in 13omple-
ment of laser experiments a 13ompa13t pulsed power generator might drive
astrophysi13ally relevant sho13ks in low pressure noble gases with a higher
availability and at a rather modest 13apital 13ost The longitudinal a1313elera-
tion of a plasma sheath in a low ba13king gas pressure has been demonstrated
in 13oaxial plasma guns as early as in the 70s even leading to speeds of 100
kms in light gas as H2 whi13h were measured and modeled A 13hoi13e was
made to revisit this type of devi13e At present the ele13tromagneti13 13oaxial
plasma guns are quite only 13onsidered as the initiators of a strong radial
plasma implosion plasma fo13us devi13es (PFD) due to a good ability to pro-
du13e a pin13h plasma and emission of energeti13 parti13les and hard X-ray For
that reason it was ne13essary to adapt the prin13iple of a PFD with two major
13onstraints in13rease the 13onversion of stored energy into the kineti13 energy
of a plasma sheath prevent totally any pin13hing
1
Institute for Plasma Fo13us Studies resour13e website httpplasmafo13usnet last
13onne13tion in 2016
139
That obje13tive was rea13hed by dening the ele13tri13al and geometri13al
parameters of a low-indu13tan13e and 13ompa13t pulsed power generator An
optimization pro13ess has been established to mat13h a13hievable ele13tri13al pa-
rameters of the 13ir13uit (13apa13itan13e indu13tan13e peaking 13urrent intensity)
with a plasma motion in the a1313elerating tube over few 13m leading to a nal
speed of 10-30 kms A model was built up to des13ribe the dynami13s of an
RLC 13ir13uit with a varying element 13omposed of the moving plasma sheath
a1313elerated by the self generated magneti13 pressure The 13hange in plasma
position along the axis is in13reasing the indu13tan13e and the mass a1313retion
is in13reasing the plasma mass It was shown that a 13hange in the ele13trode
geometry namely a 30deg 13oni13al shape was in13reasing the energy 13onversion
and preventing any fo13using The further step in 13ontrast with PFD is the
merging of the plasma ring into a rather planar one whi13h is inje13ted in a
drift tube
Considering Ar or Xe at gas pressure in the range 01 10 mbar and
a highly 13oaxial 1-kJ devi13e the simple model was su13ient to predi13t
performan13es agreeing with the only work quoted in the literature Basi13
observations of the moving plasma using side-on dete13tion of the visible
emission give features 13oherent with the model Ma13h numbers from 20
to 60 are obtained tting with the obje13tive of strong sho13k formation A
rened 3-D MHD numeri13al simulation indi13ates very promising features of
the sho13k in view of exploring mm-size sho13ks in a regime 13omplementary
to laser driven ones and on a table-top devi13e allowing a higher repetition
rate
140 CHAPTER 8 THESIS SUMMARY
Appendi13es
141
Appendix A
Visible Interferometry
Laser interferometry is used in plasmas to dedu13e the ele13tron density through
the variation of the refra13tive index 13aused by the ionisation of the matter
In this appendix rstly the experimental setup of Ma13h Zehnder interfero-
metry as well as the underlying physi13al prin13iples are presented
A01 Refra13tive index of a plasma
In order to derive the refra13tive index of a gas one starts with the set of
Maxwells equations for a mono13hromati13 plane wave propagating in the
plasma (see for instan13e equation 411 of referen13e (Hut13hinson 2002))
Free ele13trons and ions are supposed to be distributed uniformly and the net
total 13harge density is equal to zero
For harmoni13 waves propagating in the dire13tion of +x and with the
pulsation ω one obvious solution is given by
E(x t) = E0ei(kxminusωt)
(A1)
where E0 and k are 13omplex quantities
The relation between k and ω is given by the dispersion relation
k2 =εmicroω2
c2(1 + i
4πσ
ωε) (A2)
where micro is the ele13tri13 permitivity and ε is permeability
The free ele13trons are a1313elerated by the ele13tri13 eld and therefore equa-
tion their motion is given as
medv
dt= minuseE0e
minusiωt(A3)
The obvious solution for the ele13tron velo13ity v is
v = minusie
meωE (A4)
and thus the ele13trons 13arry a 13urrent with a density J
J = Neev = iNee
2
meωE (A5)
where Ne is the ele13tron density Due to the larger mass of the ions the
13orresponding 13urrent is negligible
143
144 APPENDIX A VISIBLE INTERFEROMETRY
The previous equation allows to derive the plasma 13ondu13tivity σ whi13his equal to iNee
2ωme Putting this value in the equation (A2) and assu-
ming a thin medium where ε asymp 1 and micro = 1 one obtains nally
k2 =ω2
c2(1minus
ω2p
ω2) (A6)
where ωp is the plasma frequen13y dened as
ω2p =
4πNee2
me(A7)
Equation A6 shows that a ele13tromagneti13 wave with a pulsation larger than
ωp 13ant propagate in the plasma This allows to derive a 13riti13al ele13tron
density Nc in 13m
minus3as
Nc =meω
2
4πe2=
4πmec2
e2λ2(A8)
For instan13e for a radiation of 1 microm wavelength this upper limit is equal
to Nc = 446times 1021cmminus3
Therefore the index of refra13tion is given by the expression
n =
radic
1minusω2p
ω2=
radic
1minus Ne
Nc(A9)
Finally one obtains
E(x t) = E0ei(kxminusωt) with k = n
ω
c(A10)
As a 13onsequen13e of the variation of the refra13tive index in the dierent
plasma layers the light is refra13ted Indeed the Snell laws stipulate that if a
beam is in13ident on a plane interfa13e between layers (of refra13tive index n1
and n2) it undergoes a bending from the angle of in13iden13e The in13ident
beam the refra13ted beam and the normal to the interfa13e lie in the same
plane and one has n1 sinθ1 = n2 sinθ2 where θ1 is the angle of in13iden13e
and θ2 is the angle of refra13tion Thus a mono13hromati13 beam propagating
in a plasma with in13reasing density be13omes more and more dee13ted
A02 Absorption of the laser beam
Beside refra13tion the light is absorbed by the plasma and the absorption
also in13reases rapidly near the 13riti13al density In the absen13e of any ato-
mi13 bound-bound or bound-free transition this absorption is due to the
inverse Bremsstrahlung pro13ess (ion-ele13tron 13ollision (Pfalzner 2006)) by
145
the plasma free ele13trons The absorption 13oe13ient κ (in 13m
minus1) is then
given by (NRL formulary)
κ(Ne Tev) = 317 10minus7LnΛ lt Z gt
(
Ne
ω
)2 1
T32ev (1minusNeNc)12
(A11)
where Tev is the temperature in eV Ne is the ele13tron density in 13m
minus3
and LnΛ is the Coulomb Logarithm dened as Λ = Max (2 Min(XY))
with (NRL formulary)
X = 23minus log
(
lt Z gtN
12e
T32ev
)
and Y = 24minus log
(
N12e
Tev
)
(A12)
The transmission T (d) through a layer of thi13kness d is given by
T (d) = exp(minusint d
0κ(l)dl) (A13)
For instan13e for λ = 527 nm taking Ne= 101913mminus3 ltZgt= 10 T = 10 eV
whi13h are typi13al values for the post-sho13k in our experiments in Xenon and
supposing a homogenous plasma thi13kness d = 600 microm the transmission is
about 09 whereas at 5times 1019cmminus3 it falls to 01
A03 Prin13iple of interferometry
In the opti13al interferometry te13hnique (Ovsyannikov and Zhukov 2000) the
interferen13e of two plane waves is performed by addition of the two 13orre-
sponding 13oherent ele13tri13 elds The analyzed medium (here the plasma)
in whi13h one of the beam is propagating indu13es a deshaping between the
beams whi13h ae13ts the resulting intensity depending if the elds interfere
in phase or out of phase ie 13onstru13tively or destru13tively
Two te13hniques are used the wave front and the amplitude division
The wave front division is obtained by using two portions (Fresnel mir-
rors Youngs double slit Lloyds mirror prisms et13) of the original wave
front whi13h are then superimposed to produ13e interferen13e
In the amplitude division the two beams are separated by division of
the amplitude over the same se13tion of the wave front Mi13helson and Ma13h
Zehnder interferometers et13 are the typi13al examples of this te13hnique of
amplitude division
The prin13iple of interferen13e of two 13oherent mono13hromati13 traveling
waves 1 and 2 with ele13tri13 elds E1 and E2 propagating in the same
dire13tion and polarized in the same plane 13an be understood as follows for
a given time t at the point r where these waves superpose the amplitude of
the ele13tri13 eld is the ve13tor addition of the two 13orresponding elementary
amplitudes ie
146 APPENDIX A VISIBLE INTERFEROMETRY
E(r t) = E1(r t) + E2(r t)
= a1exp(ikr + ωt+ iφ1) + a2exp(ikr + ωt+ iφ2)
where φ1 and φ2 are the phases of the two waves at r The total intensity
I(r t) = |E1 + E2|2 (A14)
One gets
I = I1 + I2 + 2radic
(I1I2)cos∆φ (A15)
where I1 = a12and I2 = a2
2are the intensities at the point of interferen13e
due to the two waves a13ting independently and ∆φ = φ1minusφ2 represents the
phase dieren13e between the two waves In the 13ase of equal intensities I1one gets
I = 4I1cos2(∆φ2) (A16)
and the interferometri13 pattern 13onsists of dark (∆φ = π + 2nπ) and
bright patterns (∆φ = 2nπ)In general the two beams are only partially 13oherent and the minimum
of the fringe intensity is not equal to zero The fringe 13ontrast
FC =Imax minus Imin
Imax + Imin(A17)
is then a measure of the interferen13e quality FC is maximum and equal to
1 in the 13ase of fully 13oherent beams of the same intensity I1 = I2
A04 Ma13h Zehnder Interferometer
The Ma13h Zehnder Interferometer is a devi13e 13ommonly and e13iently used
to estimate the 13hange in refra13tive index by the plasma Su13h 13hange is in
most the 13ases related to variations in the ele13tron density We employed
this interferometri13 setup in our experiment
The prin13iple of this interferometer is based on the division of the in13ident
light beam into two beams with the help of a beam-splitter whi13h 13an be
a 13oated glass plate or a 13ube After this division one of the beams (probe
beam) is allowed to pass through the medium of interest (a plasma in our
13ase) and then made to interfere with the other unae13ted part of the beam
(referen13e beam) with the help of se13ond beam-splitter as may be noted
from Fig A1
In order to understand the prin13iple of interferen13e and its use in esti-
mating plasma properties let us 13onsider l1 and l2 whi13h are the total path
lengths (in13luding the path in the opti13al elements) for the light travelling
from the sour13e to the dete13tor for the upper and lower paths (13f Figure A1)
respe13tively Suppose for the moment that the sample is removed from beam
147
Figure A1 S13hemati13s of Ma13h Zehnder interferometer with two mirrors two
50 beam splitters and the dete13tor A plasma slab is pla13ed in one of the two
arms of the interferometer
1 (see Figure A1) the two opti13al paths of the probe and referen13e beams
dier in terms of phases Simplify to supposing that the beams propagate
in air (n=1) these phases are then equal to the 2πl1λ and 2πl2λThus a1313ounting for the phase dieren13e δ between the two beams tra-
velled up to dete13tor A 13an be given as
2πl1 minus l2
λ= δ (A18)
If the two beams are perfe13tly parallel at the position of the dete13tor the
interferometri13 gure presents a at intensity whi13h passes through minima
and maxima when one length of the 2 arms is slightly 13hanged Usually one
introdu13es a small tilt between the two dire13tions of propagations whi13h
leads to the apparition of parallel linear dark and bright fringes
In order to obtain interferen13e patterns it is important to make the paths
l1 and l2 as 13lose as possible 13ompared to the 13oheren13e length of the laser
For the multimode Evolution 15 laser used in the PALS experiment this
length is approximately 1 13m (value 13ommuni13ated by the 13onstru13tor)
If we introdu13e the sample in the probe beam 1 this sample will introdu13e
an additional phase denoted by δsample and the net phase shift is then δ +δsample
The prin13iple of the dedu13tion of the ele13tron density with the Ma13h
Zehnder interferometer is to measure the refra13tive index of the test plasma
slab whi13h is pla13ed in the probe beam We assume for the moment that
148 APPENDIX A VISIBLE INTERFEROMETRY
this plasma slab is a phase obje13t for the in13oming beam whi13h means that
it does not signi13antly ae13t the intensity of the probe beam while it only
13hanges the phase of the wave [5]The additive phase dieren13e in the probe arm introdu13ed by the plasma
13an be dened as
φ =
int d
0kplasmadl =
int d
0nω
cdl (A19)
where n is the lo13al refra13tive index at position l of the plasma slab and
the integral is performed other the slab thi13kness d Therefore the phase
dieren13e introdu13ed by the plasma relative to the propagation into va13uum
of the referen13e beam is equal to
∆φ =
int d
0(kplasma minus k0)dl (A20)
where k0 = ωc = 2πλ is the wave ve13tor of the beam propagating in
va13uum
As long as Ne lt Nc the beam propagates in the plasma the plasma
refra13tive index is given by equation A9 and one has
∆φ =ω
c
int d
0
[
(
(1minus Ne
Nc
)12
minus 1]
dl (A21)
At very small ele13tron density (Ne ≪ Nc) the phase shift 13an be written as
∆φ asymp minus ω
2cNc
int d
0Nedl = minus πd
λNclt Ne gt (A22)
where
lt Ne gt=
int d
0
Ne(z t)dy
d(A23)
is the ele13tron density averaged over the path dIn pra13ti13e a referen13e re13ord is rst taken without plasma To dedu13e
the density one has to analyze how the fringes depart from their referen13e
positions
The spa13e between two fringes (inter-fringe) 13orresponds to a phase va-
riation of 2 π A shift of the fringe by 2 π from its referen13e position (where
Ne = 0) 13orresponds to an ele13tron density variation equal to lt Ne gt = 2
λNcd For λ = 527 nm and supposing an homogeneous plasma of thi13kness
600 microm this gives lt Ne gt= 7 times 10
1813m
minus3
Appendix B
Opa13ities and mean 13harge
The variations in logarithmi13 s13ale with the temperature of the PROPA-
CEOS Rosseland and Plan13k opa13ities (se13tion of 13hapter 5 see for intense
Fig 511)as also of the mean 13harge of the plasma are reported for two
densities 16 times 10
minus3g13m
3and 51 times 10
minus4g13m
3for three noble gases Ar
Kr and Xe in Fig B1 and B2
ρ = 51 times 10
minus4g13m
313orrespond to typi13al density of the pre13ursor
For this density Xenon and Krypton have quite similar variations for the
Rosseland opa13ity with a bump of 7000 13m
2g at 3 eV for Xe and 12000
13m
2g at 4 eV for Krypton At 10 eV the two opa13ities de13rease respe13tively
to 600 and 900 13m
2g In 13omparison with these two 13ases the opa13ity of
Argon is broader and more regular Its opa13ity peaks at 6 eV to 13000 13m
2g
At 10 eV it have de13reased to 7000 13m
2g whi13h is higher than for the two
previous gases
The mean 13harge variations in13rease regularly with the temperature At
10 eV these mean 13harges are respe13tively equal to 65 55 and 5 for Xe Kr
and Ar
149
150 APPENDIX B OPACITIES AND MEAN CHARGE
(a)
(b)
Figure B1 Rosseland and Plan13k opa13ity for three gases Xe Kr and Ar at mass
densities 51 times 10
minus4g13m
3(a) and 16 times 10
minus3g13m
3(b)
151
(a)
(b)
Figure B2 Mean 13harge for three gases Xe Kr and Ar at mass densities 16 times10
minus3g13m
3(a) and 51 times 10
minus4g13m
3(b)
152 APPENDIX B OPACITIES AND MEAN CHARGE
Referen13es
Batani D Strati F Telaro B Loumlwer T Hall T Benuzzi-Mounaix A
Koenig M 2003 Produ13tion of high quality sho13ks for equation of state
experiments European Physi13al Journal D 23 99 DOI ADS
Bernard A 2002 Plasma fo13us et appli13ations Le Vide 57(306)
Bett TH Jinks P Stevenson RM Danson CN Pepler DA Ross
IN 1995 Binary phase zone-plate arrays for laser-beam spatial-intensity
distribution 13onversion Applied Opti13s 34 4025 DOI ADS
Bouquet S Teyssier R Chieze JP 2000 Analyti13al study and stru13-
ture of a stationary radiative sho13k The Astrophysi13al Journal Supplement
Series 127(2) 245
Bouquet S Stehleacute C Koenig M Chiegraveze J Benuzzi-Mounaix A Ba-
tani D Leygna13 S Fleury X Merdji H Mi13haut C et al 2004
Observation of laser driven super13riti13al radiative sho13k pre13ursors Phy-
si13al review letters 92 225001
Bozier J Thiell G Le Breton J Azra S De13roisette M S13hirmann
D 1986 Experimental observation of a radiative wave generated in xenon
by a laser-driven super13riti13al sho13k Physi13al review letters 57(11) 1304
Burhop E Spitzer L 1957 Physi13s of fully ionized gases inters13ien13e
tra13ts on physi13s and astronomy no 3 JSTOR
Chaulagain U 2015 Radiative sho13ks expriments modellin and links to
Astrophysi13s PhD thesis Pierre and Marie Curie University
Chaulagain U Stehleacute C de Saacute L Larour J Auvray P Kozlova M
Krus M Dostal J Propupek J Suzuki-Vidal F et al 2012 Labo-
ratory experiments of radiative sho13ks in the 13ontext of stellar a1313retion
In SF2A-2012 Pro13eedings of the Annual meeting of the Fren13h So13iety
of Astronomy and Astrophysi13s 1 305
Chaulagain U Stehleacute C Larour J Kozlovaacute M Suzuki-Vidal F Bar-
roso P Cotelo M Velarde P Rodriguez R Gil JM Ciardi A
A13ef O Nejdl J de Saacute L Singh RL Ibgui L Champion N 2015
Stru13ture of a laser-driven radiative sho13k High Energy Density Physi13s
17 106 DOI ADS
Chevalier RA 1977 The intera13tion of supernovae with the interstellar
medium Annual review of astronomy and astrophysi13s 15 175 DOI ADS
153
154 REFERENCES
Chittenden JP Lebedev SV Jennings CA Bland SN Ciardi A
2004 X-ray generation me13hanisms in three-dimensional simulations of
wire array Z-pin13hesthe evolution of magneti13 tower jets in the laboratory
Plasma Physi13s and Controlled Fusion 46 B457 DOI ADS
Ciardi A 2014 private 13ommuni13ation
Ciardi A Lebedev SV Frank A Bla13kman EG Chittenden JP
Jennings CJ Ampleford DJ Bland SN Bott SC Rapley J Hall
GN Suzuki-Vidal FA Maro1313hino A Lery T Stehle C 2007 The
evolution of magneti13 tower jets in the laboratory Physi13s of Plasmas
14(5) 056501 DOI ADS
Clayson T Suzuki-Vidal F Lebedev SV Swadling GF Burdiak GC
Patankar S Smith RA Foster J Skidmore J Gumbrell E Gra-
ham P Danson C Stehleacute C Singh RL Chaulagain U Larour
J Kozlova M Spindloe C 2016 Counter-propagating radiative sho13k
experiments on the Orion laser fa13ility In APS Meeting Abstra13ts DOI
ADS
Clayson T Suzuki-Vidal F Lebedev S Swadling G Stehleacute C Bur-
diak G Foster J Skidmore J Graham P Gumbrell E et al 2017
Counter-propagating radiative sho13k experiments on the orion laser and
the formation of radiative pre13ursors High Energy Density Physi13s 23
60
Cotelo M Velarde P de la Varga AG Portillo D Stehleacute C Chaul-
again U Kozlova M Larour J Suzuki-Vidal F 2015 Simulation
of radiative sho13k waves in Xe of last PALS experiments High Energy
Density Physi13s 17 68 DOI ADS
de la Varga AG Velarde P Cotelo M de Gaufridy F
Zeitoun P 2011 Radiative properties for warm and hot
dense matter High Energy Density Physi13s 7(3) 163 DOI
httpwwws13ien13edire13t13oms13ien13earti13lepiiS1574181811000498
Diziegravere A Mi13haut C Koenig M Gregory C Ravasio A Sakawa
Y Kuramitsu Y Morita T Ide T Tanji H et al 2011 Highly
radiative sho13k experiments driven by gekko xii Astrophysi13s and Spa13e
S13ien13e 336(1) 213
Dobbs-Dixon I Cumming A Lin DNC 2010 Radiative Hydrodyna-
mi13 Simulations of HD209458b Temporal Variability The Astrophysi13al
Journal 710 1395 DOI ADS
Doss FW Robey HF Drake RP Kuranz CC 2009 Wall sho13ks in
high-energy-density sho13k tube experiments Physi13s of Plasmas 16(11)
112705
REFERENCES 155
Drake RP 2006 High-energy-density physi13s Fundamentals inertial fu-
sion and experimental astrophysi13s Sho13k wave and high pressure phe-
nomena Springer Berlin Heidelberg New York ISBN 3-540-29314-0
httpopa13inriafrre13ord=b1128433
Drake R Doss F M13Clarren R Adams M Amato N Bingham D
Chou C DiStefano C Fidkowski K Fryxell B et al 2011 Radiative
ee13ts in radiative sho13ks in sho13k tubes High Energy Density Physi13s
7(3) 130
Ensman L 1994 Test problems for radiation and radiation-hydrodynami13s
13odes The Astrophysi13al Journal 424 275 DOI ADS
Ensman L Burrows A 1992 Sho13k breakout in sn 1987a The Astrophy-
si13al Journal 393 742
Fadeyev YA Gillet D 2001 The stru13ture of radiative sho13k waves
III The model grid for partially ionized hydrogen gas Astronomy and
Astrophysi13s 368 901 DOI ADS
Fleury X Bouquet S Stehleacute C Koenig M Batani D Benuzzi-
Mounaix A Chieze J-P Grandjouan N Grenier J Hall T et al
2002 A laser experiment for studying radiative sho13ks in astrophysi13s
Laser and Parti13le Beams 20(02) 263
Fryxell B Olson K Ri13ker P Timmes FX Zingale M Lamb DQ
Ma13Nei13e P Rosner R Truran JW Tufo H 2000 FLASH An
Adaptive Mesh Hydrodynami13s Code for Modeling Astrophysi13al Ther-
monu13lear Flasheshera13les a three-dimensional radiation hydrodynami13s
13ode Astrophysi13al Journal Supplement 131 273 DOI ADS
Gonzalez JH Brollo FR Clausse A 2009 Modeling of the Dynami13
Plasma Pin13h in Plasma Fo13us Dis13harges Based in Von Karman Approx-
imations IEEE Transa13tions on Plasma S13ien13e 37 2178 DOI ADS
Gonzalez JH Clausse A Bruzzone H Florido PCMZ 2004 A Lum-
ped Parameter Model of Plasma Fo13us IEEE Transa13tions on Plasma
S13ien13e 32 1383 DOI ADS
Gonzaacutelez M Audit E Huynh P 2007 HERACLES a three-dimensional
radiation hydrodynami13s 13ode Astronomy and Astrophysi13s 464 429
DOI ADS
Gonzaacutelez M Audit E Huynh P 2007 Hera13les a three-dimensional
radiation hydrodynami13s 13ode Astronomy and Astrophysi13s 464(2) 429
Gonzaacutelez M Audit E Stehleacute C 2009 2D numeri13al study of the ra-
diation inuen13e on sho13k stru13ture relevant to laboratory astrophysi13s
Astronomy and Astrophysi13s 497 27 DOI ADS
156 REFERENCES
Gonzaacutelez M Stehleacute C Rus B Busquet M Thais F A13ef O Barroso
P Bar-Shalom A Bauduin D Kozlova M Lery T Madouri A To-
mas M Polan J 2006 Astrophysi13al radiative sho13ks From modeling
to laboratory experiments Laser and Parti13le Beams 24 535540 DOI
ADS
Gonzalez M StehlE C Audit E Busquet M Rus B Thais F A13ef
O Barroso P Bar-Shalom A Bauduin D et al 2006 Astrophysi13al
radiative sho13ks From modeling to laboratory experiments Laser and
Parti13le Beams-Pulse Power and High Energy Densities 24(4) 535
Gonzaacutelez M Vaytet N Commerccedilon B Masson J 2015 Multigroup
radiation hydrodynami13s with ux-limited diusion and adaptive mesh
renement Astronomy and Astrophysi13s 578 A12 DOI ADS
Griem HR 2005 Prin13iples of Plasma Spe13tros13opy 386 ADS
Gu MF 2008 The exible atomi13 13ode Canadian Journal of Physi13s
86(5) 675
Harilal SS Tilla13k MS 2004 Laser plasma
density measurements using interferometry
httplinkapsorgdoi101103PhysRevE64066412
Hartigan P 1989 The visibility of the Ma13h disk and the bow sho13k of a
stellar jet Astrophysi13al Journal 339 987 DOI ADS
Hartigan P 2003 Sho13k Waves in Outows from Young Stars Astrophysi13s
and Spa13e S13ien13e 287 111 DOI ADS
Hartigan P Morse JA Reipurth B Heath13ote S Bally J 2001 Pro-
per Motions of the HH 111 Jet Observed with the Hubble Spa13e Teles13ope
Astrophysi13al Journal Letters 559 L157 DOI ADS
Henke BL Gullikson E Davis JC 1993 X-ray intera13tions Photoab-
sorption s13attering transmission and ree13tion atlt igt eltigt= 50-30000
evlt igt zltigt= 1-92 Atomi13 data and nu13lear data tables 54(2) 181
Hugoniot PH 1887 Meacutemoire sur la propagation du mouvement dans les
13orps et ples speacute13ialement dans les gaz parfaits Journal de matheacutematiques
pures et appliqueacutees 3 477
Hugoniot PH 1889 Meacutemoire sur la propagation des mouvements dans les
13orps et speacute13ialement dans les gaz parfaits (deuxiegraveme partie) Journal de
lEacute13ole Polyte13hnique 58 1
Hut13hinson IH 2002 Prin13iples of Plasma Diagnosti13s 458 ADS
REFERENCES 157
Jungwirth K Cejnarova A Juha L Kralikova B Krasa J Krousky
E Krupi13kova P Laska L Masek K Mo13ek T et al 2001 The
prague asterix laser system Physi13s of Plasmas 8 2495
Keiter P Drake R Perry T Robey H Remington B Iglesias C
Walla13e R Knauer J 2002 Observation of a hydrodynami13ally driven
radiative-pre13ursor sho13k Physi13al review letters 89(16) 165003
Koenig M Faral B Boudenne JM Batani D Benuzzi A Bossi S
1994 Opti13al smoothing te13hniques for sho13k wave generation in laser-
produ13ed plasmas Physi13al Review E 50 R3314 DOI ADS
Kondo K Nakajima M Kawamura T Horioka K 2006 Compa13t pulse
power devi13e for generation of one-dimensional strong sho13k waves Review
of S13ienti13 Instruments 77(3) 036104 DOI ADS
Kondo K Nakajima M Kawamura T Horioka K 2008 Relaxation
layer in ele13tro-magneti13ally driven strong sho13ks Journal of Physi13s Con-
feren13e Series 112(4) 042028 DOI ADS
Kondo K Nakajima M Kawamura T Horioka K 2009a Atomi13 pro-
13ess in ele13tro-magneti13ally driven strong sho13k wave Nu13lear Instruments
and Methods in Physi13s Resear13h A 606 223 DOI ADS
Kondo K Nakajima M Kawamura T Horioka K 2009b Radiation ef-
fe13t and relaxation layer in ele13tro-magneti13ally driven strong sho13k waves
In APS Meeting Abstra13ts ADS
Kunze H-J 2009 Springer series on atomi13 opti13al and
plasma physi13s 125 Springer Berlin Heidelberg New York
httpwwwspringer13omseries411
Kwek K Tou T Lee S 1990 Current sheath stru13ture of the plasma
fo13us in the run-down phase IEEE Transa13tions on Plasma S13ien13e 18(5)
826
Larour J Singh RL Stehleacute C Ciardi A Chaulagain U Suzuki-Vidal
F 2015 Optimization of an ele13tromagneti13 generator for strong sho13ks
in low pressure gas High Energy Density Physi13s 17 129 DOI ADS
Lee S 1969 Transverse ionizing sho13k waves in a planar ele13tromagneti13
sho13k tube PhD thesis Australian National University
Lee S Serban A 1996 Dimensions and lifetime of the plasma fo13us pin13h
IEEE Transa13tions on plasma s13ien13e 24(3) 1101
Leygna13 S Boireau L Mi13haut C Lanz T Stehleacute C Clique C
Bouquet S 2006 Modeling multidimensional ee13ts in the propagation
of radiative sho13ks Physi13s of Plasmas (1994-present) 13(11) 113301
158 REFERENCES
Ma13Farlane JJ Golovkin IE Woodru PR 2006 HELIOS-CR A 1-
D radiation-magnetohydrodynami13s 13ode with inline atomi13 kineti13s mo-
deling Journal of Quantitative Spe13tros13opy and Radiative Transfer 99
381 DOI ADS
Ma13quorn Rankine WJ 1870 On the Thermodynami13 Theory of Waves of
Finite Longitudinal Disturban13e Philosophi13al Transa13tions of the Royal
So13iety of London Series I 160 277 ADS
Mahmood S Ur-Rehman H 2009 Ele13trostati13 solitons in unmagnetized
hot ele13tron-positron-ion plasmas Physi13s Letters A 373 2255 DOI ADS
Martin TH Williams M Kristiansen M 1996 J13 martin on pulsed
power 3 Springer S13ien13e amp Business Media
M13Clarren RG Drake RP Morel JE Holloway JP 2010 Theory of
radiative sho13ks in the mixed opti13ally thi13k-thin 13ase Physi13s of Plasmas
17(9) 093301
Mi13haut C Stehleacute C Leygna13 S Lanz T Boireau L 2004 Jump
13onditions in hypersoni13 sho13ks Quantitative ee13ts of ioni13 ex13itation
and radiation European Physi13al Journal D 28 381 DOI ADS
Mihalas D 1978 Stellar atmospheres 2nd edition ADS
Mihalas D Mihalas BW 1984 Foundations of radiation hydrodynami13s
ADS
Mihalas D Mihalas BW 1999 Foundations of radiation hydrodynami13s
More R Warren K Young D Zimmerman G 1988 A new quotidian
equation of state (qeos) for hot dense matter Physi13s of Fluids (1958-
1988) 31(10) 3059
Ogando F Velarde P 2001 Development of a radiation transport uid
dynami13 13ode under amr s13heme Journal of Quantitative Spe13tros13opy and
Radiative Transfer 71(2) 541
Olson GL Auer LH Hall ML 2000 Diusion P1 and other approx-
imate forms of radiation transport Journal of Quantitative Spe13tros13opy
amp Radiative Transfer 64 619 DOI ADS
Orban C Fatenejad M Chawla S Wilks SC Lamb DQ 2013 A
Radiation-Hydrodynami13s Code Comparison for Laser-Produ13ed Plasmas
FLASH versus HYDRA and the Results of Validation Experiments ArXiv
e-prints ADS
REFERENCES 159
Orlando S Bonito R Argiro C Reale F Peres G Mi13eli M Mat-
sakos T Stehleacute C Ibgui L de Saacute L Chiegraveze J-P Lanz T 2013
Radiative a1313retion sho13ks along nonuniform stellar magneti13 elds in 13las-
si13al T Tauri stars Astronomy amp Astrophysi13s
Ovsyannikov AA Zhukov MF 2000 Plasma diagnos-
ti13s Cambridge International S13ien13e Publishing Cambridge
httppubli13eblib13om13hoi13epubli13fullre13ordaspxp=258189
Pfalzner S 2006 An introdu13tion to inertial 13onnement fusion 1st
edn Series in Plasma Physi13s Taylor amp Fran13is Taylor amp Fran13is
httpgenlibruse13bookindexphpmd5=EACF38292FD79113F78BB532AB5301B8
Potter DE 1971 Numeri13al Studies of the Plasma Fo13us Physi13s of Fluids
14 1911 DOI ADS
Proakis JG 2001 Digital signal pro13essins Wiley Online Library
Raga A Mellema G Arthur S Binette L Ferruit P Steen W 1999
3d transfer of the diuse ionizing radiation in ism ows and the preioni-
zation of a herbig-haro working surfa13e Revista Mexi13ana de Astronomia
y Astrosi13a 35 123
Reighard AB 2007 Collapsing radiative sho13k experiments on the omega
laser PhD thesis The University of Mi13higan
Reighard A Drake R Dannenberg K Kremer D Grosskopf M Har-
ding E Leibrandt D Glendinning S Perry T Remington B et al
2006 Observation of 13ollapsing radiative sho13ks in laboratory experiments
Physi13s of Plasmas (1994-present) 13(8) 082901
Rodriguez R Gil JM Florido R Rubiano JG Mendoza MA Mar-
tel P Minguez E Symes DR Hohenberger M Smith RA 2011
Determination of the average ionization and thermodynami13 regimes of
xenon plasmas with an appli13ation to the 13hara13terization of blast waves
laun13hed in xenon 13lusters High Energy Density Physi13s 7(2) 71 DOI
httpwwws13ien13edire13t13oms13ien13earti13lepiiS157418181100022X
Rodriacuteguez R Espinosa G Gil JM Stehleacute C Suzuki-Vidal F Rubi-
ano JG Martel P Miacutenguez E 2015 Mi13ros13opi13 properties of xenon
plasmas for density and temperature regimes of laboratory astrophysi13s
experiments on radiative sho13ks Physi13al Review E 91(5) 053106 DOI
ADS
Ross IN Pepler DA Danson CN 1995 Binary phase zone plate de-
signs using 13al13ulations of far-eld distributions Opti13s Communi13ations
116 55 DOI ADS
160 REFERENCES
Seaton MJ 1987 Atomi13 data for opa13ity 13al13ulations I General des13rip-
tion Journal of Physi13s B Atomi13 and Mole13ular Physi13s 20(23) 6363
Serban A 1995 Anode geometry and fo13us 13hara13teristi13s PhD thesis
Nanyang Te13hnology University httphdlhandlenet104971568
Serban A Lee S 1995 A simple opti13al ber axial speed dete13tor Review
of S13ienti13 Instruments 66 4958 DOI ADS
Serban A Lee S 1995 A simple opti13al ber axial speed dete13tor Review
of s13ienti13 instruments 66(10) 4958
Singh JP Thakur SN (eds) 2007 Laser-indu13ed breakdown
spe13tros13opy Elsevier s13ien13edire13t
Singh R Stehleacute C Suzuki-Vidal F Kozlova M Larour J Chaulagain
U Clayson T Rodriguez R Gil J Nejdl J et al 2017 Experimental
study of the intera13tion of two laser-driven radiative sho13ks at the pals
laser High Energy Density Physi13s 23 20
Stahler SW Palla F Salpeter EE 1986 Primordial stellar evolution -
The protostar phase The Astrophysi13al Journal 302 590 DOI ADS
Stehleacute C Gonzaacutelez M Kozlova M Rus B Mo13ek T A13ef O Colom-
bier JP Lanz T Champion N Jakub13zak K et al 2010 Experi-
mental study of radiative sho13ks at pals fa13ility Laser and parti13le beams
28(02) 253
Stehleacute C Kozlovaacute M Larour J Nejdl J Champion N Barroso P
Suzuki-Vidal F A13ef O Delattre P-A Dostaacutel J et al 2012 New
probing te13hniques of radiative sho13ks Opti13s Communi13ations 285(1)
64
Stehleacute C Lefegravevre R Chaulagain U Champion N Barroso P Reix
F Jagourel P Larour J Melt13hakov E Mer13ier R et al 2014a
Criti13al 13omponents for xuv probing of laser driven sho13ks In X-Ray
Lasers 2012 Springer 239
Stehleacute C Kozlovaacute M Larour J Nejdl J Suzuki-Vidal F Cohen M
Chaulagain UP Champion N Barroso P A13ef O et al 2014b
Reply on the 13omment of the paper new probing te13hniques of radiative
sho13ks Opti13s Communi13ations 318 226
Suzuki-Vidal F Lebedev S Pi13kworth L Swadling G Burdiak G
Skidmore J Hall G Bennett M Bland S Chittenden J et al
2014 Sho13k formation in 13ounter-streaming jets on the magpie pulsed-
power generator In APS Meeting Abstra13ts 1 5005
REFERENCES 161
Suzuki-Vidal F Clayson T Swadling GF Patankar S Burdiak GC
Lebedev SV Smith RA Stehle C Chaulagain U Singh RL La-
rour J Kozlova M Spindloe C Foster J Skidmore J Gumbrell E
Graham P Danson C 2016 Counter-streaming radiative sho13k experi-
ments on the Orion laser In APS Meeting Abstra13ts ADS
Toledo-Roy JC Velaacutezquez PF de Colle F Gonzaacutelez RF Reynoso
EM Kurtz SE Reyes-Iturbide J 2009 Numeri13al model for the SNR
DEM L316 simulated X-ray emission Monthly Noti13es of the RAS 395
351 DOI ADS
Tou T Lee S Kwek K 1989 Nonperturbing plasma-fo13us measurements
in the run-down phase IEEE Transa13tions on Plasma S13ien13e 17(2) 311
Vaytet NMH Audit E Dubro13a B Delahaye F 2011 A numeri13al
model for multigroup radiation hydrodynami13s Journal of Quantitiative
Spe13tros13opy and Radiative Transfer 112 1323 DOI ADS
Vaytet N Chabrier G Audit E Commerccedilon B Masson J Ferguson
J Delahaye F 2013 Simulations of protostellar 13ollapse using mul-
tigroup radiation hydrodynami13s ii the se13ond 13ollapse arXiv preprint
arXiv13071010
Vaytet N Gonzaacutelez M Audit E Chabrier G 2013 The inuen13e
of frequen13y-dependent radiative transfer on the stru13tures of radiative
sho13ks Journal of Quantitiative Spe13tros13opy and Radiative Transfer 125
105 DOI ADS
Velarde P Gar13iacutea-Senz D Bravo E Ogando F Relantildeo A Gar13iacutea C
Oliva E 2006 Intera13tion of supernova remnants From the 13ir13umstellar
medium to the terrestrial laboratory Physi13s of Plasmas 13(9) 092901
DOI ADS
Veloso F Moreno J Tarifentildeo-Saldivia A Pavez C Zambra M Soto
L 2012a Non-intrusive plasma diagnosti13s for measuring sheath kine-
mati13s in plasma fo13us dis13harges Measurement S13ien13e and Te13hnology
23(8) 087002
Veloso F Tarifentildeo-Saldivia A Pavez C Moreno J Zambra M Soto
L 2012b Plasma sheath kinemati13s and some impli13ations on the mo-
deling of very low energy plasma fo13us devi13es Plasma Physi13s and Con-
trolled Fusion 54(9) 095007
Vis13o A Drake R Glenzer S Doumlppner T Gregori G Froula D
Grosskopf M 2012 Measurement of radiative sho13k properties by x-ray
thomson s13attering Physi13al review letters 108(14) 145001
162 REFERENCES
Williams RM Chu Y-H Di13kel JR Beyer R Petre R Smith RC
Milne DK 1997 Supernova Remnants in the Magellani13 Clouds I The
Colliding Remnants DEM L316 Astrophysi13al Journal 480 618 ADS
Williams RM Chu Y-H Di13kel JR Gruendl RA Seward FD
Guerrero MA Hobbs G 2005 Supernova Remnants in the Magella-
ni13 Clouds V The Complex Interior Stru13ture of the N206 Supernova
Remnant Astrophysi13al Journal 628 704 DOI ADS
Zambra M Kalise D Fernandez J Hernandez E Pas-
ten D Munoz V 2009 Current sheet thi13kness in the
plasma fo13us snowplow model J Plasma Fusion 8(879-882)
httpspeopleri13amoeawa13atdkalisefilesplasmapdf
Zeldovi13h IB Zeldovi13h YB Raizer YP 2002 Physi13s of sho13k waves
and high-temperature hydrodynami13 phenomena Courier Dover Publi13ati-
ons London New York Sydney
REFERENCES 163
ACKNOWLEDGEMENTS
First and foremost I would like pay my sin13ere gratitude to Dr Jean La-
rour and Dr Chantal Stehleacute my thesis supervisors who have introdu13ed
me to the marvelous world of Laboratory Astrophysi13s I would express my
thanks for their 13ontinuous support during my PhD journey for en13ouraging
me to learn 13hallenging things Working under their guidan13e was a unique
experien13e and a great joy I am parti13ularly grateful for their serenity and
patien13e that helped me in developing my spa13e and style in resear13h Co-
ming all the way from India to a non-English speaking 13ommunity had been
presented by them to me like a se13ond home through their benign love and
ae13tion
I have imparted a signi13ant time during my PhD for 13ondu13ting expe-
riments in PALS laser fa13ility at Prague through LASERLAB a1313ess Being
newbie in su13h a large s13ale experimental fa13ility has never been a problem
for whi13h I am thankful to all the team members for their help and guidan13e
from Paris PALS Prague and Imperial College London Spe13ial thanks to
Dr M Kozlova Dr M Krus Dr J Nejdl Dr J Dostal and all other
sta members at the PALS laser fa13ility and the target fabri13ation to Mr
P Barrasso and team at the Observatoire de Paris without whom none of
these experiments 13ould have been possible Spe13ial a13knowledgement to
Dr F Suzuki Vidal Mr T Clayson P Barroso and Dr U Chaulagain for
sharing their insight and explanations during the experiments
I espe13ially appre13iate the 13onstru13tive help and 13omments of Dr F
Suzuki Vidal Dr M Kozlova Dr M Cotelo Dr R Rodriacuteguez and Dr U
Chaulagain in the data analysis arti13le and thesis writing
I am grateful to all the jury members of my PhD 13ommittee Dr Sylvaine
Tur13k-Chieze Dr Katedegina Falk Dr Caterina Ri13onda Dr Matthias Gon-
zalez Dr Chantal Stehleacute and Dr Jean Larour for taking time to attend
my PhD defense Thanking very mu13h to Dr Tur13k-Chieze and Dr Falk for
reviewing my thesis and providing valuable 13omments and feedba13k whi13h
have surely improved my thesis
I would like to express sin13ere thanks to the dire13tors of LPP Dr P
Chabert and of LERMA Dr D Lis for allowing me to work on the thesis
in su13h a fabulous university ambien13e It would be unjust if I do not men-
tion the administrative help re13eived from Dr D Zahorski former Proje13t
13hief of PlasPar as well as Dr X Fresquet the present proje13t 13hief of
PlasPar I am grateful to PlasPar Labex for funding my PhD studies in
Fran13e
Next I would like to thank Dr U Chaulagain for helping me in Paris
during my stay from the very rst day I am also grateful to Dr L de Sagrave
Dr L Ibgui Dr M Drouin Dr A Stan Mr B Khiar Mr L Ni13olas
Dr J Freundli13h for their help and support during my stay in Fran13e
My thanks is due to my friends for 13reating a homely atmosphere during
my thesis in Fran13e In parti13ular I would like to thank Dr R Mourya
and Mr A Ranjan for their 13ontinuous support in the administrative for-
malities and others whi13h helped me a lot to 13on13entrate and speed up my
thesis writing I would also like to mention the names of my friends Miss T
Bhowmik Miss V Shaw and Dr A Gupta for their support
Good 13ompany plays a very important role in the life In this I am par-
ti13ularly thankful to Dr S C Tripathi Dr J Pal Dr R Kumar Dr M
Dave Dr A Bharadwaj Mr K Singh Miss S Mishra and Miss S Pandey
for helping me in keeping pa13e with resear13h and daily life developments I
am indebted to my brother Dr A Awasthi for his sin13ere guidan13e en13oura-
gement and 13ooperation during my entire do13toral study Also I would like
to take this opportunity to thank my 13hildhood friend Miss R Yadav for her
love and en13ouragement during my thesis
Finally last but not the least I express my whole hearted gratitude to the
support I re13eived in the form of love and ae13tion from my grandparents
parents sisters brothers and my other family members This is something
beyond a13knowledgements They were always with me and the en13ourage-
ment I re13eived from them will never fade away
Raj Laxmi Singh
Strong radiative sho13ks relevant for stellar
environments experimental study and numeri13al
approa13h
ABSTRACT
Strong sho13ks are present in various astrophysi13al phenomena Su13h
sho13ks are strongly inuen13ed by the radiation through its 13oupling with
hydrodynami13s Thus their topology and dynami13s are quite 13omplex Ge-
nerating su13h hypersoni13 sho13ks in the laboratory with 13ontrolled 13ondi-
tions is thus an adequate tool to study the inuen13e of radiation and to
13ompare them with numeri13al simulations Su13h sho13ks 13an be generated
by intense lasers and ele13tromagneti13 devi13es The rst part of this dis-
sertation 13on13erns the numeri13al and experimental study of the intera13tion
of two 13ounter propagating laser-driven sho13ks The experiments perfor-
med at the kJ PALS laser fa13ility allowed to generate sho13ks with dierent
speeds (sim 30-55 kms and 10-25 kms) in noble gases and low pressure
(less than 1 bar) Several diagnosti13s were implemented visible interfero-
metry time- and spa13e-resolved visible spe13tros13opy and time integrated
XUV spe13tros13opy Our experiment shows a strong intera13tion of one radi-
ative pre13ursor onto the se13ond one The physi13al parameters of the plasma
were dedu13ed from the diagnosti13s and 13ompared with 1-D simulation re-
sults The se13ond part is devoted to the design of an experiment where the
sho13k is generated ele13tromagneti13ally The optimization of this generator is
presented and also the full experimental set up whi13h allows studying sho13k
sim 30 kms in noble gas at sim 1 mbar
Keywords Radiative sho13k hydrodynami13s laser-plasmas visible
and XUV spe13tros13opy
numeri13al simulation laboratory astrophysi13s high energy density
physi13s
Cho13s forts et radiatifs dinteacuterecirct pour les
environnements stellaires eacutetude expeacuterimentale et
appro13he numeacuterique
RESUME
Les 13ho13s forts sont preacutesents dans des pheacutenomegravenes astrophysiques varieacutes
De tels 13ho13s sont fortement inuen13eacutes par le rayonnement par son 13ouplage
ave13 lhydrodynamique Par suite leur topologie et leur dynamique sont
assez 13omplexes Geacuteneacuterer de tels 13ho13s hypersoniques en laboratoire dans
des 13onditions 13ontrleacutees est ainsi un outil pertinent pour eacutetudier linuen13e
du rayonnement et pour 13omparer aux reacutesultats des simulations numeacuteriques
Ces 13ho13s sont geacuteneacutereacutes par des lasers intenses et par des moyens eacutele13tromag-
neacutetiques La premiegravere partie du texte est 13onsa13reacute agrave leacutetude numeacuterique et
expeacuterimentale de lintera13tion de deux 13ho13s induits par laser se propageant
en sens 13ontraires Les expeacuterien13es ont eacuteteacute meneacutees sur linstallation laser kJ
PALS qui permet de former deux 13ho13s ave13 des vitesses propres dieacuteren-
tes (sim 30-55 et 10-25 kms respe13tivement) dans des gaz rares agrave pression
faible (moins de 1 bar) Des diagnosti13s ont eacuteteacute installeacutes interfeacuteromeacutetrie
visible spe13tros13opie visible agrave reacutesolution spatiale et temporelle spe13tros13o-
pie XUV inteacutegreacutee en temps Nos expeacuterien13es montrent une forte intera13tion
entre les deux preacute13urseurs radiatifs Les paramegravetres physiques du plasma
ont eacuteteacute deacuteduits de 13es diagnosti13s et 13ompareacutes aux reacutesultats de simulations
monodimensionnelles La se13onde partie est 13onsa13reacutee agrave la 13on13eption dune
expeacuterien13e ougrave le 13ho13 est geacuteneacutereacute de faccedilon eacutele13tromagneacutetique Loptimisation
de 13e geacuteneacuterateur est preacutesenteacutee ainsi que lenvironnement expeacuterimental per-
mettant deacutetudier des 13ho13s jusquagrave 30 kms dans des gaz rares peu denses
(1 mbar)
Mots 13leacutes 13ho13s radiatifs plasmas laser hydrodynamique spe13tros13opie
visible et XUV simulation numeacuterique astrophysique de laboratoire
physique agrave haute densiteacute deacutenergie
Reacutesumeacute long
En introdu13tion (13hapitre 1) les 13ho13s radiatifs (CR) sont preacutesenteacutes
dans de nombreux environnements astrophysiques en parti13ulier dans les
eacutetoiles Ce sont des 13ho13s forts ave13 un nombre de Ma13h eacuteleveacute (M1) et une
tregraves haute tempeacuterature 13e qui induit un rayonnement intense Alors que
lobservation de 13ette signature ave13 une reacutesolution spatiale est tregraves di13ile
en astrophysique une appro13he innovante 13elle dite des plasmas astrophy-
siques de laboratoire fournit un puissant moyen deacutetude des CR sur Terre
Depuis plus dune deacute13ennie 13es 13ho13s sont eacutetudieacutes en laboratoire prin13ipale-
ment sur de grandes installations laser dans les gaz rares et sous dieacuterentes
geacuteomeacutetries Lintera13tion entre un laser et une feuille min13e produit une
forte ablation et par eet fuseacutee a1313eacutelegravere la feuille qui se propage dans le
gaz agrave plusieurs dizaines de kms Pour des irradiations laser entre 10
14and
10
15W13m
2 on enregistre des vitesses de 13ho13s entre 40 et 150 kms En
parallegravele une des13ription ne est permise par des simulations numeacuteriques
Au deacutebut de 13e travail les expeacuterien13es eacutetaient 13on13entreacutees sur des CR
simples alors que la situation astrophysique est 13elle dun CR interagissant
ave13 un milieu plus dense donnant des 13ho13s reacuteeacute13his et transmis Des
exemples repreacutesentatifs de 13es pheacutenomegravenes sont la propagation de restes de
supernovae dans des nuages moleacute13ulaires denses les 13ho13s da1313reacutetion sur
la photosphegravere des eacutetoiles T-Tauri ou en13ore les 13ho13s deacutetrave agrave lavant
des jets stellaires La 13ollision (ou lintera13tion) de deux CR est bien sucircr un
eacutevegravenement rare en astrophysique mais un exemple est fourni par lintera13tion
de deacutebris de la supernova DEM L316 (13f Fig 1 de Williams et al (1997))
mecircme si lhypothegravese est 13ontesteacutee (Velarde et al (2006))
Il y a don13 une forte motivation pour reacutealiser une expeacuterien13e de labora-
toire pour eacutetudier la propagation et lintera13tion de deux 13ho13s fa13e agrave fa13e
et interpreacuteter les signatures observeacutees Cette appro13he 13onstitue une grande
part de mon travail En 13ompleacutement aux expeacuterien13es laser jai eacutetudieacute un
geacuteneacuterateur 13ompa13t de puissan13e pulseacutee alimentant un 13anon agrave plasma 13oax-
ial pour 13reacuteer des 13ho13s dinteacuterecirct astrophysique dans les gaz agrave basse pression
Les 13ho13s ainsi 13reacuteeacutes sont de plus grande taille et peuvent ecirctre eacutetudieacutes plus
fa13ilement (Kondo et al 2008) Pour 13ette thegravese je me suis atta13heacutee agrave preacute-
senter les 13ho13s radiatifs dans le 13ontexte de lastrophysique de laboratoire
en suivant 13es deux appro13hes
Le deuxiegraveme 13hapitre traite de la physique des CR Ils 13omposent
une 13lasse de 13ho13s supersoniques qui sont 13haueacutes agrave haute tempeacuterature
et sont par 13onseacutequent sour13es dun rayonnement intense En retour 13e
rayonnement modie la dynamique et la stru13ture du 13ho13 13e qui 13omplexie
la des13ription Les CR se ren13ontrent dans des situations astrophysiques
13omme les 13ho13s da1313reacutetion lors de la formation des protoeacutetoiles (Stahler
Palla et Salpeter 1986) lexplosion de supernovae et lintera13tion de leur
restes ave13 le milieu interstellaire dense (Chevalier 1977) ou en13ore les 13ho13s
deacutetrave en tecircte des jets stellaires (Hartigan et al 2001) Les CR sont
maintenant eacutetudieacutes en laboratoire 13e qui permet de 13omparer les reacutesultats
aux modegraveles de la litteacuterature et de veacuterier la possibiliteacute de les retrouver ave13
les simulations numeacuteriques disponibles (Bouquet et al (2004) Leygna13 et
al (2006)) Leacutetude expeacuterimentale des CR est 13ru13iale notamment pour
13omprendre les pro13essus eacutenergeacutetiques au sein des plasmas astrophysiques
Dans le 13as des gaz parfaits le saut au niveau dun 13ho13 est deacute13rit par les
eacutequations de Rankine-Hugoniot qui relient les quantiteacutes thermodynamiques
de part et dautre de la dis13ontinuiteacute La solution monodimensionnelle dun
13ho13 se propageant agrave la vitesse us dans un gaz au repos est usuellement
deacute13rite dans le repegravere lieacute au 13ho13 et on distingue la reacutegion amont (ou preacute-
13ho13) agrave la vitesse u1= - us et en arriegravere du 13ho13 (reacutegion post-13ho13 ou aval)
la vitesse est u2 selon la Fig 21 Pour les 13ho13s forts ougrave le nombre de Ma13h
M est tregraves grand devant 1 les sauts des quantiteacutes thermodynamiques dun
gaz parfait sont donneacutes par les eacuteq 21 agrave 24 Pour un gaz monoatomique
le rapport de 13ompression est de 4 et la tempeacuterature de la zone post-13ho13
est proportionnelle au 13arreacute de la vitesse du 13ho13 et agrave la masse atomique
La situation est plus 13omplexe pour un gaz reacuteel 13ar une part de leacutenergie
13ineacutetique sert agrave ex13iter et agrave ioniser le gaz du post 13ho13 La tempeacuterature
attendue est don13 infeacuterieure agrave 13elle du 13as ideacuteal en outre la pression est
modieacutee par lionisation (Mi13haut et al 2004) Un nouveau jeu deacutequations
in13orpore la 13harge ee13tive Z des ions (eacuteq 25 agrave 28) Ces modegraveles ignorent
le 13hauage et le refroidissement radiatifs alors que le rayonnement eacutemis
ae13te la stru13ture dun 13ho13 fort Cette des13ription demande dajouter aux
eacutequations de 13onservation de la masse quantiteacute de mouvement et eacutenergie les
13ontributions radiatives (ux eacutenergie et pression Le 13as est plus 13omplexe
et les eacutequations dhydrodynamique radiative ont eacuteteacute exprimeacutees dans le repegravere
du 13ho13 (Mihalas and Mihalas 1999 Drake 2006) selon les eacuteq 29 agrave 211
Les 13ontributions de la pression radiative et du 13hamp de rayonnement
sont importantes agrave tregraves haute tempeacuterature (vitesse) mais deacutejagrave pour nos 13ho13s
le ux radiatif est dominant Selon lopa13iteacute le rayonnement du 13ho13 peut
ecirctre absorbeacute par la region preacute-13ho13 induisant un 13hauage loin en avant de
la dis13ontinuiteacute Egalement la zone 13hoqueacutee pro13he de la dis13ontinuiteacute est
ae13teacutee par la 13ompeacutetition entre leacutemission (refroidissement) et labsorption
(13hauage) Ainsi leacutepaisseur optique lo13ale (eacuteq 211) devient le paramegravetre
pertinent Comme on raisonne sur le ux radiatif moyenneacute en freacutequen13e
13est lopa13iteacute de Rosseland qui est utiliseacutee et on dieacuteren13ie les reacutegimes op-
tiquement eacutepais et optiquement min13e 13e qui permet de 13lasser les 13ho13s
radiatifs
Notre inteacuterecirct sest 13on13entreacute sur les 13ho13s preacutesentant un preacute13urseur radi-
atif En laboratoire une intensiteacute laser au dessus de 10
14W13m
2le permet
et les installations sont rappeleacutees dans le tableau 21 Il est aussi possible de
lan13er des 13ho13s moins rapides don13 peu radiatifs (sim 10-30 kms) ave13 un
piston eacutele13tromagneacutetique
Le 13hapitre 3 est 13onsa13reacute agrave lanalyse numeacuterique des CR en deacute13rivant
les pro13essus physiques jouant un rle dans 13es systegravemes physiques De mecircme
que les observations les expeacuterien13es en laboratoire ne sont pas dire13tement
interpreacutetables et la simulation numeacuterique devient un outil preacute13ieux Diverses
appro13hes numeacuteriques sont utiliseacutees pour simuler les plasmas de laboratoire
On 13onsidegravere soit des grilles xes soit des grilles adaptables (Adaptative
Mesh Renement) pour maintenir une reacutesolution susante sur 13haque zone
du plasma Une autre appro13he utilise des grilles qui suivent le uide dans
son mouvement (modegravele Lagrangien) don13 sans passage de matiegravere dune
13ellule agrave la voisine Les eacutequations sont alors dieacuterentes (Orban et al 2013)
Pour deacute13rire les 13ollisions de 13ho13s jai utiliseacute le 13ode 13ommer13ial HE-
LIOS un 13ode Lagrangien mono-dimensionnel pouvant geacuterer le rayonnement
et lhydrodynamique (Ma13Farlane Golovkin et Woodru 2006) Assez fa13ile
agrave prendre en main il a le gros avantage de pouvoir simuler notre 13ongura-
tion deux 13ho13s de vitesses opposeacutees lan13eacutes par deux lasers HELIOS deacute13rit
les plasmas hors deacutequilibre thermodynamique (hors ETL) gracirc13e agrave un mo-
degravele 13ollisionnel-radiatif (HELIOS-CR) de 13al13ul des populations atomiques
agrave 13haque pas de simulation hydrodynamique De fait jai utiliseacute HELIOS
dans une approximation ETL qui 13orrespond aux 13onditions expeacuterimentales
(Rodriguez et al 2011) et 13ela est dis13uteacute en deacutetails
Le 13ode reacutesout les eacutequations uides en introduisant les pressions des eacutele13-
trons des ions et du rayonnement Ele13trons et ions sont deacute13rits par deux
uides en intera13tion de tempeacuteratures respe13tives Te et Ti La 13ondu13tion
thermique est geacutereacutee par 13elle des eacutele13trons agrave partir de la 13ondu13tiviteacute de
Spitzer et le deacutept deacutenergie laser par un modegravele de Bremsstrahlung inverse
Leacutemission et labsorption du rayonnement sont introduites dans leacutequation
deacutenergie des eacutele13trons et dans 13elles du transport radiatif Une meacutethode
utilise un modegravele de diusion du rayonnement multi-groupes et agrave ux li-
miteacute ougrave le ux radiatif est proportionnel au gradient de leacutenergie radiative
et inversement proportionnel agrave lopa13iteacute de Rosseland ave13 une pondeacuteration
pour obtenir une bonne des13ription du 13as optiquement min13e selon Olson
Auer et Hall (2000) Une se13onde meacutethode utilise un s13heacutema deacutependant du
temps agrave 13ourte distan13e et multi-angles Dans notre 13as ougrave rayonnement
et hydrodynamique sont fortement 13oupleacutes et aussi en variation rapide on
utilise le modegravele de diusion et les opa13iteacutes ETL multi-groupes de Plan13k et
Rosseland
En outre notre version dHELIOS utilise PROPACEOS (Ma13Farlane
Golovkin et Woodru 2006) une table deacutequation deacutetat et dopa13iteacute multi-
groupes geacuteneacutereacutee par le 13ode sans possibiliteacute dintroduire une autre table La
13onnaissan13e des 13onditions thermodynamiques est neacute13essaire pour la simu-
lation du plasma dans nos 13as masse volumique entre 10
minus4et 10
minus1g13m
3
tempeacuterature entre 0 et 50 eV Une eacutevaluation preacute13ise par Rodriguez et al
(13f Fig 2 de Rodriguez et al 2011) pour le Xeacutenon indique que les 13onditi-
ons thermodynamiques de nos CR 13orrespondent au reacutegime ETL Leacutequation
deacutetat relie pression ionisation et eacutenergie interne agrave la masse volumique et agrave
la tempeacuterature Notre version dHELIOS pour plasma ETL utilise PROPA-
CEOS qui est peu do13umenteacutee elle utilise le modegravele de More et al (1988)
en reacutegime de 13ouplage fort (haute densiteacute et basse tempeacuterature) et un modegravele
datome isoleacute pour un 13ouplage faible Des interpolations sont in13luses pour
passer au 13ouplage fort (httpwwwprism-13s13omSoftwarePROPACEOS)
Jai reacutealiseacute des simulations HELIOS 1D pour divers 13as de CR 13ho13 seul
ou 13ho13s en 13ollision qui sont deacute13rites dans 13e 13hapitre 3 Les CR se propa-
geant dans le Xe agrave 01 bar et agrave 50 kms sont 13ara13teacuteriseacutes par un preacute13urseur
radiatif eacutetendu Une 13ompression eacuteleveacutee (38) dans le post-13ho13 est attribueacutee
agrave lionisation du gaz et au refroidissement radiatif Les tempeacuteratures du
post-13ho13 et du preacute-13ho13 de part et dautre du pi13 sont identiques signe
dun 13ho13 super13ritique Le rle spe13ta13ulaire de lopa13iteacute a eacuteteacute remarqueacute
mais agrave 13onsideacuterer lin13ertitude sur lopa13iteacute du Xeacutenon et le traitement 1D
jai deacute13ideacute de ne pas 13her13her agrave raner la simulation en termes de nombre
de groupes
Leacutetude numeacuterique permet de retrouver les 13ara13teacuteristiques de lintera13tion
de deux 13ho13s opposeacutes ave13 des vitesses respe13tives 50-50 kms et 50-20 kms
Dans tous les 13as la prin13ipale signature de lintera13tion (don13 avant la 13ol-
lision) est le re13ouvrement des preacute13urseurs agrave t0+8 ns pour 50-50 kms et
agrave t0+15 ns pour 50-20 kms Ensuite on assiste agrave une remonteacutee reacuteguliegravere
de la densiteacute et de la tempeacuterature eacutele13troniques Linstant de la 13ollision
voit une augmentation soudaine de la densiteacute eacutele13tronique (par un ordre de
grandeur) qui atteint 66 times 10
21et 3 times 10
2113m
minus3respe13tivement alors que
la tempeacuterature monte agrave 39 et 28 eV
Dans le quatriegraveme 13hapitre je preacutesente linstallation laser PALS puis
la 13ellule dintera13tion son implantation et les diagnosti13s asso13ieacutes Rappe-
lons le prin13ipe expeacuterimental Quand une impulsion bregraveve de lumiegravere laser
est fo13aliseacutee sur une feuille min13e une part importante deacutenergie est trans-
feacutereacutee et le 13hauage du solide 13onduit agrave son ablation Un plasma 13oronal
13haud et peu dense est eacuteje13teacute vers larriegravere et un 13ho13 vers lavant se forme
dans la feuille par eet fuseacutee Comme 13ette feuille limite le tube le 13ho13
se transmet au gaz la feuille agissant 13omme un piston Des expeacuterien13es
ont eacuteteacute meneacutees sur le Prague Asterix Laser System (PALS) au printemps
2015 (5 semaines 20 avril - 22 mai) ave13 pour obje13tif la premiegravere eacutetude
de lintera13tion puis de la 13ollision de deux CR sous la13tion de deux lasers
au niveau de 10
14W13m
2 Le Prague Asterix Laser System (PALS) repose
sur un laser agrave iode infrarouge (Asterix IV Jungwirth et al 2001) Apregraves
ampli13ation il deacutelivre jusquagrave 1 kJ en 03 ns agrave la longueur donde fonda-
mentale de 1315 nm Des fais13eaux auxiliaires sont disponibles agrave freacutequen13e
doubleacutee (λ = 657 nm) ou tripleacutee (438nm) PALS deacutelivre deux tirs par heure
agrave haute eacutenergie et dans de bonnes 13onditions dhomogeacuteneacuteiteacute du fais13eau
Pour nos expeacuterien13es nous utilisons deux fais13eaux Le fais13eau fondamental
est diviseacute apregraves le 4egraveme ampli13ateur en deux fais13eaux deacutenergie 60 et
40 (voir Fig 42) Le plus puissant est inje13teacute dans le 5
meampli13ateur et
sa freacutequen13e est tripleacutee Ce fais13eau agrave 438 nm est nommeacute MAIN Le se13ond
fais13eau est utiliseacute sans modi13ation don13 agrave 1315 nm et est nommeacute AUX
Le s13heacutema de distribution est en Fig 42 et les proprieacuteteacutes de MAIN and
AUX laser sont rappeleacutees Tdans le ableau 41
PALS propose deux 13hambres agrave vide spheacuterique et 13ylindrique respe13ti-
vement nous avons utiliseacute la 13hambre spheacuterique (Fig 43) Dun diamegravetre
de 100 13m elle est a1313essible aux fais13eaux MAIN et AUX par deux hublots
de 80 13m et 50 13m Des portes sont ameacutenageacutees ainsi que de nombreux ports
A linteacuterieur la 13ellule et des diagnosti13s sont monteacutes sur une table optique
deacute13oupleacutee meacute13aniquement de la 13hambre Les 13ibles (Fig 45) 13omprennent
un petit tube long de 4 mm fermeacute agrave ses deux extreacutemiteacutes par des feuilles min-
13es de parylegravene-N doreacute de 11 microm sur lesquelles les deux lasers sont fo13aliseacutes
au niveau de 10
14W13m
minus2 Le tube est rempli du gaz dans lequel le 13ho13
se propagera agrave une vitesse attendue entre 30 et 60 kms Les feuilles min13es
assurent la 13onversion de leacutenergie laser en eacutenergie 13ineacutetique via lablation et
la geacuteneacuteration de 13ho13 Des 13ellules speacute13iques en dural massif permettent
lalignement et la mesure du diamegravetre du fais13eau Toutes les 13ellules sont
reacutealiseacutees par le Ple instrumental de lObservatoire de Paris
Les 13ellules agrave gaz sont remplies in situ agrave une fra13tion de bar ave13 Xe
Ar Xe90He10 et He permettant ainsi de bien 13onnaicirctre le milieu ougrave se
propagent les 13ho13s Le systegraveme de remplissage (Fig 410) permet de limiter
leacute13art de pression sur les fenecirctres de la 13ellule et permet de suivre la pression
jusquau moment du tir Cest 13ritique pour ne pas 13asser la fenecirctre ultra
ne de Si3N4 pour la spe13tros13opie XUV et pour geacuterer une fuite eacuteventuelle
Le reacuteglage de la position et de la fo13alisation permet dobtenir des ta13hes
laser 13entreacutees de diamegravetres 450 - 500 microm et 250 - 300 microm pour MAIN et
AUX respe13tivement Ce 13ontrle est reacutepeacuteteacute 13haque jour Les impa13ts sont
toujours suivis par des 13ameacuteras X au keV mecircme si limpa13t sur la feuille de
parylegravene-N donne un signal X faible 13ompareacute agrave 13elui du tir sur Al
Les diagnosti13s du plasma sont prin13ipalement linterfeacuteromeacutetrie visible et
la spe13tros13opie XUV et visible Lalignement est suivi par des 13ameacuteras dans
le visible Pour linterfeacuteromeacutetrie visible preacute13iseacutement on doit prendre une
image de reacutefeacuteren13e (sans franges) de 13haque 13ible (Fig 417a) Le fais13eau de
reacutefeacuteren13e de linterfeacuteromegravetre Ma13h-Zehnder est bloqueacute et le fais13eau sonde est
utiliseacute seul La fente est ensuite reacuteduite agrave 200 microm pour ne sonder que la zone
13entrale du tube mais seacutetendant le long de laxe jusquaux pistons pour avoir
une reacutefeacuteren13e des positions initiales Ensuite en mode interfeacuterogramme on
13reacutee un systegraveme de franges perpendi13ulaires agrave la fente don13 perpendi13ulaires
agrave la dire13tion de propagation du 13ho13 (13f Fig 417b et Fig 413) Le tube
de 13ho13 est imageacute sur la 13ameacutera ave13 un grandissement susant (54 mm) et
il est possible dassurer un balayage eacutele13tronique de la fente sur le deacutete13teur
jusquagrave 200 ns Un interfeacuterogramme dit streak permet ainsi de suivre limpa13t
des 13ho13s sur les franges en fon13tion du temps
Leacutemission XUV du plasma est enregistreacutee ave13 une inteacutegration spatiale
et temporelle sur un spe13trographe agrave 13hamp plan muni dun reacuteseau 13on13ave
Il est installeacute dans la partie haute de la 13hambre (13f Fig 418) et enregistre
leacutemission XUV traversant la fenecirctre de Si3N4 Une spe13tros13opie visible
reacutesolue en temps et en position longitudinale a eacuteteacute installeacutee (Fig 419a)
mais na pas en13ore pu ecirctre exploiteacutee 13omplegravetement
Le 13hapitre 5 preacutesente les reacutesultats des deux diagnosti13s surtout linterfeacute-
romeacutetrie qui a eacuteteacute 13omplegravetement analyseacutee Jai deacuteveloppeacute une pro13eacutedure
de deacutepouillement ave13 des outils numeacuteriques ad ho13 de 13al13ul et danalyse
dimage Certes linterfeacuteromeacutetrie ne permet pas de sonder des plasmas plus
denses que la densiteacute 13ritique imposeacutee par le laser (4 times 10
2113m
minus3agrave 627 nm)
et on 13onstate que le post-13ho13 est opaque Au 13ontraire le rayonnement
XUV est preacutesent dans toute la stru13ture et la spe13tros13opie XUV inteacutegreacutee
en temps et en espa13e permet dexplorer agrave la fois le post-13ho13 et le preacute13ur-
seur Lanalyse des interfeacuterogrammes streak en visible part dune eacutevaluation
de leet de lindi13e de reacutefra13tion des eacutele13trons deacutetailleacutee dans lappendi13e A
mais qui se heurte assez vite agrave une perte de 13ontraste et agrave une disparition
des franges bien en dessous de la valeur de la densiteacute 13ritique deacutejagrave menti-
onneacutee Les deacutephasages a1313essibles ave13 preacute13ision ne deacutepassent guegravere 2 ou 3
fois 2π La re13her13he de maxima des franges est semi automatiseacutee et permet
darriver agrave une densiteacute eacutele13tronique moyenneacutee sur les 600 microm de leacutepaisseur
du tube de 13ho13 Le prol non plan du 13ho13 13onduit agrave penser que la moyenne
sous-estime dun fa13teur 2 voire plus la densiteacute dans le preacute13urseur La dis-
parition des franges est attribueacutee agrave la preacutesen13e du front de 13ho13 Par suite
les interfeacuterogrammes permettent de mesurer des vitesses de 13ho13s de 30-55
et 10-30 kms pour MAIN et AUX respe13tivement ainsi que des densiteacutes
eacutele13tronique dans les preacute13urseurs entre 10
17et 10
1913m
minus3
Nous avons ainsi deacutemontreacute lintera13tion entre deux preacute13urseurs radiatifs
dans Xe at 01 bar pour deux 13ho13s agrave 54 et 23 kms Lintera13tion est
13lairement 13ara13teacuteriseacutee par laugmentation de londe dionisation puis par la
superposition des deux preacute13urseurs agrave t0+20ns La 13ollision est enregistreacutee agrave
t0+47 ns un reacutesultat retrouveacute par la simulation
Nous avons reacutepeacuteteacute mais agrave plus haute pression 02 bar et trouveacute des
vitesses de sim 41 kms pour MAIN et sim 18 kms pour AUX Cependant
nous navons pas de signature du preacute13urseur radiatif pour AUX et le temps de
13ollision na pas eacuteteacute a1313essible agrave lenregistrement Le preacute13urseur 13teacute MAIN
nest pas inuen13eacute par AUX jusquagrave la limite de t0+48 ns (Fig 59(b) et (13))
La simulation preacutedit un tregraves faible preacute13urseur 13teacute AUX et une intera13tion
des preacute13urseurs agrave t0+49 ns
Linterfeacuteromeacutetrie transverse agrave 02 bar ave13 des vitesses sim 40 et 20 kms
indique que le preacute13urseur 13teacute MAIN a une extension lateacuterale de sim 600 microm13ontre 300 microm pour AUX Le preacute13urseur de MAIN est leacutegegraverement ae13teacute
pregraves des parois alors que 13elui de AUX est fortement 13ourbeacute Les eets 2D
sont don13 pronon13eacutes pour AUX et faibles pour MAIN En outre pour le
Xeacutenon nous disposons de la tempeacuterature et de la 13harge moyenne par le
spe13tre XUV inteacutegreacute (tir48143 Xeacutenon 06 bar) On peut en 13on13lure que
la 13harge moyenne est au moins eacutegale agrave 6 et quon a atteint une tempeacuterature
de 15 eV
Nos simulations donnent une des13ription qualitative des CR en inte-
ra13tion agrave la reacuteserve pregraves que nous introduisions dans HELIOS une uen13e
laser ajusteacutee pour donner la bonne vitesse Cependant il est maintenant
admis que les simulations 2D (ave13 les opa13iteacutes 13orre13tes) deacute13rivent bien les
expeacuterien13es (Gonzaacutelez Audit et Stehleacute 2009 Leygna13 et al 2006 Stehleacute
et al 2010) A la mecircme eacutenergie laser le 13al13ul 2D 13onduit agrave diminuer la
vitesse du 13ho13 13ompare au 13al13ul 1D et aussi agrave une baisse de la densiteacute
eacutele13tronique Par exemple pour un 13ho13 lan13eacute agrave PALS par un laser agrave 1315
nm dans le Xeacutenon agrave 03 bar ave13 une uen13e de 85000 J13m
2 ARWEN 2D
donne une vitesse de 44 kms en a1313ord ave13 lexpeacuterien13e (Cotelo et al
2015) La simulation 1D demanderait 30000 J13m
2pour obtenir la mecircme
vitesse
De mecircme la spe13tros13opie XUV inteacutegreacutee agrave 06 bar pour des vitesses reacuteel-
les de sim 39 et 18 kms indique que la tempeacuterature a atteint 15 eV et que
la 13harge moyenne a atteint 6 ou 7 alors que la simulation 1D preacutedit 10-30
eV et 5-10 respe13tivement (Fig 518) On peut en 13on13lure quune eacutetude
deacutetailleacutee baseacutee sur des simulations 2D et un post-traitement du transfert de
rayonnement sont neacute13essaires pour raner lanalyse Pour les autres gaz
rares (Ar Kr) on a observeacute que agrave eacutenergie laser donneacutee la vitesse de CR est
une fon13tion deacute13roissante de la masse volumique Ce13i 13onrme que pour
une densiteacute et une vitesse les eets radiatifs augmentent ave13 le numeacutero
atomique Pour le Krypton on a observeacute un petit preacute13urseur mais au13une
intera13tion Celle-13i ne serait possible quen augmentant nettement la vi-
tesse don13 leacutenergie du laser Ce13i a eacuteteacute rendu possible sur linstallation
laser Orion (AWE Aldermaston G-B) ougrave la 13ollision de deux CR deacutegale
vitesse a eacuteteacute obtenue agrave sim 80 kms pour une uen13e laser sim 6 times 10
14W13m
2
(Clayson et al 2016 Suzuki-Vidal et al 2016) pour une large gamme de
gaz rares et des pressions entre 01 et 1 bar Pour eacutetudier agrave la fois les 13ho13s
et les preacute13urseurs radiatifs de nombreux diagnosti13s eacutetaient installeacutes radi-
ographie X imagerie optique en 13ameacutera agrave balayage de fente interfeacuteromeacutetrie
multi-vues et agrave balayage de fente Bien que je naie pas pu parti13iper aux
13ampagnes jai fait des simulations 1D pour interpreacuteter les reacutesultats Cet
aspe13t nest pas deacute13rit dans 13ette thegravese mais a eacuteteacute publieacute tregraves reacute13emment
(Clayson et al 2017)
Apregraves des 13hapitres 13onsa13reacutes aux 13ho13s radiatifs geacuteneacutereacutes par laser le
13hapitre 6 deacute13rit un moyen alternatif passant par la voie eacutele13trique agrave haute
puissan13e Les 13hires de la voie laser sont tregraves eacuteleveacutes en uen13e plus de
10
14W13m
2 mais repreacutesentent des eacutenergies modeacutereacutees dans limpulsion (120
et 60 J pour MAIN et AUX au PALS) Or un systegraveme eacutele13trique devant
deacutelivrer 100 J reste modeste pour sa partie sto13kage Les vitesses de 50
kms seront probablement di13iles agrave atteindre mais on peut tabler sur un
reacutegime 13ompleacutementaire des 13ho13s radiatifs ave13 plus de 10 kms si le ren-
dement de 13onversion en eacutenergie 13ineacutetique reste 13orre13t Dougrave un neacute13essaire
travail doptimisation Des vitesses jusquagrave 100 kms avaient eacuteteacute mesureacutees
tregraves tt dans des 13anons agrave plasma dhydrogegravene agrave basse pression (Lee 1969)
mais 13e13i a eacuteteacute exploiteacute ensuite pour dautres naliteacutes La litteacuterature ne
mentionne quun 13as de 13anon eacutele13tromagneacutetique 13oaxial deacutedieacute aux 13ho13s
dinteacuterecirct astrophysique (Kondo et al 2006) Il est signaleacute que le lan13eur
eacutele13tromagneacutetique produit des 13ho13s plans de grande dimension transverse
(1 ordre de grandeur par rapport agrave la voie laser) et don13 plus fa13iles agrave in-
vestiguer (Kondo et al 2008) Le prin13ipe de fon13tionnement est le suivant
i) 13reacuteation dun 13laquage haute tension initial sur une surfa13e dieacutele13trique
fa13e agrave du gaz ii) 13hauage ohmique du plasma par le 13ourant en 13roissan13e
rapide iii) a1313eacuteleacuteration de la 13ou13he de plasma sous la pression magneacuteti-
que auto-geacuteneacutereacutee par le 13ourant (dizaines de kA) iv) maintien (1 micros) de la
pression magneacutetique pour 13ommuniquer une impulsion importante Des tra-
vaux sur le Z-pin13h ou le plasma fo13us permettent de postuler des geacuteomeacutetries
13onvenables pour a1313eacuteleacuterer une masse de lordre de 10
minus510
minus4g
Apregraves avoir eacutetudieacute le fon13tionnement dun 13ir13uit eacutele13trique rapide jai
listeacute les points forts et les points faibles en fon13tion des obje13tifs et jai pro-
poseacute une geacuteomeacutetrie 13oaxiale 13ompa13te Loptimisation du 13anon agrave plasma a
eacuteteacute faite en terme de vitesse maximale agrave la sortie ave13 13omme seules 13ontrain-
tes une eacutenergie sto13keacutee de 1 kJ une dimension transverse de 4 mm et une
dimension axiale dun ordre plus grande Lobje13tif eacutetait le 13ho13 fortement
supersonique 10-30 kms agrave quelques mbar dAr ou de Xe Un modegravele simple
(0D) a eacuteteacute 13ompareacute agrave un modegravele 3D MHD par le 13ode GORGON qui deacute13rit
bien les expeacuterien13es de plasmas astrophysique par puissan13e pulseacutee (Ciardi
et al 2007)
La pression magneacutetique sexprimant par B
22micro0 un 13ourant supeacuterieur agrave
100 kA apparaicirct neacute13essaire et atteignable sur un geacuteneacuterateur 1 kJ - 1 micros 13om-
pa13t de type R-L-C Ensuite il a fallu monter un modegravele de 13ir13uit deacuteformable
13omportant une se13tion formeacutee de la lame de plasma en mouvement et qui
prenne en 13ompte la1313reacutetion du gaz dans un modegravele dit snowplow (Potter
1971) Des paramegravetres geacuteomeacutetriques 13omme la forme et les dimensions du
13anon ont pu ecirctre optimiseacutes pour donner une grande vitesse de sortie du
plasma pour Ar et Xe et dans la gamme 01 - 10 mbar Un 13anon a eacuteteacute 13on-
struit et 13onne13teacute agrave un geacuteneacuterateur eacutele13trique pulseacute existant Des observations
du mouvement du plasma par la dynamique du rayonnement radial donnent
des vitesses 13oheacuterentes ave13 le modegravele ave13 la simulation 3D MHD et des
nombres de Ma13h eacuteleveacutes de 20 agrave 60 Ces travaux ont soutenu la 13on13eption
dun autre geacuteneacuterateur pour obtenir des performan13es plus eacuteleveacutees en vitesse
de 13ho13 et en taux de reacutepeacutetition
Contents
List of Figures
List of Tables
1 Introdu13tion 1
11 General Context 1
12 Outline of Chapters 2
13 My 13ontribution 3
2 Radiative sho13k waves 5
21 Rankine-Hugoniot(R-H) relations 6
211 Jump relations for a real gas 9
212 Ee13t of radiation on sho13k waves 11
213 Radiative hydrodynami13s equations 13
22 Radiative sho13k regimes 15
221 Typi13al radiative sho13k waves with a pre13ursor 16
23 Radiative sho13k waves experiments 16
231 Laser-driven sho13k experiments 18
232 Ele13tromagneti13ally laun13hed sho13k 21
3 1D Simulations 23
31 HELIOS 23
32 LTE approximation 24
321 Mean opa13ity 25
322 Equation of State 27
33 Single radiative sho13k waves 27
34 Intera13ting radiative sho13k waves 40
35 Summary 44
4 Experimental Setup 47
41 Prague Asterix Laser System (PALS) fa13ility 48
42 Targets 49
421 Massive Targets 51
422 Gaseous Targets 51
423 Target holder 55
CONTENTS
424 Target lling 56
43 Laser Fo13using 57
431 Fo13al Lenses and Phase Zone Plates 57
432 Fo13using of the MAIN and AUX laser beams 58
44 Diagnosti13s 59
441 Visible interferometry 61
442 XUV spe13tros13opy 65
443 Visible spe13tros13opy 67
45 Summary 68
5 Results and Interpretation 71
51 Visible interferometry 71
511 Longitudinal interferometry sho13k speed and ele13tron
density 74
512 Transverse interferometry lateral extension of the sho13k 91
52 XUV Spe13tros13opy 92
53 Simulations based on experimental results 93
54 Summary 97
6 Ele13tromagneti13ally laun13hed sho13k 101
61 Prin13iples of operation of a high 13urrent generator 102
62 Prin13iples of the run-down phase in a PFD 103
63 Proposed design for the plasma gun 104
64 Dynami13 13ir13uit modelling 106
65 3-D MHD simulations using GORGON 13ode 111
66 Measurements 116
67 Summary 118
7 Con13lusion 121
71 Con13lusions 121
72 Perspe13tives 123
8 Thesis summary 127
Appendi13es 141
A Visible Interferometry 143
A01 Refra13tive index of a plasma 143
A02 Absorption of the laser beam 144
A03 Prin13iple of interferometry 145
A04 Ma13h Zehnder Interferometer 146
B Opa13ities and mean 13harge 149
Referen13es 152
List of Figures
21 S13hemati13 diagram for a 1-D sho13k moving in the x dire13tion
The speed of the sho13k front is us In the frame of referen13e of
the sho13k front the pre-sho13k region has a relative speed u1=
-us density ρ1 total pressure P1 temperature T1 whereas these
quantities are respe13tively u2 ρ2 P2 and T2 for the post-sho13k
region 6
22 Proles of temperature (Te = Ti = T ) and mass density (ρ) at 10ns for a sho13k propagating in Xenon at 01 bar (ρ1= 57 times 10
minus4
g13m
minus3) with a speed of sim 45 kms It is to note that the given
pressure value are at 298 K The simulation is performed with the
HELIOS 13ode ex13luding the radiation ee13t and for a polytropi13
gas (γ=53) The initial temperature of Xenon is set to 1 eV
and with a negligible (10
minus5WmK) thermal 13ondu13tivity The
Ma13h number is then equal to 40 For this simulation the sho13k
is laun13hed by a 100 microm thi13k Aluminium piston moving at the
velo13ity of 33 kms The position x=0 13orresponds to the position
of the XeAl interfa13e at time t=0 8
23 Ele13tron temperature (Te) ion temperature (Ti) and mass density
(ρ) proles at 10 ns derived from 1-D simulation for the 13ase of
ionisation (no radiation) The sho13k propagates in Xe gas at 01
bar (ρ1= 57 times 10
minus4g13m
minus3by sim 45 kms) It is to note that the
given pressure value is at 298 K The simulation is performed with
the HELIOS (Te 6= Ti 13ase) using the PROPACEOS equation
of state without any radiation and with a negligible thermal
13ondu13tivity (10
minus05WmK) The initial temperature in Xenon
is set to 1 eV and thus the Ma13h number is equal to 40 For this
simulation the sho13k is laun13hed by a 100 microm thi13k Aluminium
piston moving at the velo13ity of 38 kms The position x=0
13orresponds to the position of the XeAl interfa13e at time t=0 10
LIST OF FIGURES
24 Figure 14 amp 17 of Mi13haut et al (2004) (a) Compression ρ2ρ1(full 13ir13les thin line) kT2 in eV (inverted full triangles thin line)
in Argon (a) Xenon (b) for initial 13onditions kT1( = 10 eV for
Ar and = 01 eV for Xe) ρ1 = 5 times 10minus4 gcm3 versus sho13k
speed in kms with (full markers full lines) and without (empty
markers dashed lines) radiation The ionization stage lt z gt2 is
plotted with the diamond symbol 12
25 Simulated ele13tron temperature (Te) ion temperature (Ti) and
mass density (ρ) proles at 10 ns of a radiative sho13k propagatingwith a 15 kms (a sub-13riti13al) and 48 kms (b super13riti13al)
The sho13k is propagating in Xenon at 01 bar (ρ = 54 times 10
minus4
g13m
minus3 at 298 K) The initial temperature in Xenon is set to
1eV The two Ma13h numbers are respe13tively equal to 13 and 43
The radiation and ionisation ee13ts are in13luded in the simulation
whi13h is performed for two dierent temperatures (Te 6= Ti) and a
negligible thermal 13ondu13tivity (10
minus05WmK) For this simula-
tion the sho13k is laun13hed by a 100 microm thi13k Aluminium piston
moving at the speed of 12 kms and 45 kms for 13ase (a) and (b)
respe13tively The position x=0 13orresponds to the position of the
XeAl interfa13e at time t=0 17
31 PROPACEOS Mono13hromati13 opa13ity versus the photon energy
in eV of Xenon at 10 eV and at two ele13tron densities respe13tively
equal to 1018 and 1020 13m
minus3 26
32 Mass density and temperature (a) ele13tron density and mean
13harge (b) at 10 ns for a radiative sho13k of speed sim 47 kms
in Xenon at 01 bar The verti13al dotted bla13k lines show the
position of the interfa13e between piston and ba13king Xenon gas
Zero at x-axis is the interfa13e of piston and Xenon at time zero
Spitzer thermal 13ondu13tivity has been used in the simulation 29
33 Mean Rosseland opa13ity (in 13m
2g
minus1) and opa13ity (13m
minus1) at 10
ns for a radiative sho13k of speed sim 47 kms in Xenon at 01
bar The verti13al dotted bla13k lines show the position of the
interfa13e between piston and ba13king Xenon gas Zero at x-axis
is the interfa13e of piston and Xenon at time zero Spitzer thermal
13ondu13tivity has been used in the simulation 30
34 (a) Mass density and temperature (b) ele13tron density and
mean 13harge at 10 ns for a radiative sho13k of speed sim 47 kms
in Xenon at 01 bar The verti13al dotted bla13k lines show the
position of the interfa13e between piston and ba13king Xenon gas
Zero at x-axis is the interfa13e of piston and Xenon at time zero
A negligible 13onstant thermal 13ondu13tivity (10
minus12WmK) have
been used in the simulation for Xenon 31
LIST OF FIGURES
35 (a) Ele13tron density and ele13tron temperature (b) Mass density
and mean 13harge at 10 ns for a radiative sho13k of speed sim 48
kms in Xenon at 01 bar for the 13ase when the ele13tron and ion
temperature are dierent The verti13al dotted bla13k lines show
the position of the interfa13e between piston and ba13king Xenon
gas Zero at x-axis 13orresponds to the position of the interfa13e
between the piston and Xenon at time zero The Spitzer thermal
13ondu13tivity has been used in this simulation 32
36 (a) Mass density and ele13tron temperature (b) ele13tron density
and mean 13harge at 10 ns for a radiative sho13k of speed sim 45
kms in Xenon at 01 bar for the 13ase when the ele13tron and ion
temperature are dierent The verti13al dotted bla13k lines show
the position of the interfa13e between piston and ba13king Xenon
gas Zero at x-axis 13orresponds to the position of the interfa13e
between the piston and Xenon at time zero In this simulation
value of thermal 13ondu13tivity (10
minus12WmK) is negligible for
Xenon The peak value of ion temperature is found to be 405 eV
whereas theoreti13ally this value is expe13ted to be sim 600 eV It is
possible to a13hieve the expe13ted value by in13reasing the resolution
of the simulation 34
37 Ele13tron temperature proles for various numbers of frequen13y
groups N = 1 20 50 60 70 90 and 100 35
38 (a) Ele13tron temperature (a) mean 13harge (b) and mass density
(13) for four (1 10 30 and 40) multipliers of the Xenon opa13ity at
10 ns 36
39 Plan13k and Rosseland mono13hromati13 opa13ity proles Te for ρ=15 times 10
minus3from PROPACEOS (I) BIGBART (II) and Rodri-
guez et al (III) 38
310 Mono13hromati13 Xenon opa13ity versus the photon energy in eV
at T = 15 eV and for ρ = 15 times 10
minus3g13m
minus3for the PROPA-
CEOS (a) and Rodriacuteguez et al (2015) (Fig 16 of Rodriacuteguez et al
(2015)) (b) models 39
311 Ele13tron density Ne (a) and ele13tron temperature Te (b) versus
axial position (along a 04 13m long sho13k tube) at 3 10 20 30
and 38 ns from HELIOS simulations (with opa13ity times 20) for the
13ases of single sho13k of sim 50 kms (dotted line) and two identi-
13al 13ounter-propagating sho13ks of sim 50 kms (solid lines) The
verti13al dotted lines show the position of the interfa13e between
piston and ba13king Xenon gas 41
LIST OF FIGURES
312 Variations ofNe (a) and Te (b) versus axial position for the 13ase of
two identi13al 13ounter-propagating sho13ks (of speeds sim 50 kms)
at 3 10 20 30 35 and 40 ns as derived from HELIOS simulations
For these simulations we have negle13ted the ee13t of radiation
by keeping the Xe opa13ity equal to zero The verti13al dotted lines
show the position of the interfa13e between the piston and ba13king
Xenon gas 43
313 Variations of Ne (a) and Te (b) with axial position for the 13ase
of two non-identi13al 13ounter-propagating sho13ks (of speeds sim 50
amp 20 kms) and two single sho13ks (dotted lines) of speeds sim 50
amp 20 kms respe13tively at 3 10 30 35 and 46 ns as derived
from HELIOS simulations The verti13al dotted lines show the
position of the interfa13e between piston and ba13king Xenon gas
(with opa13ity times 20) 45
41 a Spatial prole of the PALS Laser beam b Intensity prole of
laser pulse I(t) with time 48
42 Energy distribution s13heme for MAIN and AUX laser beams 49
43 Snapshot of the spheri13al 13hamber showing the horizontal bred-
board and on the right the fo13using MAIN lens 50
44 Dimension drawing (a) and snapshot (b) of Aluminum massive
(mo13k) target 50
45 Values on the gures are in mm (a) Verti13al 13ross se13tion view
of the gaseous target (b) Horizontal 13ross se13tion view 52
46 Pi13ture of a the gaseous target on its base 53
47 S13hemati13 of the gaseous target 53
48 Transmission of 100 nm thi13k Si3N4 membrane (CXRO database) 54
49 Snapshot of a target holder with one V-shield regarding the AUX
laser whi13h will pass through the hole in this shield marked in
green in the Figure The target whi13h fa13es the two diagnosti13s
(tubes) and is lo13ated behind the V-shield is thus not visible 54
410 In-situ system for target lling and pressure monitoring 56
411 S13hemati13 of the keV 13amera 59
412 (a) Sign 13onvention for the respe13tive positions of the lens and
the target (b) keV image of a MAIN impa13t on a massive target
(lens position +1500 microm) The spot size is 475plusmn25 microm Pixel
size for MAIN keV 13amera is 56 microm (13) keV image of an AUX
impa13t (lens position -1500 microm) The spot size is sim 280plusmn20 micromon target One pixel = 66 microm on AUXs keV 13amera 60
413 First interferometri13 setup All the opti13al elements are 1 in13h
For simpli13ity the plasma slab is not reported in the probe beam 61
414 Se13ond interferometri13 setup 63
415 S13hemati13 of the Streak 13amera (from Hamamatsu noti13e) 64
LIST OF FIGURES
416 Imaging setup of the Ma13h Zehnder Interferometer setup opera-
ting in the longitudinal mode The 13hannel of the target is fully
illuminated by the probe beam In this mode the lens images
the axis of symmetry of the two sho13k waves along the slit of the
streak 13amera 65
417 (a) Referen13e 2D image of a target before the shot re13orded on
the Streak 13amera The positions of the two pistons 13losing the 4
mm long target are lo13ated at 800 and 4800 microm The dark zones
along them (between 800 to 1000 and 4600-4800 microm) 13orresponds
to the glue This glue is then visible through verti13al bla13k strips
in the interferometri13 re13ord (b) Corresponding interferometri13
time-spa13e re13ord 66
418 Typi13al ray tra13ing in a grazing in13iden13e XUV spe13trograph
The dete13tor (GMCP or image plate) is installed tangentially to
the Rowland 13ir13le 67
419 (a) Setup of the XUV spe13trometer in the spheri13al 13hamber (b)
Typi13al re13ord between sim 10 and 40 nm is shown in the bottom
panel 68
420 (a) S13hemati13 of setup employed to re13ord the time and spa13e
visible plasma emission (b) A typi13al spe13trogram 69
51 Original re13ord of shot 48055 (left) FFT of the original image
(13enter) and the frequen13y ltered image (right) 73
52 A 13ropped se13tion of the re13ord from shot the 48055 The
rst ve positions have been sele13ted manually `+ signs (in red
13olor) on ea13h fringe On this re13ord the distan13e between two
unperturbed fringes is 13orrespond to 15 pixels (ie 159 microm) 73
53 (a) Representative points of the fringes as derived from the spline
tting of the 5 manually sele13ted points (Fig 52 (II)) on ea13h
fringe (b) Positions of the fringes maxima along Y-axis for ea13h
fringe derived by lo13ating the points of maximum intensity in
X-dire13tion of the previous points obtained by spline t 75
54 Fringe maxima on the full image 76
55 Imaging setup of the Ma13h Zehnder Interferometer operating in
the longitudinal mode The lens images the axis of symmetry of
the two sho13k waves along the slit of the streak 13amera 76
56 Interferometri13 image re13orded for the shot 48055 in Xe at 01
bar The sho13k speeds for the sho13ks driven by MAIN (from left
side) land AUX (from right side) lasers are respe13tively equal to
sim 54 and 23 kms The time of laser arrival on the piston is at
146 ns The positions of the Au-Xe interfa13e on the re13ord are
respe13tively 950plusmn50 and 4950plusmn50 mi13rons 77
LIST OF FIGURES
57 (a) MAIN sho13k speed (13al13ulated by the last fringe method)
versus the MAIN energy for Xenon or XeHe mixture at dierent
pressures with the error bars (b) AUX sho13k speed versus AUX
energy also for Xenon or XeHe mixture at dierent pressures 79
58 MAIN sho13k speed (13al13ulated by the last fringe method) versus
the MAIN laser energy for Xe (at 01 bar only) Ar He and Kr
at dierent pressures with the error bars 82
59 Left panel interferometri13 re13ords 48055 in Xe at 01 bar (a)
48132 in Xe+He at 02 bar (b) and 48138 in Xe+He at 02 bar
(13) Right panel ele13tron density at 10 20 30 and 40 ns versus
distan13e for these re13ords The positions of maxima have been
identied on the re13ords in the left panel The time t = 0 13orre-
sponds to the time of laser arrival on the target and the position
x = 0 13orresponds to the interfa13e between the piston (Au layer)
and the gas Its determination is pre13ise within 100 mi13rons The
distan13es between two unperturbed fringes for re13ords 48055
48132 and 48138 are 159 244 and 244 microm respe13tively The
lt Ne gt un13ertainty (plusmn 2 pixels) is indi13ated by the error bar in
the right panels It 13orresponds respe13tively to plusmn 9 times 10
17 plusmn 6
times 10
17and plusmn 6 times 10
1713m
minus3for the gures (a) (b) and (13) The
limit of dete13tion (2 pixels) is presented by a dotted line on ea13h
gure 85
510 (a) Interferometri13 re13ord in Ar at 03 bar (48079) The esti-
mated speeds for MAIN and AUX are 49plusmn5 and 23plusmn3 kms (b)
Same re13ord where the fringes maxima are marked by points
The bins denition is as fellows bin 0 Ne le 11 times 10
1813m
minus3
(white) bin 1 11 - 18 times 10
1813m
minus3(yellow) and bin 2 39 -
57 times 10
1813m
minus3(red) The time t = 0 13orresponds to the time
of laser arrival on the target and the position x = 0 13orresponds
to the interfa13e between the piston (Au layer) and the gas Its
determination is pre13ise within 100 mi13rons 87
511 Results for temperature (a) and Rosseland opa13ity (b) obtained
from HELIOS simulation at 10 ns for two 13ounter-propagating
sho13ks at sim 50 and 18 kms for Xe (with opa13ity multiplier 1
and 20) Ar and Kr at 54 times 10
minus4g13m
minus3initial mass density
A 13omparison Ar Kr and Xe PROPACEOS opa13ity shown in
APPENDIX B 88
LIST OF FIGURES
512 (a) Raw interferometri13 re13ord in Kr at 02 bar (shot 48146)
The estimated speed for MAIN sho13k is 53plusmn2 kms The time
t = 0 13orresponds to the time of laser arrival on the target(b)
Same re13ord shown in (a) maxima is marked by 13olored points
The bins denition is as follows bin 0 Ne le 11 times 10
1813m
minus3
(white) bin 1 11 - 18 times 10
1813m
minus3(yellow) and bin 2 39 -
57 times 10
1813m
minus3(red) The time t = 0 13orresponds to the time
of laser arrival on the target and the position x = 0 13orresponds
to the interfa13e between the piston (Au layer) and the gas Its
determination is pre13ise within 100 mi13rons 90
513 S13hemati13s of the Ma13h Zehnder Interferometer setup to re13ord
transverse interferometri13 images The lens allows to make on
the slit of the 13amera the image of a se13tion perpendi13ular to
dire13tion of sho13k propagation 91
514 Transverse interferometri13 images for (a) shot48111 (MAIN
sho13k only) (b) shot48130 (AUX sho13k only) The time is
measured after an oset equal to 14 and 23 ns respe13tively after
the time of the laser arrival on the target The position zero on
the x-axis of ea13h image 13orresponds to the base of the target 92
515 Interferometri13 image for the shot48143 The time t = 0 13orre-
sponds to the time of laser arrival on the target and the position
x = 0 13orresponds to the interfa13e between the piston (Au layer)
and the gas Its determination is pre13ise within 100 mi13rons 94
516 Raw (a) and 13orre13ted XUV spe13trum (b) for the shot 48143 95
517 Re13orded ele13tron density (shot 48055) together with the HE-
LIOS results (with Xenon opa13ity times 20) at dierent times in
Xenon at 01 bar 97
518 Time evolution of the mass density (a) ele13tron temperature (b)
and mean 13harge (13) at 56 57 58 60 64 and 65 ns within the
sho13k tube derived from the HELIOS simulations (with Xenon
opa13ity multiplier = 20) for two 13ounter streaming sho13ks of sim39 and 18 kms in Xenon at 06 bar 98
61 (a) Sket13h of the sho13k generator showing the pulsed ele13tri13al
13ir13uit the set of 13oaxial 13oni13al ele13trodes with a 13onstant radial
gap and the plasti13 insulator featured in grey on whi13h a planar
surfa13e dis13harge is initiated The installation of three opti13al
bers allows looking radially at the plasma moving in the sho13k
tube (b) S13hemati13s of the plasma dynami13s inside the 13oaxial
gun in fast-pulse mode the ele13tri13al 13urrent ows in the super-
13ial layers of the two 13oaxial 13oni13al ele13trodes and through an
annular plasma layer The magneti13 pressure Pmag pushes the
dis13harge axially 105
62 Exploded view of the plasma gun 106
LIST OF FIGURES
63 Time dependen13e of the main parameters for Ar gas at 1 mbar
a1313ording to the 13ir13uit model (a) 13urrent (kA) and voltage
(V) (b) a1313reted mass (kg) and rate of a1313retion (kgs) (13)
speed (ms) and kineti13 energy (J) 108
64 Plot of the dierent output parameters (normalized with respe13t
to the peak values) obtained at the top of the inner ele13trode
For ea13h group upper panel represents the variation for Ar gas
whereas lower panel is for Xe gas Proles of output parameters
are given (a) with 13one angle (b) with a1313retion fa13tor (13) with
gas pressure and (d) with damping resistan13e 109
65 Example of mapping of the ele13tron density from a 3-D MHD
simulation (ba13kground gas Argon at 1 mbar) when the plasma
sheath is (left) at the exit of the plasma gun (right) in free
ight 13onditions (log s13ales for the false 13olors) 113
66 Time history of plasma merging and early free ight through
the mapping of the ele13tron density ele13tron temperature and
average ioni13 13harge as given by a 3-D MHD simulation (ba13k-
ground gas Argon at 1 mbar) (s13ales for the false 13olors) 114
67 Axial proles values taken in the sho13k tube along a line slightly
o-axis at a distan13e of 1 mm for mass density ion temperature
ion density average ion 13harge magneti13 eld ele13tron tempera-
ture ele13tron density and average speed at 1500 ns (ba13kground
gas Argon at 1 mbar) A maping of |B| is given with a dashed
line indi13ating sho13k front position as well 115
68 Experimental results (a) time history of the passive opti13al
re13ords 13ompared to the main 13urrent for Argon gas 13lose to the
referen13e pressure (b) time-integrated signal vs Ar pressure for
the dierent bers (13) averaged speed vs Ar ba13king pressure 117
71 Interferometri13 re13ord obtained at 546 ns for a single radiative
sho13k of speed sim 100 kms propagating in Xe+He gas at 06 bar
driven by laser at 348 nm with energy 170 J The dierent 13olors
in (b) 13orrespond to yellow le Ne le 35 times 10
1713m
minus3 13yan 37
- 74 times 10
1713m
minus3 blue74 - 15 times 10
1813m
minus3 green 15 - 18 times
10
1813m
minus3 magenta 18 - 22 times 10
1813m
minus3 red 22 - 26 times 10
18
13m
minus3 orange26 times 10
1813m
minus3- 18 times 10
1913m
minus3 124
A1 S13hemati13s of Ma13h Zehnder interferometer with two mirrors
two 50 beam splitters and the dete13tor A plasma slab is pla13ed
in one of the two arms of the interferometer 147
B1 Rosseland and Plan13k opa13ity for three gases Xe Kr and Ar at
mass densities 51 times 10
minus4g13m
3(a) and 16 times 10
minus3g13m
3(b) 150
LIST OF FIGURES
B2 Mean 13harge for three gases Xe Kr and Ar at mass densities 16
times 10
minus3g13m
3(a) and 51 times 10
minus4g13m
3(b) 151
List of Tables
21 Summary of the radiative sho13ks experiments and out13omes in
dierent laser installations All the experiments are performed in
Xe ex13ept Keiter et al (2002) and Vis13o et al (2012) In these
two experiments the rst uses SiO
lowast
2 and the se13ond uses Ar
lowastlowastas
a sho13ked medium 20
41 Nominal 13hara13teristi13s of MAIN and AUX laser beams 49
42 Spe13i13ations of the MAIN and AUX fo13al lenses and the f- num-
ber (fo13al lengthbeam diameter) is given (see table 41 for the
beam diameters) 57
43 Spe13i13ations of the two Phase Zone Plates 58
44 Spe13i13ations of the two uEye keV 13ameras (1stVsion 13ompany) 59
45 Spe13i13ations of Andor CCD 13amera 67
46 Spe13i13ations of the XUV grating 67
51 Sho13k speeds estimated from the `last fringe method `NA re-
presents the entries whi13h 13ould not be dedu13ed from the re13ord
Further entries in the bold font are dis13ussed in detail in this
13hapter 81
52 Atomi13 data and density at 01 bar (at room temperature) for
He Ar Kr and Xe 82
61 Input parameters their standard values and range of variation 110
62 Laun13hing performan13e ArXe 111
63 Values of variables in various 13onditions at rest (with seed ele13-
trons) inside the sho13k and inside post-sho13k region A star lowastdenotes the insulator surfa13e 112
Chapter 1
Introdu13tion
11 General Context
Radiative sho13ks (RS) are present in various astrophysi13al environments es-
pe13ially in stars They are then present in all the supersoni13 a1313retioneje13tion
pro13esses during the stellar evolution from the early phase of the proto-
stellar 13ollapse up to the magnetosphere a1313retion on to the already formed
T- Tauri stars up to the late stage of the supernovae sho13ks
Radiative sho13ks are strong sho13ks (ie Ma13h number Mgtgt1) whi13hrea13h high temperatures and thus are the sour13e of intense radiation Alt-
hough the observation of (ie spatially resolved ) signatures of su13h sho13ks
in the astrophysi13al environments is very di13ult an alternate approa13h na-
med laboratory plasma astrophysi13s provides a powerful tool to study them
on the Earth
These waves have been experimentally studied sin13e more than a de13ade
mostly on large-s13ale laser fa13ilities in noble gases and with dierent targets
geometries The ablation pro13ess generated by the intera13tion between the
laser and a foil indu13es a sho13k wave in the foil whi13h then propagates
in the gas at a speed of several tens of kms Numeri13al simulations 13an
des13ribe these sho13ks with an improving pre13ision With laser intensities on
target 13omprised between 1014 and 1015 W13m
2 these experiments allowed
to re13ord sho13k speeds ranging between 40 and 150 kms
All previous experimental studies have been fo13used on the 13ase of single
radiative sho13ks However in astrophysi13al 13onditions the radiative sho13k
in general intera13ts with a denser medium leading to the development of
ree13ted and transmitted sho13ks A few representative examples of su13h
phenomena are the intera13tion of supernovae remnants with dense mole13ular
13louds the a1313retion sho13ks on the photosphere of T-Tauri stars and the bow
sho13ks at the head of fast stellar jets The 13ollision (or the intera13tion) of
two radiative sho13k waves is obviously a rare astrophysi13al event and the
template 13ase of the supernova remnants DEM L316 (see Fig 1 of Williams
et al (1997)) is still the subje13t of debates (Velarde et al 2006) In this
regard the development of dedi13ated laboratory experiments to the study of
propagation and intera13tion of 13ounter-propagating sho13k waves is important
as a tool to 13hara13terize su13h events through their spe13i13 signatures
Complementary to laser experiments 13ompa13t pulsed power generators
may drive an ele13tromagneti13 13oaxial plasma gun to 13reate astrophysi13al
1
2 CHAPTER 1 INTRODUCTION
relevant sho13ks in lower pressure noble gases with a high availability and a
rather modest 13apital 13ost The ele13tromagneti13ally driven sho13k waves may
have larger s13ales than those by laser Thus they 13an be analyzed rather
easily with the adequate diagnosti13s (Kondo et al 2008)
In this thesis I am presenting radiative sho13ks in the 13ontext of labo-
ratory astrophysi13s using two experimental approa13hes laser-driven sho13ks
and ele13tromagneti13ally driven sho13ks
12 Outline of Chapters
In the rst 13hapter the physi13s of radiative sho13ks wave will be summarized
I will dis13uss the Rankine -Hugoniot (RH) for the perfe13t gas and genera-
lized RH equations for ionized gas Further the ee13ts of radiation on the
sho13k equations will be introdu13ed and I will review the dierent regimes of
the radiative sho13ks presenting a radiative pre13ursor Then I will present
a short histori13al review of laser and ele13tromagneti13ally laboratory experi-
ments on this topi13 Finally I will highlight the obje13tives of my work in
this framework
The 13hapter two is dedi13ated to the numeri13al analysis of radiative sho13k
waves and of the main physi13al pro13esses whi13h play a role in these 13omplex
pro13esses The simulations will be performed in 1D using the 13ommer13ial ra-
diation hydrodynami13 HELIOS 13ode (Ma13Farlane Golovkin and Woodru
2006) After a des13ription of the 13ode and his possibilities several simu-
lations will be performed to analyze the ee13ts of the two uids (ele13trons
and ions) approa13h 13ompared to the one uid 13ase (identi13al temperature
for both) of the thermal 13ondu13tivity and the multi-groups opa13ity des13rip-
tion Then a brief overview of single and two 13ounter-propagating radiative
sho13ks will be presented
Chapter three will start with a short overview of the Prague Asterix Laser
System fa13ility (PALS) It will be followed by the experimental setup in13lu-
ding the two high energy beams and targets des13ription These paragraphs
will be followed by the detail of the diagnosti13s
The fourth 13hapter will be dedi13ated to the data analysis the results
and the interpretation of the experimental re13ords I will rst des13ribe the
method I followed for the analysis of interferometri13 re13ords Subsequently
the results obtained from a few representative re13ords for Xe Ar and Kr
will be dis13ussed in details Complementary 1D simulations performed with
initial 13onditions similar to that in the experiment will be shown and help to
interpret the experimental results Finally the results of XUV spe13tros13opy
diagnosti13s will be presented
In the fth 13hapter I will present the 13alibration of a table top ele13tro-
magneti13 plasma generator whi13h is able to laun13h supersoni13 sho13ks up to
sim 40 kms speed at stati13 pressures of few mbar The results of this 13ali-
13 MY CONTRIBUTION 3
bration will also be presented and a 13omparison with 3-D MHD simulations
will be performed The diagnosti13s whi13h have been implemented will be
presented to illustrate the model as well as preliminary re13ords of the plasma
speed
The nal 13hapter of the thesis is dedi13ated to the summary of this work
together with perspe13tives for the follow-up of these experiments
13 My 13ontribution
I joined PhD on November 12 2013 under the supervision of Dr Jean
Larour and Dr Chantal Stehleacute My PhD thesis is 13omprised of two parts
the rst part of the work is to study the laser-driven sho13ks The se13ond
part of my thesis is to study the ele13tromagneti13ally laun13hed strong sho13ks
Con13erning the laser-driven sho13ks I have started to work on the analysis
and interpretation of XUV images re13orded previously during an experimen-
tal 13ampaign 13ondu13ted at the Prague Asterix Laser System (PALS) fa13ility
in 2011 This enabled me to estimate the lo13al opa13ity by employing Abel
inversion Although this work not presented in this thesis it shed light on
the basi13 understanding of radiative sho13k as seen in the experiments as well
as the knowledge of experimental setups Later in April - May 2015 I have
parti13ipated in a ve weeks experimental 13ampaign on PALS laser fa13ility
in Prague whi13h was dedi13ated to the study of the spe13tros13opi13 and in-
terferometri13 signatures of laser-produ13ed sho13ks Espe13ially the experiment
was designed to study the physi13s during the 13ollision of 13ounter-propagating
sho13ks In the preparation of this 13ampaign I be13ame a13quainted with the
instruments and the targets to be used as well as prepared a master re-
port 13ontaining all the 13ru13ial aspe13ts of the experiment to be referred by
all the parti13ipants during the experiments During the 13ampaign we have
performed around 55 shots 13orresponding to dierent gases (Xenon Argon
Helium and Krypton) at dierent pressure and used three types of diagnos-
ti13s ie visible interferometry visible spe13tros13opy and XUV spe13tros13opy
In this experiment parti13ularly I was in 13harge of the target sele13tion its
installation in the experimental 13hamber gas lling preparing logbook re-
13ording of the visible spe13tros13opy data from the 13ontrol room qui13k data
analysis for the spot size sho13k speed from interferometry data et13 In
addition I have worked on spe13tros13opi13 and interferometri13 data analysis
re13orded during the experiment Next in order to interpret the experimental
results I have performed several 1D hydrodynami13 simulations with varying
input 13onditions In parallel I have also performed simulations for another
experiment performed by my 13ollaborators at ORION laser fa13ility in the
United Kingdom However I have not presented this work in this thesis
Based on the results obtained from my analysis of re13ords obtained from
the PALS experiments in 2015 grabbing the opportunity we have planned
4 CHAPTER 1 INTRODUCTION
and parti13ipated to a 13omplementary experimental 13ampaign at PALS in
September 2016 The analysis of data re13orded during experiments is still
under progress and forms the future s13ope of the thesis
The se13ond part of my thesis has been 13omprised of the study of sho13k
laun13hed through the ele13tromagneti13 generator setup In this I have wor-
ked on the optimization of the design of the ele13tri13 generator in view of the
required sho13k parameters Moreover I have derived various sho13k 13hara13-
teristi13s viz radial dis13harge layer namely speed mass a1313eleration et13
at the dierent positions of its propagation In present a upgraded ele13tri13
generator with new diagnosti13s is ready to perform the new experiments
Chapter 2
Radiative sho13k waves
Contents
21 Rankine-Hugoniot(R-H) relations 6
211 Jump relations for a real gas 9
212 Ee13t of radiation on sho13k waves 11
213 Radiative hydrodynami13s equations 13
22 Radiative sho13k regimes 15
221 Typi13al radiative sho13k waves with a pre13ursor 16
23 Radiative sho13k waves experiments 16
231 Laser-driven sho13k experiments 18
232 Ele13tromagneti13ally laun13hed sho13k 21
Small-amplitude disturban13es in a medium often take the form of linear
waves On the 13ontrary strong disturban13es generally produ13ed by pro13esses
su13h as explosions lightening supersoni13 movements of bodies powerful
ele13tri13 dis13harges et13 where the uid properties 13hange rapidly generate
nonlinear waves like sho13k waves As a 13onsequen13e of sho13k waves the ow
moves faster than the sound speed A sho13k is 13hara13terized by a sudden
in13rease in the density and velo13ity of the medium The Ma13h number (M)
is 13ommonly used to 13hara13terize the strength of the sho13k This number
dened as the ratio of the sho13k speed to the speed of sound in the upstream
medium in whi13h the sho13k propagates
Radiative sho13k waves are hypersoni13 sho13k waves whi13h are heated to
high temperature and as a 13onsequen13e be13ome the sour13e of intense radi-
ation This radiation in turn modies the dynami13s and stru13ture of the
sho13k itself whi13h makes its stru13ture more 13ompli13ated Radiative sho13k
waves o1313ur in several astrophysi13al 13ir13umstan13es su13h as in the a1313retion
sho13k of protostellar formation (Stahler Palla and Salpeter 1986) the su-
pernova explosion and the intera13tion of their remnants with the dense in-
terstellar medium (Chevalier 1977) the bow sho13ks at the head of stellar
jets (Hartigan et al 2001)
Radiative sho13k waves 13an now be studied in the laboratory whi13h allows
to 13ompare the data with models existing in the literature and to 13he13k the
ability of the numeri13al 13odes to reprodu13e them (see for instan13e Bouquet
et al (2004) Leygna13 et al (2006)) The experimental study of radiative
5
6 CHAPTER 2 RADIATIVE SHOCK WAVES
sho13k 13hara13teristi13s is 13ru13ial in understanding these various energeti13 pro-
13esses o1313urring in the astrophysi13al plasmas In this 13hapter I will rst
present a short summary of the jump 13onditions (Rankine-Hugoniot rela-
tions) for the simple 13ase of an ideal gas whi13h rely on the values of the
thermodynami13al quantities on both sides of the dis13ontinuity Next I will
present the role played by the ionisation and ex13itation of the gas and then
the role played by the radiation energy and pressure The des13ription of the
dierent regimes of radiative sho13ks whi13h depend on the opa13ity of medium
and 13an be 13hara13terized by the presen13e or absen13e of a radiative pre13ursor
will follow Further I will give a brief overview of the literature fo13using on
the generation of radiative sho13ks in the laboratory and I will present the
motivations and obje13tives of my work
21 Rankine-Hugoniot(R-H) relations
Let us 13onsider a one-dimensional sho13k propagating in a gas at rest with
the speed us In the frame asso13iated with the sho13k front the pre-sho13k
(upstream) uid velo13ity is then u1 = - us while behind the sho13k (post-
sho13k or downstream region) this velo13ity is u2 as shown in the Fig 21
Figure 21 S13hemati13 diagram for a 1-D sho13k moving in the x dire13tion The
speed of the sho13k front is us In the frame of referen13e of the sho13k front the pre-
sho13k region has a relative speed u1= -us density ρ1 total pressure P1 temperature
T1 whereas these quantities are respe13tively u2 ρ2 P2 and T2 for the post-sho13k
region
Assuming that a stationary state is rea13hed the 13onditions whi13h deter-
mine the post-sho13k thermodynami13al quantities are the well known Rankine-
21 RANKINE-HUGONIOT(R-H) RELATIONS 7
Hugoniot equations (Ma13quorn Rankine 1870 Hugoniot 1887 1889) De-
du13ed from the Euler equations these 13onservation equations for the mass
momentum and energy are given by
ρ2u2 = ρ1u1 (21)
ρ2u22 + P2 = ρ1u
21 + P1 (22)
u2(ε2 +P2
ρ2) +
1
2u32 = u1(ε1 +
P1
ρ1) +
1
2u31 (23)
where ε1 and ε2 are the spe13i13 internal energies (energy per unit mass)
in the two (upstream and downstream) regions
For an ideal gas of atomi13 mass m =mPA (mP proton mass A atomi13
number) and having an adiabati13 index γ the speed of sound is given by
Cs =
radic
γP
ρ=
radic
γkBT
m(24)
and the Ma13h number (M) of the sho13k is then
M =u1Cs1
(25)
Inserting these the aforementioned form of M and Cs in the equations
21 22 and 23 we 13an derive the 13ompression pressure and temperature
ratios for the system as follows
ρ2ρ1
=u1u2
=M2(γ + 1)
2 +M2(γ minus 1)(26)
P2
P1=
2M2γ minus (γ minus 1)
(γ + 1)(27)
T2
T1=
P2
P1
ρ1ρ2
=(2M2γ minus (γ minus 1))(2 +M2(γ minus 1)
M2(γ + 1)2) (28)
Considering the 13ase of strong sho13ks where M may be 13onsidered to be
≫1 these equations be13ome
ρ2ρ1
=(γ + 1)
(γ minus 1)(29)
T2
T1=
2M2γ(γ minus 1)
(γ + 1)2(210)
kBT2 =2(γ minus 1)
(γ + 1)2mus
2(211)
8 CHAPTER 2 RADIATIVE SHOCK WAVES
For mono-atomi13 gases γ = 53 the 13ompression ratio is estimated to
be 4 and the post-sho13k temperature is
kBT2 =3
16mu1
2 =3
16mPAu1
2(212)
Temperature is thus proportional to the square of the sho13k velo13ity and
to the atomi13 mass Therefore to a13hieve a high temperature in the radiative
sho13k experiment we often use high atomi13 number and mass gases For Xe-
non (A=13129) the post-sho13k temperature is then equal to T2 = 0257u21(eV kms) whi13h for instan13e gives sim 520 eV for a sho13k propagating at
45 kms (see Fig 22)
Figure 22 Proles of temperature (Te = Ti = T ) and mass density (ρ) at 10ns for a sho13k propagating in Xenon at 01 bar (ρ1= 57 times 10
minus4g13m
minus3) with
a speed of sim 45 kms It is to note that the given pressure value are at 298 K
The simulation is performed with the HELIOS 13ode ex13luding the radiation ee13t
and for a polytropi13 gas (γ=53) The initial temperature of Xenon is set to 1
eV and with a negligible (10
minus5WmK) thermal 13ondu13tivity The Ma13h number
is then equal to 40 For this simulation the sho13k is laun13hed by a 100 microm thi13k
Aluminium piston moving at the velo13ity of 33 kms The position x=0 13orresponds
to the position of the XeAl interfa13e at time t=0
21 RANKINE-HUGONIOT(R-H) RELATIONS 9
211 Jump relations for a real gas
In 13ontrast to the dis13ussion on ideal gas presented in the previous se13tion
the 13ase of a real gas is a little more 13ompli13ated It is to note that in this
13ase a part of the kineti13 energy is used to ex13ite and ionize the post-sho13k
gas As a result its temperature is lower than that for the 13ase of ideal gas
In addition the pressure is also modied due to ionisation (Mi13haut et al
2004) as dis13ussed below
Let us now dene j as the ionization stage of the atom (varying between
0 and z) and i to be the atomi13 state in this ionisation stage We shall
denote by Pj amp Pji the fra13tional ionization of the ion and the population
of the atomi13 state i in the jth ionization state respe13tively Then the mean
ionization stage (average 13harge) per atom 13an be written as
z =
zsum
j=0
jPj (213)
and the ex13itation energy (in13luding ionization) per atom and per unit of
mass is
ǫexc =
sumzj=0
sum
i Pji Eji
mpA(214)
where Eji is the energy of the ionisation stage j in the atomi13 stage iThus the total pressure of gas 13an be written as
P = Pi + Pe (215)
where Pi and Pe are the ioni13 and ele13troni13 pressures 13ontributions
Using the 13harge neutrality 13ondition Ne = z Ni this gives
P = Ni(1 + z) kBT = ρkBT
m(1 + z) (216)
The thermal pressure and enthalpy of the real gas are given by
Pth = ρ(1 + z)
mkBT (217)
h =5
2
(1 + z)
mkBT + ǫexc (218)
Next the speed of sound in13luding the ee13t of ionization with the
average 13harge z in the medium is now modied by ionisation
Cs ≃radic
5
3
γ(z + 1)kBT
m(219)
Thus the 13ontinuity relations in the frame moving with the sho13k front
be13ome
ρ2u2 = ρ1u1 (220)
10 CHAPTER 2 RADIATIVE SHOCK WAVES
ρ2u22 + ρ2
kBT2
m(1 + z2) = ρ1u
21 + ρ1
kBT1
m(1 + z1) (221)
ρ2u2
[5
2
(1 + z2)
mkBT2 + ǫexc2 +
u222
]
= ρ1u1
[5
2
(1 + z1)
mkBT1 + ǫexc1 +
u212
]
(222)
Figure 23 Ele13tron temperature (Te) ion temperature (Ti) and mass density
(ρ) proles at 10 ns derived from 1-D simulation for the 13ase of ionisation (no
radiation) The sho13k propagates in Xe gas at 01 bar (ρ1= 57 times 10
minus4g13m
minus3by
sim 45 kms) It is to note that the given pressure value is at 298 K The simulation
is performed with the HELIOS (Te 6= Ti 13ase) using the PROPACEOS equation
of state without any radiation and with a negligible thermal 13ondu13tivity (10
minus05
WmK) The initial temperature in Xenon is set to 1 eV and thus the Ma13h
number is equal to 40 For this simulation the sho13k is laun13hed by a 100 microm thi13k
Aluminium piston moving at the velo13ity of 38 kms The position x=0 13orresponds
to the position of the XeAl interfa13e at time t=0
Similar to the Fig 22 however for the 13ase of a real gas the Fig
23 shows the variation the ele13tron and ion temperatures as well as the
mass density with the distan13e for a sho13k propagating in Xenon at sim 45
kms Now the post-sho13k ele13tron temperature peaks at 22 eV whi13h is
mu13h smaller than the maximum temperature of the ions (510 eV) The
13ompression is now equal to 10 instead of 4 as in the previous 13ase The ions
21 RANKINE-HUGONIOT(R-H) RELATIONS 11
are heated rst in the sho13k through ion-ion 13ollisions This ion temperature
de13reases and the ele13tron temperature in13reases as a 13onsequen13e of the
ele13tron-ion 13ollisions The two temperatures thus be13ome equal Sin13e the
momentum 13hange of the ions require only a few number of kineti13 13ollisions
the thi13kness of the abrupt transition between the pre-sho13k and the post-
sho13k is of the order of a mean free path of the ions
It should be noted that for sho13ks propagating with very high speed
the post-sho13k temperature is so high that the 13ompressed gas is fully ioni-
zed Therefore the ee13ts of ionisation and ex13itation be13ome negligible and
hen13e the 13ompression ratio rea13hes 4 as in the 13ase of the adiabati13 sho13k
212 Ee13t of radiation on sho13k waves
In the previous se13tion the ee13ts of radiation heating and 13ooling have
been negle13ted However the radiation emitted in a strong sho13k will ae13t
its stru13ture These ee13ts 13an be determined using mass momentum and
energy 13onservation equations whi13h now have to in13lude the 13ontributi-
ons of the radiative ux energy and pressure To this purpose and with
some simpli13ations we 13an use the same approa13h than in se13tion 21 still
assuming that the sho13k is stationary
Jump relations with radiation
Let us 13onsider a stationary sho13k propagating in an atomi13 gas far away from
the dis13ontinuity The medium is then opti13ally thi13k and we 13an negle13t
the 13ontribution of the radiative ux Let us also assume that the medium
is at LTE and that the radiation pressure and energy may be des13ribed as
follows
Prad =1
3Erad =
4
3cσT 4 =
1
3aradT
4(223)
where arad is the radiation density 13onstant and T is the sho13k radiation
temperature whi13h we shall suppose to be equal to the ele13tron temperature
Following Mi13haut et al (2004) the new jump 13onditions a13ross the sho13k
dis13ontinuity be13ome
ρ2u2 = ρ1u1 (224)
ρ2u22+ρ2
kBT2
m(1+z2)+
1
3aradT
42 = ρ1u
21+ρ1
kBT1
m(1+z1)+
1
3aradT
41 (225)
ρ2u2
(
5
2
(1 + z2)
mkBT2 + ǫexc2 +
u222
)
+4
3aradT
42 u2 =
ρ1u1
(
5
2
(1 + z1)
mkBT1 + ǫexc1 +
u212
)
+4
3aradT
41 u1 (226)
12 CHAPTER 2 RADIATIVE SHOCK WAVES
(a)
(b)
Figure 24 Figure 14 amp 17 of Mi13haut et al (2004) (a) Compression ρ2ρ1 (full13ir13les thin line) kT2 in eV (inverted full triangles thin line) in Argon (a) Xenon
(b) for initial 13onditions kT1( = 10 eV for Ar and = 01 eV for Xe) ρ1 = 5 times10minus4 gcm3
versus sho13k speed in kms with (full markers full lines) and without
(empty markers dashed lines) radiation The ionization stage lt z gt2 is plotted
with the diamond symbol
21 RANKINE-HUGONIOT(R-H) RELATIONS 13
In the radiative regime the 13ompression ratio ρ2ρ1 approa13hes to the
value of 7 at very high velo13ities (several hundred of kms) whereas it is equal
to 4 when the ee13ts of radiation energy and pressure have been negle13ted
This limit is the same for the 13ases of adiabati13 (Bouquet Teyssier and
Chieze 2000) as well as the real gas (Mi13haut et al 2004) This is due to
the fa13t that at very high velo13ities the medium be13omes fully ionized and
behaves as a perfe13t gas On the 13ontrary in the 13omparatively low velo13ity
regime the ee13t of ionisation and ex13itation dominates the radiation ee13ts
This is illustrated in Fig 24 taken from Mi13haut et al (2004) whi13h reports
the variations of the Ar and Xe 13ompression ratio versus the sho13k speed for
an initial mass density of 5 times 10
minus4g13m
3 At 50 kms the 13ompression
ratio and the temperature are respe13tively equal to (5 32eV) and (11 30
eV) for Argon and Xenon
The 13ompression ratio shows bumps for high velo13ities These bumps
may be attributed to the fa13t that as the temperature of the post-sho13k
in13reases with the velo13ity the atom ionises more and more Thus the ex-
ternal shells in its ioni13 atomi13 stru13ture are su1313essively opened indu13ing
a strong variation in the ex13itation energy
The post-sho13k temperature is very high for the very high-speed sho13ks
In su13h 13ases the 13ompressed gas is fully ionized and the ee13ts of ionisation
and ex13itation remain negligible like in the adiabati13 13ase It should be noted
that below 100 - 200 kms the ee13ts of radiation pressure remain negligible
Thus very energeti13 lasers like LMJ or NIF are required to rea13h a domain
where this ee13t 13an be observed
Thus whereas for the range of velo13ities of 20 - 50 kms 13overed by my
study the ee13ts of radiative energy and pressure 13an be negle13ted this is
not the 13ase for the radiative ux whi13h has been negle13ted in the previous
se13tions and whi13h may be13ome 13omparable to the material energy ux Its
ee13t on the system must be taken into a1313ount as will be shown below
213 Radiative hydrodynami13s equations
To in13lude the 13ontributions of radiation it is 13ompulsory to introdu13e the
relevant terms in the uid equations The propagation of the radiation in the
matter is governed by the pro13esses of absorption and emission of photons
whi13h respe13tively heats and 13ools the medium As the mean free path of
photons is always higher than the mean free path of parti13les the sho13k
stru13ture is more inuen13ed by the radiative transfer than the vis13osity of
the uid
The propagation of radiation in the matter is des13ribed by the radiative
transfer equation
(
1
c
part
partt+
part
parts
)
I(rn ν t) = η(r ν)minus χ(r ν) I(rn ν t) (227)
14 CHAPTER 2 RADIATIVE SHOCK WAVES
where χ(ρ ν T ) and η(ρ ν T ) are respe13tively the mono13hromati13 opa-
13ity and emissivity of the gas
The spe13i13 intensity I(rn ν t) (erg cmminus2 sminus1Hzminus1 srminus1) is the energy
radiated per units of surfa13e area time frequen13y and solid angle in the
dire13tion n
The radiative energy density Erad ux Frad and pressure Prad are re-
spe13tively dened as the zero rst and se13ond moments of the spe13i13 in-
tensity versus the angle after integration over the photon frequen13y ν
Erad =1
c
int ∮
I(rn ν t)dΩdν (228)
Frad =
int ∮
I (rn ν t) cos θ dΩdν (229)
Prad =1
c
int ∮
I (rn ν t) cos2 θ dΩdν (230)
(231)
Taking into a1313ount radiative ee13ts the equations of radiative hydro-
dynami13s in the 13o-moving frame of the sho13k (Mihalas and Mihalas 1999
Drake 2006) 13an be written as
partρ
partt= minusnabla(ρu) (232)
ρ
(
partu
partt+ unablau
)
= minusnabla(P + Prad) (233)
part
partt
(
ρu2
2+ ρǫ+ Erad
)
+nabla
[
ρu
(
ǫ+u2
2+
P
ρ
)
+ (Erad + Prad)u
]
= minusnablaFrad
(234)
As mentioned previously the 13ontributions linked to Prad and Erad are
ee13tive only at very high temperature (ie velo13ity) However for the
radiative sho13ks of interest in this work the 13ontribution of the radiative
ux Frad is non-negligible 13ompared to ρu3 These sho13ks are thus in the
radiation ux dominated regime
Most of the experimental sho13k waves are in this regime These hyd-
rodynami13 equations are strongly non-linear In the present work the one-
dimensional radiative-hydrodynami13 13ode `HELIOS (Ma13Farlane Golovkin
and Woodru 2006) has been employed to derive the physi13al parameters
of the sho13k in various 13onditions as presented in the following se13tion HE-
LIOS will be dis13ussed in detail in the next 13hapter
22 RADIATIVE SHOCK REGIMES 15
22 Radiative sho13k regimes
Radiative sho13ks are strong sho13ks (Ma13h number Mgtgt1) whi13h attain
high temperatures and thus are the sour13e of intense radiation (Mihalas
and Mihalas 1984 Zeldovi13h Zeldovi13h and Raizer 2002 Drake 2006)
Depending on the opa13ity the radiation emitted from the sho13k may be ab-
sorbed by the pre-sho13k region indu13ing its pre-heating Far away from the
dis13ontinuity the stru13ture of the upstream medium is determined by the
absorption On its side the stru13ture of the transition layer of the down-
stream medium 13lose to the dis13ontinuity depends on the balan13e between
the emission (13ooling) and the absorption (heating) Thus the full stru13ture
of the sho13k mainly depends on the variation of the opa13ity χ (13m
2g) The
opti13al depth (τ(s)) measured from the position of the jump is then the
relevant parameter for the absorption It is given by
τ(s) =
int s
sjump
χ(sprime)ρ(sprime) dsprime (235)
where sprime 13hara13terizes the path of the radiation As we are interested in
the mean frequen13y averaged radiative ux the relevant opa13ity here will be
the averaged Rosseland opa13ity whi13h will be dened in due 13ourse
If τ gtgt 1 the regime is referred as opti13ally thi13k while in the opposite
13ase (τ lt 1) it is opti13ally thin Flux dominated radiative sho13ks have
been 13lassied depending on the opti13al depth of upstream and downstream
regimes
An attempt of radiative sho13k 13lassi13ation has been performed by Drake
(Drake (2006)) a1313ording to the opti13al the thi13kness of the upstream and
downstream regimes resulting in the denition of four dierent regimes
namely a thin-thi13k thin-thin thi13k-thi13k thi13k-thin radiative sho13ks
Following this author a thin-thi13k type of radiative sho13k probably exists
only in theory In the 13ase of a thin-thin radiative sho13ks both upstream
and downstream regimes are opti13ally thin and the sho13k radiates away
its energy in both dire13tions This regime of radiative sho13k is present in
various astrophysi13al 13ontexts for instan13e in the internal sho13ks of stellar
jets (Hartigan 2003) It is possible to generate these type of sho13ks in the
laboratory in very low-density gases In thi13k-thi13k radiative sho13ks both
upstream and downstream zones are opti13ally thi13k This regime o1313urs
in the stellar interiors The radiation 13oming from the hot downstream
region heats the 13ooler upstream material and forms a pre13ursor The fourth
regime is the thi13k-thin radiative sho13k whi13h is 13hara13terized by an opti13ally
thi13k downstream and an opti13ally thin upstream region The radiation
from the heated downstream region passes ahead of the sho13k and 13reates a
13ooling layer in the downstream before rea13hing the nal state There is no
absorption of the radiation in the upstream region and therefore 13ontrary to
16 CHAPTER 2 RADIATIVE SHOCK WAVES
the thi13k-thi13k 13ase there is no radiative pre13ursor Sho13ks generated in the
laser experiments are usually between the thi13k-thi13k and thin-thin regimes
221 Typi13al radiative sho13k waves with a pre13ursor
The radiation from the post-sho13ked region just behind the sho13k passes
ahead of the sho13k and generates the pre13ursor However it also leads to a
radiative 13ooling layer downstream that extension depends on the opa13ity
A typi13al stru13ture of su13h sho13ks is presented in M13Clarren et al (2010)
In these 13ases the 13ompression in the 13ooling layer behind the sho13k is
mu13h higher than that in the hydrodynami13 13ase This kind of sho13k is
most 13ommonly generated in the laboratory experiments (Bozier et al 1986
Keiter et al 2002 Reighard et al 2006 Fleury et al 2002) In astrophysi13s
these sho13ks are present for instan13e in supernovae when the blast wave
emerges from the stellar interior (Ensman and Burrows 1992)
A detailed 13lassi13ation of the sho13ks presenting a radiative pre13ursor
is reported in several referen13es (Zeldovi13h Zeldovi13h and Raizer 2002
Ensman 1994 Vaytet et al 2013)
In the 13ase of sub-13riti13al sho13k the temperature just ahead of the sho13k
front is smaller than the post-sho13k temperature Su13h sho13ks are relatively
weak and hen13e the radiative ux transfer from the post-sho13k to the pre-
sho13k is small A piston moving with a 13onstant speed of 12 kms in Xenon
gas at 01 bar is able to generate su13h a sub-13riti13al radiative sho13k The
resulting mass density and ele13tron temperature proles have been presented
at 10 ns in the Fig 25a
The other regime of radiative sho13k is known as super13riti13al in whi13h
the pre-sho13k and the nal post-sho13k temperatures are the same The mass
density and ele13tron temperature proles of super13riti13al sho13k driven by a
13onstant moving piston with speed 45 kms in Xenon at 01 bar are presented
in Fig 25b
The speeds a13hieved in the PALS experiment range between sim 15 - 55
kms for pressure 13omprised between 01- 06 Our simulations indi13ate
that depending on the speeds the regimes 13over either the sub13riti13al or
super13riti13al 13ases
23 Radiative sho13k waves experiments
Radiative sho13k waves have been studied experimentally for more than a
de13ade on high-energy lasers fast magneti13 pin13h ma13hines and pulsed po-
wer fa13ilities Su13h fa13ilities are able to a13hieve strong sho13k 13onditions
(high-temperature high-pressure) over a very small volume and time Va-
rious diagnosti13s are able to re13ord the plasma 13onditions whi13h are then
13ompared with the results of numeri13al simulations The results of these
experiments are also employed to validate numeri13al 13odes some of them
23 RADIATIVE SHOCK WAVES EXPERIMENTS 17
(a)
(b)
Figure 25 Simulated ele13tron temperature (Te) ion temperature (Ti) and mass
density (ρ) proles at 10 ns of a radiative sho13k propagating with a 15 kms (a sub-
13riti13al) and 48 kms (b super13riti13al) The sho13k is propagating in Xenon at 01
bar (ρ = 54 times 10
minus4g13m
minus3 at 298 K) The initial temperature in Xenon is set to
1eV The two Ma13h numbers are respe13tively equal to 13 and 43 The radiation and
ionisation ee13ts are in13luded in the simulation whi13h is performed for two dierent
temperatures (Te 6= Ti) and a negligible thermal 13ondu13tivity (10
minus05WmK) For
this simulation the sho13k is laun13hed by a 100 microm thi13k Aluminium piston moving
at the speed of 12 kms and 45 kms for 13ase (a) and (b) respe13tively The position
x=0 13orresponds to the position of the XeAl interfa13e at time t=0
18 CHAPTER 2 RADIATIVE SHOCK WAVES
being relevant to the astrophysi13s like FLASH (Fryxell et al 2000) and
HERACLES (Gonzaacutelez Audit and Huynh 2007)
231 Laser-driven sho13k experiments
Most of laboratory experiments on radiative sho13ks have been performed
on large-s13ale laser fa13ilities (Bouquet et al 2004 Chaulagain et al 2015
Gonzaacutelez et al 2006 Reighard et al 2006 Stehleacute et al 2010 Doss et al
2009 Drake et al 2011 Diziegravere et al 2011 Stehleacute et al 2012) in noble
gases at pressures between 01 and 1 bar With laser intensities on the
target 13omprised between 1014 and 1015 W13m
2 these experiments allowed
to re13ord sho13k speeds ranging between 40 and 150 kms
Bozier et al (1986) for the rst time experimentally eviden13ed the ge-
neration of a radiative wave propagating ahead of a super13riti13al sho13k in
xenon (pressure up to 6 bars) at the CEAs Limeil laser fa13ilities (irradian13e
of 5times 10
14W13m
minus2) In this experiment the laser beam (wavelength of 106
microm duration 1 ns) of 40 J energy was fo13used on an aluminum foil a13ting
as a piston A sho13k of speed sim 50 kms was produ13ed in the gas A visible
streak 13amera was used to re13ord the pre13ursor emissivity over 10 ns
Later Keiter et al (2002) performed a similar experiment at the OMEGA
laser fa13ility (25 kJ 1ns fo13al spot diameter 600 microm) still in a planar
geometry The sho13k propagated here in a low-density SiO2 aerogel foam
(ρ sim 96 mg13m
3) In this experiment X-ray absorption spe13tros13opi13 data
were re13orded to qualitatively predi13t the temperature of the pre13ursor
In the year 2000 a new radiative sho13k experiment was performed in
Xenon at 02 bar at the LULI laser fa13ility (sim 100 J λ = 053 microm 4-
6times 10
13W13m
minus2) (Fleury et al (2002) Bouquet et al (2004)) This expe-
riment used a three-layered piston 13omposed of a thin polystyrene a13ting
as ablator (2 microm thi13k) a titanium foil (3microm) a13ting as x-ray shield and a
layer of a polyethylene foam (25 microm) as an a1313elerator Further a VISAR
(velo13ity interferometer system for any ree13tor) and a Ma13h - Zehnder in-
terferometer were used as a diagnosti13s to measure the front sho13k speed and
the pre13ursor ele13tron density Two streak 13ameras re13orded the sho13k pro-
pagation in longitudinal and transverse dire13tions This was the rst attempt
to estimate ele13tron density within the pre13ursor Their measurements indi-
13ate an ele13tron density ranging between 1018minus1020cm3 The experimental
results were in qualitative agreements with 1D numeri13al MULTI and FCI
simulations However the speed of the pre13ursor in these 1D simulations
was too large This was then explained later when multidimensional ee13ts
were in13luded in the simulations (Leygna13 et al 2006 Gonzaacutelez Audit and
Stehleacute 2009)
Complementary experiments were then performed by Gonzaacutelez et al
(2006) at the PALS laser fa13ility (60 J 035 ns 438 nm spot diameter 600microm
sim 7times 10
13W13m
minus2) in Xe gas The goal was to study the sho13ks over longer
23 RADIATIVE SHOCK WAVES EXPERIMENTS 19
times and to inspe13t the ee13ts of the walls albedo A doubled layered pis-
ton made of 10microm polystyrene and 05 microm gold was used The pre13ursor
edge was re13orded by shadowgraphy using a visible streak 13amera The ex-
perimental results were 13ompared with 2D numeri13al simulations performed
with HERACLES (Gonzaacutelez Audit and Huynh 2007) This 13omparison
allowed quantifying the albedo of the walls It also showed that the radia-
tive losses at the walls lead to a small 13urvature of the ionization front and
to a redu13tion of its longitudinal extension (Leygna13 et al 2006 Gonzaacutelez
Audit and Stehleacute 2009)
In parallel another experiment was performed at the OMEGA laser fa-
13ility (035microm spot sim 700 - 800microm sim 2times 10
15W13m
minus2) by Reighardt et
al (Reighard et al (2006) Reighard (2007)) The sho13k was imaged by an
X-ray ba13k-lighter at dierent times whi13h enabled to inspe13t for the rst
time the shape of the sho13k front At higher speeds (sim 100 kms) X-ray
radiography pointed out a 13ollapse of the post-sho13k (Reighard et al 2006)
whi13h was attributed to the radiation losses trough the sho13k front For
these high-speed 13onditions the wall heating leads to the development of se-
13ondary wall sho13ks intera13ting with the primary sho13k (Doss et al 2009)
Su13h wall sho13ks have not yet been observed at lower speeds
Subsequently an experiment by Stehleacute et al (2010) was performed at
PALS laser fa13ility with the similar laser parameters as taken in Gonzalez
et al (2006) In this experiment a Ma13h-Zehnder interferometer and a
streak 13amera were used to re13ord the sho13k propagation over 50 ns Time-
integrated XUV spe13tros13opi13 analysis between 16 and 22 nm of sho13k was
performed The experiment showed that of a quasi-stationary regime was
rea13hed after 20 ns
The rst experimental study in Argon gas was reported in 2012 by Vis13o
et al (2012) at OMEGA fa13ility (035microm spotsim 600 microm sim 7times 10
14W13m
minus2)
The ele13tron temperatures in the radiative pre13ursor and sho13k-front were
measured to be 34 eV and 60 eV respe13tively using X-ray Thomson s13atte-
ring diagnosti13
In addition a new experiment at PALS (Jungwirth et al 2001) was
presented by Chaulagain et al (2015) using for the rst time a Zn x-ray
laser for XUV imaging at 21 nm This diagnosti13 allowed imaging both the
pre13ursor and post-sho13k stru13ture of radiative sho13ks in Xe gas at 03 bar
In this experiment where the sho13k wave did not ll the tube 2D ee13ts
be13ame pronoun13ed The experimental results were su1313essfully 13ompared
with the results of 2D ARWEN simulations (Cotelo et al 2015)
A summary of these dierent experiments and their prin13ipal diagnosti13s
is presented in table 21
20
CHAPTER2RADIATIVESHOCKWAVES
Experiment Laser fa13ility I (times 1013) λ amp τ ρ0 u Prin13ipal diagnosti13s
W13m
minus2 microm ns mg 13m
minus3
km s
minus1
Bozier et al (1986) CEAs Limeil 20-50 0351 1 10 -30 50-80 Visible streak Camera
Fleury et al (2002) LULI 5 053 05 1 60-65 Visible streak Camera
Visible interferometry
VISAR
Keiter et al (2002) OMEGA 35- 85 0351 1 5 -15
lowast
100 x-ray spe13tros13opy
Gonzaacutelez et al (2006) PALS 10-15 0438 035 1 65 Visible streak 13amera
Visible interferometry
Reighard (2007) OMEGA 70-100 035 1 6 100-140 x-ray ba13k-lighting
VISAR
Stehleacute et al (2010) PALS 10-20 1315 035 1 60 Visible Interferometry
XUV spe13trometer
Stehleacute et al (2012) PALS 10-20 1315 035 15 50-55 XUV imaging
XUV fast Si diodes
Vis13o et al (2012) OMEGA 70 035 1 196
lowastlowast
100-150 x-ray Thomson S13atte-
ring
Chaulagain et al (2015) PALS 10-20 1315 035 15 50 XUV fast Si diodes
Table 21 Summary of the radiative sho13ks experiments and out13omes in dierent laser installations All the experiments are performed
in Xe ex13ept Keiter et al (2002) and Vis13o et al (2012) In these two experiments the rst uses SiO
lowast
2 and the se13ond uses Ar
lowastlowast
as a
sho13ked medium
23 RADIATIVE SHOCK WAVES EXPERIMENTS 21
Laser driven radiative sho13ks Goal of my thesis
All these experimental studies have been fo13used on the 13ase of single ra-
diative sho13ks propagating in a tube However in astrophysi13al 13onditions
radiative sho13ks often intera13t with a denser medium leading to the deve-
lopment of ree13ted and transmitted sho13ks A few representative examples
of su13h phenomena are the intera13tion of supernovae remnants with dense
mole13ular 13louds (Gonzaacutelez Audit and Stehleacute 2009) the a1313retion sho13ks
on the photosphere of T-Tauri stars (Orlando et al 2013) and the bow
sho13ks at the head of stellar jets (Hartigan 1989 Raga et al 1999) The
13ollision (or the intera13tion) of two radiative sho13k waves is obviously a rare
astrophysi13al event and the template 13ase of supernova remnant DEM L316
(see Fig 1 of Williams et al (1997)) is still the subje13t of debates (Williams
et al 2005 Toledo-Roy et al 2009 Velarde et al 2006) as the observation
of these two dierent sho13ks 13an be also interpreted as the superposition of
two blast waves in the eld of view of the teles13ope In this regard the de-
velopment of dedi13ated laboratory experiments to the study of propagation
and intera13tion of 13ounter-propagating sho13k waves is important as a tool to
13hara13terize su13h events through their spe13i13 signatures
In this thesis I will present the results of experiments performed at the
Prague Asterix Laser System (PALS) fa13ility (Jungwirth et al (2001)) on
the study of the intera13tion of two radiative sho13k waves The experimental
obje13tives were
To generate two sho13ks of dierent speeds propagating in opposite
dire13tions in the tube lled with Xenon gas at low pressure (lt 1 bar)
with speeds 13omprised between 12 and 55 kms
To re13ord the ele13tron density by using Ma13h-Zehnder Interferome-
ter over a duration 13omprised between 50 and 200 ns This allowed
studying the ee13t of pre13ursor intera13tion on to the other one
To determine the temperature and the ion 13harge of gas using a time
and spa13e integrated XUV emission spe13tros13opy
The experimental results were interpreted with the help of one-dimensional
simulations using the Lagrangian radiation hydrodynami13 13ode HELIOS
232 Ele13tromagneti13ally laun13hed sho13k by a 13ompa13t pul-
sed power devi13e
As dis13ussed in the previous se13tion most of the plasma laboratory astrop-
hysi13s studies are mainly performed on large-s13ale laser fa13ilities addressing
pure hydrodynami13 radiative sho13ks at very high speed (50 - 150 kms)
and moderate pressure (01 - 1 bar) Complementary to laser experiments
22 CHAPTER 2 RADIATIVE SHOCK WAVES
dis13harge produ13ed plasma has also been a well-established method for ge-
nerating the strong sho13ks (Kondo et al 2008 2009ab Larour et al 2015)
Su13h devi13e presents a high exibility and a larger repetition rate whi13h al-
lows a deeper understanding of sho13k physi13s in 13onditions of astrophysi13al
interest for instan13e in the 13ontext of stellar jets
In prin13iple a 13oaxial ele13trode pair lled with gas followed by an intense
ele13tri13al surfa13e dis13harge generates a thin layer of plasma (plasma sheath)
at the bottom of the ele13trode Due to the high rising 13urrent in the plasma
layer a magneti13 pressure is produ13ed whi13h a1313elerates the plasma layer
also termed as plasma fo13us (Gonzalez Brollo and Clausse 2009) During
this motion the plasma sheath a1313retes a noti13eable part of the ba13kground
gas Experiments performed by Serban and Lee (1995) in the light gasses
H2 D2 have shown fast axial sho13k waves (100 kms) but the plasma sheath
was annular
Later on 13ompa13t pulsed power ele13tri13 generators were able to laun13h
astrophysi13ally relevant strong sho13ks in low-pressure noble gases (Kondo
et al 2006) These authors reported sho13k of speed 45 kms using 13oni13al
13oaxial ele13trodes and Xenon gas (006 - 11 mbar) The ion - ele13tron energy
transfer and ionization relaxation pro13ess as also the radiation ee13t were
investigated by Kondo et al (2008 2009ab)
My work on ele13tromagneti13ally laun13hed sho13k experiment
My work fo13using on the ele13tromagneti13ally laun13hed strong sho13ks rele-
vant for astrophysi13s aimed to design and to optimize an ele13tromagneti13
generator to be able to produ13e strong sho13ks in noble gasses at low pres-
sures of few mbar The ele13tromagneti13 generator is a powerful a1313elerating
devi13e whi13h eje13ts a quasi-planar plasma sheath out of a set of 13oaxial 13o-
ni13al ele13trodes where a pulsed 100-kA 13urrent is passing
Due to the ele13tri13al dis13harge in the gas a hot and fast moving plasma
is generated whi13h travels along the ele13trodes The high-intensity 13urrents
produ13e a magneti13 eld of several Teslas whi13h a13ts as a piston a1313elerating
an annular plasma sheath and driving a strong sho13k towards the top of the
13one I have employed a simple model to optimize the operation parame-
ters With su13h optimization our ele13tromagneti13 generator should produ13e
strong sho13ks over few nanose13onds
Our preliminary experiments show that the generator is 13apable of laun-
13hing supersoni13 sho13ks in Argon in the form of a thin plasma layer with
the speed of sim 1 - 30 kms Three-dimensional MHD simulation performed
by A Ciardi is 13onsistent with the rst observations This is presented in
detail in the 13hapter 6 of the thesis
Chapter 3
1D Simulations
Contents
31 HELIOS 23
32 LTE approximation 24
321 Mean opa13ity 25
322 Equation of State 27
33 Single radiative sho13k waves 27
34 Intera13ting radiative sho13k waves 40
35 Summary 44
Radiative sho13ks are present in various astrophysi13al pro13esses implying
extreme 13onditions Laboratory experiments then allow investigating the un-
derlying physi13al pro13esses whi13h take pla13e in these 13onditions and whi13h
are not observable from the Earth due to a la13k of spatial resolution of the te-
les13opes instruments However experiments are not always straightforward
to interpret and the help of numeri13al simulations be13omes pre13ious
In this 13hapter I will present with the help of 1D radiative hydrodynami13
simulations a brief overview of the physi13s of isolated radiative sho13ks and
of two intera13ting radiative 13ounter-propagating sho13k waves
31 HELIOS
Dierent numeri13al approa13hes are used for the simulation of laboratory
plasmas Some of them use xed grids Other introdu13e the possibility of
renement (on the y) in some meshes (Adaptative Mesh Renement) to
maintain a good resolution in the interesting parts of the plasma Another
approa13h is based on meshes following the uid (Lagrangian des13ription)
whi13h means that the meshes move with the uid and no matter may go
through a mesh to another The form of the equations dier in this 13ase (see
for instan13e Orban et al (2013))
HELIOS is su13h a one-dimensional Lagrangian radiation hydrodynami13
13ommer13ial 13ode (Ma13Farlane Golovkin and Woodru 2006) and I have
used it to simulate our experiment of intera13ting sho13k waves This 13ode
is easy to handle and has the presently interesting 13apability to be able to
simulate the 13ase of two sho13k waves propagating in opposite dire13tions and
laun13hed by two dierent lasers HELIOS may des13ribe non-LTE plasmas as
23
24 CHAPTER 3 1D SIMULATIONS
it in13ludes an in-line 13ollisional radiative (HELIOS-CR) model for 13omputing
non-LTE atomi13 level populations at ea13h time step of the hydrodynami13s
simulation However I used HELIOS in the LTE approximation whi13h is
appropriate to our experimental 13onditions (Rodriguez et al 2011) as will
be dis13ussed in the next se13tion
The 13ode solves the uid equations of motion using the pressure 13ontri-
butions of ele13trons ions and radiation It may des13ribe the ele13trons and
ions as two intera13ting uids at respe13tive temperatures Te and Ti Ther-
mal 13ondu13tion is des13ribed within a ux-limited ele13tron 13ondu13tion model
based on the Spitzer 13ondu13tivity (Burhop and Spitzer 1957) and the laser
energy deposition is 13omputed with an inverse Bremsstrahlung model ()
The radiation emission and absorption terms are introdu13ed in the equa-
tion of energy for the ele13trons and in the radiation transport equations
One of the methods whi13h are proposed is the ux-limited multi-group
radiation diusion model (Gonzaacutelez et al 2015) where the radiative ux is
proportional to the gradient of the radiative energy and is inversely propor-
tional to the Rosseland opa13ity (Dobbs-Dixon Cumming and Lin 2010)
The expression is pondered by a ux-limited diusion 13oe13ient in order
to obtain the good opti13ally thin limit This 13oe13ient follows the Larsen
expression des13ribed in Olson Auer and Hall (2000) The se13ond method is
a (time independent) multi-angle short-13hara13teristi13s s13heme In our 13ase
where radiation and hydrodynami13s are strongly 13oupled and thus the ra-
diation eld varies rapidly we used the diusion model together with LTE
Plan13k and Rosseland multi-groups opa13ities
In addition our version of HELIOS uses the PROPACEOS (Ma13Farlane
Golovkin and Woodru 2006) Equation of State (EOS) and multi-group
opa13ity tables whi13h are generated by the 13ode
1
and it is not possible to
in13lude any other opa13ity or EOS table
32 Lo13al Thermodynami13 Equilibrium
The knowledge of the thermodynami13al 13onditions is required for an ade-
quate simulation of the plasma des13ription
The Lo13al Thermodynami13 Equilibrium (LTE) regime is valid at large
ele13tron densities In that 13ase the 13ollisions between the ele13trons and the
ions and atoms are so frequent that they maintain at ea13h time a steady-
state populations of ions Moreover within a given ioni13 stage the popu-
lations of the dierent energy levels follow the Boltzmann equilibrium and
the populations of the dierent ions the Saha-Boltzmann equation (Fadeyev
and Gillet 2001)
On the 13ontrary in the 13ase of Non Lo13al Thermodynami13 Equilibrium
(NLTE) regime one has to take into a1313ount the ex13itationdeex13itation
1
see http wwwprismminus cscomSoftwarePROPACEOS
32 LTE APPROXIMATION 25
indu13ed by the ele13tron 13ollisions and various radiative pro13esses This leads
to a set of dierential time dependent equations (rate equations) whi13h has
to be solved through and adequate Collisional Radiative model (CR) (Kunze
2009)
Therefore it is 13ru13ial to determine if the regime is LTE or NLTE A
13riterium is given by Griem (2005)
In our experiment the mass density ranges between sim 10
minus4and 10
minus1
g13m
3 while the temperature values are 13omprised between andsim 0 to 50 eV
The pre13ise 13al13ulations performed by Rodriguez et al (Fig2 of Rodriguez
et al (2011)) for Xenon indi13ate that the thermodynami13al 13onditions of
our radiative sho13k experiments 13orrespond to the LTE regime
321 Mean opa13ity
The opa13ity is a fundamental ingredient of the radiative transfer equations
whi13h provide the spe13i13 intensity and its moments (radiative ux energy
and pressure) These last ones enter in the uid equations (see se13tion 213
in 13hapter 2) The 13oupling of the uid equations with the radiative transfer
equation is demanding in terms of 13omputational resour13es and most of the
13odes use simplied radiation transport equations whi13h require the know-
ledge of adequate frequen13y averaged opa13ity 13oe13ients The form of the
average of the opa13ity is not obvious as the average has to be done on the
radiative ux or energy whi13h are unknown before the 13omputation Then
the 13ommonly used opa13ities are the Plan13k and Rosseland mean opa13ities
(Mihalas 1978 Seaton 1987)
The Rosseland mean opa13ity (κR in 13m
minus1and χR = κRρ in 13m
2g) isderived by 13onsidering harmoni13 weighting over the temperature derivative
of the Plan13k fun13tion It gives the 13orre13t radiative ux in the opti13ally
thi13k diusion limit The average is performed over the total mono13hromati13
opa13ity κtotν = κabsν +κscatν (in 13m
minus1) where in the present 13ase the s13attering
13ontribution 13omes from the ele13tron Thomson s13attering
1
κR=
int
infin
01
κtotν
dBν
dT dνint
infin
0dBν
dT dν(31)
The Plan13k mean opa13ity (κP in 13m
minus1and χP = κP ρ in 13m
2g) employs
the normalized Plan13k bla13kbody energy density distribution as a weighting
fa13tor It will give the 13orre13t value for the integrated thermal LTE emission
of an opti13ally thin plasma and is expressed as
κP =
int
infin
0 κabsν Bνdνint
infin
0 Bνdν(32)
In the multi-groups des13ription the radiation transport takes into a13-
13ount the propagation of radiation in N dierent wavelength bands (νk minusνk+1 k = 1 N) Hen13e for instan13e the medium may be transparent for
26 CHAPTER 3 1D SIMULATIONS
some wavelengths and opaque for other ones The radiation transport equa-
tions split into N 13ontributions ea13h of them being asso13iated to adequate
opa13ity 13oe13ients The expression of the 13oe13ients is similar to the pre-
vious grey 13ase (N =1) ex13ept that the integrals are performed between
the boundary frequen13ies of the group
In HELIOS the mono13hromati13 opa13ities κν are 13omputed for the die-
rent radiation groups using the ATBASE atomi13 suite of 13odes Energy levels
and other atomi13 data relevant for bound-bound and bound-free transitions
are 13omputed within a 13onguration intera13tion model using Hartree-Fo13k
wave-fun13tions (Ma13Farlane Golovkin and Woodru 2006)
Dividing the opa13ity by the mass density redu13es the variation of this
quantity with ρ The literature then often uses the mass absorption 13oe-
13ient (also termed as opa13ity) whi13h is dened as
χν = κνρ (33)
Figure 31 PROPACEOS Mono13hromati13 opa13ity versus the photon energy in
eV of Xenon at 10 eV and at two ele13tron densities respe13tively equal to 1018 and
1020 13mminus3
An example of the variations of the PROPACEOS opa13ity χν versus the
frequen13y in eV is given in Fig 310a for Xenon at 10 eV and two ele13tron
densities respe13tively equal to 1018 and 1020 13m
minus3 whi13h do 13orrespond to
typi13al 13onditions of the pre13ursor and post-sho13k of our laser generated
radiative sho13ks
33 SINGLE RADIATIVE SHOCK WAVES 27
322 Equation of State
The equation of state (EOS) denes the dependen13e of the pressure ionisa-
tion internal energy with the mass density and temperature Several models
do exist in the literature
Our version of HELIOS for LTE plasmas may handle polytropi13 or PRO-
PACEOS equation of state Unfortunately there are not many details about
this last method The authors of the 13ode mention that it uses a QEOS-type
model (More et al 1988) in the strong 13oupling regime (ie at relatively
high densities and low temperatures) and an isolated atom model whi13h
uses a detailed atomi13 stru13ture modeling in the weak 13oupling region The
properties between the weak and strong 13oupling regimes are said to be obtai-
ned by interpolating in a thermodynami13ally 13onsistent manner This EOS
in13ludes the 13ontributions from the translation of ions and ele13trons ele13-
tron degenera13y atom ionization and ex13itation and Coulomb intera13tions
(Debye - Hu13kel 13orre13tion)
Comparisons on spe13i13 13ases have been performed by the authors of the
13ode with Los Alamos SESAME data for CH Al and Au showing a good
agreement when using these two equations of state
In this 13hapter HELIOS 13ode will then be employed to simulate the
detailed 13hara13teristi13s of single and 13ounter-propagating radiative sho13ks
driven by one or two lasers in the 13onditions of our experiments (Chapter 5)
The target 13ell (4-mm length) is lled with Xenon gas at 01 bar This valueof pressure is representative and 13orrespond to standard ambient tempera-
ture ie 298 K and not the initial temperature 13onsidered in the simulations
This terminology is maintained throughout the thesis Two lasers are inte-
ra13ting at both ends with two gold-13oated `CH foils 11microm CH and 06 micromAu) ea13h of them resulting in a piston of total thi13kness 116 microm 13losing the
13ell
33 Single radiative sho13k waves
Typi13al simulations of the radiative sho13ks espe13ially multidimensional ones
whi13h are time-13onsuming use a frequen13y independent (grey or one group)
opa13ities However then the opa13ity of the gas under investigation presents
strong variations with the frequen13y a multi-group approa13h may be suitable
(Vaytet et al 2011 2013) When the number N of groups tends to innity
ea13h group opa13ity κkR and κkP tends to the lo13al mono13hromati13 opa13ity κν whi13h solves the question of the adequate opa13ity average
In the following I will investigate the inuen13e of the opa13ity and the
number of groups on the stru13ture of a sho13k wave propagating at sim 50 kms
in Xenon at 01 bar To this purpose I have performed HELIOS simulations
for a laser of uen13e 28000 Jcm2whi13h is able to drive a sho13k with the
relevant velo13ity In addition I have also analyzed the ee13ts of taking into
28 CHAPTER 3 1D SIMULATIONS
a1313ount the dierent ion and ele13tron temperatures
Referen13e simulation (1 group Te = Ti = T )
A result of the simulation is reported at 10 ns in the Fig 32 whi13h reports
the variations of ele13tron density (Ne) temperature (T ) mean 13harge (Z)and mass density (ρ) The initial 13onditions are 45 times 10
minus4g13m
3and T =
1 eV
The sho13k front is lo13ated at sim 0047 13m (speed of 47 kms) At this
position the temperature peaks at 22 eV The bla13k verti13al dotted line shows
the position of the interfa13e between the piston and the ba13king Xenon gas
This interfa13e is lo13ated at sim 20 microm from to sho13k front The post-sho13k gas
mass density pressure ele13tron density and ion 13harge are respe13tively equal
to 002 g13m
3 11times 10
4bars 52times 10
2013m
3and 56 The high 13ompression
is due to ionisation and radiation 13ooling (13hapter 2) Its temperature of 12
eV is the same than in the pre13ursor where it is 13onstant over the 2 mm of the
simulation The pre13ursor ele13tron density and mean 13harge are respe13tively
equal to 18 times 10
1913m
minus3and 7 The pre13ursor is not 13ompressed
The variations of the Rosseland opa13ities (13m
2g
minus1and 13m
minus1) with the
distan13e are reported in the Fig 33 The Rosseland opa13ity is low in the
pre13ursor Hen13e 600 13m
2g
minus113orresponds to 05 13m
minus1 whi13h means that
an opti13al depth of 1 is rea13hed at 2 13m from the sho13k front This explains
the at prole of the pre13ursor With a typi13al value in the post-sho13k of
2577 in 13m
2g
minus1 this opa13ity is now 53 13m
minus1 and an opti13al depth of 1 is
here rea13hed at 200 microm from the sho13k front (whi13h is mu13h smaller than
the post sho13k extension)
This simulation was performed using the Spitzer thermal 13ondu13tivity
To 13he13k the 13ondu13tivity ee13t I performed another simulation with a
negligible thermal 13ondu13tivity (ie 10
minus12WmK) for the Xenon layers
only The results are reported in the Fig 34 The interfa13e between the
piston and the ba13king Xenon gas is still 13lose to the sho13k front Ex13ept
for the peak of temperature whi13h is higher (sim 29 eV instead of 22 eV) and
thiner the sho13k stru13ture is similar to the previous 13ase
Two uids with dierent temperatures Te and Ti (1 group)
In the previous se13tion HELIOS simulations have been 13arried out for the
13ase of equal ion and ele13tron temperatures I will now investigate the ee13t
of su13h approximation on the simulated results employing 1) Spitzer and 2)
negligible 13onstant thermal 13ondu13tivity in the Xenon layers It is important
to note that this two uids approximation 13an not be restri13ted to the Xenon
layers only and thus it is valid from the piston (CHAu) up to the Xenon
gas Thus the dynami13s and 13onditions of the piston are also modied whi13h
may slightly modify the sho13k velo13ity
33 SINGLE RADIATIVE SHOCK WAVES 29
(a)
(b)
Figure 32 Mass density and temperature (a) ele13tron density and mean 13harge
(b) at 10 ns for a radiative sho13k of speed sim 47 kms in Xenon at 01 bar The
verti13al dotted bla13k lines show the position of the interfa13e between piston and
ba13king Xenon gas Zero at x-axis is the interfa13e of piston and Xenon at time zero
Spitzer thermal 13ondu13tivity has been used in the simulation
30 CHAPTER 3 1D SIMULATIONS
Figure 33 Mean Rosseland opa13ity (in 13m
2g
minus1) and opa13ity (13m
minus1) at 10 ns
for a radiative sho13k of speed sim 47 kms in Xenon at 01 bar The verti13al dotted
bla13k lines show the position of the interfa13e between piston and ba13king Xenon gas
Zero at x-axis is the interfa13e of piston and Xenon at time zero Spitzer thermal
13ondu13tivity has been used in the simulation
33 SINGLE RADIATIVE SHOCK WAVES 31
(a)
(b)
Figure 34 (a) Mass density and temperature (b) ele13tron density and mean
13harge at 10 ns for a radiative sho13k of speed sim 47 kms in Xenon at 01 bar The
verti13al dotted bla13k lines show the position of the interfa13e between piston and
ba13king Xenon gas Zero at x-axis is the interfa13e of piston and Xenon at time zero
A negligible 13onstant thermal 13ondu13tivity (10
minus12WmK) have been used in the
simulation for Xenon
32 CHAPTER 3 1D SIMULATIONS
(a)
(b)
Figure 35 (a) Ele13tron density and ele13tron temperature (b) Mass density and
mean 13harge at 10 ns for a radiative sho13k of speed sim 48 kms in Xenon at 01
bar for the 13ase when the ele13tron and ion temperature are dierent The verti13al
dotted bla13k lines show the position of the interfa13e between piston and ba13king
Xenon gas Zero at x-axis 13orresponds to the position of the interfa13e between the
piston and Xenon at time zero The Spitzer thermal 13ondu13tivity has been used in
this simulation
33 SINGLE RADIATIVE SHOCK WAVES 33
The Fig 35a reports the sho13k stru13ture at 10 ns for the 13ase of Spitzer
thermal 13ondu13tivity The sho13k is lo13ated at 0048 13m The ion temperature
peaks at 53 eV and the ele13tron temperature peaks at the same value of 20
eV than previously Ex13ept for the ion temperature the sho13k stru13ture is
not signi13antly 13hanged 13ompared to the 13ase of equal temperatures
Let us now inspe13t the role played by the thermal 13ondu13tivity (see Fig
36) In the 13ase of a negligible thermal 13ondu13tivity of (10
minus12WmK) for
the Xenon layers only the ele13tron temperature (and as a 13onsequen13e the
mean 13harge) peaks at higher value 25 eV (instead of 20 eV with the Spitzer
13ondu13tivity) and the ion temperature rea13hes to 405 eV (instead of 53 eV)
The rest of the sho13k is not ae13ted by this 13hange in the 13ondu13tivity This
13onrms that the width and height of the ion peak temperature are a dire13t
13onsequen13e of the thermal 13ondu13tivity
Several groups for the opa13ity one temperature
As mentioned earlier the number of frequen13y groups inuen13es the dyn-
ami13s and morphology of radiative sho13ks A set of simulations has been
performed 13onsidering (only for Xenon) dierent numbers (N) of frequen13y
group varying between 1 and 100 for Te = Ti The distribution in energy
of the groups is logarithmi13 and the grid is set up with approximately 85
of the groups having photon energies between 01 eV and 3 KeV while the
remaining 15 lie between 3 KeV and 1 MeV
The Figures 37 (a) and (b) show the variations of the ele13tron tempe-
rature with the number of groups (from 1 to 100) for the pre13ursor (a) and
the front (b) In13reasing the number of groups ae13ts mostly the pre13ursor
A similar behavior is found in Vaytet et al (2013)
It should be noted that this multigroup opa13ity treatment is applied
to all the layers (ex13ept gold only one group opa13ity is provided by the
13ompany) in the simulation This 13ould ae13t slightly the dynami13s of the
piston Con13erning the pre13ursor at this time of the simulation and taking
an a1313ountable variation in temperature of 10 the 13onvergen13e is a13hieved
after 20 groups
Variation with the opa13ity
In order to estimate the ee13t of the opa13ity I have performed several simu-
lations by in13reasing the Plan13k and Rosseland opa13ities in Xenon (Stehleacute
et al 2014b) using a 13ommon multiplier ranging between 1 to 40 Su13h
pro13edure is often used to redu13e the gap between the available and more
sophisti13ated opa13ities To ex13lude the impa13ts of the radiation 13oming from
the post-sho13k to the piston layers the opa13ity of the piston (CH and Au)
layers have been set to zero
I present in Fig 38 the results (proles of temperature mass density and
34 CHAPTER 3 1D SIMULATIONS
(a)
(b)
Figure 36 (a) Mass density and ele13tron temperature (b) ele13tron density and
mean 13harge at 10 ns for a radiative sho13k of speed sim 45 kms in Xenon at 01 bar
for the 13ase when the ele13tron and ion temperature are dierent The verti13al dotted
bla13k lines show the position of the interfa13e between piston and ba13king Xenon
gas Zero at x-axis 13orresponds to the position of the interfa13e between the piston
and Xenon at time zero In this simulation value of thermal 13ondu13tivity (10
minus12
WmK) is negligible for Xenon The peak value of ion temperature is found to be
405 eV whereas theoreti13ally this value is expe13ted to be sim 600 eV It is possible
to a13hieve the expe13ted value by in13reasing the resolution of the simulation
33 SINGLE RADIATIVE SHOCK WAVES 35
Figure 37 Ele13tron temperature proles for various numbers of frequen13y groups
N = 1 20 50 60 70 90 and 100
mean 13harge) of four 13ases 13orresponding to an opa13ity multiplier equal to
1 10 30 and 40 The ele13tron and ion temperatures are taken to be equal
In13reasing the opa13ity in13reases the absorption in the pre13ursor and thus
de13reases its length as the photons emitted by the sho13k are more rapidly
absorbed by the 13old pre-sho13k Taking as a referen13e the previous value of
the Rosseland opa13ity of 600 13m
2g
minus1and a multipli13ative fa13tor of 40 the
opti13al depth of 1 will now be rea13hed at 005 13m (instead of 2 13m without
any multiplying fa13tor)
The temperature peak value at the sho13k front also de13reases I do believe
that this 13an be attributed to the in13rease of the radiative 13ooling of the peak
of temperature with the multiplying fa13tor
I adopted here an oversimplied method in the sele13tion of the groups
boundaries A more realisti13 one would be to take into a1313ount the variation
of the opa13ity in the dierent layers with the wavelength In the 13ase of
xenon for instan13e it would be then more adapted to rene the des13ription
of the opa13ity between 5 and 100 eV (see Figure 310a)
Opa13ities 13omparison
The previous study illustrates the role played by the opa13ity for the radi-
ative sho13ks waves with a pre13ursor It seams then logi13al to 13ompare the
36 CHAPTER 3 1D SIMULATIONS
(a)
(b)
(13)
Figure 38 (a) Ele13tron temperature (a) mean 13harge (b) and mass density (13)
for four (1 10 30 and 40) multipliers of the Xenon opa13ity at 10 ns
33 SINGLE RADIATIVE SHOCK WAVES 37
PROPACEOS opa13ity whi13h is used by HELIOS with more sophisti13ated
13al13ulations
The Figures 39a and 39b report the variations of the Plan13k and Ros-
seland grey opa13ities 13omputed for the density of the pre13ursor ρ = 15 times10
minus3g13m
3by two dierent 13odes PROPACEOS (Ma13Farlane Golovkin
and Woodru 2006) and BiGBART (de la Varga et al 2011 Ogando and
Velarde 2001) PROPACEOS opa13ities and EOS used in HELIOS simula-
tions 13omputes frequen13y-dependent opa13ities along with spe13i13 internal
energies and pressures The atomi13 13ode BiGBART able to 13al13ulate two
type of opa13ities dependent on atomi13 stru13ture and frequen13y dependent
The opa13ities 13omputed by Rodriacuteguez et al (2015) are also reported for 13om-
parison The two last 13al13ulations use the FAC (Flexible Atomi13 Code) (Gu
2008) for the 13al13ulation of the atomi13 data
We see that for a temperature equal to 10 eV the Rosseland opa13ity is
equal respe13tively to 800 15000 and 8000 g13m
2 whereas the Plan13k opa13ity
is equal to 18000 65000 370000 g13m
2 This means that the PROPACEOS
opa13ity whi13h is used in HELIOS is smaller by more than one order of
magnitude than the two last opa13ities As it is not possible to 13hange the
opa13ity model within HELIOS we then de13ided to use the PROPACEOS
opa13ity with a multipli13ative fa13tor whi13h we have set equal to 20
The dis13repan13ies in the average opa13ity are a 13onsequen13e of important
dieren13es in the mono13hromati13 opa13ities This is illustrated in the Fig
310 whi13h 13ompares the mono13hromati13 opa13ities given by PROPACEOS
and Rodriacuteguez et al (2015) (see Fig 16 of Rodriacuteguez et al (2015)) at 15
eV and 15 times 10
minus3g13m
minus3) This gure indi13ates a noti13eable dieren13e
between 5 and 150 eV with more bb transitions for the last method This
explains the dieren13es observed in the 13orresponding averages
Synthesis
These dierent simulations for a sho13k propagating at sim 50 kms in Xenon
at 01 bar show that the simulations performed with only one temperature
are adequate for the des13ription of the sho13k
The opa13ity strongly ae13ts the pre13ursor extension whi13h de13reases from
more than 4 mm for 1 group to 3 mm then the number of groups is larger
than 20 The 13onvergen13e versus the number of groups is rea13hed after 20
groups
To t with more sophisti13ated opa13ities and as it was impossible to
in13lude them in HELIOS I de13ided to multiply the opa13ities of Xenon by a
fa13tor of 20 This redu13es strongly the pre13ursor extension up to sim 1 mm
at 10 ns Taking into a1313ount the present huge un13ertainty in the opa13ity
I 13onsidered that it was not ne13essary to use multi-group opa13ities This
also means that our 1D simulations will be used - only - for a qualitative
interpretation of the experimental results A quantitative 13omparison should
38 CHAPTER 3 1D SIMULATIONS
(a)
(b)
Figure 39 Plan13k and Rosseland mono13hromati13 opa13ity proles Te for ρ =15
times 10
minus3from PROPACEOS (I) BIGBART (II) and Rodriguez et al (III)
33 SINGLE RADIATIVE SHOCK WAVES 39
(a)
(b)
Figure 310 Mono13hromati13 Xenon opa13ity versus the photon energy in eV at T
= 15 eV and for ρ = 15 times 10
minus3g13m
minus3for the PROPACEOS (a) and Rodriacuteguez
et al (2015) (Fig 16 of Rodriacuteguez et al (2015)) (b) models
40 CHAPTER 3 1D SIMULATIONS
be in the future based on adequate opa13ities and multi-groups modeling
34 Intera13ting radiative sho13k waves
In order to pre13ise what is expe13ted from the experimental study I will now
investigate the 13hara13teristi13 parameters of two 13ounter-streaming sho13ks
propagating in Xenon at 01 bar (ρ = 54 times 10
minus4g13m
minus3) with equal and
dierent velo13ities To highlight the ee13t of the intera13tion I will 13ompare
the results with those obtained with 1D simulations All the simulations will
be performed with HELIOS For the Xenon opa13ity a multiplier times 20 will
be used The number of groups will be set to be 1 and the ele13tron and ion
temperatures will be supposed to be equal
Three representative sets will be 13onsidered
Case(I) two identi13al radiative sho13ks (2RS) at sim 50 kms propaga-
ting in opposite dire13tions (ie starting from the left and right end
respe13tively)
Case(II) same 13onditions but without any 13oupling with radiation
Case(III) two 13ounter-propagating radiative sho13ks propagating with
dierent speeds sim 50 kms from the left side of the 13ell and sim 20
kms from the right side
To a13hieve the aforementioned speeds on the left and right sides the
uen13es of the two laser beams have been adjusted The pulse duration is
set to 03 ns (peak at 015 ns) to reprodu13e the experimental 13onditions
detailed later in the manus13ript
Case(I) 13ounter propagating sho13ks with the same speed
This a13ademi13al 13ase is fully symmetri13al and it is equivalent to the 13ase of
one RS with a fully ree13tive boundary (for hydrodynami13s and radiation)
in the middle of the tube The Fig 311 shows the variations of the ele13tron
density (Ne) and temperature (Te) in the Xenon layers The two sho13ks
appear in Xenon at sim 2 ns and the 13ollision o1313urs at sim 38 ns At 3 ns the
pre13ursor extension is sim 008 13m whereas the post-sho13k ele13tron density
and ele13tron temperature are 78 times 10
2013m
minus3and 16 eV respe13tively The
length of pre13ursor in13reases rapidly with time and the two pre13ursors merge
suddenly at sim 8 ns After this time the merging ee13t in13reases signi13antly
It is 13hara13terized by a at 13ommon pre13ursor those ele13tron density and
temperature are in13reasing with time At the time of the 13ollision (sim 38
ns) the post-sho13k mass and ele13tron density in13rease from 0011 to 014
g13m
minus3and 67 times 10
20to 66 times 10
2113m
minus3 whereas the temperature rises
up to 39 eV The 13ollision leads to the development of two reverse sho13k
34 INTERACTING RADIATIVE SHOCK WAVES 41
(a)
(b)
Figure 311 Ele13tron density Ne (a) and ele13tron temperature Te (b) versus axial
position (along a 04 13m long sho13k tube) at 3 10 20 30 and 38 ns from HELIOS
simulations (with opa13ity times 20) for the 13ases of single sho13k of sim 50 kms (dotted
line) and two identi13al 13ounter-propagating sho13ks of sim 50 kms (solid lines) The
verti13al dotted lines show the position of the interfa13e between piston and ba13king
Xenon gas
42 CHAPTER 3 1D SIMULATIONS
waves propagating ba13k with a speed of sim 15 kms rst in Xenon and later
on in the dierent piston layers (not shown in the Figure) These reverse
sho13ks lead to a dense plasma (Ne gt 10
2113m
minus3) whi13h is not a1313essible to
the experimental diagnosti13s and will not be detailed here
To investigate the ee13ts of the intera13tion I have reported in Fig 311
the 13ase of 1RS moving from the left to the right dire13tion in the 13ell (dotted
lines) The wave propagates identi13ally to 2RS until 10 ns After this time
the proles of the temperature and ele13tron density dier strongly from the
previous 13ase and their values are lower than for the 2RS The post-sho13k
extension is slightly smaller than for the 2RS and the sho13k speed is thus
slightly smaller too This last ee13t is due to the fa13t that for the two inte-
ra13ting waves the pre13ursor is at a higher temperature than for the isolated
sho13k and thus the sho13k propagates in a warmer medium then modifying
the sho13k 13onditions (sound speed opa13ity et13 ) A similar ee13t in the post
- sho13k extension may be seen for an isolated sho13k wave propagating in a
warmer pre-sho13k gas
Case (II) Identi13al 13ounter-propagating sho13ks for an ideal gas
In order to highlight the ee13t of the radiation another simulation has been
performed with the same set of parameters as in the 13ase I however putting
the Xenon opa13ity equal to zero I still use here the PROPACEOS equation
of state The result of the simulation is presented in Fig312 The 13ollision
time is now 40 ns instead of 38 ns (thus the sho13k speed is slightly smaller)
The post-sho13k is no more 13ompressed by radiation 13ooling Its 13ompression
at 10 ns is 10 instead of 35 This 13ompression is due to the vis13ous sho13k
(fa13tor 4) followed by the ionisationex13itation of the gasThere is obviously
no radiative pre13ursor Moreover there are no dieren13es in the Ne and Tproles of the single sho13k and that of the two 13ounter-propagating sho13ks
before the 13ollision time
Case (III) Two sho13ks at dierent speeds
The spatial and temporal variations of Ne and Te are plotted at times 3
10 20 30 38 and 49 ns in Fig 313a and 313b The left and the right
sho13ks appear in Xenon at sim 2 and 3 ns respe13tively Later at 10 ns
the two pre13ursor extensions are respe13tively equal to 018 and 0034 13m
The merging of the two pre13ursors starts at sim 15 ns As expe13ted from the
values of sho13k speeds the 13ollision time o1313urs at 49 ns whi13h is delayed
in 13omparison to the 13ase (I)
In 2RS the two radiative pre13ursors merging results in a plateau for the
ele13tron density and the temperature The temperature at 13ollision time is
now 28 eV instead of 39 eV in the 13ase I and the ele13tron density rea13hes
up to 31 times 10
2113m
minus3instead of 66 times 10
2113m
minus3
34 INTERACTING RADIATIVE SHOCK WAVES 43
(a)
(b)
Figure 312 Variations of Ne (a) and Te (b) versus axial position for the 13ase
of two identi13al 13ounter-propagating sho13ks (of speeds sim 50 kms) at 3 10 20
30 35 and 40 ns as derived from HELIOS simulations For these simulations we
have negle13ted the ee13t of radiation by keeping the Xe opa13ity equal to zero
The verti13al dotted lines show the position of the interfa13e between the piston and
ba13king Xenon gas
44 CHAPTER 3 1D SIMULATIONS
It may be noted that up to this 13ollision time the post-sho13k density
13onditions and speed are identi13al for the 13ase of an isolated RS propagating
from the left (1RSL) at 50 kms and this present left sho13k This reveals
that in the 13ase of 2RS there is no noti13eable ee13t of the right sho13k with
speed 20 kms on the left post-sho13k of 50 kms
On the 13ontrary we note a dieren13e in extension of the right post sho13k
from the 2RS 13ompared with the 13ase of an isolated sho13k (1RSR) propa-
gating at 20 kms from the right whi13h indi13ates that the left sho13k of the
2RS has an inuen13e on the right post-sho13k (Fig 313a)
35 Summary
Radiative sho13k waves propagating in Xenon at 01 bar with a velo13ity of 50
kms are 13hara13terized by the development of an extended radiative pre13ur-
sor The huge 13ompression of 38 in the post-sho13k is a 13onsequen13e of both
the sho13k and the gas ionisation (fa13tor of 10) as also from the radiative
13ooling The post-sho13k and pre-sho13k temperature on both side of the peak
are identi13al indi13ating that the sho13k is super13riti13al
The spe13ta13ular role of the opa13ity has been highlighted Due to the
un13ertainty in the opa13ity used for Xenon in our simulations I de13ided that
at this stage it was not ne13essary to rene the simulation in terms of group
numbers within the 1D des13ription of the sho13k wave
Our numeri13al study gives the main 13hara13teristi13s of the intera13tion of
two 13ounter-propagating sho13k waves with speeds equal to 50-50 kms and
50-20 kms The 13ase of identi13al speeds is simpler due to the symmetry of
the problem However whatever the speeds the most important signature
of the intera13tion is the merging of the pre13ursor at 8 ns for 50-50 kms
and at 15 ns for 50-20 kms This merging is followed by a regular in13rease
with time of the ele13tron density and the temperature The 13ollision time
is 13hara13terized by a sudden in13rease of the ele13tron density by an order of
magnitude rea13hing 66 times 10
21and 3 times 10
2113m
minus3respe13tively whereas the
temperature in13reases up to 39 and 28 eV
35 SUMMARY 45
(a)
(b)
Figure 313 Variations of Ne (a) and Te (b) with axial position for the 13ase of
two non-identi13al 13ounter-propagating sho13ks (of speeds sim 50 amp 20 kms) and two
single sho13ks (dotted lines) of speeds sim 50 amp 20 kms respe13tively at 3 10 30 35
and 46 ns as derived from HELIOS simulations The verti13al dotted lines show the
position of the interfa13e between piston and ba13king Xenon gas (with opa13ity times20)
46 CHAPTER 3 1D SIMULATIONS
Chapter 4
Laser-driven radiative sho13k
Experimental Setup
Contents
41 Prague Asterix Laser System (PALS) fa13ility 48
42 Targets 49
421 Massive Targets 51
422 Gaseous Targets 51
423 Target holder 55
424 Target lling 56
43 Laser Fo13using 57
431 Fo13al Lenses and Phase Zone Plates 57
432 Fo13using of the MAIN and AUX laser beams 58
44 Diagnosti13s 59
441 Visible interferometry 61
442 XUV spe13tros13opy 65
443 Visible spe13tros13opy 67
45 Summary 68
When a short pulse of a high-power laser beam is fo13ussed on a thin foil
it transfers a huge amount of energy to the foil material Almost instantane-
ously the laser absorption heats it up to the ablation A hot and low density
13orona is generated ba13kwards by this pro13ess Beyond the ablation surfa13e
a sho13k is formed in the foil as a 13onsequen13e of the momentum 13onservation
(ro13ket ee13t) The sho13k moves inward in the foil Our 13ase of the experi-
mental setup as the foil 13loses a tube lled with gas the sho13k propagates
then in the gas where it is studied through various kinds of diagnosti13s The
foil then a13ts as a piston
We performed the experiments at the Prague Asterix Laser System (PALS)
in the Spring 2015 (ve weeks 20th April - 22th May) The obje13tive of the
experimental 13ampaign was to study the evolution of two 13ounter streaming
radiative sho13ks driven by two 13ounter-propagating lasers with an irradian13e
in the range of 10
14W13m
minus2 and therefore to 13hara13terize their intera13tion
and 13ollision In this 13hapter I will rst present a brief des13ription of the
PALS laser fa13ility and of the laser beams used in our experiment This will
47
48 CHAPTER 4 EXPERIMENTAL SETUP
be followed by a presentation of the targets design and an overview of the
general setup and of our main diagnosti13s
41 Prague Asterix Laser System (PALS) fa13ility
The Prague Asterix Laser System (PALS) is a laser fa13ility based on an
Infrared high-power iodine laser system (Asterix IV) (Jungwirth et al 2001)
Using dierent amplifying stages the laser fa13ility is able to deliver energy
up to 1 kJ in 03 ns at the fundamental wavelength 1315 nm The output
laser beam 13an be further subdivided in few auxiliary beams All auxiliary
beams may be frequen13y doubled (λ = 657 nm) or tripled (438 nm) The
PALS laser fa13ility is 13apable of ring up to two high energy laser shots per
hour Compared with solid state lasers this gas laser is known to deliver a
quite homogenous beam intensity without hot spots A spatial prole of the
PALS laser beam is shown in the Fig 41a (re13orded by PALS team during
the experimental 13ampaign) while its pulse 13hara13teristi13s are plotted with
time in the Fig 41b
(a)
(b)
Figure 41 a Spatial prole of the PALS Laser beam b Intensity prole of
laser pulse I(t) with time
For our experiments we used two independent laser beams to drive the
two 13ounter streaming sho13ks in the gaseous target For this purpose the
fundamental PALS laser beam has been subdivided after the fourth amplier
(V4) into two beams with respe13tive energy fra13tions of 60 and 40 (see
Fig 42) The most energeti13 of these laser beams is then inje13ted through
the fth (V5) amplier and its frequen13y are then tripled This beam at 438
nm will be hereafter termed termed as the MAIN laser beam
The se13ond of the two previous laser beams propagates then without any
modi13ation It will be hereafter 13alled the AUX laser beam (1315 nm)
The s13heme of the energy distribution is shown in the Fig 42 and the
13hara13teristi13s of MAIN and AUX laser are presented in Table 41
For the experiments PALS oers two va13uum 13hambers with a spheri13al
42 TARGETS 49
Figure 42 Energy distribution s13heme for MAIN and AUX laser beams
LASER MAIN AUX
Typi13al laser energy (J) 120 60
Beam diameter (mm) 290 148
Wavelength (nm) 1315 438
Pulse duration (ps) 350 350
Table 41 Nominal 13hara13teristi13s of MAIN and AUX laser beams
and 13ylindri13al geometry respe13tively The spheri13al 13hamber assembly used
in our experiments is shown in Fig 43 This 13hamber having a diameter
of 100 13m is 13losed by two entran13e 13ir13ular windows of respe13tive diameters
80 13m and 50 13m to allow the MAIN and AUX beams to penetrate in the
13hamber and to rea13h the target Ea13h of these windows is outtted with a
hinged door Several 13ir13ular ports of various diameters also allow for the
dierent diagnosti13s The target holder and some diagnosti13s are mounted
on an opti13al ben13h whi13h is de13oupled from the 13hamber walls
42 Targets
Our targets s13hemati13ally 13onsist in small tubes of 4 mm length 13losed on
both sides by two spe13i13 foils of thi13kness 11 microm on whi13h the laser beams
are fo13used (one laser per foil) with an irradian13e of about 10
1413m
minus3 The
tube is lled with gas in whi13h the radiative sho13k propagates with a velo-
13ity of 30 - 60 kms The two foils 13losing the target insure the 13onversion
through ablation and sho13k generation of the laser energy into me13hani13al
energy
In our experiments we used spe13i13 targets for alignment and 13hara13te-
rization of the beam size These targets were manufa13tured at the Pole
instrumental of Observatoire de Paris
50 CHAPTER 4 EXPERIMENTAL SETUP
Figure 43 Snapshot of the spheri13al 13hamber showing the horizontal bredboard
and on the right the fo13using MAIN lens
(a) (b)
Figure 44 Dimension drawing (a) and snapshot (b) of Aluminum massive
(mo13k) target
42 TARGETS 51
421 Massive Targets
For the laser alignment on the target we have used spe13ial targets also
termed as mo13k targets A mo13k target (Fig 44) is an aluminum blo13k
with a base Three 13uts made on it help to fo13us the laser beam at the
desired position The 13enter of ea13h of the two laser spots should be lo13ated
at the interse13tion of the verti13al and horizontal marks and the pre13ision of
this alignment is less than 20 microm
Beside alignment purpose su13h targets are used to 13hara13terize the MAIN
and AUX spot diameters in the planes whi13h 13orrespond to the lo13ation of
the two Parylene foils Hen13e the laser beam fo13ussed on these targets
generates a 13rater (Chaulagain 2015) The detailed analysis of the shape
and size of these 13raters provides a quantitative information about the ho-
mogeneity of the in13ident beam and the size of the fo13al spot A detailed
dis13ussion about the fo13alisation of the laser with a massive target will be
presented in the se13tion 432
422 Gaseous Targets
The 13ore of the gaseous targets (Stehleacute et al 2014a) 13onsists of a 13hannel of
a parallel pipe shape having the dimension of 09 times 06 times 4 mm pla13ed at
the top of an aluminum stru13ture The pipe in whi13h the sho13ks propagate
is dened by two 500 microm thi13k verti13al lateral sides made with 13oated fused
sili13a (SiO2) windows (09 times 4 mm) as shown in the Fig 45 Su13h windows
are suitable for the visible interferometry diagnosti13 whi13h uses an auxiliary
Q-swit13hed NdYLF green Evolution laser (see Se13tion 441)
The 13hannel is 13losed at the top with a window made of a 100 nm thi13k
Si3N4 membrane whi13h is itself supported is by a 200 microm thi13k sili13on frame
(26 times 4 mm) made by SILSON 13ompany Su13h membrane is transparent
to XUV radiation whi13h is ne13essary for the XUV spe13tros13opi13 diagnosti13
used in the experiment The transmission of the aforementioned material is
obtained from the 13enter for X-ray opti13s (CXRO) database
1
and is plotted
between 0 and 40 nm in Fig 48
The pistons 13losing the tube from opposite sides and whi13h will laun13h
the two sho13ks are made of Parylene-N (11 microm) 13oated by Au (06 microm)
(SCITECH 13ompany) They are glued on 01 mm thi13k Ni13kel disks with
external diameter 5 mm and internal diameter 1 mm This disk helps in
assembling the targets It also 13ontributes together with a larger V-shield
(se13tion 49) to prevent hard X-ray emitted at the laser impa13t on the foil
to rea13h the gas in the tube The Parylene layer is fa13ing the laser and plays
the role of an ablator The gold layer whi13h is 13oated on the Parylene and
is lo13ated between the Parylene and the gas aims at blo13king the X-rays
generated by the laser plasma intera13tion to rea13h and preheat the gas in
1
httphenkelblgovopti13al_13onstants
52 CHAPTER 4 EXPERIMENTAL SETUP
(a)
(b)
Figure 45 Values on the gures are in mm (a) Verti13al 13ross se13tion view of
the gaseous target (b) Horizontal 13ross se13tion view
42 TARGETS 53
Figure 46 Pi13ture of a the gaseous target on its base
Figure 47 S13hemati13 of the gaseous target
54 CHAPTER 4 EXPERIMENTAL SETUP
Figure 48 Transmission of 100 nm thi13k Si3N4 membrane (CXRO database)
Figure 49 Snapshot of a target holder with one V-shield regarding the AUX
laser whi13h will pass through the hole in this shield marked in green in the Figure
The target whi13h fa13es the two diagnosti13s (tubes) and is lo13ated behind the
V-shield is thus not visible
42 TARGETS 55
the 13ell These various elements are manually glued on the base of the target
to built the sho13k 13hannel Ea13h target has been tested against leaks before
the experiment and before ea13h shot The axis of symmetry of the sho13k
waves is required to be lo13ated at 03 mm from the base and at 03 mm from
the two verti13al SiO2 windows S13hemati13s of verti13al and horizontal 13ross
se13tion views of the target are presented in Fig 45b The axis of symmetry
is visible in the verti13al view while the position of the laser fo13us point (`+
sign) in the horizontal view This is a13hieved during the alignment pro13edure
employing mo13k targets as dis13ussed in the se13tion 432 The base of the
target 13onsists of a mi13ro-ma13hined aluminum stru13ture A gas pipe at the
bottom of the stru13ture allows for in situ gas lling at a pressure of a fra13tion
of a bar
A total of 55 targets were manufa13tured for the experiment Among
them 31 gave valuable results
423 Target holder
We x the targets on a stainless holder and then pla13e the system into the
va13uum 13hamber The target holder has been designed and manufa13tured
at Observatoire de Paris It allows a very reprodu13ible positioning of the
target on the va13uum 13hamber breadboard The main element of the target
holder is a base where ea13h target is rst xed by s13rews and removed after
ea13h shot to be repla13ed by a new one It also in13ludes a diagnosti13s support
(Fig 46) whi13h is atta13hed on it from the top with s13rews One of these
diagnosti13s 13onsists in a fast XUV Si diode with a pinhole The diode is
atta13hed to the bla13k tube whi13h is visible on the left part the target holder
in Fig 47 This last diagnosti13 aimed for sho13k timing as the pinhole-diode
system allows to re13ord the sho13k passing at the imaged position of the tube
(Stehleacute et al 2012 Chaulagain 2015) Unfortunately this diagnosti13 did
not work well and will not be mentioned further in the thesis
The se13ond diagnosti13 is a visible time-and-spa13e-resolved visible spe13tro-
meter On the target holder this diagnosti13 13ontains a lens whi13h is lo13ated
inside the bla13k tube on the right part of Fig 49 The lens allows imaging
the sho13k on a linear bundle of 21 bers 13onne13ted to the visible spe13trometer
through a feedthrough (se13tion 443)
A snapshot of the target holder is shown in the gure 49 The red arrows
in the Fig 49 shows the distribution of the various diagnosti13s
In addition we used two 3 mm thi13k stainless steel V-shape shields on
both sides of the target whi13h were xed to the holder These V-shields have
a hole (sim25 mm) shaped entran13e to allow laser in13iden13e on the target (in
green on the Figure 49) The role of the V-shields is to prevent the target
from the extra laser or another kind of hard X-ray plasma self-emission linked
to the ablation pro13ess
56 CHAPTER 4 EXPERIMENTAL SETUP
424 Target lling
The gaseous targets were lled in situ at a fra13tion of 1 bar with a gas (viz
Xe Ar a mixture of Xe and He) whi13h thus provide the medium for the
sho13k propagation We 13hose an in-situ lling system (Fig 410) to redu13e
the dieren13e of pressures inside and outside of the tube This is 13riti13al as
a large pressure dieren13e between the target 13ell and the va13uum 13hamber
13an result in the breaking of the ultra-thin Si3N4 windows of the target
To 13ontrol the pressure we used a piezo-resistive pressure transdu13er
(Swagelok PTI-S-AC6-32) whi13h has 10 kΩ bridge resistan13e and works
with 14-30 V biasing voltage A sensitive Bourdan manometer and a gauge
allowed monitoring the pressure inside the target 13ell To read the pressure
remotely (from the 13ontrol room) until the last se13ond before the shot we
13onne13ted the system with a long 13able
The transdu13er has been operated with a +17 V DC biasing voltage The
pressure is read using an industrial pressure transdu13er from Swagelok (0 to
10 Volts -10 bar to 06 bar 05 pre13ision) and reading is made with a
voltmeter at better than 05 This system generally took approximately
one minute to stabilize A s13hemati13 of the aforementioned gas lling system
is presented in the Figure 410
The pro13edure of the target lling is presented as follows
Figure 410 In-situ system for target lling and pressure monitoring
The target is pumped during the 13hamber eva13uation
The target is isolated at the end of the pumping
The target is lled to the desired pressure
43 LASER FOCUSING 57
The target is isolated from the gas bottle to avoid any leakage in
13hamber
Then the pressure 13an be read from the 13ontrol room till the last se13ond
before the shot
The gases are taken from high grade bottles 99995 purity The Xe+He
mixture is 1007plusmn050 He molar fra13tion The lling pro13edure is pre-
venting the presen13e of impurities like air When 13losing the 13hamber and
during the primary pumping the lling 13ir13uit 13ommuni13ates with the 13ham-
ber thus the whole tubing and the target are eva13uated down to 001 mbar
After the turbo pump is a13tivated the 13onne13tion to 13hamber is 13losed and
the gas in inje13ted When the target pressure is rea13hed the 13ell is isolated
from the pressure regulator In 13ase there is an una1313eptable overpressure
it is possible to re13onne13t to 13hamber to eva13uate and to restart the lling
The pressure reading is possible in the 13ontrol room still at pre13ision better
than 1 and the value kept for the re13ords is the one just before the shot
whi13h happens normally 1-2 minutes after 13losing
43 Laser Fo13using
431 Fo13al Lenses and Phase Zone Plates
Two fo13al lenses are used to fo13us the laser beams on the target piston foil for
MAIN and AUX respe13tively The MAIN lens is lo13ated inside the va13uum
13hamber whereas the AUX lens is outside this 13hamber The spe13i13ations
of these lenses are given in the Table 42
laser Diameter (mm) fo13al length (mm) f-number
MAIN 340 564 at 3ω 19
AUX 150 1022 at ω 69
Table 42 Spe13i13ations of the MAIN and AUX fo13al lenses and the f- number
(fo13al lengthbeam diameter) is given (see table 41 for the beam diameters)
Whereas these lenses provide a smooth intensity at the fo13us this is not
su13ient to laun13h a planar sho13k wave To this purpose phase zone plates
(PZP) have been used between the output laser beam and the lens
A PZP 13omprises several phase zone elements whi13h divide the input
beam into several beamlets whi13h 13ombined with the main fo13al lens provi-
ding a uniform intensity distribution on the target with a spe13ied diameter
(Ross Pepler and Danson 1995 Bett et al 1995) The PZP are 13ommonly
used in EOS laser experiments whi13h require a uniform intensity distribution
on the target (Koenig et al 1994 Batani et al 2003)
Two PZP as dis13ussed below were used in the experiment
58 CHAPTER 4 EXPERIMENTAL SETUP
The rst one made by COLSICOAT was already used at PALS for
radiative sho13k experiments (Stehleacute et al 2010) This PZP used for
MAIN (438 nm) is made on a 13oated BK7 substrate of a diameter
of 310 mm and se13tion of 8 mm It is designed to provide a uniform
13ir13ular fo13al spot over a diameter of 05 mm
The se13ond plate made by SCITECH has been used for AUX (1315
nm) The plate is made on a 63 mm thi13k 13oated Viosil substrate
with a square 153 x 153 mm shape It is designed to provide a uniform
13ir13ular fo13al spot over a diameter of 0250 mm
Laser Spe13ied spot diameter (mm)
MAIN 05
AUX 025
Table 43 Spe13i13ations of the two Phase Zone Plates
In terms of energy distribution sim 80 of the laser energy is in the laser
spot (C Spindloe private 13omm) The energy of the laser whi13h is given
by the PALS installation should be 13orre13ted by this fa13tor of 80 For
AUX another 13orre13tion of sim 10 has to be added whi13h is linked to the
transmission of the entran13e window of the va13uum 13hamber As a result
for nominal energies of the MAIN and AUX lasers equal to 120 and 60 J (see
table 41) the uen13es on the target are equal to sim 48800 and 59500 J13m
2
respe13tively Taking into a1313ount the 03 ns laser duration of the lasers this
results in an intensity of sim 1-2 times 10
14W13m
2
432 Fo13using of the MAIN and AUX laser beams
The determinations of the fo13i of the two lenses were performed with a
Hartmann setup using a photographi13 paper and without any PZP plate
The determination of the size of the beam at the impa13t is important both to
know the initial 13onditions for the sho13k waves (laser irradian13e whi13h should
be few 10
14W13m
minus2and se13tion of the sho13k wave) The sele13tion of the
best spots size on the piston was done with mo13k targets and by translating
the two laser lenses The in13iden13e of a laser beam on a solid planar target
generates a 13rater in the foil and an ablated plume propagating ba13kward
at high temperature (Mahmood and Ur-Rehman 2009 Singh and Thakur
2007) The 13rater is the result of boiling and ablation of the material The
ablated plasma whi13h expands towards the in13ident laser beam emits in the
keV range
A standard pro13edure whi13h works well at low energy (sim 10 J) 13onsists
of the estimation of the spot size by measuring the 13rater on the bulk target
using a standard mi13ros13ope At higher energy the boiling ee13t makes
44 DIAGNOSTICS 59
this estimation impre13ise and we preferred to use a keV imaging diagnosti13s
(Chaulagain et al 2012) whi13h is based on a pinhole (25 microm for MAIN keV
and 20 microm for AUX) and a keV 13amera These two keV 13ameras have been
used during ea13h laser shot to image the impa13ts of the MAIN and AUX
laser beams on the two respe13tive pistons of the target Ea13h keV 13amera
is a CMOS dete13tor without any glass It is prote13ted by an Aluminium
lter (200 microm for MAIN and 42 microm for AUX) to blo13k the visible light and
to prote13t the ship from the debris The spe13i13ations of the 13ameras are
presented in the Table 44 and the keV 13amera system is shown in the Fig
411 The two PALS laser lenses were translated up to a13hieve the suitable
diameters on the target whi13h were nally set to 450 - 500 microm and 250 -
300 microm for the MAIN and AUX beams
The size of the impa13t was 13ontrolled ea13h day on mo13k targets before
using the laser beams for real shots on gaseous targets It is worth to pre13ise
that the keV diagnosti13 was still in pla13e for the shots on the gaseous targets
In that 13ase the beams impa13t Parylene-N instead of Aluminium The keV
signal is then weaker than for the mo13k aluminum targets
Figure 411 S13hemati13 of the keV 13amera
KeV 13amera KeV 13amera
MAIN laser AUX laser
Model UI-122xLE UI-164xLE
Resolution 752(H) x 480(V) 1280(H) x 1024(V)
Pixel size (microm x microm) 60 x 60 36 x 36
A13tive Area (mm x mm) 451 (H) x 288 (V) 46 (H) x 37 (V)
Magni13ation 107 068
Pinhole Material Ta Pt
Table 44 Spe13i13ations of the two uEye keV 13ameras (1stVsion 13ompany)
44 Diagnosti13s
Various kinds of diagnosti13s were implemented namely visible interfero-
metry XUV and visible spe13tros13opy For the same two keV 13ameras are
indu13ted to monitor the impa13t and two visible 13ameras for target align-
60 CHAPTER 4 EXPERIMENTAL SETUP
(a)
(b)
(13)
Figure 412 (a) Sign 13onvention for the respe13tive positions of the lens and the
target (b) keV image of a MAIN impa13t on a massive target (lens position +1500
microm) The spot size is 475plusmn25 microm Pixel size for MAIN keV 13amera is 56 microm (13)
keV image of an AUX impa13t (lens position -1500 microm) The spot size is sim 280plusmn20microm on target One pixel = 66 microm on AUXs keV 13amera
44 DIAGNOSTICS 61
ment In this se13tion I will des13ribe the setups of the visible interferometry
XUV spe13tros13opy and visible spe13tros13opy The analysis of the re13ords of
visible interferometry and XUV spe13tros13opy will be dis13ussed in subsequent
13hapters
441 Visible Interferometry
A Ma13h-Zehnder Interferometer (MZI) has been employed to perform visible
interferometry The re13orded interferometri13 images have been used to study
the sho13k speed and the ele13tron density during the sho13ks propagation The
experimental setup of this interferometer is presented below
Ma13h Zehnder Interferometer
We pla13ed the target in whi13h sho13k is produ13ed in one of the two arms
of the interferometer The probe laser wavelength (527 nm) is supposed to
be far away of any atomi13 absorption resonan13es in the plasma and thus the
13ontributions to the refra13tive index 13ome from the plasma free ele13trons and
not from bound ele13trons (Harilal and Tilla13k 2004)
Figure 413 First interferometri13 setup All the opti13al elements are 1 in13h For
simpli13ity the plasma slab is not reported in the probe beam
The rst experimental setup shown in Fig 413 was used until 04052015
62 CHAPTER 4 EXPERIMENTAL SETUP
In this setup
the 300 ns long probe laser of wavelength 527nm (beam diameter asymp 1
in13h energy on the target few tens of mi13rojoules) rst passes through
an alignment pinhole PH1
the non-polarizing beam splitter BS1 transmits 50 of the intensity
of the in13ident beam and the remaining 50 of intensity in the per-
pendi13ular dire13tion At this stage the two arms of the interferometer
namely the referen13e and probe beams are produ13ed
two half wave plates HWP1 and HWP2 are pla13ed respe13tively in
the path of the probe and referen13e beams They allow to play with
the polarization in order to optimize the fringes 13ontrast as it will be
explained below
the verti13ally polarized probe beam passes through the target (plasma
slab) Due to its polarization it is fully ree13ted by the polarizing 13ube
beam splitter PBS1 towards the polarizing 13ube beam splitter PBS2
the referen13e beam is ree13ted by the mirror M4 on to this 13ube splitter
PBS2
At the output of PBS2 the two beams overlap but they have ortho-
gonal polarizations and 13ant interfere They then travel through a
polarizing beam splitter PBS3 those axis makes an angle of 45
with
the 13ommon dire13tion of in13iden13e of the two beams This rotates
the polarizations of the referen13e (horizontally polarized) and probe
(verti13ally polarized) beams along the same axis
At this stage the beams share the same polarization and interfere
A last non-polarizing beamsplitter plate (90-10 ) BS2 is pla13ed after
PBS3 to enable the propagation of the two beams on the same axis as
the laser beam at the entran13e of the interferometer
The fringes are re13orded on a HAMAMATSU C7700 VIS Streak Camera
lo13ated after the last alignment pinhole PH4 To this purpose an imaging
setup makes the image of the target longitudinal axis (ie along the dire13tion
of the sho13k propagation) on the streak slit This is done with an a13hromati13
spheri13al doublet of fo13al length equal to 1000 mm The magni13ation is
measured using an AGAR grid to 226 whi13h means that 106 microm on target
are re13orded by one pixel (size of the pixel 24 microm) on the streak on both
dire13tions (Fig 413)
44 DIAGNOSTICS 63
Figure 414 Se13ond interferometri13 setup
Role of the polarizing devi13es
Let us suppose that the polarization state of the in13oming beam is unknown
The polarization of the probe beam is then xed to be verti13al by the pola-
rizer HWP1 Ree13ted by the polarizing 13ube PBS2 it remains verti13al at
the entran13e of PBS3 On this side the polarization of the referen13e beam
is set to be horizontal after the 13ube BS1 using the polariser HWP2 and is
transmitted as horizontal by PBS2 The rotation of PBS3 allows to proje13t
part of the two beams on a 13ommon polarization axis and to interfere
This setup presents the following interest the intensity of the laser beam
in the two arms of the interferometer 13an be modied (and thus the 13ontrast
of the fringes) 13an be optimized using the two polarizing plates HPW1 and
HPW2 However as the multimode laser is not fully 13oherent the dark
fringe (zero net intensity) has never been obtained
Moreover the setup is designed to also allow alignment and adjustment
of the imaging setup with a green HeNe laser whi13h shares the same path
as the probe beam up to PBS1
A more optimized setup was used after 05052015 (ie shot number
48033) where the mirror M1 in the referen13e beam was repla13ed by a
polarizing beam splitter PBS4 ree13ting the probe beam like the mirror M1
(Fig 414) This was done in order to redu13e the dieren13e between the
64 CHAPTER 4 EXPERIMENTAL SETUP
two paths l1 and l2 and then to optimize the fringe 13ontrast This was more
suitable due to the short 13oheren13e length (1- 2 13m) of the probe laser
Streak 13amera
Figure 415 S13hemati13 of the Streak 13amera (from Hamamatsu noti13e)
A streak 13amera is a setup whi13h allows measuring the temporal varia-
tion of a fast signal whi13h is re13orded on its slit As indi13ated in the noti13e)
of the Hamamatsu C770 13amera the light pulse rst passes through the slit
then it is fo13used on the photo13athode (size 7times17 mm) of the streak by an
input opti13s The visible photons are 13onverted into a number of ele13trons
whi13h is proportional to the intensity of the in13ident light These ele13trons
are a1313elerated and 13ondu13ted towards the phosphor s13reen when a high-
speed voltage whi13h is syn13hronized to the in13ident light is applied As a
result the ele13trons are swept at high speed from the top to the bottom
after whi13h they are bombarded against the phosphor s13reen of the streak
tube and 13onverted through an output opti13s to an opti13al image on a dual
mode 13ooled CCD 13amera with 512 times 512 pixels (Fig 415) In the PALS
experiment the sweeping times were kept either 50 or 200 ns The CCD
pixel size is 24 x 24 microm
The streak 13amera 13an be operated with time swap mode with the slit
of the 13amera almost 13losed The re13ord 13onsisted then in images (position
along the slit versus time) as dis13ussed above or in a stati13 mode (no time
swap) whatever the slit aperture In that 13ase the re13ord is a usual 2D
spatial image of the obje13t
During the experiment we have taken referen13e images (without any
fringe) for every target as shown in the Fig 417a for one target example
In order to re13ord su13h images we blo13ked the referen13e beam and let the
probe beam only to illuminate the target in a stati13 mode and with the slit
open Then we redu13ed the slit width to its nominal value (200 microm) This
enabled us to have the visualization of the portion of the target whi13h was
44 DIAGNOSTICS 65
Figure 416 Imaging setup of the Ma13h Zehnder Interferometer setup operating
in the longitudinal mode The 13hannel of the target is fully illuminated by the
probe beam In this mode the lens images the axis of symmetry of the two sho13k
waves along the slit of the streak 13amera
imaged in the slit on the target and also to lo13ate the position of the pistons
before the shot This image is termed as the referen13e target image Then
the system was pla13ed in the interferometri13 mode and the fringe pattern
was optimized to be perpendi13ular to the slit and thus in the longitudinal
mode perpendi13ular to the dire13tion of the sho13k propagation
The Fig 417b reports su13h a typi13al interferogram re13orded during our
experiment (13f Fig 413) The horizontal axis 13orresponds to the dire13tion
of the sho13k propagation (with a total s13ale of 54 mm on the target) and
the verti13al axis 13orresponds to time (s13aling 200 ns)
442 XUV Spe13tros13opy
The time and spa13e integrated XUV plasma emission are re13orded with a
Flat Field XUV spe13trometer (grazing in13iden13e XUV spe13trometer) using
a 13on13ave grating Typi13al ray tra13ing a grazing in13iden13e XUV spe13tro-
meter is shown in Fig 418 In pra13ti13al the entran13e slit was removed
as the plasma extension was small In the Fig 419 the drawing of the
XUV spe13trometer setup s13heme (red 13olor) is presented together with the
spheri13al 13hamber and the target The XUV spe13trometer is installed on
the top of the spheri13al 13hamber whi13h allows the XUV emission passing
through the Si3N4 membrane on the top of the target (see se13tion 422) to
be re13orded
A 13ooled Andor DX440 CCD (without glass prote13tion) is atta13hed to
the spe13trometer to re13ord the spe13trum of the XUV radiation between 12
66 CHAPTER 4 EXPERIMENTAL SETUP
(a)
(b)
Figure 417 (a) Referen13e 2D image of a target before the shot re13orded on the
Streak 13amera The positions of the two pistons 13losing the 4 mm long target are
lo13ated at 800 and 4800 microm The dark zones along them (between 800 to 1000 and
4600-4800 microm) 13orresponds to the glue This glue is then visible through verti13al
bla13k strips in the interferometri13 re13ord (b) Corresponding interferometri13 time-
spa13e re13ord
44 DIAGNOSTICS 67
Figure 418 Typi13al ray tra13ing in a grazing in13iden13e XUV spe13trograph The
dete13tor (GMCP or image plate) is installed tangentially to the Rowland 13ir13le
and 40 nm (31- 100 eV) Its 13hara13teristi13s are given in the Table 45 Two Al
lters of thi13kness 08 and 16 microm prote13t the grating and the CCD 13amera
from debris and visible light The XUV images re13orded on the CCD are
time (300 micros) and spa13e (sim 3 mm) integrated
Andor 13amera (DX440)
Pixel Size 135 x 135 micromArea 276 (H) x 69 (V) mm
A13tive pixels 2048 x 512
Table 45 Spe13i13ations of Andor CCD 13amera
The 13urved grating (Table 46) used to fo13us the XUV emission on to
the CCD 13hip has a variable growes spa13ing as shown in the bottom panel
of the Fig 419 A typi13al output re13ord is also presented for illustration
Grating
Type Dira13tion
Growes per mm 1200
Dimensions 30 x 50 x 10 mmRadius of 13urvature 5649 mm
Blazing angle 37 degree
Blazing wavelength 16 nm
Table 46 Spe13i13ations of the XUV grating
443 Visible spe13tros13opy
In addition to the interferometry and XUV spe13tros13opy we have also im-
plemented a time- and spa13e-resolved visible spe13tros13opi13 plasma emission
diagnosti13
68 CHAPTER 4 EXPERIMENTAL SETUP
Figure 419 (a) Setup of the XUV spe13trometer in the spheri13al 13hamber (b)
Typi13al re13ord between sim 10 and 40 nm is shown in the bottom panel
The 13ore of the diagnosti13s is a Czerny-Turner 300 mm Spe13trometer
2
lo13ated outside the va13uum 13hamber The light of the plasma is inje13ted on
the slit of this spe13trometer through a bers bundle This bundle 13onsists of
three sets of an aligned seven individual bers looking at dierent se13tions
of the target 13hannel through an imaging lens lo13ated in the bla13k tube of
Fig 49 The time resolution is performed through an iCCD 13amera lo13ated
at the exit port of the spe13trometer (Fig 420a)
A typi13al re13ord is reported in Fig 420b The results of this diagnosti13s
show several signatures whi13h dier along the tube Most of them are in
absorption and 13ould be linked to the heating of the target sili13a window
Their interpretation is not yet a13hieved and will not be presented in this
study
45 Summary
In this 13hapter I have presented the details of our experimental setup and
diagnosti13s The next 13hapter will be dedi13ated to the presentation of our
re13ords results and their analysis
2
see http wwwandorcomspectrographshamrock minus spectrographminus series
45 SUMMARY 69
(a)
(b)
Figure 420 (a) S13hemati13 of setup employed to re13ord the time and spa13e
visible plasma emission (b) A typi13al spe13trogram
70 CHAPTER 4 EXPERIMENTAL SETUP
Chapter 5
Results and interpretation
Contents
51 Visible interferometry 71
511 Longitudinal interferometry sho13k speed and
ele13tron density 74
512 Transverse interferometry lateral extension of
the sho13k 91
52 XUV Spe13tros13opy 92
53 Simulations based on experimental results 93
54 Summary 97
In the previous 13hapter I presented a brief des13ription of our experimen-
tal setup This setup in13ludes two main diagnosti13s namely visible interfe-
rometry and XUV spe13tros13opy The visible interferometry is an adequate
tool to probe the ele13tron density of the radiative pre13ursor up to the 13riti-
13al density (4 times 10
2113m
minus3at the wavelength 527 nm of the probing laser)
whereas the denser post-sho13k region is opaque to the visible light XUV
radiation is present in the whole sho13k stru13ture and thus the time- and
spa13e-integrated XUV spe13tros13opy may allow exploring the self-emission
13oming from the post-sho13k and pre13ursor regions
In this 13hapter
1
I present the results of the both diagnosti13s with a
parti13ular fo13us on the interferometri13 data whi13h I have extensively studied
with the help of a spe13i13 data analysis pro13edure that I have developed I
will then present the main results of the XUV spe13tros13opi13 diagnosti13s The
results will then be interpreted with the help of numeri13al simulations
51 Visible interferometry
In the visible interferometry the interferen13es between the main and refe-
ren13e beams overlapping on the dete13tor produ13e a pattern of fringes whi13h
follow the relative phase variations between the probe (whi13h passes through
the plasma) and the referen13e beams (see se13tion 441 of the previous 13hapter
for the experimental setup) The phase variation between two 13onse13utive
fringes of the unperturbed beams (ie without any plasma) is equal to 2π
1
Most of results presented in this 13hapter has been published in Singh et al (2017)
71
72 CHAPTER 5 RESULTS AND INTERPRETATION
Then in the presen13e of the plasma in the probe beam these fringes be-
13ome shifted due to the modi13ation of the refra13tive index More details
about the prin13iple of the interferometry and the Ma13h-Zehnder interfero-
metri13 setup may be found in the APPENDIX A and we re13all here only
the expression of the phase shift
∆φ asymp minus πd
λNclt Ne gt (51)
where λ = 527 nm is the wavelength of the probing laser Nc = 4 times 10
21
13m
minus3the 13riti13al density at this wavelength and lt Ne gt is the ele13tron
density averaged over the laser path d in the 13ell and is dened as
lt Ne gt=
int d
0
Ne(z t)dy
d(52)
In order to dedu13e the ele13tron density it is ne13essary to follow the
evolution of the positions of the maxima (respe13tively minima) of the fringes
versus time To this purpose I have developed a spe13i13 data treatment
pipeline in the intera13tive data language (IDL)
2
To improve the visibility of
the fringes obtained in the interferograms I pro13essed the images with the
Fast Fourier Transform (FFT) (Proakis 2001) This treatment transforms
the image from the spatial domain (ie our re13ords) into the frequen13y
domain It then allows to dene spe13i13 bands of frequen13y whi13h represent
noise for the image and to 13lean them The 13omplete pro13edure is des13ribed
below
First the FFT of the (512 times 512) interferometri13 image is derived In
the next step the low and high-frequen13y lterings within the FFT of the
image are done using low-pass Fminus and high-pass F+ Butterworth frequen13y
lters (Proakis 2001) as follows
Fminus(u) = 1[1 + C(uu0)2n] with C = 1 n = 1
F+(u) = 1[1 + C(u0u)2n] with C = 1 n = 1
(53)
where u represents the position in the FFT image and u0 is the nominal
lter 13uto frequen13y (represented as the width of the region in pixels) For
the low-pass Butterworth frequen13y lter u0 is 13onsidered to be 55 whereas
for high-pass Butterworth frequen13y lter it is taken to be 45 The values of
u0 are dened manually by a trial method with the aim to result in better
noise redu13tion
After this step the inverse FFT of the ltered FFT image is performed
whi13h results in a 13lean image in the spatial domain In Fig 51 the original
re13ord for the shot number 48055 (left) the FFT of this image (13enter)
and the nal frequen13y ltered image (right) are presented showing how the
2
httpwwwast13ama13uk~vasilyidlidl_introdu13tionpdf
51 VISIBLE INTERFEROMETRY 73
frequen13y ltering improves the fringes 13ontrast and thus fa13ilitates their
analysis
Figure 51 Original re13ord of shot 48055 (left) FFT of the original image
(13enter) and the frequen13y ltered image (right)
Next the 13ru13ial task is to identify and to follow the fringes with the
help of a pre13ise determination of the intensity (pixel 13ounts) maxima In
order to lo13ate these maxima I have adopted the following steps
Figure 52 A 13ropped se13tion of the re13ord from shot the 48055 The rst ve
positions have been sele13ted manually `+ signs (in red 13olor) on ea13h fringe On
this re13ord the distan13e between two unperturbed fringes is 13orrespond to 15 pixels
(ie 159 microm)
Firstly I have manually 13hosen a se13tion of a typi13al re13ord with the
aim to sele13t only the area asso13iated with the sho13k dynami13s
74 CHAPTER 5 RESULTS AND INTERPRETATION
Next ve representative points are sele13ted visually (`+ marks in red
13olor) on ea13h fringe 13overing the important lo13ations on the fringe
(13f Fig 52) They serve as the input to the spline interpolation for
estimating all intermediate pixels lo13ations along the fringe (see Fig
53a)
In order to obtain the lo13ations representing the fringe maxima the
spline interpolated lo13ation points are further rened by sele13ting the
pixel of maximum intensity within plusmn5 pixels of the respe13tive spline
lo13ations in the X-dire13tion (see Fig 53b)
The position of fringe maxima for the 13ropped image obtained in an
aforesaid manner is shifted so as to represent the same points however
on the full-image (see Fig 54)
The re13ords obtained from the experiments have been pro13essed in this
way to estimate the sho13k se13tion speed ele13tron temperature and density
511 Longitudinal interferometry sho13k speed and ele13tron
density
In the longitudinal interferometry we perform on the slit of the streak 13a-
mera the image of the axis of symmetry of the sho13k propagation along the
tube to analyse the sho13k propagation in this dire13tion
The interferometri13 images have been pro13essed as explained previously
to enhan13e the fringes 13ontrast The lo13ations of the maximum intensity in
ea13h fringe are then used to derive the sho13k speed and the average ele13tron
density as will be presented below
Sho13k speed dedu13ed from the last fringe method
An estimation of the sho13k speed 13an be obtained through the interferometri13
image In this se13tion I will present the method to derive sho13k speed
based on the last visible end points of the fringes also 13alled last fringe
method This determination provides rapidly a qualitative estimation of the
sho13k speed whi13h is based on the absorption behavior of the plasma (see
Equation A11 in appendix A) and not on the real position of the front
dis13ontinuity The front dis13ontinuity is not seen in the re13ords due to the
strong absorption
At the positions of the last visible end points of the fringes the ele13tron
density rea13hes the maximum value a1313essible to the diagnosti13 The frin-
ges are strongly bent and the absorption of the visible light be13omes also
important Therefore it is believed that the sho13k front is 13lose from this
lo13ation
The lo13ations of the fringe maxima have been already dedu13ed following
the analysis presented in previous se13tions In this regard the best visible end
51 VISIBLE INTERFEROMETRY 75
(a)
(b)
Figure 53 (a) Representative points of the fringes as derived from the spline
tting of the 5 manually sele13ted points (Fig 52 (II)) on ea13h fringe (b) Positions
of the fringes maxima along Y-axis for ea13h fringe derived by lo13ating the points
of maximum intensity in X-dire13tion of the previous points obtained by spline t
76 CHAPTER 5 RESULTS AND INTERPRETATION
Figure 54 Fringe maxima on the full image
Figure 55 Imaging setup of the Ma13h Zehnder Interferometer operating in the
longitudinal mode The lens images the axis of symmetry of the two sho13k waves
along the slit of the streak 13amera
51 VISIBLE INTERFEROMETRY 77
point of ea13h respe13tive fringe is 13onsidered for the average speed estimation
for the sho13k
At very initial time the in13rease in sho13k speed is non-linear This non-
linear trend may result in non-physi13al speed estimations Therefore I dis-
13arded the end points whi13h appear to be within the ve nanose13onds after
the sho13k laun13hing time (eg t = 145 ns on Fig 56)
However the above pro13edure may result in un13ertainties as the sele13ted
end points may have dierent intensity (13ounts) values In this regard I
have evaluated the un13ertainty in the determination of the average speeds
To this purpose I rst sele13ted the best visible end-point of a random fringe
Su13h an end point gives a referen13e intensity whi13h was then used to obtain
on the rest of fringes the end points having 13lose intensity This pro13edure
was performed thri13e in order to dedu13e three sets of su13h end-points (see
Fig 56) Ea13h of these three sets of points was then used to estimate the
speed through a linear-t method (lines in white red and green) It may be
noted that these sets of points and tted lines are very 13lose to ea13h other
Therefore it is tough to distinguish them on the Fig 56 These three speed
determinations enable us to derive an average value with an un13ertainty
whi13h is equal to the standard deviation of these three values
Figure 56 Interferometri13 image re13orded for the shot 48055 in Xe at 01 bar
The sho13k speeds for the sho13ks driven by MAIN (from left side) land AUX (from
right side) lasers are respe13tively equal to sim 54 and 23 kms The time of laser
arrival on the piston is at 146 ns The positions of the Au-Xe interfa13e on the
re13ord are respe13tively 950plusmn50 and 4950plusmn50 mi13rons
This estimation of the sho13k average speed is satisfa13tory for the re13ords
in whi13h the sho13k speed is almost 13onstant throughout its propagation Ho-
wever the variation with the time of the speed may be substantial for some
re13ords In su13h 13ases the un13ertainty in the average speed will be in prin-
13iple higher than the previous estimations Therefore I introdu13ed another
78 CHAPTER 5 RESULTS AND INTERPRETATION
method to determine the average velo13ity this velo13ity is then estimated
during the initial and in nal durations of the sho13k propagation by deriving
the slope of points on the rst middle and last few fringes respe13tively
The standard deviation estimated by this method is then termed as the
un13ertainty in the average speed of respe13tive re13ord
Using these two methods I have obtained two values of un13ertainties for
all the MAIN and AUX sho13ks observed in the experimental re13ords Finally
the largest un13ertainty value of the two is 13onsidered as the nal un13ertainty
in the estimation of the respe13tive speed and is shown by error bars in Fig
57a and 57b 13orresponding to the MAIN and AUX sho13ks respe13tively
The table 51 reports for dierent noble gases the values of the velo13ities
re13orded in several shots for the 13ounter-propagating sho13k waves In addi-
tion one re13ord (48131) 13orresponds to the 13ase of an isolated sho13k wave
laun13hed by the MAIN laser The values of the MAIN and AUX laser ener-
gies are also reported In some 13ases the glue on the window prevented the
probe laser to pass through the 13hannel Then it be13ame impossible to derive
any value for the velo13ity In su13h 13ases I put `NA for the 13orresponding
sho13k velo13ity
It may be noted that the sho13k speeds generated by the MAIN laser vary
in the range of 30-55 kms while for the sho13k originated from AUX laser
the speeds vary between 10 and 30 kms
Variations with laser energy and pressure the 13ase of Xenon
When the number of re13ords is su13ient to make a statisti13al analysis we
may analyse the variations of the sho13k speeds with the laser energy of
the MAIN and AUX lasers This is the 13ase for Xe or XeHe (90-10)
mixture and for gas pressures ranging between 01 and 02 bar At 01 bar
all the measurements 13on13ern Xenon (ρ= 54 times 10
minus4g13m
minus3) At 02 bar
the re13ords 13on13ern a mixture XeHe (90-10 ρ=10 times 10
minus3g13m
minus3)
ex13ept for one point at 121 J whi13h 13orrespond to the 13ase of pure Xe (ρ =
108 times 10
minus3g13m
minus3) We do believe that taking into a1313ount the pre13ision
of our re13ord the introdu13tion of tra13es of Helium do not ae13t mu13h the
dynami13s of the sho13k wave as the 13orresponding variation of mass density
is negligible
As expe13ted the sho13k speed in13reases with the laser energy and de13rea-
ses with the pressure (Fig 57a) A linear t gives the following dependen13e
of the speed in kms with the MAIN laser energy (in J) at 01 and 02 bar
respe13tively
3
v01bar = 1423 + 030E (54)
3
Two outlier points have been dis13arded from the set 13orresponding to 01 bar They
are shown for information on the Figure
51 VISIBLE INTERFEROMETRY 79
(a)
(b)
Figure 57 (a) MAIN sho13k speed (13al13ulated by the last fringe method) versus
the MAIN energy for Xenon or XeHe mixture at dierent pressures with the error
bars (b) AUX sho13k speed versus AUX energy also for Xenon or XeHe mixture
at dierent pressures
80 CHAPTER 5 RESULTS AND INTERPRETATION
v02bar = minus1405 + 047E (55)
The speeds generated by the AUX laser are lower than the previous ones
Although the re13ords are more sparse we note that the speeds at 01 bar
also in13rease with the laser energy However at 02 bar I 13ould not nd
similar trend due to the la13k of re13ords
Variations with the gas
As 13an be seen from the Table 51 we have also performed several shots in
Ar at 01 02 03 and 08 bar one shot in He at 05 and one in Kr at 02
bar
To 13ompare the velo13ities obtained for the dierent gases one 13an either
keep the pressure 13onstant (ie the number of atoms) or the mass density
For a given mass density and laser energy the velo13ity should be the same
in the adiabati13 limit However due to the radiation 13ooling and ionisation
this 13an not be the 13ase Nonetheless if we 13ompare the two re13ords of Ar
at 03 bar (ρ = 493 times 10
minus4g13m
minus3 E = 121 J) with Xe at 01 bar (54
times 10
minus4g13m
minus3 E = 133 J) we measure similar speeds (49 and 54 kms)
These two re13ords are thus interesting to 13ompare This will be done later
on in this 13hapter
More generally we note that the speed of Helium at 05 bar is higher by
sim 40 than for Ar at the same energy (see Fig 58) This is not surprising
for this lighter element (Table 52) However it is important to note that
the la13k of shots prevents us to make any pre13ise 13on13lusion
Ex13ept for this 13ase and for the unique re13ord of Ar at 02 bar whi13h
gives also a higher velo13ity the velo13ities of the other re13ords follow more or
less the variation with the energy than Xenon
To 13ompare the radiative ee13ts it seems justied to perform the 13om-
parison at a given sho13k speed independent of the laser energy and for mass
densities whi13h are 13lose to ea13h other In the following we shall thus 13om-
pare the results of Xe at 01 bar (54 kms 54times 10
minus4g13m
minus3) Ar at 03
bar (49 kms 49times 10
minus4g13m
minus3) and Kr at 02 bar (55 kms 68 times 10
minus4
g13m
minus3)
Line averaged ele13tron density lt Ne gt
I have estimated the ele13tron density lt Ne gt of the plasma averaged along
the path of the probe laser beam by analyzing the interferogram re13ords
assuming the same se13tion d of the plasma layer of 600 microm for both MAIN
and AUX side sho13ks
51 VISIBLE INTERFEROMETRY 81
Shot Gas Pressure E3ω MAIN sho13k Eω AUX sho13k
(bar) (J) speed (kms) (J) speed (kms)
at 298 K
48033 Air 03 124 54plusmn1 66 20plusmn248034 Air 03 131 52plusmn1 74 27plusmn248076 Ar 05 100 41plusmn1 66 18plusmn148077 Ar 01 115 42plusmn1 67 18plusmn148078 Ar 03 112 46plusmn3 65 25plusmn248079 Ar 03 121 49plusmn5 67 23plusmn348080 Ar 08 103 38plusmn2 62 21plusmn248081 Ar 08 113 38plusmn1 68 NA
48082 Ar 08 107 36plusmn1 65 NA
48141 Ar 02 111 63plusmn1 57 NA
48083 He 05 106 57plusmn3 63 NA
48146 Kr 02 125 55plusmn2 53 NA
48043 Xe 03 138 57plusmn1 72 25plusmn148051 Xe 01 123 58plusmn1 67 30plusmn148055 Xe 01 133 54plusmn3 68 22plusmn348057 Xe 01 127 53plusmn1 68 23plusmn148058 Xe 01 115 48plusmn3 63 18plusmn148059 Xe 01 116 50plusmn1 67 21plusmn148061 Xe 01 127 53plusmn1 67 17plusmn248065 Xe 01 122 52plusmn4 68 23plusmn348066 Xe 01 114 50plusmn3 67 15plusmn248067 Xe 05 115 39plusmn2 65 NA
48068 Xe 05 109 36plusmn2 65 12plusmn248070 Xe 05 109 33plusmn3 65 NA
48138 Xe 02 121 45plusmn5 0 0
48131 Xe +He 02 112 38plusmn1 0 0
48132 Xe +He 02 118 41plusmn4 56 18plusmn248133 Xe +He 02 112 41plusmn3 56 NA
48134 Xe +He 02 111 38plusmn1 60 NA
48136 Xe +He 02 115 39plusmn3 59 14plusmn148143 Xe +He 06 123 39plusmn4 63 18plusmn548144 Xe+He 02 133 45plusmn3 66 NA
Table 51 Sho13k speeds estimated from the `last fringe method `NA represents
the entries whi13h 13ould not be dedu13ed from the re13ord Further entries in the
bold font are dis13ussed in detail in this 13hapter
82 CHAPTER 5 RESULTS AND INTERPRETATION
Helium Argon Krypton Xenon
Atomi13 Number 2 18 36 54
Atomi13 Mass 4 3995 8380 13129
First Ionization Energy (eV) 246 157 14 12
Density (10
minus4g13m
minus3) at 01 bar 016 164 344 539
Table 52 Atomi13 data and density at 01 bar (at room temperature) for He Ar
Kr and Xe
Figure 58 MAIN sho13k speed (13al13ulated by the last fringe method) versus the
MAIN laser energy for Xe (at 01 bar only) Ar He and Kr at dierent pressures
with the error bars
51 VISIBLE INTERFEROMETRY 83
It should be noted that the beam se13tions dMAIN and dAUX of MAIN
and AUX lasers are approximately 600 and 300 microm respe13tively on the two
pistons and that the transverse horizontal se13tion of the sho13k tube is equal
to dtube = 600 microm Thus the value of lt Ne gt 13omputed for the sho13k wave
laun13hed by MAIN should be 13lose to the physi13al lo13al value Ne supposing
that the plasma is homogeneous along the transverse se13tion However
the value obtained for AUX is 13ertainly larger by a fa13tor whi13h 13an be
estimated at a rst step as the fra13tion of dtubedAUX giving a fa13tor of
about 2
As explained in the previous se13tion the pixels representing the positions
of the fringe maxima have been already determined with the best possible
pre13ision Therefore it is possible to 13al13ulate the relative phase shift (with
respe13t to zero time) variation along y-axis (time axis) at ea13h fringe maxima
Zero time is time of MAIN and AUX laser rival on the target With these
phase shifts estimated at ea13h pixel of ea13h fringe I then derived lt Ne gtusing Equation A22
Let us take the 13ase of three shots in Xenon for illustrating the method
whi13h will be followed (see the left panel of the Figure 59) The maximum
density is estimated to be sim11 times 1019cmminus3(13orresponding to phase shift sim
16) The number of subdivisions is taken to 5 bins with the following phase
(∆φ) ranges (in the units of 2π) and average ele13tron densities (ltNegt)
bin 1 ∆φ le 06 lt Ne gtle 39 10
1813m
minus3(white)
bin 2 06 - 08 39 - 57 10
1813m
minus3(red)
bin 3 08 - 11 57 - 75 10
1813m
minus3(blue)
bin 4 11 - 13 75 - 93 10
1813m
minus3(green)
bin 5 gt 13 gt 93 10
1813m
minus3(magenta)
All the re13ords shown in the Table 51 have been pro13essed using this
method The limit of dete13tion for the phase shift 13orresponds to 2 pixels
giving a threshold for the Ne measurement lt Nemin gt This threshold
diers from one re13ord to another due to the variation in the number of
fringes (and thus the distan13e between two unperturbed fringes)
In order to make a 13omparative investigation of the sho13ks propagation
and intera13tion in dierent gases as already mentioned previously I sele13ted
three 13ases (shot48055 shot48132 and shot48138) relative to Xe (or Xe-
He mixture) one 13ase relative to Ar (shot48079) and Kr (shot48146)
The 13orresponding experimental 13onditions are noted in bold in the Table
51
84 CHAPTER 5 RESULTS AND INTERPRETATION
lt Ne gt in Xenon
The Fig 59 reports the interferometri13 re13ords for the three Xenon 13ases
The top and the middle panels 13orrespond to the propagation of two 13ounter-
streaming radiative sho13k waves at 01 (shot48055) and 02 bar (48132)
For 13omparison one re13ord (48138 bottom panel) is dedi13ated to the pro-
pagation of single sho13k (MAIN) at 02 bar The Xe-He mixture (90 - 10
in numbers of atoms) is used for the investigation in the 13ase of 02 bar
pressure while it is only Xe in the 13ase of 01 bar As mentioned previously
at the pre13ision of our re13ords we 13onsider that this impurity 13on13entration
has a negligible ee13t on the sho13k speed and the pre13ursor ele13tron density
The limit of dete13tion of lt Ne gt over the se13tion of the tube (06 mm) is
13orresponds respe13tively to 9 times 10
17 6 times 10
17and 6 times 10
1713m
minus3for the
Figures 59(a) (b) and (13)
The variations of lt Ne gt with the distan13e along the sho13k tube (ie
parallel to the dire13tion of the sho13ks) are reported in the right panel of Fig
59 at 10 ns (in red) 20 ns (in blue) 30 ns (in green) and 40 ns (in magenta)
The intera13tion between the two pre13ursors is 13learly visible at 01 bar
(Fig 59(a)) at 10 ns the intera13tion of the 13ounter-propagating sho13ks
has either not yet started or is below the sensitivity of this diagnosti13 The
intera13tion o1313urs at later times with a typi13al signature whi13h is as follows
the slope of lt Ne gt is de13reasing from the left (MAIN pre13ursor) passes
through a minimum and in13reases at the right (AUX) The minimum itself
in13reases with time up to 7 times 10
1813m
minus3at 40 ns
At 02 bar we have not been able to re13ord the 13ollision in the temporal
range (50 ns) of the streak However the two re13ords (with MAIN only and
with the two sho13k waves) indi13ate a pre13ursor for MAIN The two gures
(Fig 59(b)) and (Fig 59(13)) show that the pre13ursor of the MAIN sho13k
wave is not inuen13ed by the presen13e of AUX sho13k wave up to 45 ns
There is no obvious indi13ation about a pre13ursor for AUX in the 13ase of
two 13ounter-propagating sho13k waves (Fig 59(b)) At this pressure and
13ompared with the previous 13ase at 01 bar the absen13e of pre13ursor for
AUX may be attributed to i) a low sho13k speed (18 kms) 13ombined with a
larger pressure (hen13e for a given gas the pre13ursor extension in13reases with
the speed and de13reases with pressure) ii) a too small longitudinal extension
of the eventual pre13ursor (see Fig 59(b)) 13ompared with the resolution of
20 mi13rons (2 pixels) Our 1D numeri13al simulations with Xenon opa13ity
multiplier times 20 (not presented here) indi13ate a small pre13ursor for AUX
sho13k At 15 ns its extension is 50 microm (900 microm for MAIN sho13k) with
a typi13al ele13tron density sim 35 times 10
1913m
minus3(23 times 10
1913m
minus3for MAIN
sho13k) whi13h does not agree with the re13ord At 42 ns the pre13ursor of MAIN
rea13hes the AUX sho13k front and the prole is similar to the prole at 20 ns
shown in Fig 313a at 01 bar with a plateau of almost 13onstant ele13tron
density between the two fronts This might be 13ompatible with small bending
51 VISIBLE INTERFEROMETRY 85
Figure 59 Left panel interferometri13 re13ords 48055 in Xe at 01 bar (a)
48132 in Xe+He at 02 bar (b) and 48138 in Xe+He at 02 bar (13) Right
panel ele13tron density at 10 20 30 and 40 ns versus distan13e for these re13ords
The positions of maxima have been identied on the re13ords in the left panel
The time t = 0 13orresponds to the time of laser arrival on the target and the
position x = 0 13orresponds to the interfa13e between the piston (Au layer) and the
gas Its determination is pre13ise within 100 mi13rons The distan13es between two
unperturbed fringes for re13ords 48055 48132 and 48138 are 159 244 and 244
microm respe13tively The lt Ne gt un13ertainty (plusmn 2 pixels) is indi13ated by the error bar
in the right panels It 13orresponds respe13tively to plusmn 9 times 10
17 plusmn 6 times 10
17and plusmn
6 times 10
1713m
minus3for the gures (a) (b) and (13) The limit of dete13tion (2 pixels) is
presented by a dotted line on ea13h gure
86 CHAPTER 5 RESULTS AND INTERPRETATION
of the 4
thfringe (from the right) between 45 and 50 ns As 1D simulations
are known to overestimate the pre13ursor ele13tron density 2D simulations are
ne13essary for a more pre13ise interpretation of the experimental result
lt Ne gt in Argon and in Krypton
As seen in the Fig 58 the variation with the laser energy of the sho13k speed
in Argon at 03 bar (in the red dashed 13ir13le) is in good agreement with the
13orresponding one of Xenon at 01 bar (blue linet) As these two 13ases have
a 13omparable mass density of sim 5 times 10
minus4g13m
3 this qualitative agreement
is expe13ted
Taking into a1313ount the larger pressure (and thus density) the slope of
the variation of the slower sho13k speeds in Ar at 08 bar (in a red dash-
dot 13ir13le) with the laser intensity is 13ompatible with the previous trends
However any 13on13lusion 13an not be derived for Argon at 02 and 05 bar
owing to the s13ar13ity of the re13ords
In Fig 510 the interferometri13 re13ord for Ar at 03 bar (shot 48079
49 times 10
minus4g13m
minus3) is presented The estimated speeds of MAIN and AUX
sho13ks dedu13ed from the last fringe method are respe13tively equal to 49 and
23 kms whi13h are 13lose to the speeds of 54 and 23 kms re13orded in Xe
at 01 bar (shot 48055 539 times 10
minus4g13m
minus3 Fig 59(a)) Contrarily to
the Xenon 13ase we do not see any signi13ant bending in the fringes The
maximum fringe shift is reported in red on the fringe 1 The 13orresponding
ele13tron density is equal to 38 times 10
1813m
minus3
Thus obviously the radiation ee13ts are less important for Argon than
for Xenon at the same density and speed Part of this dieren13e 13ould be
attributed to the huge dieren13e in the atomi13 numbers (40 and 131) of the
two gases whi13h leads to dierent peak temperatures for the ions as expe13ted
from equation 84 of 13hapter 2 However the post-sho13k temperatures dedu-
13ed from the jump relations for a real gas (Chapter 2) without radiation are
very 13lose (sim 25 eV) and thus this 13ould be not the relevant explanation
To 13larify this I performed HELIOSPROPACEOS simulations for two
13ounter-propagating sho13ks at 53 and 23 kms respe13tively in Ar and Xe
at the same initial mass density (54 times 10
minus4g13m
minus3) without any opa13ity
multiplier I have also reported for information the 13ase of Xenon with
an opa13ity multiplier equal to 20 The simulated temperature proles are
shown in the Fig 511 One notes immediately that the pre13ursor extension
is smaller for Ar than for Xe The peak temperatures of the MAIN sho13k
are respe13tively sim 21 eV (16 eV for opa13ity multiplier=20) for Xe and 21
eV for Ar In all the 13ases the post-sho13k temperatures are equal to sim 11
eV This indi13ates that the temperature is not the main explanation of the
dieren13es in the pre13ursor length
More interesting are the dieren13es in the Rosseland opa13ity (here per
unit of length) whi13h is reported in the Fig 511b We see that 13lose to the
51 VISIBLE INTERFEROMETRY 87
(a)
(b)
Figure 510 (a) Interferometri13 re13ord in Ar at 03 bar (48079) The estimated
speeds for MAIN and AUX are 49plusmn5 and 23plusmn3 kms (b) Same re13ord where the
fringes maxima are marked by points The bins denition is as fellows bin 0 Ne le11 times 10
1813m
minus3(white) bin 1 11 - 18 times 10
1813m
minus3(yellow) and bin 2 39 - 57
times 10
1813m
minus3(red) The time t = 0 13orresponds to the time of laser arrival on the
target and the position x = 0 13orresponds to the interfa13e between the piston (Au
layer) and the gas Its determination is pre13ise within 100 mi13rons
88 CHAPTER 5 RESULTS AND INTERPRETATION
(a)
(b)
Figure 511 Results for temperature (a) and Rosseland opa13ity (b) obtained from
HELIOS simulation at 10 ns for two 13ounter-propagating sho13ks at sim 50 and 18
kms for Xe (with opa13ity multiplier 1 and 20) Ar and Kr at 54 times 10
minus4g13m
minus3
initial mass density A 13omparison Ar Kr and Xe PROPACEOS opa13ity shown in
APPENDIX B
51 VISIBLE INTERFEROMETRY 89
front the Argon opa13ity is 13lose to 4 13m
minus1 whi13h means that the radiation
from the sho13k is absorbed within 25 mm This has to be 13ompared with the
Xenon 13ase (no multiplier) where it is equal to sim 03 13m
minus1 whi13h means
that the hot pre13ursor is almost transparent over the length of the tube
Thus the dierent behaviors of the opa13ity in the upstream gas explain the
quantitative dieren13e in the development of the pre13ursor The sharp peak
of the Xenon opa13ity with the multiplier of times 20 is due to the strong rise
of the opa13ity when the temperature de13reases below 5 eV whi13h marks the
end of the pre13ursor
I will present now the 13ase of two 13ounter-propagating sho13ks in Kr
(48146 in Fig 512) at 02 bar (69 times 10
minus4g13m
minus3) In this 13ase unfortu-
nately the AUX sho13k 13ould not be imaged 13ompletely due to the presen13e
of glue at the right edge of the target 13ell but the 13ollision was re13orded
at sim 40 ns The speed of the MAIN sho13k is estimated to be 55 kms
Thus despite a slightly higher initial mass density this sho13k wave may be
13ompared with the shot (48055) in Xe at 01 bar (54 times 10
minus4g13m
minus3 54
kms)
Despite the poor quality of the re13ord and the inadequate temporal range
(150 ns) sele13ted on the streak 13amera we note some bending in all the
fringes indi13ating the presen13e of pre13ursor The maximum fringe deviation
(063 of the distan13e between two fringes) is noted at the end of fringe 2
whi13h 13orresponds to the lt Ne gt value 45 times 10
1813m
minus3 The pre13ursor
seams to be more developed here than for the previous Ar 13ase
The result of the simulation in Xe and Kr with equal initial mass density
54 times 10
minus4g13m
minus3 and for the two respe13tive sho13k speeds of 50 and 20
kms is reported in the Figure 511 The peak temperatures of the MAIN
sho13k are respe13tively sim 21 eV (16 eV for opa13ity multiplier = 20) for Xe
and 22 eV for Kr The post-sho13k temperatures (sim 10-11 eV) are 13lose from
ea13h other
The pre13ursor extension in Krypton is smaller than in Xenon whi13h is lo-
gi13al if we 13onsider the opa13ity values (Fig 511b) However if we take into
a1313ount the multiplying fa13tor for the Xenon opa13ity the Krypton pre13ursor
is now more extended than for xenon whi13h is in 13ontradi13tion with the ex-
periment This probably means that either the Kr opa13ity is under estimated
by PROPACEOS or that our multiplier for Xenon is overestimated
Synthesis
The previous experimental investigation 13onrms that the sho13k velo13ity in-
13reases with the laser energy and de13reases with the mass density Moreover
for our given sho13k velo13ity and mass density the extension of the radiative
pre13ursor in the experiment de13reases with the atomi13 number Among Ar
Kr and Xe this last one appears to be the most adapted for the investiga-
90 CHAPTER 5 RESULTS AND INTERPRETATION
(a)
(b)
Figure 512 (a) Raw interferometri13 re13ord in Kr at 02 bar (shot 48146) The
estimated speed for MAIN sho13k is 53plusmn2 kms The time t = 0 13orresponds to the
time of laser arrival on the target(b) Same re13ord shown in (a) maxima is marked
by 13olored points The bins denition is as follows bin 0 Ne le 11 times 10
1813m
minus3
(white) bin 1 11 - 18 times 10
1813m
minus3(yellow) and bin 2 39 - 57 times 10
1813m
minus3
(red) The time t = 0 13orresponds to the time of laser arrival on the target and
the position x = 0 13orresponds to the interfa13e between the piston (Au layer) and
the gas Its determination is pre13ise within 100 mi13rons
51 VISIBLE INTERFEROMETRY 91
tion of the radiative pre13ursor A similar behaviour is found also at ORION
with higher speeds (Clayson et al 2017)
512 Transverse interferometry lateral extension of the sho13k
The transverse interferometry is a dierent geometri13al imaging setup of the
Ma13h-Zehnder interferometer whi13h provides qualitative information about
the 13urvature lo13alisation and transverse extension of the radiative pre13ur-
sor and thus of the sho13k
In the transverse interferometry we perform the image of a transverse
se13tion (on the slit of the streak 13amera) of the tube whi13h is lo13ated at a
distan13e dslit equal to 3 mm from the initial position of the MAIN piston
As the setup is originally adapted for the longitudinal interferometry we use
a Dove prism between the va13uum 13hamber and the streak 13amera to rotate
the image by 90 degrees
Figure 513 S13hemati13s of the Ma13h Zehnder Interferometer setup to re13ord
transverse interferometri13 images The lens allows to make on the slit of the
13amera the image of a se13tion perpendi13ular to dire13tion of sho13k propagation
A transverse interferometri13 re13ord for the MAIN sho13k alone in Xe at
02 bar is reported in Fig514(a) The streak 13amera rst re13ords the unper-
turbed plasma on the se13tion dslit in the tube Then it re13ords su1313essively
the pre13ursor and the post-sho13k (whi13h is opaque to visible radiation and
appears as a dark zone in the re13ords) Taking into a1313ount the oset of 14
ns the time of sho13k arrival is re13orded at 72 ns after the time t0 of laser
arrival on the target and the sho13k speed is estimated to be sim 35 kms Due
to glue on one lateral window (on the right part of the gure) only 6 fringes
92 CHAPTER 5 RESULTS AND INTERPRETATION
Figure 514 Transverse interferometri13 images for (a) shot48111 (MAIN sho13k
only) (b) shot48130 (AUX sho13k only) The time is measured after an oset
equal to 14 and 23 ns respe13tively after the time of the laser arrival on the target
The position zero on the x-axis of ea13h image 13orresponds to the base of the target
are visible The lateral extension of the sho13k stru13ture at this time is deri-
ved to be sim 570plusmn30 microm in qualitative agreement with the spe13i13ations of
the MAIN phase plate and the shape of the pre13ursor is relatively at The
axis of symmetry of the sho13k is determined to be at sim 350 mi13rons from
the base of the target (ie 50 microm above the nominal value of 300 microm)
A re13ord for the AUX sho13k alone is shown in Fig 514(b) whi13h 13orre-
sponds to a gate opening of 50 ns The start time of the image has an oset
of +23 ns from t0 and the distan13e dslit is set to 700 mi13rons from the initial
position of the AUX piston The AUX sho13k duration extends from 30 ns
to at least 34 ns after t0 The sho13k speed is then estimated to be ranging
between 23 and 20 kms The shape of the pre13ursor is strongly bent and
we note a tiny shift of the se13ond and third fringes (starting from the left)
at sim 8 ns It may further be noted that the lateral spread of the opaque
sho13k is ranging between 275plusmn25 microm (whi13h is also in agreement with the
spe13i13ations of the AUX phase plate) and that the axis of symmetry of the
sho13k system is also lo13ated at about 350 microm from the bottom of the 13ell
52 XUV Spe13tros13opy
XUV emission originates from the 13omparatively hot portions of the plasma
In the present 13ase it 13orresponds to the post-sho13k and the pre13ursor re-
53 SIMULATIONS BASED ON EXPERIMENTAL RESULTS 93
gion whi13h is 13lose to the sho13k front The spe13trum re13orded by the XUV
spe13trometer enables us to analyze the spe13tros13opi13 signatures of the radi-
ative sho13k One of the aims of this diagnosti13s was to identify the sho13ks
13ollision whi13h leads to higher temperatures through 13omparing the spe13tra
obtained for the 13ases of single and 13ounter-propagating sho13k waves Howe-
ver only a few re13ords were possible to be obtained during the experiment
and unfortunately the 13omparison with the 13ase of an isolated sho13k wave
was not performed
Among the shots re13orded the XUV spe13trum of the shot 48143 is
presented herewith in detail This shot was performed for [Xe (90)+ He
(10) mixture at 06 bar with laser energies of 123 J for MAIN and 63 J
for AUX The interferometri13 re13ord of this shot is shown in Fig 515 The
MAIN sho13k speed has been estimated to be sim 39 plusmn 4 kms The estimated
AUX sho13k speed (18 plusmn 5 kms) is not pre13ise due to the presen13e of glue
on the right se13tion of the re13ord (note Fig 515) In this interferometri13
re13ord we have not been able to retrieve the 13ollision time However an
extrapolation of the slope 13orresponding to the speeds sim 39 kms (MAIN)
and sim 18 kms (AUX) enables us to approximately determine the 13ollision
time to be between 60 to 65 ns
The raw spe13trum (shown in Fig 516a) re13orded for the wavelength
range of 15-35 nm (35-82 eV) shows the `L edge of Aluminum at 17 nm
(34 nm in se13ond order) in the rst and se13ond (34 nm)orders These two
wavelengths will be used for the wavelength 13alibration The net spe13trum
13orre13tion for the transmission (Henke Gullikson and Davis 1993) of the
100 nm thi13k Si3N4 window (3 mm times 04 mm) is introdu13ed in Fig 516b
A remarkable feature is a strong absorption dip between 19 and 22 nm (56-65
eV) This absorption probably 13omes from the 13older layers (thi13kness 300
microm) between the sho13k heated plasma and the Si3N4 window Few lines
of Xe VII-VIII are identied through NIST database
4
as also Oxygen IV
and V lines Lyman lines of He II (from 1-2 to 1-7) are also present in the
spe13trum This information will be useful for the estimation of the ele13tron
temperature
53 Simulations based on experimental results
In this se13tion I will 13ompare the experimental sho13k 13hara13teristi13s with the
results of HELIOS simulations using the PROPACEOS equation of state and
opa13ity (limited to 1 group) As indi13ated in the 13hapter 3 this opa13ity has
been multiplied by 20 for the Xenon only As our interest is to understand the
sho13k stru13ture in Xenon and not the laser matter intera13tion on the piston
we performed several simulations with Xe gas for various sets of uen13es for
4
[httpphysi13snistgovPhysRefDataASDlines_formhtml
94 CHAPTER 5 RESULTS AND INTERPRETATION
Figure 515 Interferometri13 image for the shot48143 The time t = 0 13orre-
sponds to the time of laser arrival on the target and the position x = 0 13orresponds
to the interfa13e between the piston (Au layer) and the gas Its determination is
pre13ise within 100 mi13rons
53 SIMULATIONS BASED ON EXPERIMENTAL RESULTS 95
(a)
(b)
Figure 516 Raw (a) and 13orre13ted XUV spe13trum (b) for the shot 48143
96 CHAPTER 5 RESULTS AND INTERPRETATION
the MAIN and AUX beams in order to obtain the best agreement between
the simulated and measured sho13k speeds
To analyse the results from the shot 48055 (Fig 59(a)) we set the u-
en13es to 32000 amp 7500 J13m
2 This allows produ13ing the experimental sho13k
speeds 54 and 23 kms in Xenon at 01 bar for the MAIN and AUX beams
respe13tively The two sho13ks appear in Xenon at 2 and 3 ns respe13tively for
MAIN and AUX The merging of the two pre13ursors starts at sim 15 ns and
the sho13k 13ollision time o1313urs at 47 ns In Fig 517 we present the ele13tron
density proles from the simulation (dotted lines) and the experiment (solid
lines) at 10 20 30 and 40 ns
At 10 ns the two simulated pre13ursor extensions are 0165 and 0022 13m
for MAIN and AUX respe13tively The ele13tron density is larger by a fa13tor
of 4 than in the experiment The shapes of the pre13ursors are also very dif-
ferent However this 1D pi13ture supposes the plasma to be uniform within
the tube In reality in the transverse dire13tion Ne is de13reasing from the
13enter to the walls whi13h results in a lower estimation of the average value
(lt Ne gt) 13ompared to the value at the target 13enter and in a smoother pro-
le near 02 amp 035 13m at 10 ns It is also important to note that for AUX
sho13k the average lt Ne gt value underestimates the lo13al one by a fa13tor
of about 2 (as it is averaged over 06 mm instead of 03 mm) Moreover
our 1D simulation suers from an inexa13t opa13ity and 2D ee13ts are pro-
bably important espe13ially for AUX Thus we have here only a qualitative
interpretation of the experimental results
The intera13tion between the two HELIOS radiative pre13ursors starts be-
tween 10 and 20 ns like in the experiment However the shape as well as
absolute values of the simulated ele13tron density 13urves are not in agreement
with the experimental results and the intera13tion is stronger in the simulation
than in the experiment
In order to interpret the spe13tros13opi13 data presented in se13tion 52 we
performed another 1D simulation in Xenon at 06 bar and adapted the
uen13es to generate two 13ounter-propagating sho13ks with the speeds 36 and
18 kms 13lose to the experiment The time evolutions of the ele13tron density
mean 13harge and ele13tron temperature at 56 57 58 60 64 and 65 ns are
presented in Fig 518 The two sho13ks appear in Xenon at 2 and 3 ns
respe13tively for MAIN and AUX Con13erning AUX the 13ombination of a
small speed and a relative high pressure does not allow to develop a radiative
pre13ursor in agreement with the experimental results (Fig 515) whereas
the MAIN sho13k has a pre13ursor and its length is in13reasing with time
The post sho13k temperature of the MAIN is sim 21 eV and the ion 13harge
sim 9 At 57 ns the pre13ursor of MAIN rea13hes the AUX sho13k front This
time is out of our re13ord (see Fig 515) whi13h means that the intera13tion
ee13t is either absent or o1313urs at later times The stru13ture of the AUX
post sho13k is modied by the intera13tion with the MAIN pre13ursor (Fig
518b) The sho13k 13ollision o1313urs at 65 ns (Fig 518a) resulting in the
54 SUMMARY 97
Figure 517 Re13orded ele13tron density (shot 48055) together with the HELIOS
results (with Xenon opa13ity times 20) at dierent times in Xenon at 01 bar
development of two reserve sho13k waves At the 13ollision time the ele13tron
density mass density ele13tron temperature and ion 13harge rea13h respe13tively
sim 14 times 10
2113m
minus3 0034 g13m
minus3 26 eV and 10 Mean 13harge is varying
between 5-10 whi13h 13ompatible with the presen13e of lines of Xe VII-VIII in
the experimental re13ord shown in Fig 516b
Moreover in order to interpret XUV spe13tros13opi13 results shown in se13tion
52 Rodriguez performed qualitative preliminary 13omputations (as des13ribed
in Rodriacuteguez et al (2015)) of the XUV spe13tra emerging from a 600 microm thi13k
plasma with two representative values of the mass density ρ = 32 times 10
minus2
and 33 times 10
minus3g13m
3 They show that the lines of HeII 13an only be ob-
served at a temperature of sim 15 eV and for the lowest density ie in the
radiative pre13ursor
54 Summary
In this 13hapter I have presented an extensive data analysis of few representa-
tive interferometri13 and spe13tros13opi13 re13ords The average sho13k speed and
ele13tron density have been estimated from the interferograms The sho13k
speeds of the MAIN and AUX radiative sho13k waves vary between sim 30-55
and 10-30 kms respe13tively and the averaged pre13ursor ele13tron density
ranges between 10
17and 10
1913m
minus3during the sho13ks propagation
We have demonstrated the intera13tion ee13t between the two radiative
pre13ursors in the 13ase of Xe at 01 bar at 54 and 23 kms The intera13tion is
13learly 13hara13terized in the experiment by the enhan13ement of the ionisation
wave followed by the merging of the two radiative pre13ursors at 20 ns The
13ollision time is re13orded at 47 ns Su13h behavior is reprodu13ed by the
98 CHAPTER 5 RESULTS AND INTERPRETATION
(a)
(b)
(13)
Figure 518 Time evolution of the mass density (a) ele13tron temperature (b) and
mean 13harge (13) at 56 57 58 60 64 and 65 ns within the sho13k tube derived from
the HELIOS simulations (with Xenon opa13ity multiplier = 20) for two 13ounter
streaming sho13ks of sim 39 and 18 kms in Xenon at 06 bar
54 SUMMARY 99
simulation
We have investigated this intera13tion at a larger pressure 02 bar with
the following speeds sim 41 kms for the MAIN and sim 18 kms for the AUX
sho13k waves We do not re13ord any experimental signature of the radiative
pre13ursor for AUX Further we have not been able to 13at13h experimentally
the 13ollision time The re13orded pre13ursor of MAIN is not inuen13ed by AUX
wave up to 48 ns (Fig 59(b) and (13) in data analysis 13hapter) whi13h is the
limit of the re13ord On its side the 1D simulation predi13ts a tiny pre13ursor
for AUX and that both pre13ursors start to intera13t at 49 ns This plausible
sho13ks intera13tion o1313urring at times whi13h are outside of the re13ord 13an
not be 13onrmed by our experiment
The results of the transverse interferometry at 02 bar with speeds of sim40 and 20 kms indi13ate that the MAIN pre13ursor has a lateral extension of
sim 600 microm whereas it is 300 microm for AUX The pre13ursor of MAIN is almost
at with a probable small bending at the edges of the tube whereas the
AUX pre13ursor is more 13urved This means that the 2D ee13ts are more
important for AUX than for MAIN Still in the 13ase of Xenon we have
obtained information about the temperature and the mean 13harge of gas
through our time integrated XUV spe13tra (shot48143) in Xenon at 06
bar From these results we may 13on13lude that the mean ion 13harge is at
least equal to 6 and that the temperature has rea13hed 15 eV
Our simulations give a qualitative des13ription of the sho13ks intera13tion
when the laser uen13e is adjusted to give the 13orre13t sho13k velo13ities Howe-
ver it is now well known that 2D simulations (together with state of the art
opa13ities) t better with experiments (Gonzaacutelez Audit and Stehleacute 2009
Leygna13 et al 2006 Stehleacute et al 2010) For the same laser energy the 2D
simulations lead to a diminution of the sho13k speed 13ompared to 1D as also
to a diminution of the ele13tron density For instan13e in the 13ase of a sho13k
wave laun13hed by a laser beam at 1315 nm in Xenon at 03 bar at PALS and
with a laser uen13e of 85000 J13m
2 ARWEN 2D simulations give a sho13k
speed of 44 kms in agreement with the experimental one (Cotelo et al
2015) 1D simulation would require in this 13ase a uen13e of 30000 J13m
2to
a13hieve the same velo13ity
The spa13e and time integrated XUV re13ords at 06 bar for respe13tive
speeds whi13h are equal to sim 39 and 18 kms indi13ate that the temperature
of the sho13k has rea13hed values up to 15 eV and that the Xenon mean ion
13harge has also rea13hed values of 6 - 7 whereas 1D simulations predi13t ele13tron
temperature 10-30 eV and ion 13harge 5-10 (Fig 51813) A more detailed
study based on 2D simulation and radiative transfer post-pro13essing will be
ne13essary to rene the analysis
We have investigated the 13ase of other noble gases (Ar Kr) and we have
observed that for a given laser energy the sho13k velo13ity de13reases with the
mass density This study has 13onrmed that for a given density and sho13k
velo13ity the radiative ee13ts in13rease with the atomi13 number For Kr we
100 CHAPTER 5 RESULTS AND INTERPRETATION
noti13ed a tiny pre13ursor without any intera13tion To get su13h ee13t it will
be ne13essary to in13rease the sho13k velo13ity and thus the laser energy
This has been done at the ORION laser fa13ility in UK where the 13ol-
lision of two 13ounter-propagating sho13ks at equal speed sim 80 kms with
laser uen13e sim 6 times 10
14W13m
2has been performed (Clayson et al 2017
Suzuki-Vidal et al 2016) for dierent noble gases and for pressures 13ompri-
sed between 01 and 1 bar A number of diagnosti13s setup X-ray ba13klig-
hting opti13al self-emission streak imaging and interferometry (multi-frame
and streak imaging) were used to study both the post-sho13k and the radia-
tive pre13ursor Although I have not parti13ipated in the experiments I have
performed 1D simulations to interpret ORION experimental results This
work is not presented in this thesis
Chapter 6
Optimization of an
ele13tromagneti13 generator for
strong sho13ks in low pressure
gas
Contents
61 Prin13iples of operation of a high 13urrent generator 102
62 Prin13iples of the run-down phase in a PFD 103
63 Proposed design for the plasma gun 104
64 Dynami13 13ir13uit modelling 106
65 3-D MHD simulations using GORGON 13ode 111
66 Measurements 116
67 Summary 118
The previous 13hapters were dedi13ated to the study of laser generated
radiative sho13ks With irradian13e more than 10
14W13m
2 we were able to
study sho13k waves propagating at velo13ity up to 50 kms in noble gases with
an initial mass density 13omprised between 5 times 10
minus4- 3 times 10
minus3g13m
minus3
Complementary to laser experiments 13ompa13t pulsed power generators may
drive an ele13tromagneti13 13oaxial plasma gun to 13reate astrophysi13al relevant
sho13ks in lower pressure noble gases (Kondo et al 2006) with a high avai-
lability and a rather modest 13apital 13ost The ele13tromagneti13ally driven
sho13k waves may have larger s13ales than those by laser thus they 13an be
observed rather easily (Kondo et al 2008)
Su13h ele13tri13al pulsed power devi13es may then i) 13reate a high-voltage
breakdown through a gas or more easily at the surfa13e of a diele13tri13 ii)
produ13e relatively hot plasma by ohmi13 heating when the rising 13urrent in the
devi13e is passing through a portion of gas iii) a1313elerate plasma layers under
the magneti13 pressure asso13iated with the self-generated magneti13 eld when
high ele13tri13al 13urrent (say 10s of kA) is maintained for a short but su13ient
time (say 1 micros) Su13h ionization and a1313eleration are present in Z-pin13h
plasmas and in plasma fo13us devi13es (PFD) providing a 13lever geometry is
employed and the mass under 13onsideration (say 1 mg) is 13ompatible with
the stored energy (Martin Williams and Kristiansen 1996)
101
102 CHAPTER 6 ELECTROMAGNETICALLY LAUNCHED SHOCK
In this 13hapter we will see rst how a fast ele13tri13al 13ir13uit works then
we will dis13uss the prin13iples of a 13oaxial plasma gun and of PFD We will
see what is in favor of our obje13tive and what has to be avoided A geometry
13apable to rea13h our obje13tive will be proposed
The optimization of the 13oaxial gun (in term of plasma speed) will be
performed and some typi13al gures will be given in the 13ase of a generator
with a stored energy around 1 kJ and a plasma slab of 4-mm diameter ie
quite 10 times the transverse dimension of a laser driven RS The operating
13onditions are supersoni13 sho13ks up to 10-30 kms speed at stati13 pressures
of few mbar in heavy rare gases (Ar Xe) The results of this 0-D model will
be then 13ompared to those obtained with 3-D MHD simulations performed
with the 13ode GORGON (by Andrea Ciardi at LERMA) whi13h has been
used su1313essfully to des13ribe other pulse-power driven plasma experiments
(Chittenden et al 2004) as well as laboratory plasma astrophysi13s experi-
ments (Ciardi et al 2007) The diagnosti13s whi13h have been implemented
will be presented to illustrate the model as well as preliminary re13ords of the
plasma speed
61 Prin13iples of operation of a high 13urrent gene-
rator
Our aim being to a1313elerate a plasma slab using the magneti13 pressure it
is obviously needed to drive a high intensity 13urrent be13ause the magneti13
pressure is expressed by B
22micro0 in the region where exists an indu13tion B
The pressure in bars is simply 4B
2 with B in Teslas and in a 13ylindri13al
geometry 1 Tesla is the eld around a 50-kA 13urrent at a radius of 1 13m
Thus we should 13ount on roughly 100 kA delivered by the generator but
su13h a high 13urrent is delivered only in a pulsed mode by a laboratory s13ale
devi13e
There are numbers of te13hni13al solutions to a13hieve su13h ele13tri13al pa-
rameters one may use rather slow generators at moderate high voltage like
13apa13itors bank with the advantage of well know te13hniques but with limited
adjustments A1313ording to an abundant literature
1
a 13onvenient devi13e is a
medium-energy 13apa13itor bank feeding a low-indu13tan13e 13ir13uit An R-L-C
13ir13uit is a well-known 13ombination delivering a high peak 13urrent in the
pseudo-periodi13 mode For a 13apa13itor C initially 13harged under U0 key
performan13es are as follows
Current intensity I(t) is a damped sinusoid
Pseudo pulsation ω =radic[1(LC)minusR2(4L2)]
1
Institute for Plasma Fo13us Studies resour13e website httpplasmafo13usnet last
13onne13tion in 2016
62 PRINCIPLES OF THE RUN-DOWN PHASE IN A PFD 103
First 13urrent peak Ipeak = U0[radic(LC) + 08R]
Time of 13urrent peaking T4 = π(2ω)
Equivalent impedan13e Z =radic(LC)
For a safe handling in air a voltage not higher than 30 kV is re13ommen-
ded Taking a total 13apa13itan13e of 6 microF 2700 J are stored under 30 kV For
an indu13tan13e of 240 nH the expe13ted peak 13urrent might be 150 kA at 19
micros This set of values will be a guideline for the development presented in
further se13tions
62 Prin13iples of the run-down phase in a PFD
Our aim is thus to 13reate a fast moving plasma sheath with quite a one-
dimension geometry The issues are to initiate this plasma at the best then
to maintain even improve its stru13ture during the a1313eleration nally to
laun13h it
Instead of 13reating a gas breakdown in volume whi13h would lead ra-
pidly to 13on13entrate the 13urrent in an ar13 the idea for PFD operation is to
start from a surfa13e dis13harge (Lee 1969 Bernard 2002) In the 13lassi13al
13oaxial geometry of the Mather-type PFD the 13entral ele13trode (usually the
anode) is a 13ut metal 13ylinder and the outer one is a se13ond metal 13ylin-
der of the same length or better a squirrel 13age allowing many viewpoints
to the plasma Both ele13trodes are atta13hed to a diele13tri13 bottom plate
made of polymer or 13erami13 When the 13apa13itor bank is swit13hed on the
high voltage is applied to this 13m-size gap and a radial surfa13e dis13harge is
initiated at the interfa13e with the gas forming a quite uniform ring-shaped
layer of thi13kness around 1 mm An insulating sleeve is adjusted around
the 13entral ele13trode to for13e the plasma to ow rapidly upwards under the
magneti13 pressure in this region The result is an elongation the dis13harge
path preventing an ar13ing in the plasma layer
At later times as the 13urrent is growing the plasma layer is strongly
inuen13ed it is pushed upwards by the magneti13 pressure jtimesB it is heatedby the ohmi13 ee13t then its ionization degree in13reases it be13omes denser
as a per13entage (10-40) of the heavy parti13les from the swept volume are
a1313reted (Potter 1971) This ele13tromagneti13ally driven motion is 13alled the
rundown phase Due to the a1313retion it is des13ribed by a so-13alled snowplow
model whi13h will be dis13ussed later in detail In quite all the designs even
if very high speeds are reported (Lee 1969 Serban 1995) the plasma is bell
shaped whi13h does not fulll our requirements
At the end of the rundown phase whi13h mat13hes roughly with the 13urrent
peak the plasma stays 13onne13ting the upper ends of the ele13trodes and the
radial 13omponent of the magneti13 for13es grows rapidly Then the plasma is
104 CHAPTER 6 ELECTROMAGNETICALLY LAUNCHED SHOCK
strongly pushed to the axis giving the so-13alled Z-pin13h This stage made
the PFD popular be13ause the pin13hed plasma is a sour13e of fast ele13trons
fast ions hard radiation and possibly neutron beams it must be avoided
in our 13ase It was also quoted (Lee and Serban 1996) that optimal PFD
are mat13hing a universal fa13tor 13omprising anode radius peak 13urrent and
gas density meaning a robustness of the design when a geometry has been
13hosen
63 Proposed design for the plasma gun
The previous des13ription has shown the positive inuen13e of a surfa13e dis-
13harge the modi13ation of the shape by the sleeve and the issue of the
pin13hing ee13t A1313ordingly the following design is proposed following a
work by Kondo et al (Kondo et al 2006 2008)
To ensure a rapid dis13harge with the values given in the se13tion 61 a
highly 13oaxial 13ir13uit is 13hosen for all the large parts the energy bank the
13losing swit13h and all the 13onne13tions The initial phase is kept with two
13oaxial ele13trodes pressed on a at insulator Homogeneity of the plasma
sheath is expe13ted by 13hoosing a rather small radial gap of 25 mm The ee13t
of magneti13 for13es during the rundown phase must be enhan13ed espe13ially
due to the in13rease of the mass of the plasma as des13ribed above As the
13urrent 13urve is the rising part of a sinusoid a tri13k is used to in13rease
the lo13al magneti13 eld B around a 13ondu13tor varies as the inverse of the
distan13e to axis so the proposed shape for the anode is a 13oneHowever a
nal divergen13e must be avoided and that is obtained by a rounded tip In
order to keep 13onstant the plasma length along the 13urrent path the radial
gap is kept 13onstant so the outer ele13trode is a hollow 13one with the same
angle Above the 13one the a1313elerator is 13onne13ted to a 13ondu13ting tube
where the plasma 13an propagate freely in the ba13kground gas The distan13e
of the plasma sheath to the 13one tip is in13reasing rapidly and the main
13urrent whi13h is still high will pass preferably through the diuse plasma
remaining between the ele13trodes Thus there will be no magneti13 pressure
anymore a13ting on the sheath whi13h 13an propagate freely The resulting
devi13e is des13ribed in Fig 61a and the a13ting magneti13 for13es shown in Fig
61b
Te13hni13ally su13h a 13onguration is 13onvenient even at 15 kV be13ause the
sharp edge of the anode tou13hing the insulator forms a so-13alled triple point
where the ele13tri13 eld is enhan13ed at the surfa13e of the insulator whi13h
triggers e13iently a dis13harge in presen13e of gas at low pressure (01 10
mbar) as proposed by Kondo et al (Kondo et al 2006) Polya13etal is a
13onvenient material for diele13tri13 and me13hani13al performan13es
The dire13tion and the orientation of the magneti13 for13e are other points to
be dis13ussed The high-intensity 13urrent (up to sim 150 kA) generates a strong
63 PROPOSED DESIGN FOR THE PLASMA GUN 105
(a)
(b)
Figure 61 (a) Sket13h of the sho13k generator showing the pulsed ele13tri13al 13ir13uit
the set of 13oaxial 13oni13al ele13trodes with a 13onstant radial gap and the plasti13
insulator featured in grey on whi13h a planar surfa13e dis13harge is initiated The
installation of three opti13al bers allows looking radially at the plasma moving in
the sho13k tube (b) S13hemati13s of the plasma dynami13s inside the 13oaxial gun
in fast-pulse mode the ele13tri13al 13urrent ows in the super13ial layers of the two
13oaxial 13oni13al ele13trodes and through an annular plasma layer The magneti13
pressure Pmag pushes the dis13harge axially
106 CHAPTER 6 ELECTROMAGNETICALLY LAUNCHED SHOCK
Figure 62 Exploded view of the plasma gun
azimuthal magneti13 eld between the ele13trodes thus the magneti13 pressure
jtimesB whi13h a1313elerates the annular plasma sheath stays perpendi13ular to the
sheath lo13ally (Fig 61b) The more planar is the initial sheath the more
axial will be the magneti13 pressure Finally if the roles of the ele13trodes are
ex13hanged the dire13tion of the 13urrent will be inverted and the same for B
so the pressure will stay oriented in order to expel the plasma Compared
with the Mather-type plasma fo13us (Potter 1971 Zambra et al 2009) the
plasma sheath 13onsidered here is quite planar 13lose to the insulating surfa13e
and later is expe13ted to stay planar and perpendi13ular to the axis As in a
PFD the thi13kness of the plasma sheath in13reases gradually (Zambra et al
2009) but it is assumed to stay in the mm-range We employ 13oaxial 13oni13al
brass ele13trodes ea13h 42-mm high with a 13onstant gap of 25 mm At the
bottom level internal and external radii of ele13trodes are 125 mm and 15
mm respe13tively This small-size dis13harge 13ell on top of the generator will
be modelled in the following
64 Dynami13 13ir13uit modelling
We optimize the ele13tri13 generator for various gases namely Ar and Xe
with the motivation to produ13e plasma sho13ks with speeds sim 1 - 30 kms
ie Ma13h numbers up to 200 For that we design the ele13trodes and set
64 DYNAMIC CIRCUIT MODELLING 107
the ele13tri13al 13ir13uit parameters to produ13e 1-micros pulses in the gas 13hamber
The geometry of the ele13trodes is as mentioned above The half angle of the
13oni13al ele13trode is 13hosen to be α =15deg and thus the height of the 13one is h
sim 42 mm
Eleven (N = 11) 13apa13itors ea13h with 13apa13itan13e C0 of 06 microF are13onne13ted in parallel giving an equivalent 13apa13itan13e of the bank C = 66
microF 13harging voltage U0 of the bank is set to 15 kV Thus the available
ele13tri13al energy is 750 J The 13ir13uit resistan13e is used to be 1 m Ω and
damping resistan13e is set to zero whi13h yields a total resistan13e (R = Rcir+
Rdamp N ) of 1 mΩ and a pseudo-periodi13 regime is expe13ted
Based on the above geometri13al and ele13tri13al parameters we have esti-
mated the time evolution of various parameters of the radial dis13harge layer
namely speed mass a1313eleration et13 at a dierent position along the axis
from z = 0 to h For this 13al13ulation we need to solve a set of three equa-
tions whi13h give the main 13urrent (- dqdt) supposed to ow only through
the plasma layer the mass a1313retion rate (dMdt) and the velo13ity v of
the plasma These equations together with initial 13onditions are explained
below The ele13tri13al equation is written 13lassi13ally
Ld2qdt2 +Rdqdt+ qC = 0 (61)
When the R L and C parameters are not time-dependent the 13ir13uit
equation 61 has an analyti13al solution a damped sinusoid for the 13urrent I
= -dqdt and that was used as a test of the iterative solving pro13ess and to
13onrm the parameters of the external 13ir13uit with a short 13ir13uit load The
indu13tan13e of the external 13ir13uit L0 is 13omputed from the ringing frequen13y
in short 13ir13uit (66 nH) whi13h results in a 13ir13uit impedan13e Z0 =
radic
L0Cto be 01 Ω In the plasma gun mode the sheath 13arrying the 13urrent (see
Fig 61b) is a1313elerated by the magneti13 pressure and it behaves as the de-
formable part of the 13ir13uit then L is time-dependent and a numeri13al solver
is ne13essary for equation 61 We used a lumped-parameter model whi13h has
proven to des13ribe similar 13ir13uits su1313essfully (Potter 1971 Zambra et al
2009) Gonzalez (Gonzalez et al 2004) showed that the mass and momen-
tum equations for a 13urrent sheath in the shape of an annular piston moving
forward in the axial dire13tion 13an be given by
dMdt = ερ0π(R2e minusR2
i)v (62)
where ε is a1313retion fa13tor and that the transformation of magneti13 to
kineti13 energy is expressed by
d(Mv)dt = lI22 (63)
where l is the linear indu13tan13e of the gun and 13an be expressed as l =
(micro04π) ln (ReRi) the radii being estimated where the plasma sheath is
13urrently lo13ated The equations 62 and 63 have been introdu13ed in the
108 CHAPTER 6 ELECTROMAGNETICALLY LAUNCHED SHOCK
(a)
(b)
(13)
Figure 63 Time dependen13e of the main parameters for Ar gas at 1 mbar a1313or-
ding to the 13ir13uit model (a) 13urrent (kA) and voltage (V) (b) a1313reted mass
(kg) and rate of a1313retion (kgs) (13) speed (ms) and kineti13 energy (J)
64 DYNAMIC CIRCUIT MODELLING 109
Figure 64 Plot of the dierent output parameters (normalized with respe13t to
the peak values) obtained at the top of the inner ele13trode For ea13h group upper
panel represents the variation for Ar gas whereas lower panel is for Xe gas Proles
of output parameters are given (a) with 13one angle (b) with a1313retion fa13tor (13)
with gas pressure and (d) with damping resistan13e
110 CHAPTER 6 ELECTROMAGNETICALLY LAUNCHED SHOCK
time-dependent iterative solving of equation 61 Therefore from equations
62 and 63 we get
dvdt =[ lI
2
2 minus ερ0π(R2e minusR2
i)v2]
M(64)
From equation 64 we may note that initial value of mass M(t0) 13annot
be set to zero as it would lead to a nonphysi13al divergen13e of the a1313eleration
In this regard we have 13onsidered an initial thi13kness of the 13urrent sheath
of 1 mm and integrated the gas density at rest leading to M(t0) equal to
35 times 10
minus10kg for Ar and 13 times 10
minus9kg for Xe A higher limit has to be
put too M(t0) must be small 13ompared with the mass at the end of the
laun13hing phase that is the mass 13ontained in the gap times the a1313retion
fa13tor The above value for M(t0) follows that 13riterion and within these
limits the 13hoi13e of initial mass is not inuen13ing the results
Parameter Referen13e Value Variation
Half angle of the 13one 15
13
14
15
16
middot middot middot 24A1313retion fa13tor ǫ 04 01 02 03 middot middot middot 09
Pressure P 100 Pa 10 20 50 100 200 500 1000
Damping resistan13e Rdamp 0 ohm 0 01 02
Table 61 Input parameters their standard values and range of variation
As the motion is mainly along the z-axis the speed v is the derivative
of the plasma position v = dzdt We solved these equations by a time-
dependent method where the initial 13ondition is that at the onset time t0
the 13harge q is set to CU0 Further the 13urrent i = -dqdt is taken to be zero
at t0 Thus using equation 61 the initial rate of 13hange of the 13urrent is
d
2qdt
2= - qL0C Then all the rst time-derivatives are updated at the rst
time step t0+∆t with ∆t=043 ns using equations 61-64 The quantities
like q M v z are updated at the next step then the iteration is pursued
Initially position a1313eleration and velo13ity of the sheath are set to be zero
as well as the rate of 13hange of the mass
As we initiate the ele13tri13al 13urrent into the gas inside the 13hamber the
plasma sheath is a1313elerated with respe13t to its initial mass as seen above
At later times the mass a1313retion is masking the inuen13e of this arbitrary
value We have 13al13ulated the plasma evolution in the dis13harge for dierent
sets of initial 13onditions whi13h are reported in Table 61
Fig 63 shows the behavior for the standard values reported in Table 61
in the 13ase of Argon at 1 mbar The 13orresponding values of 13harge mass
and velo13ity will be hereafter 13alled the referen13e values In Fig 63a the
variation of 13urrent and voltage in the rst half-period is 13lassi13al showing
that the pulse generator departs slightly from a pure RLC 13ir13uit Fig
63b represents the a1313reted mass whi13h in13reases in time as well whereas
65 3-D MHD SIMULATIONS USING GORGON CODE 111
the rate of mass a1313retion 13ulminates with a 13ompetition of speed growth
and de13rease of the swept volume per time step be13ause it depends on the
squared radii In Fig 6313 plotted are the speed and kineti13 energy of the
plasma sheath First the speed of the plasma is in13reasing slowly with time
quite linearly as the mass and the kineti13 energy is roughly a third power
of the time
That leads to 13hoose a design where the a1313eleration is imposed for a
su13iently long time However the estimated speed of plasma is sim 140 kms
whi13h is a very optimisti13 value The reason may 13ome from two reasons
(i) a part of the 13urrent ows through the large sho13ked volume redu13ing
the magneti13 pressure a13ting on the plasma sheath (ii) energy transfer from
magneti13 to thermal energy and ionization is redu13ing the kineti13 one The
observation of a very bright emission from the plasma will be a 13lear eviden13e
of this redistribution of the energy
Further we made a 13omparative study of the parameters indi13ated in
Table 61 and applied to two gases namely Ar and Xe In order to 13he13k the
ee13ts of the initial 13onditions (13one angle a1313retion fa13tor pressure and
damping resistan13e) we have performed a systemati13 study by varying only
one parameter and keeping the others 13onstant and equal to the referen13e
values given by the se13ond 13olumn of Table 61 The tested range is listed
in the third 13olumn of Table 61 Similarly for ea13h set the variation of the
plasma 13onditions at the top of the muzzle is normalized to the referen13e
value Fig 64 summarizes the variation of the sele13ted dierent output
parameters the time tm to rea13h the muzzle (z = h) the ele13tri13al 13urrent
and the axial speed at muzzle the laun13hed mass M the kineti13 energy and
the average speed at the top of the ele13trode 13al13ulated from the initial
point v=h( tm - t0)
Table 62 summarizes the respe13tive ratios for Ar and Xe at the top of
the 13oaxial ele13trodes As the values of interest are obtained at 13onstant
stored energy in the 13apa13itors there is a benet to work with a lighter gas
like Argon However the results obtained using this 13ir13uit model show that
the referen13e values are ensuring a satisfa13tory behavior for both gases
Speed Kineti13 energy Time to laun13h Mass
222 151 66 31
Table 62 Laun13hing performan13e ArXe
65 3-D MHD simulations using GORGON 13ode
Our simplied model (13f se13tion 64) is 13onvenient to perform qui13kly some
parametri13 tests in the a1313eleration phase and to help designing the devi13e
but it is not 13apable to treat the sho13k dynami13s in detail nor to give valuable
112 CHAPTER 6 ELECTROMAGNETICALLY LAUNCHED SHOCK
information on the plasma parameters in the sho13k region So rened nume-
ri13al simulations of the experiments were performed in 13ollaboration with A
Ciardi (Ciardi 2014) using the GORGON 13ode (Chittenden et al 2004 Ci-
ardi et al 2007 Suzuki-Vidal et al 2014) GORGON is an expli13it parallel
13ode designed to solve the resistive MHD equations on a three-dimensional
(3-D) Cartesian grid employing a Van Leer type algorithm The 13ode treats
the plasma as a single uid but solves separately the energy equations for
ions and ele13trons allowing dierent temperatures for the two spe13ies Both
thermal 13ondu13tion and resistive diusion are treated using Braginskii-like
transport 13oe13ients
Variables At rest Sho13k Post-sho13k
Ar density ρ (g13m
3) 18E-6 25E-5 10E-5
Ele13tron density Ne (13m3) 10 E+15 20E+18 15E+18
Ele13tron Temperature Te (eV) 003 (1lowast) 10 15-18
Ion density ni (13m3) 30E+16 35E+17 2-3E+7
Ion Temperature Ti 0 50 15-20
Average Speed ltVgt kms 0 50 20-40
ltzgt 02 6 7-9
Table 63 Values of variables in various 13onditions at rest (with seed ele13trons)
inside the sho13k and inside post-sho13k region A star lowast denotes the insulator surfa13e
This 13ode des13ribes the 13reation of the surfa13e dis13harge then its a1313ele-
ration by the magneti13 pressure It provides the mapping of all the plasma
parameters and lo13al B-eld in the laun13hing 13one as well as in the free ight
region The 3-D simulation was performed for Argon gas at 1 mbar only the
dis13harge 13urrent law being taken from experiment The Table 63 presents
the ranges of all the variables in the regions of interest
As an example Fig 65 shows an enlarged mapping of log (ne) at two
times in the laun13hing phase then when the sho13k has deeply penetrated
in the sho13k tube Taking the zero time as a referen13e the averaged speed
zt is 40 kms in this simulation When the two snapshots are 13onsidered
the estimate of the instantaneous speed zt is higher at 56 kms The
time history of plasma merging and early free ight into the sho13k tube
is presented in Fig 66 During the a1313eleration (inside the double 13one)
the plasma is well lo13alized on a planar annular sheath whi13h is 13arrying
the 13urrent as assumed in the lumped parameter model We see that the
merging is well a13hieved at the muzzle shortly before 1250 ns When the
sho13k has penetrated inside the tube (1500 ns) the 13omputed B-eld is
negligible at this lo13ation and the propagation is quite free as expe13ted for
the 13hosen design of the devi13e However a high 13urrent is still passing
through the gas at that time a su13ient ele13tron density remaining in the
13oni13al gap for that
65 3-D MHD SIMULATIONS USING GORGON CODE 113
Figure 65 Example of mapping of the ele13tron density from a 3-D MHD simu-
lation (ba13kground gas Argon at 1 mbar) when the plasma sheath is (left) at
the exit of the plasma gun (right) in free ight 13onditions (log s13ales for the false
13olors)
114 CHAPTER 6 ELECTROMAGNETICALLY LAUNCHED SHOCK
Figure 66 Time history of plasma merging and early free ight through the
mapping of the ele13tron density ele13tron temperature and average ioni13 13harge as
given by a 3-D MHD simulation (ba13kground gas Argon at 1 mbar) (s13ales for the
false 13olors)
65 3-D MHD SIMULATIONS USING GORGON CODE 115
Figure 67 Axial proles values taken in the sho13k tube along a line slightly o-
axis at a distan13e of 1 mm for mass density ion temperature ion density average
ion 13harge magneti13 eld ele13tron temperature ele13tron density and average speed
at 1500 ns (ba13kground gas Argon at 1 mbar) A maping of |B| is given with a
dashed line indi13ating sho13k front position as well
116 CHAPTER 6 ELECTROMAGNETICALLY LAUNCHED SHOCK
To 13onrm the steep stru13ture of the sho13k Fig 67 presents the axial
proles of ele13tron temperature Te number density Ne mass density ρ and
average ion 13harge ltZgt at 1500 ns inside the tube with an enlarged axial
position s13ale Steep fronts are observed for mass density ele13tron density
and temperature as in the earliest simulations (Potter 1971 Kondo et al
2008) The sho13k region is at 10 eV and sim2times1018 ele13trons13m
minus3 the post-
sho13k region is hotter Argon ionization stages of 5 (sho13k) to 9 (post-sho13k)
are obtained fore13asting an emission spe13trum in the UV or harder A wider
view of the plasma is given by other quantities at the sho13k front and behind
ion temperature average speed and lo13al magneti13 eld in Fig 67 We get
the 13onrmation of the extin13tion of B after the rst 5 mm inside the sho13k
tube as well as the sho13k speed at 50 kms 13oherent with the 13onstant speed
of 56 kms mentioned just above At present these parameters are guiding
the 13hoi13e of diagnosti13s like ultra-fast interferometry or UV-spe13tros13opy
66 Measurements
The aim of these tests was to show the 13onsisten13e of the above model by me-
asuring the plasma speed 13lose to the plasma gun muzzle The parameters of
the devi13e are the referen13e ones as dened in Table 61 A rather simple and
noise-free method for diagnosing a plasma gun during the rundown phase
is to re13ord the light emitted by the plasma sheath with a spatial resolution
13ompatible with the plasma stru13ture If we 13onsider a line of sight whi13h
is radial or tangential ie perpendi13ular to the dire13tion of propagation
one expe13ts to see a sudden rise when the plasma enters the dete13tion vo-
lume possibly a plateau when the plasma travels inside the volume then a
slow de13rease when the hotter plasma leaves the volume but when the sho13k
remnants are still present So the expe13ted signal is a triangular asymme-
tri13 pulse (Serban and Lee 1995) or some kind of double exponential pulse
(Stehleacute et al 2012) A peak indi13ates the time of ight for the given posi-
tion allowing to 13ompute a mean speed providing the laun13hing time (t=0)
and all positions are known An instant speed is a1313essible providing two
13lose dete13tors are 13onsidered That has been proposed and tested su1313ess-
fully in the 13ase of radiative sho13ks driven by one PALS laser beam (Stehleacute
et al 2012) the dete13tors being 13ollimated and ltered sili13on diodes For
13ooler plasmas the visible emission is more relevant Inside a squirrel 13age
plasma fo13us Serban has installed a set of opti13al ber looking tangentially
at the drift zone (Serban and Lee 1995) Despite a 13oni13al eld of view the
results were 13on13lusive at speeds as high as 100 kms in D2 An improved
version with 13ollimated opti13al bers (a1313eptan13e angle 2deg) was developed
too (Serban and Lee 1995) A non intrusive method was developed (Veloso
et al 2012ba) by imaging the plasma on a set of opti13al bers allowing to
play with the magni13ation
66 MEASUREMENTS 117
(a)
(b)
Figure 68 Experimental results (a) time history of the passive opti13al re13ords
13ompared to the main 13urrent for Argon gas 13lose to the referen13e pressure (b)
time-integrated signal vs Ar pressure for the dierent bers (13) averaged speed vs
Ar ba13king pressure
118 CHAPTER 6 ELECTROMAGNETICALLY LAUNCHED SHOCK
In our 13ase (see Fig 61a) three 1-mm PMMA (Poly(methyl metha13ry-
late)) opti13al bers are installed radially along a transparent se13tion of the
sho13k tube at z sim 625 mm 70 mm and 775 mm and fast DET10A dete13tors
re13ord the transient presen13e of the plasma sheath in the respe13tive lines of
sight The re13orded spe13tral range is limited to the whole visible in a preli-
minary stage a band-pass ltering might give information on the presen13e of
spe13i13 ioni13 spe13ies 13hara13teristi13 of a plasma temperature level however
the amount of light was not su13ient to do so Fig 68a presents the time
evolution of the ele13tri13al 13urrent with a period of 4 micros and the delayed
side-on observational data of the moving plasma inside the sho13k tube The
behavior of the ele13tri13al 13ir13uit is following well the expe13ted ringing of a
weakly damped RLC 13ir13uit the zero-time being xed at the onset of the
13urrent On opti13al 13hannels a steep rise and a peak are the signature of a
mm-thi13k fast-sho13k stru13ture (Serban and Lee 1995) 13onsistent with early
numeri13al simulations (Potter 1971 Tou Lee and Kwek 1989 Kwek Tou
and Lee 1990 Veloso et al 2012b) On ea13h ber signal se13ondary peaks
namely the one delayed by 4-5 micros are attributed to 13urrent restrikes at the
pseudoperiod To avoid the mixing of sho13ks further experiments should be
13ondu13ted in an aperiodi13 regime with a heavy-duty damping resistor added
in series to ea13h 13apa13itor Rdamp sim NZ
On Fig 68b and 13 are presented the exploitation of opti13al re13ords with
the Ar ba13kground pressure this parameter varying in the widest range (b)
the integrated emission whi13h presents a maximum between low pressure side
when the temperature is high but the number of emitters is low and the high
pressure side when the temperature is mu13h lower (13) the averaged speed
ziti for the rst peak on ea13h ber The relative intensities re13orded in the
three lo13ation s are not signi13ant be13ause of a strong 13opper deposition
13loser to the muzzle whi13h ae13ted the dete13tion The long-life noise on
the opti13al signals was making the lo13al speed 13al13ulated as (zi-zj)(ti-tj) a
non-reliable output One 13an note Ma13h numbers in the range 20 60 with
a pressure dependen13e similar to previous work (Kondo et al 2006) The
reported observations are 13onsistent with a high-speed millimetri13 planar
plasma travelling in the sho13k tube The speed is lower than dedu13ed from
the simplied model possibly due to the la13k of dissipation terms whi13h
is overestimating the transfer to kineti13 energy At the 13ontrary a realisti13
simulation as the one performed using GORGON is in better agreement
67 Summary
For this part of the work the obje13tive was to show that in 13omplement of
laser experiments a 13ompa13t pulsed power generator might drive astrophy-
si13ally relevant sho13ks in low pressure noble gases with a higher availability
and at a rather modest 13apital 13ost The longitudinal a1313eleration of a
67 SUMMARY 119
plasma sheath in a low ba13king gas pressure has been demonstrated in 13o-
axial plasma guns as early as in the 70s even leading to speeds of 100 kms
in light gas as H2 whi13h were measured and modeled A 13hoi13e was made
to revisit this type of devi13e At present the ele13tromagneti13 13oaxial plasma
guns are quite only 13onsidered as the initiators of a strong radial plasma
implosion plasma fo13us devi13es (PFD) due to a good ability to produ13e a
pin13h plasma and emission of energeti13 parti13les and hard X-ray For that
reason it was ne13essary to adapt the prin13iple of a PFD with two major
13onstraints in13rease the 13onversion of stored energy into the kineti13 energy
of a plasma sheath prevent totally any pin13hing
That obje13tive was rea13hed by dening the ele13tri13al and geometri13al
parameters of a low-indu13tan13e and 13ompa13t pulsed power generator An
optimization pro13ess has been established to mat13h a13hievable ele13tri13al pa-
rameters of the 13ir13uit (13apa13itan13e indu13tan13e peaking 13urrent intensity)
with a plasma motion in the a1313elerating tube over few 13m leading to a nal
speed of 10-30 kms A model was built up to des13ribe the dynami13s of an
RLC 13ir13uit with a varying element 13omposed of the moving plasma sheath
a1313elerated by the self generated magneti13 pressure The 13hange in plasma
position along the axis is in13reasing the indu13tan13e and the mass a1313retion
is in13reasing the plasma mass It was shown that a 13hange in the ele13trode
geometry namely a 30deg 13oni13al shape was in13reasing the energy 13onversion
and preventing any fo13using The further step in 13ontrast with PFD is the
merging of the plasma ring into a rather planar one whi13h is inje13ted in a
drift tube
Considering Ar or Xe at gas pressure in the range 01 10 mbar and
a highly 13oaxial 1-kJ devi13e the simple model was su13ient to predi13t
performan13es agreeing with the only work quoted in the literature Basi13
observations of the moving plasma using side-on dete13tion of the visible
emission give features 13oherent with the model Ma13h numbers from 20
to 60 are obtained tting with the obje13tive of strong sho13k formation A
rened 3-D MHD numeri13al simulation indi13ates very promising features of
the sho13k in view of exploring mm-size sho13ks in a regime 13omplementary
to laser driven ones and on a table-top devi13e allowing a higher repetition
rate
120 CHAPTER 6 ELECTROMAGNETICALLY LAUNCHED SHOCK
Chapter 7
Con13lusions and perspe13tives
71 Con13lusions
Although ubiquitous in the astrophysi13al environments radiative sho13ks are
13omplex phenomena whi13h still require an important eort to understand
them Beside observations whi13h suer from a la13k of angular resolution
laboratory experiments provide today an interesting approa13h to improve
our present knowledge In this 13ontext my thesis work is dedi13ated to the
experimental and numeri13al study of the 13hara13teristi13s of two 13ounter pro-
pagating radiative sho13ks propagating at dierent velo13ities (20-50 kms)
Most of the work presented here is the out13ome of the rst-of-its-kind ex-
perimental 13ampaign held in year 2015 at Prague and Laser Asterix system
(PALS) laser fa13ility
In this manus13ript I presented the setup for the aforesaid sho13k expe-
riment Following this the data analysis results interpretation as well as
relevant numeri13al simulations 13on13erning various diagnosti13s viz visible in-
terferometry and XUV spe13tros13opy have been presented Although a large
fra13tion of the experimental re13ords were performed for Xenon at various
pressures I have also studied the sho13k 13hara13teristi13s of dierent gases like
Ar Kr and He Our results are 13omplementary to those obtained at ORION
laser fa13ility also in 2015 relative to the 13ollision of identi13al radiative sho13k
waves at higher velo13ities (80 kms) In parallel in this thesis I have also
worked on the optimization of a setup dedi13ated to ele13tromagneti13ally ge-
nerated strong sho13ks at lower velo13ities In the following I briey present
a brief summary of the main results of my thesis work
In the third 13hapter I have presented a series of 1D hydrodynami13 simu-
lations (realized with the 13ommer13ial 13ode HELIOS) on isolated and 13ounter-
propagating sho13k waves at equal and dierent velo13ities The simulations
13onrm that a single radiative sho13k propagating in Xenon gas at 01 bar
is 13hara13terized by an extended pre13ursor and a large 13ompression of 30 in
the post sho13k At 50 kms the temperature evolution with the distan13e
shows identi13al post-sho13k and pre-sho13k values This indi13ates that the
numeri13al sho13k is of super13riti13al nature However I nd that there is an
important un13ertainty in the Xenon opa13ity whi13h makes the renement of
the simulation unne13essary espe13ially in terms of group numbers Next I
have investigated the intera13tion two 13ounter propagating sho13ks for iden-
ti13al (50-50 kms) and non identi13al (50-20 kms) sho13k speeds For the
121
122 CHAPTER 7 CONCLUSION
13ase of identi13al speeds the pre13ursors merge together at around 10 ns The
intera13tion is then 13hara13terized by a regular in13rease of the ele13tron density
and the temperature with the time The sho13ks 13ollision at 38 ns leads to a
jump in the ele13tron density (6 times 10
2113m
minus3) and temperature (39 eV) On
the other hand for the 13ase of non-identi13al sho13k speeds (13ase representing
our experiments) the pre13ursors intera13tion starts later than in the former
13ase The sho13ks 13ollision o1313urs at 49 ns and it is 13hara13terized by a sudden
in13rease of the ele13tron density also by an order of magnitude (4 times 10
21
13m
minus3) whereas the temperature in13reases up to 29 eV
Although the data analysis of all the experimental re13ords obtained du-
ring the PALS experimental 13ampaign has been 13arried out only a few repre-
sentative interferometri13 and spe13tros13opi13 re13ords for Xe gas are dis13ussed
in details in this thesis The interferometri13 re13ords allowed me to estimate
the average sho13k speed and time variation of the ele13tron density during the
sho13k propagation The sho13k speeds of the MAIN and AUX radiative sho13k
waves are found to be ranging between sim 30-55 and 12-25 kms respe13tively
whereas the averaged pre13ursor ele13tron density is varying between 10
17and
10
1913m
minus3during the sho13ks propagation
From the interferometri13 re13ords I have investigated the ee13t of the
intera13tion between the two radiative pre13ursors for the sho13ks propagating
in Xenon at 01 bar with respe13tive speeds of 50plusmn3 and 23plusmn3 kms The
intera13tion starts at 20 ns and is followed by the merging of the two ra-
diative pre13ursors This pre13ursor intera13tion is 13hara13terized through the
enhan13ement of the ionisation wave The sho13ks 13ollision is re13orded at 50
ns On the 13ontrary the same intera13tion behavior is not seen at 02 bar
(sho13k speeds sim 38plusmn4 and 18plusmn2 kms for the MAIN and AUX sho13k waves
respe13tively) Moreover in this 13ase there is no signature of a radiative pre-
13ursor for the AUX sho13k The pre13ursors intera13tion if any should then
13ould o1313ur at times whi13h are outside of the re13ord and 13an not be 13on-
rmed by the experiment This indi13ates that for a given gas and sho13k
speed the radiative ee13ts de13reases with the initial mass density
The investigation of the lateral extension of the sho13k has been made
through the analysis of transverse interferograms At 02 bar the MAIN
sho13k with a speed sim 40 kms has a lateral pre13ursor extension of sim570plusmn30microm whereas it is 275plusmn25 microm for AUX sho13k whi13h is propagating
with a speed of 20 kms The MAIN sho13k pre13ursor is almost of a at
spatial prole whereas the AUX pre13ursor is more 13urved suggesting that
the 2D ee13ts are mu13h more important for AUX than for MAIN One of
the explanations is that the spot size of AUX laser on the target is smaller
than the target width whereas the MAIN laser has a spot size whi13h is equal
to the target width
In order to interpret the experimental results with appropriate 1D simu-
lations I have optimized the lasers uen13es to numeri13ally obtain a sho13k
speed equal to that re13orded in the experiment In the 13orresponding simu-
72 PERSPECTIVES 123
lations for Xenon at 01 bar I note the intera13tion of the two pre13ursors
However I do not nd a good quantitative agreement for the ele13tron den-
sity Beside the question of the impre13ise opa13ities this disagreement might
also be attributed to 2D ee13ts (Gonzaacutelez Audit and Stehleacute 2009 Leygna13
et al 2006 Stehleacute et al 2010)(Cotelo et al 2015)
The analysis of spa13e- and time-integrated XUV re13ord at 06 bar is
presented for two 13ounter propagating sho13ks of speeds sim 36plusmn4 and 18plusmn5kms for MAIN and AUX respe13tively The presen13e of HeII Balmer lines
and Xenon lines tends to indi13ate a temperature of the sho13k of about 15
eV and a Xenon mean ion 13harge around 6 - 7 The 1D simulations predi13t
ele13tron temperature in the range of 10-30 eV while ion 13harge to be ranging
between 5 and 10
In addition to this study whi13h was fo13used on the Xe 13ase I have
investigated the sho13k 13hara13teristi13s for Ar Kr at 13lose mass density (sim 6
times 10
minus4g13m3) whi13h 13orrespond to 03 bar and Kr gas 02 bar Almost no
pre13ursor is found to be present in 13ase of Ar whereas a tiny pre13ursor have
been observed for Kr This indi13ates that for a given density and sho13k
speed the radiative ee13ts in13rease with the atomi13 number
In parallel I have worked on the optimization of the design of an experi-
mental setup where the sho13k is generated ele13tromagneti13ally This setup
allows studying sho13k sim 30 kms in noble gas at sim1 mbar The parameters
of a low-indu13tan13e and 13ompa13t pulsed power devi13e have been optimized
with a simple model in view of building-up and a1313elerating a plasma sheath
in Ar and Xe at gas pressure in the range 01-10 mbar Simple observations
of the moving plasma using side-on dete13tion of the visible emission give
features 13oherent with the model Sho13ks 13orresponding to ma13h numbers
ranging from 20 to 60 are obtained
72 Perspe13tives
The experimental results presented in this thesis 13orrespond to the 13ase of
two 13ounter-propagating radiative sho13ks We have been able to study the
time variations of the pre13ursor intera13tion and the ee13t of gas pressure
with 1D time dependent interferometri13 re13ords obtained with a streak 13a-
mera Together with the help of XUV spe13tros13opy we 13ould estimate the
sho13k speed ele13tron density and sho13k temperature However no informa-
tion is provided about the lateral extension of the pre13ursor and we were
not able to re13ord XUV data for all the shots Therefore in order to make
these results more 13omprehensive we have designed and parti13ipated in a
follow-up experimental 13ampaign in September 2016 at PALS This time we
fo13ussed on single sho13ks in 13lose experimental 13onditions but with a wider
range in terms of speeds (45-100 kms) We repla13ed the time dependent
streak 13amera interferometri13 study by an instantaneous 2D interferometri13
124 CHAPTER 7 CONCLUSION
image (GOI) obtained with a short pulse sub pi13ose13ond laser and 13ameras
XUV spe13tra were also re13orded The re13ords were performed at three dif-
ferent times of the sho13k propagation allowing to dedu13e the sho13k speed
and the time evolution of the pre13ursor morphology This follow up experi-
ment will allow to 13omplete the analysis of the laser generated sho13k waves
presented in this thesis
The preliminary analysis I have performed on these new re13ords is very
promising An example of these interferometri13 re13ords is given in the Figure
71a for the shot 30364 in Xe+He at 06 bar The high speed of the sho13k is
related to the energy of the laser whi13h is here 170 J whereas it was less than
120 J in the previous experiment The 13omparison with the interferometri13
re13ord (not shown) before the shot allows to dedu13e the ele13tron density
whi13h rea13hes 18 times 10
1913m
3 The high quality re13ord shows 13learly 2D
ee13ts for this sho13k wave
(a)
(b)
Figure 71 Interferometri13 re13ord obtained at 546 ns for a single radiative sho13k
of speed sim 100 kms propagating in Xe+He gas at 06 bar driven by laser at 348
nm with energy 170 J The dierent 13olors in (b) 13orrespond to yellow le Ne le35 times 10
1713m
minus3 13yan 37 - 74 times 10
1713m
minus3 blue74 - 15 times 10
1813m
minus3 green
15 - 18 times 10
1813m
minus3 magenta 18 - 22 times 10
1813m
minus3 red 22 - 26 times 10
1813m
minus3
orange26 times 10
1813m
minus3- 18 times 10
1913m
minus3
This analysis is now ongoing and the results of this experimental 13am-
72 PERSPECTIVES 125
paign will be 13ompared with that obtained from the analysis of re13ords from
2015 presented in this manus13ript This instantaneous imaging interferome-
tri13 diagnosti13 does not follow the sho13k 13ontinuously with the time but it
gives a pre13ise map of the ele13tron density whi13h 13an be inverted through
Abel inversion to obtain the lo13al estimation of the ele13tron density
In parallel in the ele13tromagneti13ally laun13hed sho13ks we have upgra-
ded our table top setup to a13hieve higher speeds For this we have used 12
13apa13itors ea13h of 1 microF at the pla13e of the previous 11 13apa13itors ea13h of
06 microF We have implemented new diagnosti13s namely a visible interferome-
ter working in the Gated Opti13al Imaging mode and time and spa13e visible
spe13tros13opy similar to the one des13ribed in the 13hapter 4 For the visible in-
terferometry we have Ma13h Zehnder Interferometer setup and we will image
the sho13k at dierent times with the help of a SL300 pi13ose13ond high energy
NdYAG laser (λ = 532 nm and energy = 024 J) The experiment will soon
be performed on this new setup and the analysis of su13h re13ords will also be
undertaken in future allowing a 13omparison of ele13tromagneti13ally laun13hed
sho13ks with the laser-driven sho13k whi13h should make the bridge between
thin and opti13ally thi13ker sho13ks
126 CHAPTER 7 CONCLUSION
Chapter 8
Thesis summary
Radiative sho13ks (RS) are present in various astrophysi13al environments es-
pe13ially in stars They are then present in all the supersoni13 a1313retioneje13tion
pro13esses during the stellar evolution from the early phase of the proto-
stellar 13ollapse up to the magnetosphere a1313retion on to the already formed
T- Tauri stars up to the late stage of the supernovae sho13ks
Radiative sho13ks are strong sho13ks (ie Ma13h number Mgtgt1) whi13hrea13h high temperatures and thus are the sour13e of intense radiation Alt-
hough the observation of (ie spatially resolved ) signatures of su13h sho13ks
in the astrophysi13al environments is very di13ult an alternate approa13h na-
med laboratory plasma astrophysi13s provides a powerful tool to study them
on the Earth
These waves have been experimentally studied sin13e more than a de13ade
mostly on large-s13ale laser fa13ilities in noble gases and with dierent targets
geometries The ablation pro13ess generated by the intera13tion between the
laser and a foil indu13es a sho13k wave in the foil whi13h then propagates
in the gas at a speed of several tens of kms Numeri13al simulations 13an
des13ribe these sho13ks with an improving pre13ision With laser intensities on
target 13omprised between 1014 and 1015 W13m
2 these experiments allowed
to re13ord sho13k speeds ranging between 40 and 150 kms
All previous experimental studies have been fo13used on the 13ase of single
radiative sho13ks However in astrophysi13al 13onditions the radiative sho13k
in general intera13ts with a denser medium leading to the development of
ree13ted and transmitted sho13ks A few representative examples of su13h
phenomena are the intera13tion of supernovae remnants with dense mole13ular
13louds the a1313retion sho13ks on the photosphere of T-Tauri stars and the bow
sho13ks at the head of fast stellar jets The 13ollision (or the intera13tion) of
two radiative sho13k waves is obviously a rare astrophysi13al event and the
template 13ase of the supernova remnants DEM L316 (see Fig 1 of Williams
et al (1997)) is still the subje13t of debates Williams et al (2005) Toledo-
Roy et al (2009) Velarde et al (2006) In this regard the development of
dedi13ated laboratory experiments to the study of propagation and intera13tion
of 13ounter-propagating sho13k waves is important as a tool to 13hara13terize
su13h events through their spe13i13 signatures
Complementary to laser experiments 13ompa13t pulsed power generators
may drive an ele13tromagneti13 13oaxial plasma gun to 13reate astrophysi13al
relevant sho13ks in lower pressure noble gases with a high availability and a
127
128 CHAPTER 8 THESIS SUMMARY
rather modest 13apital 13ost The ele13tromagneti13ally driven sho13k waves may
have larger s13ales than those by laser Thus they 13an be analyzed rather
easily with the adequate diagnosti13s (Kondo et al 2008)
In this thesis I am presenting radiative sho13ks in the 13ontext of labo-
ratory astrophysi13s using two experimental approa13hes laser-driven sho13ks
and ele13tromagneti13ally driven sho13ks
First 13hapter of my thesis is Radiative sho13k waves in whi13h the physi13s
of radiative sho13ks have been studded Radiative sho13k waves are hypersoni13
sho13k waves whi13h are heated to high temperature and as a 13onsequen13e
be13ome the sour13e of intense radiation This radiation in turn modies
the dynami13s and stru13ture of the sho13k itself whi13h makes its stru13ture
more 13ompli13ated Radiative sho13k waves o1313ur in several astrophysi13al 13ir-
13umstan13es su13h as in the a1313retion sho13k of protostellar formation (Stahler
Palla and Salpeter 1986) the supernova explosion and the intera13tion of
their remnants with the dense interstellar medium (Chevalier 1977) the bow
sho13ks at the head of stellar jets (Hartigan et al 2001)
Radiative sho13k waves 13an now be studied in the laboratory whi13h allows
to 13ompare the data with models existing in the literature and to 13he13k the
ability of the numeri13al 13odes to reprodu13e them (see for instan13e Bouquet
et al (2004) Leygna13 et al (2006)) The experimental study of radiative
sho13k 13hara13teristi13s is 13ru13ial in understanding these various energeti13 pro-
13esses o1313urring in the astrophysi13al plasmas
The jump 13onditions (Rankine-Hugoniot relations) for the simple 13ase
of an ideal gas whi13h rely on the values of the thermodynami13al quantities
on both sides of the dis13ontinuity Let us 13onsider a one-dimensional sho13k
propagating in a gas at rest with the speed us In the frame asso13iated with
the sho13k front the pre-sho13k (upstream) uid velo13ity is then u1 = - us
while behind the sho13k (post-sho13k or downstream region) this velo13ity is
u2 as shown in the Fig 21 Considering the 13ase of strong sho13ks where
M may be 13onsidered to be ≫1 jump 13onditions for ideal gas are
ρ2ρ1
=(γ + 1)
(γ minus 1)(81)
T2
T1=
2M2γ(γ minus 1)
(γ + 1)2(82)
kBT2 =2(γ minus 1)
(γ + 1)2mus
2(83)
For mono-atomi13 gases γ = 53 the 13ompression ratio is estimated to
be 4 and the post-sho13k temperature is
kBT2 =3
16mu1
2 =3
16mPAu1
2(84)
129
This temperature is thus proportional to the square of the sho13k velo13ity
and to the atomi13 mass In 13ontrast to the dis13ussion on ideal gas in the
13ase of a real gas is a little more 13ompli13ate It is to note that in this 13ase
a part of the kineti13 energy is used to ex13ite and ionize the post-sho13k gas
As a result its temperature is lower than that for the 13ase of ideal gas
In addition the pressure is also modied due to ionisation (Mi13haut et al
2004) In13luding the ee13t of ionization with the average 13harge z in the
medium for the 13ase real gas jump relations for real gas are
ρ2u2 = ρ1u1 (85)
ρ2u22 + ρ2
kBT2
m(1 + z2) = ρ1u
21 + ρ1
kBT1
m(1 + z1) (86)
ρ2u2
[5
2
(1 + z2)
mkBT2 + ǫexc2 +
u222
]
= ρ1u1
[5
2
(1 + z1)
mkBT1 + ǫexc1 +
u212
]
(87)
In the previous both 13ases the ee13ts of radiation heating and 13ooling
have been negle13ted However the radiation emitted in a strong sho13k will
ae13t its stru13ture These ee13ts 13an be determined using mass momentum
and energy 13onservation equations whi13h now have to in13lude the 13ontribu-
tions of the radiative ux energy and pressure This 13ase is more 13omplex
now Taking into a1313ount radiative ee13ts the equations of radiative hydro-
dynami13s in the 13o-moving frame of the sho13k (Mihalas and Mihalas 1999
Drake 2006) 13an be written as
partρ
partt= minusnabla(ρu) (88)
ρ
(
partu
partt+ unablau
)
= minusnabla(P + Prad) (89)
part
partt
(
ρu2
2+ ρǫ+ Erad
)
+nabla
[
ρu
(
ǫ+u2
2+
P
ρ
)
+ (Erad + Prad)u
]
= minusnablaFrad
(810)
The 13ontributions linked to Prad and Erad are ee13tive only at very high
temperature (ie velo13ity) However for the radiative sho13ks of interest in
our experiments the 13ontribution of the radiative ux Frad is non-negligible
13ompared to ρu3 These sho13ks are thus in the radiation ux dominated
regime Most of the experimental sho13k waves are in this regime These
hydrodynami13 equations are strongly non-linear
Depending on the opa13ity the radiation emitted from the sho13k may be
absorbed by the pre-sho13k region indu13ing its pre-heating Far away from
the dis13ontinuity the stru13ture of the upstream medium is determined by
the absorption On its side the stru13ture of the transition layer of the down-
stream medium 13lose to the dis13ontinuity depends on the balan13e between
130 CHAPTER 8 THESIS SUMMARY
the emission (13ooling) and the absorption (heating) Thus the full stru13ture
of the sho13k mainly depends on the variation of the opa13ity χ (13m
2g) The
opti13al depth (τ(s)) measured from the position of the jump is then the
relevant parameter for the absorption It is given by
τ(s) =
int s
sjump
χ(sprime)ρ(sprime) dsprime (811)
where sprime 13hara13terizes the path of the radiation As we are interested in
the mean frequen13y averaged radiative ux the relevant opa13ity here will be
the averaged Rosseland opa13ity whi13h will be dened in due 13ourse
If τ gtgt 1 the regime is referred as opti13ally thi13k while in the opposite
13ase (τ lt 1) it is opti13ally thin Flux dominated radiative sho13ks have
been 13lassied depending on the opti13al depth of upstream and downstream
regimes
We are more interested in the kind of sho13k have a radiative pre13ursor It
is possible to produ13e a radiative sho13k at the high energy laser fa13ilities with
laser intensity of sim 10
14W13m
2 A short summary of number of experiment
performed at high power laser fa13ilities have been listed in table 21 Other
than laser driven radiative sho13k it is possible to laun13h 13omparatively slower
radiative sho13ks (sim 10-30 kms) using ele13tromagneti13 generators
The 13hapter two is dedi13ated to the numeri13al analysis of radiative sho13k
waves and of the main physi13al pro13esses whi13h play a role in these 13omplex
pro13esses Radiative sho13ks are present in various astrophysi13al pro13esses
implying extreme 13onditions Laboratory experiments then allow investiga-
ting the underlying physi13al pro13esses whi13h take pla13e in these 13onditions
and whi13h are not observable from the Earth due to a la13k of spatial resolu-
tion However experiments are not always straightforward to interpret and
the help of numeri13al simulations be13omes pre13ious
Dierent numeri13al approa13hes are used for the simulation of laboratory
plasmas Some of them use xed grids Other introdu13e the possibility of
renement (on the y) in some meshes (Adaptative Mesh Renement) to
maintain a good resolution in the interesting parts of the plasma Another
approa13h is based on meshes following the uid (Lagrangian des13ription)
whi13h means that the meshes move with the uid and no matter may go
through a mesh to another The form of the equations dier in this 13ase (see
for instan13e Orban et al (2013))
HELIOS is su13h a one-dimensional Lagrangian radiation hydrodynami13
13ommer13ial 13ode (Ma13Farlane Golovkin and Woodru 2006) and I have
used it to simulate our experiment of intera13ting sho13k waves This 13ode
is easy to handle and has the presently interesting 13apability to be able to
simulate the 13ase of two sho13k waves propagating in opposite dire13tions and
laun13hed by two dierent lasers HELIOS may des13ribe non-LTE plasmas as
it in13ludes an in-line 13ollisional radiative (HELIOS-CR) model for 13omputing
131
non-LTE atomi13 level populations at ea13h time step of the hydrodynami13s
simulation However I used HELIOS in the LTE approximation whi13h is
appropriate to our experimental 13onditions (Rodriguez et al 2011) as will
be dis13ussed in the next se13tion
The 13ode solves the uid equations of motion using the pressure 13ontri-
butions of ele13trons ions and radiation It may des13ribe the ele13trons and
ions as two intera13ting uids at respe13tive temperatures Te and Ti Thermal
13ondu13tion is des13ribed within a ux-limited ele13tron 13ondu13tion model ba-
sed on the Spitzer 13ondu13tivity and the laser energy deposition is 13omputed
with an inverse Bremsstrahlung model
The radiation emission and absorption terms are introdu13ed in the equa-
tion of energy for the ele13trons and in the radiation transport equations One
of the methods whi13h are proposed is the ux-limited multi-group radia-
tion diusion model where the radiative ux is proportional to the gradient
of the radiative energy and is inversely proportional to the Rosseland opa-
13ity The expression is pondered by a ux-limited diusion 13oe13ient in
order to obtain the good opti13ally thin limit This 13oe13ient follows the
Larsen expression des13ribed in Olson Auer and Hall (2000) The se13ond
method is a (time independent) multi-angle short-13hara13teristi13s s13heme In
our 13ase where radiation and hydrodynami13s are strongly 13oupled and thus
the radiation eld varies rapidly we used the diusion model together with
LTE Plan13k and Rosseland multi-groups opa13ities
In addition our version of HELIOS uses the PROPACEOS (Ma13Farlane
Golovkin and Woodru 2006) Equation of State (EOS) and multi-group
opa13ity tables whi13h are generated by the 13ode
1
and it is not possible
to in13lude any other opa13ity or EOS table The knowledge of the thermo-
dynami13al 13onditions is required for an adequate simulation of the plasma
des13ription In our experiment the mass density ranges between sim 10
minus4and
10
minus1g13m
3 while the temperature values are 13omprised between and sim 0 to
50 eV The pre13ise 13al13ulations performed by Rodriguez et al (Fig2 of Rodri-
guez et al (2011)) for Xenon indi13ate that the thermodynami13al 13onditions
of our radiative sho13k experiments 13orrespond to the LTE regime
The equation of state (EOS) denes the dependen13e of the pressure
ionisation internal energy with the mass density and temperature Several
models do exist in the literature
Our version of HELIOS for LTE plasmas may handle polytropi13 or PRO-
PACEOS equation of state Unfortunately there are not many details about
this last method The authors of the 13ode mention that it uses a QEOS-type
model (More et al 1988) in the strong 13oupling regime (ie at relatively
high densities and low temperatures) and an isolated atom model whi13h
uses a detailed atomi13 stru13ture modeling in the weak 13oupling region The
properties between the weak and strong 13oupling regimes are said to be obtai-
1
see http wwwprism minus cscomSoftwarePROPACEOS
132 CHAPTER 8 THESIS SUMMARY
ned by interpolating in a thermodynami13ally 13onsistent manner This EOS
in13ludes the 13ontributions from the translation of ions and ele13trons ele13-
tron degenera13y atom ionization and ex13itation and Coulomb intera13tions
(Debye - Hu13kel 13orre13tion)
I have performed HELIOS 1D simulations for dierent 13ases for single
and 13ounter propagating sho13ks in 13hapter 2 of my thesis Radiative sho13k
waves propagating in Xenon at 01 bar with a velo13ity of 50 kms are 13ha-
ra13terized by the development of an extended radiative pre13ursor The huge
13ompression of 38 in the post-sho13k is a 13onsequen13e of both the sho13k and
the gas ionisation (fa13tor of 10) as also from the radiative 13ooling The post-
sho13k and pre-sho13k temperature on both side of the peak are identi13al
indi13ating that the sho13k is super13riti13al
The spe13ta13ular role of the opa13ity has been highlighted Due to the
un13ertainty in the opa13ity used for Xenon in our simulations I de13ided that
at this stage it was not ne13essary to rene the simulation in terms of group
numbers within the 1D des13ription of the sho13k wave
Our numeri13al study gives the main 13hara13teristi13s of the intera13tion of
two 13ounter-propagating sho13k waves with speeds equal to 50-50 kms and
50-20 kms The 13ase of identi13al speeds is simpler due to the symmetry of
the problem However whatever the speeds the most important signature
of the intera13tion is the merging of the pre13ursor at 8 ns for 50-50 kms
and at 15 ns for 50-20 kms This merging is followed by a regular in13rease
with time of the ele13tron density and the temperature The 13ollision time
is 13hara13terized by a sudden in13rease of the ele13tron density by an order of
magnitude rea13hing 66 times 10
21and 3 times 10
2113m
minus3respe13tively whereas the
temperature in13reases up to 39 and 28 eV
When a short pulse of a high-power laser beam is fo13ussed on a thin foil
it transfers a huge amount of energy to the foil material Almost instantane-
ously the laser absorption heats it up to the ablation A hot and low density
13orona is generated ba13kwards by this pro13ess Beyond the ablation surfa13e
a sho13k is formed in the foil as a 13onsequen13e of the momentum 13onservation
(ro13ket ee13t) The sho13k moves inward in the foil Our 13ase of the experi-
mental setup as the foil 13loses a tube lled with gas the sho13k propagates
then in the gas where it is studied through various kinds of diagnosti13s The
foil then a13ts as a piston
We performed the experiments at Prague Asterix Laser System (PALS)
in the Spring 2015 (ve weeks 20th April - 22th May) The obje13tive of the
experimental 13ampaign was to study the evolution of two 13ounter streaming
radiative sho13ks driven by two 13ounter-propagating lasers with an irradian13e
in the range of 10
14W13m
minus2 and therefore to 13hara13terize their intera13tion
and 13ollision In the 13hapter third 13hapter I have rst presented a brief
des13ription of the PALS laser fa13ility and of the laser beams used in our
experiment This followed by a presentation of the targets design and an
overview of the general setup and of our main diagnosti13s
133
The Prague Asterix Laser System (PALS) is a laser fa13ility based on
an Infrared high-power iodine laser system (Asterix IV) (Jungwirth et al
2001) Using dierent amplifying stages the laser fa13ility is able to deliver
energy up to 1 kJ in 03 ns at the fundamental wavelength 1315 nm The
output laser beam 13an be further subdivided in few auxiliary beams All
auxiliary beams may be frequen13y doubled (λ = 657 nm) or tripled (438
nm) The PALS laser fa13ility is 13apable of ring up to two high energy laser
shots per hour Compared with solid states lasers this gas laser is known to
deliver a quite homogenous beam intensity without hot spots
For our experiments we used two independent laser beams to drive the
two 13ounter streaming sho13ks in the gaseous target For this purpose the
fundamental PALS laser beam has been subdivided after the fourth amplier
(V4) into two beams with respe13tive energy fra13tions of 60 and 40 (see
Fig 42) The most energeti13 of these laser beams is then inje13ted through
the fth (V5) amplier and its frequen13y are then tripled This beam at 438
nm will be hereafter termed termed as the MAIN laser beam
The se13ond of the two previous laser beams propagates then without any
modi13ation It will be hereafter 13alled the AUX laser beam (1315 nm)
The s13heme of the energy distribution is shown in the Fig 42 and the
13hara13teristi13s of MAIN and AUX laser are presented in Table 41
For the experiments PALS oers two va13uum 13hambers with a spheri13al
and 13ylindri13al geometry respe13tively The spheri13al 13hamber assembly used
in our experiments is shown in Fig 43 This 13hamber having a diameter
of 100 13m is 13losed by two entran13e 13ir13ular windows of respe13tive diameters
80 13m and 50 13m to allow the MAIN and AUX beams to penetrate in the
13hamber and to rea13h the target Ea13h of these windows is outtted with a
hinged door Several 13ir13ular ports of various diameters also allow for the
dierent diagnosti13s The target holder and some diagnosti13s are mounted
on an opti13al ben13h whi13h is de13oupled from the 13hamber walls
Our targets (see Fig 45) s13hemati13ally 13onsist in small tubes of 4 mm
length 13losed on both sides by two spe13i13 foils of thi13kness 11 microm on whi13h
the laser beams are fo13used (one laser per foil) with an irradian13e of about
10
1413m
minus3 The tube is lled with gas in whi13h the radiative sho13k propa-
gates with a velo13ity of 30 - 60 kms The two foils 13losing the target insure
the 13onversion through ablation and sho13k generation of the laser energy
into me13hani13al energy
In our experiments we used spe13i13 targets for alignment and 13hara13te-
rization of the beam size These targets were manufa13tured at the Pole
instrumental of Observatoire de Paris
The gaseous targets were lled in situ at a fra13tion of 1 bar with a gas
(viz Xe Ar a mixture of Xe and He) whi13h thus provide the medium for the
sho13k propagation We 13hose an in-situ lling system (Fig 410) to redu13e
the dieren13e of pressures inside and outside of the tube This is 13riti13al as
a large pressure dieren13e between the target 13ell and the va13uum 13hamber
134 CHAPTER 8 THESIS SUMMARY
13an result in the breaking of the ultra-thin SiN windows of the target
First of all we performed fo13alization of MAIN and AUX laser beam
on target by using Al-massive targets The two PALS laser lenses were
translated up to a13hieve the suitable diameters on the target whi13h were
nally set to 450 - 500 microm and 250 - 300 microm for the MAIN and AUX beams
The size of the impa13t was 13ontrolled ea13h day on mo13k targets before using
the laser beams for real shots on gaseous targets It is worth to pre13ise that
the keV diagnosti13 was still in pla13e for the shots on the gaseous targets
In that 13ase the beams impa13t Parylene-N instead of Aluminium The keV
signal is then weaker than for the mo13k aluminum targets
Various kinds of diagnosti13s were implemented namely visible interfero-
metry XUV and visible spe13tros13opy For the same two keV 13ameras are
indu13ted to monitor the impa13t and two visible 13ameras for target alignment
For visible interferometory we have taken referen13e images (without any
fringe) for every target as shown in the Fig 417a for one target example
In order to re13ord su13h images we blo13ked the referen13e beam and let the
probe beam only to illuminate the target in a stati13 mode and with the slit
open Then we redu13ed the slit width to its nominal value (200 microm) This
enabled us to have the visualization of the portion of the target whi13h was
imaged in the slit on the target and also to lo13ate the position of the pistons
before the shot This image is termed as the referen13e target image Then
the system was pla13ed in the interferometri13 mode and the fringe pattern
was optimized to be perpendi13ular to the slit and thus in the longitudinal
mode perpendi13ular to the dire13tion of the sho13k propagation The Fig
417b reports su13h a typi13al interferogram re13orded during our experiment
(13f Fig 413) The horizontal axis 13orresponds to the dire13tion of the sho13k
propagation (with a total s13ale of 54 mm on the target) and the verti13al
axis 13orresponds to time (s13aling 200 ns)
The time and spa13e integrated XUV plasma emission are re13orded with
a Flat Field XUV spe13trometer using a 13on13ave grating In the Fig 419
the drawing of the XUV spe13trometer setup s13heme (red 13olor) is presented
together with the spheri13al 13hamber and the target The XUV spe13trometer
is installed on the top of the spheri13al 13hamber whi13h allows the XUV
emission passing through the Si3N4 membrane on the top of the target (see
se13tion 422) to be re13orded
In addition to the interferometry and XUV spe13tros13opy we have also im-
plemented a time- and spa13e-resolved visible spe13tros13opi13 (see Fig 420a)plasma
emission diagnosti13
In 13hapter 5 I present the results of the two diagnosti13s with a parti13u-
lar fo13us on the interferometri13 data whi13h I have extensively studied with
the help of a spe13i13 data analysis pro13edure that I have developed The
visible interferometry is an adequate tool to probe the ele13tron density of
the radiative pre13ursor up to the 13riti13al density (4 times 10
2113m
minus3at the wa-
velength 527 nm of the probing laser) whereas the denser post-sho13k region
135
is opaque to the visible light XUV radiation is present in the whole sho13k
stru13ture and thus the time- and spa13e-integrated XUV spe13tros13opy may
allow exploring the self-emission 13oming from the post-sho13k and pre13ursor
regions
In the visible interferometry the interferen13es between the main and
referen13e beams overlapping on the dete13tor produ13e a pattern of fringes
whi13h follow the relative phase variations between the probe (whi13h passes
through the plasma) and the referen13e beams (see se13tion 441 of the previ-
ous 13hapter for the experimental setup) The phase variation between two
13onse13utive fringes of the unperturbed beams (ie without any plasma) is
equal to 2π Then in the presen13e of the plasma in the probe beam these
fringes be13ome shifted due to the modi13ation of the refra13tive index More
details about the prin13iple of the interferometry and the Ma13h-Zehnder in-
terferometri13 setup may be found in the APPENDIX A and we re13all here
only the expression of the phase shift
∆φ asymp minus πd
λNclt Ne gt (812)
where λ = 527 nm is the wavelength of the probing laser Nc = 4 times 10
21
13m
minus3the 13riti13al density at this wavelength and lt Ne gt is the ele13tron
density averaged over the laser path d in the 13ell and is dened as
lt Ne gt=
int d
0
Ne(z t)dy
d(813)
In this 13hapter I have presented an extensive data analysis of few re-
presentative interferometri13 and spe13tros13opi13 re13ords I am presenting here
summary of results obtained bellow
The average sho13k speed and ele13tron density have been estimated from
the interferograms The sho13k speeds of the MAIN and AUX radiative sho13k
waves vary between sim 30-55 and 10-30 kms respe13tively and the averaged
pre13ursor ele13tron density ranges between 10
17and 10
1913m
minus3during the
sho13ks propagation
We have demonstrated the intera13tion ee13t between the two radiative
pre13ursors in the 13ase of Xe at 01 bar at 54 and 23 kms The intera13tion is
13learly 13hara13terized in the experiment by the enhan13ement of the ionisation
wave followed by the merging of the two radiative pre13ursors at 20 ns The
13ollision time is re13orded at 47 ns Su13h behavior is reprodu13ed by the
simulation
We have investigated this intera13tion at a larger pressure 02 bar with
the following speeds sim 41 kms for the MAIN and sim 18 kms for the AUX
sho13k waves We do not re13ord any experimental signature of the radiative
pre13ursor for AUX Further we have not been able to 13at13h experimentally
the 13ollision time The re13orded pre13ursor of MAIN is not inuen13ed by AUX
wave up to 48 ns (Fig 59(b) and (13) in data analysis 13hapter) whi13h is the
136 CHAPTER 8 THESIS SUMMARY
limit of the re13ord On its side the 1D simulation predi13ts a tiny pre13ursor
for AUX and that both pre13ursors start to intera13t at 49 ns This plausible
sho13ks intera13tion o1313urring at times whi13h are outside of the re13ord 13an
not be 13onrmed by our experiment
The results of the transverse interferometry at 02 bar with speeds of sim40 and 20 kms indi13ate that the MAIN pre13ursor has a lateral extension of
sim 600 microm whereas it is 300 microm for AUX The pre13ursor of MAIN is almost
at with a probable small bending at the edges of the tube whereas the
AUX pre13ursor is more 13urved This means that the 2D ee13ts are more
important for AUX than for MAIN Still in the 13ase of Xenon we have
obtained information about the temperature and the mean 13harge of gas
through our time integrated XUV spe13tra (shot48143) in Xenon at 06
bar From these results we may 13on13lude that the mean ion 13harge is at
least equal to 6 and that the temperature has rea13hed 15 eV
Our simulations give a qualitative des13ription of the sho13ks intera13tion
when the laser uen13e is adjusted to give the 13orre13t sho13k velo13ities Howe-
ver it is now well known that 2D simulations (together with state of the art
opa13ities) t better with experiments (Gonzaacutelez Audit and Stehleacute 2009
Leygna13 et al 2006 Stehleacute et al 2010) For the same laser energy the 2D
simulations lead to a diminution of the sho13k speed 13ompared to 1D as also
to a diminution of the ele13tron density For instan13e in the 13ase of a sho13k
wave laun13hed by a laser beam at 1315 nm in Xenon at 03 bar at PALS and
with a laser uen13e of 85000 J13m
2 ARWEN 2D simulations give a sho13k
speed of 44 kms in agreement with the experimental one (Cotelo et al
2015) 1D simulation would require in this 13ase a uen13e of 30000 J13m
2to
a13hieve the same velo13ity
The spa13e and time integrated XUV re13ords at 06 bar for respe13tive
speeds whi13h are equal to sim 39 and 18 kms indi13ate that the temperature
of the sho13k has rea13hed values up to 15 eV and that the Xenon mean ion
13harge has also rea13hed values of 6 - 7 whereas 1D simulations predi13t ele13tron
temperature 10-30 eV and ion 13harge 5-10 (Fig 51813) A more detailed
study based on 2D simulation and radiative transfer post-pro13essing will be
ne13essary to rene the analysis
We have investigated the 13ase of other noble gases (Ar Kr) and we have
observed that for a given laser energy the sho13k velo13ity de13reases with the
mass density This study has 13onrmed that for a given density and sho13k
velo13ity the radiative ee13ts in13rease with the atomi13 number For Kr we
noti13ed a tiny pre13ursor without any intera13tion To get su13h ee13t it will
be ne13essary to in13rease the sho13k velo13ity and thus the laser energy
This has been done at the ORION laser fa13ility in UK where the 13ol-
lision of two 13ounter-propagating sho13ks at equal speed sim 80 kms with
laser uen13e sim 6 times 10
14W13m
2has been performed (Clayson et al 2016
Suzuki-Vidal et al 2016) for dierent noble gases and for pressures 13ompri-
sed between 01 and 1 bar A number of diagnosti13s setup X-ray ba13klig-
137
hting opti13al self-emission streak imaging and interferometry (multi-frame
and streak imaging) were used to study both the post-sho13k and the radia-
tive pre13ursor Although I have not parti13ipated in the experiments I have
performed 1D simulations to interpret ORION experimental results This
work is not presented in this thesis
The previous 13hapters were dedi13ated to the study of laser generated
radiative sho13ks With irradian13e more than 10
14W13m
2 we were able to
study sho13k waves propagating at velo13ity up to 50 kms in noble gases with
an initial mass density 13omprised between 5 times 10
minus4- 3 times 10
minus3g13m
minus3
Complementary to laser experiments 13ompa13t pulsed power generators may
drive an ele13tromagneti13 13oaxial plasma gun to 13reate astrophysi13al relevant
sho13ks in lower pressure noble gases (Kondo et al 2006) with a high avai-
lability and a rather modest 13apital 13ost The ele13tromagneti13ally driven
sho13k waves may have larger s13ales than those by laser thus they 13an be
observed rather easily (Kondo et al 2008)
Su13h ele13tri13al pulsed power devi13es may then i) 13reate a high-voltage
breakdown through a gas or more easily at the surfa13e of a diele13tri13 ii)
produ13e relatively hot plasma by ohmi13 heating when the rising 13urrent in the
devi13e is passing through a portion of gas iii) a1313elerate plasma layers under
the magneti13 pressure asso13iated with the self-generated magneti13 eld when
high ele13tri13al 13urrent (say 10s of kA) is maintained for a short but su13ient
time (say 1 micros) Su13h ionization and a1313eleration are present in Z-pin13h
plasmas and in plasma fo13us devi13es (PFD) providing a 13lever geometry is
employed and the mass under 13onsideration (say 1 mg) is 13ompatible with
the stored energy (Martin Williams and Kristiansen 1996)
In this 13hapter we will see rst how a fast ele13tri13al 13ir13uit works then
we will dis13uss the prin13iples of a 13oaxial plasma gun and of PFD We will
see what is in favor of our obje13tive and what has to be avoided A geometry
13apable to rea13h our obje13tive will be proposed
The optimization of the 13oaxial gun (in term of plasma speed) will be
performed and some typi13al gures will be given in the 13ase of a generator
with a stored energy around 1 kJ and a plasma slab of 4-mm diameter ie
quite 10 times the transverse dimension of a laser driven RS The operating
13onditions are supersoni13 sho13ks up to 10-30 kms speed at stati13 pressures
of few mbar in heavy rare gases (Ar Xe) The results of this 0-D model will
be then 13ompared to those obtained with 3-D MHD simulations performed
with the 13ode GORGON (by Andrea Ciardi at LERMA) whi13h has been
used su1313essfully to des13ribe other pulse-power driven plasma experiments
(Chittenden et al 2004) as well as laboratory plasma astrophysi13s experi-
ments (Ciardi et al 2007) The diagnosti13s whi13h have been implemented
will be presented to illustrate the model as well as preliminary re13ords of the
plasma speed
Our aim being to a1313elerate a plasma slab using the magneti13 pressure
it is obviously needed to drive a high intensity 13urrent be13ause the magneti13
138 CHAPTER 8 THESIS SUMMARY
pressure is expressed by B
22micro0 in the region where exists an indu13tion B
The pressure in bars is simply 4B
2 with B in Teslas and in a 13ylindri13al
geometry 1 Tesla is the eld around a 50-kA 13urrent at a radius of 1 13m
Thus we should 13ount on roughly 100 kA delivered by the generator but
su13h a high 13urrent is delivered only in a pulsed mode by a laboratory s13ale
devi13e
There are numbers of te13hni13al solutions to a13hieve su13h ele13tri13al pa-
rameters one may use rather slow generators at moderate high voltage like
13apa13itors bank with the advantage of well know te13hniques but with limited
adjustments A1313ording to an abundant literature
1
a 13onvenient devi13e is a
medium-energy 13apa13itor bank feeding a low-indu13tan13e 13ir13uit An R-L-C
13ir13uit is a well-known 13ombination delivering a high peak 13urrent in the
pseudo-periodi13 mode For a 13apa13itor C initially 13harged under U0 key
performan13es are as follows
Current intensity I(t) is a damped sinusoid
Pseudo pulsation ω =radic[1(LC)minusR2(4L2)]
First 13urrent peak Ipeak = U0[radic(LC) + 08R]
Time of 13urrent peaking T4 = π(2ω)
Equivalent impedan13e Z =radic(LC)
For a safe handling in air a voltage not higher than 30 kV is re13ommen-
ded Taking a total 13apa13itan13e of 6 microF 2700 J are stored under 30 kV For
an indu13tan13e of 240 nH the expe13ted peak 13urrent might be 150 kA at 19
micros This set of values will be a guideline for the development presented in
further se13tions
For this part of the work the obje13tive was to show that in 13omple-
ment of laser experiments a 13ompa13t pulsed power generator might drive
astrophysi13ally relevant sho13ks in low pressure noble gases with a higher
availability and at a rather modest 13apital 13ost The longitudinal a1313elera-
tion of a plasma sheath in a low ba13king gas pressure has been demonstrated
in 13oaxial plasma guns as early as in the 70s even leading to speeds of 100
kms in light gas as H2 whi13h were measured and modeled A 13hoi13e was
made to revisit this type of devi13e At present the ele13tromagneti13 13oaxial
plasma guns are quite only 13onsidered as the initiators of a strong radial
plasma implosion plasma fo13us devi13es (PFD) due to a good ability to pro-
du13e a pin13h plasma and emission of energeti13 parti13les and hard X-ray For
that reason it was ne13essary to adapt the prin13iple of a PFD with two major
13onstraints in13rease the 13onversion of stored energy into the kineti13 energy
of a plasma sheath prevent totally any pin13hing
1
Institute for Plasma Fo13us Studies resour13e website httpplasmafo13usnet last
13onne13tion in 2016
139
That obje13tive was rea13hed by dening the ele13tri13al and geometri13al
parameters of a low-indu13tan13e and 13ompa13t pulsed power generator An
optimization pro13ess has been established to mat13h a13hievable ele13tri13al pa-
rameters of the 13ir13uit (13apa13itan13e indu13tan13e peaking 13urrent intensity)
with a plasma motion in the a1313elerating tube over few 13m leading to a nal
speed of 10-30 kms A model was built up to des13ribe the dynami13s of an
RLC 13ir13uit with a varying element 13omposed of the moving plasma sheath
a1313elerated by the self generated magneti13 pressure The 13hange in plasma
position along the axis is in13reasing the indu13tan13e and the mass a1313retion
is in13reasing the plasma mass It was shown that a 13hange in the ele13trode
geometry namely a 30deg 13oni13al shape was in13reasing the energy 13onversion
and preventing any fo13using The further step in 13ontrast with PFD is the
merging of the plasma ring into a rather planar one whi13h is inje13ted in a
drift tube
Considering Ar or Xe at gas pressure in the range 01 10 mbar and
a highly 13oaxial 1-kJ devi13e the simple model was su13ient to predi13t
performan13es agreeing with the only work quoted in the literature Basi13
observations of the moving plasma using side-on dete13tion of the visible
emission give features 13oherent with the model Ma13h numbers from 20
to 60 are obtained tting with the obje13tive of strong sho13k formation A
rened 3-D MHD numeri13al simulation indi13ates very promising features of
the sho13k in view of exploring mm-size sho13ks in a regime 13omplementary
to laser driven ones and on a table-top devi13e allowing a higher repetition
rate
140 CHAPTER 8 THESIS SUMMARY
Appendi13es
141
Appendix A
Visible Interferometry
Laser interferometry is used in plasmas to dedu13e the ele13tron density through
the variation of the refra13tive index 13aused by the ionisation of the matter
In this appendix rstly the experimental setup of Ma13h Zehnder interfero-
metry as well as the underlying physi13al prin13iples are presented
A01 Refra13tive index of a plasma
In order to derive the refra13tive index of a gas one starts with the set of
Maxwells equations for a mono13hromati13 plane wave propagating in the
plasma (see for instan13e equation 411 of referen13e (Hut13hinson 2002))
Free ele13trons and ions are supposed to be distributed uniformly and the net
total 13harge density is equal to zero
For harmoni13 waves propagating in the dire13tion of +x and with the
pulsation ω one obvious solution is given by
E(x t) = E0ei(kxminusωt)
(A1)
where E0 and k are 13omplex quantities
The relation between k and ω is given by the dispersion relation
k2 =εmicroω2
c2(1 + i
4πσ
ωε) (A2)
where micro is the ele13tri13 permitivity and ε is permeability
The free ele13trons are a1313elerated by the ele13tri13 eld and therefore equa-
tion their motion is given as
medv
dt= minuseE0e
minusiωt(A3)
The obvious solution for the ele13tron velo13ity v is
v = minusie
meωE (A4)
and thus the ele13trons 13arry a 13urrent with a density J
J = Neev = iNee
2
meωE (A5)
where Ne is the ele13tron density Due to the larger mass of the ions the
13orresponding 13urrent is negligible
143
144 APPENDIX A VISIBLE INTERFEROMETRY
The previous equation allows to derive the plasma 13ondu13tivity σ whi13his equal to iNee
2ωme Putting this value in the equation (A2) and assu-
ming a thin medium where ε asymp 1 and micro = 1 one obtains nally
k2 =ω2
c2(1minus
ω2p
ω2) (A6)
where ωp is the plasma frequen13y dened as
ω2p =
4πNee2
me(A7)
Equation A6 shows that a ele13tromagneti13 wave with a pulsation larger than
ωp 13ant propagate in the plasma This allows to derive a 13riti13al ele13tron
density Nc in 13m
minus3as
Nc =meω
2
4πe2=
4πmec2
e2λ2(A8)
For instan13e for a radiation of 1 microm wavelength this upper limit is equal
to Nc = 446times 1021cmminus3
Therefore the index of refra13tion is given by the expression
n =
radic
1minusω2p
ω2=
radic
1minus Ne
Nc(A9)
Finally one obtains
E(x t) = E0ei(kxminusωt) with k = n
ω
c(A10)
As a 13onsequen13e of the variation of the refra13tive index in the dierent
plasma layers the light is refra13ted Indeed the Snell laws stipulate that if a
beam is in13ident on a plane interfa13e between layers (of refra13tive index n1
and n2) it undergoes a bending from the angle of in13iden13e The in13ident
beam the refra13ted beam and the normal to the interfa13e lie in the same
plane and one has n1 sinθ1 = n2 sinθ2 where θ1 is the angle of in13iden13e
and θ2 is the angle of refra13tion Thus a mono13hromati13 beam propagating
in a plasma with in13reasing density be13omes more and more dee13ted
A02 Absorption of the laser beam
Beside refra13tion the light is absorbed by the plasma and the absorption
also in13reases rapidly near the 13riti13al density In the absen13e of any ato-
mi13 bound-bound or bound-free transition this absorption is due to the
inverse Bremsstrahlung pro13ess (ion-ele13tron 13ollision (Pfalzner 2006)) by
145
the plasma free ele13trons The absorption 13oe13ient κ (in 13m
minus1) is then
given by (NRL formulary)
κ(Ne Tev) = 317 10minus7LnΛ lt Z gt
(
Ne
ω
)2 1
T32ev (1minusNeNc)12
(A11)
where Tev is the temperature in eV Ne is the ele13tron density in 13m
minus3
and LnΛ is the Coulomb Logarithm dened as Λ = Max (2 Min(XY))
with (NRL formulary)
X = 23minus log
(
lt Z gtN
12e
T32ev
)
and Y = 24minus log
(
N12e
Tev
)
(A12)
The transmission T (d) through a layer of thi13kness d is given by
T (d) = exp(minusint d
0κ(l)dl) (A13)
For instan13e for λ = 527 nm taking Ne= 101913mminus3 ltZgt= 10 T = 10 eV
whi13h are typi13al values for the post-sho13k in our experiments in Xenon and
supposing a homogenous plasma thi13kness d = 600 microm the transmission is
about 09 whereas at 5times 1019cmminus3 it falls to 01
A03 Prin13iple of interferometry
In the opti13al interferometry te13hnique (Ovsyannikov and Zhukov 2000) the
interferen13e of two plane waves is performed by addition of the two 13orre-
sponding 13oherent ele13tri13 elds The analyzed medium (here the plasma)
in whi13h one of the beam is propagating indu13es a deshaping between the
beams whi13h ae13ts the resulting intensity depending if the elds interfere
in phase or out of phase ie 13onstru13tively or destru13tively
Two te13hniques are used the wave front and the amplitude division
The wave front division is obtained by using two portions (Fresnel mir-
rors Youngs double slit Lloyds mirror prisms et13) of the original wave
front whi13h are then superimposed to produ13e interferen13e
In the amplitude division the two beams are separated by division of
the amplitude over the same se13tion of the wave front Mi13helson and Ma13h
Zehnder interferometers et13 are the typi13al examples of this te13hnique of
amplitude division
The prin13iple of interferen13e of two 13oherent mono13hromati13 traveling
waves 1 and 2 with ele13tri13 elds E1 and E2 propagating in the same
dire13tion and polarized in the same plane 13an be understood as follows for
a given time t at the point r where these waves superpose the amplitude of
the ele13tri13 eld is the ve13tor addition of the two 13orresponding elementary
amplitudes ie
146 APPENDIX A VISIBLE INTERFEROMETRY
E(r t) = E1(r t) + E2(r t)
= a1exp(ikr + ωt+ iφ1) + a2exp(ikr + ωt+ iφ2)
where φ1 and φ2 are the phases of the two waves at r The total intensity
I(r t) = |E1 + E2|2 (A14)
One gets
I = I1 + I2 + 2radic
(I1I2)cos∆φ (A15)
where I1 = a12and I2 = a2
2are the intensities at the point of interferen13e
due to the two waves a13ting independently and ∆φ = φ1minusφ2 represents the
phase dieren13e between the two waves In the 13ase of equal intensities I1one gets
I = 4I1cos2(∆φ2) (A16)
and the interferometri13 pattern 13onsists of dark (∆φ = π + 2nπ) and
bright patterns (∆φ = 2nπ)In general the two beams are only partially 13oherent and the minimum
of the fringe intensity is not equal to zero The fringe 13ontrast
FC =Imax minus Imin
Imax + Imin(A17)
is then a measure of the interferen13e quality FC is maximum and equal to
1 in the 13ase of fully 13oherent beams of the same intensity I1 = I2
A04 Ma13h Zehnder Interferometer
The Ma13h Zehnder Interferometer is a devi13e 13ommonly and e13iently used
to estimate the 13hange in refra13tive index by the plasma Su13h 13hange is in
most the 13ases related to variations in the ele13tron density We employed
this interferometri13 setup in our experiment
The prin13iple of this interferometer is based on the division of the in13ident
light beam into two beams with the help of a beam-splitter whi13h 13an be
a 13oated glass plate or a 13ube After this division one of the beams (probe
beam) is allowed to pass through the medium of interest (a plasma in our
13ase) and then made to interfere with the other unae13ted part of the beam
(referen13e beam) with the help of se13ond beam-splitter as may be noted
from Fig A1
In order to understand the prin13iple of interferen13e and its use in esti-
mating plasma properties let us 13onsider l1 and l2 whi13h are the total path
lengths (in13luding the path in the opti13al elements) for the light travelling
from the sour13e to the dete13tor for the upper and lower paths (13f Figure A1)
respe13tively Suppose for the moment that the sample is removed from beam
147
Figure A1 S13hemati13s of Ma13h Zehnder interferometer with two mirrors two
50 beam splitters and the dete13tor A plasma slab is pla13ed in one of the two
arms of the interferometer
1 (see Figure A1) the two opti13al paths of the probe and referen13e beams
dier in terms of phases Simplify to supposing that the beams propagate
in air (n=1) these phases are then equal to the 2πl1λ and 2πl2λThus a1313ounting for the phase dieren13e δ between the two beams tra-
velled up to dete13tor A 13an be given as
2πl1 minus l2
λ= δ (A18)
If the two beams are perfe13tly parallel at the position of the dete13tor the
interferometri13 gure presents a at intensity whi13h passes through minima
and maxima when one length of the 2 arms is slightly 13hanged Usually one
introdu13es a small tilt between the two dire13tions of propagations whi13h
leads to the apparition of parallel linear dark and bright fringes
In order to obtain interferen13e patterns it is important to make the paths
l1 and l2 as 13lose as possible 13ompared to the 13oheren13e length of the laser
For the multimode Evolution 15 laser used in the PALS experiment this
length is approximately 1 13m (value 13ommuni13ated by the 13onstru13tor)
If we introdu13e the sample in the probe beam 1 this sample will introdu13e
an additional phase denoted by δsample and the net phase shift is then δ +δsample
The prin13iple of the dedu13tion of the ele13tron density with the Ma13h
Zehnder interferometer is to measure the refra13tive index of the test plasma
slab whi13h is pla13ed in the probe beam We assume for the moment that
148 APPENDIX A VISIBLE INTERFEROMETRY
this plasma slab is a phase obje13t for the in13oming beam whi13h means that
it does not signi13antly ae13t the intensity of the probe beam while it only
13hanges the phase of the wave [5]The additive phase dieren13e in the probe arm introdu13ed by the plasma
13an be dened as
φ =
int d
0kplasmadl =
int d
0nω
cdl (A19)
where n is the lo13al refra13tive index at position l of the plasma slab and
the integral is performed other the slab thi13kness d Therefore the phase
dieren13e introdu13ed by the plasma relative to the propagation into va13uum
of the referen13e beam is equal to
∆φ =
int d
0(kplasma minus k0)dl (A20)
where k0 = ωc = 2πλ is the wave ve13tor of the beam propagating in
va13uum
As long as Ne lt Nc the beam propagates in the plasma the plasma
refra13tive index is given by equation A9 and one has
∆φ =ω
c
int d
0
[
(
(1minus Ne
Nc
)12
minus 1]
dl (A21)
At very small ele13tron density (Ne ≪ Nc) the phase shift 13an be written as
∆φ asymp minus ω
2cNc
int d
0Nedl = minus πd
λNclt Ne gt (A22)
where
lt Ne gt=
int d
0
Ne(z t)dy
d(A23)
is the ele13tron density averaged over the path dIn pra13ti13e a referen13e re13ord is rst taken without plasma To dedu13e
the density one has to analyze how the fringes depart from their referen13e
positions
The spa13e between two fringes (inter-fringe) 13orresponds to a phase va-
riation of 2 π A shift of the fringe by 2 π from its referen13e position (where
Ne = 0) 13orresponds to an ele13tron density variation equal to lt Ne gt = 2
λNcd For λ = 527 nm and supposing an homogeneous plasma of thi13kness
600 microm this gives lt Ne gt= 7 times 10
1813m
minus3
Appendix B
Opa13ities and mean 13harge
The variations in logarithmi13 s13ale with the temperature of the PROPA-
CEOS Rosseland and Plan13k opa13ities (se13tion of 13hapter 5 see for intense
Fig 511)as also of the mean 13harge of the plasma are reported for two
densities 16 times 10
minus3g13m
3and 51 times 10
minus4g13m
3for three noble gases Ar
Kr and Xe in Fig B1 and B2
ρ = 51 times 10
minus4g13m
313orrespond to typi13al density of the pre13ursor
For this density Xenon and Krypton have quite similar variations for the
Rosseland opa13ity with a bump of 7000 13m
2g at 3 eV for Xe and 12000
13m
2g at 4 eV for Krypton At 10 eV the two opa13ities de13rease respe13tively
to 600 and 900 13m
2g In 13omparison with these two 13ases the opa13ity of
Argon is broader and more regular Its opa13ity peaks at 6 eV to 13000 13m
2g
At 10 eV it have de13reased to 7000 13m
2g whi13h is higher than for the two
previous gases
The mean 13harge variations in13rease regularly with the temperature At
10 eV these mean 13harges are respe13tively equal to 65 55 and 5 for Xe Kr
and Ar
149
150 APPENDIX B OPACITIES AND MEAN CHARGE
(a)
(b)
Figure B1 Rosseland and Plan13k opa13ity for three gases Xe Kr and Ar at mass
densities 51 times 10
minus4g13m
3(a) and 16 times 10
minus3g13m
3(b)
151
(a)
(b)
Figure B2 Mean 13harge for three gases Xe Kr and Ar at mass densities 16 times10
minus3g13m
3(a) and 51 times 10
minus4g13m
3(b)
152 APPENDIX B OPACITIES AND MEAN CHARGE
Referen13es
Batani D Strati F Telaro B Loumlwer T Hall T Benuzzi-Mounaix A
Koenig M 2003 Produ13tion of high quality sho13ks for equation of state
experiments European Physi13al Journal D 23 99 DOI ADS
Bernard A 2002 Plasma fo13us et appli13ations Le Vide 57(306)
Bett TH Jinks P Stevenson RM Danson CN Pepler DA Ross
IN 1995 Binary phase zone-plate arrays for laser-beam spatial-intensity
distribution 13onversion Applied Opti13s 34 4025 DOI ADS
Bouquet S Teyssier R Chieze JP 2000 Analyti13al study and stru13-
ture of a stationary radiative sho13k The Astrophysi13al Journal Supplement
Series 127(2) 245
Bouquet S Stehleacute C Koenig M Chiegraveze J Benuzzi-Mounaix A Ba-
tani D Leygna13 S Fleury X Merdji H Mi13haut C et al 2004
Observation of laser driven super13riti13al radiative sho13k pre13ursors Phy-
si13al review letters 92 225001
Bozier J Thiell G Le Breton J Azra S De13roisette M S13hirmann
D 1986 Experimental observation of a radiative wave generated in xenon
by a laser-driven super13riti13al sho13k Physi13al review letters 57(11) 1304
Burhop E Spitzer L 1957 Physi13s of fully ionized gases inters13ien13e
tra13ts on physi13s and astronomy no 3 JSTOR
Chaulagain U 2015 Radiative sho13ks expriments modellin and links to
Astrophysi13s PhD thesis Pierre and Marie Curie University
Chaulagain U Stehleacute C de Saacute L Larour J Auvray P Kozlova M
Krus M Dostal J Propupek J Suzuki-Vidal F et al 2012 Labo-
ratory experiments of radiative sho13ks in the 13ontext of stellar a1313retion
In SF2A-2012 Pro13eedings of the Annual meeting of the Fren13h So13iety
of Astronomy and Astrophysi13s 1 305
Chaulagain U Stehleacute C Larour J Kozlovaacute M Suzuki-Vidal F Bar-
roso P Cotelo M Velarde P Rodriguez R Gil JM Ciardi A
A13ef O Nejdl J de Saacute L Singh RL Ibgui L Champion N 2015
Stru13ture of a laser-driven radiative sho13k High Energy Density Physi13s
17 106 DOI ADS
Chevalier RA 1977 The intera13tion of supernovae with the interstellar
medium Annual review of astronomy and astrophysi13s 15 175 DOI ADS
153
154 REFERENCES
Chittenden JP Lebedev SV Jennings CA Bland SN Ciardi A
2004 X-ray generation me13hanisms in three-dimensional simulations of
wire array Z-pin13hesthe evolution of magneti13 tower jets in the laboratory
Plasma Physi13s and Controlled Fusion 46 B457 DOI ADS
Ciardi A 2014 private 13ommuni13ation
Ciardi A Lebedev SV Frank A Bla13kman EG Chittenden JP
Jennings CJ Ampleford DJ Bland SN Bott SC Rapley J Hall
GN Suzuki-Vidal FA Maro1313hino A Lery T Stehle C 2007 The
evolution of magneti13 tower jets in the laboratory Physi13s of Plasmas
14(5) 056501 DOI ADS
Clayson T Suzuki-Vidal F Lebedev SV Swadling GF Burdiak GC
Patankar S Smith RA Foster J Skidmore J Gumbrell E Gra-
ham P Danson C Stehleacute C Singh RL Chaulagain U Larour
J Kozlova M Spindloe C 2016 Counter-propagating radiative sho13k
experiments on the Orion laser fa13ility In APS Meeting Abstra13ts DOI
ADS
Clayson T Suzuki-Vidal F Lebedev S Swadling G Stehleacute C Bur-
diak G Foster J Skidmore J Graham P Gumbrell E et al 2017
Counter-propagating radiative sho13k experiments on the orion laser and
the formation of radiative pre13ursors High Energy Density Physi13s 23
60
Cotelo M Velarde P de la Varga AG Portillo D Stehleacute C Chaul-
again U Kozlova M Larour J Suzuki-Vidal F 2015 Simulation
of radiative sho13k waves in Xe of last PALS experiments High Energy
Density Physi13s 17 68 DOI ADS
de la Varga AG Velarde P Cotelo M de Gaufridy F
Zeitoun P 2011 Radiative properties for warm and hot
dense matter High Energy Density Physi13s 7(3) 163 DOI
httpwwws13ien13edire13t13oms13ien13earti13lepiiS1574181811000498
Diziegravere A Mi13haut C Koenig M Gregory C Ravasio A Sakawa
Y Kuramitsu Y Morita T Ide T Tanji H et al 2011 Highly
radiative sho13k experiments driven by gekko xii Astrophysi13s and Spa13e
S13ien13e 336(1) 213
Dobbs-Dixon I Cumming A Lin DNC 2010 Radiative Hydrodyna-
mi13 Simulations of HD209458b Temporal Variability The Astrophysi13al
Journal 710 1395 DOI ADS
Doss FW Robey HF Drake RP Kuranz CC 2009 Wall sho13ks in
high-energy-density sho13k tube experiments Physi13s of Plasmas 16(11)
112705
REFERENCES 155
Drake RP 2006 High-energy-density physi13s Fundamentals inertial fu-
sion and experimental astrophysi13s Sho13k wave and high pressure phe-
nomena Springer Berlin Heidelberg New York ISBN 3-540-29314-0
httpopa13inriafrre13ord=b1128433
Drake R Doss F M13Clarren R Adams M Amato N Bingham D
Chou C DiStefano C Fidkowski K Fryxell B et al 2011 Radiative
ee13ts in radiative sho13ks in sho13k tubes High Energy Density Physi13s
7(3) 130
Ensman L 1994 Test problems for radiation and radiation-hydrodynami13s
13odes The Astrophysi13al Journal 424 275 DOI ADS
Ensman L Burrows A 1992 Sho13k breakout in sn 1987a The Astrophy-
si13al Journal 393 742
Fadeyev YA Gillet D 2001 The stru13ture of radiative sho13k waves
III The model grid for partially ionized hydrogen gas Astronomy and
Astrophysi13s 368 901 DOI ADS
Fleury X Bouquet S Stehleacute C Koenig M Batani D Benuzzi-
Mounaix A Chieze J-P Grandjouan N Grenier J Hall T et al
2002 A laser experiment for studying radiative sho13ks in astrophysi13s
Laser and Parti13le Beams 20(02) 263
Fryxell B Olson K Ri13ker P Timmes FX Zingale M Lamb DQ
Ma13Nei13e P Rosner R Truran JW Tufo H 2000 FLASH An
Adaptive Mesh Hydrodynami13s Code for Modeling Astrophysi13al Ther-
monu13lear Flasheshera13les a three-dimensional radiation hydrodynami13s
13ode Astrophysi13al Journal Supplement 131 273 DOI ADS
Gonzalez JH Brollo FR Clausse A 2009 Modeling of the Dynami13
Plasma Pin13h in Plasma Fo13us Dis13harges Based in Von Karman Approx-
imations IEEE Transa13tions on Plasma S13ien13e 37 2178 DOI ADS
Gonzalez JH Clausse A Bruzzone H Florido PCMZ 2004 A Lum-
ped Parameter Model of Plasma Fo13us IEEE Transa13tions on Plasma
S13ien13e 32 1383 DOI ADS
Gonzaacutelez M Audit E Huynh P 2007 HERACLES a three-dimensional
radiation hydrodynami13s 13ode Astronomy and Astrophysi13s 464 429
DOI ADS
Gonzaacutelez M Audit E Huynh P 2007 Hera13les a three-dimensional
radiation hydrodynami13s 13ode Astronomy and Astrophysi13s 464(2) 429
Gonzaacutelez M Audit E Stehleacute C 2009 2D numeri13al study of the ra-
diation inuen13e on sho13k stru13ture relevant to laboratory astrophysi13s
Astronomy and Astrophysi13s 497 27 DOI ADS
156 REFERENCES
Gonzaacutelez M Stehleacute C Rus B Busquet M Thais F A13ef O Barroso
P Bar-Shalom A Bauduin D Kozlova M Lery T Madouri A To-
mas M Polan J 2006 Astrophysi13al radiative sho13ks From modeling
to laboratory experiments Laser and Parti13le Beams 24 535540 DOI
ADS
Gonzalez M StehlE C Audit E Busquet M Rus B Thais F A13ef
O Barroso P Bar-Shalom A Bauduin D et al 2006 Astrophysi13al
radiative sho13ks From modeling to laboratory experiments Laser and
Parti13le Beams-Pulse Power and High Energy Densities 24(4) 535
Gonzaacutelez M Vaytet N Commerccedilon B Masson J 2015 Multigroup
radiation hydrodynami13s with ux-limited diusion and adaptive mesh
renement Astronomy and Astrophysi13s 578 A12 DOI ADS
Griem HR 2005 Prin13iples of Plasma Spe13tros13opy 386 ADS
Gu MF 2008 The exible atomi13 13ode Canadian Journal of Physi13s
86(5) 675
Harilal SS Tilla13k MS 2004 Laser plasma
density measurements using interferometry
httplinkapsorgdoi101103PhysRevE64066412
Hartigan P 1989 The visibility of the Ma13h disk and the bow sho13k of a
stellar jet Astrophysi13al Journal 339 987 DOI ADS
Hartigan P 2003 Sho13k Waves in Outows from Young Stars Astrophysi13s
and Spa13e S13ien13e 287 111 DOI ADS
Hartigan P Morse JA Reipurth B Heath13ote S Bally J 2001 Pro-
per Motions of the HH 111 Jet Observed with the Hubble Spa13e Teles13ope
Astrophysi13al Journal Letters 559 L157 DOI ADS
Henke BL Gullikson E Davis JC 1993 X-ray intera13tions Photoab-
sorption s13attering transmission and ree13tion atlt igt eltigt= 50-30000
evlt igt zltigt= 1-92 Atomi13 data and nu13lear data tables 54(2) 181
Hugoniot PH 1887 Meacutemoire sur la propagation du mouvement dans les
13orps et ples speacute13ialement dans les gaz parfaits Journal de matheacutematiques
pures et appliqueacutees 3 477
Hugoniot PH 1889 Meacutemoire sur la propagation des mouvements dans les
13orps et speacute13ialement dans les gaz parfaits (deuxiegraveme partie) Journal de
lEacute13ole Polyte13hnique 58 1
Hut13hinson IH 2002 Prin13iples of Plasma Diagnosti13s 458 ADS
REFERENCES 157
Jungwirth K Cejnarova A Juha L Kralikova B Krasa J Krousky
E Krupi13kova P Laska L Masek K Mo13ek T et al 2001 The
prague asterix laser system Physi13s of Plasmas 8 2495
Keiter P Drake R Perry T Robey H Remington B Iglesias C
Walla13e R Knauer J 2002 Observation of a hydrodynami13ally driven
radiative-pre13ursor sho13k Physi13al review letters 89(16) 165003
Koenig M Faral B Boudenne JM Batani D Benuzzi A Bossi S
1994 Opti13al smoothing te13hniques for sho13k wave generation in laser-
produ13ed plasmas Physi13al Review E 50 R3314 DOI ADS
Kondo K Nakajima M Kawamura T Horioka K 2006 Compa13t pulse
power devi13e for generation of one-dimensional strong sho13k waves Review
of S13ienti13 Instruments 77(3) 036104 DOI ADS
Kondo K Nakajima M Kawamura T Horioka K 2008 Relaxation
layer in ele13tro-magneti13ally driven strong sho13ks Journal of Physi13s Con-
feren13e Series 112(4) 042028 DOI ADS
Kondo K Nakajima M Kawamura T Horioka K 2009a Atomi13 pro-
13ess in ele13tro-magneti13ally driven strong sho13k wave Nu13lear Instruments
and Methods in Physi13s Resear13h A 606 223 DOI ADS
Kondo K Nakajima M Kawamura T Horioka K 2009b Radiation ef-
fe13t and relaxation layer in ele13tro-magneti13ally driven strong sho13k waves
In APS Meeting Abstra13ts ADS
Kunze H-J 2009 Springer series on atomi13 opti13al and
plasma physi13s 125 Springer Berlin Heidelberg New York
httpwwwspringer13omseries411
Kwek K Tou T Lee S 1990 Current sheath stru13ture of the plasma
fo13us in the run-down phase IEEE Transa13tions on Plasma S13ien13e 18(5)
826
Larour J Singh RL Stehleacute C Ciardi A Chaulagain U Suzuki-Vidal
F 2015 Optimization of an ele13tromagneti13 generator for strong sho13ks
in low pressure gas High Energy Density Physi13s 17 129 DOI ADS
Lee S 1969 Transverse ionizing sho13k waves in a planar ele13tromagneti13
sho13k tube PhD thesis Australian National University
Lee S Serban A 1996 Dimensions and lifetime of the plasma fo13us pin13h
IEEE Transa13tions on plasma s13ien13e 24(3) 1101
Leygna13 S Boireau L Mi13haut C Lanz T Stehleacute C Clique C
Bouquet S 2006 Modeling multidimensional ee13ts in the propagation
of radiative sho13ks Physi13s of Plasmas (1994-present) 13(11) 113301
158 REFERENCES
Ma13Farlane JJ Golovkin IE Woodru PR 2006 HELIOS-CR A 1-
D radiation-magnetohydrodynami13s 13ode with inline atomi13 kineti13s mo-
deling Journal of Quantitative Spe13tros13opy and Radiative Transfer 99
381 DOI ADS
Ma13quorn Rankine WJ 1870 On the Thermodynami13 Theory of Waves of
Finite Longitudinal Disturban13e Philosophi13al Transa13tions of the Royal
So13iety of London Series I 160 277 ADS
Mahmood S Ur-Rehman H 2009 Ele13trostati13 solitons in unmagnetized
hot ele13tron-positron-ion plasmas Physi13s Letters A 373 2255 DOI ADS
Martin TH Williams M Kristiansen M 1996 J13 martin on pulsed
power 3 Springer S13ien13e amp Business Media
M13Clarren RG Drake RP Morel JE Holloway JP 2010 Theory of
radiative sho13ks in the mixed opti13ally thi13k-thin 13ase Physi13s of Plasmas
17(9) 093301
Mi13haut C Stehleacute C Leygna13 S Lanz T Boireau L 2004 Jump
13onditions in hypersoni13 sho13ks Quantitative ee13ts of ioni13 ex13itation
and radiation European Physi13al Journal D 28 381 DOI ADS
Mihalas D 1978 Stellar atmospheres 2nd edition ADS
Mihalas D Mihalas BW 1984 Foundations of radiation hydrodynami13s
ADS
Mihalas D Mihalas BW 1999 Foundations of radiation hydrodynami13s
More R Warren K Young D Zimmerman G 1988 A new quotidian
equation of state (qeos) for hot dense matter Physi13s of Fluids (1958-
1988) 31(10) 3059
Ogando F Velarde P 2001 Development of a radiation transport uid
dynami13 13ode under amr s13heme Journal of Quantitative Spe13tros13opy and
Radiative Transfer 71(2) 541
Olson GL Auer LH Hall ML 2000 Diusion P1 and other approx-
imate forms of radiation transport Journal of Quantitative Spe13tros13opy
amp Radiative Transfer 64 619 DOI ADS
Orban C Fatenejad M Chawla S Wilks SC Lamb DQ 2013 A
Radiation-Hydrodynami13s Code Comparison for Laser-Produ13ed Plasmas
FLASH versus HYDRA and the Results of Validation Experiments ArXiv
e-prints ADS
REFERENCES 159
Orlando S Bonito R Argiro C Reale F Peres G Mi13eli M Mat-
sakos T Stehleacute C Ibgui L de Saacute L Chiegraveze J-P Lanz T 2013
Radiative a1313retion sho13ks along nonuniform stellar magneti13 elds in 13las-
si13al T Tauri stars Astronomy amp Astrophysi13s
Ovsyannikov AA Zhukov MF 2000 Plasma diagnos-
ti13s Cambridge International S13ien13e Publishing Cambridge
httppubli13eblib13om13hoi13epubli13fullre13ordaspxp=258189
Pfalzner S 2006 An introdu13tion to inertial 13onnement fusion 1st
edn Series in Plasma Physi13s Taylor amp Fran13is Taylor amp Fran13is
httpgenlibruse13bookindexphpmd5=EACF38292FD79113F78BB532AB5301B8
Potter DE 1971 Numeri13al Studies of the Plasma Fo13us Physi13s of Fluids
14 1911 DOI ADS
Proakis JG 2001 Digital signal pro13essins Wiley Online Library
Raga A Mellema G Arthur S Binette L Ferruit P Steen W 1999
3d transfer of the diuse ionizing radiation in ism ows and the preioni-
zation of a herbig-haro working surfa13e Revista Mexi13ana de Astronomia
y Astrosi13a 35 123
Reighard AB 2007 Collapsing radiative sho13k experiments on the omega
laser PhD thesis The University of Mi13higan
Reighard A Drake R Dannenberg K Kremer D Grosskopf M Har-
ding E Leibrandt D Glendinning S Perry T Remington B et al
2006 Observation of 13ollapsing radiative sho13ks in laboratory experiments
Physi13s of Plasmas (1994-present) 13(8) 082901
Rodriguez R Gil JM Florido R Rubiano JG Mendoza MA Mar-
tel P Minguez E Symes DR Hohenberger M Smith RA 2011
Determination of the average ionization and thermodynami13 regimes of
xenon plasmas with an appli13ation to the 13hara13terization of blast waves
laun13hed in xenon 13lusters High Energy Density Physi13s 7(2) 71 DOI
httpwwws13ien13edire13t13oms13ien13earti13lepiiS157418181100022X
Rodriacuteguez R Espinosa G Gil JM Stehleacute C Suzuki-Vidal F Rubi-
ano JG Martel P Miacutenguez E 2015 Mi13ros13opi13 properties of xenon
plasmas for density and temperature regimes of laboratory astrophysi13s
experiments on radiative sho13ks Physi13al Review E 91(5) 053106 DOI
ADS
Ross IN Pepler DA Danson CN 1995 Binary phase zone plate de-
signs using 13al13ulations of far-eld distributions Opti13s Communi13ations
116 55 DOI ADS
160 REFERENCES
Seaton MJ 1987 Atomi13 data for opa13ity 13al13ulations I General des13rip-
tion Journal of Physi13s B Atomi13 and Mole13ular Physi13s 20(23) 6363
Serban A 1995 Anode geometry and fo13us 13hara13teristi13s PhD thesis
Nanyang Te13hnology University httphdlhandlenet104971568
Serban A Lee S 1995 A simple opti13al ber axial speed dete13tor Review
of S13ienti13 Instruments 66 4958 DOI ADS
Serban A Lee S 1995 A simple opti13al ber axial speed dete13tor Review
of s13ienti13 instruments 66(10) 4958
Singh JP Thakur SN (eds) 2007 Laser-indu13ed breakdown
spe13tros13opy Elsevier s13ien13edire13t
Singh R Stehleacute C Suzuki-Vidal F Kozlova M Larour J Chaulagain
U Clayson T Rodriguez R Gil J Nejdl J et al 2017 Experimental
study of the intera13tion of two laser-driven radiative sho13ks at the pals
laser High Energy Density Physi13s 23 20
Stahler SW Palla F Salpeter EE 1986 Primordial stellar evolution -
The protostar phase The Astrophysi13al Journal 302 590 DOI ADS
Stehleacute C Gonzaacutelez M Kozlova M Rus B Mo13ek T A13ef O Colom-
bier JP Lanz T Champion N Jakub13zak K et al 2010 Experi-
mental study of radiative sho13ks at pals fa13ility Laser and parti13le beams
28(02) 253
Stehleacute C Kozlovaacute M Larour J Nejdl J Champion N Barroso P
Suzuki-Vidal F A13ef O Delattre P-A Dostaacutel J et al 2012 New
probing te13hniques of radiative sho13ks Opti13s Communi13ations 285(1)
64
Stehleacute C Lefegravevre R Chaulagain U Champion N Barroso P Reix
F Jagourel P Larour J Melt13hakov E Mer13ier R et al 2014a
Criti13al 13omponents for xuv probing of laser driven sho13ks In X-Ray
Lasers 2012 Springer 239
Stehleacute C Kozlovaacute M Larour J Nejdl J Suzuki-Vidal F Cohen M
Chaulagain UP Champion N Barroso P A13ef O et al 2014b
Reply on the 13omment of the paper new probing te13hniques of radiative
sho13ks Opti13s Communi13ations 318 226
Suzuki-Vidal F Lebedev S Pi13kworth L Swadling G Burdiak G
Skidmore J Hall G Bennett M Bland S Chittenden J et al
2014 Sho13k formation in 13ounter-streaming jets on the magpie pulsed-
power generator In APS Meeting Abstra13ts 1 5005
REFERENCES 161
Suzuki-Vidal F Clayson T Swadling GF Patankar S Burdiak GC
Lebedev SV Smith RA Stehle C Chaulagain U Singh RL La-
rour J Kozlova M Spindloe C Foster J Skidmore J Gumbrell E
Graham P Danson C 2016 Counter-streaming radiative sho13k experi-
ments on the Orion laser In APS Meeting Abstra13ts ADS
Toledo-Roy JC Velaacutezquez PF de Colle F Gonzaacutelez RF Reynoso
EM Kurtz SE Reyes-Iturbide J 2009 Numeri13al model for the SNR
DEM L316 simulated X-ray emission Monthly Noti13es of the RAS 395
351 DOI ADS
Tou T Lee S Kwek K 1989 Nonperturbing plasma-fo13us measurements
in the run-down phase IEEE Transa13tions on Plasma S13ien13e 17(2) 311
Vaytet NMH Audit E Dubro13a B Delahaye F 2011 A numeri13al
model for multigroup radiation hydrodynami13s Journal of Quantitiative
Spe13tros13opy and Radiative Transfer 112 1323 DOI ADS
Vaytet N Chabrier G Audit E Commerccedilon B Masson J Ferguson
J Delahaye F 2013 Simulations of protostellar 13ollapse using mul-
tigroup radiation hydrodynami13s ii the se13ond 13ollapse arXiv preprint
arXiv13071010
Vaytet N Gonzaacutelez M Audit E Chabrier G 2013 The inuen13e
of frequen13y-dependent radiative transfer on the stru13tures of radiative
sho13ks Journal of Quantitiative Spe13tros13opy and Radiative Transfer 125
105 DOI ADS
Velarde P Gar13iacutea-Senz D Bravo E Ogando F Relantildeo A Gar13iacutea C
Oliva E 2006 Intera13tion of supernova remnants From the 13ir13umstellar
medium to the terrestrial laboratory Physi13s of Plasmas 13(9) 092901
DOI ADS
Veloso F Moreno J Tarifentildeo-Saldivia A Pavez C Zambra M Soto
L 2012a Non-intrusive plasma diagnosti13s for measuring sheath kine-
mati13s in plasma fo13us dis13harges Measurement S13ien13e and Te13hnology
23(8) 087002
Veloso F Tarifentildeo-Saldivia A Pavez C Moreno J Zambra M Soto
L 2012b Plasma sheath kinemati13s and some impli13ations on the mo-
deling of very low energy plasma fo13us devi13es Plasma Physi13s and Con-
trolled Fusion 54(9) 095007
Vis13o A Drake R Glenzer S Doumlppner T Gregori G Froula D
Grosskopf M 2012 Measurement of radiative sho13k properties by x-ray
thomson s13attering Physi13al review letters 108(14) 145001
162 REFERENCES
Williams RM Chu Y-H Di13kel JR Beyer R Petre R Smith RC
Milne DK 1997 Supernova Remnants in the Magellani13 Clouds I The
Colliding Remnants DEM L316 Astrophysi13al Journal 480 618 ADS
Williams RM Chu Y-H Di13kel JR Gruendl RA Seward FD
Guerrero MA Hobbs G 2005 Supernova Remnants in the Magella-
ni13 Clouds V The Complex Interior Stru13ture of the N206 Supernova
Remnant Astrophysi13al Journal 628 704 DOI ADS
Zambra M Kalise D Fernandez J Hernandez E Pas-
ten D Munoz V 2009 Current sheet thi13kness in the
plasma fo13us snowplow model J Plasma Fusion 8(879-882)
httpspeopleri13amoeawa13atdkalisefilesplasmapdf
Zeldovi13h IB Zeldovi13h YB Raizer YP 2002 Physi13s of sho13k waves
and high-temperature hydrodynami13 phenomena Courier Dover Publi13ati-
ons London New York Sydney
REFERENCES 163
Next I would like to thank Dr U Chaulagain for helping me in Paris
during my stay from the very rst day I am also grateful to Dr L de Sagrave
Dr L Ibgui Dr M Drouin Dr A Stan Mr B Khiar Mr L Ni13olas
Dr J Freundli13h for their help and support during my stay in Fran13e
My thanks is due to my friends for 13reating a homely atmosphere during
my thesis in Fran13e In parti13ular I would like to thank Dr R Mourya
and Mr A Ranjan for their 13ontinuous support in the administrative for-
malities and others whi13h helped me a lot to 13on13entrate and speed up my
thesis writing I would also like to mention the names of my friends Miss T
Bhowmik Miss V Shaw and Dr A Gupta for their support
Good 13ompany plays a very important role in the life In this I am par-
ti13ularly thankful to Dr S C Tripathi Dr J Pal Dr R Kumar Dr M
Dave Dr A Bharadwaj Mr K Singh Miss S Mishra and Miss S Pandey
for helping me in keeping pa13e with resear13h and daily life developments I
am indebted to my brother Dr A Awasthi for his sin13ere guidan13e en13oura-
gement and 13ooperation during my entire do13toral study Also I would like
to take this opportunity to thank my 13hildhood friend Miss R Yadav for her
love and en13ouragement during my thesis
Finally last but not the least I express my whole hearted gratitude to the
support I re13eived in the form of love and ae13tion from my grandparents
parents sisters brothers and my other family members This is something
beyond a13knowledgements They were always with me and the en13ourage-
ment I re13eived from them will never fade away
Raj Laxmi Singh
Strong radiative sho13ks relevant for stellar
environments experimental study and numeri13al
approa13h
ABSTRACT
Strong sho13ks are present in various astrophysi13al phenomena Su13h
sho13ks are strongly inuen13ed by the radiation through its 13oupling with
hydrodynami13s Thus their topology and dynami13s are quite 13omplex Ge-
nerating su13h hypersoni13 sho13ks in the laboratory with 13ontrolled 13ondi-
tions is thus an adequate tool to study the inuen13e of radiation and to
13ompare them with numeri13al simulations Su13h sho13ks 13an be generated
by intense lasers and ele13tromagneti13 devi13es The rst part of this dis-
sertation 13on13erns the numeri13al and experimental study of the intera13tion
of two 13ounter propagating laser-driven sho13ks The experiments perfor-
med at the kJ PALS laser fa13ility allowed to generate sho13ks with dierent
speeds (sim 30-55 kms and 10-25 kms) in noble gases and low pressure
(less than 1 bar) Several diagnosti13s were implemented visible interfero-
metry time- and spa13e-resolved visible spe13tros13opy and time integrated
XUV spe13tros13opy Our experiment shows a strong intera13tion of one radi-
ative pre13ursor onto the se13ond one The physi13al parameters of the plasma
were dedu13ed from the diagnosti13s and 13ompared with 1-D simulation re-
sults The se13ond part is devoted to the design of an experiment where the
sho13k is generated ele13tromagneti13ally The optimization of this generator is
presented and also the full experimental set up whi13h allows studying sho13k
sim 30 kms in noble gas at sim 1 mbar
Keywords Radiative sho13k hydrodynami13s laser-plasmas visible
and XUV spe13tros13opy
numeri13al simulation laboratory astrophysi13s high energy density
physi13s
Cho13s forts et radiatifs dinteacuterecirct pour les
environnements stellaires eacutetude expeacuterimentale et
appro13he numeacuterique
RESUME
Les 13ho13s forts sont preacutesents dans des pheacutenomegravenes astrophysiques varieacutes
De tels 13ho13s sont fortement inuen13eacutes par le rayonnement par son 13ouplage
ave13 lhydrodynamique Par suite leur topologie et leur dynamique sont
assez 13omplexes Geacuteneacuterer de tels 13ho13s hypersoniques en laboratoire dans
des 13onditions 13ontrleacutees est ainsi un outil pertinent pour eacutetudier linuen13e
du rayonnement et pour 13omparer aux reacutesultats des simulations numeacuteriques
Ces 13ho13s sont geacuteneacutereacutes par des lasers intenses et par des moyens eacutele13tromag-
neacutetiques La premiegravere partie du texte est 13onsa13reacute agrave leacutetude numeacuterique et
expeacuterimentale de lintera13tion de deux 13ho13s induits par laser se propageant
en sens 13ontraires Les expeacuterien13es ont eacuteteacute meneacutees sur linstallation laser kJ
PALS qui permet de former deux 13ho13s ave13 des vitesses propres dieacuteren-
tes (sim 30-55 et 10-25 kms respe13tivement) dans des gaz rares agrave pression
faible (moins de 1 bar) Des diagnosti13s ont eacuteteacute installeacutes interfeacuteromeacutetrie
visible spe13tros13opie visible agrave reacutesolution spatiale et temporelle spe13tros13o-
pie XUV inteacutegreacutee en temps Nos expeacuterien13es montrent une forte intera13tion
entre les deux preacute13urseurs radiatifs Les paramegravetres physiques du plasma
ont eacuteteacute deacuteduits de 13es diagnosti13s et 13ompareacutes aux reacutesultats de simulations
monodimensionnelles La se13onde partie est 13onsa13reacutee agrave la 13on13eption dune
expeacuterien13e ougrave le 13ho13 est geacuteneacutereacute de faccedilon eacutele13tromagneacutetique Loptimisation
de 13e geacuteneacuterateur est preacutesenteacutee ainsi que lenvironnement expeacuterimental per-
mettant deacutetudier des 13ho13s jusquagrave 30 kms dans des gaz rares peu denses
(1 mbar)
Mots 13leacutes 13ho13s radiatifs plasmas laser hydrodynamique spe13tros13opie
visible et XUV simulation numeacuterique astrophysique de laboratoire
physique agrave haute densiteacute deacutenergie
Reacutesumeacute long
En introdu13tion (13hapitre 1) les 13ho13s radiatifs (CR) sont preacutesenteacutes
dans de nombreux environnements astrophysiques en parti13ulier dans les
eacutetoiles Ce sont des 13ho13s forts ave13 un nombre de Ma13h eacuteleveacute (M1) et une
tregraves haute tempeacuterature 13e qui induit un rayonnement intense Alors que
lobservation de 13ette signature ave13 une reacutesolution spatiale est tregraves di13ile
en astrophysique une appro13he innovante 13elle dite des plasmas astrophy-
siques de laboratoire fournit un puissant moyen deacutetude des CR sur Terre
Depuis plus dune deacute13ennie 13es 13ho13s sont eacutetudieacutes en laboratoire prin13ipale-
ment sur de grandes installations laser dans les gaz rares et sous dieacuterentes
geacuteomeacutetries Lintera13tion entre un laser et une feuille min13e produit une
forte ablation et par eet fuseacutee a1313eacutelegravere la feuille qui se propage dans le
gaz agrave plusieurs dizaines de kms Pour des irradiations laser entre 10
14and
10
15W13m
2 on enregistre des vitesses de 13ho13s entre 40 et 150 kms En
parallegravele une des13ription ne est permise par des simulations numeacuteriques
Au deacutebut de 13e travail les expeacuterien13es eacutetaient 13on13entreacutees sur des CR
simples alors que la situation astrophysique est 13elle dun CR interagissant
ave13 un milieu plus dense donnant des 13ho13s reacuteeacute13his et transmis Des
exemples repreacutesentatifs de 13es pheacutenomegravenes sont la propagation de restes de
supernovae dans des nuages moleacute13ulaires denses les 13ho13s da1313reacutetion sur
la photosphegravere des eacutetoiles T-Tauri ou en13ore les 13ho13s deacutetrave agrave lavant
des jets stellaires La 13ollision (ou lintera13tion) de deux CR est bien sucircr un
eacutevegravenement rare en astrophysique mais un exemple est fourni par lintera13tion
de deacutebris de la supernova DEM L316 (13f Fig 1 de Williams et al (1997))
mecircme si lhypothegravese est 13ontesteacutee (Velarde et al (2006))
Il y a don13 une forte motivation pour reacutealiser une expeacuterien13e de labora-
toire pour eacutetudier la propagation et lintera13tion de deux 13ho13s fa13e agrave fa13e
et interpreacuteter les signatures observeacutees Cette appro13he 13onstitue une grande
part de mon travail En 13ompleacutement aux expeacuterien13es laser jai eacutetudieacute un
geacuteneacuterateur 13ompa13t de puissan13e pulseacutee alimentant un 13anon agrave plasma 13oax-
ial pour 13reacuteer des 13ho13s dinteacuterecirct astrophysique dans les gaz agrave basse pression
Les 13ho13s ainsi 13reacuteeacutes sont de plus grande taille et peuvent ecirctre eacutetudieacutes plus
fa13ilement (Kondo et al 2008) Pour 13ette thegravese je me suis atta13heacutee agrave preacute-
senter les 13ho13s radiatifs dans le 13ontexte de lastrophysique de laboratoire
en suivant 13es deux appro13hes
Le deuxiegraveme 13hapitre traite de la physique des CR Ils 13omposent
une 13lasse de 13ho13s supersoniques qui sont 13haueacutes agrave haute tempeacuterature
et sont par 13onseacutequent sour13es dun rayonnement intense En retour 13e
rayonnement modie la dynamique et la stru13ture du 13ho13 13e qui 13omplexie
la des13ription Les CR se ren13ontrent dans des situations astrophysiques
13omme les 13ho13s da1313reacutetion lors de la formation des protoeacutetoiles (Stahler
Palla et Salpeter 1986) lexplosion de supernovae et lintera13tion de leur
restes ave13 le milieu interstellaire dense (Chevalier 1977) ou en13ore les 13ho13s
deacutetrave en tecircte des jets stellaires (Hartigan et al 2001) Les CR sont
maintenant eacutetudieacutes en laboratoire 13e qui permet de 13omparer les reacutesultats
aux modegraveles de la litteacuterature et de veacuterier la possibiliteacute de les retrouver ave13
les simulations numeacuteriques disponibles (Bouquet et al (2004) Leygna13 et
al (2006)) Leacutetude expeacuterimentale des CR est 13ru13iale notamment pour
13omprendre les pro13essus eacutenergeacutetiques au sein des plasmas astrophysiques
Dans le 13as des gaz parfaits le saut au niveau dun 13ho13 est deacute13rit par les
eacutequations de Rankine-Hugoniot qui relient les quantiteacutes thermodynamiques
de part et dautre de la dis13ontinuiteacute La solution monodimensionnelle dun
13ho13 se propageant agrave la vitesse us dans un gaz au repos est usuellement
deacute13rite dans le repegravere lieacute au 13ho13 et on distingue la reacutegion amont (ou preacute-
13ho13) agrave la vitesse u1= - us et en arriegravere du 13ho13 (reacutegion post-13ho13 ou aval)
la vitesse est u2 selon la Fig 21 Pour les 13ho13s forts ougrave le nombre de Ma13h
M est tregraves grand devant 1 les sauts des quantiteacutes thermodynamiques dun
gaz parfait sont donneacutes par les eacuteq 21 agrave 24 Pour un gaz monoatomique
le rapport de 13ompression est de 4 et la tempeacuterature de la zone post-13ho13
est proportionnelle au 13arreacute de la vitesse du 13ho13 et agrave la masse atomique
La situation est plus 13omplexe pour un gaz reacuteel 13ar une part de leacutenergie
13ineacutetique sert agrave ex13iter et agrave ioniser le gaz du post 13ho13 La tempeacuterature
attendue est don13 infeacuterieure agrave 13elle du 13as ideacuteal en outre la pression est
modieacutee par lionisation (Mi13haut et al 2004) Un nouveau jeu deacutequations
in13orpore la 13harge ee13tive Z des ions (eacuteq 25 agrave 28) Ces modegraveles ignorent
le 13hauage et le refroidissement radiatifs alors que le rayonnement eacutemis
ae13te la stru13ture dun 13ho13 fort Cette des13ription demande dajouter aux
eacutequations de 13onservation de la masse quantiteacute de mouvement et eacutenergie les
13ontributions radiatives (ux eacutenergie et pression Le 13as est plus 13omplexe
et les eacutequations dhydrodynamique radiative ont eacuteteacute exprimeacutees dans le repegravere
du 13ho13 (Mihalas and Mihalas 1999 Drake 2006) selon les eacuteq 29 agrave 211
Les 13ontributions de la pression radiative et du 13hamp de rayonnement
sont importantes agrave tregraves haute tempeacuterature (vitesse) mais deacutejagrave pour nos 13ho13s
le ux radiatif est dominant Selon lopa13iteacute le rayonnement du 13ho13 peut
ecirctre absorbeacute par la region preacute-13ho13 induisant un 13hauage loin en avant de
la dis13ontinuiteacute Egalement la zone 13hoqueacutee pro13he de la dis13ontinuiteacute est
ae13teacutee par la 13ompeacutetition entre leacutemission (refroidissement) et labsorption
(13hauage) Ainsi leacutepaisseur optique lo13ale (eacuteq 211) devient le paramegravetre
pertinent Comme on raisonne sur le ux radiatif moyenneacute en freacutequen13e
13est lopa13iteacute de Rosseland qui est utiliseacutee et on dieacuteren13ie les reacutegimes op-
tiquement eacutepais et optiquement min13e 13e qui permet de 13lasser les 13ho13s
radiatifs
Notre inteacuterecirct sest 13on13entreacute sur les 13ho13s preacutesentant un preacute13urseur radi-
atif En laboratoire une intensiteacute laser au dessus de 10
14W13m
2le permet
et les installations sont rappeleacutees dans le tableau 21 Il est aussi possible de
lan13er des 13ho13s moins rapides don13 peu radiatifs (sim 10-30 kms) ave13 un
piston eacutele13tromagneacutetique
Le 13hapitre 3 est 13onsa13reacute agrave lanalyse numeacuterique des CR en deacute13rivant
les pro13essus physiques jouant un rle dans 13es systegravemes physiques De mecircme
que les observations les expeacuterien13es en laboratoire ne sont pas dire13tement
interpreacutetables et la simulation numeacuterique devient un outil preacute13ieux Diverses
appro13hes numeacuteriques sont utiliseacutees pour simuler les plasmas de laboratoire
On 13onsidegravere soit des grilles xes soit des grilles adaptables (Adaptative
Mesh Renement) pour maintenir une reacutesolution susante sur 13haque zone
du plasma Une autre appro13he utilise des grilles qui suivent le uide dans
son mouvement (modegravele Lagrangien) don13 sans passage de matiegravere dune
13ellule agrave la voisine Les eacutequations sont alors dieacuterentes (Orban et al 2013)
Pour deacute13rire les 13ollisions de 13ho13s jai utiliseacute le 13ode 13ommer13ial HE-
LIOS un 13ode Lagrangien mono-dimensionnel pouvant geacuterer le rayonnement
et lhydrodynamique (Ma13Farlane Golovkin et Woodru 2006) Assez fa13ile
agrave prendre en main il a le gros avantage de pouvoir simuler notre 13ongura-
tion deux 13ho13s de vitesses opposeacutees lan13eacutes par deux lasers HELIOS deacute13rit
les plasmas hors deacutequilibre thermodynamique (hors ETL) gracirc13e agrave un mo-
degravele 13ollisionnel-radiatif (HELIOS-CR) de 13al13ul des populations atomiques
agrave 13haque pas de simulation hydrodynamique De fait jai utiliseacute HELIOS
dans une approximation ETL qui 13orrespond aux 13onditions expeacuterimentales
(Rodriguez et al 2011) et 13ela est dis13uteacute en deacutetails
Le 13ode reacutesout les eacutequations uides en introduisant les pressions des eacutele13-
trons des ions et du rayonnement Ele13trons et ions sont deacute13rits par deux
uides en intera13tion de tempeacuteratures respe13tives Te et Ti La 13ondu13tion
thermique est geacutereacutee par 13elle des eacutele13trons agrave partir de la 13ondu13tiviteacute de
Spitzer et le deacutept deacutenergie laser par un modegravele de Bremsstrahlung inverse
Leacutemission et labsorption du rayonnement sont introduites dans leacutequation
deacutenergie des eacutele13trons et dans 13elles du transport radiatif Une meacutethode
utilise un modegravele de diusion du rayonnement multi-groupes et agrave ux li-
miteacute ougrave le ux radiatif est proportionnel au gradient de leacutenergie radiative
et inversement proportionnel agrave lopa13iteacute de Rosseland ave13 une pondeacuteration
pour obtenir une bonne des13ription du 13as optiquement min13e selon Olson
Auer et Hall (2000) Une se13onde meacutethode utilise un s13heacutema deacutependant du
temps agrave 13ourte distan13e et multi-angles Dans notre 13as ougrave rayonnement
et hydrodynamique sont fortement 13oupleacutes et aussi en variation rapide on
utilise le modegravele de diusion et les opa13iteacutes ETL multi-groupes de Plan13k et
Rosseland
En outre notre version dHELIOS utilise PROPACEOS (Ma13Farlane
Golovkin et Woodru 2006) une table deacutequation deacutetat et dopa13iteacute multi-
groupes geacuteneacutereacutee par le 13ode sans possibiliteacute dintroduire une autre table La
13onnaissan13e des 13onditions thermodynamiques est neacute13essaire pour la simu-
lation du plasma dans nos 13as masse volumique entre 10
minus4et 10
minus1g13m
3
tempeacuterature entre 0 et 50 eV Une eacutevaluation preacute13ise par Rodriguez et al
(13f Fig 2 de Rodriguez et al 2011) pour le Xeacutenon indique que les 13onditi-
ons thermodynamiques de nos CR 13orrespondent au reacutegime ETL Leacutequation
deacutetat relie pression ionisation et eacutenergie interne agrave la masse volumique et agrave
la tempeacuterature Notre version dHELIOS pour plasma ETL utilise PROPA-
CEOS qui est peu do13umenteacutee elle utilise le modegravele de More et al (1988)
en reacutegime de 13ouplage fort (haute densiteacute et basse tempeacuterature) et un modegravele
datome isoleacute pour un 13ouplage faible Des interpolations sont in13luses pour
passer au 13ouplage fort (httpwwwprism-13s13omSoftwarePROPACEOS)
Jai reacutealiseacute des simulations HELIOS 1D pour divers 13as de CR 13ho13 seul
ou 13ho13s en 13ollision qui sont deacute13rites dans 13e 13hapitre 3 Les CR se propa-
geant dans le Xe agrave 01 bar et agrave 50 kms sont 13ara13teacuteriseacutes par un preacute13urseur
radiatif eacutetendu Une 13ompression eacuteleveacutee (38) dans le post-13ho13 est attribueacutee
agrave lionisation du gaz et au refroidissement radiatif Les tempeacuteratures du
post-13ho13 et du preacute-13ho13 de part et dautre du pi13 sont identiques signe
dun 13ho13 super13ritique Le rle spe13ta13ulaire de lopa13iteacute a eacuteteacute remarqueacute
mais agrave 13onsideacuterer lin13ertitude sur lopa13iteacute du Xeacutenon et le traitement 1D
jai deacute13ideacute de ne pas 13her13her agrave raner la simulation en termes de nombre
de groupes
Leacutetude numeacuterique permet de retrouver les 13ara13teacuteristiques de lintera13tion
de deux 13ho13s opposeacutes ave13 des vitesses respe13tives 50-50 kms et 50-20 kms
Dans tous les 13as la prin13ipale signature de lintera13tion (don13 avant la 13ol-
lision) est le re13ouvrement des preacute13urseurs agrave t0+8 ns pour 50-50 kms et
agrave t0+15 ns pour 50-20 kms Ensuite on assiste agrave une remonteacutee reacuteguliegravere
de la densiteacute et de la tempeacuterature eacutele13troniques Linstant de la 13ollision
voit une augmentation soudaine de la densiteacute eacutele13tronique (par un ordre de
grandeur) qui atteint 66 times 10
21et 3 times 10
2113m
minus3respe13tivement alors que
la tempeacuterature monte agrave 39 et 28 eV
Dans le quatriegraveme 13hapitre je preacutesente linstallation laser PALS puis
la 13ellule dintera13tion son implantation et les diagnosti13s asso13ieacutes Rappe-
lons le prin13ipe expeacuterimental Quand une impulsion bregraveve de lumiegravere laser
est fo13aliseacutee sur une feuille min13e une part importante deacutenergie est trans-
feacutereacutee et le 13hauage du solide 13onduit agrave son ablation Un plasma 13oronal
13haud et peu dense est eacuteje13teacute vers larriegravere et un 13ho13 vers lavant se forme
dans la feuille par eet fuseacutee Comme 13ette feuille limite le tube le 13ho13
se transmet au gaz la feuille agissant 13omme un piston Des expeacuterien13es
ont eacuteteacute meneacutees sur le Prague Asterix Laser System (PALS) au printemps
2015 (5 semaines 20 avril - 22 mai) ave13 pour obje13tif la premiegravere eacutetude
de lintera13tion puis de la 13ollision de deux CR sous la13tion de deux lasers
au niveau de 10
14W13m
2 Le Prague Asterix Laser System (PALS) repose
sur un laser agrave iode infrarouge (Asterix IV Jungwirth et al 2001) Apregraves
ampli13ation il deacutelivre jusquagrave 1 kJ en 03 ns agrave la longueur donde fonda-
mentale de 1315 nm Des fais13eaux auxiliaires sont disponibles agrave freacutequen13e
doubleacutee (λ = 657 nm) ou tripleacutee (438nm) PALS deacutelivre deux tirs par heure
agrave haute eacutenergie et dans de bonnes 13onditions dhomogeacuteneacuteiteacute du fais13eau
Pour nos expeacuterien13es nous utilisons deux fais13eaux Le fais13eau fondamental
est diviseacute apregraves le 4egraveme ampli13ateur en deux fais13eaux deacutenergie 60 et
40 (voir Fig 42) Le plus puissant est inje13teacute dans le 5
meampli13ateur et
sa freacutequen13e est tripleacutee Ce fais13eau agrave 438 nm est nommeacute MAIN Le se13ond
fais13eau est utiliseacute sans modi13ation don13 agrave 1315 nm et est nommeacute AUX
Le s13heacutema de distribution est en Fig 42 et les proprieacuteteacutes de MAIN and
AUX laser sont rappeleacutees Tdans le ableau 41
PALS propose deux 13hambres agrave vide spheacuterique et 13ylindrique respe13ti-
vement nous avons utiliseacute la 13hambre spheacuterique (Fig 43) Dun diamegravetre
de 100 13m elle est a1313essible aux fais13eaux MAIN et AUX par deux hublots
de 80 13m et 50 13m Des portes sont ameacutenageacutees ainsi que de nombreux ports
A linteacuterieur la 13ellule et des diagnosti13s sont monteacutes sur une table optique
deacute13oupleacutee meacute13aniquement de la 13hambre Les 13ibles (Fig 45) 13omprennent
un petit tube long de 4 mm fermeacute agrave ses deux extreacutemiteacutes par des feuilles min-
13es de parylegravene-N doreacute de 11 microm sur lesquelles les deux lasers sont fo13aliseacutes
au niveau de 10
14W13m
minus2 Le tube est rempli du gaz dans lequel le 13ho13
se propagera agrave une vitesse attendue entre 30 et 60 kms Les feuilles min13es
assurent la 13onversion de leacutenergie laser en eacutenergie 13ineacutetique via lablation et
la geacuteneacuteration de 13ho13 Des 13ellules speacute13iques en dural massif permettent
lalignement et la mesure du diamegravetre du fais13eau Toutes les 13ellules sont
reacutealiseacutees par le Ple instrumental de lObservatoire de Paris
Les 13ellules agrave gaz sont remplies in situ agrave une fra13tion de bar ave13 Xe
Ar Xe90He10 et He permettant ainsi de bien 13onnaicirctre le milieu ougrave se
propagent les 13ho13s Le systegraveme de remplissage (Fig 410) permet de limiter
leacute13art de pression sur les fenecirctres de la 13ellule et permet de suivre la pression
jusquau moment du tir Cest 13ritique pour ne pas 13asser la fenecirctre ultra
ne de Si3N4 pour la spe13tros13opie XUV et pour geacuterer une fuite eacuteventuelle
Le reacuteglage de la position et de la fo13alisation permet dobtenir des ta13hes
laser 13entreacutees de diamegravetres 450 - 500 microm et 250 - 300 microm pour MAIN et
AUX respe13tivement Ce 13ontrle est reacutepeacuteteacute 13haque jour Les impa13ts sont
toujours suivis par des 13ameacuteras X au keV mecircme si limpa13t sur la feuille de
parylegravene-N donne un signal X faible 13ompareacute agrave 13elui du tir sur Al
Les diagnosti13s du plasma sont prin13ipalement linterfeacuteromeacutetrie visible et
la spe13tros13opie XUV et visible Lalignement est suivi par des 13ameacuteras dans
le visible Pour linterfeacuteromeacutetrie visible preacute13iseacutement on doit prendre une
image de reacutefeacuteren13e (sans franges) de 13haque 13ible (Fig 417a) Le fais13eau de
reacutefeacuteren13e de linterfeacuteromegravetre Ma13h-Zehnder est bloqueacute et le fais13eau sonde est
utiliseacute seul La fente est ensuite reacuteduite agrave 200 microm pour ne sonder que la zone
13entrale du tube mais seacutetendant le long de laxe jusquaux pistons pour avoir
une reacutefeacuteren13e des positions initiales Ensuite en mode interfeacuterogramme on
13reacutee un systegraveme de franges perpendi13ulaires agrave la fente don13 perpendi13ulaires
agrave la dire13tion de propagation du 13ho13 (13f Fig 417b et Fig 413) Le tube
de 13ho13 est imageacute sur la 13ameacutera ave13 un grandissement susant (54 mm) et
il est possible dassurer un balayage eacutele13tronique de la fente sur le deacutete13teur
jusquagrave 200 ns Un interfeacuterogramme dit streak permet ainsi de suivre limpa13t
des 13ho13s sur les franges en fon13tion du temps
Leacutemission XUV du plasma est enregistreacutee ave13 une inteacutegration spatiale
et temporelle sur un spe13trographe agrave 13hamp plan muni dun reacuteseau 13on13ave
Il est installeacute dans la partie haute de la 13hambre (13f Fig 418) et enregistre
leacutemission XUV traversant la fenecirctre de Si3N4 Une spe13tros13opie visible
reacutesolue en temps et en position longitudinale a eacuteteacute installeacutee (Fig 419a)
mais na pas en13ore pu ecirctre exploiteacutee 13omplegravetement
Le 13hapitre 5 preacutesente les reacutesultats des deux diagnosti13s surtout linterfeacute-
romeacutetrie qui a eacuteteacute 13omplegravetement analyseacutee Jai deacuteveloppeacute une pro13eacutedure
de deacutepouillement ave13 des outils numeacuteriques ad ho13 de 13al13ul et danalyse
dimage Certes linterfeacuteromeacutetrie ne permet pas de sonder des plasmas plus
denses que la densiteacute 13ritique imposeacutee par le laser (4 times 10
2113m
minus3agrave 627 nm)
et on 13onstate que le post-13ho13 est opaque Au 13ontraire le rayonnement
XUV est preacutesent dans toute la stru13ture et la spe13tros13opie XUV inteacutegreacutee
en temps et en espa13e permet dexplorer agrave la fois le post-13ho13 et le preacute13ur-
seur Lanalyse des interfeacuterogrammes streak en visible part dune eacutevaluation
de leet de lindi13e de reacutefra13tion des eacutele13trons deacutetailleacutee dans lappendi13e A
mais qui se heurte assez vite agrave une perte de 13ontraste et agrave une disparition
des franges bien en dessous de la valeur de la densiteacute 13ritique deacutejagrave menti-
onneacutee Les deacutephasages a1313essibles ave13 preacute13ision ne deacutepassent guegravere 2 ou 3
fois 2π La re13her13he de maxima des franges est semi automatiseacutee et permet
darriver agrave une densiteacute eacutele13tronique moyenneacutee sur les 600 microm de leacutepaisseur
du tube de 13ho13 Le prol non plan du 13ho13 13onduit agrave penser que la moyenne
sous-estime dun fa13teur 2 voire plus la densiteacute dans le preacute13urseur La dis-
parition des franges est attribueacutee agrave la preacutesen13e du front de 13ho13 Par suite
les interfeacuterogrammes permettent de mesurer des vitesses de 13ho13s de 30-55
et 10-30 kms pour MAIN et AUX respe13tivement ainsi que des densiteacutes
eacutele13tronique dans les preacute13urseurs entre 10
17et 10
1913m
minus3
Nous avons ainsi deacutemontreacute lintera13tion entre deux preacute13urseurs radiatifs
dans Xe at 01 bar pour deux 13ho13s agrave 54 et 23 kms Lintera13tion est
13lairement 13ara13teacuteriseacutee par laugmentation de londe dionisation puis par la
superposition des deux preacute13urseurs agrave t0+20ns La 13ollision est enregistreacutee agrave
t0+47 ns un reacutesultat retrouveacute par la simulation
Nous avons reacutepeacuteteacute mais agrave plus haute pression 02 bar et trouveacute des
vitesses de sim 41 kms pour MAIN et sim 18 kms pour AUX Cependant
nous navons pas de signature du preacute13urseur radiatif pour AUX et le temps de
13ollision na pas eacuteteacute a1313essible agrave lenregistrement Le preacute13urseur 13teacute MAIN
nest pas inuen13eacute par AUX jusquagrave la limite de t0+48 ns (Fig 59(b) et (13))
La simulation preacutedit un tregraves faible preacute13urseur 13teacute AUX et une intera13tion
des preacute13urseurs agrave t0+49 ns
Linterfeacuteromeacutetrie transverse agrave 02 bar ave13 des vitesses sim 40 et 20 kms
indique que le preacute13urseur 13teacute MAIN a une extension lateacuterale de sim 600 microm13ontre 300 microm pour AUX Le preacute13urseur de MAIN est leacutegegraverement ae13teacute
pregraves des parois alors que 13elui de AUX est fortement 13ourbeacute Les eets 2D
sont don13 pronon13eacutes pour AUX et faibles pour MAIN En outre pour le
Xeacutenon nous disposons de la tempeacuterature et de la 13harge moyenne par le
spe13tre XUV inteacutegreacute (tir48143 Xeacutenon 06 bar) On peut en 13on13lure que
la 13harge moyenne est au moins eacutegale agrave 6 et quon a atteint une tempeacuterature
de 15 eV
Nos simulations donnent une des13ription qualitative des CR en inte-
ra13tion agrave la reacuteserve pregraves que nous introduisions dans HELIOS une uen13e
laser ajusteacutee pour donner la bonne vitesse Cependant il est maintenant
admis que les simulations 2D (ave13 les opa13iteacutes 13orre13tes) deacute13rivent bien les
expeacuterien13es (Gonzaacutelez Audit et Stehleacute 2009 Leygna13 et al 2006 Stehleacute
et al 2010) A la mecircme eacutenergie laser le 13al13ul 2D 13onduit agrave diminuer la
vitesse du 13ho13 13ompare au 13al13ul 1D et aussi agrave une baisse de la densiteacute
eacutele13tronique Par exemple pour un 13ho13 lan13eacute agrave PALS par un laser agrave 1315
nm dans le Xeacutenon agrave 03 bar ave13 une uen13e de 85000 J13m
2 ARWEN 2D
donne une vitesse de 44 kms en a1313ord ave13 lexpeacuterien13e (Cotelo et al
2015) La simulation 1D demanderait 30000 J13m
2pour obtenir la mecircme
vitesse
De mecircme la spe13tros13opie XUV inteacutegreacutee agrave 06 bar pour des vitesses reacuteel-
les de sim 39 et 18 kms indique que la tempeacuterature a atteint 15 eV et que
la 13harge moyenne a atteint 6 ou 7 alors que la simulation 1D preacutedit 10-30
eV et 5-10 respe13tivement (Fig 518) On peut en 13on13lure quune eacutetude
deacutetailleacutee baseacutee sur des simulations 2D et un post-traitement du transfert de
rayonnement sont neacute13essaires pour raner lanalyse Pour les autres gaz
rares (Ar Kr) on a observeacute que agrave eacutenergie laser donneacutee la vitesse de CR est
une fon13tion deacute13roissante de la masse volumique Ce13i 13onrme que pour
une densiteacute et une vitesse les eets radiatifs augmentent ave13 le numeacutero
atomique Pour le Krypton on a observeacute un petit preacute13urseur mais au13une
intera13tion Celle-13i ne serait possible quen augmentant nettement la vi-
tesse don13 leacutenergie du laser Ce13i a eacuteteacute rendu possible sur linstallation
laser Orion (AWE Aldermaston G-B) ougrave la 13ollision de deux CR deacutegale
vitesse a eacuteteacute obtenue agrave sim 80 kms pour une uen13e laser sim 6 times 10
14W13m
2
(Clayson et al 2016 Suzuki-Vidal et al 2016) pour une large gamme de
gaz rares et des pressions entre 01 et 1 bar Pour eacutetudier agrave la fois les 13ho13s
et les preacute13urseurs radiatifs de nombreux diagnosti13s eacutetaient installeacutes radi-
ographie X imagerie optique en 13ameacutera agrave balayage de fente interfeacuteromeacutetrie
multi-vues et agrave balayage de fente Bien que je naie pas pu parti13iper aux
13ampagnes jai fait des simulations 1D pour interpreacuteter les reacutesultats Cet
aspe13t nest pas deacute13rit dans 13ette thegravese mais a eacuteteacute publieacute tregraves reacute13emment
(Clayson et al 2017)
Apregraves des 13hapitres 13onsa13reacutes aux 13ho13s radiatifs geacuteneacutereacutes par laser le
13hapitre 6 deacute13rit un moyen alternatif passant par la voie eacutele13trique agrave haute
puissan13e Les 13hires de la voie laser sont tregraves eacuteleveacutes en uen13e plus de
10
14W13m
2 mais repreacutesentent des eacutenergies modeacutereacutees dans limpulsion (120
et 60 J pour MAIN et AUX au PALS) Or un systegraveme eacutele13trique devant
deacutelivrer 100 J reste modeste pour sa partie sto13kage Les vitesses de 50
kms seront probablement di13iles agrave atteindre mais on peut tabler sur un
reacutegime 13ompleacutementaire des 13ho13s radiatifs ave13 plus de 10 kms si le ren-
dement de 13onversion en eacutenergie 13ineacutetique reste 13orre13t Dougrave un neacute13essaire
travail doptimisation Des vitesses jusquagrave 100 kms avaient eacuteteacute mesureacutees
tregraves tt dans des 13anons agrave plasma dhydrogegravene agrave basse pression (Lee 1969)
mais 13e13i a eacuteteacute exploiteacute ensuite pour dautres naliteacutes La litteacuterature ne
mentionne quun 13as de 13anon eacutele13tromagneacutetique 13oaxial deacutedieacute aux 13ho13s
dinteacuterecirct astrophysique (Kondo et al 2006) Il est signaleacute que le lan13eur
eacutele13tromagneacutetique produit des 13ho13s plans de grande dimension transverse
(1 ordre de grandeur par rapport agrave la voie laser) et don13 plus fa13iles agrave in-
vestiguer (Kondo et al 2008) Le prin13ipe de fon13tionnement est le suivant
i) 13reacuteation dun 13laquage haute tension initial sur une surfa13e dieacutele13trique
fa13e agrave du gaz ii) 13hauage ohmique du plasma par le 13ourant en 13roissan13e
rapide iii) a1313eacuteleacuteration de la 13ou13he de plasma sous la pression magneacuteti-
que auto-geacuteneacutereacutee par le 13ourant (dizaines de kA) iv) maintien (1 micros) de la
pression magneacutetique pour 13ommuniquer une impulsion importante Des tra-
vaux sur le Z-pin13h ou le plasma fo13us permettent de postuler des geacuteomeacutetries
13onvenables pour a1313eacuteleacuterer une masse de lordre de 10
minus510
minus4g
Apregraves avoir eacutetudieacute le fon13tionnement dun 13ir13uit eacutele13trique rapide jai
listeacute les points forts et les points faibles en fon13tion des obje13tifs et jai pro-
poseacute une geacuteomeacutetrie 13oaxiale 13ompa13te Loptimisation du 13anon agrave plasma a
eacuteteacute faite en terme de vitesse maximale agrave la sortie ave13 13omme seules 13ontrain-
tes une eacutenergie sto13keacutee de 1 kJ une dimension transverse de 4 mm et une
dimension axiale dun ordre plus grande Lobje13tif eacutetait le 13ho13 fortement
supersonique 10-30 kms agrave quelques mbar dAr ou de Xe Un modegravele simple
(0D) a eacuteteacute 13ompareacute agrave un modegravele 3D MHD par le 13ode GORGON qui deacute13rit
bien les expeacuterien13es de plasmas astrophysique par puissan13e pulseacutee (Ciardi
et al 2007)
La pression magneacutetique sexprimant par B
22micro0 un 13ourant supeacuterieur agrave
100 kA apparaicirct neacute13essaire et atteignable sur un geacuteneacuterateur 1 kJ - 1 micros 13om-
pa13t de type R-L-C Ensuite il a fallu monter un modegravele de 13ir13uit deacuteformable
13omportant une se13tion formeacutee de la lame de plasma en mouvement et qui
prenne en 13ompte la1313reacutetion du gaz dans un modegravele dit snowplow (Potter
1971) Des paramegravetres geacuteomeacutetriques 13omme la forme et les dimensions du
13anon ont pu ecirctre optimiseacutes pour donner une grande vitesse de sortie du
plasma pour Ar et Xe et dans la gamme 01 - 10 mbar Un 13anon a eacuteteacute 13on-
struit et 13onne13teacute agrave un geacuteneacuterateur eacutele13trique pulseacute existant Des observations
du mouvement du plasma par la dynamique du rayonnement radial donnent
des vitesses 13oheacuterentes ave13 le modegravele ave13 la simulation 3D MHD et des
nombres de Ma13h eacuteleveacutes de 20 agrave 60 Ces travaux ont soutenu la 13on13eption
dun autre geacuteneacuterateur pour obtenir des performan13es plus eacuteleveacutees en vitesse
de 13ho13 et en taux de reacutepeacutetition
Contents
List of Figures
List of Tables
1 Introdu13tion 1
11 General Context 1
12 Outline of Chapters 2
13 My 13ontribution 3
2 Radiative sho13k waves 5
21 Rankine-Hugoniot(R-H) relations 6
211 Jump relations for a real gas 9
212 Ee13t of radiation on sho13k waves 11
213 Radiative hydrodynami13s equations 13
22 Radiative sho13k regimes 15
221 Typi13al radiative sho13k waves with a pre13ursor 16
23 Radiative sho13k waves experiments 16
231 Laser-driven sho13k experiments 18
232 Ele13tromagneti13ally laun13hed sho13k 21
3 1D Simulations 23
31 HELIOS 23
32 LTE approximation 24
321 Mean opa13ity 25
322 Equation of State 27
33 Single radiative sho13k waves 27
34 Intera13ting radiative sho13k waves 40
35 Summary 44
4 Experimental Setup 47
41 Prague Asterix Laser System (PALS) fa13ility 48
42 Targets 49
421 Massive Targets 51
422 Gaseous Targets 51
423 Target holder 55
CONTENTS
424 Target lling 56
43 Laser Fo13using 57
431 Fo13al Lenses and Phase Zone Plates 57
432 Fo13using of the MAIN and AUX laser beams 58
44 Diagnosti13s 59
441 Visible interferometry 61
442 XUV spe13tros13opy 65
443 Visible spe13tros13opy 67
45 Summary 68
5 Results and Interpretation 71
51 Visible interferometry 71
511 Longitudinal interferometry sho13k speed and ele13tron
density 74
512 Transverse interferometry lateral extension of the sho13k 91
52 XUV Spe13tros13opy 92
53 Simulations based on experimental results 93
54 Summary 97
6 Ele13tromagneti13ally laun13hed sho13k 101
61 Prin13iples of operation of a high 13urrent generator 102
62 Prin13iples of the run-down phase in a PFD 103
63 Proposed design for the plasma gun 104
64 Dynami13 13ir13uit modelling 106
65 3-D MHD simulations using GORGON 13ode 111
66 Measurements 116
67 Summary 118
7 Con13lusion 121
71 Con13lusions 121
72 Perspe13tives 123
8 Thesis summary 127
Appendi13es 141
A Visible Interferometry 143
A01 Refra13tive index of a plasma 143
A02 Absorption of the laser beam 144
A03 Prin13iple of interferometry 145
A04 Ma13h Zehnder Interferometer 146
B Opa13ities and mean 13harge 149
Referen13es 152
List of Figures
21 S13hemati13 diagram for a 1-D sho13k moving in the x dire13tion
The speed of the sho13k front is us In the frame of referen13e of
the sho13k front the pre-sho13k region has a relative speed u1=
-us density ρ1 total pressure P1 temperature T1 whereas these
quantities are respe13tively u2 ρ2 P2 and T2 for the post-sho13k
region 6
22 Proles of temperature (Te = Ti = T ) and mass density (ρ) at 10ns for a sho13k propagating in Xenon at 01 bar (ρ1= 57 times 10
minus4
g13m
minus3) with a speed of sim 45 kms It is to note that the given
pressure value are at 298 K The simulation is performed with the
HELIOS 13ode ex13luding the radiation ee13t and for a polytropi13
gas (γ=53) The initial temperature of Xenon is set to 1 eV
and with a negligible (10
minus5WmK) thermal 13ondu13tivity The
Ma13h number is then equal to 40 For this simulation the sho13k
is laun13hed by a 100 microm thi13k Aluminium piston moving at the
velo13ity of 33 kms The position x=0 13orresponds to the position
of the XeAl interfa13e at time t=0 8
23 Ele13tron temperature (Te) ion temperature (Ti) and mass density
(ρ) proles at 10 ns derived from 1-D simulation for the 13ase of
ionisation (no radiation) The sho13k propagates in Xe gas at 01
bar (ρ1= 57 times 10
minus4g13m
minus3by sim 45 kms) It is to note that the
given pressure value is at 298 K The simulation is performed with
the HELIOS (Te 6= Ti 13ase) using the PROPACEOS equation
of state without any radiation and with a negligible thermal
13ondu13tivity (10
minus05WmK) The initial temperature in Xenon
is set to 1 eV and thus the Ma13h number is equal to 40 For this
simulation the sho13k is laun13hed by a 100 microm thi13k Aluminium
piston moving at the velo13ity of 38 kms The position x=0
13orresponds to the position of the XeAl interfa13e at time t=0 10
LIST OF FIGURES
24 Figure 14 amp 17 of Mi13haut et al (2004) (a) Compression ρ2ρ1(full 13ir13les thin line) kT2 in eV (inverted full triangles thin line)
in Argon (a) Xenon (b) for initial 13onditions kT1( = 10 eV for
Ar and = 01 eV for Xe) ρ1 = 5 times 10minus4 gcm3 versus sho13k
speed in kms with (full markers full lines) and without (empty
markers dashed lines) radiation The ionization stage lt z gt2 is
plotted with the diamond symbol 12
25 Simulated ele13tron temperature (Te) ion temperature (Ti) and
mass density (ρ) proles at 10 ns of a radiative sho13k propagatingwith a 15 kms (a sub-13riti13al) and 48 kms (b super13riti13al)
The sho13k is propagating in Xenon at 01 bar (ρ = 54 times 10
minus4
g13m
minus3 at 298 K) The initial temperature in Xenon is set to
1eV The two Ma13h numbers are respe13tively equal to 13 and 43
The radiation and ionisation ee13ts are in13luded in the simulation
whi13h is performed for two dierent temperatures (Te 6= Ti) and a
negligible thermal 13ondu13tivity (10
minus05WmK) For this simula-
tion the sho13k is laun13hed by a 100 microm thi13k Aluminium piston
moving at the speed of 12 kms and 45 kms for 13ase (a) and (b)
respe13tively The position x=0 13orresponds to the position of the
XeAl interfa13e at time t=0 17
31 PROPACEOS Mono13hromati13 opa13ity versus the photon energy
in eV of Xenon at 10 eV and at two ele13tron densities respe13tively
equal to 1018 and 1020 13m
minus3 26
32 Mass density and temperature (a) ele13tron density and mean
13harge (b) at 10 ns for a radiative sho13k of speed sim 47 kms
in Xenon at 01 bar The verti13al dotted bla13k lines show the
position of the interfa13e between piston and ba13king Xenon gas
Zero at x-axis is the interfa13e of piston and Xenon at time zero
Spitzer thermal 13ondu13tivity has been used in the simulation 29
33 Mean Rosseland opa13ity (in 13m
2g
minus1) and opa13ity (13m
minus1) at 10
ns for a radiative sho13k of speed sim 47 kms in Xenon at 01
bar The verti13al dotted bla13k lines show the position of the
interfa13e between piston and ba13king Xenon gas Zero at x-axis
is the interfa13e of piston and Xenon at time zero Spitzer thermal
13ondu13tivity has been used in the simulation 30
34 (a) Mass density and temperature (b) ele13tron density and
mean 13harge at 10 ns for a radiative sho13k of speed sim 47 kms
in Xenon at 01 bar The verti13al dotted bla13k lines show the
position of the interfa13e between piston and ba13king Xenon gas
Zero at x-axis is the interfa13e of piston and Xenon at time zero
A negligible 13onstant thermal 13ondu13tivity (10
minus12WmK) have
been used in the simulation for Xenon 31
LIST OF FIGURES
35 (a) Ele13tron density and ele13tron temperature (b) Mass density
and mean 13harge at 10 ns for a radiative sho13k of speed sim 48
kms in Xenon at 01 bar for the 13ase when the ele13tron and ion
temperature are dierent The verti13al dotted bla13k lines show
the position of the interfa13e between piston and ba13king Xenon
gas Zero at x-axis 13orresponds to the position of the interfa13e
between the piston and Xenon at time zero The Spitzer thermal
13ondu13tivity has been used in this simulation 32
36 (a) Mass density and ele13tron temperature (b) ele13tron density
and mean 13harge at 10 ns for a radiative sho13k of speed sim 45
kms in Xenon at 01 bar for the 13ase when the ele13tron and ion
temperature are dierent The verti13al dotted bla13k lines show
the position of the interfa13e between piston and ba13king Xenon
gas Zero at x-axis 13orresponds to the position of the interfa13e
between the piston and Xenon at time zero In this simulation
value of thermal 13ondu13tivity (10
minus12WmK) is negligible for
Xenon The peak value of ion temperature is found to be 405 eV
whereas theoreti13ally this value is expe13ted to be sim 600 eV It is
possible to a13hieve the expe13ted value by in13reasing the resolution
of the simulation 34
37 Ele13tron temperature proles for various numbers of frequen13y
groups N = 1 20 50 60 70 90 and 100 35
38 (a) Ele13tron temperature (a) mean 13harge (b) and mass density
(13) for four (1 10 30 and 40) multipliers of the Xenon opa13ity at
10 ns 36
39 Plan13k and Rosseland mono13hromati13 opa13ity proles Te for ρ=15 times 10
minus3from PROPACEOS (I) BIGBART (II) and Rodri-
guez et al (III) 38
310 Mono13hromati13 Xenon opa13ity versus the photon energy in eV
at T = 15 eV and for ρ = 15 times 10
minus3g13m
minus3for the PROPA-
CEOS (a) and Rodriacuteguez et al (2015) (Fig 16 of Rodriacuteguez et al
(2015)) (b) models 39
311 Ele13tron density Ne (a) and ele13tron temperature Te (b) versus
axial position (along a 04 13m long sho13k tube) at 3 10 20 30
and 38 ns from HELIOS simulations (with opa13ity times 20) for the
13ases of single sho13k of sim 50 kms (dotted line) and two identi-
13al 13ounter-propagating sho13ks of sim 50 kms (solid lines) The
verti13al dotted lines show the position of the interfa13e between
piston and ba13king Xenon gas 41
LIST OF FIGURES
312 Variations ofNe (a) and Te (b) versus axial position for the 13ase of
two identi13al 13ounter-propagating sho13ks (of speeds sim 50 kms)
at 3 10 20 30 35 and 40 ns as derived from HELIOS simulations
For these simulations we have negle13ted the ee13t of radiation
by keeping the Xe opa13ity equal to zero The verti13al dotted lines
show the position of the interfa13e between the piston and ba13king
Xenon gas 43
313 Variations of Ne (a) and Te (b) with axial position for the 13ase
of two non-identi13al 13ounter-propagating sho13ks (of speeds sim 50
amp 20 kms) and two single sho13ks (dotted lines) of speeds sim 50
amp 20 kms respe13tively at 3 10 30 35 and 46 ns as derived
from HELIOS simulations The verti13al dotted lines show the
position of the interfa13e between piston and ba13king Xenon gas
(with opa13ity times 20) 45
41 a Spatial prole of the PALS Laser beam b Intensity prole of
laser pulse I(t) with time 48
42 Energy distribution s13heme for MAIN and AUX laser beams 49
43 Snapshot of the spheri13al 13hamber showing the horizontal bred-
board and on the right the fo13using MAIN lens 50
44 Dimension drawing (a) and snapshot (b) of Aluminum massive
(mo13k) target 50
45 Values on the gures are in mm (a) Verti13al 13ross se13tion view
of the gaseous target (b) Horizontal 13ross se13tion view 52
46 Pi13ture of a the gaseous target on its base 53
47 S13hemati13 of the gaseous target 53
48 Transmission of 100 nm thi13k Si3N4 membrane (CXRO database) 54
49 Snapshot of a target holder with one V-shield regarding the AUX
laser whi13h will pass through the hole in this shield marked in
green in the Figure The target whi13h fa13es the two diagnosti13s
(tubes) and is lo13ated behind the V-shield is thus not visible 54
410 In-situ system for target lling and pressure monitoring 56
411 S13hemati13 of the keV 13amera 59
412 (a) Sign 13onvention for the respe13tive positions of the lens and
the target (b) keV image of a MAIN impa13t on a massive target
(lens position +1500 microm) The spot size is 475plusmn25 microm Pixel
size for MAIN keV 13amera is 56 microm (13) keV image of an AUX
impa13t (lens position -1500 microm) The spot size is sim 280plusmn20 micromon target One pixel = 66 microm on AUXs keV 13amera 60
413 First interferometri13 setup All the opti13al elements are 1 in13h
For simpli13ity the plasma slab is not reported in the probe beam 61
414 Se13ond interferometri13 setup 63
415 S13hemati13 of the Streak 13amera (from Hamamatsu noti13e) 64
LIST OF FIGURES
416 Imaging setup of the Ma13h Zehnder Interferometer setup opera-
ting in the longitudinal mode The 13hannel of the target is fully
illuminated by the probe beam In this mode the lens images
the axis of symmetry of the two sho13k waves along the slit of the
streak 13amera 65
417 (a) Referen13e 2D image of a target before the shot re13orded on
the Streak 13amera The positions of the two pistons 13losing the 4
mm long target are lo13ated at 800 and 4800 microm The dark zones
along them (between 800 to 1000 and 4600-4800 microm) 13orresponds
to the glue This glue is then visible through verti13al bla13k strips
in the interferometri13 re13ord (b) Corresponding interferometri13
time-spa13e re13ord 66
418 Typi13al ray tra13ing in a grazing in13iden13e XUV spe13trograph
The dete13tor (GMCP or image plate) is installed tangentially to
the Rowland 13ir13le 67
419 (a) Setup of the XUV spe13trometer in the spheri13al 13hamber (b)
Typi13al re13ord between sim 10 and 40 nm is shown in the bottom
panel 68
420 (a) S13hemati13 of setup employed to re13ord the time and spa13e
visible plasma emission (b) A typi13al spe13trogram 69
51 Original re13ord of shot 48055 (left) FFT of the original image
(13enter) and the frequen13y ltered image (right) 73
52 A 13ropped se13tion of the re13ord from shot the 48055 The
rst ve positions have been sele13ted manually `+ signs (in red
13olor) on ea13h fringe On this re13ord the distan13e between two
unperturbed fringes is 13orrespond to 15 pixels (ie 159 microm) 73
53 (a) Representative points of the fringes as derived from the spline
tting of the 5 manually sele13ted points (Fig 52 (II)) on ea13h
fringe (b) Positions of the fringes maxima along Y-axis for ea13h
fringe derived by lo13ating the points of maximum intensity in
X-dire13tion of the previous points obtained by spline t 75
54 Fringe maxima on the full image 76
55 Imaging setup of the Ma13h Zehnder Interferometer operating in
the longitudinal mode The lens images the axis of symmetry of
the two sho13k waves along the slit of the streak 13amera 76
56 Interferometri13 image re13orded for the shot 48055 in Xe at 01
bar The sho13k speeds for the sho13ks driven by MAIN (from left
side) land AUX (from right side) lasers are respe13tively equal to
sim 54 and 23 kms The time of laser arrival on the piston is at
146 ns The positions of the Au-Xe interfa13e on the re13ord are
respe13tively 950plusmn50 and 4950plusmn50 mi13rons 77
LIST OF FIGURES
57 (a) MAIN sho13k speed (13al13ulated by the last fringe method)
versus the MAIN energy for Xenon or XeHe mixture at dierent
pressures with the error bars (b) AUX sho13k speed versus AUX
energy also for Xenon or XeHe mixture at dierent pressures 79
58 MAIN sho13k speed (13al13ulated by the last fringe method) versus
the MAIN laser energy for Xe (at 01 bar only) Ar He and Kr
at dierent pressures with the error bars 82
59 Left panel interferometri13 re13ords 48055 in Xe at 01 bar (a)
48132 in Xe+He at 02 bar (b) and 48138 in Xe+He at 02 bar
(13) Right panel ele13tron density at 10 20 30 and 40 ns versus
distan13e for these re13ords The positions of maxima have been
identied on the re13ords in the left panel The time t = 0 13orre-
sponds to the time of laser arrival on the target and the position
x = 0 13orresponds to the interfa13e between the piston (Au layer)
and the gas Its determination is pre13ise within 100 mi13rons The
distan13es between two unperturbed fringes for re13ords 48055
48132 and 48138 are 159 244 and 244 microm respe13tively The
lt Ne gt un13ertainty (plusmn 2 pixels) is indi13ated by the error bar in
the right panels It 13orresponds respe13tively to plusmn 9 times 10
17 plusmn 6
times 10
17and plusmn 6 times 10
1713m
minus3for the gures (a) (b) and (13) The
limit of dete13tion (2 pixels) is presented by a dotted line on ea13h
gure 85
510 (a) Interferometri13 re13ord in Ar at 03 bar (48079) The esti-
mated speeds for MAIN and AUX are 49plusmn5 and 23plusmn3 kms (b)
Same re13ord where the fringes maxima are marked by points
The bins denition is as fellows bin 0 Ne le 11 times 10
1813m
minus3
(white) bin 1 11 - 18 times 10
1813m
minus3(yellow) and bin 2 39 -
57 times 10
1813m
minus3(red) The time t = 0 13orresponds to the time
of laser arrival on the target and the position x = 0 13orresponds
to the interfa13e between the piston (Au layer) and the gas Its
determination is pre13ise within 100 mi13rons 87
511 Results for temperature (a) and Rosseland opa13ity (b) obtained
from HELIOS simulation at 10 ns for two 13ounter-propagating
sho13ks at sim 50 and 18 kms for Xe (with opa13ity multiplier 1
and 20) Ar and Kr at 54 times 10
minus4g13m
minus3initial mass density
A 13omparison Ar Kr and Xe PROPACEOS opa13ity shown in
APPENDIX B 88
LIST OF FIGURES
512 (a) Raw interferometri13 re13ord in Kr at 02 bar (shot 48146)
The estimated speed for MAIN sho13k is 53plusmn2 kms The time
t = 0 13orresponds to the time of laser arrival on the target(b)
Same re13ord shown in (a) maxima is marked by 13olored points
The bins denition is as follows bin 0 Ne le 11 times 10
1813m
minus3
(white) bin 1 11 - 18 times 10
1813m
minus3(yellow) and bin 2 39 -
57 times 10
1813m
minus3(red) The time t = 0 13orresponds to the time
of laser arrival on the target and the position x = 0 13orresponds
to the interfa13e between the piston (Au layer) and the gas Its
determination is pre13ise within 100 mi13rons 90
513 S13hemati13s of the Ma13h Zehnder Interferometer setup to re13ord
transverse interferometri13 images The lens allows to make on
the slit of the 13amera the image of a se13tion perpendi13ular to
dire13tion of sho13k propagation 91
514 Transverse interferometri13 images for (a) shot48111 (MAIN
sho13k only) (b) shot48130 (AUX sho13k only) The time is
measured after an oset equal to 14 and 23 ns respe13tively after
the time of the laser arrival on the target The position zero on
the x-axis of ea13h image 13orresponds to the base of the target 92
515 Interferometri13 image for the shot48143 The time t = 0 13orre-
sponds to the time of laser arrival on the target and the position
x = 0 13orresponds to the interfa13e between the piston (Au layer)
and the gas Its determination is pre13ise within 100 mi13rons 94
516 Raw (a) and 13orre13ted XUV spe13trum (b) for the shot 48143 95
517 Re13orded ele13tron density (shot 48055) together with the HE-
LIOS results (with Xenon opa13ity times 20) at dierent times in
Xenon at 01 bar 97
518 Time evolution of the mass density (a) ele13tron temperature (b)
and mean 13harge (13) at 56 57 58 60 64 and 65 ns within the
sho13k tube derived from the HELIOS simulations (with Xenon
opa13ity multiplier = 20) for two 13ounter streaming sho13ks of sim39 and 18 kms in Xenon at 06 bar 98
61 (a) Sket13h of the sho13k generator showing the pulsed ele13tri13al
13ir13uit the set of 13oaxial 13oni13al ele13trodes with a 13onstant radial
gap and the plasti13 insulator featured in grey on whi13h a planar
surfa13e dis13harge is initiated The installation of three opti13al
bers allows looking radially at the plasma moving in the sho13k
tube (b) S13hemati13s of the plasma dynami13s inside the 13oaxial
gun in fast-pulse mode the ele13tri13al 13urrent ows in the super-
13ial layers of the two 13oaxial 13oni13al ele13trodes and through an
annular plasma layer The magneti13 pressure Pmag pushes the
dis13harge axially 105
62 Exploded view of the plasma gun 106
LIST OF FIGURES
63 Time dependen13e of the main parameters for Ar gas at 1 mbar
a1313ording to the 13ir13uit model (a) 13urrent (kA) and voltage
(V) (b) a1313reted mass (kg) and rate of a1313retion (kgs) (13)
speed (ms) and kineti13 energy (J) 108
64 Plot of the dierent output parameters (normalized with respe13t
to the peak values) obtained at the top of the inner ele13trode
For ea13h group upper panel represents the variation for Ar gas
whereas lower panel is for Xe gas Proles of output parameters
are given (a) with 13one angle (b) with a1313retion fa13tor (13) with
gas pressure and (d) with damping resistan13e 109
65 Example of mapping of the ele13tron density from a 3-D MHD
simulation (ba13kground gas Argon at 1 mbar) when the plasma
sheath is (left) at the exit of the plasma gun (right) in free
ight 13onditions (log s13ales for the false 13olors) 113
66 Time history of plasma merging and early free ight through
the mapping of the ele13tron density ele13tron temperature and
average ioni13 13harge as given by a 3-D MHD simulation (ba13k-
ground gas Argon at 1 mbar) (s13ales for the false 13olors) 114
67 Axial proles values taken in the sho13k tube along a line slightly
o-axis at a distan13e of 1 mm for mass density ion temperature
ion density average ion 13harge magneti13 eld ele13tron tempera-
ture ele13tron density and average speed at 1500 ns (ba13kground
gas Argon at 1 mbar) A maping of |B| is given with a dashed
line indi13ating sho13k front position as well 115
68 Experimental results (a) time history of the passive opti13al
re13ords 13ompared to the main 13urrent for Argon gas 13lose to the
referen13e pressure (b) time-integrated signal vs Ar pressure for
the dierent bers (13) averaged speed vs Ar ba13king pressure 117
71 Interferometri13 re13ord obtained at 546 ns for a single radiative
sho13k of speed sim 100 kms propagating in Xe+He gas at 06 bar
driven by laser at 348 nm with energy 170 J The dierent 13olors
in (b) 13orrespond to yellow le Ne le 35 times 10
1713m
minus3 13yan 37
- 74 times 10
1713m
minus3 blue74 - 15 times 10
1813m
minus3 green 15 - 18 times
10
1813m
minus3 magenta 18 - 22 times 10
1813m
minus3 red 22 - 26 times 10
18
13m
minus3 orange26 times 10
1813m
minus3- 18 times 10
1913m
minus3 124
A1 S13hemati13s of Ma13h Zehnder interferometer with two mirrors
two 50 beam splitters and the dete13tor A plasma slab is pla13ed
in one of the two arms of the interferometer 147
B1 Rosseland and Plan13k opa13ity for three gases Xe Kr and Ar at
mass densities 51 times 10
minus4g13m
3(a) and 16 times 10
minus3g13m
3(b) 150
LIST OF FIGURES
B2 Mean 13harge for three gases Xe Kr and Ar at mass densities 16
times 10
minus3g13m
3(a) and 51 times 10
minus4g13m
3(b) 151
List of Tables
21 Summary of the radiative sho13ks experiments and out13omes in
dierent laser installations All the experiments are performed in
Xe ex13ept Keiter et al (2002) and Vis13o et al (2012) In these
two experiments the rst uses SiO
lowast
2 and the se13ond uses Ar
lowastlowastas
a sho13ked medium 20
41 Nominal 13hara13teristi13s of MAIN and AUX laser beams 49
42 Spe13i13ations of the MAIN and AUX fo13al lenses and the f- num-
ber (fo13al lengthbeam diameter) is given (see table 41 for the
beam diameters) 57
43 Spe13i13ations of the two Phase Zone Plates 58
44 Spe13i13ations of the two uEye keV 13ameras (1stVsion 13ompany) 59
45 Spe13i13ations of Andor CCD 13amera 67
46 Spe13i13ations of the XUV grating 67
51 Sho13k speeds estimated from the `last fringe method `NA re-
presents the entries whi13h 13ould not be dedu13ed from the re13ord
Further entries in the bold font are dis13ussed in detail in this
13hapter 81
52 Atomi13 data and density at 01 bar (at room temperature) for
He Ar Kr and Xe 82
61 Input parameters their standard values and range of variation 110
62 Laun13hing performan13e ArXe 111
63 Values of variables in various 13onditions at rest (with seed ele13-
trons) inside the sho13k and inside post-sho13k region A star lowastdenotes the insulator surfa13e 112
Chapter 1
Introdu13tion
11 General Context
Radiative sho13ks (RS) are present in various astrophysi13al environments es-
pe13ially in stars They are then present in all the supersoni13 a1313retioneje13tion
pro13esses during the stellar evolution from the early phase of the proto-
stellar 13ollapse up to the magnetosphere a1313retion on to the already formed
T- Tauri stars up to the late stage of the supernovae sho13ks
Radiative sho13ks are strong sho13ks (ie Ma13h number Mgtgt1) whi13hrea13h high temperatures and thus are the sour13e of intense radiation Alt-
hough the observation of (ie spatially resolved ) signatures of su13h sho13ks
in the astrophysi13al environments is very di13ult an alternate approa13h na-
med laboratory plasma astrophysi13s provides a powerful tool to study them
on the Earth
These waves have been experimentally studied sin13e more than a de13ade
mostly on large-s13ale laser fa13ilities in noble gases and with dierent targets
geometries The ablation pro13ess generated by the intera13tion between the
laser and a foil indu13es a sho13k wave in the foil whi13h then propagates
in the gas at a speed of several tens of kms Numeri13al simulations 13an
des13ribe these sho13ks with an improving pre13ision With laser intensities on
target 13omprised between 1014 and 1015 W13m
2 these experiments allowed
to re13ord sho13k speeds ranging between 40 and 150 kms
All previous experimental studies have been fo13used on the 13ase of single
radiative sho13ks However in astrophysi13al 13onditions the radiative sho13k
in general intera13ts with a denser medium leading to the development of
ree13ted and transmitted sho13ks A few representative examples of su13h
phenomena are the intera13tion of supernovae remnants with dense mole13ular
13louds the a1313retion sho13ks on the photosphere of T-Tauri stars and the bow
sho13ks at the head of fast stellar jets The 13ollision (or the intera13tion) of
two radiative sho13k waves is obviously a rare astrophysi13al event and the
template 13ase of the supernova remnants DEM L316 (see Fig 1 of Williams
et al (1997)) is still the subje13t of debates (Velarde et al 2006) In this
regard the development of dedi13ated laboratory experiments to the study of
propagation and intera13tion of 13ounter-propagating sho13k waves is important
as a tool to 13hara13terize su13h events through their spe13i13 signatures
Complementary to laser experiments 13ompa13t pulsed power generators
may drive an ele13tromagneti13 13oaxial plasma gun to 13reate astrophysi13al
1
2 CHAPTER 1 INTRODUCTION
relevant sho13ks in lower pressure noble gases with a high availability and a
rather modest 13apital 13ost The ele13tromagneti13ally driven sho13k waves may
have larger s13ales than those by laser Thus they 13an be analyzed rather
easily with the adequate diagnosti13s (Kondo et al 2008)
In this thesis I am presenting radiative sho13ks in the 13ontext of labo-
ratory astrophysi13s using two experimental approa13hes laser-driven sho13ks
and ele13tromagneti13ally driven sho13ks
12 Outline of Chapters
In the rst 13hapter the physi13s of radiative sho13ks wave will be summarized
I will dis13uss the Rankine -Hugoniot (RH) for the perfe13t gas and genera-
lized RH equations for ionized gas Further the ee13ts of radiation on the
sho13k equations will be introdu13ed and I will review the dierent regimes of
the radiative sho13ks presenting a radiative pre13ursor Then I will present
a short histori13al review of laser and ele13tromagneti13ally laboratory experi-
ments on this topi13 Finally I will highlight the obje13tives of my work in
this framework
The 13hapter two is dedi13ated to the numeri13al analysis of radiative sho13k
waves and of the main physi13al pro13esses whi13h play a role in these 13omplex
pro13esses The simulations will be performed in 1D using the 13ommer13ial ra-
diation hydrodynami13 HELIOS 13ode (Ma13Farlane Golovkin and Woodru
2006) After a des13ription of the 13ode and his possibilities several simu-
lations will be performed to analyze the ee13ts of the two uids (ele13trons
and ions) approa13h 13ompared to the one uid 13ase (identi13al temperature
for both) of the thermal 13ondu13tivity and the multi-groups opa13ity des13rip-
tion Then a brief overview of single and two 13ounter-propagating radiative
sho13ks will be presented
Chapter three will start with a short overview of the Prague Asterix Laser
System fa13ility (PALS) It will be followed by the experimental setup in13lu-
ding the two high energy beams and targets des13ription These paragraphs
will be followed by the detail of the diagnosti13s
The fourth 13hapter will be dedi13ated to the data analysis the results
and the interpretation of the experimental re13ords I will rst des13ribe the
method I followed for the analysis of interferometri13 re13ords Subsequently
the results obtained from a few representative re13ords for Xe Ar and Kr
will be dis13ussed in details Complementary 1D simulations performed with
initial 13onditions similar to that in the experiment will be shown and help to
interpret the experimental results Finally the results of XUV spe13tros13opy
diagnosti13s will be presented
In the fth 13hapter I will present the 13alibration of a table top ele13tro-
magneti13 plasma generator whi13h is able to laun13h supersoni13 sho13ks up to
sim 40 kms speed at stati13 pressures of few mbar The results of this 13ali-
13 MY CONTRIBUTION 3
bration will also be presented and a 13omparison with 3-D MHD simulations
will be performed The diagnosti13s whi13h have been implemented will be
presented to illustrate the model as well as preliminary re13ords of the plasma
speed
The nal 13hapter of the thesis is dedi13ated to the summary of this work
together with perspe13tives for the follow-up of these experiments
13 My 13ontribution
I joined PhD on November 12 2013 under the supervision of Dr Jean
Larour and Dr Chantal Stehleacute My PhD thesis is 13omprised of two parts
the rst part of the work is to study the laser-driven sho13ks The se13ond
part of my thesis is to study the ele13tromagneti13ally laun13hed strong sho13ks
Con13erning the laser-driven sho13ks I have started to work on the analysis
and interpretation of XUV images re13orded previously during an experimen-
tal 13ampaign 13ondu13ted at the Prague Asterix Laser System (PALS) fa13ility
in 2011 This enabled me to estimate the lo13al opa13ity by employing Abel
inversion Although this work not presented in this thesis it shed light on
the basi13 understanding of radiative sho13k as seen in the experiments as well
as the knowledge of experimental setups Later in April - May 2015 I have
parti13ipated in a ve weeks experimental 13ampaign on PALS laser fa13ility
in Prague whi13h was dedi13ated to the study of the spe13tros13opi13 and in-
terferometri13 signatures of laser-produ13ed sho13ks Espe13ially the experiment
was designed to study the physi13s during the 13ollision of 13ounter-propagating
sho13ks In the preparation of this 13ampaign I be13ame a13quainted with the
instruments and the targets to be used as well as prepared a master re-
port 13ontaining all the 13ru13ial aspe13ts of the experiment to be referred by
all the parti13ipants during the experiments During the 13ampaign we have
performed around 55 shots 13orresponding to dierent gases (Xenon Argon
Helium and Krypton) at dierent pressure and used three types of diagnos-
ti13s ie visible interferometry visible spe13tros13opy and XUV spe13tros13opy
In this experiment parti13ularly I was in 13harge of the target sele13tion its
installation in the experimental 13hamber gas lling preparing logbook re-
13ording of the visible spe13tros13opy data from the 13ontrol room qui13k data
analysis for the spot size sho13k speed from interferometry data et13 In
addition I have worked on spe13tros13opi13 and interferometri13 data analysis
re13orded during the experiment Next in order to interpret the experimental
results I have performed several 1D hydrodynami13 simulations with varying
input 13onditions In parallel I have also performed simulations for another
experiment performed by my 13ollaborators at ORION laser fa13ility in the
United Kingdom However I have not presented this work in this thesis
Based on the results obtained from my analysis of re13ords obtained from
the PALS experiments in 2015 grabbing the opportunity we have planned
4 CHAPTER 1 INTRODUCTION
and parti13ipated to a 13omplementary experimental 13ampaign at PALS in
September 2016 The analysis of data re13orded during experiments is still
under progress and forms the future s13ope of the thesis
The se13ond part of my thesis has been 13omprised of the study of sho13k
laun13hed through the ele13tromagneti13 generator setup In this I have wor-
ked on the optimization of the design of the ele13tri13 generator in view of the
required sho13k parameters Moreover I have derived various sho13k 13hara13-
teristi13s viz radial dis13harge layer namely speed mass a1313eleration et13
at the dierent positions of its propagation In present a upgraded ele13tri13
generator with new diagnosti13s is ready to perform the new experiments
Chapter 2
Radiative sho13k waves
Contents
21 Rankine-Hugoniot(R-H) relations 6
211 Jump relations for a real gas 9
212 Ee13t of radiation on sho13k waves 11
213 Radiative hydrodynami13s equations 13
22 Radiative sho13k regimes 15
221 Typi13al radiative sho13k waves with a pre13ursor 16
23 Radiative sho13k waves experiments 16
231 Laser-driven sho13k experiments 18
232 Ele13tromagneti13ally laun13hed sho13k 21
Small-amplitude disturban13es in a medium often take the form of linear
waves On the 13ontrary strong disturban13es generally produ13ed by pro13esses
su13h as explosions lightening supersoni13 movements of bodies powerful
ele13tri13 dis13harges et13 where the uid properties 13hange rapidly generate
nonlinear waves like sho13k waves As a 13onsequen13e of sho13k waves the ow
moves faster than the sound speed A sho13k is 13hara13terized by a sudden
in13rease in the density and velo13ity of the medium The Ma13h number (M)
is 13ommonly used to 13hara13terize the strength of the sho13k This number
dened as the ratio of the sho13k speed to the speed of sound in the upstream
medium in whi13h the sho13k propagates
Radiative sho13k waves are hypersoni13 sho13k waves whi13h are heated to
high temperature and as a 13onsequen13e be13ome the sour13e of intense radi-
ation This radiation in turn modies the dynami13s and stru13ture of the
sho13k itself whi13h makes its stru13ture more 13ompli13ated Radiative sho13k
waves o1313ur in several astrophysi13al 13ir13umstan13es su13h as in the a1313retion
sho13k of protostellar formation (Stahler Palla and Salpeter 1986) the su-
pernova explosion and the intera13tion of their remnants with the dense in-
terstellar medium (Chevalier 1977) the bow sho13ks at the head of stellar
jets (Hartigan et al 2001)
Radiative sho13k waves 13an now be studied in the laboratory whi13h allows
to 13ompare the data with models existing in the literature and to 13he13k the
ability of the numeri13al 13odes to reprodu13e them (see for instan13e Bouquet
et al (2004) Leygna13 et al (2006)) The experimental study of radiative
5
6 CHAPTER 2 RADIATIVE SHOCK WAVES
sho13k 13hara13teristi13s is 13ru13ial in understanding these various energeti13 pro-
13esses o1313urring in the astrophysi13al plasmas In this 13hapter I will rst
present a short summary of the jump 13onditions (Rankine-Hugoniot rela-
tions) for the simple 13ase of an ideal gas whi13h rely on the values of the
thermodynami13al quantities on both sides of the dis13ontinuity Next I will
present the role played by the ionisation and ex13itation of the gas and then
the role played by the radiation energy and pressure The des13ription of the
dierent regimes of radiative sho13ks whi13h depend on the opa13ity of medium
and 13an be 13hara13terized by the presen13e or absen13e of a radiative pre13ursor
will follow Further I will give a brief overview of the literature fo13using on
the generation of radiative sho13ks in the laboratory and I will present the
motivations and obje13tives of my work
21 Rankine-Hugoniot(R-H) relations
Let us 13onsider a one-dimensional sho13k propagating in a gas at rest with
the speed us In the frame asso13iated with the sho13k front the pre-sho13k
(upstream) uid velo13ity is then u1 = - us while behind the sho13k (post-
sho13k or downstream region) this velo13ity is u2 as shown in the Fig 21
Figure 21 S13hemati13 diagram for a 1-D sho13k moving in the x dire13tion The
speed of the sho13k front is us In the frame of referen13e of the sho13k front the pre-
sho13k region has a relative speed u1= -us density ρ1 total pressure P1 temperature
T1 whereas these quantities are respe13tively u2 ρ2 P2 and T2 for the post-sho13k
region
Assuming that a stationary state is rea13hed the 13onditions whi13h deter-
mine the post-sho13k thermodynami13al quantities are the well known Rankine-
21 RANKINE-HUGONIOT(R-H) RELATIONS 7
Hugoniot equations (Ma13quorn Rankine 1870 Hugoniot 1887 1889) De-
du13ed from the Euler equations these 13onservation equations for the mass
momentum and energy are given by
ρ2u2 = ρ1u1 (21)
ρ2u22 + P2 = ρ1u
21 + P1 (22)
u2(ε2 +P2
ρ2) +
1
2u32 = u1(ε1 +
P1
ρ1) +
1
2u31 (23)
where ε1 and ε2 are the spe13i13 internal energies (energy per unit mass)
in the two (upstream and downstream) regions
For an ideal gas of atomi13 mass m =mPA (mP proton mass A atomi13
number) and having an adiabati13 index γ the speed of sound is given by
Cs =
radic
γP
ρ=
radic
γkBT
m(24)
and the Ma13h number (M) of the sho13k is then
M =u1Cs1
(25)
Inserting these the aforementioned form of M and Cs in the equations
21 22 and 23 we 13an derive the 13ompression pressure and temperature
ratios for the system as follows
ρ2ρ1
=u1u2
=M2(γ + 1)
2 +M2(γ minus 1)(26)
P2
P1=
2M2γ minus (γ minus 1)
(γ + 1)(27)
T2
T1=
P2
P1
ρ1ρ2
=(2M2γ minus (γ minus 1))(2 +M2(γ minus 1)
M2(γ + 1)2) (28)
Considering the 13ase of strong sho13ks where M may be 13onsidered to be
≫1 these equations be13ome
ρ2ρ1
=(γ + 1)
(γ minus 1)(29)
T2
T1=
2M2γ(γ minus 1)
(γ + 1)2(210)
kBT2 =2(γ minus 1)
(γ + 1)2mus
2(211)
8 CHAPTER 2 RADIATIVE SHOCK WAVES
For mono-atomi13 gases γ = 53 the 13ompression ratio is estimated to
be 4 and the post-sho13k temperature is
kBT2 =3
16mu1
2 =3
16mPAu1
2(212)
Temperature is thus proportional to the square of the sho13k velo13ity and
to the atomi13 mass Therefore to a13hieve a high temperature in the radiative
sho13k experiment we often use high atomi13 number and mass gases For Xe-
non (A=13129) the post-sho13k temperature is then equal to T2 = 0257u21(eV kms) whi13h for instan13e gives sim 520 eV for a sho13k propagating at
45 kms (see Fig 22)
Figure 22 Proles of temperature (Te = Ti = T ) and mass density (ρ) at 10ns for a sho13k propagating in Xenon at 01 bar (ρ1= 57 times 10
minus4g13m
minus3) with
a speed of sim 45 kms It is to note that the given pressure value are at 298 K
The simulation is performed with the HELIOS 13ode ex13luding the radiation ee13t
and for a polytropi13 gas (γ=53) The initial temperature of Xenon is set to 1
eV and with a negligible (10
minus5WmK) thermal 13ondu13tivity The Ma13h number
is then equal to 40 For this simulation the sho13k is laun13hed by a 100 microm thi13k
Aluminium piston moving at the velo13ity of 33 kms The position x=0 13orresponds
to the position of the XeAl interfa13e at time t=0
21 RANKINE-HUGONIOT(R-H) RELATIONS 9
211 Jump relations for a real gas
In 13ontrast to the dis13ussion on ideal gas presented in the previous se13tion
the 13ase of a real gas is a little more 13ompli13ated It is to note that in this
13ase a part of the kineti13 energy is used to ex13ite and ionize the post-sho13k
gas As a result its temperature is lower than that for the 13ase of ideal gas
In addition the pressure is also modied due to ionisation (Mi13haut et al
2004) as dis13ussed below
Let us now dene j as the ionization stage of the atom (varying between
0 and z) and i to be the atomi13 state in this ionisation stage We shall
denote by Pj amp Pji the fra13tional ionization of the ion and the population
of the atomi13 state i in the jth ionization state respe13tively Then the mean
ionization stage (average 13harge) per atom 13an be written as
z =
zsum
j=0
jPj (213)
and the ex13itation energy (in13luding ionization) per atom and per unit of
mass is
ǫexc =
sumzj=0
sum
i Pji Eji
mpA(214)
where Eji is the energy of the ionisation stage j in the atomi13 stage iThus the total pressure of gas 13an be written as
P = Pi + Pe (215)
where Pi and Pe are the ioni13 and ele13troni13 pressures 13ontributions
Using the 13harge neutrality 13ondition Ne = z Ni this gives
P = Ni(1 + z) kBT = ρkBT
m(1 + z) (216)
The thermal pressure and enthalpy of the real gas are given by
Pth = ρ(1 + z)
mkBT (217)
h =5
2
(1 + z)
mkBT + ǫexc (218)
Next the speed of sound in13luding the ee13t of ionization with the
average 13harge z in the medium is now modied by ionisation
Cs ≃radic
5
3
γ(z + 1)kBT
m(219)
Thus the 13ontinuity relations in the frame moving with the sho13k front
be13ome
ρ2u2 = ρ1u1 (220)
10 CHAPTER 2 RADIATIVE SHOCK WAVES
ρ2u22 + ρ2
kBT2
m(1 + z2) = ρ1u
21 + ρ1
kBT1
m(1 + z1) (221)
ρ2u2
[5
2
(1 + z2)
mkBT2 + ǫexc2 +
u222
]
= ρ1u1
[5
2
(1 + z1)
mkBT1 + ǫexc1 +
u212
]
(222)
Figure 23 Ele13tron temperature (Te) ion temperature (Ti) and mass density
(ρ) proles at 10 ns derived from 1-D simulation for the 13ase of ionisation (no
radiation) The sho13k propagates in Xe gas at 01 bar (ρ1= 57 times 10
minus4g13m
minus3by
sim 45 kms) It is to note that the given pressure value is at 298 K The simulation
is performed with the HELIOS (Te 6= Ti 13ase) using the PROPACEOS equation
of state without any radiation and with a negligible thermal 13ondu13tivity (10
minus05
WmK) The initial temperature in Xenon is set to 1 eV and thus the Ma13h
number is equal to 40 For this simulation the sho13k is laun13hed by a 100 microm thi13k
Aluminium piston moving at the velo13ity of 38 kms The position x=0 13orresponds
to the position of the XeAl interfa13e at time t=0
Similar to the Fig 22 however for the 13ase of a real gas the Fig
23 shows the variation the ele13tron and ion temperatures as well as the
mass density with the distan13e for a sho13k propagating in Xenon at sim 45
kms Now the post-sho13k ele13tron temperature peaks at 22 eV whi13h is
mu13h smaller than the maximum temperature of the ions (510 eV) The
13ompression is now equal to 10 instead of 4 as in the previous 13ase The ions
21 RANKINE-HUGONIOT(R-H) RELATIONS 11
are heated rst in the sho13k through ion-ion 13ollisions This ion temperature
de13reases and the ele13tron temperature in13reases as a 13onsequen13e of the
ele13tron-ion 13ollisions The two temperatures thus be13ome equal Sin13e the
momentum 13hange of the ions require only a few number of kineti13 13ollisions
the thi13kness of the abrupt transition between the pre-sho13k and the post-
sho13k is of the order of a mean free path of the ions
It should be noted that for sho13ks propagating with very high speed
the post-sho13k temperature is so high that the 13ompressed gas is fully ioni-
zed Therefore the ee13ts of ionisation and ex13itation be13ome negligible and
hen13e the 13ompression ratio rea13hes 4 as in the 13ase of the adiabati13 sho13k
212 Ee13t of radiation on sho13k waves
In the previous se13tion the ee13ts of radiation heating and 13ooling have
been negle13ted However the radiation emitted in a strong sho13k will ae13t
its stru13ture These ee13ts 13an be determined using mass momentum and
energy 13onservation equations whi13h now have to in13lude the 13ontributi-
ons of the radiative ux energy and pressure To this purpose and with
some simpli13ations we 13an use the same approa13h than in se13tion 21 still
assuming that the sho13k is stationary
Jump relations with radiation
Let us 13onsider a stationary sho13k propagating in an atomi13 gas far away from
the dis13ontinuity The medium is then opti13ally thi13k and we 13an negle13t
the 13ontribution of the radiative ux Let us also assume that the medium
is at LTE and that the radiation pressure and energy may be des13ribed as
follows
Prad =1
3Erad =
4
3cσT 4 =
1
3aradT
4(223)
where arad is the radiation density 13onstant and T is the sho13k radiation
temperature whi13h we shall suppose to be equal to the ele13tron temperature
Following Mi13haut et al (2004) the new jump 13onditions a13ross the sho13k
dis13ontinuity be13ome
ρ2u2 = ρ1u1 (224)
ρ2u22+ρ2
kBT2
m(1+z2)+
1
3aradT
42 = ρ1u
21+ρ1
kBT1
m(1+z1)+
1
3aradT
41 (225)
ρ2u2
(
5
2
(1 + z2)
mkBT2 + ǫexc2 +
u222
)
+4
3aradT
42 u2 =
ρ1u1
(
5
2
(1 + z1)
mkBT1 + ǫexc1 +
u212
)
+4
3aradT
41 u1 (226)
12 CHAPTER 2 RADIATIVE SHOCK WAVES
(a)
(b)
Figure 24 Figure 14 amp 17 of Mi13haut et al (2004) (a) Compression ρ2ρ1 (full13ir13les thin line) kT2 in eV (inverted full triangles thin line) in Argon (a) Xenon
(b) for initial 13onditions kT1( = 10 eV for Ar and = 01 eV for Xe) ρ1 = 5 times10minus4 gcm3
versus sho13k speed in kms with (full markers full lines) and without
(empty markers dashed lines) radiation The ionization stage lt z gt2 is plotted
with the diamond symbol
21 RANKINE-HUGONIOT(R-H) RELATIONS 13
In the radiative regime the 13ompression ratio ρ2ρ1 approa13hes to the
value of 7 at very high velo13ities (several hundred of kms) whereas it is equal
to 4 when the ee13ts of radiation energy and pressure have been negle13ted
This limit is the same for the 13ases of adiabati13 (Bouquet Teyssier and
Chieze 2000) as well as the real gas (Mi13haut et al 2004) This is due to
the fa13t that at very high velo13ities the medium be13omes fully ionized and
behaves as a perfe13t gas On the 13ontrary in the 13omparatively low velo13ity
regime the ee13t of ionisation and ex13itation dominates the radiation ee13ts
This is illustrated in Fig 24 taken from Mi13haut et al (2004) whi13h reports
the variations of the Ar and Xe 13ompression ratio versus the sho13k speed for
an initial mass density of 5 times 10
minus4g13m
3 At 50 kms the 13ompression
ratio and the temperature are respe13tively equal to (5 32eV) and (11 30
eV) for Argon and Xenon
The 13ompression ratio shows bumps for high velo13ities These bumps
may be attributed to the fa13t that as the temperature of the post-sho13k
in13reases with the velo13ity the atom ionises more and more Thus the ex-
ternal shells in its ioni13 atomi13 stru13ture are su1313essively opened indu13ing
a strong variation in the ex13itation energy
The post-sho13k temperature is very high for the very high-speed sho13ks
In su13h 13ases the 13ompressed gas is fully ionized and the ee13ts of ionisation
and ex13itation remain negligible like in the adiabati13 13ase It should be noted
that below 100 - 200 kms the ee13ts of radiation pressure remain negligible
Thus very energeti13 lasers like LMJ or NIF are required to rea13h a domain
where this ee13t 13an be observed
Thus whereas for the range of velo13ities of 20 - 50 kms 13overed by my
study the ee13ts of radiative energy and pressure 13an be negle13ted this is
not the 13ase for the radiative ux whi13h has been negle13ted in the previous
se13tions and whi13h may be13ome 13omparable to the material energy ux Its
ee13t on the system must be taken into a1313ount as will be shown below
213 Radiative hydrodynami13s equations
To in13lude the 13ontributions of radiation it is 13ompulsory to introdu13e the
relevant terms in the uid equations The propagation of the radiation in the
matter is governed by the pro13esses of absorption and emission of photons
whi13h respe13tively heats and 13ools the medium As the mean free path of
photons is always higher than the mean free path of parti13les the sho13k
stru13ture is more inuen13ed by the radiative transfer than the vis13osity of
the uid
The propagation of radiation in the matter is des13ribed by the radiative
transfer equation
(
1
c
part
partt+
part
parts
)
I(rn ν t) = η(r ν)minus χ(r ν) I(rn ν t) (227)
14 CHAPTER 2 RADIATIVE SHOCK WAVES
where χ(ρ ν T ) and η(ρ ν T ) are respe13tively the mono13hromati13 opa-
13ity and emissivity of the gas
The spe13i13 intensity I(rn ν t) (erg cmminus2 sminus1Hzminus1 srminus1) is the energy
radiated per units of surfa13e area time frequen13y and solid angle in the
dire13tion n
The radiative energy density Erad ux Frad and pressure Prad are re-
spe13tively dened as the zero rst and se13ond moments of the spe13i13 in-
tensity versus the angle after integration over the photon frequen13y ν
Erad =1
c
int ∮
I(rn ν t)dΩdν (228)
Frad =
int ∮
I (rn ν t) cos θ dΩdν (229)
Prad =1
c
int ∮
I (rn ν t) cos2 θ dΩdν (230)
(231)
Taking into a1313ount radiative ee13ts the equations of radiative hydro-
dynami13s in the 13o-moving frame of the sho13k (Mihalas and Mihalas 1999
Drake 2006) 13an be written as
partρ
partt= minusnabla(ρu) (232)
ρ
(
partu
partt+ unablau
)
= minusnabla(P + Prad) (233)
part
partt
(
ρu2
2+ ρǫ+ Erad
)
+nabla
[
ρu
(
ǫ+u2
2+
P
ρ
)
+ (Erad + Prad)u
]
= minusnablaFrad
(234)
As mentioned previously the 13ontributions linked to Prad and Erad are
ee13tive only at very high temperature (ie velo13ity) However for the
radiative sho13ks of interest in this work the 13ontribution of the radiative
ux Frad is non-negligible 13ompared to ρu3 These sho13ks are thus in the
radiation ux dominated regime
Most of the experimental sho13k waves are in this regime These hyd-
rodynami13 equations are strongly non-linear In the present work the one-
dimensional radiative-hydrodynami13 13ode `HELIOS (Ma13Farlane Golovkin
and Woodru 2006) has been employed to derive the physi13al parameters
of the sho13k in various 13onditions as presented in the following se13tion HE-
LIOS will be dis13ussed in detail in the next 13hapter
22 RADIATIVE SHOCK REGIMES 15
22 Radiative sho13k regimes
Radiative sho13ks are strong sho13ks (Ma13h number Mgtgt1) whi13h attain
high temperatures and thus are the sour13e of intense radiation (Mihalas
and Mihalas 1984 Zeldovi13h Zeldovi13h and Raizer 2002 Drake 2006)
Depending on the opa13ity the radiation emitted from the sho13k may be ab-
sorbed by the pre-sho13k region indu13ing its pre-heating Far away from the
dis13ontinuity the stru13ture of the upstream medium is determined by the
absorption On its side the stru13ture of the transition layer of the down-
stream medium 13lose to the dis13ontinuity depends on the balan13e between
the emission (13ooling) and the absorption (heating) Thus the full stru13ture
of the sho13k mainly depends on the variation of the opa13ity χ (13m
2g) The
opti13al depth (τ(s)) measured from the position of the jump is then the
relevant parameter for the absorption It is given by
τ(s) =
int s
sjump
χ(sprime)ρ(sprime) dsprime (235)
where sprime 13hara13terizes the path of the radiation As we are interested in
the mean frequen13y averaged radiative ux the relevant opa13ity here will be
the averaged Rosseland opa13ity whi13h will be dened in due 13ourse
If τ gtgt 1 the regime is referred as opti13ally thi13k while in the opposite
13ase (τ lt 1) it is opti13ally thin Flux dominated radiative sho13ks have
been 13lassied depending on the opti13al depth of upstream and downstream
regimes
An attempt of radiative sho13k 13lassi13ation has been performed by Drake
(Drake (2006)) a1313ording to the opti13al the thi13kness of the upstream and
downstream regimes resulting in the denition of four dierent regimes
namely a thin-thi13k thin-thin thi13k-thi13k thi13k-thin radiative sho13ks
Following this author a thin-thi13k type of radiative sho13k probably exists
only in theory In the 13ase of a thin-thin radiative sho13ks both upstream
and downstream regimes are opti13ally thin and the sho13k radiates away
its energy in both dire13tions This regime of radiative sho13k is present in
various astrophysi13al 13ontexts for instan13e in the internal sho13ks of stellar
jets (Hartigan 2003) It is possible to generate these type of sho13ks in the
laboratory in very low-density gases In thi13k-thi13k radiative sho13ks both
upstream and downstream zones are opti13ally thi13k This regime o1313urs
in the stellar interiors The radiation 13oming from the hot downstream
region heats the 13ooler upstream material and forms a pre13ursor The fourth
regime is the thi13k-thin radiative sho13k whi13h is 13hara13terized by an opti13ally
thi13k downstream and an opti13ally thin upstream region The radiation
from the heated downstream region passes ahead of the sho13k and 13reates a
13ooling layer in the downstream before rea13hing the nal state There is no
absorption of the radiation in the upstream region and therefore 13ontrary to
16 CHAPTER 2 RADIATIVE SHOCK WAVES
the thi13k-thi13k 13ase there is no radiative pre13ursor Sho13ks generated in the
laser experiments are usually between the thi13k-thi13k and thin-thin regimes
221 Typi13al radiative sho13k waves with a pre13ursor
The radiation from the post-sho13ked region just behind the sho13k passes
ahead of the sho13k and generates the pre13ursor However it also leads to a
radiative 13ooling layer downstream that extension depends on the opa13ity
A typi13al stru13ture of su13h sho13ks is presented in M13Clarren et al (2010)
In these 13ases the 13ompression in the 13ooling layer behind the sho13k is
mu13h higher than that in the hydrodynami13 13ase This kind of sho13k is
most 13ommonly generated in the laboratory experiments (Bozier et al 1986
Keiter et al 2002 Reighard et al 2006 Fleury et al 2002) In astrophysi13s
these sho13ks are present for instan13e in supernovae when the blast wave
emerges from the stellar interior (Ensman and Burrows 1992)
A detailed 13lassi13ation of the sho13ks presenting a radiative pre13ursor
is reported in several referen13es (Zeldovi13h Zeldovi13h and Raizer 2002
Ensman 1994 Vaytet et al 2013)
In the 13ase of sub-13riti13al sho13k the temperature just ahead of the sho13k
front is smaller than the post-sho13k temperature Su13h sho13ks are relatively
weak and hen13e the radiative ux transfer from the post-sho13k to the pre-
sho13k is small A piston moving with a 13onstant speed of 12 kms in Xenon
gas at 01 bar is able to generate su13h a sub-13riti13al radiative sho13k The
resulting mass density and ele13tron temperature proles have been presented
at 10 ns in the Fig 25a
The other regime of radiative sho13k is known as super13riti13al in whi13h
the pre-sho13k and the nal post-sho13k temperatures are the same The mass
density and ele13tron temperature proles of super13riti13al sho13k driven by a
13onstant moving piston with speed 45 kms in Xenon at 01 bar are presented
in Fig 25b
The speeds a13hieved in the PALS experiment range between sim 15 - 55
kms for pressure 13omprised between 01- 06 Our simulations indi13ate
that depending on the speeds the regimes 13over either the sub13riti13al or
super13riti13al 13ases
23 Radiative sho13k waves experiments
Radiative sho13k waves have been studied experimentally for more than a
de13ade on high-energy lasers fast magneti13 pin13h ma13hines and pulsed po-
wer fa13ilities Su13h fa13ilities are able to a13hieve strong sho13k 13onditions
(high-temperature high-pressure) over a very small volume and time Va-
rious diagnosti13s are able to re13ord the plasma 13onditions whi13h are then
13ompared with the results of numeri13al simulations The results of these
experiments are also employed to validate numeri13al 13odes some of them
23 RADIATIVE SHOCK WAVES EXPERIMENTS 17
(a)
(b)
Figure 25 Simulated ele13tron temperature (Te) ion temperature (Ti) and mass
density (ρ) proles at 10 ns of a radiative sho13k propagating with a 15 kms (a sub-
13riti13al) and 48 kms (b super13riti13al) The sho13k is propagating in Xenon at 01
bar (ρ = 54 times 10
minus4g13m
minus3 at 298 K) The initial temperature in Xenon is set to
1eV The two Ma13h numbers are respe13tively equal to 13 and 43 The radiation and
ionisation ee13ts are in13luded in the simulation whi13h is performed for two dierent
temperatures (Te 6= Ti) and a negligible thermal 13ondu13tivity (10
minus05WmK) For
this simulation the sho13k is laun13hed by a 100 microm thi13k Aluminium piston moving
at the speed of 12 kms and 45 kms for 13ase (a) and (b) respe13tively The position
x=0 13orresponds to the position of the XeAl interfa13e at time t=0
18 CHAPTER 2 RADIATIVE SHOCK WAVES
being relevant to the astrophysi13s like FLASH (Fryxell et al 2000) and
HERACLES (Gonzaacutelez Audit and Huynh 2007)
231 Laser-driven sho13k experiments
Most of laboratory experiments on radiative sho13ks have been performed
on large-s13ale laser fa13ilities (Bouquet et al 2004 Chaulagain et al 2015
Gonzaacutelez et al 2006 Reighard et al 2006 Stehleacute et al 2010 Doss et al
2009 Drake et al 2011 Diziegravere et al 2011 Stehleacute et al 2012) in noble
gases at pressures between 01 and 1 bar With laser intensities on the
target 13omprised between 1014 and 1015 W13m
2 these experiments allowed
to re13ord sho13k speeds ranging between 40 and 150 kms
Bozier et al (1986) for the rst time experimentally eviden13ed the ge-
neration of a radiative wave propagating ahead of a super13riti13al sho13k in
xenon (pressure up to 6 bars) at the CEAs Limeil laser fa13ilities (irradian13e
of 5times 10
14W13m
minus2) In this experiment the laser beam (wavelength of 106
microm duration 1 ns) of 40 J energy was fo13used on an aluminum foil a13ting
as a piston A sho13k of speed sim 50 kms was produ13ed in the gas A visible
streak 13amera was used to re13ord the pre13ursor emissivity over 10 ns
Later Keiter et al (2002) performed a similar experiment at the OMEGA
laser fa13ility (25 kJ 1ns fo13al spot diameter 600 microm) still in a planar
geometry The sho13k propagated here in a low-density SiO2 aerogel foam
(ρ sim 96 mg13m
3) In this experiment X-ray absorption spe13tros13opi13 data
were re13orded to qualitatively predi13t the temperature of the pre13ursor
In the year 2000 a new radiative sho13k experiment was performed in
Xenon at 02 bar at the LULI laser fa13ility (sim 100 J λ = 053 microm 4-
6times 10
13W13m
minus2) (Fleury et al (2002) Bouquet et al (2004)) This expe-
riment used a three-layered piston 13omposed of a thin polystyrene a13ting
as ablator (2 microm thi13k) a titanium foil (3microm) a13ting as x-ray shield and a
layer of a polyethylene foam (25 microm) as an a1313elerator Further a VISAR
(velo13ity interferometer system for any ree13tor) and a Ma13h - Zehnder in-
terferometer were used as a diagnosti13s to measure the front sho13k speed and
the pre13ursor ele13tron density Two streak 13ameras re13orded the sho13k pro-
pagation in longitudinal and transverse dire13tions This was the rst attempt
to estimate ele13tron density within the pre13ursor Their measurements indi-
13ate an ele13tron density ranging between 1018minus1020cm3 The experimental
results were in qualitative agreements with 1D numeri13al MULTI and FCI
simulations However the speed of the pre13ursor in these 1D simulations
was too large This was then explained later when multidimensional ee13ts
were in13luded in the simulations (Leygna13 et al 2006 Gonzaacutelez Audit and
Stehleacute 2009)
Complementary experiments were then performed by Gonzaacutelez et al
(2006) at the PALS laser fa13ility (60 J 035 ns 438 nm spot diameter 600microm
sim 7times 10
13W13m
minus2) in Xe gas The goal was to study the sho13ks over longer
23 RADIATIVE SHOCK WAVES EXPERIMENTS 19
times and to inspe13t the ee13ts of the walls albedo A doubled layered pis-
ton made of 10microm polystyrene and 05 microm gold was used The pre13ursor
edge was re13orded by shadowgraphy using a visible streak 13amera The ex-
perimental results were 13ompared with 2D numeri13al simulations performed
with HERACLES (Gonzaacutelez Audit and Huynh 2007) This 13omparison
allowed quantifying the albedo of the walls It also showed that the radia-
tive losses at the walls lead to a small 13urvature of the ionization front and
to a redu13tion of its longitudinal extension (Leygna13 et al 2006 Gonzaacutelez
Audit and Stehleacute 2009)
In parallel another experiment was performed at the OMEGA laser fa-
13ility (035microm spot sim 700 - 800microm sim 2times 10
15W13m
minus2) by Reighardt et
al (Reighard et al (2006) Reighard (2007)) The sho13k was imaged by an
X-ray ba13k-lighter at dierent times whi13h enabled to inspe13t for the rst
time the shape of the sho13k front At higher speeds (sim 100 kms) X-ray
radiography pointed out a 13ollapse of the post-sho13k (Reighard et al 2006)
whi13h was attributed to the radiation losses trough the sho13k front For
these high-speed 13onditions the wall heating leads to the development of se-
13ondary wall sho13ks intera13ting with the primary sho13k (Doss et al 2009)
Su13h wall sho13ks have not yet been observed at lower speeds
Subsequently an experiment by Stehleacute et al (2010) was performed at
PALS laser fa13ility with the similar laser parameters as taken in Gonzalez
et al (2006) In this experiment a Ma13h-Zehnder interferometer and a
streak 13amera were used to re13ord the sho13k propagation over 50 ns Time-
integrated XUV spe13tros13opi13 analysis between 16 and 22 nm of sho13k was
performed The experiment showed that of a quasi-stationary regime was
rea13hed after 20 ns
The rst experimental study in Argon gas was reported in 2012 by Vis13o
et al (2012) at OMEGA fa13ility (035microm spotsim 600 microm sim 7times 10
14W13m
minus2)
The ele13tron temperatures in the radiative pre13ursor and sho13k-front were
measured to be 34 eV and 60 eV respe13tively using X-ray Thomson s13atte-
ring diagnosti13
In addition a new experiment at PALS (Jungwirth et al 2001) was
presented by Chaulagain et al (2015) using for the rst time a Zn x-ray
laser for XUV imaging at 21 nm This diagnosti13 allowed imaging both the
pre13ursor and post-sho13k stru13ture of radiative sho13ks in Xe gas at 03 bar
In this experiment where the sho13k wave did not ll the tube 2D ee13ts
be13ame pronoun13ed The experimental results were su1313essfully 13ompared
with the results of 2D ARWEN simulations (Cotelo et al 2015)
A summary of these dierent experiments and their prin13ipal diagnosti13s
is presented in table 21
20
CHAPTER2RADIATIVESHOCKWAVES
Experiment Laser fa13ility I (times 1013) λ amp τ ρ0 u Prin13ipal diagnosti13s
W13m
minus2 microm ns mg 13m
minus3
km s
minus1
Bozier et al (1986) CEAs Limeil 20-50 0351 1 10 -30 50-80 Visible streak Camera
Fleury et al (2002) LULI 5 053 05 1 60-65 Visible streak Camera
Visible interferometry
VISAR
Keiter et al (2002) OMEGA 35- 85 0351 1 5 -15
lowast
100 x-ray spe13tros13opy
Gonzaacutelez et al (2006) PALS 10-15 0438 035 1 65 Visible streak 13amera
Visible interferometry
Reighard (2007) OMEGA 70-100 035 1 6 100-140 x-ray ba13k-lighting
VISAR
Stehleacute et al (2010) PALS 10-20 1315 035 1 60 Visible Interferometry
XUV spe13trometer
Stehleacute et al (2012) PALS 10-20 1315 035 15 50-55 XUV imaging
XUV fast Si diodes
Vis13o et al (2012) OMEGA 70 035 1 196
lowastlowast
100-150 x-ray Thomson S13atte-
ring
Chaulagain et al (2015) PALS 10-20 1315 035 15 50 XUV fast Si diodes
Table 21 Summary of the radiative sho13ks experiments and out13omes in dierent laser installations All the experiments are performed
in Xe ex13ept Keiter et al (2002) and Vis13o et al (2012) In these two experiments the rst uses SiO
lowast
2 and the se13ond uses Ar
lowastlowast
as a
sho13ked medium
23 RADIATIVE SHOCK WAVES EXPERIMENTS 21
Laser driven radiative sho13ks Goal of my thesis
All these experimental studies have been fo13used on the 13ase of single ra-
diative sho13ks propagating in a tube However in astrophysi13al 13onditions
radiative sho13ks often intera13t with a denser medium leading to the deve-
lopment of ree13ted and transmitted sho13ks A few representative examples
of su13h phenomena are the intera13tion of supernovae remnants with dense
mole13ular 13louds (Gonzaacutelez Audit and Stehleacute 2009) the a1313retion sho13ks
on the photosphere of T-Tauri stars (Orlando et al 2013) and the bow
sho13ks at the head of stellar jets (Hartigan 1989 Raga et al 1999) The
13ollision (or the intera13tion) of two radiative sho13k waves is obviously a rare
astrophysi13al event and the template 13ase of supernova remnant DEM L316
(see Fig 1 of Williams et al (1997)) is still the subje13t of debates (Williams
et al 2005 Toledo-Roy et al 2009 Velarde et al 2006) as the observation
of these two dierent sho13ks 13an be also interpreted as the superposition of
two blast waves in the eld of view of the teles13ope In this regard the de-
velopment of dedi13ated laboratory experiments to the study of propagation
and intera13tion of 13ounter-propagating sho13k waves is important as a tool to
13hara13terize su13h events through their spe13i13 signatures
In this thesis I will present the results of experiments performed at the
Prague Asterix Laser System (PALS) fa13ility (Jungwirth et al (2001)) on
the study of the intera13tion of two radiative sho13k waves The experimental
obje13tives were
To generate two sho13ks of dierent speeds propagating in opposite
dire13tions in the tube lled with Xenon gas at low pressure (lt 1 bar)
with speeds 13omprised between 12 and 55 kms
To re13ord the ele13tron density by using Ma13h-Zehnder Interferome-
ter over a duration 13omprised between 50 and 200 ns This allowed
studying the ee13t of pre13ursor intera13tion on to the other one
To determine the temperature and the ion 13harge of gas using a time
and spa13e integrated XUV emission spe13tros13opy
The experimental results were interpreted with the help of one-dimensional
simulations using the Lagrangian radiation hydrodynami13 13ode HELIOS
232 Ele13tromagneti13ally laun13hed sho13k by a 13ompa13t pul-
sed power devi13e
As dis13ussed in the previous se13tion most of the plasma laboratory astrop-
hysi13s studies are mainly performed on large-s13ale laser fa13ilities addressing
pure hydrodynami13 radiative sho13ks at very high speed (50 - 150 kms)
and moderate pressure (01 - 1 bar) Complementary to laser experiments
22 CHAPTER 2 RADIATIVE SHOCK WAVES
dis13harge produ13ed plasma has also been a well-established method for ge-
nerating the strong sho13ks (Kondo et al 2008 2009ab Larour et al 2015)
Su13h devi13e presents a high exibility and a larger repetition rate whi13h al-
lows a deeper understanding of sho13k physi13s in 13onditions of astrophysi13al
interest for instan13e in the 13ontext of stellar jets
In prin13iple a 13oaxial ele13trode pair lled with gas followed by an intense
ele13tri13al surfa13e dis13harge generates a thin layer of plasma (plasma sheath)
at the bottom of the ele13trode Due to the high rising 13urrent in the plasma
layer a magneti13 pressure is produ13ed whi13h a1313elerates the plasma layer
also termed as plasma fo13us (Gonzalez Brollo and Clausse 2009) During
this motion the plasma sheath a1313retes a noti13eable part of the ba13kground
gas Experiments performed by Serban and Lee (1995) in the light gasses
H2 D2 have shown fast axial sho13k waves (100 kms) but the plasma sheath
was annular
Later on 13ompa13t pulsed power ele13tri13 generators were able to laun13h
astrophysi13ally relevant strong sho13ks in low-pressure noble gases (Kondo
et al 2006) These authors reported sho13k of speed 45 kms using 13oni13al
13oaxial ele13trodes and Xenon gas (006 - 11 mbar) The ion - ele13tron energy
transfer and ionization relaxation pro13ess as also the radiation ee13t were
investigated by Kondo et al (2008 2009ab)
My work on ele13tromagneti13ally laun13hed sho13k experiment
My work fo13using on the ele13tromagneti13ally laun13hed strong sho13ks rele-
vant for astrophysi13s aimed to design and to optimize an ele13tromagneti13
generator to be able to produ13e strong sho13ks in noble gasses at low pres-
sures of few mbar The ele13tromagneti13 generator is a powerful a1313elerating
devi13e whi13h eje13ts a quasi-planar plasma sheath out of a set of 13oaxial 13o-
ni13al ele13trodes where a pulsed 100-kA 13urrent is passing
Due to the ele13tri13al dis13harge in the gas a hot and fast moving plasma
is generated whi13h travels along the ele13trodes The high-intensity 13urrents
produ13e a magneti13 eld of several Teslas whi13h a13ts as a piston a1313elerating
an annular plasma sheath and driving a strong sho13k towards the top of the
13one I have employed a simple model to optimize the operation parame-
ters With su13h optimization our ele13tromagneti13 generator should produ13e
strong sho13ks over few nanose13onds
Our preliminary experiments show that the generator is 13apable of laun-
13hing supersoni13 sho13ks in Argon in the form of a thin plasma layer with
the speed of sim 1 - 30 kms Three-dimensional MHD simulation performed
by A Ciardi is 13onsistent with the rst observations This is presented in
detail in the 13hapter 6 of the thesis
Chapter 3
1D Simulations
Contents
31 HELIOS 23
32 LTE approximation 24
321 Mean opa13ity 25
322 Equation of State 27
33 Single radiative sho13k waves 27
34 Intera13ting radiative sho13k waves 40
35 Summary 44
Radiative sho13ks are present in various astrophysi13al pro13esses implying
extreme 13onditions Laboratory experiments then allow investigating the un-
derlying physi13al pro13esses whi13h take pla13e in these 13onditions and whi13h
are not observable from the Earth due to a la13k of spatial resolution of the te-
les13opes instruments However experiments are not always straightforward
to interpret and the help of numeri13al simulations be13omes pre13ious
In this 13hapter I will present with the help of 1D radiative hydrodynami13
simulations a brief overview of the physi13s of isolated radiative sho13ks and
of two intera13ting radiative 13ounter-propagating sho13k waves
31 HELIOS
Dierent numeri13al approa13hes are used for the simulation of laboratory
plasmas Some of them use xed grids Other introdu13e the possibility of
renement (on the y) in some meshes (Adaptative Mesh Renement) to
maintain a good resolution in the interesting parts of the plasma Another
approa13h is based on meshes following the uid (Lagrangian des13ription)
whi13h means that the meshes move with the uid and no matter may go
through a mesh to another The form of the equations dier in this 13ase (see
for instan13e Orban et al (2013))
HELIOS is su13h a one-dimensional Lagrangian radiation hydrodynami13
13ommer13ial 13ode (Ma13Farlane Golovkin and Woodru 2006) and I have
used it to simulate our experiment of intera13ting sho13k waves This 13ode
is easy to handle and has the presently interesting 13apability to be able to
simulate the 13ase of two sho13k waves propagating in opposite dire13tions and
laun13hed by two dierent lasers HELIOS may des13ribe non-LTE plasmas as
23
24 CHAPTER 3 1D SIMULATIONS
it in13ludes an in-line 13ollisional radiative (HELIOS-CR) model for 13omputing
non-LTE atomi13 level populations at ea13h time step of the hydrodynami13s
simulation However I used HELIOS in the LTE approximation whi13h is
appropriate to our experimental 13onditions (Rodriguez et al 2011) as will
be dis13ussed in the next se13tion
The 13ode solves the uid equations of motion using the pressure 13ontri-
butions of ele13trons ions and radiation It may des13ribe the ele13trons and
ions as two intera13ting uids at respe13tive temperatures Te and Ti Ther-
mal 13ondu13tion is des13ribed within a ux-limited ele13tron 13ondu13tion model
based on the Spitzer 13ondu13tivity (Burhop and Spitzer 1957) and the laser
energy deposition is 13omputed with an inverse Bremsstrahlung model ()
The radiation emission and absorption terms are introdu13ed in the equa-
tion of energy for the ele13trons and in the radiation transport equations
One of the methods whi13h are proposed is the ux-limited multi-group
radiation diusion model (Gonzaacutelez et al 2015) where the radiative ux is
proportional to the gradient of the radiative energy and is inversely propor-
tional to the Rosseland opa13ity (Dobbs-Dixon Cumming and Lin 2010)
The expression is pondered by a ux-limited diusion 13oe13ient in order
to obtain the good opti13ally thin limit This 13oe13ient follows the Larsen
expression des13ribed in Olson Auer and Hall (2000) The se13ond method is
a (time independent) multi-angle short-13hara13teristi13s s13heme In our 13ase
where radiation and hydrodynami13s are strongly 13oupled and thus the ra-
diation eld varies rapidly we used the diusion model together with LTE
Plan13k and Rosseland multi-groups opa13ities
In addition our version of HELIOS uses the PROPACEOS (Ma13Farlane
Golovkin and Woodru 2006) Equation of State (EOS) and multi-group
opa13ity tables whi13h are generated by the 13ode
1
and it is not possible to
in13lude any other opa13ity or EOS table
32 Lo13al Thermodynami13 Equilibrium
The knowledge of the thermodynami13al 13onditions is required for an ade-
quate simulation of the plasma des13ription
The Lo13al Thermodynami13 Equilibrium (LTE) regime is valid at large
ele13tron densities In that 13ase the 13ollisions between the ele13trons and the
ions and atoms are so frequent that they maintain at ea13h time a steady-
state populations of ions Moreover within a given ioni13 stage the popu-
lations of the dierent energy levels follow the Boltzmann equilibrium and
the populations of the dierent ions the Saha-Boltzmann equation (Fadeyev
and Gillet 2001)
On the 13ontrary in the 13ase of Non Lo13al Thermodynami13 Equilibrium
(NLTE) regime one has to take into a1313ount the ex13itationdeex13itation
1
see http wwwprismminus cscomSoftwarePROPACEOS
32 LTE APPROXIMATION 25
indu13ed by the ele13tron 13ollisions and various radiative pro13esses This leads
to a set of dierential time dependent equations (rate equations) whi13h has
to be solved through and adequate Collisional Radiative model (CR) (Kunze
2009)
Therefore it is 13ru13ial to determine if the regime is LTE or NLTE A
13riterium is given by Griem (2005)
In our experiment the mass density ranges between sim 10
minus4and 10
minus1
g13m
3 while the temperature values are 13omprised between andsim 0 to 50 eV
The pre13ise 13al13ulations performed by Rodriguez et al (Fig2 of Rodriguez
et al (2011)) for Xenon indi13ate that the thermodynami13al 13onditions of
our radiative sho13k experiments 13orrespond to the LTE regime
321 Mean opa13ity
The opa13ity is a fundamental ingredient of the radiative transfer equations
whi13h provide the spe13i13 intensity and its moments (radiative ux energy
and pressure) These last ones enter in the uid equations (see se13tion 213
in 13hapter 2) The 13oupling of the uid equations with the radiative transfer
equation is demanding in terms of 13omputational resour13es and most of the
13odes use simplied radiation transport equations whi13h require the know-
ledge of adequate frequen13y averaged opa13ity 13oe13ients The form of the
average of the opa13ity is not obvious as the average has to be done on the
radiative ux or energy whi13h are unknown before the 13omputation Then
the 13ommonly used opa13ities are the Plan13k and Rosseland mean opa13ities
(Mihalas 1978 Seaton 1987)
The Rosseland mean opa13ity (κR in 13m
minus1and χR = κRρ in 13m
2g) isderived by 13onsidering harmoni13 weighting over the temperature derivative
of the Plan13k fun13tion It gives the 13orre13t radiative ux in the opti13ally
thi13k diusion limit The average is performed over the total mono13hromati13
opa13ity κtotν = κabsν +κscatν (in 13m
minus1) where in the present 13ase the s13attering
13ontribution 13omes from the ele13tron Thomson s13attering
1
κR=
int
infin
01
κtotν
dBν
dT dνint
infin
0dBν
dT dν(31)
The Plan13k mean opa13ity (κP in 13m
minus1and χP = κP ρ in 13m
2g) employs
the normalized Plan13k bla13kbody energy density distribution as a weighting
fa13tor It will give the 13orre13t value for the integrated thermal LTE emission
of an opti13ally thin plasma and is expressed as
κP =
int
infin
0 κabsν Bνdνint
infin
0 Bνdν(32)
In the multi-groups des13ription the radiation transport takes into a13-
13ount the propagation of radiation in N dierent wavelength bands (νk minusνk+1 k = 1 N) Hen13e for instan13e the medium may be transparent for
26 CHAPTER 3 1D SIMULATIONS
some wavelengths and opaque for other ones The radiation transport equa-
tions split into N 13ontributions ea13h of them being asso13iated to adequate
opa13ity 13oe13ients The expression of the 13oe13ients is similar to the pre-
vious grey 13ase (N =1) ex13ept that the integrals are performed between
the boundary frequen13ies of the group
In HELIOS the mono13hromati13 opa13ities κν are 13omputed for the die-
rent radiation groups using the ATBASE atomi13 suite of 13odes Energy levels
and other atomi13 data relevant for bound-bound and bound-free transitions
are 13omputed within a 13onguration intera13tion model using Hartree-Fo13k
wave-fun13tions (Ma13Farlane Golovkin and Woodru 2006)
Dividing the opa13ity by the mass density redu13es the variation of this
quantity with ρ The literature then often uses the mass absorption 13oe-
13ient (also termed as opa13ity) whi13h is dened as
χν = κνρ (33)
Figure 31 PROPACEOS Mono13hromati13 opa13ity versus the photon energy in
eV of Xenon at 10 eV and at two ele13tron densities respe13tively equal to 1018 and
1020 13mminus3
An example of the variations of the PROPACEOS opa13ity χν versus the
frequen13y in eV is given in Fig 310a for Xenon at 10 eV and two ele13tron
densities respe13tively equal to 1018 and 1020 13m
minus3 whi13h do 13orrespond to
typi13al 13onditions of the pre13ursor and post-sho13k of our laser generated
radiative sho13ks
33 SINGLE RADIATIVE SHOCK WAVES 27
322 Equation of State
The equation of state (EOS) denes the dependen13e of the pressure ionisa-
tion internal energy with the mass density and temperature Several models
do exist in the literature
Our version of HELIOS for LTE plasmas may handle polytropi13 or PRO-
PACEOS equation of state Unfortunately there are not many details about
this last method The authors of the 13ode mention that it uses a QEOS-type
model (More et al 1988) in the strong 13oupling regime (ie at relatively
high densities and low temperatures) and an isolated atom model whi13h
uses a detailed atomi13 stru13ture modeling in the weak 13oupling region The
properties between the weak and strong 13oupling regimes are said to be obtai-
ned by interpolating in a thermodynami13ally 13onsistent manner This EOS
in13ludes the 13ontributions from the translation of ions and ele13trons ele13-
tron degenera13y atom ionization and ex13itation and Coulomb intera13tions
(Debye - Hu13kel 13orre13tion)
Comparisons on spe13i13 13ases have been performed by the authors of the
13ode with Los Alamos SESAME data for CH Al and Au showing a good
agreement when using these two equations of state
In this 13hapter HELIOS 13ode will then be employed to simulate the
detailed 13hara13teristi13s of single and 13ounter-propagating radiative sho13ks
driven by one or two lasers in the 13onditions of our experiments (Chapter 5)
The target 13ell (4-mm length) is lled with Xenon gas at 01 bar This valueof pressure is representative and 13orrespond to standard ambient tempera-
ture ie 298 K and not the initial temperature 13onsidered in the simulations
This terminology is maintained throughout the thesis Two lasers are inte-
ra13ting at both ends with two gold-13oated `CH foils 11microm CH and 06 micromAu) ea13h of them resulting in a piston of total thi13kness 116 microm 13losing the
13ell
33 Single radiative sho13k waves
Typi13al simulations of the radiative sho13ks espe13ially multidimensional ones
whi13h are time-13onsuming use a frequen13y independent (grey or one group)
opa13ities However then the opa13ity of the gas under investigation presents
strong variations with the frequen13y a multi-group approa13h may be suitable
(Vaytet et al 2011 2013) When the number N of groups tends to innity
ea13h group opa13ity κkR and κkP tends to the lo13al mono13hromati13 opa13ity κν whi13h solves the question of the adequate opa13ity average
In the following I will investigate the inuen13e of the opa13ity and the
number of groups on the stru13ture of a sho13k wave propagating at sim 50 kms
in Xenon at 01 bar To this purpose I have performed HELIOS simulations
for a laser of uen13e 28000 Jcm2whi13h is able to drive a sho13k with the
relevant velo13ity In addition I have also analyzed the ee13ts of taking into
28 CHAPTER 3 1D SIMULATIONS
a1313ount the dierent ion and ele13tron temperatures
Referen13e simulation (1 group Te = Ti = T )
A result of the simulation is reported at 10 ns in the Fig 32 whi13h reports
the variations of ele13tron density (Ne) temperature (T ) mean 13harge (Z)and mass density (ρ) The initial 13onditions are 45 times 10
minus4g13m
3and T =
1 eV
The sho13k front is lo13ated at sim 0047 13m (speed of 47 kms) At this
position the temperature peaks at 22 eV The bla13k verti13al dotted line shows
the position of the interfa13e between the piston and the ba13king Xenon gas
This interfa13e is lo13ated at sim 20 microm from to sho13k front The post-sho13k gas
mass density pressure ele13tron density and ion 13harge are respe13tively equal
to 002 g13m
3 11times 10
4bars 52times 10
2013m
3and 56 The high 13ompression
is due to ionisation and radiation 13ooling (13hapter 2) Its temperature of 12
eV is the same than in the pre13ursor where it is 13onstant over the 2 mm of the
simulation The pre13ursor ele13tron density and mean 13harge are respe13tively
equal to 18 times 10
1913m
minus3and 7 The pre13ursor is not 13ompressed
The variations of the Rosseland opa13ities (13m
2g
minus1and 13m
minus1) with the
distan13e are reported in the Fig 33 The Rosseland opa13ity is low in the
pre13ursor Hen13e 600 13m
2g
minus113orresponds to 05 13m
minus1 whi13h means that
an opti13al depth of 1 is rea13hed at 2 13m from the sho13k front This explains
the at prole of the pre13ursor With a typi13al value in the post-sho13k of
2577 in 13m
2g
minus1 this opa13ity is now 53 13m
minus1 and an opti13al depth of 1 is
here rea13hed at 200 microm from the sho13k front (whi13h is mu13h smaller than
the post sho13k extension)
This simulation was performed using the Spitzer thermal 13ondu13tivity
To 13he13k the 13ondu13tivity ee13t I performed another simulation with a
negligible thermal 13ondu13tivity (ie 10
minus12WmK) for the Xenon layers
only The results are reported in the Fig 34 The interfa13e between the
piston and the ba13king Xenon gas is still 13lose to the sho13k front Ex13ept
for the peak of temperature whi13h is higher (sim 29 eV instead of 22 eV) and
thiner the sho13k stru13ture is similar to the previous 13ase
Two uids with dierent temperatures Te and Ti (1 group)
In the previous se13tion HELIOS simulations have been 13arried out for the
13ase of equal ion and ele13tron temperatures I will now investigate the ee13t
of su13h approximation on the simulated results employing 1) Spitzer and 2)
negligible 13onstant thermal 13ondu13tivity in the Xenon layers It is important
to note that this two uids approximation 13an not be restri13ted to the Xenon
layers only and thus it is valid from the piston (CHAu) up to the Xenon
gas Thus the dynami13s and 13onditions of the piston are also modied whi13h
may slightly modify the sho13k velo13ity
33 SINGLE RADIATIVE SHOCK WAVES 29
(a)
(b)
Figure 32 Mass density and temperature (a) ele13tron density and mean 13harge
(b) at 10 ns for a radiative sho13k of speed sim 47 kms in Xenon at 01 bar The
verti13al dotted bla13k lines show the position of the interfa13e between piston and
ba13king Xenon gas Zero at x-axis is the interfa13e of piston and Xenon at time zero
Spitzer thermal 13ondu13tivity has been used in the simulation
30 CHAPTER 3 1D SIMULATIONS
Figure 33 Mean Rosseland opa13ity (in 13m
2g
minus1) and opa13ity (13m
minus1) at 10 ns
for a radiative sho13k of speed sim 47 kms in Xenon at 01 bar The verti13al dotted
bla13k lines show the position of the interfa13e between piston and ba13king Xenon gas
Zero at x-axis is the interfa13e of piston and Xenon at time zero Spitzer thermal
13ondu13tivity has been used in the simulation
33 SINGLE RADIATIVE SHOCK WAVES 31
(a)
(b)
Figure 34 (a) Mass density and temperature (b) ele13tron density and mean
13harge at 10 ns for a radiative sho13k of speed sim 47 kms in Xenon at 01 bar The
verti13al dotted bla13k lines show the position of the interfa13e between piston and
ba13king Xenon gas Zero at x-axis is the interfa13e of piston and Xenon at time zero
A negligible 13onstant thermal 13ondu13tivity (10
minus12WmK) have been used in the
simulation for Xenon
32 CHAPTER 3 1D SIMULATIONS
(a)
(b)
Figure 35 (a) Ele13tron density and ele13tron temperature (b) Mass density and
mean 13harge at 10 ns for a radiative sho13k of speed sim 48 kms in Xenon at 01
bar for the 13ase when the ele13tron and ion temperature are dierent The verti13al
dotted bla13k lines show the position of the interfa13e between piston and ba13king
Xenon gas Zero at x-axis 13orresponds to the position of the interfa13e between the
piston and Xenon at time zero The Spitzer thermal 13ondu13tivity has been used in
this simulation
33 SINGLE RADIATIVE SHOCK WAVES 33
The Fig 35a reports the sho13k stru13ture at 10 ns for the 13ase of Spitzer
thermal 13ondu13tivity The sho13k is lo13ated at 0048 13m The ion temperature
peaks at 53 eV and the ele13tron temperature peaks at the same value of 20
eV than previously Ex13ept for the ion temperature the sho13k stru13ture is
not signi13antly 13hanged 13ompared to the 13ase of equal temperatures
Let us now inspe13t the role played by the thermal 13ondu13tivity (see Fig
36) In the 13ase of a negligible thermal 13ondu13tivity of (10
minus12WmK) for
the Xenon layers only the ele13tron temperature (and as a 13onsequen13e the
mean 13harge) peaks at higher value 25 eV (instead of 20 eV with the Spitzer
13ondu13tivity) and the ion temperature rea13hes to 405 eV (instead of 53 eV)
The rest of the sho13k is not ae13ted by this 13hange in the 13ondu13tivity This
13onrms that the width and height of the ion peak temperature are a dire13t
13onsequen13e of the thermal 13ondu13tivity
Several groups for the opa13ity one temperature
As mentioned earlier the number of frequen13y groups inuen13es the dyn-
ami13s and morphology of radiative sho13ks A set of simulations has been
performed 13onsidering (only for Xenon) dierent numbers (N) of frequen13y
group varying between 1 and 100 for Te = Ti The distribution in energy
of the groups is logarithmi13 and the grid is set up with approximately 85
of the groups having photon energies between 01 eV and 3 KeV while the
remaining 15 lie between 3 KeV and 1 MeV
The Figures 37 (a) and (b) show the variations of the ele13tron tempe-
rature with the number of groups (from 1 to 100) for the pre13ursor (a) and
the front (b) In13reasing the number of groups ae13ts mostly the pre13ursor
A similar behavior is found in Vaytet et al (2013)
It should be noted that this multigroup opa13ity treatment is applied
to all the layers (ex13ept gold only one group opa13ity is provided by the
13ompany) in the simulation This 13ould ae13t slightly the dynami13s of the
piston Con13erning the pre13ursor at this time of the simulation and taking
an a1313ountable variation in temperature of 10 the 13onvergen13e is a13hieved
after 20 groups
Variation with the opa13ity
In order to estimate the ee13t of the opa13ity I have performed several simu-
lations by in13reasing the Plan13k and Rosseland opa13ities in Xenon (Stehleacute
et al 2014b) using a 13ommon multiplier ranging between 1 to 40 Su13h
pro13edure is often used to redu13e the gap between the available and more
sophisti13ated opa13ities To ex13lude the impa13ts of the radiation 13oming from
the post-sho13k to the piston layers the opa13ity of the piston (CH and Au)
layers have been set to zero
I present in Fig 38 the results (proles of temperature mass density and
34 CHAPTER 3 1D SIMULATIONS
(a)
(b)
Figure 36 (a) Mass density and ele13tron temperature (b) ele13tron density and
mean 13harge at 10 ns for a radiative sho13k of speed sim 45 kms in Xenon at 01 bar
for the 13ase when the ele13tron and ion temperature are dierent The verti13al dotted
bla13k lines show the position of the interfa13e between piston and ba13king Xenon
gas Zero at x-axis 13orresponds to the position of the interfa13e between the piston
and Xenon at time zero In this simulation value of thermal 13ondu13tivity (10
minus12
WmK) is negligible for Xenon The peak value of ion temperature is found to be
405 eV whereas theoreti13ally this value is expe13ted to be sim 600 eV It is possible
to a13hieve the expe13ted value by in13reasing the resolution of the simulation
33 SINGLE RADIATIVE SHOCK WAVES 35
Figure 37 Ele13tron temperature proles for various numbers of frequen13y groups
N = 1 20 50 60 70 90 and 100
mean 13harge) of four 13ases 13orresponding to an opa13ity multiplier equal to
1 10 30 and 40 The ele13tron and ion temperatures are taken to be equal
In13reasing the opa13ity in13reases the absorption in the pre13ursor and thus
de13reases its length as the photons emitted by the sho13k are more rapidly
absorbed by the 13old pre-sho13k Taking as a referen13e the previous value of
the Rosseland opa13ity of 600 13m
2g
minus1and a multipli13ative fa13tor of 40 the
opti13al depth of 1 will now be rea13hed at 005 13m (instead of 2 13m without
any multiplying fa13tor)
The temperature peak value at the sho13k front also de13reases I do believe
that this 13an be attributed to the in13rease of the radiative 13ooling of the peak
of temperature with the multiplying fa13tor
I adopted here an oversimplied method in the sele13tion of the groups
boundaries A more realisti13 one would be to take into a1313ount the variation
of the opa13ity in the dierent layers with the wavelength In the 13ase of
xenon for instan13e it would be then more adapted to rene the des13ription
of the opa13ity between 5 and 100 eV (see Figure 310a)
Opa13ities 13omparison
The previous study illustrates the role played by the opa13ity for the radi-
ative sho13ks waves with a pre13ursor It seams then logi13al to 13ompare the
36 CHAPTER 3 1D SIMULATIONS
(a)
(b)
(13)
Figure 38 (a) Ele13tron temperature (a) mean 13harge (b) and mass density (13)
for four (1 10 30 and 40) multipliers of the Xenon opa13ity at 10 ns
33 SINGLE RADIATIVE SHOCK WAVES 37
PROPACEOS opa13ity whi13h is used by HELIOS with more sophisti13ated
13al13ulations
The Figures 39a and 39b report the variations of the Plan13k and Ros-
seland grey opa13ities 13omputed for the density of the pre13ursor ρ = 15 times10
minus3g13m
3by two dierent 13odes PROPACEOS (Ma13Farlane Golovkin
and Woodru 2006) and BiGBART (de la Varga et al 2011 Ogando and
Velarde 2001) PROPACEOS opa13ities and EOS used in HELIOS simula-
tions 13omputes frequen13y-dependent opa13ities along with spe13i13 internal
energies and pressures The atomi13 13ode BiGBART able to 13al13ulate two
type of opa13ities dependent on atomi13 stru13ture and frequen13y dependent
The opa13ities 13omputed by Rodriacuteguez et al (2015) are also reported for 13om-
parison The two last 13al13ulations use the FAC (Flexible Atomi13 Code) (Gu
2008) for the 13al13ulation of the atomi13 data
We see that for a temperature equal to 10 eV the Rosseland opa13ity is
equal respe13tively to 800 15000 and 8000 g13m
2 whereas the Plan13k opa13ity
is equal to 18000 65000 370000 g13m
2 This means that the PROPACEOS
opa13ity whi13h is used in HELIOS is smaller by more than one order of
magnitude than the two last opa13ities As it is not possible to 13hange the
opa13ity model within HELIOS we then de13ided to use the PROPACEOS
opa13ity with a multipli13ative fa13tor whi13h we have set equal to 20
The dis13repan13ies in the average opa13ity are a 13onsequen13e of important
dieren13es in the mono13hromati13 opa13ities This is illustrated in the Fig
310 whi13h 13ompares the mono13hromati13 opa13ities given by PROPACEOS
and Rodriacuteguez et al (2015) (see Fig 16 of Rodriacuteguez et al (2015)) at 15
eV and 15 times 10
minus3g13m
minus3) This gure indi13ates a noti13eable dieren13e
between 5 and 150 eV with more bb transitions for the last method This
explains the dieren13es observed in the 13orresponding averages
Synthesis
These dierent simulations for a sho13k propagating at sim 50 kms in Xenon
at 01 bar show that the simulations performed with only one temperature
are adequate for the des13ription of the sho13k
The opa13ity strongly ae13ts the pre13ursor extension whi13h de13reases from
more than 4 mm for 1 group to 3 mm then the number of groups is larger
than 20 The 13onvergen13e versus the number of groups is rea13hed after 20
groups
To t with more sophisti13ated opa13ities and as it was impossible to
in13lude them in HELIOS I de13ided to multiply the opa13ities of Xenon by a
fa13tor of 20 This redu13es strongly the pre13ursor extension up to sim 1 mm
at 10 ns Taking into a1313ount the present huge un13ertainty in the opa13ity
I 13onsidered that it was not ne13essary to use multi-group opa13ities This
also means that our 1D simulations will be used - only - for a qualitative
interpretation of the experimental results A quantitative 13omparison should
38 CHAPTER 3 1D SIMULATIONS
(a)
(b)
Figure 39 Plan13k and Rosseland mono13hromati13 opa13ity proles Te for ρ =15
times 10
minus3from PROPACEOS (I) BIGBART (II) and Rodriguez et al (III)
33 SINGLE RADIATIVE SHOCK WAVES 39
(a)
(b)
Figure 310 Mono13hromati13 Xenon opa13ity versus the photon energy in eV at T
= 15 eV and for ρ = 15 times 10
minus3g13m
minus3for the PROPACEOS (a) and Rodriacuteguez
et al (2015) (Fig 16 of Rodriacuteguez et al (2015)) (b) models
40 CHAPTER 3 1D SIMULATIONS
be in the future based on adequate opa13ities and multi-groups modeling
34 Intera13ting radiative sho13k waves
In order to pre13ise what is expe13ted from the experimental study I will now
investigate the 13hara13teristi13 parameters of two 13ounter-streaming sho13ks
propagating in Xenon at 01 bar (ρ = 54 times 10
minus4g13m
minus3) with equal and
dierent velo13ities To highlight the ee13t of the intera13tion I will 13ompare
the results with those obtained with 1D simulations All the simulations will
be performed with HELIOS For the Xenon opa13ity a multiplier times 20 will
be used The number of groups will be set to be 1 and the ele13tron and ion
temperatures will be supposed to be equal
Three representative sets will be 13onsidered
Case(I) two identi13al radiative sho13ks (2RS) at sim 50 kms propaga-
ting in opposite dire13tions (ie starting from the left and right end
respe13tively)
Case(II) same 13onditions but without any 13oupling with radiation
Case(III) two 13ounter-propagating radiative sho13ks propagating with
dierent speeds sim 50 kms from the left side of the 13ell and sim 20
kms from the right side
To a13hieve the aforementioned speeds on the left and right sides the
uen13es of the two laser beams have been adjusted The pulse duration is
set to 03 ns (peak at 015 ns) to reprodu13e the experimental 13onditions
detailed later in the manus13ript
Case(I) 13ounter propagating sho13ks with the same speed
This a13ademi13al 13ase is fully symmetri13al and it is equivalent to the 13ase of
one RS with a fully ree13tive boundary (for hydrodynami13s and radiation)
in the middle of the tube The Fig 311 shows the variations of the ele13tron
density (Ne) and temperature (Te) in the Xenon layers The two sho13ks
appear in Xenon at sim 2 ns and the 13ollision o1313urs at sim 38 ns At 3 ns the
pre13ursor extension is sim 008 13m whereas the post-sho13k ele13tron density
and ele13tron temperature are 78 times 10
2013m
minus3and 16 eV respe13tively The
length of pre13ursor in13reases rapidly with time and the two pre13ursors merge
suddenly at sim 8 ns After this time the merging ee13t in13reases signi13antly
It is 13hara13terized by a at 13ommon pre13ursor those ele13tron density and
temperature are in13reasing with time At the time of the 13ollision (sim 38
ns) the post-sho13k mass and ele13tron density in13rease from 0011 to 014
g13m
minus3and 67 times 10
20to 66 times 10
2113m
minus3 whereas the temperature rises
up to 39 eV The 13ollision leads to the development of two reverse sho13k
34 INTERACTING RADIATIVE SHOCK WAVES 41
(a)
(b)
Figure 311 Ele13tron density Ne (a) and ele13tron temperature Te (b) versus axial
position (along a 04 13m long sho13k tube) at 3 10 20 30 and 38 ns from HELIOS
simulations (with opa13ity times 20) for the 13ases of single sho13k of sim 50 kms (dotted
line) and two identi13al 13ounter-propagating sho13ks of sim 50 kms (solid lines) The
verti13al dotted lines show the position of the interfa13e between piston and ba13king
Xenon gas
42 CHAPTER 3 1D SIMULATIONS
waves propagating ba13k with a speed of sim 15 kms rst in Xenon and later
on in the dierent piston layers (not shown in the Figure) These reverse
sho13ks lead to a dense plasma (Ne gt 10
2113m
minus3) whi13h is not a1313essible to
the experimental diagnosti13s and will not be detailed here
To investigate the ee13ts of the intera13tion I have reported in Fig 311
the 13ase of 1RS moving from the left to the right dire13tion in the 13ell (dotted
lines) The wave propagates identi13ally to 2RS until 10 ns After this time
the proles of the temperature and ele13tron density dier strongly from the
previous 13ase and their values are lower than for the 2RS The post-sho13k
extension is slightly smaller than for the 2RS and the sho13k speed is thus
slightly smaller too This last ee13t is due to the fa13t that for the two inte-
ra13ting waves the pre13ursor is at a higher temperature than for the isolated
sho13k and thus the sho13k propagates in a warmer medium then modifying
the sho13k 13onditions (sound speed opa13ity et13 ) A similar ee13t in the post
- sho13k extension may be seen for an isolated sho13k wave propagating in a
warmer pre-sho13k gas
Case (II) Identi13al 13ounter-propagating sho13ks for an ideal gas
In order to highlight the ee13t of the radiation another simulation has been
performed with the same set of parameters as in the 13ase I however putting
the Xenon opa13ity equal to zero I still use here the PROPACEOS equation
of state The result of the simulation is presented in Fig312 The 13ollision
time is now 40 ns instead of 38 ns (thus the sho13k speed is slightly smaller)
The post-sho13k is no more 13ompressed by radiation 13ooling Its 13ompression
at 10 ns is 10 instead of 35 This 13ompression is due to the vis13ous sho13k
(fa13tor 4) followed by the ionisationex13itation of the gasThere is obviously
no radiative pre13ursor Moreover there are no dieren13es in the Ne and Tproles of the single sho13k and that of the two 13ounter-propagating sho13ks
before the 13ollision time
Case (III) Two sho13ks at dierent speeds
The spatial and temporal variations of Ne and Te are plotted at times 3
10 20 30 38 and 49 ns in Fig 313a and 313b The left and the right
sho13ks appear in Xenon at sim 2 and 3 ns respe13tively Later at 10 ns
the two pre13ursor extensions are respe13tively equal to 018 and 0034 13m
The merging of the two pre13ursors starts at sim 15 ns As expe13ted from the
values of sho13k speeds the 13ollision time o1313urs at 49 ns whi13h is delayed
in 13omparison to the 13ase (I)
In 2RS the two radiative pre13ursors merging results in a plateau for the
ele13tron density and the temperature The temperature at 13ollision time is
now 28 eV instead of 39 eV in the 13ase I and the ele13tron density rea13hes
up to 31 times 10
2113m
minus3instead of 66 times 10
2113m
minus3
34 INTERACTING RADIATIVE SHOCK WAVES 43
(a)
(b)
Figure 312 Variations of Ne (a) and Te (b) versus axial position for the 13ase
of two identi13al 13ounter-propagating sho13ks (of speeds sim 50 kms) at 3 10 20
30 35 and 40 ns as derived from HELIOS simulations For these simulations we
have negle13ted the ee13t of radiation by keeping the Xe opa13ity equal to zero
The verti13al dotted lines show the position of the interfa13e between the piston and
ba13king Xenon gas
44 CHAPTER 3 1D SIMULATIONS
It may be noted that up to this 13ollision time the post-sho13k density
13onditions and speed are identi13al for the 13ase of an isolated RS propagating
from the left (1RSL) at 50 kms and this present left sho13k This reveals
that in the 13ase of 2RS there is no noti13eable ee13t of the right sho13k with
speed 20 kms on the left post-sho13k of 50 kms
On the 13ontrary we note a dieren13e in extension of the right post sho13k
from the 2RS 13ompared with the 13ase of an isolated sho13k (1RSR) propa-
gating at 20 kms from the right whi13h indi13ates that the left sho13k of the
2RS has an inuen13e on the right post-sho13k (Fig 313a)
35 Summary
Radiative sho13k waves propagating in Xenon at 01 bar with a velo13ity of 50
kms are 13hara13terized by the development of an extended radiative pre13ur-
sor The huge 13ompression of 38 in the post-sho13k is a 13onsequen13e of both
the sho13k and the gas ionisation (fa13tor of 10) as also from the radiative
13ooling The post-sho13k and pre-sho13k temperature on both side of the peak
are identi13al indi13ating that the sho13k is super13riti13al
The spe13ta13ular role of the opa13ity has been highlighted Due to the
un13ertainty in the opa13ity used for Xenon in our simulations I de13ided that
at this stage it was not ne13essary to rene the simulation in terms of group
numbers within the 1D des13ription of the sho13k wave
Our numeri13al study gives the main 13hara13teristi13s of the intera13tion of
two 13ounter-propagating sho13k waves with speeds equal to 50-50 kms and
50-20 kms The 13ase of identi13al speeds is simpler due to the symmetry of
the problem However whatever the speeds the most important signature
of the intera13tion is the merging of the pre13ursor at 8 ns for 50-50 kms
and at 15 ns for 50-20 kms This merging is followed by a regular in13rease
with time of the ele13tron density and the temperature The 13ollision time
is 13hara13terized by a sudden in13rease of the ele13tron density by an order of
magnitude rea13hing 66 times 10
21and 3 times 10
2113m
minus3respe13tively whereas the
temperature in13reases up to 39 and 28 eV
35 SUMMARY 45
(a)
(b)
Figure 313 Variations of Ne (a) and Te (b) with axial position for the 13ase of
two non-identi13al 13ounter-propagating sho13ks (of speeds sim 50 amp 20 kms) and two
single sho13ks (dotted lines) of speeds sim 50 amp 20 kms respe13tively at 3 10 30 35
and 46 ns as derived from HELIOS simulations The verti13al dotted lines show the
position of the interfa13e between piston and ba13king Xenon gas (with opa13ity times20)
46 CHAPTER 3 1D SIMULATIONS
Chapter 4
Laser-driven radiative sho13k
Experimental Setup
Contents
41 Prague Asterix Laser System (PALS) fa13ility 48
42 Targets 49
421 Massive Targets 51
422 Gaseous Targets 51
423 Target holder 55
424 Target lling 56
43 Laser Fo13using 57
431 Fo13al Lenses and Phase Zone Plates 57
432 Fo13using of the MAIN and AUX laser beams 58
44 Diagnosti13s 59
441 Visible interferometry 61
442 XUV spe13tros13opy 65
443 Visible spe13tros13opy 67
45 Summary 68
When a short pulse of a high-power laser beam is fo13ussed on a thin foil
it transfers a huge amount of energy to the foil material Almost instantane-
ously the laser absorption heats it up to the ablation A hot and low density
13orona is generated ba13kwards by this pro13ess Beyond the ablation surfa13e
a sho13k is formed in the foil as a 13onsequen13e of the momentum 13onservation
(ro13ket ee13t) The sho13k moves inward in the foil Our 13ase of the experi-
mental setup as the foil 13loses a tube lled with gas the sho13k propagates
then in the gas where it is studied through various kinds of diagnosti13s The
foil then a13ts as a piston
We performed the experiments at the Prague Asterix Laser System (PALS)
in the Spring 2015 (ve weeks 20th April - 22th May) The obje13tive of the
experimental 13ampaign was to study the evolution of two 13ounter streaming
radiative sho13ks driven by two 13ounter-propagating lasers with an irradian13e
in the range of 10
14W13m
minus2 and therefore to 13hara13terize their intera13tion
and 13ollision In this 13hapter I will rst present a brief des13ription of the
PALS laser fa13ility and of the laser beams used in our experiment This will
47
48 CHAPTER 4 EXPERIMENTAL SETUP
be followed by a presentation of the targets design and an overview of the
general setup and of our main diagnosti13s
41 Prague Asterix Laser System (PALS) fa13ility
The Prague Asterix Laser System (PALS) is a laser fa13ility based on an
Infrared high-power iodine laser system (Asterix IV) (Jungwirth et al 2001)
Using dierent amplifying stages the laser fa13ility is able to deliver energy
up to 1 kJ in 03 ns at the fundamental wavelength 1315 nm The output
laser beam 13an be further subdivided in few auxiliary beams All auxiliary
beams may be frequen13y doubled (λ = 657 nm) or tripled (438 nm) The
PALS laser fa13ility is 13apable of ring up to two high energy laser shots per
hour Compared with solid state lasers this gas laser is known to deliver a
quite homogenous beam intensity without hot spots A spatial prole of the
PALS laser beam is shown in the Fig 41a (re13orded by PALS team during
the experimental 13ampaign) while its pulse 13hara13teristi13s are plotted with
time in the Fig 41b
(a)
(b)
Figure 41 a Spatial prole of the PALS Laser beam b Intensity prole of
laser pulse I(t) with time
For our experiments we used two independent laser beams to drive the
two 13ounter streaming sho13ks in the gaseous target For this purpose the
fundamental PALS laser beam has been subdivided after the fourth amplier
(V4) into two beams with respe13tive energy fra13tions of 60 and 40 (see
Fig 42) The most energeti13 of these laser beams is then inje13ted through
the fth (V5) amplier and its frequen13y are then tripled This beam at 438
nm will be hereafter termed termed as the MAIN laser beam
The se13ond of the two previous laser beams propagates then without any
modi13ation It will be hereafter 13alled the AUX laser beam (1315 nm)
The s13heme of the energy distribution is shown in the Fig 42 and the
13hara13teristi13s of MAIN and AUX laser are presented in Table 41
For the experiments PALS oers two va13uum 13hambers with a spheri13al
42 TARGETS 49
Figure 42 Energy distribution s13heme for MAIN and AUX laser beams
LASER MAIN AUX
Typi13al laser energy (J) 120 60
Beam diameter (mm) 290 148
Wavelength (nm) 1315 438
Pulse duration (ps) 350 350
Table 41 Nominal 13hara13teristi13s of MAIN and AUX laser beams
and 13ylindri13al geometry respe13tively The spheri13al 13hamber assembly used
in our experiments is shown in Fig 43 This 13hamber having a diameter
of 100 13m is 13losed by two entran13e 13ir13ular windows of respe13tive diameters
80 13m and 50 13m to allow the MAIN and AUX beams to penetrate in the
13hamber and to rea13h the target Ea13h of these windows is outtted with a
hinged door Several 13ir13ular ports of various diameters also allow for the
dierent diagnosti13s The target holder and some diagnosti13s are mounted
on an opti13al ben13h whi13h is de13oupled from the 13hamber walls
42 Targets
Our targets s13hemati13ally 13onsist in small tubes of 4 mm length 13losed on
both sides by two spe13i13 foils of thi13kness 11 microm on whi13h the laser beams
are fo13used (one laser per foil) with an irradian13e of about 10
1413m
minus3 The
tube is lled with gas in whi13h the radiative sho13k propagates with a velo-
13ity of 30 - 60 kms The two foils 13losing the target insure the 13onversion
through ablation and sho13k generation of the laser energy into me13hani13al
energy
In our experiments we used spe13i13 targets for alignment and 13hara13te-
rization of the beam size These targets were manufa13tured at the Pole
instrumental of Observatoire de Paris
50 CHAPTER 4 EXPERIMENTAL SETUP
Figure 43 Snapshot of the spheri13al 13hamber showing the horizontal bredboard
and on the right the fo13using MAIN lens
(a) (b)
Figure 44 Dimension drawing (a) and snapshot (b) of Aluminum massive
(mo13k) target
42 TARGETS 51
421 Massive Targets
For the laser alignment on the target we have used spe13ial targets also
termed as mo13k targets A mo13k target (Fig 44) is an aluminum blo13k
with a base Three 13uts made on it help to fo13us the laser beam at the
desired position The 13enter of ea13h of the two laser spots should be lo13ated
at the interse13tion of the verti13al and horizontal marks and the pre13ision of
this alignment is less than 20 microm
Beside alignment purpose su13h targets are used to 13hara13terize the MAIN
and AUX spot diameters in the planes whi13h 13orrespond to the lo13ation of
the two Parylene foils Hen13e the laser beam fo13ussed on these targets
generates a 13rater (Chaulagain 2015) The detailed analysis of the shape
and size of these 13raters provides a quantitative information about the ho-
mogeneity of the in13ident beam and the size of the fo13al spot A detailed
dis13ussion about the fo13alisation of the laser with a massive target will be
presented in the se13tion 432
422 Gaseous Targets
The 13ore of the gaseous targets (Stehleacute et al 2014a) 13onsists of a 13hannel of
a parallel pipe shape having the dimension of 09 times 06 times 4 mm pla13ed at
the top of an aluminum stru13ture The pipe in whi13h the sho13ks propagate
is dened by two 500 microm thi13k verti13al lateral sides made with 13oated fused
sili13a (SiO2) windows (09 times 4 mm) as shown in the Fig 45 Su13h windows
are suitable for the visible interferometry diagnosti13 whi13h uses an auxiliary
Q-swit13hed NdYLF green Evolution laser (see Se13tion 441)
The 13hannel is 13losed at the top with a window made of a 100 nm thi13k
Si3N4 membrane whi13h is itself supported is by a 200 microm thi13k sili13on frame
(26 times 4 mm) made by SILSON 13ompany Su13h membrane is transparent
to XUV radiation whi13h is ne13essary for the XUV spe13tros13opi13 diagnosti13
used in the experiment The transmission of the aforementioned material is
obtained from the 13enter for X-ray opti13s (CXRO) database
1
and is plotted
between 0 and 40 nm in Fig 48
The pistons 13losing the tube from opposite sides and whi13h will laun13h
the two sho13ks are made of Parylene-N (11 microm) 13oated by Au (06 microm)
(SCITECH 13ompany) They are glued on 01 mm thi13k Ni13kel disks with
external diameter 5 mm and internal diameter 1 mm This disk helps in
assembling the targets It also 13ontributes together with a larger V-shield
(se13tion 49) to prevent hard X-ray emitted at the laser impa13t on the foil
to rea13h the gas in the tube The Parylene layer is fa13ing the laser and plays
the role of an ablator The gold layer whi13h is 13oated on the Parylene and
is lo13ated between the Parylene and the gas aims at blo13king the X-rays
generated by the laser plasma intera13tion to rea13h and preheat the gas in
1
httphenkelblgovopti13al_13onstants
52 CHAPTER 4 EXPERIMENTAL SETUP
(a)
(b)
Figure 45 Values on the gures are in mm (a) Verti13al 13ross se13tion view of
the gaseous target (b) Horizontal 13ross se13tion view
42 TARGETS 53
Figure 46 Pi13ture of a the gaseous target on its base
Figure 47 S13hemati13 of the gaseous target
54 CHAPTER 4 EXPERIMENTAL SETUP
Figure 48 Transmission of 100 nm thi13k Si3N4 membrane (CXRO database)
Figure 49 Snapshot of a target holder with one V-shield regarding the AUX
laser whi13h will pass through the hole in this shield marked in green in the Figure
The target whi13h fa13es the two diagnosti13s (tubes) and is lo13ated behind the
V-shield is thus not visible
42 TARGETS 55
the 13ell These various elements are manually glued on the base of the target
to built the sho13k 13hannel Ea13h target has been tested against leaks before
the experiment and before ea13h shot The axis of symmetry of the sho13k
waves is required to be lo13ated at 03 mm from the base and at 03 mm from
the two verti13al SiO2 windows S13hemati13s of verti13al and horizontal 13ross
se13tion views of the target are presented in Fig 45b The axis of symmetry
is visible in the verti13al view while the position of the laser fo13us point (`+
sign) in the horizontal view This is a13hieved during the alignment pro13edure
employing mo13k targets as dis13ussed in the se13tion 432 The base of the
target 13onsists of a mi13ro-ma13hined aluminum stru13ture A gas pipe at the
bottom of the stru13ture allows for in situ gas lling at a pressure of a fra13tion
of a bar
A total of 55 targets were manufa13tured for the experiment Among
them 31 gave valuable results
423 Target holder
We x the targets on a stainless holder and then pla13e the system into the
va13uum 13hamber The target holder has been designed and manufa13tured
at Observatoire de Paris It allows a very reprodu13ible positioning of the
target on the va13uum 13hamber breadboard The main element of the target
holder is a base where ea13h target is rst xed by s13rews and removed after
ea13h shot to be repla13ed by a new one It also in13ludes a diagnosti13s support
(Fig 46) whi13h is atta13hed on it from the top with s13rews One of these
diagnosti13s 13onsists in a fast XUV Si diode with a pinhole The diode is
atta13hed to the bla13k tube whi13h is visible on the left part the target holder
in Fig 47 This last diagnosti13 aimed for sho13k timing as the pinhole-diode
system allows to re13ord the sho13k passing at the imaged position of the tube
(Stehleacute et al 2012 Chaulagain 2015) Unfortunately this diagnosti13 did
not work well and will not be mentioned further in the thesis
The se13ond diagnosti13 is a visible time-and-spa13e-resolved visible spe13tro-
meter On the target holder this diagnosti13 13ontains a lens whi13h is lo13ated
inside the bla13k tube on the right part of Fig 49 The lens allows imaging
the sho13k on a linear bundle of 21 bers 13onne13ted to the visible spe13trometer
through a feedthrough (se13tion 443)
A snapshot of the target holder is shown in the gure 49 The red arrows
in the Fig 49 shows the distribution of the various diagnosti13s
In addition we used two 3 mm thi13k stainless steel V-shape shields on
both sides of the target whi13h were xed to the holder These V-shields have
a hole (sim25 mm) shaped entran13e to allow laser in13iden13e on the target (in
green on the Figure 49) The role of the V-shields is to prevent the target
from the extra laser or another kind of hard X-ray plasma self-emission linked
to the ablation pro13ess
56 CHAPTER 4 EXPERIMENTAL SETUP
424 Target lling
The gaseous targets were lled in situ at a fra13tion of 1 bar with a gas (viz
Xe Ar a mixture of Xe and He) whi13h thus provide the medium for the
sho13k propagation We 13hose an in-situ lling system (Fig 410) to redu13e
the dieren13e of pressures inside and outside of the tube This is 13riti13al as
a large pressure dieren13e between the target 13ell and the va13uum 13hamber
13an result in the breaking of the ultra-thin Si3N4 windows of the target
To 13ontrol the pressure we used a piezo-resistive pressure transdu13er
(Swagelok PTI-S-AC6-32) whi13h has 10 kΩ bridge resistan13e and works
with 14-30 V biasing voltage A sensitive Bourdan manometer and a gauge
allowed monitoring the pressure inside the target 13ell To read the pressure
remotely (from the 13ontrol room) until the last se13ond before the shot we
13onne13ted the system with a long 13able
The transdu13er has been operated with a +17 V DC biasing voltage The
pressure is read using an industrial pressure transdu13er from Swagelok (0 to
10 Volts -10 bar to 06 bar 05 pre13ision) and reading is made with a
voltmeter at better than 05 This system generally took approximately
one minute to stabilize A s13hemati13 of the aforementioned gas lling system
is presented in the Figure 410
The pro13edure of the target lling is presented as follows
Figure 410 In-situ system for target lling and pressure monitoring
The target is pumped during the 13hamber eva13uation
The target is isolated at the end of the pumping
The target is lled to the desired pressure
43 LASER FOCUSING 57
The target is isolated from the gas bottle to avoid any leakage in
13hamber
Then the pressure 13an be read from the 13ontrol room till the last se13ond
before the shot
The gases are taken from high grade bottles 99995 purity The Xe+He
mixture is 1007plusmn050 He molar fra13tion The lling pro13edure is pre-
venting the presen13e of impurities like air When 13losing the 13hamber and
during the primary pumping the lling 13ir13uit 13ommuni13ates with the 13ham-
ber thus the whole tubing and the target are eva13uated down to 001 mbar
After the turbo pump is a13tivated the 13onne13tion to 13hamber is 13losed and
the gas in inje13ted When the target pressure is rea13hed the 13ell is isolated
from the pressure regulator In 13ase there is an una1313eptable overpressure
it is possible to re13onne13t to 13hamber to eva13uate and to restart the lling
The pressure reading is possible in the 13ontrol room still at pre13ision better
than 1 and the value kept for the re13ords is the one just before the shot
whi13h happens normally 1-2 minutes after 13losing
43 Laser Fo13using
431 Fo13al Lenses and Phase Zone Plates
Two fo13al lenses are used to fo13us the laser beams on the target piston foil for
MAIN and AUX respe13tively The MAIN lens is lo13ated inside the va13uum
13hamber whereas the AUX lens is outside this 13hamber The spe13i13ations
of these lenses are given in the Table 42
laser Diameter (mm) fo13al length (mm) f-number
MAIN 340 564 at 3ω 19
AUX 150 1022 at ω 69
Table 42 Spe13i13ations of the MAIN and AUX fo13al lenses and the f- number
(fo13al lengthbeam diameter) is given (see table 41 for the beam diameters)
Whereas these lenses provide a smooth intensity at the fo13us this is not
su13ient to laun13h a planar sho13k wave To this purpose phase zone plates
(PZP) have been used between the output laser beam and the lens
A PZP 13omprises several phase zone elements whi13h divide the input
beam into several beamlets whi13h 13ombined with the main fo13al lens provi-
ding a uniform intensity distribution on the target with a spe13ied diameter
(Ross Pepler and Danson 1995 Bett et al 1995) The PZP are 13ommonly
used in EOS laser experiments whi13h require a uniform intensity distribution
on the target (Koenig et al 1994 Batani et al 2003)
Two PZP as dis13ussed below were used in the experiment
58 CHAPTER 4 EXPERIMENTAL SETUP
The rst one made by COLSICOAT was already used at PALS for
radiative sho13k experiments (Stehleacute et al 2010) This PZP used for
MAIN (438 nm) is made on a 13oated BK7 substrate of a diameter
of 310 mm and se13tion of 8 mm It is designed to provide a uniform
13ir13ular fo13al spot over a diameter of 05 mm
The se13ond plate made by SCITECH has been used for AUX (1315
nm) The plate is made on a 63 mm thi13k 13oated Viosil substrate
with a square 153 x 153 mm shape It is designed to provide a uniform
13ir13ular fo13al spot over a diameter of 0250 mm
Laser Spe13ied spot diameter (mm)
MAIN 05
AUX 025
Table 43 Spe13i13ations of the two Phase Zone Plates
In terms of energy distribution sim 80 of the laser energy is in the laser
spot (C Spindloe private 13omm) The energy of the laser whi13h is given
by the PALS installation should be 13orre13ted by this fa13tor of 80 For
AUX another 13orre13tion of sim 10 has to be added whi13h is linked to the
transmission of the entran13e window of the va13uum 13hamber As a result
for nominal energies of the MAIN and AUX lasers equal to 120 and 60 J (see
table 41) the uen13es on the target are equal to sim 48800 and 59500 J13m
2
respe13tively Taking into a1313ount the 03 ns laser duration of the lasers this
results in an intensity of sim 1-2 times 10
14W13m
2
432 Fo13using of the MAIN and AUX laser beams
The determinations of the fo13i of the two lenses were performed with a
Hartmann setup using a photographi13 paper and without any PZP plate
The determination of the size of the beam at the impa13t is important both to
know the initial 13onditions for the sho13k waves (laser irradian13e whi13h should
be few 10
14W13m
minus2and se13tion of the sho13k wave) The sele13tion of the
best spots size on the piston was done with mo13k targets and by translating
the two laser lenses The in13iden13e of a laser beam on a solid planar target
generates a 13rater in the foil and an ablated plume propagating ba13kward
at high temperature (Mahmood and Ur-Rehman 2009 Singh and Thakur
2007) The 13rater is the result of boiling and ablation of the material The
ablated plasma whi13h expands towards the in13ident laser beam emits in the
keV range
A standard pro13edure whi13h works well at low energy (sim 10 J) 13onsists
of the estimation of the spot size by measuring the 13rater on the bulk target
using a standard mi13ros13ope At higher energy the boiling ee13t makes
44 DIAGNOSTICS 59
this estimation impre13ise and we preferred to use a keV imaging diagnosti13s
(Chaulagain et al 2012) whi13h is based on a pinhole (25 microm for MAIN keV
and 20 microm for AUX) and a keV 13amera These two keV 13ameras have been
used during ea13h laser shot to image the impa13ts of the MAIN and AUX
laser beams on the two respe13tive pistons of the target Ea13h keV 13amera
is a CMOS dete13tor without any glass It is prote13ted by an Aluminium
lter (200 microm for MAIN and 42 microm for AUX) to blo13k the visible light and
to prote13t the ship from the debris The spe13i13ations of the 13ameras are
presented in the Table 44 and the keV 13amera system is shown in the Fig
411 The two PALS laser lenses were translated up to a13hieve the suitable
diameters on the target whi13h were nally set to 450 - 500 microm and 250 -
300 microm for the MAIN and AUX beams
The size of the impa13t was 13ontrolled ea13h day on mo13k targets before
using the laser beams for real shots on gaseous targets It is worth to pre13ise
that the keV diagnosti13 was still in pla13e for the shots on the gaseous targets
In that 13ase the beams impa13t Parylene-N instead of Aluminium The keV
signal is then weaker than for the mo13k aluminum targets
Figure 411 S13hemati13 of the keV 13amera
KeV 13amera KeV 13amera
MAIN laser AUX laser
Model UI-122xLE UI-164xLE
Resolution 752(H) x 480(V) 1280(H) x 1024(V)
Pixel size (microm x microm) 60 x 60 36 x 36
A13tive Area (mm x mm) 451 (H) x 288 (V) 46 (H) x 37 (V)
Magni13ation 107 068
Pinhole Material Ta Pt
Table 44 Spe13i13ations of the two uEye keV 13ameras (1stVsion 13ompany)
44 Diagnosti13s
Various kinds of diagnosti13s were implemented namely visible interfero-
metry XUV and visible spe13tros13opy For the same two keV 13ameras are
indu13ted to monitor the impa13t and two visible 13ameras for target align-
60 CHAPTER 4 EXPERIMENTAL SETUP
(a)
(b)
(13)
Figure 412 (a) Sign 13onvention for the respe13tive positions of the lens and the
target (b) keV image of a MAIN impa13t on a massive target (lens position +1500
microm) The spot size is 475plusmn25 microm Pixel size for MAIN keV 13amera is 56 microm (13)
keV image of an AUX impa13t (lens position -1500 microm) The spot size is sim 280plusmn20microm on target One pixel = 66 microm on AUXs keV 13amera
44 DIAGNOSTICS 61
ment In this se13tion I will des13ribe the setups of the visible interferometry
XUV spe13tros13opy and visible spe13tros13opy The analysis of the re13ords of
visible interferometry and XUV spe13tros13opy will be dis13ussed in subsequent
13hapters
441 Visible Interferometry
A Ma13h-Zehnder Interferometer (MZI) has been employed to perform visible
interferometry The re13orded interferometri13 images have been used to study
the sho13k speed and the ele13tron density during the sho13ks propagation The
experimental setup of this interferometer is presented below
Ma13h Zehnder Interferometer
We pla13ed the target in whi13h sho13k is produ13ed in one of the two arms
of the interferometer The probe laser wavelength (527 nm) is supposed to
be far away of any atomi13 absorption resonan13es in the plasma and thus the
13ontributions to the refra13tive index 13ome from the plasma free ele13trons and
not from bound ele13trons (Harilal and Tilla13k 2004)
Figure 413 First interferometri13 setup All the opti13al elements are 1 in13h For
simpli13ity the plasma slab is not reported in the probe beam
The rst experimental setup shown in Fig 413 was used until 04052015
62 CHAPTER 4 EXPERIMENTAL SETUP
In this setup
the 300 ns long probe laser of wavelength 527nm (beam diameter asymp 1
in13h energy on the target few tens of mi13rojoules) rst passes through
an alignment pinhole PH1
the non-polarizing beam splitter BS1 transmits 50 of the intensity
of the in13ident beam and the remaining 50 of intensity in the per-
pendi13ular dire13tion At this stage the two arms of the interferometer
namely the referen13e and probe beams are produ13ed
two half wave plates HWP1 and HWP2 are pla13ed respe13tively in
the path of the probe and referen13e beams They allow to play with
the polarization in order to optimize the fringes 13ontrast as it will be
explained below
the verti13ally polarized probe beam passes through the target (plasma
slab) Due to its polarization it is fully ree13ted by the polarizing 13ube
beam splitter PBS1 towards the polarizing 13ube beam splitter PBS2
the referen13e beam is ree13ted by the mirror M4 on to this 13ube splitter
PBS2
At the output of PBS2 the two beams overlap but they have ortho-
gonal polarizations and 13ant interfere They then travel through a
polarizing beam splitter PBS3 those axis makes an angle of 45
with
the 13ommon dire13tion of in13iden13e of the two beams This rotates
the polarizations of the referen13e (horizontally polarized) and probe
(verti13ally polarized) beams along the same axis
At this stage the beams share the same polarization and interfere
A last non-polarizing beamsplitter plate (90-10 ) BS2 is pla13ed after
PBS3 to enable the propagation of the two beams on the same axis as
the laser beam at the entran13e of the interferometer
The fringes are re13orded on a HAMAMATSU C7700 VIS Streak Camera
lo13ated after the last alignment pinhole PH4 To this purpose an imaging
setup makes the image of the target longitudinal axis (ie along the dire13tion
of the sho13k propagation) on the streak slit This is done with an a13hromati13
spheri13al doublet of fo13al length equal to 1000 mm The magni13ation is
measured using an AGAR grid to 226 whi13h means that 106 microm on target
are re13orded by one pixel (size of the pixel 24 microm) on the streak on both
dire13tions (Fig 413)
44 DIAGNOSTICS 63
Figure 414 Se13ond interferometri13 setup
Role of the polarizing devi13es
Let us suppose that the polarization state of the in13oming beam is unknown
The polarization of the probe beam is then xed to be verti13al by the pola-
rizer HWP1 Ree13ted by the polarizing 13ube PBS2 it remains verti13al at
the entran13e of PBS3 On this side the polarization of the referen13e beam
is set to be horizontal after the 13ube BS1 using the polariser HWP2 and is
transmitted as horizontal by PBS2 The rotation of PBS3 allows to proje13t
part of the two beams on a 13ommon polarization axis and to interfere
This setup presents the following interest the intensity of the laser beam
in the two arms of the interferometer 13an be modied (and thus the 13ontrast
of the fringes) 13an be optimized using the two polarizing plates HPW1 and
HPW2 However as the multimode laser is not fully 13oherent the dark
fringe (zero net intensity) has never been obtained
Moreover the setup is designed to also allow alignment and adjustment
of the imaging setup with a green HeNe laser whi13h shares the same path
as the probe beam up to PBS1
A more optimized setup was used after 05052015 (ie shot number
48033) where the mirror M1 in the referen13e beam was repla13ed by a
polarizing beam splitter PBS4 ree13ting the probe beam like the mirror M1
(Fig 414) This was done in order to redu13e the dieren13e between the
64 CHAPTER 4 EXPERIMENTAL SETUP
two paths l1 and l2 and then to optimize the fringe 13ontrast This was more
suitable due to the short 13oheren13e length (1- 2 13m) of the probe laser
Streak 13amera
Figure 415 S13hemati13 of the Streak 13amera (from Hamamatsu noti13e)
A streak 13amera is a setup whi13h allows measuring the temporal varia-
tion of a fast signal whi13h is re13orded on its slit As indi13ated in the noti13e)
of the Hamamatsu C770 13amera the light pulse rst passes through the slit
then it is fo13used on the photo13athode (size 7times17 mm) of the streak by an
input opti13s The visible photons are 13onverted into a number of ele13trons
whi13h is proportional to the intensity of the in13ident light These ele13trons
are a1313elerated and 13ondu13ted towards the phosphor s13reen when a high-
speed voltage whi13h is syn13hronized to the in13ident light is applied As a
result the ele13trons are swept at high speed from the top to the bottom
after whi13h they are bombarded against the phosphor s13reen of the streak
tube and 13onverted through an output opti13s to an opti13al image on a dual
mode 13ooled CCD 13amera with 512 times 512 pixels (Fig 415) In the PALS
experiment the sweeping times were kept either 50 or 200 ns The CCD
pixel size is 24 x 24 microm
The streak 13amera 13an be operated with time swap mode with the slit
of the 13amera almost 13losed The re13ord 13onsisted then in images (position
along the slit versus time) as dis13ussed above or in a stati13 mode (no time
swap) whatever the slit aperture In that 13ase the re13ord is a usual 2D
spatial image of the obje13t
During the experiment we have taken referen13e images (without any
fringe) for every target as shown in the Fig 417a for one target example
In order to re13ord su13h images we blo13ked the referen13e beam and let the
probe beam only to illuminate the target in a stati13 mode and with the slit
open Then we redu13ed the slit width to its nominal value (200 microm) This
enabled us to have the visualization of the portion of the target whi13h was
44 DIAGNOSTICS 65
Figure 416 Imaging setup of the Ma13h Zehnder Interferometer setup operating
in the longitudinal mode The 13hannel of the target is fully illuminated by the
probe beam In this mode the lens images the axis of symmetry of the two sho13k
waves along the slit of the streak 13amera
imaged in the slit on the target and also to lo13ate the position of the pistons
before the shot This image is termed as the referen13e target image Then
the system was pla13ed in the interferometri13 mode and the fringe pattern
was optimized to be perpendi13ular to the slit and thus in the longitudinal
mode perpendi13ular to the dire13tion of the sho13k propagation
The Fig 417b reports su13h a typi13al interferogram re13orded during our
experiment (13f Fig 413) The horizontal axis 13orresponds to the dire13tion
of the sho13k propagation (with a total s13ale of 54 mm on the target) and
the verti13al axis 13orresponds to time (s13aling 200 ns)
442 XUV Spe13tros13opy
The time and spa13e integrated XUV plasma emission are re13orded with a
Flat Field XUV spe13trometer (grazing in13iden13e XUV spe13trometer) using
a 13on13ave grating Typi13al ray tra13ing a grazing in13iden13e XUV spe13tro-
meter is shown in Fig 418 In pra13ti13al the entran13e slit was removed
as the plasma extension was small In the Fig 419 the drawing of the
XUV spe13trometer setup s13heme (red 13olor) is presented together with the
spheri13al 13hamber and the target The XUV spe13trometer is installed on
the top of the spheri13al 13hamber whi13h allows the XUV emission passing
through the Si3N4 membrane on the top of the target (see se13tion 422) to
be re13orded
A 13ooled Andor DX440 CCD (without glass prote13tion) is atta13hed to
the spe13trometer to re13ord the spe13trum of the XUV radiation between 12
66 CHAPTER 4 EXPERIMENTAL SETUP
(a)
(b)
Figure 417 (a) Referen13e 2D image of a target before the shot re13orded on the
Streak 13amera The positions of the two pistons 13losing the 4 mm long target are
lo13ated at 800 and 4800 microm The dark zones along them (between 800 to 1000 and
4600-4800 microm) 13orresponds to the glue This glue is then visible through verti13al
bla13k strips in the interferometri13 re13ord (b) Corresponding interferometri13 time-
spa13e re13ord
44 DIAGNOSTICS 67
Figure 418 Typi13al ray tra13ing in a grazing in13iden13e XUV spe13trograph The
dete13tor (GMCP or image plate) is installed tangentially to the Rowland 13ir13le
and 40 nm (31- 100 eV) Its 13hara13teristi13s are given in the Table 45 Two Al
lters of thi13kness 08 and 16 microm prote13t the grating and the CCD 13amera
from debris and visible light The XUV images re13orded on the CCD are
time (300 micros) and spa13e (sim 3 mm) integrated
Andor 13amera (DX440)
Pixel Size 135 x 135 micromArea 276 (H) x 69 (V) mm
A13tive pixels 2048 x 512
Table 45 Spe13i13ations of Andor CCD 13amera
The 13urved grating (Table 46) used to fo13us the XUV emission on to
the CCD 13hip has a variable growes spa13ing as shown in the bottom panel
of the Fig 419 A typi13al output re13ord is also presented for illustration
Grating
Type Dira13tion
Growes per mm 1200
Dimensions 30 x 50 x 10 mmRadius of 13urvature 5649 mm
Blazing angle 37 degree
Blazing wavelength 16 nm
Table 46 Spe13i13ations of the XUV grating
443 Visible spe13tros13opy
In addition to the interferometry and XUV spe13tros13opy we have also im-
plemented a time- and spa13e-resolved visible spe13tros13opi13 plasma emission
diagnosti13
68 CHAPTER 4 EXPERIMENTAL SETUP
Figure 419 (a) Setup of the XUV spe13trometer in the spheri13al 13hamber (b)
Typi13al re13ord between sim 10 and 40 nm is shown in the bottom panel
The 13ore of the diagnosti13s is a Czerny-Turner 300 mm Spe13trometer
2
lo13ated outside the va13uum 13hamber The light of the plasma is inje13ted on
the slit of this spe13trometer through a bers bundle This bundle 13onsists of
three sets of an aligned seven individual bers looking at dierent se13tions
of the target 13hannel through an imaging lens lo13ated in the bla13k tube of
Fig 49 The time resolution is performed through an iCCD 13amera lo13ated
at the exit port of the spe13trometer (Fig 420a)
A typi13al re13ord is reported in Fig 420b The results of this diagnosti13s
show several signatures whi13h dier along the tube Most of them are in
absorption and 13ould be linked to the heating of the target sili13a window
Their interpretation is not yet a13hieved and will not be presented in this
study
45 Summary
In this 13hapter I have presented the details of our experimental setup and
diagnosti13s The next 13hapter will be dedi13ated to the presentation of our
re13ords results and their analysis
2
see http wwwandorcomspectrographshamrock minus spectrographminus series
45 SUMMARY 69
(a)
(b)
Figure 420 (a) S13hemati13 of setup employed to re13ord the time and spa13e
visible plasma emission (b) A typi13al spe13trogram
70 CHAPTER 4 EXPERIMENTAL SETUP
Chapter 5
Results and interpretation
Contents
51 Visible interferometry 71
511 Longitudinal interferometry sho13k speed and
ele13tron density 74
512 Transverse interferometry lateral extension of
the sho13k 91
52 XUV Spe13tros13opy 92
53 Simulations based on experimental results 93
54 Summary 97
In the previous 13hapter I presented a brief des13ription of our experimen-
tal setup This setup in13ludes two main diagnosti13s namely visible interfe-
rometry and XUV spe13tros13opy The visible interferometry is an adequate
tool to probe the ele13tron density of the radiative pre13ursor up to the 13riti-
13al density (4 times 10
2113m
minus3at the wavelength 527 nm of the probing laser)
whereas the denser post-sho13k region is opaque to the visible light XUV
radiation is present in the whole sho13k stru13ture and thus the time- and
spa13e-integrated XUV spe13tros13opy may allow exploring the self-emission
13oming from the post-sho13k and pre13ursor regions
In this 13hapter
1
I present the results of the both diagnosti13s with a
parti13ular fo13us on the interferometri13 data whi13h I have extensively studied
with the help of a spe13i13 data analysis pro13edure that I have developed I
will then present the main results of the XUV spe13tros13opi13 diagnosti13s The
results will then be interpreted with the help of numeri13al simulations
51 Visible interferometry
In the visible interferometry the interferen13es between the main and refe-
ren13e beams overlapping on the dete13tor produ13e a pattern of fringes whi13h
follow the relative phase variations between the probe (whi13h passes through
the plasma) and the referen13e beams (see se13tion 441 of the previous 13hapter
for the experimental setup) The phase variation between two 13onse13utive
fringes of the unperturbed beams (ie without any plasma) is equal to 2π
1
Most of results presented in this 13hapter has been published in Singh et al (2017)
71
72 CHAPTER 5 RESULTS AND INTERPRETATION
Then in the presen13e of the plasma in the probe beam these fringes be-
13ome shifted due to the modi13ation of the refra13tive index More details
about the prin13iple of the interferometry and the Ma13h-Zehnder interfero-
metri13 setup may be found in the APPENDIX A and we re13all here only
the expression of the phase shift
∆φ asymp minus πd
λNclt Ne gt (51)
where λ = 527 nm is the wavelength of the probing laser Nc = 4 times 10
21
13m
minus3the 13riti13al density at this wavelength and lt Ne gt is the ele13tron
density averaged over the laser path d in the 13ell and is dened as
lt Ne gt=
int d
0
Ne(z t)dy
d(52)
In order to dedu13e the ele13tron density it is ne13essary to follow the
evolution of the positions of the maxima (respe13tively minima) of the fringes
versus time To this purpose I have developed a spe13i13 data treatment
pipeline in the intera13tive data language (IDL)
2
To improve the visibility of
the fringes obtained in the interferograms I pro13essed the images with the
Fast Fourier Transform (FFT) (Proakis 2001) This treatment transforms
the image from the spatial domain (ie our re13ords) into the frequen13y
domain It then allows to dene spe13i13 bands of frequen13y whi13h represent
noise for the image and to 13lean them The 13omplete pro13edure is des13ribed
below
First the FFT of the (512 times 512) interferometri13 image is derived In
the next step the low and high-frequen13y lterings within the FFT of the
image are done using low-pass Fminus and high-pass F+ Butterworth frequen13y
lters (Proakis 2001) as follows
Fminus(u) = 1[1 + C(uu0)2n] with C = 1 n = 1
F+(u) = 1[1 + C(u0u)2n] with C = 1 n = 1
(53)
where u represents the position in the FFT image and u0 is the nominal
lter 13uto frequen13y (represented as the width of the region in pixels) For
the low-pass Butterworth frequen13y lter u0 is 13onsidered to be 55 whereas
for high-pass Butterworth frequen13y lter it is taken to be 45 The values of
u0 are dened manually by a trial method with the aim to result in better
noise redu13tion
After this step the inverse FFT of the ltered FFT image is performed
whi13h results in a 13lean image in the spatial domain In Fig 51 the original
re13ord for the shot number 48055 (left) the FFT of this image (13enter)
and the nal frequen13y ltered image (right) are presented showing how the
2
httpwwwast13ama13uk~vasilyidlidl_introdu13tionpdf
51 VISIBLE INTERFEROMETRY 73
frequen13y ltering improves the fringes 13ontrast and thus fa13ilitates their
analysis
Figure 51 Original re13ord of shot 48055 (left) FFT of the original image
(13enter) and the frequen13y ltered image (right)
Next the 13ru13ial task is to identify and to follow the fringes with the
help of a pre13ise determination of the intensity (pixel 13ounts) maxima In
order to lo13ate these maxima I have adopted the following steps
Figure 52 A 13ropped se13tion of the re13ord from shot the 48055 The rst ve
positions have been sele13ted manually `+ signs (in red 13olor) on ea13h fringe On
this re13ord the distan13e between two unperturbed fringes is 13orrespond to 15 pixels
(ie 159 microm)
Firstly I have manually 13hosen a se13tion of a typi13al re13ord with the
aim to sele13t only the area asso13iated with the sho13k dynami13s
74 CHAPTER 5 RESULTS AND INTERPRETATION
Next ve representative points are sele13ted visually (`+ marks in red
13olor) on ea13h fringe 13overing the important lo13ations on the fringe
(13f Fig 52) They serve as the input to the spline interpolation for
estimating all intermediate pixels lo13ations along the fringe (see Fig
53a)
In order to obtain the lo13ations representing the fringe maxima the
spline interpolated lo13ation points are further rened by sele13ting the
pixel of maximum intensity within plusmn5 pixels of the respe13tive spline
lo13ations in the X-dire13tion (see Fig 53b)
The position of fringe maxima for the 13ropped image obtained in an
aforesaid manner is shifted so as to represent the same points however
on the full-image (see Fig 54)
The re13ords obtained from the experiments have been pro13essed in this
way to estimate the sho13k se13tion speed ele13tron temperature and density
511 Longitudinal interferometry sho13k speed and ele13tron
density
In the longitudinal interferometry we perform on the slit of the streak 13a-
mera the image of the axis of symmetry of the sho13k propagation along the
tube to analyse the sho13k propagation in this dire13tion
The interferometri13 images have been pro13essed as explained previously
to enhan13e the fringes 13ontrast The lo13ations of the maximum intensity in
ea13h fringe are then used to derive the sho13k speed and the average ele13tron
density as will be presented below
Sho13k speed dedu13ed from the last fringe method
An estimation of the sho13k speed 13an be obtained through the interferometri13
image In this se13tion I will present the method to derive sho13k speed
based on the last visible end points of the fringes also 13alled last fringe
method This determination provides rapidly a qualitative estimation of the
sho13k speed whi13h is based on the absorption behavior of the plasma (see
Equation A11 in appendix A) and not on the real position of the front
dis13ontinuity The front dis13ontinuity is not seen in the re13ords due to the
strong absorption
At the positions of the last visible end points of the fringes the ele13tron
density rea13hes the maximum value a1313essible to the diagnosti13 The frin-
ges are strongly bent and the absorption of the visible light be13omes also
important Therefore it is believed that the sho13k front is 13lose from this
lo13ation
The lo13ations of the fringe maxima have been already dedu13ed following
the analysis presented in previous se13tions In this regard the best visible end
51 VISIBLE INTERFEROMETRY 75
(a)
(b)
Figure 53 (a) Representative points of the fringes as derived from the spline
tting of the 5 manually sele13ted points (Fig 52 (II)) on ea13h fringe (b) Positions
of the fringes maxima along Y-axis for ea13h fringe derived by lo13ating the points
of maximum intensity in X-dire13tion of the previous points obtained by spline t
76 CHAPTER 5 RESULTS AND INTERPRETATION
Figure 54 Fringe maxima on the full image
Figure 55 Imaging setup of the Ma13h Zehnder Interferometer operating in the
longitudinal mode The lens images the axis of symmetry of the two sho13k waves
along the slit of the streak 13amera
51 VISIBLE INTERFEROMETRY 77
point of ea13h respe13tive fringe is 13onsidered for the average speed estimation
for the sho13k
At very initial time the in13rease in sho13k speed is non-linear This non-
linear trend may result in non-physi13al speed estimations Therefore I dis-
13arded the end points whi13h appear to be within the ve nanose13onds after
the sho13k laun13hing time (eg t = 145 ns on Fig 56)
However the above pro13edure may result in un13ertainties as the sele13ted
end points may have dierent intensity (13ounts) values In this regard I
have evaluated the un13ertainty in the determination of the average speeds
To this purpose I rst sele13ted the best visible end-point of a random fringe
Su13h an end point gives a referen13e intensity whi13h was then used to obtain
on the rest of fringes the end points having 13lose intensity This pro13edure
was performed thri13e in order to dedu13e three sets of su13h end-points (see
Fig 56) Ea13h of these three sets of points was then used to estimate the
speed through a linear-t method (lines in white red and green) It may be
noted that these sets of points and tted lines are very 13lose to ea13h other
Therefore it is tough to distinguish them on the Fig 56 These three speed
determinations enable us to derive an average value with an un13ertainty
whi13h is equal to the standard deviation of these three values
Figure 56 Interferometri13 image re13orded for the shot 48055 in Xe at 01 bar
The sho13k speeds for the sho13ks driven by MAIN (from left side) land AUX (from
right side) lasers are respe13tively equal to sim 54 and 23 kms The time of laser
arrival on the piston is at 146 ns The positions of the Au-Xe interfa13e on the
re13ord are respe13tively 950plusmn50 and 4950plusmn50 mi13rons
This estimation of the sho13k average speed is satisfa13tory for the re13ords
in whi13h the sho13k speed is almost 13onstant throughout its propagation Ho-
wever the variation with the time of the speed may be substantial for some
re13ords In su13h 13ases the un13ertainty in the average speed will be in prin-
13iple higher than the previous estimations Therefore I introdu13ed another
78 CHAPTER 5 RESULTS AND INTERPRETATION
method to determine the average velo13ity this velo13ity is then estimated
during the initial and in nal durations of the sho13k propagation by deriving
the slope of points on the rst middle and last few fringes respe13tively
The standard deviation estimated by this method is then termed as the
un13ertainty in the average speed of respe13tive re13ord
Using these two methods I have obtained two values of un13ertainties for
all the MAIN and AUX sho13ks observed in the experimental re13ords Finally
the largest un13ertainty value of the two is 13onsidered as the nal un13ertainty
in the estimation of the respe13tive speed and is shown by error bars in Fig
57a and 57b 13orresponding to the MAIN and AUX sho13ks respe13tively
The table 51 reports for dierent noble gases the values of the velo13ities
re13orded in several shots for the 13ounter-propagating sho13k waves In addi-
tion one re13ord (48131) 13orresponds to the 13ase of an isolated sho13k wave
laun13hed by the MAIN laser The values of the MAIN and AUX laser ener-
gies are also reported In some 13ases the glue on the window prevented the
probe laser to pass through the 13hannel Then it be13ame impossible to derive
any value for the velo13ity In su13h 13ases I put `NA for the 13orresponding
sho13k velo13ity
It may be noted that the sho13k speeds generated by the MAIN laser vary
in the range of 30-55 kms while for the sho13k originated from AUX laser
the speeds vary between 10 and 30 kms
Variations with laser energy and pressure the 13ase of Xenon
When the number of re13ords is su13ient to make a statisti13al analysis we
may analyse the variations of the sho13k speeds with the laser energy of
the MAIN and AUX lasers This is the 13ase for Xe or XeHe (90-10)
mixture and for gas pressures ranging between 01 and 02 bar At 01 bar
all the measurements 13on13ern Xenon (ρ= 54 times 10
minus4g13m
minus3) At 02 bar
the re13ords 13on13ern a mixture XeHe (90-10 ρ=10 times 10
minus3g13m
minus3)
ex13ept for one point at 121 J whi13h 13orrespond to the 13ase of pure Xe (ρ =
108 times 10
minus3g13m
minus3) We do believe that taking into a1313ount the pre13ision
of our re13ord the introdu13tion of tra13es of Helium do not ae13t mu13h the
dynami13s of the sho13k wave as the 13orresponding variation of mass density
is negligible
As expe13ted the sho13k speed in13reases with the laser energy and de13rea-
ses with the pressure (Fig 57a) A linear t gives the following dependen13e
of the speed in kms with the MAIN laser energy (in J) at 01 and 02 bar
respe13tively
3
v01bar = 1423 + 030E (54)
3
Two outlier points have been dis13arded from the set 13orresponding to 01 bar They
are shown for information on the Figure
51 VISIBLE INTERFEROMETRY 79
(a)
(b)
Figure 57 (a) MAIN sho13k speed (13al13ulated by the last fringe method) versus
the MAIN energy for Xenon or XeHe mixture at dierent pressures with the error
bars (b) AUX sho13k speed versus AUX energy also for Xenon or XeHe mixture
at dierent pressures
80 CHAPTER 5 RESULTS AND INTERPRETATION
v02bar = minus1405 + 047E (55)
The speeds generated by the AUX laser are lower than the previous ones
Although the re13ords are more sparse we note that the speeds at 01 bar
also in13rease with the laser energy However at 02 bar I 13ould not nd
similar trend due to the la13k of re13ords
Variations with the gas
As 13an be seen from the Table 51 we have also performed several shots in
Ar at 01 02 03 and 08 bar one shot in He at 05 and one in Kr at 02
bar
To 13ompare the velo13ities obtained for the dierent gases one 13an either
keep the pressure 13onstant (ie the number of atoms) or the mass density
For a given mass density and laser energy the velo13ity should be the same
in the adiabati13 limit However due to the radiation 13ooling and ionisation
this 13an not be the 13ase Nonetheless if we 13ompare the two re13ords of Ar
at 03 bar (ρ = 493 times 10
minus4g13m
minus3 E = 121 J) with Xe at 01 bar (54
times 10
minus4g13m
minus3 E = 133 J) we measure similar speeds (49 and 54 kms)
These two re13ords are thus interesting to 13ompare This will be done later
on in this 13hapter
More generally we note that the speed of Helium at 05 bar is higher by
sim 40 than for Ar at the same energy (see Fig 58) This is not surprising
for this lighter element (Table 52) However it is important to note that
the la13k of shots prevents us to make any pre13ise 13on13lusion
Ex13ept for this 13ase and for the unique re13ord of Ar at 02 bar whi13h
gives also a higher velo13ity the velo13ities of the other re13ords follow more or
less the variation with the energy than Xenon
To 13ompare the radiative ee13ts it seems justied to perform the 13om-
parison at a given sho13k speed independent of the laser energy and for mass
densities whi13h are 13lose to ea13h other In the following we shall thus 13om-
pare the results of Xe at 01 bar (54 kms 54times 10
minus4g13m
minus3) Ar at 03
bar (49 kms 49times 10
minus4g13m
minus3) and Kr at 02 bar (55 kms 68 times 10
minus4
g13m
minus3)
Line averaged ele13tron density lt Ne gt
I have estimated the ele13tron density lt Ne gt of the plasma averaged along
the path of the probe laser beam by analyzing the interferogram re13ords
assuming the same se13tion d of the plasma layer of 600 microm for both MAIN
and AUX side sho13ks
51 VISIBLE INTERFEROMETRY 81
Shot Gas Pressure E3ω MAIN sho13k Eω AUX sho13k
(bar) (J) speed (kms) (J) speed (kms)
at 298 K
48033 Air 03 124 54plusmn1 66 20plusmn248034 Air 03 131 52plusmn1 74 27plusmn248076 Ar 05 100 41plusmn1 66 18plusmn148077 Ar 01 115 42plusmn1 67 18plusmn148078 Ar 03 112 46plusmn3 65 25plusmn248079 Ar 03 121 49plusmn5 67 23plusmn348080 Ar 08 103 38plusmn2 62 21plusmn248081 Ar 08 113 38plusmn1 68 NA
48082 Ar 08 107 36plusmn1 65 NA
48141 Ar 02 111 63plusmn1 57 NA
48083 He 05 106 57plusmn3 63 NA
48146 Kr 02 125 55plusmn2 53 NA
48043 Xe 03 138 57plusmn1 72 25plusmn148051 Xe 01 123 58plusmn1 67 30plusmn148055 Xe 01 133 54plusmn3 68 22plusmn348057 Xe 01 127 53plusmn1 68 23plusmn148058 Xe 01 115 48plusmn3 63 18plusmn148059 Xe 01 116 50plusmn1 67 21plusmn148061 Xe 01 127 53plusmn1 67 17plusmn248065 Xe 01 122 52plusmn4 68 23plusmn348066 Xe 01 114 50plusmn3 67 15plusmn248067 Xe 05 115 39plusmn2 65 NA
48068 Xe 05 109 36plusmn2 65 12plusmn248070 Xe 05 109 33plusmn3 65 NA
48138 Xe 02 121 45plusmn5 0 0
48131 Xe +He 02 112 38plusmn1 0 0
48132 Xe +He 02 118 41plusmn4 56 18plusmn248133 Xe +He 02 112 41plusmn3 56 NA
48134 Xe +He 02 111 38plusmn1 60 NA
48136 Xe +He 02 115 39plusmn3 59 14plusmn148143 Xe +He 06 123 39plusmn4 63 18plusmn548144 Xe+He 02 133 45plusmn3 66 NA
Table 51 Sho13k speeds estimated from the `last fringe method `NA represents
the entries whi13h 13ould not be dedu13ed from the re13ord Further entries in the
bold font are dis13ussed in detail in this 13hapter
82 CHAPTER 5 RESULTS AND INTERPRETATION
Helium Argon Krypton Xenon
Atomi13 Number 2 18 36 54
Atomi13 Mass 4 3995 8380 13129
First Ionization Energy (eV) 246 157 14 12
Density (10
minus4g13m
minus3) at 01 bar 016 164 344 539
Table 52 Atomi13 data and density at 01 bar (at room temperature) for He Ar
Kr and Xe
Figure 58 MAIN sho13k speed (13al13ulated by the last fringe method) versus the
MAIN laser energy for Xe (at 01 bar only) Ar He and Kr at dierent pressures
with the error bars
51 VISIBLE INTERFEROMETRY 83
It should be noted that the beam se13tions dMAIN and dAUX of MAIN
and AUX lasers are approximately 600 and 300 microm respe13tively on the two
pistons and that the transverse horizontal se13tion of the sho13k tube is equal
to dtube = 600 microm Thus the value of lt Ne gt 13omputed for the sho13k wave
laun13hed by MAIN should be 13lose to the physi13al lo13al value Ne supposing
that the plasma is homogeneous along the transverse se13tion However
the value obtained for AUX is 13ertainly larger by a fa13tor whi13h 13an be
estimated at a rst step as the fra13tion of dtubedAUX giving a fa13tor of
about 2
As explained in the previous se13tion the pixels representing the positions
of the fringe maxima have been already determined with the best possible
pre13ision Therefore it is possible to 13al13ulate the relative phase shift (with
respe13t to zero time) variation along y-axis (time axis) at ea13h fringe maxima
Zero time is time of MAIN and AUX laser rival on the target With these
phase shifts estimated at ea13h pixel of ea13h fringe I then derived lt Ne gtusing Equation A22
Let us take the 13ase of three shots in Xenon for illustrating the method
whi13h will be followed (see the left panel of the Figure 59) The maximum
density is estimated to be sim11 times 1019cmminus3(13orresponding to phase shift sim
16) The number of subdivisions is taken to 5 bins with the following phase
(∆φ) ranges (in the units of 2π) and average ele13tron densities (ltNegt)
bin 1 ∆φ le 06 lt Ne gtle 39 10
1813m
minus3(white)
bin 2 06 - 08 39 - 57 10
1813m
minus3(red)
bin 3 08 - 11 57 - 75 10
1813m
minus3(blue)
bin 4 11 - 13 75 - 93 10
1813m
minus3(green)
bin 5 gt 13 gt 93 10
1813m
minus3(magenta)
All the re13ords shown in the Table 51 have been pro13essed using this
method The limit of dete13tion for the phase shift 13orresponds to 2 pixels
giving a threshold for the Ne measurement lt Nemin gt This threshold
diers from one re13ord to another due to the variation in the number of
fringes (and thus the distan13e between two unperturbed fringes)
In order to make a 13omparative investigation of the sho13ks propagation
and intera13tion in dierent gases as already mentioned previously I sele13ted
three 13ases (shot48055 shot48132 and shot48138) relative to Xe (or Xe-
He mixture) one 13ase relative to Ar (shot48079) and Kr (shot48146)
The 13orresponding experimental 13onditions are noted in bold in the Table
51
84 CHAPTER 5 RESULTS AND INTERPRETATION
lt Ne gt in Xenon
The Fig 59 reports the interferometri13 re13ords for the three Xenon 13ases
The top and the middle panels 13orrespond to the propagation of two 13ounter-
streaming radiative sho13k waves at 01 (shot48055) and 02 bar (48132)
For 13omparison one re13ord (48138 bottom panel) is dedi13ated to the pro-
pagation of single sho13k (MAIN) at 02 bar The Xe-He mixture (90 - 10
in numbers of atoms) is used for the investigation in the 13ase of 02 bar
pressure while it is only Xe in the 13ase of 01 bar As mentioned previously
at the pre13ision of our re13ords we 13onsider that this impurity 13on13entration
has a negligible ee13t on the sho13k speed and the pre13ursor ele13tron density
The limit of dete13tion of lt Ne gt over the se13tion of the tube (06 mm) is
13orresponds respe13tively to 9 times 10
17 6 times 10
17and 6 times 10
1713m
minus3for the
Figures 59(a) (b) and (13)
The variations of lt Ne gt with the distan13e along the sho13k tube (ie
parallel to the dire13tion of the sho13ks) are reported in the right panel of Fig
59 at 10 ns (in red) 20 ns (in blue) 30 ns (in green) and 40 ns (in magenta)
The intera13tion between the two pre13ursors is 13learly visible at 01 bar
(Fig 59(a)) at 10 ns the intera13tion of the 13ounter-propagating sho13ks
has either not yet started or is below the sensitivity of this diagnosti13 The
intera13tion o1313urs at later times with a typi13al signature whi13h is as follows
the slope of lt Ne gt is de13reasing from the left (MAIN pre13ursor) passes
through a minimum and in13reases at the right (AUX) The minimum itself
in13reases with time up to 7 times 10
1813m
minus3at 40 ns
At 02 bar we have not been able to re13ord the 13ollision in the temporal
range (50 ns) of the streak However the two re13ords (with MAIN only and
with the two sho13k waves) indi13ate a pre13ursor for MAIN The two gures
(Fig 59(b)) and (Fig 59(13)) show that the pre13ursor of the MAIN sho13k
wave is not inuen13ed by the presen13e of AUX sho13k wave up to 45 ns
There is no obvious indi13ation about a pre13ursor for AUX in the 13ase of
two 13ounter-propagating sho13k waves (Fig 59(b)) At this pressure and
13ompared with the previous 13ase at 01 bar the absen13e of pre13ursor for
AUX may be attributed to i) a low sho13k speed (18 kms) 13ombined with a
larger pressure (hen13e for a given gas the pre13ursor extension in13reases with
the speed and de13reases with pressure) ii) a too small longitudinal extension
of the eventual pre13ursor (see Fig 59(b)) 13ompared with the resolution of
20 mi13rons (2 pixels) Our 1D numeri13al simulations with Xenon opa13ity
multiplier times 20 (not presented here) indi13ate a small pre13ursor for AUX
sho13k At 15 ns its extension is 50 microm (900 microm for MAIN sho13k) with
a typi13al ele13tron density sim 35 times 10
1913m
minus3(23 times 10
1913m
minus3for MAIN
sho13k) whi13h does not agree with the re13ord At 42 ns the pre13ursor of MAIN
rea13hes the AUX sho13k front and the prole is similar to the prole at 20 ns
shown in Fig 313a at 01 bar with a plateau of almost 13onstant ele13tron
density between the two fronts This might be 13ompatible with small bending
51 VISIBLE INTERFEROMETRY 85
Figure 59 Left panel interferometri13 re13ords 48055 in Xe at 01 bar (a)
48132 in Xe+He at 02 bar (b) and 48138 in Xe+He at 02 bar (13) Right
panel ele13tron density at 10 20 30 and 40 ns versus distan13e for these re13ords
The positions of maxima have been identied on the re13ords in the left panel
The time t = 0 13orresponds to the time of laser arrival on the target and the
position x = 0 13orresponds to the interfa13e between the piston (Au layer) and the
gas Its determination is pre13ise within 100 mi13rons The distan13es between two
unperturbed fringes for re13ords 48055 48132 and 48138 are 159 244 and 244
microm respe13tively The lt Ne gt un13ertainty (plusmn 2 pixels) is indi13ated by the error bar
in the right panels It 13orresponds respe13tively to plusmn 9 times 10
17 plusmn 6 times 10
17and plusmn
6 times 10
1713m
minus3for the gures (a) (b) and (13) The limit of dete13tion (2 pixels) is
presented by a dotted line on ea13h gure
86 CHAPTER 5 RESULTS AND INTERPRETATION
of the 4
thfringe (from the right) between 45 and 50 ns As 1D simulations
are known to overestimate the pre13ursor ele13tron density 2D simulations are
ne13essary for a more pre13ise interpretation of the experimental result
lt Ne gt in Argon and in Krypton
As seen in the Fig 58 the variation with the laser energy of the sho13k speed
in Argon at 03 bar (in the red dashed 13ir13le) is in good agreement with the
13orresponding one of Xenon at 01 bar (blue linet) As these two 13ases have
a 13omparable mass density of sim 5 times 10
minus4g13m
3 this qualitative agreement
is expe13ted
Taking into a1313ount the larger pressure (and thus density) the slope of
the variation of the slower sho13k speeds in Ar at 08 bar (in a red dash-
dot 13ir13le) with the laser intensity is 13ompatible with the previous trends
However any 13on13lusion 13an not be derived for Argon at 02 and 05 bar
owing to the s13ar13ity of the re13ords
In Fig 510 the interferometri13 re13ord for Ar at 03 bar (shot 48079
49 times 10
minus4g13m
minus3) is presented The estimated speeds of MAIN and AUX
sho13ks dedu13ed from the last fringe method are respe13tively equal to 49 and
23 kms whi13h are 13lose to the speeds of 54 and 23 kms re13orded in Xe
at 01 bar (shot 48055 539 times 10
minus4g13m
minus3 Fig 59(a)) Contrarily to
the Xenon 13ase we do not see any signi13ant bending in the fringes The
maximum fringe shift is reported in red on the fringe 1 The 13orresponding
ele13tron density is equal to 38 times 10
1813m
minus3
Thus obviously the radiation ee13ts are less important for Argon than
for Xenon at the same density and speed Part of this dieren13e 13ould be
attributed to the huge dieren13e in the atomi13 numbers (40 and 131) of the
two gases whi13h leads to dierent peak temperatures for the ions as expe13ted
from equation 84 of 13hapter 2 However the post-sho13k temperatures dedu-
13ed from the jump relations for a real gas (Chapter 2) without radiation are
very 13lose (sim 25 eV) and thus this 13ould be not the relevant explanation
To 13larify this I performed HELIOSPROPACEOS simulations for two
13ounter-propagating sho13ks at 53 and 23 kms respe13tively in Ar and Xe
at the same initial mass density (54 times 10
minus4g13m
minus3) without any opa13ity
multiplier I have also reported for information the 13ase of Xenon with
an opa13ity multiplier equal to 20 The simulated temperature proles are
shown in the Fig 511 One notes immediately that the pre13ursor extension
is smaller for Ar than for Xe The peak temperatures of the MAIN sho13k
are respe13tively sim 21 eV (16 eV for opa13ity multiplier=20) for Xe and 21
eV for Ar In all the 13ases the post-sho13k temperatures are equal to sim 11
eV This indi13ates that the temperature is not the main explanation of the
dieren13es in the pre13ursor length
More interesting are the dieren13es in the Rosseland opa13ity (here per
unit of length) whi13h is reported in the Fig 511b We see that 13lose to the
51 VISIBLE INTERFEROMETRY 87
(a)
(b)
Figure 510 (a) Interferometri13 re13ord in Ar at 03 bar (48079) The estimated
speeds for MAIN and AUX are 49plusmn5 and 23plusmn3 kms (b) Same re13ord where the
fringes maxima are marked by points The bins denition is as fellows bin 0 Ne le11 times 10
1813m
minus3(white) bin 1 11 - 18 times 10
1813m
minus3(yellow) and bin 2 39 - 57
times 10
1813m
minus3(red) The time t = 0 13orresponds to the time of laser arrival on the
target and the position x = 0 13orresponds to the interfa13e between the piston (Au
layer) and the gas Its determination is pre13ise within 100 mi13rons
88 CHAPTER 5 RESULTS AND INTERPRETATION
(a)
(b)
Figure 511 Results for temperature (a) and Rosseland opa13ity (b) obtained from
HELIOS simulation at 10 ns for two 13ounter-propagating sho13ks at sim 50 and 18
kms for Xe (with opa13ity multiplier 1 and 20) Ar and Kr at 54 times 10
minus4g13m
minus3
initial mass density A 13omparison Ar Kr and Xe PROPACEOS opa13ity shown in
APPENDIX B
51 VISIBLE INTERFEROMETRY 89
front the Argon opa13ity is 13lose to 4 13m
minus1 whi13h means that the radiation
from the sho13k is absorbed within 25 mm This has to be 13ompared with the
Xenon 13ase (no multiplier) where it is equal to sim 03 13m
minus1 whi13h means
that the hot pre13ursor is almost transparent over the length of the tube
Thus the dierent behaviors of the opa13ity in the upstream gas explain the
quantitative dieren13e in the development of the pre13ursor The sharp peak
of the Xenon opa13ity with the multiplier of times 20 is due to the strong rise
of the opa13ity when the temperature de13reases below 5 eV whi13h marks the
end of the pre13ursor
I will present now the 13ase of two 13ounter-propagating sho13ks in Kr
(48146 in Fig 512) at 02 bar (69 times 10
minus4g13m
minus3) In this 13ase unfortu-
nately the AUX sho13k 13ould not be imaged 13ompletely due to the presen13e
of glue at the right edge of the target 13ell but the 13ollision was re13orded
at sim 40 ns The speed of the MAIN sho13k is estimated to be 55 kms
Thus despite a slightly higher initial mass density this sho13k wave may be
13ompared with the shot (48055) in Xe at 01 bar (54 times 10
minus4g13m
minus3 54
kms)
Despite the poor quality of the re13ord and the inadequate temporal range
(150 ns) sele13ted on the streak 13amera we note some bending in all the
fringes indi13ating the presen13e of pre13ursor The maximum fringe deviation
(063 of the distan13e between two fringes) is noted at the end of fringe 2
whi13h 13orresponds to the lt Ne gt value 45 times 10
1813m
minus3 The pre13ursor
seams to be more developed here than for the previous Ar 13ase
The result of the simulation in Xe and Kr with equal initial mass density
54 times 10
minus4g13m
minus3 and for the two respe13tive sho13k speeds of 50 and 20
kms is reported in the Figure 511 The peak temperatures of the MAIN
sho13k are respe13tively sim 21 eV (16 eV for opa13ity multiplier = 20) for Xe
and 22 eV for Kr The post-sho13k temperatures (sim 10-11 eV) are 13lose from
ea13h other
The pre13ursor extension in Krypton is smaller than in Xenon whi13h is lo-
gi13al if we 13onsider the opa13ity values (Fig 511b) However if we take into
a1313ount the multiplying fa13tor for the Xenon opa13ity the Krypton pre13ursor
is now more extended than for xenon whi13h is in 13ontradi13tion with the ex-
periment This probably means that either the Kr opa13ity is under estimated
by PROPACEOS or that our multiplier for Xenon is overestimated
Synthesis
The previous experimental investigation 13onrms that the sho13k velo13ity in-
13reases with the laser energy and de13reases with the mass density Moreover
for our given sho13k velo13ity and mass density the extension of the radiative
pre13ursor in the experiment de13reases with the atomi13 number Among Ar
Kr and Xe this last one appears to be the most adapted for the investiga-
90 CHAPTER 5 RESULTS AND INTERPRETATION
(a)
(b)
Figure 512 (a) Raw interferometri13 re13ord in Kr at 02 bar (shot 48146) The
estimated speed for MAIN sho13k is 53plusmn2 kms The time t = 0 13orresponds to the
time of laser arrival on the target(b) Same re13ord shown in (a) maxima is marked
by 13olored points The bins denition is as follows bin 0 Ne le 11 times 10
1813m
minus3
(white) bin 1 11 - 18 times 10
1813m
minus3(yellow) and bin 2 39 - 57 times 10
1813m
minus3
(red) The time t = 0 13orresponds to the time of laser arrival on the target and
the position x = 0 13orresponds to the interfa13e between the piston (Au layer) and
the gas Its determination is pre13ise within 100 mi13rons
51 VISIBLE INTERFEROMETRY 91
tion of the radiative pre13ursor A similar behaviour is found also at ORION
with higher speeds (Clayson et al 2017)
512 Transverse interferometry lateral extension of the sho13k
The transverse interferometry is a dierent geometri13al imaging setup of the
Ma13h-Zehnder interferometer whi13h provides qualitative information about
the 13urvature lo13alisation and transverse extension of the radiative pre13ur-
sor and thus of the sho13k
In the transverse interferometry we perform the image of a transverse
se13tion (on the slit of the streak 13amera) of the tube whi13h is lo13ated at a
distan13e dslit equal to 3 mm from the initial position of the MAIN piston
As the setup is originally adapted for the longitudinal interferometry we use
a Dove prism between the va13uum 13hamber and the streak 13amera to rotate
the image by 90 degrees
Figure 513 S13hemati13s of the Ma13h Zehnder Interferometer setup to re13ord
transverse interferometri13 images The lens allows to make on the slit of the
13amera the image of a se13tion perpendi13ular to dire13tion of sho13k propagation
A transverse interferometri13 re13ord for the MAIN sho13k alone in Xe at
02 bar is reported in Fig514(a) The streak 13amera rst re13ords the unper-
turbed plasma on the se13tion dslit in the tube Then it re13ords su1313essively
the pre13ursor and the post-sho13k (whi13h is opaque to visible radiation and
appears as a dark zone in the re13ords) Taking into a1313ount the oset of 14
ns the time of sho13k arrival is re13orded at 72 ns after the time t0 of laser
arrival on the target and the sho13k speed is estimated to be sim 35 kms Due
to glue on one lateral window (on the right part of the gure) only 6 fringes
92 CHAPTER 5 RESULTS AND INTERPRETATION
Figure 514 Transverse interferometri13 images for (a) shot48111 (MAIN sho13k
only) (b) shot48130 (AUX sho13k only) The time is measured after an oset
equal to 14 and 23 ns respe13tively after the time of the laser arrival on the target
The position zero on the x-axis of ea13h image 13orresponds to the base of the target
are visible The lateral extension of the sho13k stru13ture at this time is deri-
ved to be sim 570plusmn30 microm in qualitative agreement with the spe13i13ations of
the MAIN phase plate and the shape of the pre13ursor is relatively at The
axis of symmetry of the sho13k is determined to be at sim 350 mi13rons from
the base of the target (ie 50 microm above the nominal value of 300 microm)
A re13ord for the AUX sho13k alone is shown in Fig 514(b) whi13h 13orre-
sponds to a gate opening of 50 ns The start time of the image has an oset
of +23 ns from t0 and the distan13e dslit is set to 700 mi13rons from the initial
position of the AUX piston The AUX sho13k duration extends from 30 ns
to at least 34 ns after t0 The sho13k speed is then estimated to be ranging
between 23 and 20 kms The shape of the pre13ursor is strongly bent and
we note a tiny shift of the se13ond and third fringes (starting from the left)
at sim 8 ns It may further be noted that the lateral spread of the opaque
sho13k is ranging between 275plusmn25 microm (whi13h is also in agreement with the
spe13i13ations of the AUX phase plate) and that the axis of symmetry of the
sho13k system is also lo13ated at about 350 microm from the bottom of the 13ell
52 XUV Spe13tros13opy
XUV emission originates from the 13omparatively hot portions of the plasma
In the present 13ase it 13orresponds to the post-sho13k and the pre13ursor re-
53 SIMULATIONS BASED ON EXPERIMENTAL RESULTS 93
gion whi13h is 13lose to the sho13k front The spe13trum re13orded by the XUV
spe13trometer enables us to analyze the spe13tros13opi13 signatures of the radi-
ative sho13k One of the aims of this diagnosti13s was to identify the sho13ks
13ollision whi13h leads to higher temperatures through 13omparing the spe13tra
obtained for the 13ases of single and 13ounter-propagating sho13k waves Howe-
ver only a few re13ords were possible to be obtained during the experiment
and unfortunately the 13omparison with the 13ase of an isolated sho13k wave
was not performed
Among the shots re13orded the XUV spe13trum of the shot 48143 is
presented herewith in detail This shot was performed for [Xe (90)+ He
(10) mixture at 06 bar with laser energies of 123 J for MAIN and 63 J
for AUX The interferometri13 re13ord of this shot is shown in Fig 515 The
MAIN sho13k speed has been estimated to be sim 39 plusmn 4 kms The estimated
AUX sho13k speed (18 plusmn 5 kms) is not pre13ise due to the presen13e of glue
on the right se13tion of the re13ord (note Fig 515) In this interferometri13
re13ord we have not been able to retrieve the 13ollision time However an
extrapolation of the slope 13orresponding to the speeds sim 39 kms (MAIN)
and sim 18 kms (AUX) enables us to approximately determine the 13ollision
time to be between 60 to 65 ns
The raw spe13trum (shown in Fig 516a) re13orded for the wavelength
range of 15-35 nm (35-82 eV) shows the `L edge of Aluminum at 17 nm
(34 nm in se13ond order) in the rst and se13ond (34 nm)orders These two
wavelengths will be used for the wavelength 13alibration The net spe13trum
13orre13tion for the transmission (Henke Gullikson and Davis 1993) of the
100 nm thi13k Si3N4 window (3 mm times 04 mm) is introdu13ed in Fig 516b
A remarkable feature is a strong absorption dip between 19 and 22 nm (56-65
eV) This absorption probably 13omes from the 13older layers (thi13kness 300
microm) between the sho13k heated plasma and the Si3N4 window Few lines
of Xe VII-VIII are identied through NIST database
4
as also Oxygen IV
and V lines Lyman lines of He II (from 1-2 to 1-7) are also present in the
spe13trum This information will be useful for the estimation of the ele13tron
temperature
53 Simulations based on experimental results
In this se13tion I will 13ompare the experimental sho13k 13hara13teristi13s with the
results of HELIOS simulations using the PROPACEOS equation of state and
opa13ity (limited to 1 group) As indi13ated in the 13hapter 3 this opa13ity has
been multiplied by 20 for the Xenon only As our interest is to understand the
sho13k stru13ture in Xenon and not the laser matter intera13tion on the piston
we performed several simulations with Xe gas for various sets of uen13es for
4
[httpphysi13snistgovPhysRefDataASDlines_formhtml
94 CHAPTER 5 RESULTS AND INTERPRETATION
Figure 515 Interferometri13 image for the shot48143 The time t = 0 13orre-
sponds to the time of laser arrival on the target and the position x = 0 13orresponds
to the interfa13e between the piston (Au layer) and the gas Its determination is
pre13ise within 100 mi13rons
53 SIMULATIONS BASED ON EXPERIMENTAL RESULTS 95
(a)
(b)
Figure 516 Raw (a) and 13orre13ted XUV spe13trum (b) for the shot 48143
96 CHAPTER 5 RESULTS AND INTERPRETATION
the MAIN and AUX beams in order to obtain the best agreement between
the simulated and measured sho13k speeds
To analyse the results from the shot 48055 (Fig 59(a)) we set the u-
en13es to 32000 amp 7500 J13m
2 This allows produ13ing the experimental sho13k
speeds 54 and 23 kms in Xenon at 01 bar for the MAIN and AUX beams
respe13tively The two sho13ks appear in Xenon at 2 and 3 ns respe13tively for
MAIN and AUX The merging of the two pre13ursors starts at sim 15 ns and
the sho13k 13ollision time o1313urs at 47 ns In Fig 517 we present the ele13tron
density proles from the simulation (dotted lines) and the experiment (solid
lines) at 10 20 30 and 40 ns
At 10 ns the two simulated pre13ursor extensions are 0165 and 0022 13m
for MAIN and AUX respe13tively The ele13tron density is larger by a fa13tor
of 4 than in the experiment The shapes of the pre13ursors are also very dif-
ferent However this 1D pi13ture supposes the plasma to be uniform within
the tube In reality in the transverse dire13tion Ne is de13reasing from the
13enter to the walls whi13h results in a lower estimation of the average value
(lt Ne gt) 13ompared to the value at the target 13enter and in a smoother pro-
le near 02 amp 035 13m at 10 ns It is also important to note that for AUX
sho13k the average lt Ne gt value underestimates the lo13al one by a fa13tor
of about 2 (as it is averaged over 06 mm instead of 03 mm) Moreover
our 1D simulation suers from an inexa13t opa13ity and 2D ee13ts are pro-
bably important espe13ially for AUX Thus we have here only a qualitative
interpretation of the experimental results
The intera13tion between the two HELIOS radiative pre13ursors starts be-
tween 10 and 20 ns like in the experiment However the shape as well as
absolute values of the simulated ele13tron density 13urves are not in agreement
with the experimental results and the intera13tion is stronger in the simulation
than in the experiment
In order to interpret the spe13tros13opi13 data presented in se13tion 52 we
performed another 1D simulation in Xenon at 06 bar and adapted the
uen13es to generate two 13ounter-propagating sho13ks with the speeds 36 and
18 kms 13lose to the experiment The time evolutions of the ele13tron density
mean 13harge and ele13tron temperature at 56 57 58 60 64 and 65 ns are
presented in Fig 518 The two sho13ks appear in Xenon at 2 and 3 ns
respe13tively for MAIN and AUX Con13erning AUX the 13ombination of a
small speed and a relative high pressure does not allow to develop a radiative
pre13ursor in agreement with the experimental results (Fig 515) whereas
the MAIN sho13k has a pre13ursor and its length is in13reasing with time
The post sho13k temperature of the MAIN is sim 21 eV and the ion 13harge
sim 9 At 57 ns the pre13ursor of MAIN rea13hes the AUX sho13k front This
time is out of our re13ord (see Fig 515) whi13h means that the intera13tion
ee13t is either absent or o1313urs at later times The stru13ture of the AUX
post sho13k is modied by the intera13tion with the MAIN pre13ursor (Fig
518b) The sho13k 13ollision o1313urs at 65 ns (Fig 518a) resulting in the
54 SUMMARY 97
Figure 517 Re13orded ele13tron density (shot 48055) together with the HELIOS
results (with Xenon opa13ity times 20) at dierent times in Xenon at 01 bar
development of two reserve sho13k waves At the 13ollision time the ele13tron
density mass density ele13tron temperature and ion 13harge rea13h respe13tively
sim 14 times 10
2113m
minus3 0034 g13m
minus3 26 eV and 10 Mean 13harge is varying
between 5-10 whi13h 13ompatible with the presen13e of lines of Xe VII-VIII in
the experimental re13ord shown in Fig 516b
Moreover in order to interpret XUV spe13tros13opi13 results shown in se13tion
52 Rodriguez performed qualitative preliminary 13omputations (as des13ribed
in Rodriacuteguez et al (2015)) of the XUV spe13tra emerging from a 600 microm thi13k
plasma with two representative values of the mass density ρ = 32 times 10
minus2
and 33 times 10
minus3g13m
3 They show that the lines of HeII 13an only be ob-
served at a temperature of sim 15 eV and for the lowest density ie in the
radiative pre13ursor
54 Summary
In this 13hapter I have presented an extensive data analysis of few representa-
tive interferometri13 and spe13tros13opi13 re13ords The average sho13k speed and
ele13tron density have been estimated from the interferograms The sho13k
speeds of the MAIN and AUX radiative sho13k waves vary between sim 30-55
and 10-30 kms respe13tively and the averaged pre13ursor ele13tron density
ranges between 10
17and 10
1913m
minus3during the sho13ks propagation
We have demonstrated the intera13tion ee13t between the two radiative
pre13ursors in the 13ase of Xe at 01 bar at 54 and 23 kms The intera13tion is
13learly 13hara13terized in the experiment by the enhan13ement of the ionisation
wave followed by the merging of the two radiative pre13ursors at 20 ns The
13ollision time is re13orded at 47 ns Su13h behavior is reprodu13ed by the
98 CHAPTER 5 RESULTS AND INTERPRETATION
(a)
(b)
(13)
Figure 518 Time evolution of the mass density (a) ele13tron temperature (b) and
mean 13harge (13) at 56 57 58 60 64 and 65 ns within the sho13k tube derived from
the HELIOS simulations (with Xenon opa13ity multiplier = 20) for two 13ounter
streaming sho13ks of sim 39 and 18 kms in Xenon at 06 bar
54 SUMMARY 99
simulation
We have investigated this intera13tion at a larger pressure 02 bar with
the following speeds sim 41 kms for the MAIN and sim 18 kms for the AUX
sho13k waves We do not re13ord any experimental signature of the radiative
pre13ursor for AUX Further we have not been able to 13at13h experimentally
the 13ollision time The re13orded pre13ursor of MAIN is not inuen13ed by AUX
wave up to 48 ns (Fig 59(b) and (13) in data analysis 13hapter) whi13h is the
limit of the re13ord On its side the 1D simulation predi13ts a tiny pre13ursor
for AUX and that both pre13ursors start to intera13t at 49 ns This plausible
sho13ks intera13tion o1313urring at times whi13h are outside of the re13ord 13an
not be 13onrmed by our experiment
The results of the transverse interferometry at 02 bar with speeds of sim40 and 20 kms indi13ate that the MAIN pre13ursor has a lateral extension of
sim 600 microm whereas it is 300 microm for AUX The pre13ursor of MAIN is almost
at with a probable small bending at the edges of the tube whereas the
AUX pre13ursor is more 13urved This means that the 2D ee13ts are more
important for AUX than for MAIN Still in the 13ase of Xenon we have
obtained information about the temperature and the mean 13harge of gas
through our time integrated XUV spe13tra (shot48143) in Xenon at 06
bar From these results we may 13on13lude that the mean ion 13harge is at
least equal to 6 and that the temperature has rea13hed 15 eV
Our simulations give a qualitative des13ription of the sho13ks intera13tion
when the laser uen13e is adjusted to give the 13orre13t sho13k velo13ities Howe-
ver it is now well known that 2D simulations (together with state of the art
opa13ities) t better with experiments (Gonzaacutelez Audit and Stehleacute 2009
Leygna13 et al 2006 Stehleacute et al 2010) For the same laser energy the 2D
simulations lead to a diminution of the sho13k speed 13ompared to 1D as also
to a diminution of the ele13tron density For instan13e in the 13ase of a sho13k
wave laun13hed by a laser beam at 1315 nm in Xenon at 03 bar at PALS and
with a laser uen13e of 85000 J13m
2 ARWEN 2D simulations give a sho13k
speed of 44 kms in agreement with the experimental one (Cotelo et al
2015) 1D simulation would require in this 13ase a uen13e of 30000 J13m
2to
a13hieve the same velo13ity
The spa13e and time integrated XUV re13ords at 06 bar for respe13tive
speeds whi13h are equal to sim 39 and 18 kms indi13ate that the temperature
of the sho13k has rea13hed values up to 15 eV and that the Xenon mean ion
13harge has also rea13hed values of 6 - 7 whereas 1D simulations predi13t ele13tron
temperature 10-30 eV and ion 13harge 5-10 (Fig 51813) A more detailed
study based on 2D simulation and radiative transfer post-pro13essing will be
ne13essary to rene the analysis
We have investigated the 13ase of other noble gases (Ar Kr) and we have
observed that for a given laser energy the sho13k velo13ity de13reases with the
mass density This study has 13onrmed that for a given density and sho13k
velo13ity the radiative ee13ts in13rease with the atomi13 number For Kr we
100 CHAPTER 5 RESULTS AND INTERPRETATION
noti13ed a tiny pre13ursor without any intera13tion To get su13h ee13t it will
be ne13essary to in13rease the sho13k velo13ity and thus the laser energy
This has been done at the ORION laser fa13ility in UK where the 13ol-
lision of two 13ounter-propagating sho13ks at equal speed sim 80 kms with
laser uen13e sim 6 times 10
14W13m
2has been performed (Clayson et al 2017
Suzuki-Vidal et al 2016) for dierent noble gases and for pressures 13ompri-
sed between 01 and 1 bar A number of diagnosti13s setup X-ray ba13klig-
hting opti13al self-emission streak imaging and interferometry (multi-frame
and streak imaging) were used to study both the post-sho13k and the radia-
tive pre13ursor Although I have not parti13ipated in the experiments I have
performed 1D simulations to interpret ORION experimental results This
work is not presented in this thesis
Chapter 6
Optimization of an
ele13tromagneti13 generator for
strong sho13ks in low pressure
gas
Contents
61 Prin13iples of operation of a high 13urrent generator 102
62 Prin13iples of the run-down phase in a PFD 103
63 Proposed design for the plasma gun 104
64 Dynami13 13ir13uit modelling 106
65 3-D MHD simulations using GORGON 13ode 111
66 Measurements 116
67 Summary 118
The previous 13hapters were dedi13ated to the study of laser generated
radiative sho13ks With irradian13e more than 10
14W13m
2 we were able to
study sho13k waves propagating at velo13ity up to 50 kms in noble gases with
an initial mass density 13omprised between 5 times 10
minus4- 3 times 10
minus3g13m
minus3
Complementary to laser experiments 13ompa13t pulsed power generators may
drive an ele13tromagneti13 13oaxial plasma gun to 13reate astrophysi13al relevant
sho13ks in lower pressure noble gases (Kondo et al 2006) with a high avai-
lability and a rather modest 13apital 13ost The ele13tromagneti13ally driven
sho13k waves may have larger s13ales than those by laser thus they 13an be
observed rather easily (Kondo et al 2008)
Su13h ele13tri13al pulsed power devi13es may then i) 13reate a high-voltage
breakdown through a gas or more easily at the surfa13e of a diele13tri13 ii)
produ13e relatively hot plasma by ohmi13 heating when the rising 13urrent in the
devi13e is passing through a portion of gas iii) a1313elerate plasma layers under
the magneti13 pressure asso13iated with the self-generated magneti13 eld when
high ele13tri13al 13urrent (say 10s of kA) is maintained for a short but su13ient
time (say 1 micros) Su13h ionization and a1313eleration are present in Z-pin13h
plasmas and in plasma fo13us devi13es (PFD) providing a 13lever geometry is
employed and the mass under 13onsideration (say 1 mg) is 13ompatible with
the stored energy (Martin Williams and Kristiansen 1996)
101
102 CHAPTER 6 ELECTROMAGNETICALLY LAUNCHED SHOCK
In this 13hapter we will see rst how a fast ele13tri13al 13ir13uit works then
we will dis13uss the prin13iples of a 13oaxial plasma gun and of PFD We will
see what is in favor of our obje13tive and what has to be avoided A geometry
13apable to rea13h our obje13tive will be proposed
The optimization of the 13oaxial gun (in term of plasma speed) will be
performed and some typi13al gures will be given in the 13ase of a generator
with a stored energy around 1 kJ and a plasma slab of 4-mm diameter ie
quite 10 times the transverse dimension of a laser driven RS The operating
13onditions are supersoni13 sho13ks up to 10-30 kms speed at stati13 pressures
of few mbar in heavy rare gases (Ar Xe) The results of this 0-D model will
be then 13ompared to those obtained with 3-D MHD simulations performed
with the 13ode GORGON (by Andrea Ciardi at LERMA) whi13h has been
used su1313essfully to des13ribe other pulse-power driven plasma experiments
(Chittenden et al 2004) as well as laboratory plasma astrophysi13s experi-
ments (Ciardi et al 2007) The diagnosti13s whi13h have been implemented
will be presented to illustrate the model as well as preliminary re13ords of the
plasma speed
61 Prin13iples of operation of a high 13urrent gene-
rator
Our aim being to a1313elerate a plasma slab using the magneti13 pressure it
is obviously needed to drive a high intensity 13urrent be13ause the magneti13
pressure is expressed by B
22micro0 in the region where exists an indu13tion B
The pressure in bars is simply 4B
2 with B in Teslas and in a 13ylindri13al
geometry 1 Tesla is the eld around a 50-kA 13urrent at a radius of 1 13m
Thus we should 13ount on roughly 100 kA delivered by the generator but
su13h a high 13urrent is delivered only in a pulsed mode by a laboratory s13ale
devi13e
There are numbers of te13hni13al solutions to a13hieve su13h ele13tri13al pa-
rameters one may use rather slow generators at moderate high voltage like
13apa13itors bank with the advantage of well know te13hniques but with limited
adjustments A1313ording to an abundant literature
1
a 13onvenient devi13e is a
medium-energy 13apa13itor bank feeding a low-indu13tan13e 13ir13uit An R-L-C
13ir13uit is a well-known 13ombination delivering a high peak 13urrent in the
pseudo-periodi13 mode For a 13apa13itor C initially 13harged under U0 key
performan13es are as follows
Current intensity I(t) is a damped sinusoid
Pseudo pulsation ω =radic[1(LC)minusR2(4L2)]
1
Institute for Plasma Fo13us Studies resour13e website httpplasmafo13usnet last
13onne13tion in 2016
62 PRINCIPLES OF THE RUN-DOWN PHASE IN A PFD 103
First 13urrent peak Ipeak = U0[radic(LC) + 08R]
Time of 13urrent peaking T4 = π(2ω)
Equivalent impedan13e Z =radic(LC)
For a safe handling in air a voltage not higher than 30 kV is re13ommen-
ded Taking a total 13apa13itan13e of 6 microF 2700 J are stored under 30 kV For
an indu13tan13e of 240 nH the expe13ted peak 13urrent might be 150 kA at 19
micros This set of values will be a guideline for the development presented in
further se13tions
62 Prin13iples of the run-down phase in a PFD
Our aim is thus to 13reate a fast moving plasma sheath with quite a one-
dimension geometry The issues are to initiate this plasma at the best then
to maintain even improve its stru13ture during the a1313eleration nally to
laun13h it
Instead of 13reating a gas breakdown in volume whi13h would lead ra-
pidly to 13on13entrate the 13urrent in an ar13 the idea for PFD operation is to
start from a surfa13e dis13harge (Lee 1969 Bernard 2002) In the 13lassi13al
13oaxial geometry of the Mather-type PFD the 13entral ele13trode (usually the
anode) is a 13ut metal 13ylinder and the outer one is a se13ond metal 13ylin-
der of the same length or better a squirrel 13age allowing many viewpoints
to the plasma Both ele13trodes are atta13hed to a diele13tri13 bottom plate
made of polymer or 13erami13 When the 13apa13itor bank is swit13hed on the
high voltage is applied to this 13m-size gap and a radial surfa13e dis13harge is
initiated at the interfa13e with the gas forming a quite uniform ring-shaped
layer of thi13kness around 1 mm An insulating sleeve is adjusted around
the 13entral ele13trode to for13e the plasma to ow rapidly upwards under the
magneti13 pressure in this region The result is an elongation the dis13harge
path preventing an ar13ing in the plasma layer
At later times as the 13urrent is growing the plasma layer is strongly
inuen13ed it is pushed upwards by the magneti13 pressure jtimesB it is heatedby the ohmi13 ee13t then its ionization degree in13reases it be13omes denser
as a per13entage (10-40) of the heavy parti13les from the swept volume are
a1313reted (Potter 1971) This ele13tromagneti13ally driven motion is 13alled the
rundown phase Due to the a1313retion it is des13ribed by a so-13alled snowplow
model whi13h will be dis13ussed later in detail In quite all the designs even
if very high speeds are reported (Lee 1969 Serban 1995) the plasma is bell
shaped whi13h does not fulll our requirements
At the end of the rundown phase whi13h mat13hes roughly with the 13urrent
peak the plasma stays 13onne13ting the upper ends of the ele13trodes and the
radial 13omponent of the magneti13 for13es grows rapidly Then the plasma is
104 CHAPTER 6 ELECTROMAGNETICALLY LAUNCHED SHOCK
strongly pushed to the axis giving the so-13alled Z-pin13h This stage made
the PFD popular be13ause the pin13hed plasma is a sour13e of fast ele13trons
fast ions hard radiation and possibly neutron beams it must be avoided
in our 13ase It was also quoted (Lee and Serban 1996) that optimal PFD
are mat13hing a universal fa13tor 13omprising anode radius peak 13urrent and
gas density meaning a robustness of the design when a geometry has been
13hosen
63 Proposed design for the plasma gun
The previous des13ription has shown the positive inuen13e of a surfa13e dis-
13harge the modi13ation of the shape by the sleeve and the issue of the
pin13hing ee13t A1313ordingly the following design is proposed following a
work by Kondo et al (Kondo et al 2006 2008)
To ensure a rapid dis13harge with the values given in the se13tion 61 a
highly 13oaxial 13ir13uit is 13hosen for all the large parts the energy bank the
13losing swit13h and all the 13onne13tions The initial phase is kept with two
13oaxial ele13trodes pressed on a at insulator Homogeneity of the plasma
sheath is expe13ted by 13hoosing a rather small radial gap of 25 mm The ee13t
of magneti13 for13es during the rundown phase must be enhan13ed espe13ially
due to the in13rease of the mass of the plasma as des13ribed above As the
13urrent 13urve is the rising part of a sinusoid a tri13k is used to in13rease
the lo13al magneti13 eld B around a 13ondu13tor varies as the inverse of the
distan13e to axis so the proposed shape for the anode is a 13oneHowever a
nal divergen13e must be avoided and that is obtained by a rounded tip In
order to keep 13onstant the plasma length along the 13urrent path the radial
gap is kept 13onstant so the outer ele13trode is a hollow 13one with the same
angle Above the 13one the a1313elerator is 13onne13ted to a 13ondu13ting tube
where the plasma 13an propagate freely in the ba13kground gas The distan13e
of the plasma sheath to the 13one tip is in13reasing rapidly and the main
13urrent whi13h is still high will pass preferably through the diuse plasma
remaining between the ele13trodes Thus there will be no magneti13 pressure
anymore a13ting on the sheath whi13h 13an propagate freely The resulting
devi13e is des13ribed in Fig 61a and the a13ting magneti13 for13es shown in Fig
61b
Te13hni13ally su13h a 13onguration is 13onvenient even at 15 kV be13ause the
sharp edge of the anode tou13hing the insulator forms a so-13alled triple point
where the ele13tri13 eld is enhan13ed at the surfa13e of the insulator whi13h
triggers e13iently a dis13harge in presen13e of gas at low pressure (01 10
mbar) as proposed by Kondo et al (Kondo et al 2006) Polya13etal is a
13onvenient material for diele13tri13 and me13hani13al performan13es
The dire13tion and the orientation of the magneti13 for13e are other points to
be dis13ussed The high-intensity 13urrent (up to sim 150 kA) generates a strong
63 PROPOSED DESIGN FOR THE PLASMA GUN 105
(a)
(b)
Figure 61 (a) Sket13h of the sho13k generator showing the pulsed ele13tri13al 13ir13uit
the set of 13oaxial 13oni13al ele13trodes with a 13onstant radial gap and the plasti13
insulator featured in grey on whi13h a planar surfa13e dis13harge is initiated The
installation of three opti13al bers allows looking radially at the plasma moving in
the sho13k tube (b) S13hemati13s of the plasma dynami13s inside the 13oaxial gun
in fast-pulse mode the ele13tri13al 13urrent ows in the super13ial layers of the two
13oaxial 13oni13al ele13trodes and through an annular plasma layer The magneti13
pressure Pmag pushes the dis13harge axially
106 CHAPTER 6 ELECTROMAGNETICALLY LAUNCHED SHOCK
Figure 62 Exploded view of the plasma gun
azimuthal magneti13 eld between the ele13trodes thus the magneti13 pressure
jtimesB whi13h a1313elerates the annular plasma sheath stays perpendi13ular to the
sheath lo13ally (Fig 61b) The more planar is the initial sheath the more
axial will be the magneti13 pressure Finally if the roles of the ele13trodes are
ex13hanged the dire13tion of the 13urrent will be inverted and the same for B
so the pressure will stay oriented in order to expel the plasma Compared
with the Mather-type plasma fo13us (Potter 1971 Zambra et al 2009) the
plasma sheath 13onsidered here is quite planar 13lose to the insulating surfa13e
and later is expe13ted to stay planar and perpendi13ular to the axis As in a
PFD the thi13kness of the plasma sheath in13reases gradually (Zambra et al
2009) but it is assumed to stay in the mm-range We employ 13oaxial 13oni13al
brass ele13trodes ea13h 42-mm high with a 13onstant gap of 25 mm At the
bottom level internal and external radii of ele13trodes are 125 mm and 15
mm respe13tively This small-size dis13harge 13ell on top of the generator will
be modelled in the following
64 Dynami13 13ir13uit modelling
We optimize the ele13tri13 generator for various gases namely Ar and Xe
with the motivation to produ13e plasma sho13ks with speeds sim 1 - 30 kms
ie Ma13h numbers up to 200 For that we design the ele13trodes and set
64 DYNAMIC CIRCUIT MODELLING 107
the ele13tri13al 13ir13uit parameters to produ13e 1-micros pulses in the gas 13hamber
The geometry of the ele13trodes is as mentioned above The half angle of the
13oni13al ele13trode is 13hosen to be α =15deg and thus the height of the 13one is h
sim 42 mm
Eleven (N = 11) 13apa13itors ea13h with 13apa13itan13e C0 of 06 microF are13onne13ted in parallel giving an equivalent 13apa13itan13e of the bank C = 66
microF 13harging voltage U0 of the bank is set to 15 kV Thus the available
ele13tri13al energy is 750 J The 13ir13uit resistan13e is used to be 1 m Ω and
damping resistan13e is set to zero whi13h yields a total resistan13e (R = Rcir+
Rdamp N ) of 1 mΩ and a pseudo-periodi13 regime is expe13ted
Based on the above geometri13al and ele13tri13al parameters we have esti-
mated the time evolution of various parameters of the radial dis13harge layer
namely speed mass a1313eleration et13 at a dierent position along the axis
from z = 0 to h For this 13al13ulation we need to solve a set of three equa-
tions whi13h give the main 13urrent (- dqdt) supposed to ow only through
the plasma layer the mass a1313retion rate (dMdt) and the velo13ity v of
the plasma These equations together with initial 13onditions are explained
below The ele13tri13al equation is written 13lassi13ally
Ld2qdt2 +Rdqdt+ qC = 0 (61)
When the R L and C parameters are not time-dependent the 13ir13uit
equation 61 has an analyti13al solution a damped sinusoid for the 13urrent I
= -dqdt and that was used as a test of the iterative solving pro13ess and to
13onrm the parameters of the external 13ir13uit with a short 13ir13uit load The
indu13tan13e of the external 13ir13uit L0 is 13omputed from the ringing frequen13y
in short 13ir13uit (66 nH) whi13h results in a 13ir13uit impedan13e Z0 =
radic
L0Cto be 01 Ω In the plasma gun mode the sheath 13arrying the 13urrent (see
Fig 61b) is a1313elerated by the magneti13 pressure and it behaves as the de-
formable part of the 13ir13uit then L is time-dependent and a numeri13al solver
is ne13essary for equation 61 We used a lumped-parameter model whi13h has
proven to des13ribe similar 13ir13uits su1313essfully (Potter 1971 Zambra et al
2009) Gonzalez (Gonzalez et al 2004) showed that the mass and momen-
tum equations for a 13urrent sheath in the shape of an annular piston moving
forward in the axial dire13tion 13an be given by
dMdt = ερ0π(R2e minusR2
i)v (62)
where ε is a1313retion fa13tor and that the transformation of magneti13 to
kineti13 energy is expressed by
d(Mv)dt = lI22 (63)
where l is the linear indu13tan13e of the gun and 13an be expressed as l =
(micro04π) ln (ReRi) the radii being estimated where the plasma sheath is
13urrently lo13ated The equations 62 and 63 have been introdu13ed in the
108 CHAPTER 6 ELECTROMAGNETICALLY LAUNCHED SHOCK
(a)
(b)
(13)
Figure 63 Time dependen13e of the main parameters for Ar gas at 1 mbar a1313or-
ding to the 13ir13uit model (a) 13urrent (kA) and voltage (V) (b) a1313reted mass
(kg) and rate of a1313retion (kgs) (13) speed (ms) and kineti13 energy (J)
64 DYNAMIC CIRCUIT MODELLING 109
Figure 64 Plot of the dierent output parameters (normalized with respe13t to
the peak values) obtained at the top of the inner ele13trode For ea13h group upper
panel represents the variation for Ar gas whereas lower panel is for Xe gas Proles
of output parameters are given (a) with 13one angle (b) with a1313retion fa13tor (13)
with gas pressure and (d) with damping resistan13e
110 CHAPTER 6 ELECTROMAGNETICALLY LAUNCHED SHOCK
time-dependent iterative solving of equation 61 Therefore from equations
62 and 63 we get
dvdt =[ lI
2
2 minus ερ0π(R2e minusR2
i)v2]
M(64)
From equation 64 we may note that initial value of mass M(t0) 13annot
be set to zero as it would lead to a nonphysi13al divergen13e of the a1313eleration
In this regard we have 13onsidered an initial thi13kness of the 13urrent sheath
of 1 mm and integrated the gas density at rest leading to M(t0) equal to
35 times 10
minus10kg for Ar and 13 times 10
minus9kg for Xe A higher limit has to be
put too M(t0) must be small 13ompared with the mass at the end of the
laun13hing phase that is the mass 13ontained in the gap times the a1313retion
fa13tor The above value for M(t0) follows that 13riterion and within these
limits the 13hoi13e of initial mass is not inuen13ing the results
Parameter Referen13e Value Variation
Half angle of the 13one 15
13
14
15
16
middot middot middot 24A1313retion fa13tor ǫ 04 01 02 03 middot middot middot 09
Pressure P 100 Pa 10 20 50 100 200 500 1000
Damping resistan13e Rdamp 0 ohm 0 01 02
Table 61 Input parameters their standard values and range of variation
As the motion is mainly along the z-axis the speed v is the derivative
of the plasma position v = dzdt We solved these equations by a time-
dependent method where the initial 13ondition is that at the onset time t0
the 13harge q is set to CU0 Further the 13urrent i = -dqdt is taken to be zero
at t0 Thus using equation 61 the initial rate of 13hange of the 13urrent is
d
2qdt
2= - qL0C Then all the rst time-derivatives are updated at the rst
time step t0+∆t with ∆t=043 ns using equations 61-64 The quantities
like q M v z are updated at the next step then the iteration is pursued
Initially position a1313eleration and velo13ity of the sheath are set to be zero
as well as the rate of 13hange of the mass
As we initiate the ele13tri13al 13urrent into the gas inside the 13hamber the
plasma sheath is a1313elerated with respe13t to its initial mass as seen above
At later times the mass a1313retion is masking the inuen13e of this arbitrary
value We have 13al13ulated the plasma evolution in the dis13harge for dierent
sets of initial 13onditions whi13h are reported in Table 61
Fig 63 shows the behavior for the standard values reported in Table 61
in the 13ase of Argon at 1 mbar The 13orresponding values of 13harge mass
and velo13ity will be hereafter 13alled the referen13e values In Fig 63a the
variation of 13urrent and voltage in the rst half-period is 13lassi13al showing
that the pulse generator departs slightly from a pure RLC 13ir13uit Fig
63b represents the a1313reted mass whi13h in13reases in time as well whereas
65 3-D MHD SIMULATIONS USING GORGON CODE 111
the rate of mass a1313retion 13ulminates with a 13ompetition of speed growth
and de13rease of the swept volume per time step be13ause it depends on the
squared radii In Fig 6313 plotted are the speed and kineti13 energy of the
plasma sheath First the speed of the plasma is in13reasing slowly with time
quite linearly as the mass and the kineti13 energy is roughly a third power
of the time
That leads to 13hoose a design where the a1313eleration is imposed for a
su13iently long time However the estimated speed of plasma is sim 140 kms
whi13h is a very optimisti13 value The reason may 13ome from two reasons
(i) a part of the 13urrent ows through the large sho13ked volume redu13ing
the magneti13 pressure a13ting on the plasma sheath (ii) energy transfer from
magneti13 to thermal energy and ionization is redu13ing the kineti13 one The
observation of a very bright emission from the plasma will be a 13lear eviden13e
of this redistribution of the energy
Further we made a 13omparative study of the parameters indi13ated in
Table 61 and applied to two gases namely Ar and Xe In order to 13he13k the
ee13ts of the initial 13onditions (13one angle a1313retion fa13tor pressure and
damping resistan13e) we have performed a systemati13 study by varying only
one parameter and keeping the others 13onstant and equal to the referen13e
values given by the se13ond 13olumn of Table 61 The tested range is listed
in the third 13olumn of Table 61 Similarly for ea13h set the variation of the
plasma 13onditions at the top of the muzzle is normalized to the referen13e
value Fig 64 summarizes the variation of the sele13ted dierent output
parameters the time tm to rea13h the muzzle (z = h) the ele13tri13al 13urrent
and the axial speed at muzzle the laun13hed mass M the kineti13 energy and
the average speed at the top of the ele13trode 13al13ulated from the initial
point v=h( tm - t0)
Table 62 summarizes the respe13tive ratios for Ar and Xe at the top of
the 13oaxial ele13trodes As the values of interest are obtained at 13onstant
stored energy in the 13apa13itors there is a benet to work with a lighter gas
like Argon However the results obtained using this 13ir13uit model show that
the referen13e values are ensuring a satisfa13tory behavior for both gases
Speed Kineti13 energy Time to laun13h Mass
222 151 66 31
Table 62 Laun13hing performan13e ArXe
65 3-D MHD simulations using GORGON 13ode
Our simplied model (13f se13tion 64) is 13onvenient to perform qui13kly some
parametri13 tests in the a1313eleration phase and to help designing the devi13e
but it is not 13apable to treat the sho13k dynami13s in detail nor to give valuable
112 CHAPTER 6 ELECTROMAGNETICALLY LAUNCHED SHOCK
information on the plasma parameters in the sho13k region So rened nume-
ri13al simulations of the experiments were performed in 13ollaboration with A
Ciardi (Ciardi 2014) using the GORGON 13ode (Chittenden et al 2004 Ci-
ardi et al 2007 Suzuki-Vidal et al 2014) GORGON is an expli13it parallel
13ode designed to solve the resistive MHD equations on a three-dimensional
(3-D) Cartesian grid employing a Van Leer type algorithm The 13ode treats
the plasma as a single uid but solves separately the energy equations for
ions and ele13trons allowing dierent temperatures for the two spe13ies Both
thermal 13ondu13tion and resistive diusion are treated using Braginskii-like
transport 13oe13ients
Variables At rest Sho13k Post-sho13k
Ar density ρ (g13m
3) 18E-6 25E-5 10E-5
Ele13tron density Ne (13m3) 10 E+15 20E+18 15E+18
Ele13tron Temperature Te (eV) 003 (1lowast) 10 15-18
Ion density ni (13m3) 30E+16 35E+17 2-3E+7
Ion Temperature Ti 0 50 15-20
Average Speed ltVgt kms 0 50 20-40
ltzgt 02 6 7-9
Table 63 Values of variables in various 13onditions at rest (with seed ele13trons)
inside the sho13k and inside post-sho13k region A star lowast denotes the insulator surfa13e
This 13ode des13ribes the 13reation of the surfa13e dis13harge then its a1313ele-
ration by the magneti13 pressure It provides the mapping of all the plasma
parameters and lo13al B-eld in the laun13hing 13one as well as in the free ight
region The 3-D simulation was performed for Argon gas at 1 mbar only the
dis13harge 13urrent law being taken from experiment The Table 63 presents
the ranges of all the variables in the regions of interest
As an example Fig 65 shows an enlarged mapping of log (ne) at two
times in the laun13hing phase then when the sho13k has deeply penetrated
in the sho13k tube Taking the zero time as a referen13e the averaged speed
zt is 40 kms in this simulation When the two snapshots are 13onsidered
the estimate of the instantaneous speed zt is higher at 56 kms The
time history of plasma merging and early free ight into the sho13k tube
is presented in Fig 66 During the a1313eleration (inside the double 13one)
the plasma is well lo13alized on a planar annular sheath whi13h is 13arrying
the 13urrent as assumed in the lumped parameter model We see that the
merging is well a13hieved at the muzzle shortly before 1250 ns When the
sho13k has penetrated inside the tube (1500 ns) the 13omputed B-eld is
negligible at this lo13ation and the propagation is quite free as expe13ted for
the 13hosen design of the devi13e However a high 13urrent is still passing
through the gas at that time a su13ient ele13tron density remaining in the
13oni13al gap for that
65 3-D MHD SIMULATIONS USING GORGON CODE 113
Figure 65 Example of mapping of the ele13tron density from a 3-D MHD simu-
lation (ba13kground gas Argon at 1 mbar) when the plasma sheath is (left) at
the exit of the plasma gun (right) in free ight 13onditions (log s13ales for the false
13olors)
114 CHAPTER 6 ELECTROMAGNETICALLY LAUNCHED SHOCK
Figure 66 Time history of plasma merging and early free ight through the
mapping of the ele13tron density ele13tron temperature and average ioni13 13harge as
given by a 3-D MHD simulation (ba13kground gas Argon at 1 mbar) (s13ales for the
false 13olors)
65 3-D MHD SIMULATIONS USING GORGON CODE 115
Figure 67 Axial proles values taken in the sho13k tube along a line slightly o-
axis at a distan13e of 1 mm for mass density ion temperature ion density average
ion 13harge magneti13 eld ele13tron temperature ele13tron density and average speed
at 1500 ns (ba13kground gas Argon at 1 mbar) A maping of |B| is given with a
dashed line indi13ating sho13k front position as well
116 CHAPTER 6 ELECTROMAGNETICALLY LAUNCHED SHOCK
To 13onrm the steep stru13ture of the sho13k Fig 67 presents the axial
proles of ele13tron temperature Te number density Ne mass density ρ and
average ion 13harge ltZgt at 1500 ns inside the tube with an enlarged axial
position s13ale Steep fronts are observed for mass density ele13tron density
and temperature as in the earliest simulations (Potter 1971 Kondo et al
2008) The sho13k region is at 10 eV and sim2times1018 ele13trons13m
minus3 the post-
sho13k region is hotter Argon ionization stages of 5 (sho13k) to 9 (post-sho13k)
are obtained fore13asting an emission spe13trum in the UV or harder A wider
view of the plasma is given by other quantities at the sho13k front and behind
ion temperature average speed and lo13al magneti13 eld in Fig 67 We get
the 13onrmation of the extin13tion of B after the rst 5 mm inside the sho13k
tube as well as the sho13k speed at 50 kms 13oherent with the 13onstant speed
of 56 kms mentioned just above At present these parameters are guiding
the 13hoi13e of diagnosti13s like ultra-fast interferometry or UV-spe13tros13opy
66 Measurements
The aim of these tests was to show the 13onsisten13e of the above model by me-
asuring the plasma speed 13lose to the plasma gun muzzle The parameters of
the devi13e are the referen13e ones as dened in Table 61 A rather simple and
noise-free method for diagnosing a plasma gun during the rundown phase
is to re13ord the light emitted by the plasma sheath with a spatial resolution
13ompatible with the plasma stru13ture If we 13onsider a line of sight whi13h
is radial or tangential ie perpendi13ular to the dire13tion of propagation
one expe13ts to see a sudden rise when the plasma enters the dete13tion vo-
lume possibly a plateau when the plasma travels inside the volume then a
slow de13rease when the hotter plasma leaves the volume but when the sho13k
remnants are still present So the expe13ted signal is a triangular asymme-
tri13 pulse (Serban and Lee 1995) or some kind of double exponential pulse
(Stehleacute et al 2012) A peak indi13ates the time of ight for the given posi-
tion allowing to 13ompute a mean speed providing the laun13hing time (t=0)
and all positions are known An instant speed is a1313essible providing two
13lose dete13tors are 13onsidered That has been proposed and tested su1313ess-
fully in the 13ase of radiative sho13ks driven by one PALS laser beam (Stehleacute
et al 2012) the dete13tors being 13ollimated and ltered sili13on diodes For
13ooler plasmas the visible emission is more relevant Inside a squirrel 13age
plasma fo13us Serban has installed a set of opti13al ber looking tangentially
at the drift zone (Serban and Lee 1995) Despite a 13oni13al eld of view the
results were 13on13lusive at speeds as high as 100 kms in D2 An improved
version with 13ollimated opti13al bers (a1313eptan13e angle 2deg) was developed
too (Serban and Lee 1995) A non intrusive method was developed (Veloso
et al 2012ba) by imaging the plasma on a set of opti13al bers allowing to
play with the magni13ation
66 MEASUREMENTS 117
(a)
(b)
Figure 68 Experimental results (a) time history of the passive opti13al re13ords
13ompared to the main 13urrent for Argon gas 13lose to the referen13e pressure (b)
time-integrated signal vs Ar pressure for the dierent bers (13) averaged speed vs
Ar ba13king pressure
118 CHAPTER 6 ELECTROMAGNETICALLY LAUNCHED SHOCK
In our 13ase (see Fig 61a) three 1-mm PMMA (Poly(methyl metha13ry-
late)) opti13al bers are installed radially along a transparent se13tion of the
sho13k tube at z sim 625 mm 70 mm and 775 mm and fast DET10A dete13tors
re13ord the transient presen13e of the plasma sheath in the respe13tive lines of
sight The re13orded spe13tral range is limited to the whole visible in a preli-
minary stage a band-pass ltering might give information on the presen13e of
spe13i13 ioni13 spe13ies 13hara13teristi13 of a plasma temperature level however
the amount of light was not su13ient to do so Fig 68a presents the time
evolution of the ele13tri13al 13urrent with a period of 4 micros and the delayed
side-on observational data of the moving plasma inside the sho13k tube The
behavior of the ele13tri13al 13ir13uit is following well the expe13ted ringing of a
weakly damped RLC 13ir13uit the zero-time being xed at the onset of the
13urrent On opti13al 13hannels a steep rise and a peak are the signature of a
mm-thi13k fast-sho13k stru13ture (Serban and Lee 1995) 13onsistent with early
numeri13al simulations (Potter 1971 Tou Lee and Kwek 1989 Kwek Tou
and Lee 1990 Veloso et al 2012b) On ea13h ber signal se13ondary peaks
namely the one delayed by 4-5 micros are attributed to 13urrent restrikes at the
pseudoperiod To avoid the mixing of sho13ks further experiments should be
13ondu13ted in an aperiodi13 regime with a heavy-duty damping resistor added
in series to ea13h 13apa13itor Rdamp sim NZ
On Fig 68b and 13 are presented the exploitation of opti13al re13ords with
the Ar ba13kground pressure this parameter varying in the widest range (b)
the integrated emission whi13h presents a maximum between low pressure side
when the temperature is high but the number of emitters is low and the high
pressure side when the temperature is mu13h lower (13) the averaged speed
ziti for the rst peak on ea13h ber The relative intensities re13orded in the
three lo13ation s are not signi13ant be13ause of a strong 13opper deposition
13loser to the muzzle whi13h ae13ted the dete13tion The long-life noise on
the opti13al signals was making the lo13al speed 13al13ulated as (zi-zj)(ti-tj) a
non-reliable output One 13an note Ma13h numbers in the range 20 60 with
a pressure dependen13e similar to previous work (Kondo et al 2006) The
reported observations are 13onsistent with a high-speed millimetri13 planar
plasma travelling in the sho13k tube The speed is lower than dedu13ed from
the simplied model possibly due to the la13k of dissipation terms whi13h
is overestimating the transfer to kineti13 energy At the 13ontrary a realisti13
simulation as the one performed using GORGON is in better agreement
67 Summary
For this part of the work the obje13tive was to show that in 13omplement of
laser experiments a 13ompa13t pulsed power generator might drive astrophy-
si13ally relevant sho13ks in low pressure noble gases with a higher availability
and at a rather modest 13apital 13ost The longitudinal a1313eleration of a
67 SUMMARY 119
plasma sheath in a low ba13king gas pressure has been demonstrated in 13o-
axial plasma guns as early as in the 70s even leading to speeds of 100 kms
in light gas as H2 whi13h were measured and modeled A 13hoi13e was made
to revisit this type of devi13e At present the ele13tromagneti13 13oaxial plasma
guns are quite only 13onsidered as the initiators of a strong radial plasma
implosion plasma fo13us devi13es (PFD) due to a good ability to produ13e a
pin13h plasma and emission of energeti13 parti13les and hard X-ray For that
reason it was ne13essary to adapt the prin13iple of a PFD with two major
13onstraints in13rease the 13onversion of stored energy into the kineti13 energy
of a plasma sheath prevent totally any pin13hing
That obje13tive was rea13hed by dening the ele13tri13al and geometri13al
parameters of a low-indu13tan13e and 13ompa13t pulsed power generator An
optimization pro13ess has been established to mat13h a13hievable ele13tri13al pa-
rameters of the 13ir13uit (13apa13itan13e indu13tan13e peaking 13urrent intensity)
with a plasma motion in the a1313elerating tube over few 13m leading to a nal
speed of 10-30 kms A model was built up to des13ribe the dynami13s of an
RLC 13ir13uit with a varying element 13omposed of the moving plasma sheath
a1313elerated by the self generated magneti13 pressure The 13hange in plasma
position along the axis is in13reasing the indu13tan13e and the mass a1313retion
is in13reasing the plasma mass It was shown that a 13hange in the ele13trode
geometry namely a 30deg 13oni13al shape was in13reasing the energy 13onversion
and preventing any fo13using The further step in 13ontrast with PFD is the
merging of the plasma ring into a rather planar one whi13h is inje13ted in a
drift tube
Considering Ar or Xe at gas pressure in the range 01 10 mbar and
a highly 13oaxial 1-kJ devi13e the simple model was su13ient to predi13t
performan13es agreeing with the only work quoted in the literature Basi13
observations of the moving plasma using side-on dete13tion of the visible
emission give features 13oherent with the model Ma13h numbers from 20
to 60 are obtained tting with the obje13tive of strong sho13k formation A
rened 3-D MHD numeri13al simulation indi13ates very promising features of
the sho13k in view of exploring mm-size sho13ks in a regime 13omplementary
to laser driven ones and on a table-top devi13e allowing a higher repetition
rate
120 CHAPTER 6 ELECTROMAGNETICALLY LAUNCHED SHOCK
Chapter 7
Con13lusions and perspe13tives
71 Con13lusions
Although ubiquitous in the astrophysi13al environments radiative sho13ks are
13omplex phenomena whi13h still require an important eort to understand
them Beside observations whi13h suer from a la13k of angular resolution
laboratory experiments provide today an interesting approa13h to improve
our present knowledge In this 13ontext my thesis work is dedi13ated to the
experimental and numeri13al study of the 13hara13teristi13s of two 13ounter pro-
pagating radiative sho13ks propagating at dierent velo13ities (20-50 kms)
Most of the work presented here is the out13ome of the rst-of-its-kind ex-
perimental 13ampaign held in year 2015 at Prague and Laser Asterix system
(PALS) laser fa13ility
In this manus13ript I presented the setup for the aforesaid sho13k expe-
riment Following this the data analysis results interpretation as well as
relevant numeri13al simulations 13on13erning various diagnosti13s viz visible in-
terferometry and XUV spe13tros13opy have been presented Although a large
fra13tion of the experimental re13ords were performed for Xenon at various
pressures I have also studied the sho13k 13hara13teristi13s of dierent gases like
Ar Kr and He Our results are 13omplementary to those obtained at ORION
laser fa13ility also in 2015 relative to the 13ollision of identi13al radiative sho13k
waves at higher velo13ities (80 kms) In parallel in this thesis I have also
worked on the optimization of a setup dedi13ated to ele13tromagneti13ally ge-
nerated strong sho13ks at lower velo13ities In the following I briey present
a brief summary of the main results of my thesis work
In the third 13hapter I have presented a series of 1D hydrodynami13 simu-
lations (realized with the 13ommer13ial 13ode HELIOS) on isolated and 13ounter-
propagating sho13k waves at equal and dierent velo13ities The simulations
13onrm that a single radiative sho13k propagating in Xenon gas at 01 bar
is 13hara13terized by an extended pre13ursor and a large 13ompression of 30 in
the post sho13k At 50 kms the temperature evolution with the distan13e
shows identi13al post-sho13k and pre-sho13k values This indi13ates that the
numeri13al sho13k is of super13riti13al nature However I nd that there is an
important un13ertainty in the Xenon opa13ity whi13h makes the renement of
the simulation unne13essary espe13ially in terms of group numbers Next I
have investigated the intera13tion two 13ounter propagating sho13ks for iden-
ti13al (50-50 kms) and non identi13al (50-20 kms) sho13k speeds For the
121
122 CHAPTER 7 CONCLUSION
13ase of identi13al speeds the pre13ursors merge together at around 10 ns The
intera13tion is then 13hara13terized by a regular in13rease of the ele13tron density
and the temperature with the time The sho13ks 13ollision at 38 ns leads to a
jump in the ele13tron density (6 times 10
2113m
minus3) and temperature (39 eV) On
the other hand for the 13ase of non-identi13al sho13k speeds (13ase representing
our experiments) the pre13ursors intera13tion starts later than in the former
13ase The sho13ks 13ollision o1313urs at 49 ns and it is 13hara13terized by a sudden
in13rease of the ele13tron density also by an order of magnitude (4 times 10
21
13m
minus3) whereas the temperature in13reases up to 29 eV
Although the data analysis of all the experimental re13ords obtained du-
ring the PALS experimental 13ampaign has been 13arried out only a few repre-
sentative interferometri13 and spe13tros13opi13 re13ords for Xe gas are dis13ussed
in details in this thesis The interferometri13 re13ords allowed me to estimate
the average sho13k speed and time variation of the ele13tron density during the
sho13k propagation The sho13k speeds of the MAIN and AUX radiative sho13k
waves are found to be ranging between sim 30-55 and 12-25 kms respe13tively
whereas the averaged pre13ursor ele13tron density is varying between 10
17and
10
1913m
minus3during the sho13ks propagation
From the interferometri13 re13ords I have investigated the ee13t of the
intera13tion between the two radiative pre13ursors for the sho13ks propagating
in Xenon at 01 bar with respe13tive speeds of 50plusmn3 and 23plusmn3 kms The
intera13tion starts at 20 ns and is followed by the merging of the two ra-
diative pre13ursors This pre13ursor intera13tion is 13hara13terized through the
enhan13ement of the ionisation wave The sho13ks 13ollision is re13orded at 50
ns On the 13ontrary the same intera13tion behavior is not seen at 02 bar
(sho13k speeds sim 38plusmn4 and 18plusmn2 kms for the MAIN and AUX sho13k waves
respe13tively) Moreover in this 13ase there is no signature of a radiative pre-
13ursor for the AUX sho13k The pre13ursors intera13tion if any should then
13ould o1313ur at times whi13h are outside of the re13ord and 13an not be 13on-
rmed by the experiment This indi13ates that for a given gas and sho13k
speed the radiative ee13ts de13reases with the initial mass density
The investigation of the lateral extension of the sho13k has been made
through the analysis of transverse interferograms At 02 bar the MAIN
sho13k with a speed sim 40 kms has a lateral pre13ursor extension of sim570plusmn30microm whereas it is 275plusmn25 microm for AUX sho13k whi13h is propagating
with a speed of 20 kms The MAIN sho13k pre13ursor is almost of a at
spatial prole whereas the AUX pre13ursor is more 13urved suggesting that
the 2D ee13ts are mu13h more important for AUX than for MAIN One of
the explanations is that the spot size of AUX laser on the target is smaller
than the target width whereas the MAIN laser has a spot size whi13h is equal
to the target width
In order to interpret the experimental results with appropriate 1D simu-
lations I have optimized the lasers uen13es to numeri13ally obtain a sho13k
speed equal to that re13orded in the experiment In the 13orresponding simu-
72 PERSPECTIVES 123
lations for Xenon at 01 bar I note the intera13tion of the two pre13ursors
However I do not nd a good quantitative agreement for the ele13tron den-
sity Beside the question of the impre13ise opa13ities this disagreement might
also be attributed to 2D ee13ts (Gonzaacutelez Audit and Stehleacute 2009 Leygna13
et al 2006 Stehleacute et al 2010)(Cotelo et al 2015)
The analysis of spa13e- and time-integrated XUV re13ord at 06 bar is
presented for two 13ounter propagating sho13ks of speeds sim 36plusmn4 and 18plusmn5kms for MAIN and AUX respe13tively The presen13e of HeII Balmer lines
and Xenon lines tends to indi13ate a temperature of the sho13k of about 15
eV and a Xenon mean ion 13harge around 6 - 7 The 1D simulations predi13t
ele13tron temperature in the range of 10-30 eV while ion 13harge to be ranging
between 5 and 10
In addition to this study whi13h was fo13used on the Xe 13ase I have
investigated the sho13k 13hara13teristi13s for Ar Kr at 13lose mass density (sim 6
times 10
minus4g13m3) whi13h 13orrespond to 03 bar and Kr gas 02 bar Almost no
pre13ursor is found to be present in 13ase of Ar whereas a tiny pre13ursor have
been observed for Kr This indi13ates that for a given density and sho13k
speed the radiative ee13ts in13rease with the atomi13 number
In parallel I have worked on the optimization of the design of an experi-
mental setup where the sho13k is generated ele13tromagneti13ally This setup
allows studying sho13k sim 30 kms in noble gas at sim1 mbar The parameters
of a low-indu13tan13e and 13ompa13t pulsed power devi13e have been optimized
with a simple model in view of building-up and a1313elerating a plasma sheath
in Ar and Xe at gas pressure in the range 01-10 mbar Simple observations
of the moving plasma using side-on dete13tion of the visible emission give
features 13oherent with the model Sho13ks 13orresponding to ma13h numbers
ranging from 20 to 60 are obtained
72 Perspe13tives
The experimental results presented in this thesis 13orrespond to the 13ase of
two 13ounter-propagating radiative sho13ks We have been able to study the
time variations of the pre13ursor intera13tion and the ee13t of gas pressure
with 1D time dependent interferometri13 re13ords obtained with a streak 13a-
mera Together with the help of XUV spe13tros13opy we 13ould estimate the
sho13k speed ele13tron density and sho13k temperature However no informa-
tion is provided about the lateral extension of the pre13ursor and we were
not able to re13ord XUV data for all the shots Therefore in order to make
these results more 13omprehensive we have designed and parti13ipated in a
follow-up experimental 13ampaign in September 2016 at PALS This time we
fo13ussed on single sho13ks in 13lose experimental 13onditions but with a wider
range in terms of speeds (45-100 kms) We repla13ed the time dependent
streak 13amera interferometri13 study by an instantaneous 2D interferometri13
124 CHAPTER 7 CONCLUSION
image (GOI) obtained with a short pulse sub pi13ose13ond laser and 13ameras
XUV spe13tra were also re13orded The re13ords were performed at three dif-
ferent times of the sho13k propagation allowing to dedu13e the sho13k speed
and the time evolution of the pre13ursor morphology This follow up experi-
ment will allow to 13omplete the analysis of the laser generated sho13k waves
presented in this thesis
The preliminary analysis I have performed on these new re13ords is very
promising An example of these interferometri13 re13ords is given in the Figure
71a for the shot 30364 in Xe+He at 06 bar The high speed of the sho13k is
related to the energy of the laser whi13h is here 170 J whereas it was less than
120 J in the previous experiment The 13omparison with the interferometri13
re13ord (not shown) before the shot allows to dedu13e the ele13tron density
whi13h rea13hes 18 times 10
1913m
3 The high quality re13ord shows 13learly 2D
ee13ts for this sho13k wave
(a)
(b)
Figure 71 Interferometri13 re13ord obtained at 546 ns for a single radiative sho13k
of speed sim 100 kms propagating in Xe+He gas at 06 bar driven by laser at 348
nm with energy 170 J The dierent 13olors in (b) 13orrespond to yellow le Ne le35 times 10
1713m
minus3 13yan 37 - 74 times 10
1713m
minus3 blue74 - 15 times 10
1813m
minus3 green
15 - 18 times 10
1813m
minus3 magenta 18 - 22 times 10
1813m
minus3 red 22 - 26 times 10
1813m
minus3
orange26 times 10
1813m
minus3- 18 times 10
1913m
minus3
This analysis is now ongoing and the results of this experimental 13am-
72 PERSPECTIVES 125
paign will be 13ompared with that obtained from the analysis of re13ords from
2015 presented in this manus13ript This instantaneous imaging interferome-
tri13 diagnosti13 does not follow the sho13k 13ontinuously with the time but it
gives a pre13ise map of the ele13tron density whi13h 13an be inverted through
Abel inversion to obtain the lo13al estimation of the ele13tron density
In parallel in the ele13tromagneti13ally laun13hed sho13ks we have upgra-
ded our table top setup to a13hieve higher speeds For this we have used 12
13apa13itors ea13h of 1 microF at the pla13e of the previous 11 13apa13itors ea13h of
06 microF We have implemented new diagnosti13s namely a visible interferome-
ter working in the Gated Opti13al Imaging mode and time and spa13e visible
spe13tros13opy similar to the one des13ribed in the 13hapter 4 For the visible in-
terferometry we have Ma13h Zehnder Interferometer setup and we will image
the sho13k at dierent times with the help of a SL300 pi13ose13ond high energy
NdYAG laser (λ = 532 nm and energy = 024 J) The experiment will soon
be performed on this new setup and the analysis of su13h re13ords will also be
undertaken in future allowing a 13omparison of ele13tromagneti13ally laun13hed
sho13ks with the laser-driven sho13k whi13h should make the bridge between
thin and opti13ally thi13ker sho13ks
126 CHAPTER 7 CONCLUSION
Chapter 8
Thesis summary
Radiative sho13ks (RS) are present in various astrophysi13al environments es-
pe13ially in stars They are then present in all the supersoni13 a1313retioneje13tion
pro13esses during the stellar evolution from the early phase of the proto-
stellar 13ollapse up to the magnetosphere a1313retion on to the already formed
T- Tauri stars up to the late stage of the supernovae sho13ks
Radiative sho13ks are strong sho13ks (ie Ma13h number Mgtgt1) whi13hrea13h high temperatures and thus are the sour13e of intense radiation Alt-
hough the observation of (ie spatially resolved ) signatures of su13h sho13ks
in the astrophysi13al environments is very di13ult an alternate approa13h na-
med laboratory plasma astrophysi13s provides a powerful tool to study them
on the Earth
These waves have been experimentally studied sin13e more than a de13ade
mostly on large-s13ale laser fa13ilities in noble gases and with dierent targets
geometries The ablation pro13ess generated by the intera13tion between the
laser and a foil indu13es a sho13k wave in the foil whi13h then propagates
in the gas at a speed of several tens of kms Numeri13al simulations 13an
des13ribe these sho13ks with an improving pre13ision With laser intensities on
target 13omprised between 1014 and 1015 W13m
2 these experiments allowed
to re13ord sho13k speeds ranging between 40 and 150 kms
All previous experimental studies have been fo13used on the 13ase of single
radiative sho13ks However in astrophysi13al 13onditions the radiative sho13k
in general intera13ts with a denser medium leading to the development of
ree13ted and transmitted sho13ks A few representative examples of su13h
phenomena are the intera13tion of supernovae remnants with dense mole13ular
13louds the a1313retion sho13ks on the photosphere of T-Tauri stars and the bow
sho13ks at the head of fast stellar jets The 13ollision (or the intera13tion) of
two radiative sho13k waves is obviously a rare astrophysi13al event and the
template 13ase of the supernova remnants DEM L316 (see Fig 1 of Williams
et al (1997)) is still the subje13t of debates Williams et al (2005) Toledo-
Roy et al (2009) Velarde et al (2006) In this regard the development of
dedi13ated laboratory experiments to the study of propagation and intera13tion
of 13ounter-propagating sho13k waves is important as a tool to 13hara13terize
su13h events through their spe13i13 signatures
Complementary to laser experiments 13ompa13t pulsed power generators
may drive an ele13tromagneti13 13oaxial plasma gun to 13reate astrophysi13al
relevant sho13ks in lower pressure noble gases with a high availability and a
127
128 CHAPTER 8 THESIS SUMMARY
rather modest 13apital 13ost The ele13tromagneti13ally driven sho13k waves may
have larger s13ales than those by laser Thus they 13an be analyzed rather
easily with the adequate diagnosti13s (Kondo et al 2008)
In this thesis I am presenting radiative sho13ks in the 13ontext of labo-
ratory astrophysi13s using two experimental approa13hes laser-driven sho13ks
and ele13tromagneti13ally driven sho13ks
First 13hapter of my thesis is Radiative sho13k waves in whi13h the physi13s
of radiative sho13ks have been studded Radiative sho13k waves are hypersoni13
sho13k waves whi13h are heated to high temperature and as a 13onsequen13e
be13ome the sour13e of intense radiation This radiation in turn modies
the dynami13s and stru13ture of the sho13k itself whi13h makes its stru13ture
more 13ompli13ated Radiative sho13k waves o1313ur in several astrophysi13al 13ir-
13umstan13es su13h as in the a1313retion sho13k of protostellar formation (Stahler
Palla and Salpeter 1986) the supernova explosion and the intera13tion of
their remnants with the dense interstellar medium (Chevalier 1977) the bow
sho13ks at the head of stellar jets (Hartigan et al 2001)
Radiative sho13k waves 13an now be studied in the laboratory whi13h allows
to 13ompare the data with models existing in the literature and to 13he13k the
ability of the numeri13al 13odes to reprodu13e them (see for instan13e Bouquet
et al (2004) Leygna13 et al (2006)) The experimental study of radiative
sho13k 13hara13teristi13s is 13ru13ial in understanding these various energeti13 pro-
13esses o1313urring in the astrophysi13al plasmas
The jump 13onditions (Rankine-Hugoniot relations) for the simple 13ase
of an ideal gas whi13h rely on the values of the thermodynami13al quantities
on both sides of the dis13ontinuity Let us 13onsider a one-dimensional sho13k
propagating in a gas at rest with the speed us In the frame asso13iated with
the sho13k front the pre-sho13k (upstream) uid velo13ity is then u1 = - us
while behind the sho13k (post-sho13k or downstream region) this velo13ity is
u2 as shown in the Fig 21 Considering the 13ase of strong sho13ks where
M may be 13onsidered to be ≫1 jump 13onditions for ideal gas are
ρ2ρ1
=(γ + 1)
(γ minus 1)(81)
T2
T1=
2M2γ(γ minus 1)
(γ + 1)2(82)
kBT2 =2(γ minus 1)
(γ + 1)2mus
2(83)
For mono-atomi13 gases γ = 53 the 13ompression ratio is estimated to
be 4 and the post-sho13k temperature is
kBT2 =3
16mu1
2 =3
16mPAu1
2(84)
129
This temperature is thus proportional to the square of the sho13k velo13ity
and to the atomi13 mass In 13ontrast to the dis13ussion on ideal gas in the
13ase of a real gas is a little more 13ompli13ate It is to note that in this 13ase
a part of the kineti13 energy is used to ex13ite and ionize the post-sho13k gas
As a result its temperature is lower than that for the 13ase of ideal gas
In addition the pressure is also modied due to ionisation (Mi13haut et al
2004) In13luding the ee13t of ionization with the average 13harge z in the
medium for the 13ase real gas jump relations for real gas are
ρ2u2 = ρ1u1 (85)
ρ2u22 + ρ2
kBT2
m(1 + z2) = ρ1u
21 + ρ1
kBT1
m(1 + z1) (86)
ρ2u2
[5
2
(1 + z2)
mkBT2 + ǫexc2 +
u222
]
= ρ1u1
[5
2
(1 + z1)
mkBT1 + ǫexc1 +
u212
]
(87)
In the previous both 13ases the ee13ts of radiation heating and 13ooling
have been negle13ted However the radiation emitted in a strong sho13k will
ae13t its stru13ture These ee13ts 13an be determined using mass momentum
and energy 13onservation equations whi13h now have to in13lude the 13ontribu-
tions of the radiative ux energy and pressure This 13ase is more 13omplex
now Taking into a1313ount radiative ee13ts the equations of radiative hydro-
dynami13s in the 13o-moving frame of the sho13k (Mihalas and Mihalas 1999
Drake 2006) 13an be written as
partρ
partt= minusnabla(ρu) (88)
ρ
(
partu
partt+ unablau
)
= minusnabla(P + Prad) (89)
part
partt
(
ρu2
2+ ρǫ+ Erad
)
+nabla
[
ρu
(
ǫ+u2
2+
P
ρ
)
+ (Erad + Prad)u
]
= minusnablaFrad
(810)
The 13ontributions linked to Prad and Erad are ee13tive only at very high
temperature (ie velo13ity) However for the radiative sho13ks of interest in
our experiments the 13ontribution of the radiative ux Frad is non-negligible
13ompared to ρu3 These sho13ks are thus in the radiation ux dominated
regime Most of the experimental sho13k waves are in this regime These
hydrodynami13 equations are strongly non-linear
Depending on the opa13ity the radiation emitted from the sho13k may be
absorbed by the pre-sho13k region indu13ing its pre-heating Far away from
the dis13ontinuity the stru13ture of the upstream medium is determined by
the absorption On its side the stru13ture of the transition layer of the down-
stream medium 13lose to the dis13ontinuity depends on the balan13e between
130 CHAPTER 8 THESIS SUMMARY
the emission (13ooling) and the absorption (heating) Thus the full stru13ture
of the sho13k mainly depends on the variation of the opa13ity χ (13m
2g) The
opti13al depth (τ(s)) measured from the position of the jump is then the
relevant parameter for the absorption It is given by
τ(s) =
int s
sjump
χ(sprime)ρ(sprime) dsprime (811)
where sprime 13hara13terizes the path of the radiation As we are interested in
the mean frequen13y averaged radiative ux the relevant opa13ity here will be
the averaged Rosseland opa13ity whi13h will be dened in due 13ourse
If τ gtgt 1 the regime is referred as opti13ally thi13k while in the opposite
13ase (τ lt 1) it is opti13ally thin Flux dominated radiative sho13ks have
been 13lassied depending on the opti13al depth of upstream and downstream
regimes
We are more interested in the kind of sho13k have a radiative pre13ursor It
is possible to produ13e a radiative sho13k at the high energy laser fa13ilities with
laser intensity of sim 10
14W13m
2 A short summary of number of experiment
performed at high power laser fa13ilities have been listed in table 21 Other
than laser driven radiative sho13k it is possible to laun13h 13omparatively slower
radiative sho13ks (sim 10-30 kms) using ele13tromagneti13 generators
The 13hapter two is dedi13ated to the numeri13al analysis of radiative sho13k
waves and of the main physi13al pro13esses whi13h play a role in these 13omplex
pro13esses Radiative sho13ks are present in various astrophysi13al pro13esses
implying extreme 13onditions Laboratory experiments then allow investiga-
ting the underlying physi13al pro13esses whi13h take pla13e in these 13onditions
and whi13h are not observable from the Earth due to a la13k of spatial resolu-
tion However experiments are not always straightforward to interpret and
the help of numeri13al simulations be13omes pre13ious
Dierent numeri13al approa13hes are used for the simulation of laboratory
plasmas Some of them use xed grids Other introdu13e the possibility of
renement (on the y) in some meshes (Adaptative Mesh Renement) to
maintain a good resolution in the interesting parts of the plasma Another
approa13h is based on meshes following the uid (Lagrangian des13ription)
whi13h means that the meshes move with the uid and no matter may go
through a mesh to another The form of the equations dier in this 13ase (see
for instan13e Orban et al (2013))
HELIOS is su13h a one-dimensional Lagrangian radiation hydrodynami13
13ommer13ial 13ode (Ma13Farlane Golovkin and Woodru 2006) and I have
used it to simulate our experiment of intera13ting sho13k waves This 13ode
is easy to handle and has the presently interesting 13apability to be able to
simulate the 13ase of two sho13k waves propagating in opposite dire13tions and
laun13hed by two dierent lasers HELIOS may des13ribe non-LTE plasmas as
it in13ludes an in-line 13ollisional radiative (HELIOS-CR) model for 13omputing
131
non-LTE atomi13 level populations at ea13h time step of the hydrodynami13s
simulation However I used HELIOS in the LTE approximation whi13h is
appropriate to our experimental 13onditions (Rodriguez et al 2011) as will
be dis13ussed in the next se13tion
The 13ode solves the uid equations of motion using the pressure 13ontri-
butions of ele13trons ions and radiation It may des13ribe the ele13trons and
ions as two intera13ting uids at respe13tive temperatures Te and Ti Thermal
13ondu13tion is des13ribed within a ux-limited ele13tron 13ondu13tion model ba-
sed on the Spitzer 13ondu13tivity and the laser energy deposition is 13omputed
with an inverse Bremsstrahlung model
The radiation emission and absorption terms are introdu13ed in the equa-
tion of energy for the ele13trons and in the radiation transport equations One
of the methods whi13h are proposed is the ux-limited multi-group radia-
tion diusion model where the radiative ux is proportional to the gradient
of the radiative energy and is inversely proportional to the Rosseland opa-
13ity The expression is pondered by a ux-limited diusion 13oe13ient in
order to obtain the good opti13ally thin limit This 13oe13ient follows the
Larsen expression des13ribed in Olson Auer and Hall (2000) The se13ond
method is a (time independent) multi-angle short-13hara13teristi13s s13heme In
our 13ase where radiation and hydrodynami13s are strongly 13oupled and thus
the radiation eld varies rapidly we used the diusion model together with
LTE Plan13k and Rosseland multi-groups opa13ities
In addition our version of HELIOS uses the PROPACEOS (Ma13Farlane
Golovkin and Woodru 2006) Equation of State (EOS) and multi-group
opa13ity tables whi13h are generated by the 13ode
1
and it is not possible
to in13lude any other opa13ity or EOS table The knowledge of the thermo-
dynami13al 13onditions is required for an adequate simulation of the plasma
des13ription In our experiment the mass density ranges between sim 10
minus4and
10
minus1g13m
3 while the temperature values are 13omprised between and sim 0 to
50 eV The pre13ise 13al13ulations performed by Rodriguez et al (Fig2 of Rodri-
guez et al (2011)) for Xenon indi13ate that the thermodynami13al 13onditions
of our radiative sho13k experiments 13orrespond to the LTE regime
The equation of state (EOS) denes the dependen13e of the pressure
ionisation internal energy with the mass density and temperature Several
models do exist in the literature
Our version of HELIOS for LTE plasmas may handle polytropi13 or PRO-
PACEOS equation of state Unfortunately there are not many details about
this last method The authors of the 13ode mention that it uses a QEOS-type
model (More et al 1988) in the strong 13oupling regime (ie at relatively
high densities and low temperatures) and an isolated atom model whi13h
uses a detailed atomi13 stru13ture modeling in the weak 13oupling region The
properties between the weak and strong 13oupling regimes are said to be obtai-
1
see http wwwprism minus cscomSoftwarePROPACEOS
132 CHAPTER 8 THESIS SUMMARY
ned by interpolating in a thermodynami13ally 13onsistent manner This EOS
in13ludes the 13ontributions from the translation of ions and ele13trons ele13-
tron degenera13y atom ionization and ex13itation and Coulomb intera13tions
(Debye - Hu13kel 13orre13tion)
I have performed HELIOS 1D simulations for dierent 13ases for single
and 13ounter propagating sho13ks in 13hapter 2 of my thesis Radiative sho13k
waves propagating in Xenon at 01 bar with a velo13ity of 50 kms are 13ha-
ra13terized by the development of an extended radiative pre13ursor The huge
13ompression of 38 in the post-sho13k is a 13onsequen13e of both the sho13k and
the gas ionisation (fa13tor of 10) as also from the radiative 13ooling The post-
sho13k and pre-sho13k temperature on both side of the peak are identi13al
indi13ating that the sho13k is super13riti13al
The spe13ta13ular role of the opa13ity has been highlighted Due to the
un13ertainty in the opa13ity used for Xenon in our simulations I de13ided that
at this stage it was not ne13essary to rene the simulation in terms of group
numbers within the 1D des13ription of the sho13k wave
Our numeri13al study gives the main 13hara13teristi13s of the intera13tion of
two 13ounter-propagating sho13k waves with speeds equal to 50-50 kms and
50-20 kms The 13ase of identi13al speeds is simpler due to the symmetry of
the problem However whatever the speeds the most important signature
of the intera13tion is the merging of the pre13ursor at 8 ns for 50-50 kms
and at 15 ns for 50-20 kms This merging is followed by a regular in13rease
with time of the ele13tron density and the temperature The 13ollision time
is 13hara13terized by a sudden in13rease of the ele13tron density by an order of
magnitude rea13hing 66 times 10
21and 3 times 10
2113m
minus3respe13tively whereas the
temperature in13reases up to 39 and 28 eV
When a short pulse of a high-power laser beam is fo13ussed on a thin foil
it transfers a huge amount of energy to the foil material Almost instantane-
ously the laser absorption heats it up to the ablation A hot and low density
13orona is generated ba13kwards by this pro13ess Beyond the ablation surfa13e
a sho13k is formed in the foil as a 13onsequen13e of the momentum 13onservation
(ro13ket ee13t) The sho13k moves inward in the foil Our 13ase of the experi-
mental setup as the foil 13loses a tube lled with gas the sho13k propagates
then in the gas where it is studied through various kinds of diagnosti13s The
foil then a13ts as a piston
We performed the experiments at Prague Asterix Laser System (PALS)
in the Spring 2015 (ve weeks 20th April - 22th May) The obje13tive of the
experimental 13ampaign was to study the evolution of two 13ounter streaming
radiative sho13ks driven by two 13ounter-propagating lasers with an irradian13e
in the range of 10
14W13m
minus2 and therefore to 13hara13terize their intera13tion
and 13ollision In the 13hapter third 13hapter I have rst presented a brief
des13ription of the PALS laser fa13ility and of the laser beams used in our
experiment This followed by a presentation of the targets design and an
overview of the general setup and of our main diagnosti13s
133
The Prague Asterix Laser System (PALS) is a laser fa13ility based on
an Infrared high-power iodine laser system (Asterix IV) (Jungwirth et al
2001) Using dierent amplifying stages the laser fa13ility is able to deliver
energy up to 1 kJ in 03 ns at the fundamental wavelength 1315 nm The
output laser beam 13an be further subdivided in few auxiliary beams All
auxiliary beams may be frequen13y doubled (λ = 657 nm) or tripled (438
nm) The PALS laser fa13ility is 13apable of ring up to two high energy laser
shots per hour Compared with solid states lasers this gas laser is known to
deliver a quite homogenous beam intensity without hot spots
For our experiments we used two independent laser beams to drive the
two 13ounter streaming sho13ks in the gaseous target For this purpose the
fundamental PALS laser beam has been subdivided after the fourth amplier
(V4) into two beams with respe13tive energy fra13tions of 60 and 40 (see
Fig 42) The most energeti13 of these laser beams is then inje13ted through
the fth (V5) amplier and its frequen13y are then tripled This beam at 438
nm will be hereafter termed termed as the MAIN laser beam
The se13ond of the two previous laser beams propagates then without any
modi13ation It will be hereafter 13alled the AUX laser beam (1315 nm)
The s13heme of the energy distribution is shown in the Fig 42 and the
13hara13teristi13s of MAIN and AUX laser are presented in Table 41
For the experiments PALS oers two va13uum 13hambers with a spheri13al
and 13ylindri13al geometry respe13tively The spheri13al 13hamber assembly used
in our experiments is shown in Fig 43 This 13hamber having a diameter
of 100 13m is 13losed by two entran13e 13ir13ular windows of respe13tive diameters
80 13m and 50 13m to allow the MAIN and AUX beams to penetrate in the
13hamber and to rea13h the target Ea13h of these windows is outtted with a
hinged door Several 13ir13ular ports of various diameters also allow for the
dierent diagnosti13s The target holder and some diagnosti13s are mounted
on an opti13al ben13h whi13h is de13oupled from the 13hamber walls
Our targets (see Fig 45) s13hemati13ally 13onsist in small tubes of 4 mm
length 13losed on both sides by two spe13i13 foils of thi13kness 11 microm on whi13h
the laser beams are fo13used (one laser per foil) with an irradian13e of about
10
1413m
minus3 The tube is lled with gas in whi13h the radiative sho13k propa-
gates with a velo13ity of 30 - 60 kms The two foils 13losing the target insure
the 13onversion through ablation and sho13k generation of the laser energy
into me13hani13al energy
In our experiments we used spe13i13 targets for alignment and 13hara13te-
rization of the beam size These targets were manufa13tured at the Pole
instrumental of Observatoire de Paris
The gaseous targets were lled in situ at a fra13tion of 1 bar with a gas
(viz Xe Ar a mixture of Xe and He) whi13h thus provide the medium for the
sho13k propagation We 13hose an in-situ lling system (Fig 410) to redu13e
the dieren13e of pressures inside and outside of the tube This is 13riti13al as
a large pressure dieren13e between the target 13ell and the va13uum 13hamber
134 CHAPTER 8 THESIS SUMMARY
13an result in the breaking of the ultra-thin SiN windows of the target
First of all we performed fo13alization of MAIN and AUX laser beam
on target by using Al-massive targets The two PALS laser lenses were
translated up to a13hieve the suitable diameters on the target whi13h were
nally set to 450 - 500 microm and 250 - 300 microm for the MAIN and AUX beams
The size of the impa13t was 13ontrolled ea13h day on mo13k targets before using
the laser beams for real shots on gaseous targets It is worth to pre13ise that
the keV diagnosti13 was still in pla13e for the shots on the gaseous targets
In that 13ase the beams impa13t Parylene-N instead of Aluminium The keV
signal is then weaker than for the mo13k aluminum targets
Various kinds of diagnosti13s were implemented namely visible interfero-
metry XUV and visible spe13tros13opy For the same two keV 13ameras are
indu13ted to monitor the impa13t and two visible 13ameras for target alignment
For visible interferometory we have taken referen13e images (without any
fringe) for every target as shown in the Fig 417a for one target example
In order to re13ord su13h images we blo13ked the referen13e beam and let the
probe beam only to illuminate the target in a stati13 mode and with the slit
open Then we redu13ed the slit width to its nominal value (200 microm) This
enabled us to have the visualization of the portion of the target whi13h was
imaged in the slit on the target and also to lo13ate the position of the pistons
before the shot This image is termed as the referen13e target image Then
the system was pla13ed in the interferometri13 mode and the fringe pattern
was optimized to be perpendi13ular to the slit and thus in the longitudinal
mode perpendi13ular to the dire13tion of the sho13k propagation The Fig
417b reports su13h a typi13al interferogram re13orded during our experiment
(13f Fig 413) The horizontal axis 13orresponds to the dire13tion of the sho13k
propagation (with a total s13ale of 54 mm on the target) and the verti13al
axis 13orresponds to time (s13aling 200 ns)
The time and spa13e integrated XUV plasma emission are re13orded with
a Flat Field XUV spe13trometer using a 13on13ave grating In the Fig 419
the drawing of the XUV spe13trometer setup s13heme (red 13olor) is presented
together with the spheri13al 13hamber and the target The XUV spe13trometer
is installed on the top of the spheri13al 13hamber whi13h allows the XUV
emission passing through the Si3N4 membrane on the top of the target (see
se13tion 422) to be re13orded
In addition to the interferometry and XUV spe13tros13opy we have also im-
plemented a time- and spa13e-resolved visible spe13tros13opi13 (see Fig 420a)plasma
emission diagnosti13
In 13hapter 5 I present the results of the two diagnosti13s with a parti13u-
lar fo13us on the interferometri13 data whi13h I have extensively studied with
the help of a spe13i13 data analysis pro13edure that I have developed The
visible interferometry is an adequate tool to probe the ele13tron density of
the radiative pre13ursor up to the 13riti13al density (4 times 10
2113m
minus3at the wa-
velength 527 nm of the probing laser) whereas the denser post-sho13k region
135
is opaque to the visible light XUV radiation is present in the whole sho13k
stru13ture and thus the time- and spa13e-integrated XUV spe13tros13opy may
allow exploring the self-emission 13oming from the post-sho13k and pre13ursor
regions
In the visible interferometry the interferen13es between the main and
referen13e beams overlapping on the dete13tor produ13e a pattern of fringes
whi13h follow the relative phase variations between the probe (whi13h passes
through the plasma) and the referen13e beams (see se13tion 441 of the previ-
ous 13hapter for the experimental setup) The phase variation between two
13onse13utive fringes of the unperturbed beams (ie without any plasma) is
equal to 2π Then in the presen13e of the plasma in the probe beam these
fringes be13ome shifted due to the modi13ation of the refra13tive index More
details about the prin13iple of the interferometry and the Ma13h-Zehnder in-
terferometri13 setup may be found in the APPENDIX A and we re13all here
only the expression of the phase shift
∆φ asymp minus πd
λNclt Ne gt (812)
where λ = 527 nm is the wavelength of the probing laser Nc = 4 times 10
21
13m
minus3the 13riti13al density at this wavelength and lt Ne gt is the ele13tron
density averaged over the laser path d in the 13ell and is dened as
lt Ne gt=
int d
0
Ne(z t)dy
d(813)
In this 13hapter I have presented an extensive data analysis of few re-
presentative interferometri13 and spe13tros13opi13 re13ords I am presenting here
summary of results obtained bellow
The average sho13k speed and ele13tron density have been estimated from
the interferograms The sho13k speeds of the MAIN and AUX radiative sho13k
waves vary between sim 30-55 and 10-30 kms respe13tively and the averaged
pre13ursor ele13tron density ranges between 10
17and 10
1913m
minus3during the
sho13ks propagation
We have demonstrated the intera13tion ee13t between the two radiative
pre13ursors in the 13ase of Xe at 01 bar at 54 and 23 kms The intera13tion is
13learly 13hara13terized in the experiment by the enhan13ement of the ionisation
wave followed by the merging of the two radiative pre13ursors at 20 ns The
13ollision time is re13orded at 47 ns Su13h behavior is reprodu13ed by the
simulation
We have investigated this intera13tion at a larger pressure 02 bar with
the following speeds sim 41 kms for the MAIN and sim 18 kms for the AUX
sho13k waves We do not re13ord any experimental signature of the radiative
pre13ursor for AUX Further we have not been able to 13at13h experimentally
the 13ollision time The re13orded pre13ursor of MAIN is not inuen13ed by AUX
wave up to 48 ns (Fig 59(b) and (13) in data analysis 13hapter) whi13h is the
136 CHAPTER 8 THESIS SUMMARY
limit of the re13ord On its side the 1D simulation predi13ts a tiny pre13ursor
for AUX and that both pre13ursors start to intera13t at 49 ns This plausible
sho13ks intera13tion o1313urring at times whi13h are outside of the re13ord 13an
not be 13onrmed by our experiment
The results of the transverse interferometry at 02 bar with speeds of sim40 and 20 kms indi13ate that the MAIN pre13ursor has a lateral extension of
sim 600 microm whereas it is 300 microm for AUX The pre13ursor of MAIN is almost
at with a probable small bending at the edges of the tube whereas the
AUX pre13ursor is more 13urved This means that the 2D ee13ts are more
important for AUX than for MAIN Still in the 13ase of Xenon we have
obtained information about the temperature and the mean 13harge of gas
through our time integrated XUV spe13tra (shot48143) in Xenon at 06
bar From these results we may 13on13lude that the mean ion 13harge is at
least equal to 6 and that the temperature has rea13hed 15 eV
Our simulations give a qualitative des13ription of the sho13ks intera13tion
when the laser uen13e is adjusted to give the 13orre13t sho13k velo13ities Howe-
ver it is now well known that 2D simulations (together with state of the art
opa13ities) t better with experiments (Gonzaacutelez Audit and Stehleacute 2009
Leygna13 et al 2006 Stehleacute et al 2010) For the same laser energy the 2D
simulations lead to a diminution of the sho13k speed 13ompared to 1D as also
to a diminution of the ele13tron density For instan13e in the 13ase of a sho13k
wave laun13hed by a laser beam at 1315 nm in Xenon at 03 bar at PALS and
with a laser uen13e of 85000 J13m
2 ARWEN 2D simulations give a sho13k
speed of 44 kms in agreement with the experimental one (Cotelo et al
2015) 1D simulation would require in this 13ase a uen13e of 30000 J13m
2to
a13hieve the same velo13ity
The spa13e and time integrated XUV re13ords at 06 bar for respe13tive
speeds whi13h are equal to sim 39 and 18 kms indi13ate that the temperature
of the sho13k has rea13hed values up to 15 eV and that the Xenon mean ion
13harge has also rea13hed values of 6 - 7 whereas 1D simulations predi13t ele13tron
temperature 10-30 eV and ion 13harge 5-10 (Fig 51813) A more detailed
study based on 2D simulation and radiative transfer post-pro13essing will be
ne13essary to rene the analysis
We have investigated the 13ase of other noble gases (Ar Kr) and we have
observed that for a given laser energy the sho13k velo13ity de13reases with the
mass density This study has 13onrmed that for a given density and sho13k
velo13ity the radiative ee13ts in13rease with the atomi13 number For Kr we
noti13ed a tiny pre13ursor without any intera13tion To get su13h ee13t it will
be ne13essary to in13rease the sho13k velo13ity and thus the laser energy
This has been done at the ORION laser fa13ility in UK where the 13ol-
lision of two 13ounter-propagating sho13ks at equal speed sim 80 kms with
laser uen13e sim 6 times 10
14W13m
2has been performed (Clayson et al 2016
Suzuki-Vidal et al 2016) for dierent noble gases and for pressures 13ompri-
sed between 01 and 1 bar A number of diagnosti13s setup X-ray ba13klig-
137
hting opti13al self-emission streak imaging and interferometry (multi-frame
and streak imaging) were used to study both the post-sho13k and the radia-
tive pre13ursor Although I have not parti13ipated in the experiments I have
performed 1D simulations to interpret ORION experimental results This
work is not presented in this thesis
The previous 13hapters were dedi13ated to the study of laser generated
radiative sho13ks With irradian13e more than 10
14W13m
2 we were able to
study sho13k waves propagating at velo13ity up to 50 kms in noble gases with
an initial mass density 13omprised between 5 times 10
minus4- 3 times 10
minus3g13m
minus3
Complementary to laser experiments 13ompa13t pulsed power generators may
drive an ele13tromagneti13 13oaxial plasma gun to 13reate astrophysi13al relevant
sho13ks in lower pressure noble gases (Kondo et al 2006) with a high avai-
lability and a rather modest 13apital 13ost The ele13tromagneti13ally driven
sho13k waves may have larger s13ales than those by laser thus they 13an be
observed rather easily (Kondo et al 2008)
Su13h ele13tri13al pulsed power devi13es may then i) 13reate a high-voltage
breakdown through a gas or more easily at the surfa13e of a diele13tri13 ii)
produ13e relatively hot plasma by ohmi13 heating when the rising 13urrent in the
devi13e is passing through a portion of gas iii) a1313elerate plasma layers under
the magneti13 pressure asso13iated with the self-generated magneti13 eld when
high ele13tri13al 13urrent (say 10s of kA) is maintained for a short but su13ient
time (say 1 micros) Su13h ionization and a1313eleration are present in Z-pin13h
plasmas and in plasma fo13us devi13es (PFD) providing a 13lever geometry is
employed and the mass under 13onsideration (say 1 mg) is 13ompatible with
the stored energy (Martin Williams and Kristiansen 1996)
In this 13hapter we will see rst how a fast ele13tri13al 13ir13uit works then
we will dis13uss the prin13iples of a 13oaxial plasma gun and of PFD We will
see what is in favor of our obje13tive and what has to be avoided A geometry
13apable to rea13h our obje13tive will be proposed
The optimization of the 13oaxial gun (in term of plasma speed) will be
performed and some typi13al gures will be given in the 13ase of a generator
with a stored energy around 1 kJ and a plasma slab of 4-mm diameter ie
quite 10 times the transverse dimension of a laser driven RS The operating
13onditions are supersoni13 sho13ks up to 10-30 kms speed at stati13 pressures
of few mbar in heavy rare gases (Ar Xe) The results of this 0-D model will
be then 13ompared to those obtained with 3-D MHD simulations performed
with the 13ode GORGON (by Andrea Ciardi at LERMA) whi13h has been
used su1313essfully to des13ribe other pulse-power driven plasma experiments
(Chittenden et al 2004) as well as laboratory plasma astrophysi13s experi-
ments (Ciardi et al 2007) The diagnosti13s whi13h have been implemented
will be presented to illustrate the model as well as preliminary re13ords of the
plasma speed
Our aim being to a1313elerate a plasma slab using the magneti13 pressure
it is obviously needed to drive a high intensity 13urrent be13ause the magneti13
138 CHAPTER 8 THESIS SUMMARY
pressure is expressed by B
22micro0 in the region where exists an indu13tion B
The pressure in bars is simply 4B
2 with B in Teslas and in a 13ylindri13al
geometry 1 Tesla is the eld around a 50-kA 13urrent at a radius of 1 13m
Thus we should 13ount on roughly 100 kA delivered by the generator but
su13h a high 13urrent is delivered only in a pulsed mode by a laboratory s13ale
devi13e
There are numbers of te13hni13al solutions to a13hieve su13h ele13tri13al pa-
rameters one may use rather slow generators at moderate high voltage like
13apa13itors bank with the advantage of well know te13hniques but with limited
adjustments A1313ording to an abundant literature
1
a 13onvenient devi13e is a
medium-energy 13apa13itor bank feeding a low-indu13tan13e 13ir13uit An R-L-C
13ir13uit is a well-known 13ombination delivering a high peak 13urrent in the
pseudo-periodi13 mode For a 13apa13itor C initially 13harged under U0 key
performan13es are as follows
Current intensity I(t) is a damped sinusoid
Pseudo pulsation ω =radic[1(LC)minusR2(4L2)]
First 13urrent peak Ipeak = U0[radic(LC) + 08R]
Time of 13urrent peaking T4 = π(2ω)
Equivalent impedan13e Z =radic(LC)
For a safe handling in air a voltage not higher than 30 kV is re13ommen-
ded Taking a total 13apa13itan13e of 6 microF 2700 J are stored under 30 kV For
an indu13tan13e of 240 nH the expe13ted peak 13urrent might be 150 kA at 19
micros This set of values will be a guideline for the development presented in
further se13tions
For this part of the work the obje13tive was to show that in 13omple-
ment of laser experiments a 13ompa13t pulsed power generator might drive
astrophysi13ally relevant sho13ks in low pressure noble gases with a higher
availability and at a rather modest 13apital 13ost The longitudinal a1313elera-
tion of a plasma sheath in a low ba13king gas pressure has been demonstrated
in 13oaxial plasma guns as early as in the 70s even leading to speeds of 100
kms in light gas as H2 whi13h were measured and modeled A 13hoi13e was
made to revisit this type of devi13e At present the ele13tromagneti13 13oaxial
plasma guns are quite only 13onsidered as the initiators of a strong radial
plasma implosion plasma fo13us devi13es (PFD) due to a good ability to pro-
du13e a pin13h plasma and emission of energeti13 parti13les and hard X-ray For
that reason it was ne13essary to adapt the prin13iple of a PFD with two major
13onstraints in13rease the 13onversion of stored energy into the kineti13 energy
of a plasma sheath prevent totally any pin13hing
1
Institute for Plasma Fo13us Studies resour13e website httpplasmafo13usnet last
13onne13tion in 2016
139
That obje13tive was rea13hed by dening the ele13tri13al and geometri13al
parameters of a low-indu13tan13e and 13ompa13t pulsed power generator An
optimization pro13ess has been established to mat13h a13hievable ele13tri13al pa-
rameters of the 13ir13uit (13apa13itan13e indu13tan13e peaking 13urrent intensity)
with a plasma motion in the a1313elerating tube over few 13m leading to a nal
speed of 10-30 kms A model was built up to des13ribe the dynami13s of an
RLC 13ir13uit with a varying element 13omposed of the moving plasma sheath
a1313elerated by the self generated magneti13 pressure The 13hange in plasma
position along the axis is in13reasing the indu13tan13e and the mass a1313retion
is in13reasing the plasma mass It was shown that a 13hange in the ele13trode
geometry namely a 30deg 13oni13al shape was in13reasing the energy 13onversion
and preventing any fo13using The further step in 13ontrast with PFD is the
merging of the plasma ring into a rather planar one whi13h is inje13ted in a
drift tube
Considering Ar or Xe at gas pressure in the range 01 10 mbar and
a highly 13oaxial 1-kJ devi13e the simple model was su13ient to predi13t
performan13es agreeing with the only work quoted in the literature Basi13
observations of the moving plasma using side-on dete13tion of the visible
emission give features 13oherent with the model Ma13h numbers from 20
to 60 are obtained tting with the obje13tive of strong sho13k formation A
rened 3-D MHD numeri13al simulation indi13ates very promising features of
the sho13k in view of exploring mm-size sho13ks in a regime 13omplementary
to laser driven ones and on a table-top devi13e allowing a higher repetition
rate
140 CHAPTER 8 THESIS SUMMARY
Appendi13es
141
Appendix A
Visible Interferometry
Laser interferometry is used in plasmas to dedu13e the ele13tron density through
the variation of the refra13tive index 13aused by the ionisation of the matter
In this appendix rstly the experimental setup of Ma13h Zehnder interfero-
metry as well as the underlying physi13al prin13iples are presented
A01 Refra13tive index of a plasma
In order to derive the refra13tive index of a gas one starts with the set of
Maxwells equations for a mono13hromati13 plane wave propagating in the
plasma (see for instan13e equation 411 of referen13e (Hut13hinson 2002))
Free ele13trons and ions are supposed to be distributed uniformly and the net
total 13harge density is equal to zero
For harmoni13 waves propagating in the dire13tion of +x and with the
pulsation ω one obvious solution is given by
E(x t) = E0ei(kxminusωt)
(A1)
where E0 and k are 13omplex quantities
The relation between k and ω is given by the dispersion relation
k2 =εmicroω2
c2(1 + i
4πσ
ωε) (A2)
where micro is the ele13tri13 permitivity and ε is permeability
The free ele13trons are a1313elerated by the ele13tri13 eld and therefore equa-
tion their motion is given as
medv
dt= minuseE0e
minusiωt(A3)
The obvious solution for the ele13tron velo13ity v is
v = minusie
meωE (A4)
and thus the ele13trons 13arry a 13urrent with a density J
J = Neev = iNee
2
meωE (A5)
where Ne is the ele13tron density Due to the larger mass of the ions the
13orresponding 13urrent is negligible
143
144 APPENDIX A VISIBLE INTERFEROMETRY
The previous equation allows to derive the plasma 13ondu13tivity σ whi13his equal to iNee
2ωme Putting this value in the equation (A2) and assu-
ming a thin medium where ε asymp 1 and micro = 1 one obtains nally
k2 =ω2
c2(1minus
ω2p
ω2) (A6)
where ωp is the plasma frequen13y dened as
ω2p =
4πNee2
me(A7)
Equation A6 shows that a ele13tromagneti13 wave with a pulsation larger than
ωp 13ant propagate in the plasma This allows to derive a 13riti13al ele13tron
density Nc in 13m
minus3as
Nc =meω
2
4πe2=
4πmec2
e2λ2(A8)
For instan13e for a radiation of 1 microm wavelength this upper limit is equal
to Nc = 446times 1021cmminus3
Therefore the index of refra13tion is given by the expression
n =
radic
1minusω2p
ω2=
radic
1minus Ne
Nc(A9)
Finally one obtains
E(x t) = E0ei(kxminusωt) with k = n
ω
c(A10)
As a 13onsequen13e of the variation of the refra13tive index in the dierent
plasma layers the light is refra13ted Indeed the Snell laws stipulate that if a
beam is in13ident on a plane interfa13e between layers (of refra13tive index n1
and n2) it undergoes a bending from the angle of in13iden13e The in13ident
beam the refra13ted beam and the normal to the interfa13e lie in the same
plane and one has n1 sinθ1 = n2 sinθ2 where θ1 is the angle of in13iden13e
and θ2 is the angle of refra13tion Thus a mono13hromati13 beam propagating
in a plasma with in13reasing density be13omes more and more dee13ted
A02 Absorption of the laser beam
Beside refra13tion the light is absorbed by the plasma and the absorption
also in13reases rapidly near the 13riti13al density In the absen13e of any ato-
mi13 bound-bound or bound-free transition this absorption is due to the
inverse Bremsstrahlung pro13ess (ion-ele13tron 13ollision (Pfalzner 2006)) by
145
the plasma free ele13trons The absorption 13oe13ient κ (in 13m
minus1) is then
given by (NRL formulary)
κ(Ne Tev) = 317 10minus7LnΛ lt Z gt
(
Ne
ω
)2 1
T32ev (1minusNeNc)12
(A11)
where Tev is the temperature in eV Ne is the ele13tron density in 13m
minus3
and LnΛ is the Coulomb Logarithm dened as Λ = Max (2 Min(XY))
with (NRL formulary)
X = 23minus log
(
lt Z gtN
12e
T32ev
)
and Y = 24minus log
(
N12e
Tev
)
(A12)
The transmission T (d) through a layer of thi13kness d is given by
T (d) = exp(minusint d
0κ(l)dl) (A13)
For instan13e for λ = 527 nm taking Ne= 101913mminus3 ltZgt= 10 T = 10 eV
whi13h are typi13al values for the post-sho13k in our experiments in Xenon and
supposing a homogenous plasma thi13kness d = 600 microm the transmission is
about 09 whereas at 5times 1019cmminus3 it falls to 01
A03 Prin13iple of interferometry
In the opti13al interferometry te13hnique (Ovsyannikov and Zhukov 2000) the
interferen13e of two plane waves is performed by addition of the two 13orre-
sponding 13oherent ele13tri13 elds The analyzed medium (here the plasma)
in whi13h one of the beam is propagating indu13es a deshaping between the
beams whi13h ae13ts the resulting intensity depending if the elds interfere
in phase or out of phase ie 13onstru13tively or destru13tively
Two te13hniques are used the wave front and the amplitude division
The wave front division is obtained by using two portions (Fresnel mir-
rors Youngs double slit Lloyds mirror prisms et13) of the original wave
front whi13h are then superimposed to produ13e interferen13e
In the amplitude division the two beams are separated by division of
the amplitude over the same se13tion of the wave front Mi13helson and Ma13h
Zehnder interferometers et13 are the typi13al examples of this te13hnique of
amplitude division
The prin13iple of interferen13e of two 13oherent mono13hromati13 traveling
waves 1 and 2 with ele13tri13 elds E1 and E2 propagating in the same
dire13tion and polarized in the same plane 13an be understood as follows for
a given time t at the point r where these waves superpose the amplitude of
the ele13tri13 eld is the ve13tor addition of the two 13orresponding elementary
amplitudes ie
146 APPENDIX A VISIBLE INTERFEROMETRY
E(r t) = E1(r t) + E2(r t)
= a1exp(ikr + ωt+ iφ1) + a2exp(ikr + ωt+ iφ2)
where φ1 and φ2 are the phases of the two waves at r The total intensity
I(r t) = |E1 + E2|2 (A14)
One gets
I = I1 + I2 + 2radic
(I1I2)cos∆φ (A15)
where I1 = a12and I2 = a2
2are the intensities at the point of interferen13e
due to the two waves a13ting independently and ∆φ = φ1minusφ2 represents the
phase dieren13e between the two waves In the 13ase of equal intensities I1one gets
I = 4I1cos2(∆φ2) (A16)
and the interferometri13 pattern 13onsists of dark (∆φ = π + 2nπ) and
bright patterns (∆φ = 2nπ)In general the two beams are only partially 13oherent and the minimum
of the fringe intensity is not equal to zero The fringe 13ontrast
FC =Imax minus Imin
Imax + Imin(A17)
is then a measure of the interferen13e quality FC is maximum and equal to
1 in the 13ase of fully 13oherent beams of the same intensity I1 = I2
A04 Ma13h Zehnder Interferometer
The Ma13h Zehnder Interferometer is a devi13e 13ommonly and e13iently used
to estimate the 13hange in refra13tive index by the plasma Su13h 13hange is in
most the 13ases related to variations in the ele13tron density We employed
this interferometri13 setup in our experiment
The prin13iple of this interferometer is based on the division of the in13ident
light beam into two beams with the help of a beam-splitter whi13h 13an be
a 13oated glass plate or a 13ube After this division one of the beams (probe
beam) is allowed to pass through the medium of interest (a plasma in our
13ase) and then made to interfere with the other unae13ted part of the beam
(referen13e beam) with the help of se13ond beam-splitter as may be noted
from Fig A1
In order to understand the prin13iple of interferen13e and its use in esti-
mating plasma properties let us 13onsider l1 and l2 whi13h are the total path
lengths (in13luding the path in the opti13al elements) for the light travelling
from the sour13e to the dete13tor for the upper and lower paths (13f Figure A1)
respe13tively Suppose for the moment that the sample is removed from beam
147
Figure A1 S13hemati13s of Ma13h Zehnder interferometer with two mirrors two
50 beam splitters and the dete13tor A plasma slab is pla13ed in one of the two
arms of the interferometer
1 (see Figure A1) the two opti13al paths of the probe and referen13e beams
dier in terms of phases Simplify to supposing that the beams propagate
in air (n=1) these phases are then equal to the 2πl1λ and 2πl2λThus a1313ounting for the phase dieren13e δ between the two beams tra-
velled up to dete13tor A 13an be given as
2πl1 minus l2
λ= δ (A18)
If the two beams are perfe13tly parallel at the position of the dete13tor the
interferometri13 gure presents a at intensity whi13h passes through minima
and maxima when one length of the 2 arms is slightly 13hanged Usually one
introdu13es a small tilt between the two dire13tions of propagations whi13h
leads to the apparition of parallel linear dark and bright fringes
In order to obtain interferen13e patterns it is important to make the paths
l1 and l2 as 13lose as possible 13ompared to the 13oheren13e length of the laser
For the multimode Evolution 15 laser used in the PALS experiment this
length is approximately 1 13m (value 13ommuni13ated by the 13onstru13tor)
If we introdu13e the sample in the probe beam 1 this sample will introdu13e
an additional phase denoted by δsample and the net phase shift is then δ +δsample
The prin13iple of the dedu13tion of the ele13tron density with the Ma13h
Zehnder interferometer is to measure the refra13tive index of the test plasma
slab whi13h is pla13ed in the probe beam We assume for the moment that
148 APPENDIX A VISIBLE INTERFEROMETRY
this plasma slab is a phase obje13t for the in13oming beam whi13h means that
it does not signi13antly ae13t the intensity of the probe beam while it only
13hanges the phase of the wave [5]The additive phase dieren13e in the probe arm introdu13ed by the plasma
13an be dened as
φ =
int d
0kplasmadl =
int d
0nω
cdl (A19)
where n is the lo13al refra13tive index at position l of the plasma slab and
the integral is performed other the slab thi13kness d Therefore the phase
dieren13e introdu13ed by the plasma relative to the propagation into va13uum
of the referen13e beam is equal to
∆φ =
int d
0(kplasma minus k0)dl (A20)
where k0 = ωc = 2πλ is the wave ve13tor of the beam propagating in
va13uum
As long as Ne lt Nc the beam propagates in the plasma the plasma
refra13tive index is given by equation A9 and one has
∆φ =ω
c
int d
0
[
(
(1minus Ne
Nc
)12
minus 1]
dl (A21)
At very small ele13tron density (Ne ≪ Nc) the phase shift 13an be written as
∆φ asymp minus ω
2cNc
int d
0Nedl = minus πd
λNclt Ne gt (A22)
where
lt Ne gt=
int d
0
Ne(z t)dy
d(A23)
is the ele13tron density averaged over the path dIn pra13ti13e a referen13e re13ord is rst taken without plasma To dedu13e
the density one has to analyze how the fringes depart from their referen13e
positions
The spa13e between two fringes (inter-fringe) 13orresponds to a phase va-
riation of 2 π A shift of the fringe by 2 π from its referen13e position (where
Ne = 0) 13orresponds to an ele13tron density variation equal to lt Ne gt = 2
λNcd For λ = 527 nm and supposing an homogeneous plasma of thi13kness
600 microm this gives lt Ne gt= 7 times 10
1813m
minus3
Appendix B
Opa13ities and mean 13harge
The variations in logarithmi13 s13ale with the temperature of the PROPA-
CEOS Rosseland and Plan13k opa13ities (se13tion of 13hapter 5 see for intense
Fig 511)as also of the mean 13harge of the plasma are reported for two
densities 16 times 10
minus3g13m
3and 51 times 10
minus4g13m
3for three noble gases Ar
Kr and Xe in Fig B1 and B2
ρ = 51 times 10
minus4g13m
313orrespond to typi13al density of the pre13ursor
For this density Xenon and Krypton have quite similar variations for the
Rosseland opa13ity with a bump of 7000 13m
2g at 3 eV for Xe and 12000
13m
2g at 4 eV for Krypton At 10 eV the two opa13ities de13rease respe13tively
to 600 and 900 13m
2g In 13omparison with these two 13ases the opa13ity of
Argon is broader and more regular Its opa13ity peaks at 6 eV to 13000 13m
2g
At 10 eV it have de13reased to 7000 13m
2g whi13h is higher than for the two
previous gases
The mean 13harge variations in13rease regularly with the temperature At
10 eV these mean 13harges are respe13tively equal to 65 55 and 5 for Xe Kr
and Ar
149
150 APPENDIX B OPACITIES AND MEAN CHARGE
(a)
(b)
Figure B1 Rosseland and Plan13k opa13ity for three gases Xe Kr and Ar at mass
densities 51 times 10
minus4g13m
3(a) and 16 times 10
minus3g13m
3(b)
151
(a)
(b)
Figure B2 Mean 13harge for three gases Xe Kr and Ar at mass densities 16 times10
minus3g13m
3(a) and 51 times 10
minus4g13m
3(b)
152 APPENDIX B OPACITIES AND MEAN CHARGE
Referen13es
Batani D Strati F Telaro B Loumlwer T Hall T Benuzzi-Mounaix A
Koenig M 2003 Produ13tion of high quality sho13ks for equation of state
experiments European Physi13al Journal D 23 99 DOI ADS
Bernard A 2002 Plasma fo13us et appli13ations Le Vide 57(306)
Bett TH Jinks P Stevenson RM Danson CN Pepler DA Ross
IN 1995 Binary phase zone-plate arrays for laser-beam spatial-intensity
distribution 13onversion Applied Opti13s 34 4025 DOI ADS
Bouquet S Teyssier R Chieze JP 2000 Analyti13al study and stru13-
ture of a stationary radiative sho13k The Astrophysi13al Journal Supplement
Series 127(2) 245
Bouquet S Stehleacute C Koenig M Chiegraveze J Benuzzi-Mounaix A Ba-
tani D Leygna13 S Fleury X Merdji H Mi13haut C et al 2004
Observation of laser driven super13riti13al radiative sho13k pre13ursors Phy-
si13al review letters 92 225001
Bozier J Thiell G Le Breton J Azra S De13roisette M S13hirmann
D 1986 Experimental observation of a radiative wave generated in xenon
by a laser-driven super13riti13al sho13k Physi13al review letters 57(11) 1304
Burhop E Spitzer L 1957 Physi13s of fully ionized gases inters13ien13e
tra13ts on physi13s and astronomy no 3 JSTOR
Chaulagain U 2015 Radiative sho13ks expriments modellin and links to
Astrophysi13s PhD thesis Pierre and Marie Curie University
Chaulagain U Stehleacute C de Saacute L Larour J Auvray P Kozlova M
Krus M Dostal J Propupek J Suzuki-Vidal F et al 2012 Labo-
ratory experiments of radiative sho13ks in the 13ontext of stellar a1313retion
In SF2A-2012 Pro13eedings of the Annual meeting of the Fren13h So13iety
of Astronomy and Astrophysi13s 1 305
Chaulagain U Stehleacute C Larour J Kozlovaacute M Suzuki-Vidal F Bar-
roso P Cotelo M Velarde P Rodriguez R Gil JM Ciardi A
A13ef O Nejdl J de Saacute L Singh RL Ibgui L Champion N 2015
Stru13ture of a laser-driven radiative sho13k High Energy Density Physi13s
17 106 DOI ADS
Chevalier RA 1977 The intera13tion of supernovae with the interstellar
medium Annual review of astronomy and astrophysi13s 15 175 DOI ADS
153
154 REFERENCES
Chittenden JP Lebedev SV Jennings CA Bland SN Ciardi A
2004 X-ray generation me13hanisms in three-dimensional simulations of
wire array Z-pin13hesthe evolution of magneti13 tower jets in the laboratory
Plasma Physi13s and Controlled Fusion 46 B457 DOI ADS
Ciardi A 2014 private 13ommuni13ation
Ciardi A Lebedev SV Frank A Bla13kman EG Chittenden JP
Jennings CJ Ampleford DJ Bland SN Bott SC Rapley J Hall
GN Suzuki-Vidal FA Maro1313hino A Lery T Stehle C 2007 The
evolution of magneti13 tower jets in the laboratory Physi13s of Plasmas
14(5) 056501 DOI ADS
Clayson T Suzuki-Vidal F Lebedev SV Swadling GF Burdiak GC
Patankar S Smith RA Foster J Skidmore J Gumbrell E Gra-
ham P Danson C Stehleacute C Singh RL Chaulagain U Larour
J Kozlova M Spindloe C 2016 Counter-propagating radiative sho13k
experiments on the Orion laser fa13ility In APS Meeting Abstra13ts DOI
ADS
Clayson T Suzuki-Vidal F Lebedev S Swadling G Stehleacute C Bur-
diak G Foster J Skidmore J Graham P Gumbrell E et al 2017
Counter-propagating radiative sho13k experiments on the orion laser and
the formation of radiative pre13ursors High Energy Density Physi13s 23
60
Cotelo M Velarde P de la Varga AG Portillo D Stehleacute C Chaul-
again U Kozlova M Larour J Suzuki-Vidal F 2015 Simulation
of radiative sho13k waves in Xe of last PALS experiments High Energy
Density Physi13s 17 68 DOI ADS
de la Varga AG Velarde P Cotelo M de Gaufridy F
Zeitoun P 2011 Radiative properties for warm and hot
dense matter High Energy Density Physi13s 7(3) 163 DOI
httpwwws13ien13edire13t13oms13ien13earti13lepiiS1574181811000498
Diziegravere A Mi13haut C Koenig M Gregory C Ravasio A Sakawa
Y Kuramitsu Y Morita T Ide T Tanji H et al 2011 Highly
radiative sho13k experiments driven by gekko xii Astrophysi13s and Spa13e
S13ien13e 336(1) 213
Dobbs-Dixon I Cumming A Lin DNC 2010 Radiative Hydrodyna-
mi13 Simulations of HD209458b Temporal Variability The Astrophysi13al
Journal 710 1395 DOI ADS
Doss FW Robey HF Drake RP Kuranz CC 2009 Wall sho13ks in
high-energy-density sho13k tube experiments Physi13s of Plasmas 16(11)
112705
REFERENCES 155
Drake RP 2006 High-energy-density physi13s Fundamentals inertial fu-
sion and experimental astrophysi13s Sho13k wave and high pressure phe-
nomena Springer Berlin Heidelberg New York ISBN 3-540-29314-0
httpopa13inriafrre13ord=b1128433
Drake R Doss F M13Clarren R Adams M Amato N Bingham D
Chou C DiStefano C Fidkowski K Fryxell B et al 2011 Radiative
ee13ts in radiative sho13ks in sho13k tubes High Energy Density Physi13s
7(3) 130
Ensman L 1994 Test problems for radiation and radiation-hydrodynami13s
13odes The Astrophysi13al Journal 424 275 DOI ADS
Ensman L Burrows A 1992 Sho13k breakout in sn 1987a The Astrophy-
si13al Journal 393 742
Fadeyev YA Gillet D 2001 The stru13ture of radiative sho13k waves
III The model grid for partially ionized hydrogen gas Astronomy and
Astrophysi13s 368 901 DOI ADS
Fleury X Bouquet S Stehleacute C Koenig M Batani D Benuzzi-
Mounaix A Chieze J-P Grandjouan N Grenier J Hall T et al
2002 A laser experiment for studying radiative sho13ks in astrophysi13s
Laser and Parti13le Beams 20(02) 263
Fryxell B Olson K Ri13ker P Timmes FX Zingale M Lamb DQ
Ma13Nei13e P Rosner R Truran JW Tufo H 2000 FLASH An
Adaptive Mesh Hydrodynami13s Code for Modeling Astrophysi13al Ther-
monu13lear Flasheshera13les a three-dimensional radiation hydrodynami13s
13ode Astrophysi13al Journal Supplement 131 273 DOI ADS
Gonzalez JH Brollo FR Clausse A 2009 Modeling of the Dynami13
Plasma Pin13h in Plasma Fo13us Dis13harges Based in Von Karman Approx-
imations IEEE Transa13tions on Plasma S13ien13e 37 2178 DOI ADS
Gonzalez JH Clausse A Bruzzone H Florido PCMZ 2004 A Lum-
ped Parameter Model of Plasma Fo13us IEEE Transa13tions on Plasma
S13ien13e 32 1383 DOI ADS
Gonzaacutelez M Audit E Huynh P 2007 HERACLES a three-dimensional
radiation hydrodynami13s 13ode Astronomy and Astrophysi13s 464 429
DOI ADS
Gonzaacutelez M Audit E Huynh P 2007 Hera13les a three-dimensional
radiation hydrodynami13s 13ode Astronomy and Astrophysi13s 464(2) 429
Gonzaacutelez M Audit E Stehleacute C 2009 2D numeri13al study of the ra-
diation inuen13e on sho13k stru13ture relevant to laboratory astrophysi13s
Astronomy and Astrophysi13s 497 27 DOI ADS
156 REFERENCES
Gonzaacutelez M Stehleacute C Rus B Busquet M Thais F A13ef O Barroso
P Bar-Shalom A Bauduin D Kozlova M Lery T Madouri A To-
mas M Polan J 2006 Astrophysi13al radiative sho13ks From modeling
to laboratory experiments Laser and Parti13le Beams 24 535540 DOI
ADS
Gonzalez M StehlE C Audit E Busquet M Rus B Thais F A13ef
O Barroso P Bar-Shalom A Bauduin D et al 2006 Astrophysi13al
radiative sho13ks From modeling to laboratory experiments Laser and
Parti13le Beams-Pulse Power and High Energy Densities 24(4) 535
Gonzaacutelez M Vaytet N Commerccedilon B Masson J 2015 Multigroup
radiation hydrodynami13s with ux-limited diusion and adaptive mesh
renement Astronomy and Astrophysi13s 578 A12 DOI ADS
Griem HR 2005 Prin13iples of Plasma Spe13tros13opy 386 ADS
Gu MF 2008 The exible atomi13 13ode Canadian Journal of Physi13s
86(5) 675
Harilal SS Tilla13k MS 2004 Laser plasma
density measurements using interferometry
httplinkapsorgdoi101103PhysRevE64066412
Hartigan P 1989 The visibility of the Ma13h disk and the bow sho13k of a
stellar jet Astrophysi13al Journal 339 987 DOI ADS
Hartigan P 2003 Sho13k Waves in Outows from Young Stars Astrophysi13s
and Spa13e S13ien13e 287 111 DOI ADS
Hartigan P Morse JA Reipurth B Heath13ote S Bally J 2001 Pro-
per Motions of the HH 111 Jet Observed with the Hubble Spa13e Teles13ope
Astrophysi13al Journal Letters 559 L157 DOI ADS
Henke BL Gullikson E Davis JC 1993 X-ray intera13tions Photoab-
sorption s13attering transmission and ree13tion atlt igt eltigt= 50-30000
evlt igt zltigt= 1-92 Atomi13 data and nu13lear data tables 54(2) 181
Hugoniot PH 1887 Meacutemoire sur la propagation du mouvement dans les
13orps et ples speacute13ialement dans les gaz parfaits Journal de matheacutematiques
pures et appliqueacutees 3 477
Hugoniot PH 1889 Meacutemoire sur la propagation des mouvements dans les
13orps et speacute13ialement dans les gaz parfaits (deuxiegraveme partie) Journal de
lEacute13ole Polyte13hnique 58 1
Hut13hinson IH 2002 Prin13iples of Plasma Diagnosti13s 458 ADS
REFERENCES 157
Jungwirth K Cejnarova A Juha L Kralikova B Krasa J Krousky
E Krupi13kova P Laska L Masek K Mo13ek T et al 2001 The
prague asterix laser system Physi13s of Plasmas 8 2495
Keiter P Drake R Perry T Robey H Remington B Iglesias C
Walla13e R Knauer J 2002 Observation of a hydrodynami13ally driven
radiative-pre13ursor sho13k Physi13al review letters 89(16) 165003
Koenig M Faral B Boudenne JM Batani D Benuzzi A Bossi S
1994 Opti13al smoothing te13hniques for sho13k wave generation in laser-
produ13ed plasmas Physi13al Review E 50 R3314 DOI ADS
Kondo K Nakajima M Kawamura T Horioka K 2006 Compa13t pulse
power devi13e for generation of one-dimensional strong sho13k waves Review
of S13ienti13 Instruments 77(3) 036104 DOI ADS
Kondo K Nakajima M Kawamura T Horioka K 2008 Relaxation
layer in ele13tro-magneti13ally driven strong sho13ks Journal of Physi13s Con-
feren13e Series 112(4) 042028 DOI ADS
Kondo K Nakajima M Kawamura T Horioka K 2009a Atomi13 pro-
13ess in ele13tro-magneti13ally driven strong sho13k wave Nu13lear Instruments
and Methods in Physi13s Resear13h A 606 223 DOI ADS
Kondo K Nakajima M Kawamura T Horioka K 2009b Radiation ef-
fe13t and relaxation layer in ele13tro-magneti13ally driven strong sho13k waves
In APS Meeting Abstra13ts ADS
Kunze H-J 2009 Springer series on atomi13 opti13al and
plasma physi13s 125 Springer Berlin Heidelberg New York
httpwwwspringer13omseries411
Kwek K Tou T Lee S 1990 Current sheath stru13ture of the plasma
fo13us in the run-down phase IEEE Transa13tions on Plasma S13ien13e 18(5)
826
Larour J Singh RL Stehleacute C Ciardi A Chaulagain U Suzuki-Vidal
F 2015 Optimization of an ele13tromagneti13 generator for strong sho13ks
in low pressure gas High Energy Density Physi13s 17 129 DOI ADS
Lee S 1969 Transverse ionizing sho13k waves in a planar ele13tromagneti13
sho13k tube PhD thesis Australian National University
Lee S Serban A 1996 Dimensions and lifetime of the plasma fo13us pin13h
IEEE Transa13tions on plasma s13ien13e 24(3) 1101
Leygna13 S Boireau L Mi13haut C Lanz T Stehleacute C Clique C
Bouquet S 2006 Modeling multidimensional ee13ts in the propagation
of radiative sho13ks Physi13s of Plasmas (1994-present) 13(11) 113301
158 REFERENCES
Ma13Farlane JJ Golovkin IE Woodru PR 2006 HELIOS-CR A 1-
D radiation-magnetohydrodynami13s 13ode with inline atomi13 kineti13s mo-
deling Journal of Quantitative Spe13tros13opy and Radiative Transfer 99
381 DOI ADS
Ma13quorn Rankine WJ 1870 On the Thermodynami13 Theory of Waves of
Finite Longitudinal Disturban13e Philosophi13al Transa13tions of the Royal
So13iety of London Series I 160 277 ADS
Mahmood S Ur-Rehman H 2009 Ele13trostati13 solitons in unmagnetized
hot ele13tron-positron-ion plasmas Physi13s Letters A 373 2255 DOI ADS
Martin TH Williams M Kristiansen M 1996 J13 martin on pulsed
power 3 Springer S13ien13e amp Business Media
M13Clarren RG Drake RP Morel JE Holloway JP 2010 Theory of
radiative sho13ks in the mixed opti13ally thi13k-thin 13ase Physi13s of Plasmas
17(9) 093301
Mi13haut C Stehleacute C Leygna13 S Lanz T Boireau L 2004 Jump
13onditions in hypersoni13 sho13ks Quantitative ee13ts of ioni13 ex13itation
and radiation European Physi13al Journal D 28 381 DOI ADS
Mihalas D 1978 Stellar atmospheres 2nd edition ADS
Mihalas D Mihalas BW 1984 Foundations of radiation hydrodynami13s
ADS
Mihalas D Mihalas BW 1999 Foundations of radiation hydrodynami13s
More R Warren K Young D Zimmerman G 1988 A new quotidian
equation of state (qeos) for hot dense matter Physi13s of Fluids (1958-
1988) 31(10) 3059
Ogando F Velarde P 2001 Development of a radiation transport uid
dynami13 13ode under amr s13heme Journal of Quantitative Spe13tros13opy and
Radiative Transfer 71(2) 541
Olson GL Auer LH Hall ML 2000 Diusion P1 and other approx-
imate forms of radiation transport Journal of Quantitative Spe13tros13opy
amp Radiative Transfer 64 619 DOI ADS
Orban C Fatenejad M Chawla S Wilks SC Lamb DQ 2013 A
Radiation-Hydrodynami13s Code Comparison for Laser-Produ13ed Plasmas
FLASH versus HYDRA and the Results of Validation Experiments ArXiv
e-prints ADS
REFERENCES 159
Orlando S Bonito R Argiro C Reale F Peres G Mi13eli M Mat-
sakos T Stehleacute C Ibgui L de Saacute L Chiegraveze J-P Lanz T 2013
Radiative a1313retion sho13ks along nonuniform stellar magneti13 elds in 13las-
si13al T Tauri stars Astronomy amp Astrophysi13s
Ovsyannikov AA Zhukov MF 2000 Plasma diagnos-
ti13s Cambridge International S13ien13e Publishing Cambridge
httppubli13eblib13om13hoi13epubli13fullre13ordaspxp=258189
Pfalzner S 2006 An introdu13tion to inertial 13onnement fusion 1st
edn Series in Plasma Physi13s Taylor amp Fran13is Taylor amp Fran13is
httpgenlibruse13bookindexphpmd5=EACF38292FD79113F78BB532AB5301B8
Potter DE 1971 Numeri13al Studies of the Plasma Fo13us Physi13s of Fluids
14 1911 DOI ADS
Proakis JG 2001 Digital signal pro13essins Wiley Online Library
Raga A Mellema G Arthur S Binette L Ferruit P Steen W 1999
3d transfer of the diuse ionizing radiation in ism ows and the preioni-
zation of a herbig-haro working surfa13e Revista Mexi13ana de Astronomia
y Astrosi13a 35 123
Reighard AB 2007 Collapsing radiative sho13k experiments on the omega
laser PhD thesis The University of Mi13higan
Reighard A Drake R Dannenberg K Kremer D Grosskopf M Har-
ding E Leibrandt D Glendinning S Perry T Remington B et al
2006 Observation of 13ollapsing radiative sho13ks in laboratory experiments
Physi13s of Plasmas (1994-present) 13(8) 082901
Rodriguez R Gil JM Florido R Rubiano JG Mendoza MA Mar-
tel P Minguez E Symes DR Hohenberger M Smith RA 2011
Determination of the average ionization and thermodynami13 regimes of
xenon plasmas with an appli13ation to the 13hara13terization of blast waves
laun13hed in xenon 13lusters High Energy Density Physi13s 7(2) 71 DOI
httpwwws13ien13edire13t13oms13ien13earti13lepiiS157418181100022X
Rodriacuteguez R Espinosa G Gil JM Stehleacute C Suzuki-Vidal F Rubi-
ano JG Martel P Miacutenguez E 2015 Mi13ros13opi13 properties of xenon
plasmas for density and temperature regimes of laboratory astrophysi13s
experiments on radiative sho13ks Physi13al Review E 91(5) 053106 DOI
ADS
Ross IN Pepler DA Danson CN 1995 Binary phase zone plate de-
signs using 13al13ulations of far-eld distributions Opti13s Communi13ations
116 55 DOI ADS
160 REFERENCES
Seaton MJ 1987 Atomi13 data for opa13ity 13al13ulations I General des13rip-
tion Journal of Physi13s B Atomi13 and Mole13ular Physi13s 20(23) 6363
Serban A 1995 Anode geometry and fo13us 13hara13teristi13s PhD thesis
Nanyang Te13hnology University httphdlhandlenet104971568
Serban A Lee S 1995 A simple opti13al ber axial speed dete13tor Review
of S13ienti13 Instruments 66 4958 DOI ADS
Serban A Lee S 1995 A simple opti13al ber axial speed dete13tor Review
of s13ienti13 instruments 66(10) 4958
Singh JP Thakur SN (eds) 2007 Laser-indu13ed breakdown
spe13tros13opy Elsevier s13ien13edire13t
Singh R Stehleacute C Suzuki-Vidal F Kozlova M Larour J Chaulagain
U Clayson T Rodriguez R Gil J Nejdl J et al 2017 Experimental
study of the intera13tion of two laser-driven radiative sho13ks at the pals
laser High Energy Density Physi13s 23 20
Stahler SW Palla F Salpeter EE 1986 Primordial stellar evolution -
The protostar phase The Astrophysi13al Journal 302 590 DOI ADS
Stehleacute C Gonzaacutelez M Kozlova M Rus B Mo13ek T A13ef O Colom-
bier JP Lanz T Champion N Jakub13zak K et al 2010 Experi-
mental study of radiative sho13ks at pals fa13ility Laser and parti13le beams
28(02) 253
Stehleacute C Kozlovaacute M Larour J Nejdl J Champion N Barroso P
Suzuki-Vidal F A13ef O Delattre P-A Dostaacutel J et al 2012 New
probing te13hniques of radiative sho13ks Opti13s Communi13ations 285(1)
64
Stehleacute C Lefegravevre R Chaulagain U Champion N Barroso P Reix
F Jagourel P Larour J Melt13hakov E Mer13ier R et al 2014a
Criti13al 13omponents for xuv probing of laser driven sho13ks In X-Ray
Lasers 2012 Springer 239
Stehleacute C Kozlovaacute M Larour J Nejdl J Suzuki-Vidal F Cohen M
Chaulagain UP Champion N Barroso P A13ef O et al 2014b
Reply on the 13omment of the paper new probing te13hniques of radiative
sho13ks Opti13s Communi13ations 318 226
Suzuki-Vidal F Lebedev S Pi13kworth L Swadling G Burdiak G
Skidmore J Hall G Bennett M Bland S Chittenden J et al
2014 Sho13k formation in 13ounter-streaming jets on the magpie pulsed-
power generator In APS Meeting Abstra13ts 1 5005
REFERENCES 161
Suzuki-Vidal F Clayson T Swadling GF Patankar S Burdiak GC
Lebedev SV Smith RA Stehle C Chaulagain U Singh RL La-
rour J Kozlova M Spindloe C Foster J Skidmore J Gumbrell E
Graham P Danson C 2016 Counter-streaming radiative sho13k experi-
ments on the Orion laser In APS Meeting Abstra13ts ADS
Toledo-Roy JC Velaacutezquez PF de Colle F Gonzaacutelez RF Reynoso
EM Kurtz SE Reyes-Iturbide J 2009 Numeri13al model for the SNR
DEM L316 simulated X-ray emission Monthly Noti13es of the RAS 395
351 DOI ADS
Tou T Lee S Kwek K 1989 Nonperturbing plasma-fo13us measurements
in the run-down phase IEEE Transa13tions on Plasma S13ien13e 17(2) 311
Vaytet NMH Audit E Dubro13a B Delahaye F 2011 A numeri13al
model for multigroup radiation hydrodynami13s Journal of Quantitiative
Spe13tros13opy and Radiative Transfer 112 1323 DOI ADS
Vaytet N Chabrier G Audit E Commerccedilon B Masson J Ferguson
J Delahaye F 2013 Simulations of protostellar 13ollapse using mul-
tigroup radiation hydrodynami13s ii the se13ond 13ollapse arXiv preprint
arXiv13071010
Vaytet N Gonzaacutelez M Audit E Chabrier G 2013 The inuen13e
of frequen13y-dependent radiative transfer on the stru13tures of radiative
sho13ks Journal of Quantitiative Spe13tros13opy and Radiative Transfer 125
105 DOI ADS
Velarde P Gar13iacutea-Senz D Bravo E Ogando F Relantildeo A Gar13iacutea C
Oliva E 2006 Intera13tion of supernova remnants From the 13ir13umstellar
medium to the terrestrial laboratory Physi13s of Plasmas 13(9) 092901
DOI ADS
Veloso F Moreno J Tarifentildeo-Saldivia A Pavez C Zambra M Soto
L 2012a Non-intrusive plasma diagnosti13s for measuring sheath kine-
mati13s in plasma fo13us dis13harges Measurement S13ien13e and Te13hnology
23(8) 087002
Veloso F Tarifentildeo-Saldivia A Pavez C Moreno J Zambra M Soto
L 2012b Plasma sheath kinemati13s and some impli13ations on the mo-
deling of very low energy plasma fo13us devi13es Plasma Physi13s and Con-
trolled Fusion 54(9) 095007
Vis13o A Drake R Glenzer S Doumlppner T Gregori G Froula D
Grosskopf M 2012 Measurement of radiative sho13k properties by x-ray
thomson s13attering Physi13al review letters 108(14) 145001
162 REFERENCES
Williams RM Chu Y-H Di13kel JR Beyer R Petre R Smith RC
Milne DK 1997 Supernova Remnants in the Magellani13 Clouds I The
Colliding Remnants DEM L316 Astrophysi13al Journal 480 618 ADS
Williams RM Chu Y-H Di13kel JR Gruendl RA Seward FD
Guerrero MA Hobbs G 2005 Supernova Remnants in the Magella-
ni13 Clouds V The Complex Interior Stru13ture of the N206 Supernova
Remnant Astrophysi13al Journal 628 704 DOI ADS
Zambra M Kalise D Fernandez J Hernandez E Pas-
ten D Munoz V 2009 Current sheet thi13kness in the
plasma fo13us snowplow model J Plasma Fusion 8(879-882)
httpspeopleri13amoeawa13atdkalisefilesplasmapdf
Zeldovi13h IB Zeldovi13h YB Raizer YP 2002 Physi13s of sho13k waves
and high-temperature hydrodynami13 phenomena Courier Dover Publi13ati-
ons London New York Sydney
REFERENCES 163
Strong radiative sho13ks relevant for stellar
environments experimental study and numeri13al
approa13h
ABSTRACT
Strong sho13ks are present in various astrophysi13al phenomena Su13h
sho13ks are strongly inuen13ed by the radiation through its 13oupling with
hydrodynami13s Thus their topology and dynami13s are quite 13omplex Ge-
nerating su13h hypersoni13 sho13ks in the laboratory with 13ontrolled 13ondi-
tions is thus an adequate tool to study the inuen13e of radiation and to
13ompare them with numeri13al simulations Su13h sho13ks 13an be generated
by intense lasers and ele13tromagneti13 devi13es The rst part of this dis-
sertation 13on13erns the numeri13al and experimental study of the intera13tion
of two 13ounter propagating laser-driven sho13ks The experiments perfor-
med at the kJ PALS laser fa13ility allowed to generate sho13ks with dierent
speeds (sim 30-55 kms and 10-25 kms) in noble gases and low pressure
(less than 1 bar) Several diagnosti13s were implemented visible interfero-
metry time- and spa13e-resolved visible spe13tros13opy and time integrated
XUV spe13tros13opy Our experiment shows a strong intera13tion of one radi-
ative pre13ursor onto the se13ond one The physi13al parameters of the plasma
were dedu13ed from the diagnosti13s and 13ompared with 1-D simulation re-
sults The se13ond part is devoted to the design of an experiment where the
sho13k is generated ele13tromagneti13ally The optimization of this generator is
presented and also the full experimental set up whi13h allows studying sho13k
sim 30 kms in noble gas at sim 1 mbar
Keywords Radiative sho13k hydrodynami13s laser-plasmas visible
and XUV spe13tros13opy
numeri13al simulation laboratory astrophysi13s high energy density
physi13s
Cho13s forts et radiatifs dinteacuterecirct pour les
environnements stellaires eacutetude expeacuterimentale et
appro13he numeacuterique
RESUME
Les 13ho13s forts sont preacutesents dans des pheacutenomegravenes astrophysiques varieacutes
De tels 13ho13s sont fortement inuen13eacutes par le rayonnement par son 13ouplage
ave13 lhydrodynamique Par suite leur topologie et leur dynamique sont
assez 13omplexes Geacuteneacuterer de tels 13ho13s hypersoniques en laboratoire dans
des 13onditions 13ontrleacutees est ainsi un outil pertinent pour eacutetudier linuen13e
du rayonnement et pour 13omparer aux reacutesultats des simulations numeacuteriques
Ces 13ho13s sont geacuteneacutereacutes par des lasers intenses et par des moyens eacutele13tromag-
neacutetiques La premiegravere partie du texte est 13onsa13reacute agrave leacutetude numeacuterique et
expeacuterimentale de lintera13tion de deux 13ho13s induits par laser se propageant
en sens 13ontraires Les expeacuterien13es ont eacuteteacute meneacutees sur linstallation laser kJ
PALS qui permet de former deux 13ho13s ave13 des vitesses propres dieacuteren-
tes (sim 30-55 et 10-25 kms respe13tivement) dans des gaz rares agrave pression
faible (moins de 1 bar) Des diagnosti13s ont eacuteteacute installeacutes interfeacuteromeacutetrie
visible spe13tros13opie visible agrave reacutesolution spatiale et temporelle spe13tros13o-
pie XUV inteacutegreacutee en temps Nos expeacuterien13es montrent une forte intera13tion
entre les deux preacute13urseurs radiatifs Les paramegravetres physiques du plasma
ont eacuteteacute deacuteduits de 13es diagnosti13s et 13ompareacutes aux reacutesultats de simulations
monodimensionnelles La se13onde partie est 13onsa13reacutee agrave la 13on13eption dune
expeacuterien13e ougrave le 13ho13 est geacuteneacutereacute de faccedilon eacutele13tromagneacutetique Loptimisation
de 13e geacuteneacuterateur est preacutesenteacutee ainsi que lenvironnement expeacuterimental per-
mettant deacutetudier des 13ho13s jusquagrave 30 kms dans des gaz rares peu denses
(1 mbar)
Mots 13leacutes 13ho13s radiatifs plasmas laser hydrodynamique spe13tros13opie
visible et XUV simulation numeacuterique astrophysique de laboratoire
physique agrave haute densiteacute deacutenergie
Reacutesumeacute long
En introdu13tion (13hapitre 1) les 13ho13s radiatifs (CR) sont preacutesenteacutes
dans de nombreux environnements astrophysiques en parti13ulier dans les
eacutetoiles Ce sont des 13ho13s forts ave13 un nombre de Ma13h eacuteleveacute (M1) et une
tregraves haute tempeacuterature 13e qui induit un rayonnement intense Alors que
lobservation de 13ette signature ave13 une reacutesolution spatiale est tregraves di13ile
en astrophysique une appro13he innovante 13elle dite des plasmas astrophy-
siques de laboratoire fournit un puissant moyen deacutetude des CR sur Terre
Depuis plus dune deacute13ennie 13es 13ho13s sont eacutetudieacutes en laboratoire prin13ipale-
ment sur de grandes installations laser dans les gaz rares et sous dieacuterentes
geacuteomeacutetries Lintera13tion entre un laser et une feuille min13e produit une
forte ablation et par eet fuseacutee a1313eacutelegravere la feuille qui se propage dans le
gaz agrave plusieurs dizaines de kms Pour des irradiations laser entre 10
14and
10
15W13m
2 on enregistre des vitesses de 13ho13s entre 40 et 150 kms En
parallegravele une des13ription ne est permise par des simulations numeacuteriques
Au deacutebut de 13e travail les expeacuterien13es eacutetaient 13on13entreacutees sur des CR
simples alors que la situation astrophysique est 13elle dun CR interagissant
ave13 un milieu plus dense donnant des 13ho13s reacuteeacute13his et transmis Des
exemples repreacutesentatifs de 13es pheacutenomegravenes sont la propagation de restes de
supernovae dans des nuages moleacute13ulaires denses les 13ho13s da1313reacutetion sur
la photosphegravere des eacutetoiles T-Tauri ou en13ore les 13ho13s deacutetrave agrave lavant
des jets stellaires La 13ollision (ou lintera13tion) de deux CR est bien sucircr un
eacutevegravenement rare en astrophysique mais un exemple est fourni par lintera13tion
de deacutebris de la supernova DEM L316 (13f Fig 1 de Williams et al (1997))
mecircme si lhypothegravese est 13ontesteacutee (Velarde et al (2006))
Il y a don13 une forte motivation pour reacutealiser une expeacuterien13e de labora-
toire pour eacutetudier la propagation et lintera13tion de deux 13ho13s fa13e agrave fa13e
et interpreacuteter les signatures observeacutees Cette appro13he 13onstitue une grande
part de mon travail En 13ompleacutement aux expeacuterien13es laser jai eacutetudieacute un
geacuteneacuterateur 13ompa13t de puissan13e pulseacutee alimentant un 13anon agrave plasma 13oax-
ial pour 13reacuteer des 13ho13s dinteacuterecirct astrophysique dans les gaz agrave basse pression
Les 13ho13s ainsi 13reacuteeacutes sont de plus grande taille et peuvent ecirctre eacutetudieacutes plus
fa13ilement (Kondo et al 2008) Pour 13ette thegravese je me suis atta13heacutee agrave preacute-
senter les 13ho13s radiatifs dans le 13ontexte de lastrophysique de laboratoire
en suivant 13es deux appro13hes
Le deuxiegraveme 13hapitre traite de la physique des CR Ils 13omposent
une 13lasse de 13ho13s supersoniques qui sont 13haueacutes agrave haute tempeacuterature
et sont par 13onseacutequent sour13es dun rayonnement intense En retour 13e
rayonnement modie la dynamique et la stru13ture du 13ho13 13e qui 13omplexie
la des13ription Les CR se ren13ontrent dans des situations astrophysiques
13omme les 13ho13s da1313reacutetion lors de la formation des protoeacutetoiles (Stahler
Palla et Salpeter 1986) lexplosion de supernovae et lintera13tion de leur
restes ave13 le milieu interstellaire dense (Chevalier 1977) ou en13ore les 13ho13s
deacutetrave en tecircte des jets stellaires (Hartigan et al 2001) Les CR sont
maintenant eacutetudieacutes en laboratoire 13e qui permet de 13omparer les reacutesultats
aux modegraveles de la litteacuterature et de veacuterier la possibiliteacute de les retrouver ave13
les simulations numeacuteriques disponibles (Bouquet et al (2004) Leygna13 et
al (2006)) Leacutetude expeacuterimentale des CR est 13ru13iale notamment pour
13omprendre les pro13essus eacutenergeacutetiques au sein des plasmas astrophysiques
Dans le 13as des gaz parfaits le saut au niveau dun 13ho13 est deacute13rit par les
eacutequations de Rankine-Hugoniot qui relient les quantiteacutes thermodynamiques
de part et dautre de la dis13ontinuiteacute La solution monodimensionnelle dun
13ho13 se propageant agrave la vitesse us dans un gaz au repos est usuellement
deacute13rite dans le repegravere lieacute au 13ho13 et on distingue la reacutegion amont (ou preacute-
13ho13) agrave la vitesse u1= - us et en arriegravere du 13ho13 (reacutegion post-13ho13 ou aval)
la vitesse est u2 selon la Fig 21 Pour les 13ho13s forts ougrave le nombre de Ma13h
M est tregraves grand devant 1 les sauts des quantiteacutes thermodynamiques dun
gaz parfait sont donneacutes par les eacuteq 21 agrave 24 Pour un gaz monoatomique
le rapport de 13ompression est de 4 et la tempeacuterature de la zone post-13ho13
est proportionnelle au 13arreacute de la vitesse du 13ho13 et agrave la masse atomique
La situation est plus 13omplexe pour un gaz reacuteel 13ar une part de leacutenergie
13ineacutetique sert agrave ex13iter et agrave ioniser le gaz du post 13ho13 La tempeacuterature
attendue est don13 infeacuterieure agrave 13elle du 13as ideacuteal en outre la pression est
modieacutee par lionisation (Mi13haut et al 2004) Un nouveau jeu deacutequations
in13orpore la 13harge ee13tive Z des ions (eacuteq 25 agrave 28) Ces modegraveles ignorent
le 13hauage et le refroidissement radiatifs alors que le rayonnement eacutemis
ae13te la stru13ture dun 13ho13 fort Cette des13ription demande dajouter aux
eacutequations de 13onservation de la masse quantiteacute de mouvement et eacutenergie les
13ontributions radiatives (ux eacutenergie et pression Le 13as est plus 13omplexe
et les eacutequations dhydrodynamique radiative ont eacuteteacute exprimeacutees dans le repegravere
du 13ho13 (Mihalas and Mihalas 1999 Drake 2006) selon les eacuteq 29 agrave 211
Les 13ontributions de la pression radiative et du 13hamp de rayonnement
sont importantes agrave tregraves haute tempeacuterature (vitesse) mais deacutejagrave pour nos 13ho13s
le ux radiatif est dominant Selon lopa13iteacute le rayonnement du 13ho13 peut
ecirctre absorbeacute par la region preacute-13ho13 induisant un 13hauage loin en avant de
la dis13ontinuiteacute Egalement la zone 13hoqueacutee pro13he de la dis13ontinuiteacute est
ae13teacutee par la 13ompeacutetition entre leacutemission (refroidissement) et labsorption
(13hauage) Ainsi leacutepaisseur optique lo13ale (eacuteq 211) devient le paramegravetre
pertinent Comme on raisonne sur le ux radiatif moyenneacute en freacutequen13e
13est lopa13iteacute de Rosseland qui est utiliseacutee et on dieacuteren13ie les reacutegimes op-
tiquement eacutepais et optiquement min13e 13e qui permet de 13lasser les 13ho13s
radiatifs
Notre inteacuterecirct sest 13on13entreacute sur les 13ho13s preacutesentant un preacute13urseur radi-
atif En laboratoire une intensiteacute laser au dessus de 10
14W13m
2le permet
et les installations sont rappeleacutees dans le tableau 21 Il est aussi possible de
lan13er des 13ho13s moins rapides don13 peu radiatifs (sim 10-30 kms) ave13 un
piston eacutele13tromagneacutetique
Le 13hapitre 3 est 13onsa13reacute agrave lanalyse numeacuterique des CR en deacute13rivant
les pro13essus physiques jouant un rle dans 13es systegravemes physiques De mecircme
que les observations les expeacuterien13es en laboratoire ne sont pas dire13tement
interpreacutetables et la simulation numeacuterique devient un outil preacute13ieux Diverses
appro13hes numeacuteriques sont utiliseacutees pour simuler les plasmas de laboratoire
On 13onsidegravere soit des grilles xes soit des grilles adaptables (Adaptative
Mesh Renement) pour maintenir une reacutesolution susante sur 13haque zone
du plasma Une autre appro13he utilise des grilles qui suivent le uide dans
son mouvement (modegravele Lagrangien) don13 sans passage de matiegravere dune
13ellule agrave la voisine Les eacutequations sont alors dieacuterentes (Orban et al 2013)
Pour deacute13rire les 13ollisions de 13ho13s jai utiliseacute le 13ode 13ommer13ial HE-
LIOS un 13ode Lagrangien mono-dimensionnel pouvant geacuterer le rayonnement
et lhydrodynamique (Ma13Farlane Golovkin et Woodru 2006) Assez fa13ile
agrave prendre en main il a le gros avantage de pouvoir simuler notre 13ongura-
tion deux 13ho13s de vitesses opposeacutees lan13eacutes par deux lasers HELIOS deacute13rit
les plasmas hors deacutequilibre thermodynamique (hors ETL) gracirc13e agrave un mo-
degravele 13ollisionnel-radiatif (HELIOS-CR) de 13al13ul des populations atomiques
agrave 13haque pas de simulation hydrodynamique De fait jai utiliseacute HELIOS
dans une approximation ETL qui 13orrespond aux 13onditions expeacuterimentales
(Rodriguez et al 2011) et 13ela est dis13uteacute en deacutetails
Le 13ode reacutesout les eacutequations uides en introduisant les pressions des eacutele13-
trons des ions et du rayonnement Ele13trons et ions sont deacute13rits par deux
uides en intera13tion de tempeacuteratures respe13tives Te et Ti La 13ondu13tion
thermique est geacutereacutee par 13elle des eacutele13trons agrave partir de la 13ondu13tiviteacute de
Spitzer et le deacutept deacutenergie laser par un modegravele de Bremsstrahlung inverse
Leacutemission et labsorption du rayonnement sont introduites dans leacutequation
deacutenergie des eacutele13trons et dans 13elles du transport radiatif Une meacutethode
utilise un modegravele de diusion du rayonnement multi-groupes et agrave ux li-
miteacute ougrave le ux radiatif est proportionnel au gradient de leacutenergie radiative
et inversement proportionnel agrave lopa13iteacute de Rosseland ave13 une pondeacuteration
pour obtenir une bonne des13ription du 13as optiquement min13e selon Olson
Auer et Hall (2000) Une se13onde meacutethode utilise un s13heacutema deacutependant du
temps agrave 13ourte distan13e et multi-angles Dans notre 13as ougrave rayonnement
et hydrodynamique sont fortement 13oupleacutes et aussi en variation rapide on
utilise le modegravele de diusion et les opa13iteacutes ETL multi-groupes de Plan13k et
Rosseland
En outre notre version dHELIOS utilise PROPACEOS (Ma13Farlane
Golovkin et Woodru 2006) une table deacutequation deacutetat et dopa13iteacute multi-
groupes geacuteneacutereacutee par le 13ode sans possibiliteacute dintroduire une autre table La
13onnaissan13e des 13onditions thermodynamiques est neacute13essaire pour la simu-
lation du plasma dans nos 13as masse volumique entre 10
minus4et 10
minus1g13m
3
tempeacuterature entre 0 et 50 eV Une eacutevaluation preacute13ise par Rodriguez et al
(13f Fig 2 de Rodriguez et al 2011) pour le Xeacutenon indique que les 13onditi-
ons thermodynamiques de nos CR 13orrespondent au reacutegime ETL Leacutequation
deacutetat relie pression ionisation et eacutenergie interne agrave la masse volumique et agrave
la tempeacuterature Notre version dHELIOS pour plasma ETL utilise PROPA-
CEOS qui est peu do13umenteacutee elle utilise le modegravele de More et al (1988)
en reacutegime de 13ouplage fort (haute densiteacute et basse tempeacuterature) et un modegravele
datome isoleacute pour un 13ouplage faible Des interpolations sont in13luses pour
passer au 13ouplage fort (httpwwwprism-13s13omSoftwarePROPACEOS)
Jai reacutealiseacute des simulations HELIOS 1D pour divers 13as de CR 13ho13 seul
ou 13ho13s en 13ollision qui sont deacute13rites dans 13e 13hapitre 3 Les CR se propa-
geant dans le Xe agrave 01 bar et agrave 50 kms sont 13ara13teacuteriseacutes par un preacute13urseur
radiatif eacutetendu Une 13ompression eacuteleveacutee (38) dans le post-13ho13 est attribueacutee
agrave lionisation du gaz et au refroidissement radiatif Les tempeacuteratures du
post-13ho13 et du preacute-13ho13 de part et dautre du pi13 sont identiques signe
dun 13ho13 super13ritique Le rle spe13ta13ulaire de lopa13iteacute a eacuteteacute remarqueacute
mais agrave 13onsideacuterer lin13ertitude sur lopa13iteacute du Xeacutenon et le traitement 1D
jai deacute13ideacute de ne pas 13her13her agrave raner la simulation en termes de nombre
de groupes
Leacutetude numeacuterique permet de retrouver les 13ara13teacuteristiques de lintera13tion
de deux 13ho13s opposeacutes ave13 des vitesses respe13tives 50-50 kms et 50-20 kms
Dans tous les 13as la prin13ipale signature de lintera13tion (don13 avant la 13ol-
lision) est le re13ouvrement des preacute13urseurs agrave t0+8 ns pour 50-50 kms et
agrave t0+15 ns pour 50-20 kms Ensuite on assiste agrave une remonteacutee reacuteguliegravere
de la densiteacute et de la tempeacuterature eacutele13troniques Linstant de la 13ollision
voit une augmentation soudaine de la densiteacute eacutele13tronique (par un ordre de
grandeur) qui atteint 66 times 10
21et 3 times 10
2113m
minus3respe13tivement alors que
la tempeacuterature monte agrave 39 et 28 eV
Dans le quatriegraveme 13hapitre je preacutesente linstallation laser PALS puis
la 13ellule dintera13tion son implantation et les diagnosti13s asso13ieacutes Rappe-
lons le prin13ipe expeacuterimental Quand une impulsion bregraveve de lumiegravere laser
est fo13aliseacutee sur une feuille min13e une part importante deacutenergie est trans-
feacutereacutee et le 13hauage du solide 13onduit agrave son ablation Un plasma 13oronal
13haud et peu dense est eacuteje13teacute vers larriegravere et un 13ho13 vers lavant se forme
dans la feuille par eet fuseacutee Comme 13ette feuille limite le tube le 13ho13
se transmet au gaz la feuille agissant 13omme un piston Des expeacuterien13es
ont eacuteteacute meneacutees sur le Prague Asterix Laser System (PALS) au printemps
2015 (5 semaines 20 avril - 22 mai) ave13 pour obje13tif la premiegravere eacutetude
de lintera13tion puis de la 13ollision de deux CR sous la13tion de deux lasers
au niveau de 10
14W13m
2 Le Prague Asterix Laser System (PALS) repose
sur un laser agrave iode infrarouge (Asterix IV Jungwirth et al 2001) Apregraves
ampli13ation il deacutelivre jusquagrave 1 kJ en 03 ns agrave la longueur donde fonda-
mentale de 1315 nm Des fais13eaux auxiliaires sont disponibles agrave freacutequen13e
doubleacutee (λ = 657 nm) ou tripleacutee (438nm) PALS deacutelivre deux tirs par heure
agrave haute eacutenergie et dans de bonnes 13onditions dhomogeacuteneacuteiteacute du fais13eau
Pour nos expeacuterien13es nous utilisons deux fais13eaux Le fais13eau fondamental
est diviseacute apregraves le 4egraveme ampli13ateur en deux fais13eaux deacutenergie 60 et
40 (voir Fig 42) Le plus puissant est inje13teacute dans le 5
meampli13ateur et
sa freacutequen13e est tripleacutee Ce fais13eau agrave 438 nm est nommeacute MAIN Le se13ond
fais13eau est utiliseacute sans modi13ation don13 agrave 1315 nm et est nommeacute AUX
Le s13heacutema de distribution est en Fig 42 et les proprieacuteteacutes de MAIN and
AUX laser sont rappeleacutees Tdans le ableau 41
PALS propose deux 13hambres agrave vide spheacuterique et 13ylindrique respe13ti-
vement nous avons utiliseacute la 13hambre spheacuterique (Fig 43) Dun diamegravetre
de 100 13m elle est a1313essible aux fais13eaux MAIN et AUX par deux hublots
de 80 13m et 50 13m Des portes sont ameacutenageacutees ainsi que de nombreux ports
A linteacuterieur la 13ellule et des diagnosti13s sont monteacutes sur une table optique
deacute13oupleacutee meacute13aniquement de la 13hambre Les 13ibles (Fig 45) 13omprennent
un petit tube long de 4 mm fermeacute agrave ses deux extreacutemiteacutes par des feuilles min-
13es de parylegravene-N doreacute de 11 microm sur lesquelles les deux lasers sont fo13aliseacutes
au niveau de 10
14W13m
minus2 Le tube est rempli du gaz dans lequel le 13ho13
se propagera agrave une vitesse attendue entre 30 et 60 kms Les feuilles min13es
assurent la 13onversion de leacutenergie laser en eacutenergie 13ineacutetique via lablation et
la geacuteneacuteration de 13ho13 Des 13ellules speacute13iques en dural massif permettent
lalignement et la mesure du diamegravetre du fais13eau Toutes les 13ellules sont
reacutealiseacutees par le Ple instrumental de lObservatoire de Paris
Les 13ellules agrave gaz sont remplies in situ agrave une fra13tion de bar ave13 Xe
Ar Xe90He10 et He permettant ainsi de bien 13onnaicirctre le milieu ougrave se
propagent les 13ho13s Le systegraveme de remplissage (Fig 410) permet de limiter
leacute13art de pression sur les fenecirctres de la 13ellule et permet de suivre la pression
jusquau moment du tir Cest 13ritique pour ne pas 13asser la fenecirctre ultra
ne de Si3N4 pour la spe13tros13opie XUV et pour geacuterer une fuite eacuteventuelle
Le reacuteglage de la position et de la fo13alisation permet dobtenir des ta13hes
laser 13entreacutees de diamegravetres 450 - 500 microm et 250 - 300 microm pour MAIN et
AUX respe13tivement Ce 13ontrle est reacutepeacuteteacute 13haque jour Les impa13ts sont
toujours suivis par des 13ameacuteras X au keV mecircme si limpa13t sur la feuille de
parylegravene-N donne un signal X faible 13ompareacute agrave 13elui du tir sur Al
Les diagnosti13s du plasma sont prin13ipalement linterfeacuteromeacutetrie visible et
la spe13tros13opie XUV et visible Lalignement est suivi par des 13ameacuteras dans
le visible Pour linterfeacuteromeacutetrie visible preacute13iseacutement on doit prendre une
image de reacutefeacuteren13e (sans franges) de 13haque 13ible (Fig 417a) Le fais13eau de
reacutefeacuteren13e de linterfeacuteromegravetre Ma13h-Zehnder est bloqueacute et le fais13eau sonde est
utiliseacute seul La fente est ensuite reacuteduite agrave 200 microm pour ne sonder que la zone
13entrale du tube mais seacutetendant le long de laxe jusquaux pistons pour avoir
une reacutefeacuteren13e des positions initiales Ensuite en mode interfeacuterogramme on
13reacutee un systegraveme de franges perpendi13ulaires agrave la fente don13 perpendi13ulaires
agrave la dire13tion de propagation du 13ho13 (13f Fig 417b et Fig 413) Le tube
de 13ho13 est imageacute sur la 13ameacutera ave13 un grandissement susant (54 mm) et
il est possible dassurer un balayage eacutele13tronique de la fente sur le deacutete13teur
jusquagrave 200 ns Un interfeacuterogramme dit streak permet ainsi de suivre limpa13t
des 13ho13s sur les franges en fon13tion du temps
Leacutemission XUV du plasma est enregistreacutee ave13 une inteacutegration spatiale
et temporelle sur un spe13trographe agrave 13hamp plan muni dun reacuteseau 13on13ave
Il est installeacute dans la partie haute de la 13hambre (13f Fig 418) et enregistre
leacutemission XUV traversant la fenecirctre de Si3N4 Une spe13tros13opie visible
reacutesolue en temps et en position longitudinale a eacuteteacute installeacutee (Fig 419a)
mais na pas en13ore pu ecirctre exploiteacutee 13omplegravetement
Le 13hapitre 5 preacutesente les reacutesultats des deux diagnosti13s surtout linterfeacute-
romeacutetrie qui a eacuteteacute 13omplegravetement analyseacutee Jai deacuteveloppeacute une pro13eacutedure
de deacutepouillement ave13 des outils numeacuteriques ad ho13 de 13al13ul et danalyse
dimage Certes linterfeacuteromeacutetrie ne permet pas de sonder des plasmas plus
denses que la densiteacute 13ritique imposeacutee par le laser (4 times 10
2113m
minus3agrave 627 nm)
et on 13onstate que le post-13ho13 est opaque Au 13ontraire le rayonnement
XUV est preacutesent dans toute la stru13ture et la spe13tros13opie XUV inteacutegreacutee
en temps et en espa13e permet dexplorer agrave la fois le post-13ho13 et le preacute13ur-
seur Lanalyse des interfeacuterogrammes streak en visible part dune eacutevaluation
de leet de lindi13e de reacutefra13tion des eacutele13trons deacutetailleacutee dans lappendi13e A
mais qui se heurte assez vite agrave une perte de 13ontraste et agrave une disparition
des franges bien en dessous de la valeur de la densiteacute 13ritique deacutejagrave menti-
onneacutee Les deacutephasages a1313essibles ave13 preacute13ision ne deacutepassent guegravere 2 ou 3
fois 2π La re13her13he de maxima des franges est semi automatiseacutee et permet
darriver agrave une densiteacute eacutele13tronique moyenneacutee sur les 600 microm de leacutepaisseur
du tube de 13ho13 Le prol non plan du 13ho13 13onduit agrave penser que la moyenne
sous-estime dun fa13teur 2 voire plus la densiteacute dans le preacute13urseur La dis-
parition des franges est attribueacutee agrave la preacutesen13e du front de 13ho13 Par suite
les interfeacuterogrammes permettent de mesurer des vitesses de 13ho13s de 30-55
et 10-30 kms pour MAIN et AUX respe13tivement ainsi que des densiteacutes
eacutele13tronique dans les preacute13urseurs entre 10
17et 10
1913m
minus3
Nous avons ainsi deacutemontreacute lintera13tion entre deux preacute13urseurs radiatifs
dans Xe at 01 bar pour deux 13ho13s agrave 54 et 23 kms Lintera13tion est
13lairement 13ara13teacuteriseacutee par laugmentation de londe dionisation puis par la
superposition des deux preacute13urseurs agrave t0+20ns La 13ollision est enregistreacutee agrave
t0+47 ns un reacutesultat retrouveacute par la simulation
Nous avons reacutepeacuteteacute mais agrave plus haute pression 02 bar et trouveacute des
vitesses de sim 41 kms pour MAIN et sim 18 kms pour AUX Cependant
nous navons pas de signature du preacute13urseur radiatif pour AUX et le temps de
13ollision na pas eacuteteacute a1313essible agrave lenregistrement Le preacute13urseur 13teacute MAIN
nest pas inuen13eacute par AUX jusquagrave la limite de t0+48 ns (Fig 59(b) et (13))
La simulation preacutedit un tregraves faible preacute13urseur 13teacute AUX et une intera13tion
des preacute13urseurs agrave t0+49 ns
Linterfeacuteromeacutetrie transverse agrave 02 bar ave13 des vitesses sim 40 et 20 kms
indique que le preacute13urseur 13teacute MAIN a une extension lateacuterale de sim 600 microm13ontre 300 microm pour AUX Le preacute13urseur de MAIN est leacutegegraverement ae13teacute
pregraves des parois alors que 13elui de AUX est fortement 13ourbeacute Les eets 2D
sont don13 pronon13eacutes pour AUX et faibles pour MAIN En outre pour le
Xeacutenon nous disposons de la tempeacuterature et de la 13harge moyenne par le
spe13tre XUV inteacutegreacute (tir48143 Xeacutenon 06 bar) On peut en 13on13lure que
la 13harge moyenne est au moins eacutegale agrave 6 et quon a atteint une tempeacuterature
de 15 eV
Nos simulations donnent une des13ription qualitative des CR en inte-
ra13tion agrave la reacuteserve pregraves que nous introduisions dans HELIOS une uen13e
laser ajusteacutee pour donner la bonne vitesse Cependant il est maintenant
admis que les simulations 2D (ave13 les opa13iteacutes 13orre13tes) deacute13rivent bien les
expeacuterien13es (Gonzaacutelez Audit et Stehleacute 2009 Leygna13 et al 2006 Stehleacute
et al 2010) A la mecircme eacutenergie laser le 13al13ul 2D 13onduit agrave diminuer la
vitesse du 13ho13 13ompare au 13al13ul 1D et aussi agrave une baisse de la densiteacute
eacutele13tronique Par exemple pour un 13ho13 lan13eacute agrave PALS par un laser agrave 1315
nm dans le Xeacutenon agrave 03 bar ave13 une uen13e de 85000 J13m
2 ARWEN 2D
donne une vitesse de 44 kms en a1313ord ave13 lexpeacuterien13e (Cotelo et al
2015) La simulation 1D demanderait 30000 J13m
2pour obtenir la mecircme
vitesse
De mecircme la spe13tros13opie XUV inteacutegreacutee agrave 06 bar pour des vitesses reacuteel-
les de sim 39 et 18 kms indique que la tempeacuterature a atteint 15 eV et que
la 13harge moyenne a atteint 6 ou 7 alors que la simulation 1D preacutedit 10-30
eV et 5-10 respe13tivement (Fig 518) On peut en 13on13lure quune eacutetude
deacutetailleacutee baseacutee sur des simulations 2D et un post-traitement du transfert de
rayonnement sont neacute13essaires pour raner lanalyse Pour les autres gaz
rares (Ar Kr) on a observeacute que agrave eacutenergie laser donneacutee la vitesse de CR est
une fon13tion deacute13roissante de la masse volumique Ce13i 13onrme que pour
une densiteacute et une vitesse les eets radiatifs augmentent ave13 le numeacutero
atomique Pour le Krypton on a observeacute un petit preacute13urseur mais au13une
intera13tion Celle-13i ne serait possible quen augmentant nettement la vi-
tesse don13 leacutenergie du laser Ce13i a eacuteteacute rendu possible sur linstallation
laser Orion (AWE Aldermaston G-B) ougrave la 13ollision de deux CR deacutegale
vitesse a eacuteteacute obtenue agrave sim 80 kms pour une uen13e laser sim 6 times 10
14W13m
2
(Clayson et al 2016 Suzuki-Vidal et al 2016) pour une large gamme de
gaz rares et des pressions entre 01 et 1 bar Pour eacutetudier agrave la fois les 13ho13s
et les preacute13urseurs radiatifs de nombreux diagnosti13s eacutetaient installeacutes radi-
ographie X imagerie optique en 13ameacutera agrave balayage de fente interfeacuteromeacutetrie
multi-vues et agrave balayage de fente Bien que je naie pas pu parti13iper aux
13ampagnes jai fait des simulations 1D pour interpreacuteter les reacutesultats Cet
aspe13t nest pas deacute13rit dans 13ette thegravese mais a eacuteteacute publieacute tregraves reacute13emment
(Clayson et al 2017)
Apregraves des 13hapitres 13onsa13reacutes aux 13ho13s radiatifs geacuteneacutereacutes par laser le
13hapitre 6 deacute13rit un moyen alternatif passant par la voie eacutele13trique agrave haute
puissan13e Les 13hires de la voie laser sont tregraves eacuteleveacutes en uen13e plus de
10
14W13m
2 mais repreacutesentent des eacutenergies modeacutereacutees dans limpulsion (120
et 60 J pour MAIN et AUX au PALS) Or un systegraveme eacutele13trique devant
deacutelivrer 100 J reste modeste pour sa partie sto13kage Les vitesses de 50
kms seront probablement di13iles agrave atteindre mais on peut tabler sur un
reacutegime 13ompleacutementaire des 13ho13s radiatifs ave13 plus de 10 kms si le ren-
dement de 13onversion en eacutenergie 13ineacutetique reste 13orre13t Dougrave un neacute13essaire
travail doptimisation Des vitesses jusquagrave 100 kms avaient eacuteteacute mesureacutees
tregraves tt dans des 13anons agrave plasma dhydrogegravene agrave basse pression (Lee 1969)
mais 13e13i a eacuteteacute exploiteacute ensuite pour dautres naliteacutes La litteacuterature ne
mentionne quun 13as de 13anon eacutele13tromagneacutetique 13oaxial deacutedieacute aux 13ho13s
dinteacuterecirct astrophysique (Kondo et al 2006) Il est signaleacute que le lan13eur
eacutele13tromagneacutetique produit des 13ho13s plans de grande dimension transverse
(1 ordre de grandeur par rapport agrave la voie laser) et don13 plus fa13iles agrave in-
vestiguer (Kondo et al 2008) Le prin13ipe de fon13tionnement est le suivant
i) 13reacuteation dun 13laquage haute tension initial sur une surfa13e dieacutele13trique
fa13e agrave du gaz ii) 13hauage ohmique du plasma par le 13ourant en 13roissan13e
rapide iii) a1313eacuteleacuteration de la 13ou13he de plasma sous la pression magneacuteti-
que auto-geacuteneacutereacutee par le 13ourant (dizaines de kA) iv) maintien (1 micros) de la
pression magneacutetique pour 13ommuniquer une impulsion importante Des tra-
vaux sur le Z-pin13h ou le plasma fo13us permettent de postuler des geacuteomeacutetries
13onvenables pour a1313eacuteleacuterer une masse de lordre de 10
minus510
minus4g
Apregraves avoir eacutetudieacute le fon13tionnement dun 13ir13uit eacutele13trique rapide jai
listeacute les points forts et les points faibles en fon13tion des obje13tifs et jai pro-
poseacute une geacuteomeacutetrie 13oaxiale 13ompa13te Loptimisation du 13anon agrave plasma a
eacuteteacute faite en terme de vitesse maximale agrave la sortie ave13 13omme seules 13ontrain-
tes une eacutenergie sto13keacutee de 1 kJ une dimension transverse de 4 mm et une
dimension axiale dun ordre plus grande Lobje13tif eacutetait le 13ho13 fortement
supersonique 10-30 kms agrave quelques mbar dAr ou de Xe Un modegravele simple
(0D) a eacuteteacute 13ompareacute agrave un modegravele 3D MHD par le 13ode GORGON qui deacute13rit
bien les expeacuterien13es de plasmas astrophysique par puissan13e pulseacutee (Ciardi
et al 2007)
La pression magneacutetique sexprimant par B
22micro0 un 13ourant supeacuterieur agrave
100 kA apparaicirct neacute13essaire et atteignable sur un geacuteneacuterateur 1 kJ - 1 micros 13om-
pa13t de type R-L-C Ensuite il a fallu monter un modegravele de 13ir13uit deacuteformable
13omportant une se13tion formeacutee de la lame de plasma en mouvement et qui
prenne en 13ompte la1313reacutetion du gaz dans un modegravele dit snowplow (Potter
1971) Des paramegravetres geacuteomeacutetriques 13omme la forme et les dimensions du
13anon ont pu ecirctre optimiseacutes pour donner une grande vitesse de sortie du
plasma pour Ar et Xe et dans la gamme 01 - 10 mbar Un 13anon a eacuteteacute 13on-
struit et 13onne13teacute agrave un geacuteneacuterateur eacutele13trique pulseacute existant Des observations
du mouvement du plasma par la dynamique du rayonnement radial donnent
des vitesses 13oheacuterentes ave13 le modegravele ave13 la simulation 3D MHD et des
nombres de Ma13h eacuteleveacutes de 20 agrave 60 Ces travaux ont soutenu la 13on13eption
dun autre geacuteneacuterateur pour obtenir des performan13es plus eacuteleveacutees en vitesse
de 13ho13 et en taux de reacutepeacutetition
Contents
List of Figures
List of Tables
1 Introdu13tion 1
11 General Context 1
12 Outline of Chapters 2
13 My 13ontribution 3
2 Radiative sho13k waves 5
21 Rankine-Hugoniot(R-H) relations 6
211 Jump relations for a real gas 9
212 Ee13t of radiation on sho13k waves 11
213 Radiative hydrodynami13s equations 13
22 Radiative sho13k regimes 15
221 Typi13al radiative sho13k waves with a pre13ursor 16
23 Radiative sho13k waves experiments 16
231 Laser-driven sho13k experiments 18
232 Ele13tromagneti13ally laun13hed sho13k 21
3 1D Simulations 23
31 HELIOS 23
32 LTE approximation 24
321 Mean opa13ity 25
322 Equation of State 27
33 Single radiative sho13k waves 27
34 Intera13ting radiative sho13k waves 40
35 Summary 44
4 Experimental Setup 47
41 Prague Asterix Laser System (PALS) fa13ility 48
42 Targets 49
421 Massive Targets 51
422 Gaseous Targets 51
423 Target holder 55
CONTENTS
424 Target lling 56
43 Laser Fo13using 57
431 Fo13al Lenses and Phase Zone Plates 57
432 Fo13using of the MAIN and AUX laser beams 58
44 Diagnosti13s 59
441 Visible interferometry 61
442 XUV spe13tros13opy 65
443 Visible spe13tros13opy 67
45 Summary 68
5 Results and Interpretation 71
51 Visible interferometry 71
511 Longitudinal interferometry sho13k speed and ele13tron
density 74
512 Transverse interferometry lateral extension of the sho13k 91
52 XUV Spe13tros13opy 92
53 Simulations based on experimental results 93
54 Summary 97
6 Ele13tromagneti13ally laun13hed sho13k 101
61 Prin13iples of operation of a high 13urrent generator 102
62 Prin13iples of the run-down phase in a PFD 103
63 Proposed design for the plasma gun 104
64 Dynami13 13ir13uit modelling 106
65 3-D MHD simulations using GORGON 13ode 111
66 Measurements 116
67 Summary 118
7 Con13lusion 121
71 Con13lusions 121
72 Perspe13tives 123
8 Thesis summary 127
Appendi13es 141
A Visible Interferometry 143
A01 Refra13tive index of a plasma 143
A02 Absorption of the laser beam 144
A03 Prin13iple of interferometry 145
A04 Ma13h Zehnder Interferometer 146
B Opa13ities and mean 13harge 149
Referen13es 152
List of Figures
21 S13hemati13 diagram for a 1-D sho13k moving in the x dire13tion
The speed of the sho13k front is us In the frame of referen13e of
the sho13k front the pre-sho13k region has a relative speed u1=
-us density ρ1 total pressure P1 temperature T1 whereas these
quantities are respe13tively u2 ρ2 P2 and T2 for the post-sho13k
region 6
22 Proles of temperature (Te = Ti = T ) and mass density (ρ) at 10ns for a sho13k propagating in Xenon at 01 bar (ρ1= 57 times 10
minus4
g13m
minus3) with a speed of sim 45 kms It is to note that the given
pressure value are at 298 K The simulation is performed with the
HELIOS 13ode ex13luding the radiation ee13t and for a polytropi13
gas (γ=53) The initial temperature of Xenon is set to 1 eV
and with a negligible (10
minus5WmK) thermal 13ondu13tivity The
Ma13h number is then equal to 40 For this simulation the sho13k
is laun13hed by a 100 microm thi13k Aluminium piston moving at the
velo13ity of 33 kms The position x=0 13orresponds to the position
of the XeAl interfa13e at time t=0 8
23 Ele13tron temperature (Te) ion temperature (Ti) and mass density
(ρ) proles at 10 ns derived from 1-D simulation for the 13ase of
ionisation (no radiation) The sho13k propagates in Xe gas at 01
bar (ρ1= 57 times 10
minus4g13m
minus3by sim 45 kms) It is to note that the
given pressure value is at 298 K The simulation is performed with
the HELIOS (Te 6= Ti 13ase) using the PROPACEOS equation
of state without any radiation and with a negligible thermal
13ondu13tivity (10
minus05WmK) The initial temperature in Xenon
is set to 1 eV and thus the Ma13h number is equal to 40 For this
simulation the sho13k is laun13hed by a 100 microm thi13k Aluminium
piston moving at the velo13ity of 38 kms The position x=0
13orresponds to the position of the XeAl interfa13e at time t=0 10
LIST OF FIGURES
24 Figure 14 amp 17 of Mi13haut et al (2004) (a) Compression ρ2ρ1(full 13ir13les thin line) kT2 in eV (inverted full triangles thin line)
in Argon (a) Xenon (b) for initial 13onditions kT1( = 10 eV for
Ar and = 01 eV for Xe) ρ1 = 5 times 10minus4 gcm3 versus sho13k
speed in kms with (full markers full lines) and without (empty
markers dashed lines) radiation The ionization stage lt z gt2 is
plotted with the diamond symbol 12
25 Simulated ele13tron temperature (Te) ion temperature (Ti) and
mass density (ρ) proles at 10 ns of a radiative sho13k propagatingwith a 15 kms (a sub-13riti13al) and 48 kms (b super13riti13al)
The sho13k is propagating in Xenon at 01 bar (ρ = 54 times 10
minus4
g13m
minus3 at 298 K) The initial temperature in Xenon is set to
1eV The two Ma13h numbers are respe13tively equal to 13 and 43
The radiation and ionisation ee13ts are in13luded in the simulation
whi13h is performed for two dierent temperatures (Te 6= Ti) and a
negligible thermal 13ondu13tivity (10
minus05WmK) For this simula-
tion the sho13k is laun13hed by a 100 microm thi13k Aluminium piston
moving at the speed of 12 kms and 45 kms for 13ase (a) and (b)
respe13tively The position x=0 13orresponds to the position of the
XeAl interfa13e at time t=0 17
31 PROPACEOS Mono13hromati13 opa13ity versus the photon energy
in eV of Xenon at 10 eV and at two ele13tron densities respe13tively
equal to 1018 and 1020 13m
minus3 26
32 Mass density and temperature (a) ele13tron density and mean
13harge (b) at 10 ns for a radiative sho13k of speed sim 47 kms
in Xenon at 01 bar The verti13al dotted bla13k lines show the
position of the interfa13e between piston and ba13king Xenon gas
Zero at x-axis is the interfa13e of piston and Xenon at time zero
Spitzer thermal 13ondu13tivity has been used in the simulation 29
33 Mean Rosseland opa13ity (in 13m
2g
minus1) and opa13ity (13m
minus1) at 10
ns for a radiative sho13k of speed sim 47 kms in Xenon at 01
bar The verti13al dotted bla13k lines show the position of the
interfa13e between piston and ba13king Xenon gas Zero at x-axis
is the interfa13e of piston and Xenon at time zero Spitzer thermal
13ondu13tivity has been used in the simulation 30
34 (a) Mass density and temperature (b) ele13tron density and
mean 13harge at 10 ns for a radiative sho13k of speed sim 47 kms
in Xenon at 01 bar The verti13al dotted bla13k lines show the
position of the interfa13e between piston and ba13king Xenon gas
Zero at x-axis is the interfa13e of piston and Xenon at time zero
A negligible 13onstant thermal 13ondu13tivity (10
minus12WmK) have
been used in the simulation for Xenon 31
LIST OF FIGURES
35 (a) Ele13tron density and ele13tron temperature (b) Mass density
and mean 13harge at 10 ns for a radiative sho13k of speed sim 48
kms in Xenon at 01 bar for the 13ase when the ele13tron and ion
temperature are dierent The verti13al dotted bla13k lines show
the position of the interfa13e between piston and ba13king Xenon
gas Zero at x-axis 13orresponds to the position of the interfa13e
between the piston and Xenon at time zero The Spitzer thermal
13ondu13tivity has been used in this simulation 32
36 (a) Mass density and ele13tron temperature (b) ele13tron density
and mean 13harge at 10 ns for a radiative sho13k of speed sim 45
kms in Xenon at 01 bar for the 13ase when the ele13tron and ion
temperature are dierent The verti13al dotted bla13k lines show
the position of the interfa13e between piston and ba13king Xenon
gas Zero at x-axis 13orresponds to the position of the interfa13e
between the piston and Xenon at time zero In this simulation
value of thermal 13ondu13tivity (10
minus12WmK) is negligible for
Xenon The peak value of ion temperature is found to be 405 eV
whereas theoreti13ally this value is expe13ted to be sim 600 eV It is
possible to a13hieve the expe13ted value by in13reasing the resolution
of the simulation 34
37 Ele13tron temperature proles for various numbers of frequen13y
groups N = 1 20 50 60 70 90 and 100 35
38 (a) Ele13tron temperature (a) mean 13harge (b) and mass density
(13) for four (1 10 30 and 40) multipliers of the Xenon opa13ity at
10 ns 36
39 Plan13k and Rosseland mono13hromati13 opa13ity proles Te for ρ=15 times 10
minus3from PROPACEOS (I) BIGBART (II) and Rodri-
guez et al (III) 38
310 Mono13hromati13 Xenon opa13ity versus the photon energy in eV
at T = 15 eV and for ρ = 15 times 10
minus3g13m
minus3for the PROPA-
CEOS (a) and Rodriacuteguez et al (2015) (Fig 16 of Rodriacuteguez et al
(2015)) (b) models 39
311 Ele13tron density Ne (a) and ele13tron temperature Te (b) versus
axial position (along a 04 13m long sho13k tube) at 3 10 20 30
and 38 ns from HELIOS simulations (with opa13ity times 20) for the
13ases of single sho13k of sim 50 kms (dotted line) and two identi-
13al 13ounter-propagating sho13ks of sim 50 kms (solid lines) The
verti13al dotted lines show the position of the interfa13e between
piston and ba13king Xenon gas 41
LIST OF FIGURES
312 Variations ofNe (a) and Te (b) versus axial position for the 13ase of
two identi13al 13ounter-propagating sho13ks (of speeds sim 50 kms)
at 3 10 20 30 35 and 40 ns as derived from HELIOS simulations
For these simulations we have negle13ted the ee13t of radiation
by keeping the Xe opa13ity equal to zero The verti13al dotted lines
show the position of the interfa13e between the piston and ba13king
Xenon gas 43
313 Variations of Ne (a) and Te (b) with axial position for the 13ase
of two non-identi13al 13ounter-propagating sho13ks (of speeds sim 50
amp 20 kms) and two single sho13ks (dotted lines) of speeds sim 50
amp 20 kms respe13tively at 3 10 30 35 and 46 ns as derived
from HELIOS simulations The verti13al dotted lines show the
position of the interfa13e between piston and ba13king Xenon gas
(with opa13ity times 20) 45
41 a Spatial prole of the PALS Laser beam b Intensity prole of
laser pulse I(t) with time 48
42 Energy distribution s13heme for MAIN and AUX laser beams 49
43 Snapshot of the spheri13al 13hamber showing the horizontal bred-
board and on the right the fo13using MAIN lens 50
44 Dimension drawing (a) and snapshot (b) of Aluminum massive
(mo13k) target 50
45 Values on the gures are in mm (a) Verti13al 13ross se13tion view
of the gaseous target (b) Horizontal 13ross se13tion view 52
46 Pi13ture of a the gaseous target on its base 53
47 S13hemati13 of the gaseous target 53
48 Transmission of 100 nm thi13k Si3N4 membrane (CXRO database) 54
49 Snapshot of a target holder with one V-shield regarding the AUX
laser whi13h will pass through the hole in this shield marked in
green in the Figure The target whi13h fa13es the two diagnosti13s
(tubes) and is lo13ated behind the V-shield is thus not visible 54
410 In-situ system for target lling and pressure monitoring 56
411 S13hemati13 of the keV 13amera 59
412 (a) Sign 13onvention for the respe13tive positions of the lens and
the target (b) keV image of a MAIN impa13t on a massive target
(lens position +1500 microm) The spot size is 475plusmn25 microm Pixel
size for MAIN keV 13amera is 56 microm (13) keV image of an AUX
impa13t (lens position -1500 microm) The spot size is sim 280plusmn20 micromon target One pixel = 66 microm on AUXs keV 13amera 60
413 First interferometri13 setup All the opti13al elements are 1 in13h
For simpli13ity the plasma slab is not reported in the probe beam 61
414 Se13ond interferometri13 setup 63
415 S13hemati13 of the Streak 13amera (from Hamamatsu noti13e) 64
LIST OF FIGURES
416 Imaging setup of the Ma13h Zehnder Interferometer setup opera-
ting in the longitudinal mode The 13hannel of the target is fully
illuminated by the probe beam In this mode the lens images
the axis of symmetry of the two sho13k waves along the slit of the
streak 13amera 65
417 (a) Referen13e 2D image of a target before the shot re13orded on
the Streak 13amera The positions of the two pistons 13losing the 4
mm long target are lo13ated at 800 and 4800 microm The dark zones
along them (between 800 to 1000 and 4600-4800 microm) 13orresponds
to the glue This glue is then visible through verti13al bla13k strips
in the interferometri13 re13ord (b) Corresponding interferometri13
time-spa13e re13ord 66
418 Typi13al ray tra13ing in a grazing in13iden13e XUV spe13trograph
The dete13tor (GMCP or image plate) is installed tangentially to
the Rowland 13ir13le 67
419 (a) Setup of the XUV spe13trometer in the spheri13al 13hamber (b)
Typi13al re13ord between sim 10 and 40 nm is shown in the bottom
panel 68
420 (a) S13hemati13 of setup employed to re13ord the time and spa13e
visible plasma emission (b) A typi13al spe13trogram 69
51 Original re13ord of shot 48055 (left) FFT of the original image
(13enter) and the frequen13y ltered image (right) 73
52 A 13ropped se13tion of the re13ord from shot the 48055 The
rst ve positions have been sele13ted manually `+ signs (in red
13olor) on ea13h fringe On this re13ord the distan13e between two
unperturbed fringes is 13orrespond to 15 pixels (ie 159 microm) 73
53 (a) Representative points of the fringes as derived from the spline
tting of the 5 manually sele13ted points (Fig 52 (II)) on ea13h
fringe (b) Positions of the fringes maxima along Y-axis for ea13h
fringe derived by lo13ating the points of maximum intensity in
X-dire13tion of the previous points obtained by spline t 75
54 Fringe maxima on the full image 76
55 Imaging setup of the Ma13h Zehnder Interferometer operating in
the longitudinal mode The lens images the axis of symmetry of
the two sho13k waves along the slit of the streak 13amera 76
56 Interferometri13 image re13orded for the shot 48055 in Xe at 01
bar The sho13k speeds for the sho13ks driven by MAIN (from left
side) land AUX (from right side) lasers are respe13tively equal to
sim 54 and 23 kms The time of laser arrival on the piston is at
146 ns The positions of the Au-Xe interfa13e on the re13ord are
respe13tively 950plusmn50 and 4950plusmn50 mi13rons 77
LIST OF FIGURES
57 (a) MAIN sho13k speed (13al13ulated by the last fringe method)
versus the MAIN energy for Xenon or XeHe mixture at dierent
pressures with the error bars (b) AUX sho13k speed versus AUX
energy also for Xenon or XeHe mixture at dierent pressures 79
58 MAIN sho13k speed (13al13ulated by the last fringe method) versus
the MAIN laser energy for Xe (at 01 bar only) Ar He and Kr
at dierent pressures with the error bars 82
59 Left panel interferometri13 re13ords 48055 in Xe at 01 bar (a)
48132 in Xe+He at 02 bar (b) and 48138 in Xe+He at 02 bar
(13) Right panel ele13tron density at 10 20 30 and 40 ns versus
distan13e for these re13ords The positions of maxima have been
identied on the re13ords in the left panel The time t = 0 13orre-
sponds to the time of laser arrival on the target and the position
x = 0 13orresponds to the interfa13e between the piston (Au layer)
and the gas Its determination is pre13ise within 100 mi13rons The
distan13es between two unperturbed fringes for re13ords 48055
48132 and 48138 are 159 244 and 244 microm respe13tively The
lt Ne gt un13ertainty (plusmn 2 pixels) is indi13ated by the error bar in
the right panels It 13orresponds respe13tively to plusmn 9 times 10
17 plusmn 6
times 10
17and plusmn 6 times 10
1713m
minus3for the gures (a) (b) and (13) The
limit of dete13tion (2 pixels) is presented by a dotted line on ea13h
gure 85
510 (a) Interferometri13 re13ord in Ar at 03 bar (48079) The esti-
mated speeds for MAIN and AUX are 49plusmn5 and 23plusmn3 kms (b)
Same re13ord where the fringes maxima are marked by points
The bins denition is as fellows bin 0 Ne le 11 times 10
1813m
minus3
(white) bin 1 11 - 18 times 10
1813m
minus3(yellow) and bin 2 39 -
57 times 10
1813m
minus3(red) The time t = 0 13orresponds to the time
of laser arrival on the target and the position x = 0 13orresponds
to the interfa13e between the piston (Au layer) and the gas Its
determination is pre13ise within 100 mi13rons 87
511 Results for temperature (a) and Rosseland opa13ity (b) obtained
from HELIOS simulation at 10 ns for two 13ounter-propagating
sho13ks at sim 50 and 18 kms for Xe (with opa13ity multiplier 1
and 20) Ar and Kr at 54 times 10
minus4g13m
minus3initial mass density
A 13omparison Ar Kr and Xe PROPACEOS opa13ity shown in
APPENDIX B 88
LIST OF FIGURES
512 (a) Raw interferometri13 re13ord in Kr at 02 bar (shot 48146)
The estimated speed for MAIN sho13k is 53plusmn2 kms The time
t = 0 13orresponds to the time of laser arrival on the target(b)
Same re13ord shown in (a) maxima is marked by 13olored points
The bins denition is as follows bin 0 Ne le 11 times 10
1813m
minus3
(white) bin 1 11 - 18 times 10
1813m
minus3(yellow) and bin 2 39 -
57 times 10
1813m
minus3(red) The time t = 0 13orresponds to the time
of laser arrival on the target and the position x = 0 13orresponds
to the interfa13e between the piston (Au layer) and the gas Its
determination is pre13ise within 100 mi13rons 90
513 S13hemati13s of the Ma13h Zehnder Interferometer setup to re13ord
transverse interferometri13 images The lens allows to make on
the slit of the 13amera the image of a se13tion perpendi13ular to
dire13tion of sho13k propagation 91
514 Transverse interferometri13 images for (a) shot48111 (MAIN
sho13k only) (b) shot48130 (AUX sho13k only) The time is
measured after an oset equal to 14 and 23 ns respe13tively after
the time of the laser arrival on the target The position zero on
the x-axis of ea13h image 13orresponds to the base of the target 92
515 Interferometri13 image for the shot48143 The time t = 0 13orre-
sponds to the time of laser arrival on the target and the position
x = 0 13orresponds to the interfa13e between the piston (Au layer)
and the gas Its determination is pre13ise within 100 mi13rons 94
516 Raw (a) and 13orre13ted XUV spe13trum (b) for the shot 48143 95
517 Re13orded ele13tron density (shot 48055) together with the HE-
LIOS results (with Xenon opa13ity times 20) at dierent times in
Xenon at 01 bar 97
518 Time evolution of the mass density (a) ele13tron temperature (b)
and mean 13harge (13) at 56 57 58 60 64 and 65 ns within the
sho13k tube derived from the HELIOS simulations (with Xenon
opa13ity multiplier = 20) for two 13ounter streaming sho13ks of sim39 and 18 kms in Xenon at 06 bar 98
61 (a) Sket13h of the sho13k generator showing the pulsed ele13tri13al
13ir13uit the set of 13oaxial 13oni13al ele13trodes with a 13onstant radial
gap and the plasti13 insulator featured in grey on whi13h a planar
surfa13e dis13harge is initiated The installation of three opti13al
bers allows looking radially at the plasma moving in the sho13k
tube (b) S13hemati13s of the plasma dynami13s inside the 13oaxial
gun in fast-pulse mode the ele13tri13al 13urrent ows in the super-
13ial layers of the two 13oaxial 13oni13al ele13trodes and through an
annular plasma layer The magneti13 pressure Pmag pushes the
dis13harge axially 105
62 Exploded view of the plasma gun 106
LIST OF FIGURES
63 Time dependen13e of the main parameters for Ar gas at 1 mbar
a1313ording to the 13ir13uit model (a) 13urrent (kA) and voltage
(V) (b) a1313reted mass (kg) and rate of a1313retion (kgs) (13)
speed (ms) and kineti13 energy (J) 108
64 Plot of the dierent output parameters (normalized with respe13t
to the peak values) obtained at the top of the inner ele13trode
For ea13h group upper panel represents the variation for Ar gas
whereas lower panel is for Xe gas Proles of output parameters
are given (a) with 13one angle (b) with a1313retion fa13tor (13) with
gas pressure and (d) with damping resistan13e 109
65 Example of mapping of the ele13tron density from a 3-D MHD
simulation (ba13kground gas Argon at 1 mbar) when the plasma
sheath is (left) at the exit of the plasma gun (right) in free
ight 13onditions (log s13ales for the false 13olors) 113
66 Time history of plasma merging and early free ight through
the mapping of the ele13tron density ele13tron temperature and
average ioni13 13harge as given by a 3-D MHD simulation (ba13k-
ground gas Argon at 1 mbar) (s13ales for the false 13olors) 114
67 Axial proles values taken in the sho13k tube along a line slightly
o-axis at a distan13e of 1 mm for mass density ion temperature
ion density average ion 13harge magneti13 eld ele13tron tempera-
ture ele13tron density and average speed at 1500 ns (ba13kground
gas Argon at 1 mbar) A maping of |B| is given with a dashed
line indi13ating sho13k front position as well 115
68 Experimental results (a) time history of the passive opti13al
re13ords 13ompared to the main 13urrent for Argon gas 13lose to the
referen13e pressure (b) time-integrated signal vs Ar pressure for
the dierent bers (13) averaged speed vs Ar ba13king pressure 117
71 Interferometri13 re13ord obtained at 546 ns for a single radiative
sho13k of speed sim 100 kms propagating in Xe+He gas at 06 bar
driven by laser at 348 nm with energy 170 J The dierent 13olors
in (b) 13orrespond to yellow le Ne le 35 times 10
1713m
minus3 13yan 37
- 74 times 10
1713m
minus3 blue74 - 15 times 10
1813m
minus3 green 15 - 18 times
10
1813m
minus3 magenta 18 - 22 times 10
1813m
minus3 red 22 - 26 times 10
18
13m
minus3 orange26 times 10
1813m
minus3- 18 times 10
1913m
minus3 124
A1 S13hemati13s of Ma13h Zehnder interferometer with two mirrors
two 50 beam splitters and the dete13tor A plasma slab is pla13ed
in one of the two arms of the interferometer 147
B1 Rosseland and Plan13k opa13ity for three gases Xe Kr and Ar at
mass densities 51 times 10
minus4g13m
3(a) and 16 times 10
minus3g13m
3(b) 150
LIST OF FIGURES
B2 Mean 13harge for three gases Xe Kr and Ar at mass densities 16
times 10
minus3g13m
3(a) and 51 times 10
minus4g13m
3(b) 151
List of Tables
21 Summary of the radiative sho13ks experiments and out13omes in
dierent laser installations All the experiments are performed in
Xe ex13ept Keiter et al (2002) and Vis13o et al (2012) In these
two experiments the rst uses SiO
lowast
2 and the se13ond uses Ar
lowastlowastas
a sho13ked medium 20
41 Nominal 13hara13teristi13s of MAIN and AUX laser beams 49
42 Spe13i13ations of the MAIN and AUX fo13al lenses and the f- num-
ber (fo13al lengthbeam diameter) is given (see table 41 for the
beam diameters) 57
43 Spe13i13ations of the two Phase Zone Plates 58
44 Spe13i13ations of the two uEye keV 13ameras (1stVsion 13ompany) 59
45 Spe13i13ations of Andor CCD 13amera 67
46 Spe13i13ations of the XUV grating 67
51 Sho13k speeds estimated from the `last fringe method `NA re-
presents the entries whi13h 13ould not be dedu13ed from the re13ord
Further entries in the bold font are dis13ussed in detail in this
13hapter 81
52 Atomi13 data and density at 01 bar (at room temperature) for
He Ar Kr and Xe 82
61 Input parameters their standard values and range of variation 110
62 Laun13hing performan13e ArXe 111
63 Values of variables in various 13onditions at rest (with seed ele13-
trons) inside the sho13k and inside post-sho13k region A star lowastdenotes the insulator surfa13e 112
Chapter 1
Introdu13tion
11 General Context
Radiative sho13ks (RS) are present in various astrophysi13al environments es-
pe13ially in stars They are then present in all the supersoni13 a1313retioneje13tion
pro13esses during the stellar evolution from the early phase of the proto-
stellar 13ollapse up to the magnetosphere a1313retion on to the already formed
T- Tauri stars up to the late stage of the supernovae sho13ks
Radiative sho13ks are strong sho13ks (ie Ma13h number Mgtgt1) whi13hrea13h high temperatures and thus are the sour13e of intense radiation Alt-
hough the observation of (ie spatially resolved ) signatures of su13h sho13ks
in the astrophysi13al environments is very di13ult an alternate approa13h na-
med laboratory plasma astrophysi13s provides a powerful tool to study them
on the Earth
These waves have been experimentally studied sin13e more than a de13ade
mostly on large-s13ale laser fa13ilities in noble gases and with dierent targets
geometries The ablation pro13ess generated by the intera13tion between the
laser and a foil indu13es a sho13k wave in the foil whi13h then propagates
in the gas at a speed of several tens of kms Numeri13al simulations 13an
des13ribe these sho13ks with an improving pre13ision With laser intensities on
target 13omprised between 1014 and 1015 W13m
2 these experiments allowed
to re13ord sho13k speeds ranging between 40 and 150 kms
All previous experimental studies have been fo13used on the 13ase of single
radiative sho13ks However in astrophysi13al 13onditions the radiative sho13k
in general intera13ts with a denser medium leading to the development of
ree13ted and transmitted sho13ks A few representative examples of su13h
phenomena are the intera13tion of supernovae remnants with dense mole13ular
13louds the a1313retion sho13ks on the photosphere of T-Tauri stars and the bow
sho13ks at the head of fast stellar jets The 13ollision (or the intera13tion) of
two radiative sho13k waves is obviously a rare astrophysi13al event and the
template 13ase of the supernova remnants DEM L316 (see Fig 1 of Williams
et al (1997)) is still the subje13t of debates (Velarde et al 2006) In this
regard the development of dedi13ated laboratory experiments to the study of
propagation and intera13tion of 13ounter-propagating sho13k waves is important
as a tool to 13hara13terize su13h events through their spe13i13 signatures
Complementary to laser experiments 13ompa13t pulsed power generators
may drive an ele13tromagneti13 13oaxial plasma gun to 13reate astrophysi13al
1
2 CHAPTER 1 INTRODUCTION
relevant sho13ks in lower pressure noble gases with a high availability and a
rather modest 13apital 13ost The ele13tromagneti13ally driven sho13k waves may
have larger s13ales than those by laser Thus they 13an be analyzed rather
easily with the adequate diagnosti13s (Kondo et al 2008)
In this thesis I am presenting radiative sho13ks in the 13ontext of labo-
ratory astrophysi13s using two experimental approa13hes laser-driven sho13ks
and ele13tromagneti13ally driven sho13ks
12 Outline of Chapters
In the rst 13hapter the physi13s of radiative sho13ks wave will be summarized
I will dis13uss the Rankine -Hugoniot (RH) for the perfe13t gas and genera-
lized RH equations for ionized gas Further the ee13ts of radiation on the
sho13k equations will be introdu13ed and I will review the dierent regimes of
the radiative sho13ks presenting a radiative pre13ursor Then I will present
a short histori13al review of laser and ele13tromagneti13ally laboratory experi-
ments on this topi13 Finally I will highlight the obje13tives of my work in
this framework
The 13hapter two is dedi13ated to the numeri13al analysis of radiative sho13k
waves and of the main physi13al pro13esses whi13h play a role in these 13omplex
pro13esses The simulations will be performed in 1D using the 13ommer13ial ra-
diation hydrodynami13 HELIOS 13ode (Ma13Farlane Golovkin and Woodru
2006) After a des13ription of the 13ode and his possibilities several simu-
lations will be performed to analyze the ee13ts of the two uids (ele13trons
and ions) approa13h 13ompared to the one uid 13ase (identi13al temperature
for both) of the thermal 13ondu13tivity and the multi-groups opa13ity des13rip-
tion Then a brief overview of single and two 13ounter-propagating radiative
sho13ks will be presented
Chapter three will start with a short overview of the Prague Asterix Laser
System fa13ility (PALS) It will be followed by the experimental setup in13lu-
ding the two high energy beams and targets des13ription These paragraphs
will be followed by the detail of the diagnosti13s
The fourth 13hapter will be dedi13ated to the data analysis the results
and the interpretation of the experimental re13ords I will rst des13ribe the
method I followed for the analysis of interferometri13 re13ords Subsequently
the results obtained from a few representative re13ords for Xe Ar and Kr
will be dis13ussed in details Complementary 1D simulations performed with
initial 13onditions similar to that in the experiment will be shown and help to
interpret the experimental results Finally the results of XUV spe13tros13opy
diagnosti13s will be presented
In the fth 13hapter I will present the 13alibration of a table top ele13tro-
magneti13 plasma generator whi13h is able to laun13h supersoni13 sho13ks up to
sim 40 kms speed at stati13 pressures of few mbar The results of this 13ali-
13 MY CONTRIBUTION 3
bration will also be presented and a 13omparison with 3-D MHD simulations
will be performed The diagnosti13s whi13h have been implemented will be
presented to illustrate the model as well as preliminary re13ords of the plasma
speed
The nal 13hapter of the thesis is dedi13ated to the summary of this work
together with perspe13tives for the follow-up of these experiments
13 My 13ontribution
I joined PhD on November 12 2013 under the supervision of Dr Jean
Larour and Dr Chantal Stehleacute My PhD thesis is 13omprised of two parts
the rst part of the work is to study the laser-driven sho13ks The se13ond
part of my thesis is to study the ele13tromagneti13ally laun13hed strong sho13ks
Con13erning the laser-driven sho13ks I have started to work on the analysis
and interpretation of XUV images re13orded previously during an experimen-
tal 13ampaign 13ondu13ted at the Prague Asterix Laser System (PALS) fa13ility
in 2011 This enabled me to estimate the lo13al opa13ity by employing Abel
inversion Although this work not presented in this thesis it shed light on
the basi13 understanding of radiative sho13k as seen in the experiments as well
as the knowledge of experimental setups Later in April - May 2015 I have
parti13ipated in a ve weeks experimental 13ampaign on PALS laser fa13ility
in Prague whi13h was dedi13ated to the study of the spe13tros13opi13 and in-
terferometri13 signatures of laser-produ13ed sho13ks Espe13ially the experiment
was designed to study the physi13s during the 13ollision of 13ounter-propagating
sho13ks In the preparation of this 13ampaign I be13ame a13quainted with the
instruments and the targets to be used as well as prepared a master re-
port 13ontaining all the 13ru13ial aspe13ts of the experiment to be referred by
all the parti13ipants during the experiments During the 13ampaign we have
performed around 55 shots 13orresponding to dierent gases (Xenon Argon
Helium and Krypton) at dierent pressure and used three types of diagnos-
ti13s ie visible interferometry visible spe13tros13opy and XUV spe13tros13opy
In this experiment parti13ularly I was in 13harge of the target sele13tion its
installation in the experimental 13hamber gas lling preparing logbook re-
13ording of the visible spe13tros13opy data from the 13ontrol room qui13k data
analysis for the spot size sho13k speed from interferometry data et13 In
addition I have worked on spe13tros13opi13 and interferometri13 data analysis
re13orded during the experiment Next in order to interpret the experimental
results I have performed several 1D hydrodynami13 simulations with varying
input 13onditions In parallel I have also performed simulations for another
experiment performed by my 13ollaborators at ORION laser fa13ility in the
United Kingdom However I have not presented this work in this thesis
Based on the results obtained from my analysis of re13ords obtained from
the PALS experiments in 2015 grabbing the opportunity we have planned
4 CHAPTER 1 INTRODUCTION
and parti13ipated to a 13omplementary experimental 13ampaign at PALS in
September 2016 The analysis of data re13orded during experiments is still
under progress and forms the future s13ope of the thesis
The se13ond part of my thesis has been 13omprised of the study of sho13k
laun13hed through the ele13tromagneti13 generator setup In this I have wor-
ked on the optimization of the design of the ele13tri13 generator in view of the
required sho13k parameters Moreover I have derived various sho13k 13hara13-
teristi13s viz radial dis13harge layer namely speed mass a1313eleration et13
at the dierent positions of its propagation In present a upgraded ele13tri13
generator with new diagnosti13s is ready to perform the new experiments
Chapter 2
Radiative sho13k waves
Contents
21 Rankine-Hugoniot(R-H) relations 6
211 Jump relations for a real gas 9
212 Ee13t of radiation on sho13k waves 11
213 Radiative hydrodynami13s equations 13
22 Radiative sho13k regimes 15
221 Typi13al radiative sho13k waves with a pre13ursor 16
23 Radiative sho13k waves experiments 16
231 Laser-driven sho13k experiments 18
232 Ele13tromagneti13ally laun13hed sho13k 21
Small-amplitude disturban13es in a medium often take the form of linear
waves On the 13ontrary strong disturban13es generally produ13ed by pro13esses
su13h as explosions lightening supersoni13 movements of bodies powerful
ele13tri13 dis13harges et13 where the uid properties 13hange rapidly generate
nonlinear waves like sho13k waves As a 13onsequen13e of sho13k waves the ow
moves faster than the sound speed A sho13k is 13hara13terized by a sudden
in13rease in the density and velo13ity of the medium The Ma13h number (M)
is 13ommonly used to 13hara13terize the strength of the sho13k This number
dened as the ratio of the sho13k speed to the speed of sound in the upstream
medium in whi13h the sho13k propagates
Radiative sho13k waves are hypersoni13 sho13k waves whi13h are heated to
high temperature and as a 13onsequen13e be13ome the sour13e of intense radi-
ation This radiation in turn modies the dynami13s and stru13ture of the
sho13k itself whi13h makes its stru13ture more 13ompli13ated Radiative sho13k
waves o1313ur in several astrophysi13al 13ir13umstan13es su13h as in the a1313retion
sho13k of protostellar formation (Stahler Palla and Salpeter 1986) the su-
pernova explosion and the intera13tion of their remnants with the dense in-
terstellar medium (Chevalier 1977) the bow sho13ks at the head of stellar
jets (Hartigan et al 2001)
Radiative sho13k waves 13an now be studied in the laboratory whi13h allows
to 13ompare the data with models existing in the literature and to 13he13k the
ability of the numeri13al 13odes to reprodu13e them (see for instan13e Bouquet
et al (2004) Leygna13 et al (2006)) The experimental study of radiative
5
6 CHAPTER 2 RADIATIVE SHOCK WAVES
sho13k 13hara13teristi13s is 13ru13ial in understanding these various energeti13 pro-
13esses o1313urring in the astrophysi13al plasmas In this 13hapter I will rst
present a short summary of the jump 13onditions (Rankine-Hugoniot rela-
tions) for the simple 13ase of an ideal gas whi13h rely on the values of the
thermodynami13al quantities on both sides of the dis13ontinuity Next I will
present the role played by the ionisation and ex13itation of the gas and then
the role played by the radiation energy and pressure The des13ription of the
dierent regimes of radiative sho13ks whi13h depend on the opa13ity of medium
and 13an be 13hara13terized by the presen13e or absen13e of a radiative pre13ursor
will follow Further I will give a brief overview of the literature fo13using on
the generation of radiative sho13ks in the laboratory and I will present the
motivations and obje13tives of my work
21 Rankine-Hugoniot(R-H) relations
Let us 13onsider a one-dimensional sho13k propagating in a gas at rest with
the speed us In the frame asso13iated with the sho13k front the pre-sho13k
(upstream) uid velo13ity is then u1 = - us while behind the sho13k (post-
sho13k or downstream region) this velo13ity is u2 as shown in the Fig 21
Figure 21 S13hemati13 diagram for a 1-D sho13k moving in the x dire13tion The
speed of the sho13k front is us In the frame of referen13e of the sho13k front the pre-
sho13k region has a relative speed u1= -us density ρ1 total pressure P1 temperature
T1 whereas these quantities are respe13tively u2 ρ2 P2 and T2 for the post-sho13k
region
Assuming that a stationary state is rea13hed the 13onditions whi13h deter-
mine the post-sho13k thermodynami13al quantities are the well known Rankine-
21 RANKINE-HUGONIOT(R-H) RELATIONS 7
Hugoniot equations (Ma13quorn Rankine 1870 Hugoniot 1887 1889) De-
du13ed from the Euler equations these 13onservation equations for the mass
momentum and energy are given by
ρ2u2 = ρ1u1 (21)
ρ2u22 + P2 = ρ1u
21 + P1 (22)
u2(ε2 +P2
ρ2) +
1
2u32 = u1(ε1 +
P1
ρ1) +
1
2u31 (23)
where ε1 and ε2 are the spe13i13 internal energies (energy per unit mass)
in the two (upstream and downstream) regions
For an ideal gas of atomi13 mass m =mPA (mP proton mass A atomi13
number) and having an adiabati13 index γ the speed of sound is given by
Cs =
radic
γP
ρ=
radic
γkBT
m(24)
and the Ma13h number (M) of the sho13k is then
M =u1Cs1
(25)
Inserting these the aforementioned form of M and Cs in the equations
21 22 and 23 we 13an derive the 13ompression pressure and temperature
ratios for the system as follows
ρ2ρ1
=u1u2
=M2(γ + 1)
2 +M2(γ minus 1)(26)
P2
P1=
2M2γ minus (γ minus 1)
(γ + 1)(27)
T2
T1=
P2
P1
ρ1ρ2
=(2M2γ minus (γ minus 1))(2 +M2(γ minus 1)
M2(γ + 1)2) (28)
Considering the 13ase of strong sho13ks where M may be 13onsidered to be
≫1 these equations be13ome
ρ2ρ1
=(γ + 1)
(γ minus 1)(29)
T2
T1=
2M2γ(γ minus 1)
(γ + 1)2(210)
kBT2 =2(γ minus 1)
(γ + 1)2mus
2(211)
8 CHAPTER 2 RADIATIVE SHOCK WAVES
For mono-atomi13 gases γ = 53 the 13ompression ratio is estimated to
be 4 and the post-sho13k temperature is
kBT2 =3
16mu1
2 =3
16mPAu1
2(212)
Temperature is thus proportional to the square of the sho13k velo13ity and
to the atomi13 mass Therefore to a13hieve a high temperature in the radiative
sho13k experiment we often use high atomi13 number and mass gases For Xe-
non (A=13129) the post-sho13k temperature is then equal to T2 = 0257u21(eV kms) whi13h for instan13e gives sim 520 eV for a sho13k propagating at
45 kms (see Fig 22)
Figure 22 Proles of temperature (Te = Ti = T ) and mass density (ρ) at 10ns for a sho13k propagating in Xenon at 01 bar (ρ1= 57 times 10
minus4g13m
minus3) with
a speed of sim 45 kms It is to note that the given pressure value are at 298 K
The simulation is performed with the HELIOS 13ode ex13luding the radiation ee13t
and for a polytropi13 gas (γ=53) The initial temperature of Xenon is set to 1
eV and with a negligible (10
minus5WmK) thermal 13ondu13tivity The Ma13h number
is then equal to 40 For this simulation the sho13k is laun13hed by a 100 microm thi13k
Aluminium piston moving at the velo13ity of 33 kms The position x=0 13orresponds
to the position of the XeAl interfa13e at time t=0
21 RANKINE-HUGONIOT(R-H) RELATIONS 9
211 Jump relations for a real gas
In 13ontrast to the dis13ussion on ideal gas presented in the previous se13tion
the 13ase of a real gas is a little more 13ompli13ated It is to note that in this
13ase a part of the kineti13 energy is used to ex13ite and ionize the post-sho13k
gas As a result its temperature is lower than that for the 13ase of ideal gas
In addition the pressure is also modied due to ionisation (Mi13haut et al
2004) as dis13ussed below
Let us now dene j as the ionization stage of the atom (varying between
0 and z) and i to be the atomi13 state in this ionisation stage We shall
denote by Pj amp Pji the fra13tional ionization of the ion and the population
of the atomi13 state i in the jth ionization state respe13tively Then the mean
ionization stage (average 13harge) per atom 13an be written as
z =
zsum
j=0
jPj (213)
and the ex13itation energy (in13luding ionization) per atom and per unit of
mass is
ǫexc =
sumzj=0
sum
i Pji Eji
mpA(214)
where Eji is the energy of the ionisation stage j in the atomi13 stage iThus the total pressure of gas 13an be written as
P = Pi + Pe (215)
where Pi and Pe are the ioni13 and ele13troni13 pressures 13ontributions
Using the 13harge neutrality 13ondition Ne = z Ni this gives
P = Ni(1 + z) kBT = ρkBT
m(1 + z) (216)
The thermal pressure and enthalpy of the real gas are given by
Pth = ρ(1 + z)
mkBT (217)
h =5
2
(1 + z)
mkBT + ǫexc (218)
Next the speed of sound in13luding the ee13t of ionization with the
average 13harge z in the medium is now modied by ionisation
Cs ≃radic
5
3
γ(z + 1)kBT
m(219)
Thus the 13ontinuity relations in the frame moving with the sho13k front
be13ome
ρ2u2 = ρ1u1 (220)
10 CHAPTER 2 RADIATIVE SHOCK WAVES
ρ2u22 + ρ2
kBT2
m(1 + z2) = ρ1u
21 + ρ1
kBT1
m(1 + z1) (221)
ρ2u2
[5
2
(1 + z2)
mkBT2 + ǫexc2 +
u222
]
= ρ1u1
[5
2
(1 + z1)
mkBT1 + ǫexc1 +
u212
]
(222)
Figure 23 Ele13tron temperature (Te) ion temperature (Ti) and mass density
(ρ) proles at 10 ns derived from 1-D simulation for the 13ase of ionisation (no
radiation) The sho13k propagates in Xe gas at 01 bar (ρ1= 57 times 10
minus4g13m
minus3by
sim 45 kms) It is to note that the given pressure value is at 298 K The simulation
is performed with the HELIOS (Te 6= Ti 13ase) using the PROPACEOS equation
of state without any radiation and with a negligible thermal 13ondu13tivity (10
minus05
WmK) The initial temperature in Xenon is set to 1 eV and thus the Ma13h
number is equal to 40 For this simulation the sho13k is laun13hed by a 100 microm thi13k
Aluminium piston moving at the velo13ity of 38 kms The position x=0 13orresponds
to the position of the XeAl interfa13e at time t=0
Similar to the Fig 22 however for the 13ase of a real gas the Fig
23 shows the variation the ele13tron and ion temperatures as well as the
mass density with the distan13e for a sho13k propagating in Xenon at sim 45
kms Now the post-sho13k ele13tron temperature peaks at 22 eV whi13h is
mu13h smaller than the maximum temperature of the ions (510 eV) The
13ompression is now equal to 10 instead of 4 as in the previous 13ase The ions
21 RANKINE-HUGONIOT(R-H) RELATIONS 11
are heated rst in the sho13k through ion-ion 13ollisions This ion temperature
de13reases and the ele13tron temperature in13reases as a 13onsequen13e of the
ele13tron-ion 13ollisions The two temperatures thus be13ome equal Sin13e the
momentum 13hange of the ions require only a few number of kineti13 13ollisions
the thi13kness of the abrupt transition between the pre-sho13k and the post-
sho13k is of the order of a mean free path of the ions
It should be noted that for sho13ks propagating with very high speed
the post-sho13k temperature is so high that the 13ompressed gas is fully ioni-
zed Therefore the ee13ts of ionisation and ex13itation be13ome negligible and
hen13e the 13ompression ratio rea13hes 4 as in the 13ase of the adiabati13 sho13k
212 Ee13t of radiation on sho13k waves
In the previous se13tion the ee13ts of radiation heating and 13ooling have
been negle13ted However the radiation emitted in a strong sho13k will ae13t
its stru13ture These ee13ts 13an be determined using mass momentum and
energy 13onservation equations whi13h now have to in13lude the 13ontributi-
ons of the radiative ux energy and pressure To this purpose and with
some simpli13ations we 13an use the same approa13h than in se13tion 21 still
assuming that the sho13k is stationary
Jump relations with radiation
Let us 13onsider a stationary sho13k propagating in an atomi13 gas far away from
the dis13ontinuity The medium is then opti13ally thi13k and we 13an negle13t
the 13ontribution of the radiative ux Let us also assume that the medium
is at LTE and that the radiation pressure and energy may be des13ribed as
follows
Prad =1
3Erad =
4
3cσT 4 =
1
3aradT
4(223)
where arad is the radiation density 13onstant and T is the sho13k radiation
temperature whi13h we shall suppose to be equal to the ele13tron temperature
Following Mi13haut et al (2004) the new jump 13onditions a13ross the sho13k
dis13ontinuity be13ome
ρ2u2 = ρ1u1 (224)
ρ2u22+ρ2
kBT2
m(1+z2)+
1
3aradT
42 = ρ1u
21+ρ1
kBT1
m(1+z1)+
1
3aradT
41 (225)
ρ2u2
(
5
2
(1 + z2)
mkBT2 + ǫexc2 +
u222
)
+4
3aradT
42 u2 =
ρ1u1
(
5
2
(1 + z1)
mkBT1 + ǫexc1 +
u212
)
+4
3aradT
41 u1 (226)
12 CHAPTER 2 RADIATIVE SHOCK WAVES
(a)
(b)
Figure 24 Figure 14 amp 17 of Mi13haut et al (2004) (a) Compression ρ2ρ1 (full13ir13les thin line) kT2 in eV (inverted full triangles thin line) in Argon (a) Xenon
(b) for initial 13onditions kT1( = 10 eV for Ar and = 01 eV for Xe) ρ1 = 5 times10minus4 gcm3
versus sho13k speed in kms with (full markers full lines) and without
(empty markers dashed lines) radiation The ionization stage lt z gt2 is plotted
with the diamond symbol
21 RANKINE-HUGONIOT(R-H) RELATIONS 13
In the radiative regime the 13ompression ratio ρ2ρ1 approa13hes to the
value of 7 at very high velo13ities (several hundred of kms) whereas it is equal
to 4 when the ee13ts of radiation energy and pressure have been negle13ted
This limit is the same for the 13ases of adiabati13 (Bouquet Teyssier and
Chieze 2000) as well as the real gas (Mi13haut et al 2004) This is due to
the fa13t that at very high velo13ities the medium be13omes fully ionized and
behaves as a perfe13t gas On the 13ontrary in the 13omparatively low velo13ity
regime the ee13t of ionisation and ex13itation dominates the radiation ee13ts
This is illustrated in Fig 24 taken from Mi13haut et al (2004) whi13h reports
the variations of the Ar and Xe 13ompression ratio versus the sho13k speed for
an initial mass density of 5 times 10
minus4g13m
3 At 50 kms the 13ompression
ratio and the temperature are respe13tively equal to (5 32eV) and (11 30
eV) for Argon and Xenon
The 13ompression ratio shows bumps for high velo13ities These bumps
may be attributed to the fa13t that as the temperature of the post-sho13k
in13reases with the velo13ity the atom ionises more and more Thus the ex-
ternal shells in its ioni13 atomi13 stru13ture are su1313essively opened indu13ing
a strong variation in the ex13itation energy
The post-sho13k temperature is very high for the very high-speed sho13ks
In su13h 13ases the 13ompressed gas is fully ionized and the ee13ts of ionisation
and ex13itation remain negligible like in the adiabati13 13ase It should be noted
that below 100 - 200 kms the ee13ts of radiation pressure remain negligible
Thus very energeti13 lasers like LMJ or NIF are required to rea13h a domain
where this ee13t 13an be observed
Thus whereas for the range of velo13ities of 20 - 50 kms 13overed by my
study the ee13ts of radiative energy and pressure 13an be negle13ted this is
not the 13ase for the radiative ux whi13h has been negle13ted in the previous
se13tions and whi13h may be13ome 13omparable to the material energy ux Its
ee13t on the system must be taken into a1313ount as will be shown below
213 Radiative hydrodynami13s equations
To in13lude the 13ontributions of radiation it is 13ompulsory to introdu13e the
relevant terms in the uid equations The propagation of the radiation in the
matter is governed by the pro13esses of absorption and emission of photons
whi13h respe13tively heats and 13ools the medium As the mean free path of
photons is always higher than the mean free path of parti13les the sho13k
stru13ture is more inuen13ed by the radiative transfer than the vis13osity of
the uid
The propagation of radiation in the matter is des13ribed by the radiative
transfer equation
(
1
c
part
partt+
part
parts
)
I(rn ν t) = η(r ν)minus χ(r ν) I(rn ν t) (227)
14 CHAPTER 2 RADIATIVE SHOCK WAVES
where χ(ρ ν T ) and η(ρ ν T ) are respe13tively the mono13hromati13 opa-
13ity and emissivity of the gas
The spe13i13 intensity I(rn ν t) (erg cmminus2 sminus1Hzminus1 srminus1) is the energy
radiated per units of surfa13e area time frequen13y and solid angle in the
dire13tion n
The radiative energy density Erad ux Frad and pressure Prad are re-
spe13tively dened as the zero rst and se13ond moments of the spe13i13 in-
tensity versus the angle after integration over the photon frequen13y ν
Erad =1
c
int ∮
I(rn ν t)dΩdν (228)
Frad =
int ∮
I (rn ν t) cos θ dΩdν (229)
Prad =1
c
int ∮
I (rn ν t) cos2 θ dΩdν (230)
(231)
Taking into a1313ount radiative ee13ts the equations of radiative hydro-
dynami13s in the 13o-moving frame of the sho13k (Mihalas and Mihalas 1999
Drake 2006) 13an be written as
partρ
partt= minusnabla(ρu) (232)
ρ
(
partu
partt+ unablau
)
= minusnabla(P + Prad) (233)
part
partt
(
ρu2
2+ ρǫ+ Erad
)
+nabla
[
ρu
(
ǫ+u2
2+
P
ρ
)
+ (Erad + Prad)u
]
= minusnablaFrad
(234)
As mentioned previously the 13ontributions linked to Prad and Erad are
ee13tive only at very high temperature (ie velo13ity) However for the
radiative sho13ks of interest in this work the 13ontribution of the radiative
ux Frad is non-negligible 13ompared to ρu3 These sho13ks are thus in the
radiation ux dominated regime
Most of the experimental sho13k waves are in this regime These hyd-
rodynami13 equations are strongly non-linear In the present work the one-
dimensional radiative-hydrodynami13 13ode `HELIOS (Ma13Farlane Golovkin
and Woodru 2006) has been employed to derive the physi13al parameters
of the sho13k in various 13onditions as presented in the following se13tion HE-
LIOS will be dis13ussed in detail in the next 13hapter
22 RADIATIVE SHOCK REGIMES 15
22 Radiative sho13k regimes
Radiative sho13ks are strong sho13ks (Ma13h number Mgtgt1) whi13h attain
high temperatures and thus are the sour13e of intense radiation (Mihalas
and Mihalas 1984 Zeldovi13h Zeldovi13h and Raizer 2002 Drake 2006)
Depending on the opa13ity the radiation emitted from the sho13k may be ab-
sorbed by the pre-sho13k region indu13ing its pre-heating Far away from the
dis13ontinuity the stru13ture of the upstream medium is determined by the
absorption On its side the stru13ture of the transition layer of the down-
stream medium 13lose to the dis13ontinuity depends on the balan13e between
the emission (13ooling) and the absorption (heating) Thus the full stru13ture
of the sho13k mainly depends on the variation of the opa13ity χ (13m
2g) The
opti13al depth (τ(s)) measured from the position of the jump is then the
relevant parameter for the absorption It is given by
τ(s) =
int s
sjump
χ(sprime)ρ(sprime) dsprime (235)
where sprime 13hara13terizes the path of the radiation As we are interested in
the mean frequen13y averaged radiative ux the relevant opa13ity here will be
the averaged Rosseland opa13ity whi13h will be dened in due 13ourse
If τ gtgt 1 the regime is referred as opti13ally thi13k while in the opposite
13ase (τ lt 1) it is opti13ally thin Flux dominated radiative sho13ks have
been 13lassied depending on the opti13al depth of upstream and downstream
regimes
An attempt of radiative sho13k 13lassi13ation has been performed by Drake
(Drake (2006)) a1313ording to the opti13al the thi13kness of the upstream and
downstream regimes resulting in the denition of four dierent regimes
namely a thin-thi13k thin-thin thi13k-thi13k thi13k-thin radiative sho13ks
Following this author a thin-thi13k type of radiative sho13k probably exists
only in theory In the 13ase of a thin-thin radiative sho13ks both upstream
and downstream regimes are opti13ally thin and the sho13k radiates away
its energy in both dire13tions This regime of radiative sho13k is present in
various astrophysi13al 13ontexts for instan13e in the internal sho13ks of stellar
jets (Hartigan 2003) It is possible to generate these type of sho13ks in the
laboratory in very low-density gases In thi13k-thi13k radiative sho13ks both
upstream and downstream zones are opti13ally thi13k This regime o1313urs
in the stellar interiors The radiation 13oming from the hot downstream
region heats the 13ooler upstream material and forms a pre13ursor The fourth
regime is the thi13k-thin radiative sho13k whi13h is 13hara13terized by an opti13ally
thi13k downstream and an opti13ally thin upstream region The radiation
from the heated downstream region passes ahead of the sho13k and 13reates a
13ooling layer in the downstream before rea13hing the nal state There is no
absorption of the radiation in the upstream region and therefore 13ontrary to
16 CHAPTER 2 RADIATIVE SHOCK WAVES
the thi13k-thi13k 13ase there is no radiative pre13ursor Sho13ks generated in the
laser experiments are usually between the thi13k-thi13k and thin-thin regimes
221 Typi13al radiative sho13k waves with a pre13ursor
The radiation from the post-sho13ked region just behind the sho13k passes
ahead of the sho13k and generates the pre13ursor However it also leads to a
radiative 13ooling layer downstream that extension depends on the opa13ity
A typi13al stru13ture of su13h sho13ks is presented in M13Clarren et al (2010)
In these 13ases the 13ompression in the 13ooling layer behind the sho13k is
mu13h higher than that in the hydrodynami13 13ase This kind of sho13k is
most 13ommonly generated in the laboratory experiments (Bozier et al 1986
Keiter et al 2002 Reighard et al 2006 Fleury et al 2002) In astrophysi13s
these sho13ks are present for instan13e in supernovae when the blast wave
emerges from the stellar interior (Ensman and Burrows 1992)
A detailed 13lassi13ation of the sho13ks presenting a radiative pre13ursor
is reported in several referen13es (Zeldovi13h Zeldovi13h and Raizer 2002
Ensman 1994 Vaytet et al 2013)
In the 13ase of sub-13riti13al sho13k the temperature just ahead of the sho13k
front is smaller than the post-sho13k temperature Su13h sho13ks are relatively
weak and hen13e the radiative ux transfer from the post-sho13k to the pre-
sho13k is small A piston moving with a 13onstant speed of 12 kms in Xenon
gas at 01 bar is able to generate su13h a sub-13riti13al radiative sho13k The
resulting mass density and ele13tron temperature proles have been presented
at 10 ns in the Fig 25a
The other regime of radiative sho13k is known as super13riti13al in whi13h
the pre-sho13k and the nal post-sho13k temperatures are the same The mass
density and ele13tron temperature proles of super13riti13al sho13k driven by a
13onstant moving piston with speed 45 kms in Xenon at 01 bar are presented
in Fig 25b
The speeds a13hieved in the PALS experiment range between sim 15 - 55
kms for pressure 13omprised between 01- 06 Our simulations indi13ate
that depending on the speeds the regimes 13over either the sub13riti13al or
super13riti13al 13ases
23 Radiative sho13k waves experiments
Radiative sho13k waves have been studied experimentally for more than a
de13ade on high-energy lasers fast magneti13 pin13h ma13hines and pulsed po-
wer fa13ilities Su13h fa13ilities are able to a13hieve strong sho13k 13onditions
(high-temperature high-pressure) over a very small volume and time Va-
rious diagnosti13s are able to re13ord the plasma 13onditions whi13h are then
13ompared with the results of numeri13al simulations The results of these
experiments are also employed to validate numeri13al 13odes some of them
23 RADIATIVE SHOCK WAVES EXPERIMENTS 17
(a)
(b)
Figure 25 Simulated ele13tron temperature (Te) ion temperature (Ti) and mass
density (ρ) proles at 10 ns of a radiative sho13k propagating with a 15 kms (a sub-
13riti13al) and 48 kms (b super13riti13al) The sho13k is propagating in Xenon at 01
bar (ρ = 54 times 10
minus4g13m
minus3 at 298 K) The initial temperature in Xenon is set to
1eV The two Ma13h numbers are respe13tively equal to 13 and 43 The radiation and
ionisation ee13ts are in13luded in the simulation whi13h is performed for two dierent
temperatures (Te 6= Ti) and a negligible thermal 13ondu13tivity (10
minus05WmK) For
this simulation the sho13k is laun13hed by a 100 microm thi13k Aluminium piston moving
at the speed of 12 kms and 45 kms for 13ase (a) and (b) respe13tively The position
x=0 13orresponds to the position of the XeAl interfa13e at time t=0
18 CHAPTER 2 RADIATIVE SHOCK WAVES
being relevant to the astrophysi13s like FLASH (Fryxell et al 2000) and
HERACLES (Gonzaacutelez Audit and Huynh 2007)
231 Laser-driven sho13k experiments
Most of laboratory experiments on radiative sho13ks have been performed
on large-s13ale laser fa13ilities (Bouquet et al 2004 Chaulagain et al 2015
Gonzaacutelez et al 2006 Reighard et al 2006 Stehleacute et al 2010 Doss et al
2009 Drake et al 2011 Diziegravere et al 2011 Stehleacute et al 2012) in noble
gases at pressures between 01 and 1 bar With laser intensities on the
target 13omprised between 1014 and 1015 W13m
2 these experiments allowed
to re13ord sho13k speeds ranging between 40 and 150 kms
Bozier et al (1986) for the rst time experimentally eviden13ed the ge-
neration of a radiative wave propagating ahead of a super13riti13al sho13k in
xenon (pressure up to 6 bars) at the CEAs Limeil laser fa13ilities (irradian13e
of 5times 10
14W13m
minus2) In this experiment the laser beam (wavelength of 106
microm duration 1 ns) of 40 J energy was fo13used on an aluminum foil a13ting
as a piston A sho13k of speed sim 50 kms was produ13ed in the gas A visible
streak 13amera was used to re13ord the pre13ursor emissivity over 10 ns
Later Keiter et al (2002) performed a similar experiment at the OMEGA
laser fa13ility (25 kJ 1ns fo13al spot diameter 600 microm) still in a planar
geometry The sho13k propagated here in a low-density SiO2 aerogel foam
(ρ sim 96 mg13m
3) In this experiment X-ray absorption spe13tros13opi13 data
were re13orded to qualitatively predi13t the temperature of the pre13ursor
In the year 2000 a new radiative sho13k experiment was performed in
Xenon at 02 bar at the LULI laser fa13ility (sim 100 J λ = 053 microm 4-
6times 10
13W13m
minus2) (Fleury et al (2002) Bouquet et al (2004)) This expe-
riment used a three-layered piston 13omposed of a thin polystyrene a13ting
as ablator (2 microm thi13k) a titanium foil (3microm) a13ting as x-ray shield and a
layer of a polyethylene foam (25 microm) as an a1313elerator Further a VISAR
(velo13ity interferometer system for any ree13tor) and a Ma13h - Zehnder in-
terferometer were used as a diagnosti13s to measure the front sho13k speed and
the pre13ursor ele13tron density Two streak 13ameras re13orded the sho13k pro-
pagation in longitudinal and transverse dire13tions This was the rst attempt
to estimate ele13tron density within the pre13ursor Their measurements indi-
13ate an ele13tron density ranging between 1018minus1020cm3 The experimental
results were in qualitative agreements with 1D numeri13al MULTI and FCI
simulations However the speed of the pre13ursor in these 1D simulations
was too large This was then explained later when multidimensional ee13ts
were in13luded in the simulations (Leygna13 et al 2006 Gonzaacutelez Audit and
Stehleacute 2009)
Complementary experiments were then performed by Gonzaacutelez et al
(2006) at the PALS laser fa13ility (60 J 035 ns 438 nm spot diameter 600microm
sim 7times 10
13W13m
minus2) in Xe gas The goal was to study the sho13ks over longer
23 RADIATIVE SHOCK WAVES EXPERIMENTS 19
times and to inspe13t the ee13ts of the walls albedo A doubled layered pis-
ton made of 10microm polystyrene and 05 microm gold was used The pre13ursor
edge was re13orded by shadowgraphy using a visible streak 13amera The ex-
perimental results were 13ompared with 2D numeri13al simulations performed
with HERACLES (Gonzaacutelez Audit and Huynh 2007) This 13omparison
allowed quantifying the albedo of the walls It also showed that the radia-
tive losses at the walls lead to a small 13urvature of the ionization front and
to a redu13tion of its longitudinal extension (Leygna13 et al 2006 Gonzaacutelez
Audit and Stehleacute 2009)
In parallel another experiment was performed at the OMEGA laser fa-
13ility (035microm spot sim 700 - 800microm sim 2times 10
15W13m
minus2) by Reighardt et
al (Reighard et al (2006) Reighard (2007)) The sho13k was imaged by an
X-ray ba13k-lighter at dierent times whi13h enabled to inspe13t for the rst
time the shape of the sho13k front At higher speeds (sim 100 kms) X-ray
radiography pointed out a 13ollapse of the post-sho13k (Reighard et al 2006)
whi13h was attributed to the radiation losses trough the sho13k front For
these high-speed 13onditions the wall heating leads to the development of se-
13ondary wall sho13ks intera13ting with the primary sho13k (Doss et al 2009)
Su13h wall sho13ks have not yet been observed at lower speeds
Subsequently an experiment by Stehleacute et al (2010) was performed at
PALS laser fa13ility with the similar laser parameters as taken in Gonzalez
et al (2006) In this experiment a Ma13h-Zehnder interferometer and a
streak 13amera were used to re13ord the sho13k propagation over 50 ns Time-
integrated XUV spe13tros13opi13 analysis between 16 and 22 nm of sho13k was
performed The experiment showed that of a quasi-stationary regime was
rea13hed after 20 ns
The rst experimental study in Argon gas was reported in 2012 by Vis13o
et al (2012) at OMEGA fa13ility (035microm spotsim 600 microm sim 7times 10
14W13m
minus2)
The ele13tron temperatures in the radiative pre13ursor and sho13k-front were
measured to be 34 eV and 60 eV respe13tively using X-ray Thomson s13atte-
ring diagnosti13
In addition a new experiment at PALS (Jungwirth et al 2001) was
presented by Chaulagain et al (2015) using for the rst time a Zn x-ray
laser for XUV imaging at 21 nm This diagnosti13 allowed imaging both the
pre13ursor and post-sho13k stru13ture of radiative sho13ks in Xe gas at 03 bar
In this experiment where the sho13k wave did not ll the tube 2D ee13ts
be13ame pronoun13ed The experimental results were su1313essfully 13ompared
with the results of 2D ARWEN simulations (Cotelo et al 2015)
A summary of these dierent experiments and their prin13ipal diagnosti13s
is presented in table 21
20
CHAPTER2RADIATIVESHOCKWAVES
Experiment Laser fa13ility I (times 1013) λ amp τ ρ0 u Prin13ipal diagnosti13s
W13m
minus2 microm ns mg 13m
minus3
km s
minus1
Bozier et al (1986) CEAs Limeil 20-50 0351 1 10 -30 50-80 Visible streak Camera
Fleury et al (2002) LULI 5 053 05 1 60-65 Visible streak Camera
Visible interferometry
VISAR
Keiter et al (2002) OMEGA 35- 85 0351 1 5 -15
lowast
100 x-ray spe13tros13opy
Gonzaacutelez et al (2006) PALS 10-15 0438 035 1 65 Visible streak 13amera
Visible interferometry
Reighard (2007) OMEGA 70-100 035 1 6 100-140 x-ray ba13k-lighting
VISAR
Stehleacute et al (2010) PALS 10-20 1315 035 1 60 Visible Interferometry
XUV spe13trometer
Stehleacute et al (2012) PALS 10-20 1315 035 15 50-55 XUV imaging
XUV fast Si diodes
Vis13o et al (2012) OMEGA 70 035 1 196
lowastlowast
100-150 x-ray Thomson S13atte-
ring
Chaulagain et al (2015) PALS 10-20 1315 035 15 50 XUV fast Si diodes
Table 21 Summary of the radiative sho13ks experiments and out13omes in dierent laser installations All the experiments are performed
in Xe ex13ept Keiter et al (2002) and Vis13o et al (2012) In these two experiments the rst uses SiO
lowast
2 and the se13ond uses Ar
lowastlowast
as a
sho13ked medium
23 RADIATIVE SHOCK WAVES EXPERIMENTS 21
Laser driven radiative sho13ks Goal of my thesis
All these experimental studies have been fo13used on the 13ase of single ra-
diative sho13ks propagating in a tube However in astrophysi13al 13onditions
radiative sho13ks often intera13t with a denser medium leading to the deve-
lopment of ree13ted and transmitted sho13ks A few representative examples
of su13h phenomena are the intera13tion of supernovae remnants with dense
mole13ular 13louds (Gonzaacutelez Audit and Stehleacute 2009) the a1313retion sho13ks
on the photosphere of T-Tauri stars (Orlando et al 2013) and the bow
sho13ks at the head of stellar jets (Hartigan 1989 Raga et al 1999) The
13ollision (or the intera13tion) of two radiative sho13k waves is obviously a rare
astrophysi13al event and the template 13ase of supernova remnant DEM L316
(see Fig 1 of Williams et al (1997)) is still the subje13t of debates (Williams
et al 2005 Toledo-Roy et al 2009 Velarde et al 2006) as the observation
of these two dierent sho13ks 13an be also interpreted as the superposition of
two blast waves in the eld of view of the teles13ope In this regard the de-
velopment of dedi13ated laboratory experiments to the study of propagation
and intera13tion of 13ounter-propagating sho13k waves is important as a tool to
13hara13terize su13h events through their spe13i13 signatures
In this thesis I will present the results of experiments performed at the
Prague Asterix Laser System (PALS) fa13ility (Jungwirth et al (2001)) on
the study of the intera13tion of two radiative sho13k waves The experimental
obje13tives were
To generate two sho13ks of dierent speeds propagating in opposite
dire13tions in the tube lled with Xenon gas at low pressure (lt 1 bar)
with speeds 13omprised between 12 and 55 kms
To re13ord the ele13tron density by using Ma13h-Zehnder Interferome-
ter over a duration 13omprised between 50 and 200 ns This allowed
studying the ee13t of pre13ursor intera13tion on to the other one
To determine the temperature and the ion 13harge of gas using a time
and spa13e integrated XUV emission spe13tros13opy
The experimental results were interpreted with the help of one-dimensional
simulations using the Lagrangian radiation hydrodynami13 13ode HELIOS
232 Ele13tromagneti13ally laun13hed sho13k by a 13ompa13t pul-
sed power devi13e
As dis13ussed in the previous se13tion most of the plasma laboratory astrop-
hysi13s studies are mainly performed on large-s13ale laser fa13ilities addressing
pure hydrodynami13 radiative sho13ks at very high speed (50 - 150 kms)
and moderate pressure (01 - 1 bar) Complementary to laser experiments
22 CHAPTER 2 RADIATIVE SHOCK WAVES
dis13harge produ13ed plasma has also been a well-established method for ge-
nerating the strong sho13ks (Kondo et al 2008 2009ab Larour et al 2015)
Su13h devi13e presents a high exibility and a larger repetition rate whi13h al-
lows a deeper understanding of sho13k physi13s in 13onditions of astrophysi13al
interest for instan13e in the 13ontext of stellar jets
In prin13iple a 13oaxial ele13trode pair lled with gas followed by an intense
ele13tri13al surfa13e dis13harge generates a thin layer of plasma (plasma sheath)
at the bottom of the ele13trode Due to the high rising 13urrent in the plasma
layer a magneti13 pressure is produ13ed whi13h a1313elerates the plasma layer
also termed as plasma fo13us (Gonzalez Brollo and Clausse 2009) During
this motion the plasma sheath a1313retes a noti13eable part of the ba13kground
gas Experiments performed by Serban and Lee (1995) in the light gasses
H2 D2 have shown fast axial sho13k waves (100 kms) but the plasma sheath
was annular
Later on 13ompa13t pulsed power ele13tri13 generators were able to laun13h
astrophysi13ally relevant strong sho13ks in low-pressure noble gases (Kondo
et al 2006) These authors reported sho13k of speed 45 kms using 13oni13al
13oaxial ele13trodes and Xenon gas (006 - 11 mbar) The ion - ele13tron energy
transfer and ionization relaxation pro13ess as also the radiation ee13t were
investigated by Kondo et al (2008 2009ab)
My work on ele13tromagneti13ally laun13hed sho13k experiment
My work fo13using on the ele13tromagneti13ally laun13hed strong sho13ks rele-
vant for astrophysi13s aimed to design and to optimize an ele13tromagneti13
generator to be able to produ13e strong sho13ks in noble gasses at low pres-
sures of few mbar The ele13tromagneti13 generator is a powerful a1313elerating
devi13e whi13h eje13ts a quasi-planar plasma sheath out of a set of 13oaxial 13o-
ni13al ele13trodes where a pulsed 100-kA 13urrent is passing
Due to the ele13tri13al dis13harge in the gas a hot and fast moving plasma
is generated whi13h travels along the ele13trodes The high-intensity 13urrents
produ13e a magneti13 eld of several Teslas whi13h a13ts as a piston a1313elerating
an annular plasma sheath and driving a strong sho13k towards the top of the
13one I have employed a simple model to optimize the operation parame-
ters With su13h optimization our ele13tromagneti13 generator should produ13e
strong sho13ks over few nanose13onds
Our preliminary experiments show that the generator is 13apable of laun-
13hing supersoni13 sho13ks in Argon in the form of a thin plasma layer with
the speed of sim 1 - 30 kms Three-dimensional MHD simulation performed
by A Ciardi is 13onsistent with the rst observations This is presented in
detail in the 13hapter 6 of the thesis
Chapter 3
1D Simulations
Contents
31 HELIOS 23
32 LTE approximation 24
321 Mean opa13ity 25
322 Equation of State 27
33 Single radiative sho13k waves 27
34 Intera13ting radiative sho13k waves 40
35 Summary 44
Radiative sho13ks are present in various astrophysi13al pro13esses implying
extreme 13onditions Laboratory experiments then allow investigating the un-
derlying physi13al pro13esses whi13h take pla13e in these 13onditions and whi13h
are not observable from the Earth due to a la13k of spatial resolution of the te-
les13opes instruments However experiments are not always straightforward
to interpret and the help of numeri13al simulations be13omes pre13ious
In this 13hapter I will present with the help of 1D radiative hydrodynami13
simulations a brief overview of the physi13s of isolated radiative sho13ks and
of two intera13ting radiative 13ounter-propagating sho13k waves
31 HELIOS
Dierent numeri13al approa13hes are used for the simulation of laboratory
plasmas Some of them use xed grids Other introdu13e the possibility of
renement (on the y) in some meshes (Adaptative Mesh Renement) to
maintain a good resolution in the interesting parts of the plasma Another
approa13h is based on meshes following the uid (Lagrangian des13ription)
whi13h means that the meshes move with the uid and no matter may go
through a mesh to another The form of the equations dier in this 13ase (see
for instan13e Orban et al (2013))
HELIOS is su13h a one-dimensional Lagrangian radiation hydrodynami13
13ommer13ial 13ode (Ma13Farlane Golovkin and Woodru 2006) and I have
used it to simulate our experiment of intera13ting sho13k waves This 13ode
is easy to handle and has the presently interesting 13apability to be able to
simulate the 13ase of two sho13k waves propagating in opposite dire13tions and
laun13hed by two dierent lasers HELIOS may des13ribe non-LTE plasmas as
23
24 CHAPTER 3 1D SIMULATIONS
it in13ludes an in-line 13ollisional radiative (HELIOS-CR) model for 13omputing
non-LTE atomi13 level populations at ea13h time step of the hydrodynami13s
simulation However I used HELIOS in the LTE approximation whi13h is
appropriate to our experimental 13onditions (Rodriguez et al 2011) as will
be dis13ussed in the next se13tion
The 13ode solves the uid equations of motion using the pressure 13ontri-
butions of ele13trons ions and radiation It may des13ribe the ele13trons and
ions as two intera13ting uids at respe13tive temperatures Te and Ti Ther-
mal 13ondu13tion is des13ribed within a ux-limited ele13tron 13ondu13tion model
based on the Spitzer 13ondu13tivity (Burhop and Spitzer 1957) and the laser
energy deposition is 13omputed with an inverse Bremsstrahlung model ()
The radiation emission and absorption terms are introdu13ed in the equa-
tion of energy for the ele13trons and in the radiation transport equations
One of the methods whi13h are proposed is the ux-limited multi-group
radiation diusion model (Gonzaacutelez et al 2015) where the radiative ux is
proportional to the gradient of the radiative energy and is inversely propor-
tional to the Rosseland opa13ity (Dobbs-Dixon Cumming and Lin 2010)
The expression is pondered by a ux-limited diusion 13oe13ient in order
to obtain the good opti13ally thin limit This 13oe13ient follows the Larsen
expression des13ribed in Olson Auer and Hall (2000) The se13ond method is
a (time independent) multi-angle short-13hara13teristi13s s13heme In our 13ase
where radiation and hydrodynami13s are strongly 13oupled and thus the ra-
diation eld varies rapidly we used the diusion model together with LTE
Plan13k and Rosseland multi-groups opa13ities
In addition our version of HELIOS uses the PROPACEOS (Ma13Farlane
Golovkin and Woodru 2006) Equation of State (EOS) and multi-group
opa13ity tables whi13h are generated by the 13ode
1
and it is not possible to
in13lude any other opa13ity or EOS table
32 Lo13al Thermodynami13 Equilibrium
The knowledge of the thermodynami13al 13onditions is required for an ade-
quate simulation of the plasma des13ription
The Lo13al Thermodynami13 Equilibrium (LTE) regime is valid at large
ele13tron densities In that 13ase the 13ollisions between the ele13trons and the
ions and atoms are so frequent that they maintain at ea13h time a steady-
state populations of ions Moreover within a given ioni13 stage the popu-
lations of the dierent energy levels follow the Boltzmann equilibrium and
the populations of the dierent ions the Saha-Boltzmann equation (Fadeyev
and Gillet 2001)
On the 13ontrary in the 13ase of Non Lo13al Thermodynami13 Equilibrium
(NLTE) regime one has to take into a1313ount the ex13itationdeex13itation
1
see http wwwprismminus cscomSoftwarePROPACEOS
32 LTE APPROXIMATION 25
indu13ed by the ele13tron 13ollisions and various radiative pro13esses This leads
to a set of dierential time dependent equations (rate equations) whi13h has
to be solved through and adequate Collisional Radiative model (CR) (Kunze
2009)
Therefore it is 13ru13ial to determine if the regime is LTE or NLTE A
13riterium is given by Griem (2005)
In our experiment the mass density ranges between sim 10
minus4and 10
minus1
g13m
3 while the temperature values are 13omprised between andsim 0 to 50 eV
The pre13ise 13al13ulations performed by Rodriguez et al (Fig2 of Rodriguez
et al (2011)) for Xenon indi13ate that the thermodynami13al 13onditions of
our radiative sho13k experiments 13orrespond to the LTE regime
321 Mean opa13ity
The opa13ity is a fundamental ingredient of the radiative transfer equations
whi13h provide the spe13i13 intensity and its moments (radiative ux energy
and pressure) These last ones enter in the uid equations (see se13tion 213
in 13hapter 2) The 13oupling of the uid equations with the radiative transfer
equation is demanding in terms of 13omputational resour13es and most of the
13odes use simplied radiation transport equations whi13h require the know-
ledge of adequate frequen13y averaged opa13ity 13oe13ients The form of the
average of the opa13ity is not obvious as the average has to be done on the
radiative ux or energy whi13h are unknown before the 13omputation Then
the 13ommonly used opa13ities are the Plan13k and Rosseland mean opa13ities
(Mihalas 1978 Seaton 1987)
The Rosseland mean opa13ity (κR in 13m
minus1and χR = κRρ in 13m
2g) isderived by 13onsidering harmoni13 weighting over the temperature derivative
of the Plan13k fun13tion It gives the 13orre13t radiative ux in the opti13ally
thi13k diusion limit The average is performed over the total mono13hromati13
opa13ity κtotν = κabsν +κscatν (in 13m
minus1) where in the present 13ase the s13attering
13ontribution 13omes from the ele13tron Thomson s13attering
1
κR=
int
infin
01
κtotν
dBν
dT dνint
infin
0dBν
dT dν(31)
The Plan13k mean opa13ity (κP in 13m
minus1and χP = κP ρ in 13m
2g) employs
the normalized Plan13k bla13kbody energy density distribution as a weighting
fa13tor It will give the 13orre13t value for the integrated thermal LTE emission
of an opti13ally thin plasma and is expressed as
κP =
int
infin
0 κabsν Bνdνint
infin
0 Bνdν(32)
In the multi-groups des13ription the radiation transport takes into a13-
13ount the propagation of radiation in N dierent wavelength bands (νk minusνk+1 k = 1 N) Hen13e for instan13e the medium may be transparent for
26 CHAPTER 3 1D SIMULATIONS
some wavelengths and opaque for other ones The radiation transport equa-
tions split into N 13ontributions ea13h of them being asso13iated to adequate
opa13ity 13oe13ients The expression of the 13oe13ients is similar to the pre-
vious grey 13ase (N =1) ex13ept that the integrals are performed between
the boundary frequen13ies of the group
In HELIOS the mono13hromati13 opa13ities κν are 13omputed for the die-
rent radiation groups using the ATBASE atomi13 suite of 13odes Energy levels
and other atomi13 data relevant for bound-bound and bound-free transitions
are 13omputed within a 13onguration intera13tion model using Hartree-Fo13k
wave-fun13tions (Ma13Farlane Golovkin and Woodru 2006)
Dividing the opa13ity by the mass density redu13es the variation of this
quantity with ρ The literature then often uses the mass absorption 13oe-
13ient (also termed as opa13ity) whi13h is dened as
χν = κνρ (33)
Figure 31 PROPACEOS Mono13hromati13 opa13ity versus the photon energy in
eV of Xenon at 10 eV and at two ele13tron densities respe13tively equal to 1018 and
1020 13mminus3
An example of the variations of the PROPACEOS opa13ity χν versus the
frequen13y in eV is given in Fig 310a for Xenon at 10 eV and two ele13tron
densities respe13tively equal to 1018 and 1020 13m
minus3 whi13h do 13orrespond to
typi13al 13onditions of the pre13ursor and post-sho13k of our laser generated
radiative sho13ks
33 SINGLE RADIATIVE SHOCK WAVES 27
322 Equation of State
The equation of state (EOS) denes the dependen13e of the pressure ionisa-
tion internal energy with the mass density and temperature Several models
do exist in the literature
Our version of HELIOS for LTE plasmas may handle polytropi13 or PRO-
PACEOS equation of state Unfortunately there are not many details about
this last method The authors of the 13ode mention that it uses a QEOS-type
model (More et al 1988) in the strong 13oupling regime (ie at relatively
high densities and low temperatures) and an isolated atom model whi13h
uses a detailed atomi13 stru13ture modeling in the weak 13oupling region The
properties between the weak and strong 13oupling regimes are said to be obtai-
ned by interpolating in a thermodynami13ally 13onsistent manner This EOS
in13ludes the 13ontributions from the translation of ions and ele13trons ele13-
tron degenera13y atom ionization and ex13itation and Coulomb intera13tions
(Debye - Hu13kel 13orre13tion)
Comparisons on spe13i13 13ases have been performed by the authors of the
13ode with Los Alamos SESAME data for CH Al and Au showing a good
agreement when using these two equations of state
In this 13hapter HELIOS 13ode will then be employed to simulate the
detailed 13hara13teristi13s of single and 13ounter-propagating radiative sho13ks
driven by one or two lasers in the 13onditions of our experiments (Chapter 5)
The target 13ell (4-mm length) is lled with Xenon gas at 01 bar This valueof pressure is representative and 13orrespond to standard ambient tempera-
ture ie 298 K and not the initial temperature 13onsidered in the simulations
This terminology is maintained throughout the thesis Two lasers are inte-
ra13ting at both ends with two gold-13oated `CH foils 11microm CH and 06 micromAu) ea13h of them resulting in a piston of total thi13kness 116 microm 13losing the
13ell
33 Single radiative sho13k waves
Typi13al simulations of the radiative sho13ks espe13ially multidimensional ones
whi13h are time-13onsuming use a frequen13y independent (grey or one group)
opa13ities However then the opa13ity of the gas under investigation presents
strong variations with the frequen13y a multi-group approa13h may be suitable
(Vaytet et al 2011 2013) When the number N of groups tends to innity
ea13h group opa13ity κkR and κkP tends to the lo13al mono13hromati13 opa13ity κν whi13h solves the question of the adequate opa13ity average
In the following I will investigate the inuen13e of the opa13ity and the
number of groups on the stru13ture of a sho13k wave propagating at sim 50 kms
in Xenon at 01 bar To this purpose I have performed HELIOS simulations
for a laser of uen13e 28000 Jcm2whi13h is able to drive a sho13k with the
relevant velo13ity In addition I have also analyzed the ee13ts of taking into
28 CHAPTER 3 1D SIMULATIONS
a1313ount the dierent ion and ele13tron temperatures
Referen13e simulation (1 group Te = Ti = T )
A result of the simulation is reported at 10 ns in the Fig 32 whi13h reports
the variations of ele13tron density (Ne) temperature (T ) mean 13harge (Z)and mass density (ρ) The initial 13onditions are 45 times 10
minus4g13m
3and T =
1 eV
The sho13k front is lo13ated at sim 0047 13m (speed of 47 kms) At this
position the temperature peaks at 22 eV The bla13k verti13al dotted line shows
the position of the interfa13e between the piston and the ba13king Xenon gas
This interfa13e is lo13ated at sim 20 microm from to sho13k front The post-sho13k gas
mass density pressure ele13tron density and ion 13harge are respe13tively equal
to 002 g13m
3 11times 10
4bars 52times 10
2013m
3and 56 The high 13ompression
is due to ionisation and radiation 13ooling (13hapter 2) Its temperature of 12
eV is the same than in the pre13ursor where it is 13onstant over the 2 mm of the
simulation The pre13ursor ele13tron density and mean 13harge are respe13tively
equal to 18 times 10
1913m
minus3and 7 The pre13ursor is not 13ompressed
The variations of the Rosseland opa13ities (13m
2g
minus1and 13m
minus1) with the
distan13e are reported in the Fig 33 The Rosseland opa13ity is low in the
pre13ursor Hen13e 600 13m
2g
minus113orresponds to 05 13m
minus1 whi13h means that
an opti13al depth of 1 is rea13hed at 2 13m from the sho13k front This explains
the at prole of the pre13ursor With a typi13al value in the post-sho13k of
2577 in 13m
2g
minus1 this opa13ity is now 53 13m
minus1 and an opti13al depth of 1 is
here rea13hed at 200 microm from the sho13k front (whi13h is mu13h smaller than
the post sho13k extension)
This simulation was performed using the Spitzer thermal 13ondu13tivity
To 13he13k the 13ondu13tivity ee13t I performed another simulation with a
negligible thermal 13ondu13tivity (ie 10
minus12WmK) for the Xenon layers
only The results are reported in the Fig 34 The interfa13e between the
piston and the ba13king Xenon gas is still 13lose to the sho13k front Ex13ept
for the peak of temperature whi13h is higher (sim 29 eV instead of 22 eV) and
thiner the sho13k stru13ture is similar to the previous 13ase
Two uids with dierent temperatures Te and Ti (1 group)
In the previous se13tion HELIOS simulations have been 13arried out for the
13ase of equal ion and ele13tron temperatures I will now investigate the ee13t
of su13h approximation on the simulated results employing 1) Spitzer and 2)
negligible 13onstant thermal 13ondu13tivity in the Xenon layers It is important
to note that this two uids approximation 13an not be restri13ted to the Xenon
layers only and thus it is valid from the piston (CHAu) up to the Xenon
gas Thus the dynami13s and 13onditions of the piston are also modied whi13h
may slightly modify the sho13k velo13ity
33 SINGLE RADIATIVE SHOCK WAVES 29
(a)
(b)
Figure 32 Mass density and temperature (a) ele13tron density and mean 13harge
(b) at 10 ns for a radiative sho13k of speed sim 47 kms in Xenon at 01 bar The
verti13al dotted bla13k lines show the position of the interfa13e between piston and
ba13king Xenon gas Zero at x-axis is the interfa13e of piston and Xenon at time zero
Spitzer thermal 13ondu13tivity has been used in the simulation
30 CHAPTER 3 1D SIMULATIONS
Figure 33 Mean Rosseland opa13ity (in 13m
2g
minus1) and opa13ity (13m
minus1) at 10 ns
for a radiative sho13k of speed sim 47 kms in Xenon at 01 bar The verti13al dotted
bla13k lines show the position of the interfa13e between piston and ba13king Xenon gas
Zero at x-axis is the interfa13e of piston and Xenon at time zero Spitzer thermal
13ondu13tivity has been used in the simulation
33 SINGLE RADIATIVE SHOCK WAVES 31
(a)
(b)
Figure 34 (a) Mass density and temperature (b) ele13tron density and mean
13harge at 10 ns for a radiative sho13k of speed sim 47 kms in Xenon at 01 bar The
verti13al dotted bla13k lines show the position of the interfa13e between piston and
ba13king Xenon gas Zero at x-axis is the interfa13e of piston and Xenon at time zero
A negligible 13onstant thermal 13ondu13tivity (10
minus12WmK) have been used in the
simulation for Xenon
32 CHAPTER 3 1D SIMULATIONS
(a)
(b)
Figure 35 (a) Ele13tron density and ele13tron temperature (b) Mass density and
mean 13harge at 10 ns for a radiative sho13k of speed sim 48 kms in Xenon at 01
bar for the 13ase when the ele13tron and ion temperature are dierent The verti13al
dotted bla13k lines show the position of the interfa13e between piston and ba13king
Xenon gas Zero at x-axis 13orresponds to the position of the interfa13e between the
piston and Xenon at time zero The Spitzer thermal 13ondu13tivity has been used in
this simulation
33 SINGLE RADIATIVE SHOCK WAVES 33
The Fig 35a reports the sho13k stru13ture at 10 ns for the 13ase of Spitzer
thermal 13ondu13tivity The sho13k is lo13ated at 0048 13m The ion temperature
peaks at 53 eV and the ele13tron temperature peaks at the same value of 20
eV than previously Ex13ept for the ion temperature the sho13k stru13ture is
not signi13antly 13hanged 13ompared to the 13ase of equal temperatures
Let us now inspe13t the role played by the thermal 13ondu13tivity (see Fig
36) In the 13ase of a negligible thermal 13ondu13tivity of (10
minus12WmK) for
the Xenon layers only the ele13tron temperature (and as a 13onsequen13e the
mean 13harge) peaks at higher value 25 eV (instead of 20 eV with the Spitzer
13ondu13tivity) and the ion temperature rea13hes to 405 eV (instead of 53 eV)
The rest of the sho13k is not ae13ted by this 13hange in the 13ondu13tivity This
13onrms that the width and height of the ion peak temperature are a dire13t
13onsequen13e of the thermal 13ondu13tivity
Several groups for the opa13ity one temperature
As mentioned earlier the number of frequen13y groups inuen13es the dyn-
ami13s and morphology of radiative sho13ks A set of simulations has been
performed 13onsidering (only for Xenon) dierent numbers (N) of frequen13y
group varying between 1 and 100 for Te = Ti The distribution in energy
of the groups is logarithmi13 and the grid is set up with approximately 85
of the groups having photon energies between 01 eV and 3 KeV while the
remaining 15 lie between 3 KeV and 1 MeV
The Figures 37 (a) and (b) show the variations of the ele13tron tempe-
rature with the number of groups (from 1 to 100) for the pre13ursor (a) and
the front (b) In13reasing the number of groups ae13ts mostly the pre13ursor
A similar behavior is found in Vaytet et al (2013)
It should be noted that this multigroup opa13ity treatment is applied
to all the layers (ex13ept gold only one group opa13ity is provided by the
13ompany) in the simulation This 13ould ae13t slightly the dynami13s of the
piston Con13erning the pre13ursor at this time of the simulation and taking
an a1313ountable variation in temperature of 10 the 13onvergen13e is a13hieved
after 20 groups
Variation with the opa13ity
In order to estimate the ee13t of the opa13ity I have performed several simu-
lations by in13reasing the Plan13k and Rosseland opa13ities in Xenon (Stehleacute
et al 2014b) using a 13ommon multiplier ranging between 1 to 40 Su13h
pro13edure is often used to redu13e the gap between the available and more
sophisti13ated opa13ities To ex13lude the impa13ts of the radiation 13oming from
the post-sho13k to the piston layers the opa13ity of the piston (CH and Au)
layers have been set to zero
I present in Fig 38 the results (proles of temperature mass density and
34 CHAPTER 3 1D SIMULATIONS
(a)
(b)
Figure 36 (a) Mass density and ele13tron temperature (b) ele13tron density and
mean 13harge at 10 ns for a radiative sho13k of speed sim 45 kms in Xenon at 01 bar
for the 13ase when the ele13tron and ion temperature are dierent The verti13al dotted
bla13k lines show the position of the interfa13e between piston and ba13king Xenon
gas Zero at x-axis 13orresponds to the position of the interfa13e between the piston
and Xenon at time zero In this simulation value of thermal 13ondu13tivity (10
minus12
WmK) is negligible for Xenon The peak value of ion temperature is found to be
405 eV whereas theoreti13ally this value is expe13ted to be sim 600 eV It is possible
to a13hieve the expe13ted value by in13reasing the resolution of the simulation
33 SINGLE RADIATIVE SHOCK WAVES 35
Figure 37 Ele13tron temperature proles for various numbers of frequen13y groups
N = 1 20 50 60 70 90 and 100
mean 13harge) of four 13ases 13orresponding to an opa13ity multiplier equal to
1 10 30 and 40 The ele13tron and ion temperatures are taken to be equal
In13reasing the opa13ity in13reases the absorption in the pre13ursor and thus
de13reases its length as the photons emitted by the sho13k are more rapidly
absorbed by the 13old pre-sho13k Taking as a referen13e the previous value of
the Rosseland opa13ity of 600 13m
2g
minus1and a multipli13ative fa13tor of 40 the
opti13al depth of 1 will now be rea13hed at 005 13m (instead of 2 13m without
any multiplying fa13tor)
The temperature peak value at the sho13k front also de13reases I do believe
that this 13an be attributed to the in13rease of the radiative 13ooling of the peak
of temperature with the multiplying fa13tor
I adopted here an oversimplied method in the sele13tion of the groups
boundaries A more realisti13 one would be to take into a1313ount the variation
of the opa13ity in the dierent layers with the wavelength In the 13ase of
xenon for instan13e it would be then more adapted to rene the des13ription
of the opa13ity between 5 and 100 eV (see Figure 310a)
Opa13ities 13omparison
The previous study illustrates the role played by the opa13ity for the radi-
ative sho13ks waves with a pre13ursor It seams then logi13al to 13ompare the
36 CHAPTER 3 1D SIMULATIONS
(a)
(b)
(13)
Figure 38 (a) Ele13tron temperature (a) mean 13harge (b) and mass density (13)
for four (1 10 30 and 40) multipliers of the Xenon opa13ity at 10 ns
33 SINGLE RADIATIVE SHOCK WAVES 37
PROPACEOS opa13ity whi13h is used by HELIOS with more sophisti13ated
13al13ulations
The Figures 39a and 39b report the variations of the Plan13k and Ros-
seland grey opa13ities 13omputed for the density of the pre13ursor ρ = 15 times10
minus3g13m
3by two dierent 13odes PROPACEOS (Ma13Farlane Golovkin
and Woodru 2006) and BiGBART (de la Varga et al 2011 Ogando and
Velarde 2001) PROPACEOS opa13ities and EOS used in HELIOS simula-
tions 13omputes frequen13y-dependent opa13ities along with spe13i13 internal
energies and pressures The atomi13 13ode BiGBART able to 13al13ulate two
type of opa13ities dependent on atomi13 stru13ture and frequen13y dependent
The opa13ities 13omputed by Rodriacuteguez et al (2015) are also reported for 13om-
parison The two last 13al13ulations use the FAC (Flexible Atomi13 Code) (Gu
2008) for the 13al13ulation of the atomi13 data
We see that for a temperature equal to 10 eV the Rosseland opa13ity is
equal respe13tively to 800 15000 and 8000 g13m
2 whereas the Plan13k opa13ity
is equal to 18000 65000 370000 g13m
2 This means that the PROPACEOS
opa13ity whi13h is used in HELIOS is smaller by more than one order of
magnitude than the two last opa13ities As it is not possible to 13hange the
opa13ity model within HELIOS we then de13ided to use the PROPACEOS
opa13ity with a multipli13ative fa13tor whi13h we have set equal to 20
The dis13repan13ies in the average opa13ity are a 13onsequen13e of important
dieren13es in the mono13hromati13 opa13ities This is illustrated in the Fig
310 whi13h 13ompares the mono13hromati13 opa13ities given by PROPACEOS
and Rodriacuteguez et al (2015) (see Fig 16 of Rodriacuteguez et al (2015)) at 15
eV and 15 times 10
minus3g13m
minus3) This gure indi13ates a noti13eable dieren13e
between 5 and 150 eV with more bb transitions for the last method This
explains the dieren13es observed in the 13orresponding averages
Synthesis
These dierent simulations for a sho13k propagating at sim 50 kms in Xenon
at 01 bar show that the simulations performed with only one temperature
are adequate for the des13ription of the sho13k
The opa13ity strongly ae13ts the pre13ursor extension whi13h de13reases from
more than 4 mm for 1 group to 3 mm then the number of groups is larger
than 20 The 13onvergen13e versus the number of groups is rea13hed after 20
groups
To t with more sophisti13ated opa13ities and as it was impossible to
in13lude them in HELIOS I de13ided to multiply the opa13ities of Xenon by a
fa13tor of 20 This redu13es strongly the pre13ursor extension up to sim 1 mm
at 10 ns Taking into a1313ount the present huge un13ertainty in the opa13ity
I 13onsidered that it was not ne13essary to use multi-group opa13ities This
also means that our 1D simulations will be used - only - for a qualitative
interpretation of the experimental results A quantitative 13omparison should
38 CHAPTER 3 1D SIMULATIONS
(a)
(b)
Figure 39 Plan13k and Rosseland mono13hromati13 opa13ity proles Te for ρ =15
times 10
minus3from PROPACEOS (I) BIGBART (II) and Rodriguez et al (III)
33 SINGLE RADIATIVE SHOCK WAVES 39
(a)
(b)
Figure 310 Mono13hromati13 Xenon opa13ity versus the photon energy in eV at T
= 15 eV and for ρ = 15 times 10
minus3g13m
minus3for the PROPACEOS (a) and Rodriacuteguez
et al (2015) (Fig 16 of Rodriacuteguez et al (2015)) (b) models
40 CHAPTER 3 1D SIMULATIONS
be in the future based on adequate opa13ities and multi-groups modeling
34 Intera13ting radiative sho13k waves
In order to pre13ise what is expe13ted from the experimental study I will now
investigate the 13hara13teristi13 parameters of two 13ounter-streaming sho13ks
propagating in Xenon at 01 bar (ρ = 54 times 10
minus4g13m
minus3) with equal and
dierent velo13ities To highlight the ee13t of the intera13tion I will 13ompare
the results with those obtained with 1D simulations All the simulations will
be performed with HELIOS For the Xenon opa13ity a multiplier times 20 will
be used The number of groups will be set to be 1 and the ele13tron and ion
temperatures will be supposed to be equal
Three representative sets will be 13onsidered
Case(I) two identi13al radiative sho13ks (2RS) at sim 50 kms propaga-
ting in opposite dire13tions (ie starting from the left and right end
respe13tively)
Case(II) same 13onditions but without any 13oupling with radiation
Case(III) two 13ounter-propagating radiative sho13ks propagating with
dierent speeds sim 50 kms from the left side of the 13ell and sim 20
kms from the right side
To a13hieve the aforementioned speeds on the left and right sides the
uen13es of the two laser beams have been adjusted The pulse duration is
set to 03 ns (peak at 015 ns) to reprodu13e the experimental 13onditions
detailed later in the manus13ript
Case(I) 13ounter propagating sho13ks with the same speed
This a13ademi13al 13ase is fully symmetri13al and it is equivalent to the 13ase of
one RS with a fully ree13tive boundary (for hydrodynami13s and radiation)
in the middle of the tube The Fig 311 shows the variations of the ele13tron
density (Ne) and temperature (Te) in the Xenon layers The two sho13ks
appear in Xenon at sim 2 ns and the 13ollision o1313urs at sim 38 ns At 3 ns the
pre13ursor extension is sim 008 13m whereas the post-sho13k ele13tron density
and ele13tron temperature are 78 times 10
2013m
minus3and 16 eV respe13tively The
length of pre13ursor in13reases rapidly with time and the two pre13ursors merge
suddenly at sim 8 ns After this time the merging ee13t in13reases signi13antly
It is 13hara13terized by a at 13ommon pre13ursor those ele13tron density and
temperature are in13reasing with time At the time of the 13ollision (sim 38
ns) the post-sho13k mass and ele13tron density in13rease from 0011 to 014
g13m
minus3and 67 times 10
20to 66 times 10
2113m
minus3 whereas the temperature rises
up to 39 eV The 13ollision leads to the development of two reverse sho13k
34 INTERACTING RADIATIVE SHOCK WAVES 41
(a)
(b)
Figure 311 Ele13tron density Ne (a) and ele13tron temperature Te (b) versus axial
position (along a 04 13m long sho13k tube) at 3 10 20 30 and 38 ns from HELIOS
simulations (with opa13ity times 20) for the 13ases of single sho13k of sim 50 kms (dotted
line) and two identi13al 13ounter-propagating sho13ks of sim 50 kms (solid lines) The
verti13al dotted lines show the position of the interfa13e between piston and ba13king
Xenon gas
42 CHAPTER 3 1D SIMULATIONS
waves propagating ba13k with a speed of sim 15 kms rst in Xenon and later
on in the dierent piston layers (not shown in the Figure) These reverse
sho13ks lead to a dense plasma (Ne gt 10
2113m
minus3) whi13h is not a1313essible to
the experimental diagnosti13s and will not be detailed here
To investigate the ee13ts of the intera13tion I have reported in Fig 311
the 13ase of 1RS moving from the left to the right dire13tion in the 13ell (dotted
lines) The wave propagates identi13ally to 2RS until 10 ns After this time
the proles of the temperature and ele13tron density dier strongly from the
previous 13ase and their values are lower than for the 2RS The post-sho13k
extension is slightly smaller than for the 2RS and the sho13k speed is thus
slightly smaller too This last ee13t is due to the fa13t that for the two inte-
ra13ting waves the pre13ursor is at a higher temperature than for the isolated
sho13k and thus the sho13k propagates in a warmer medium then modifying
the sho13k 13onditions (sound speed opa13ity et13 ) A similar ee13t in the post
- sho13k extension may be seen for an isolated sho13k wave propagating in a
warmer pre-sho13k gas
Case (II) Identi13al 13ounter-propagating sho13ks for an ideal gas
In order to highlight the ee13t of the radiation another simulation has been
performed with the same set of parameters as in the 13ase I however putting
the Xenon opa13ity equal to zero I still use here the PROPACEOS equation
of state The result of the simulation is presented in Fig312 The 13ollision
time is now 40 ns instead of 38 ns (thus the sho13k speed is slightly smaller)
The post-sho13k is no more 13ompressed by radiation 13ooling Its 13ompression
at 10 ns is 10 instead of 35 This 13ompression is due to the vis13ous sho13k
(fa13tor 4) followed by the ionisationex13itation of the gasThere is obviously
no radiative pre13ursor Moreover there are no dieren13es in the Ne and Tproles of the single sho13k and that of the two 13ounter-propagating sho13ks
before the 13ollision time
Case (III) Two sho13ks at dierent speeds
The spatial and temporal variations of Ne and Te are plotted at times 3
10 20 30 38 and 49 ns in Fig 313a and 313b The left and the right
sho13ks appear in Xenon at sim 2 and 3 ns respe13tively Later at 10 ns
the two pre13ursor extensions are respe13tively equal to 018 and 0034 13m
The merging of the two pre13ursors starts at sim 15 ns As expe13ted from the
values of sho13k speeds the 13ollision time o1313urs at 49 ns whi13h is delayed
in 13omparison to the 13ase (I)
In 2RS the two radiative pre13ursors merging results in a plateau for the
ele13tron density and the temperature The temperature at 13ollision time is
now 28 eV instead of 39 eV in the 13ase I and the ele13tron density rea13hes
up to 31 times 10
2113m
minus3instead of 66 times 10
2113m
minus3
34 INTERACTING RADIATIVE SHOCK WAVES 43
(a)
(b)
Figure 312 Variations of Ne (a) and Te (b) versus axial position for the 13ase
of two identi13al 13ounter-propagating sho13ks (of speeds sim 50 kms) at 3 10 20
30 35 and 40 ns as derived from HELIOS simulations For these simulations we
have negle13ted the ee13t of radiation by keeping the Xe opa13ity equal to zero
The verti13al dotted lines show the position of the interfa13e between the piston and
ba13king Xenon gas
44 CHAPTER 3 1D SIMULATIONS
It may be noted that up to this 13ollision time the post-sho13k density
13onditions and speed are identi13al for the 13ase of an isolated RS propagating
from the left (1RSL) at 50 kms and this present left sho13k This reveals
that in the 13ase of 2RS there is no noti13eable ee13t of the right sho13k with
speed 20 kms on the left post-sho13k of 50 kms
On the 13ontrary we note a dieren13e in extension of the right post sho13k
from the 2RS 13ompared with the 13ase of an isolated sho13k (1RSR) propa-
gating at 20 kms from the right whi13h indi13ates that the left sho13k of the
2RS has an inuen13e on the right post-sho13k (Fig 313a)
35 Summary
Radiative sho13k waves propagating in Xenon at 01 bar with a velo13ity of 50
kms are 13hara13terized by the development of an extended radiative pre13ur-
sor The huge 13ompression of 38 in the post-sho13k is a 13onsequen13e of both
the sho13k and the gas ionisation (fa13tor of 10) as also from the radiative
13ooling The post-sho13k and pre-sho13k temperature on both side of the peak
are identi13al indi13ating that the sho13k is super13riti13al
The spe13ta13ular role of the opa13ity has been highlighted Due to the
un13ertainty in the opa13ity used for Xenon in our simulations I de13ided that
at this stage it was not ne13essary to rene the simulation in terms of group
numbers within the 1D des13ription of the sho13k wave
Our numeri13al study gives the main 13hara13teristi13s of the intera13tion of
two 13ounter-propagating sho13k waves with speeds equal to 50-50 kms and
50-20 kms The 13ase of identi13al speeds is simpler due to the symmetry of
the problem However whatever the speeds the most important signature
of the intera13tion is the merging of the pre13ursor at 8 ns for 50-50 kms
and at 15 ns for 50-20 kms This merging is followed by a regular in13rease
with time of the ele13tron density and the temperature The 13ollision time
is 13hara13terized by a sudden in13rease of the ele13tron density by an order of
magnitude rea13hing 66 times 10
21and 3 times 10
2113m
minus3respe13tively whereas the
temperature in13reases up to 39 and 28 eV
35 SUMMARY 45
(a)
(b)
Figure 313 Variations of Ne (a) and Te (b) with axial position for the 13ase of
two non-identi13al 13ounter-propagating sho13ks (of speeds sim 50 amp 20 kms) and two
single sho13ks (dotted lines) of speeds sim 50 amp 20 kms respe13tively at 3 10 30 35
and 46 ns as derived from HELIOS simulations The verti13al dotted lines show the
position of the interfa13e between piston and ba13king Xenon gas (with opa13ity times20)
46 CHAPTER 3 1D SIMULATIONS
Chapter 4
Laser-driven radiative sho13k
Experimental Setup
Contents
41 Prague Asterix Laser System (PALS) fa13ility 48
42 Targets 49
421 Massive Targets 51
422 Gaseous Targets 51
423 Target holder 55
424 Target lling 56
43 Laser Fo13using 57
431 Fo13al Lenses and Phase Zone Plates 57
432 Fo13using of the MAIN and AUX laser beams 58
44 Diagnosti13s 59
441 Visible interferometry 61
442 XUV spe13tros13opy 65
443 Visible spe13tros13opy 67
45 Summary 68
When a short pulse of a high-power laser beam is fo13ussed on a thin foil
it transfers a huge amount of energy to the foil material Almost instantane-
ously the laser absorption heats it up to the ablation A hot and low density
13orona is generated ba13kwards by this pro13ess Beyond the ablation surfa13e
a sho13k is formed in the foil as a 13onsequen13e of the momentum 13onservation
(ro13ket ee13t) The sho13k moves inward in the foil Our 13ase of the experi-
mental setup as the foil 13loses a tube lled with gas the sho13k propagates
then in the gas where it is studied through various kinds of diagnosti13s The
foil then a13ts as a piston
We performed the experiments at the Prague Asterix Laser System (PALS)
in the Spring 2015 (ve weeks 20th April - 22th May) The obje13tive of the
experimental 13ampaign was to study the evolution of two 13ounter streaming
radiative sho13ks driven by two 13ounter-propagating lasers with an irradian13e
in the range of 10
14W13m
minus2 and therefore to 13hara13terize their intera13tion
and 13ollision In this 13hapter I will rst present a brief des13ription of the
PALS laser fa13ility and of the laser beams used in our experiment This will
47
48 CHAPTER 4 EXPERIMENTAL SETUP
be followed by a presentation of the targets design and an overview of the
general setup and of our main diagnosti13s
41 Prague Asterix Laser System (PALS) fa13ility
The Prague Asterix Laser System (PALS) is a laser fa13ility based on an
Infrared high-power iodine laser system (Asterix IV) (Jungwirth et al 2001)
Using dierent amplifying stages the laser fa13ility is able to deliver energy
up to 1 kJ in 03 ns at the fundamental wavelength 1315 nm The output
laser beam 13an be further subdivided in few auxiliary beams All auxiliary
beams may be frequen13y doubled (λ = 657 nm) or tripled (438 nm) The
PALS laser fa13ility is 13apable of ring up to two high energy laser shots per
hour Compared with solid state lasers this gas laser is known to deliver a
quite homogenous beam intensity without hot spots A spatial prole of the
PALS laser beam is shown in the Fig 41a (re13orded by PALS team during
the experimental 13ampaign) while its pulse 13hara13teristi13s are plotted with
time in the Fig 41b
(a)
(b)
Figure 41 a Spatial prole of the PALS Laser beam b Intensity prole of
laser pulse I(t) with time
For our experiments we used two independent laser beams to drive the
two 13ounter streaming sho13ks in the gaseous target For this purpose the
fundamental PALS laser beam has been subdivided after the fourth amplier
(V4) into two beams with respe13tive energy fra13tions of 60 and 40 (see
Fig 42) The most energeti13 of these laser beams is then inje13ted through
the fth (V5) amplier and its frequen13y are then tripled This beam at 438
nm will be hereafter termed termed as the MAIN laser beam
The se13ond of the two previous laser beams propagates then without any
modi13ation It will be hereafter 13alled the AUX laser beam (1315 nm)
The s13heme of the energy distribution is shown in the Fig 42 and the
13hara13teristi13s of MAIN and AUX laser are presented in Table 41
For the experiments PALS oers two va13uum 13hambers with a spheri13al
42 TARGETS 49
Figure 42 Energy distribution s13heme for MAIN and AUX laser beams
LASER MAIN AUX
Typi13al laser energy (J) 120 60
Beam diameter (mm) 290 148
Wavelength (nm) 1315 438
Pulse duration (ps) 350 350
Table 41 Nominal 13hara13teristi13s of MAIN and AUX laser beams
and 13ylindri13al geometry respe13tively The spheri13al 13hamber assembly used
in our experiments is shown in Fig 43 This 13hamber having a diameter
of 100 13m is 13losed by two entran13e 13ir13ular windows of respe13tive diameters
80 13m and 50 13m to allow the MAIN and AUX beams to penetrate in the
13hamber and to rea13h the target Ea13h of these windows is outtted with a
hinged door Several 13ir13ular ports of various diameters also allow for the
dierent diagnosti13s The target holder and some diagnosti13s are mounted
on an opti13al ben13h whi13h is de13oupled from the 13hamber walls
42 Targets
Our targets s13hemati13ally 13onsist in small tubes of 4 mm length 13losed on
both sides by two spe13i13 foils of thi13kness 11 microm on whi13h the laser beams
are fo13used (one laser per foil) with an irradian13e of about 10
1413m
minus3 The
tube is lled with gas in whi13h the radiative sho13k propagates with a velo-
13ity of 30 - 60 kms The two foils 13losing the target insure the 13onversion
through ablation and sho13k generation of the laser energy into me13hani13al
energy
In our experiments we used spe13i13 targets for alignment and 13hara13te-
rization of the beam size These targets were manufa13tured at the Pole
instrumental of Observatoire de Paris
50 CHAPTER 4 EXPERIMENTAL SETUP
Figure 43 Snapshot of the spheri13al 13hamber showing the horizontal bredboard
and on the right the fo13using MAIN lens
(a) (b)
Figure 44 Dimension drawing (a) and snapshot (b) of Aluminum massive
(mo13k) target
42 TARGETS 51
421 Massive Targets
For the laser alignment on the target we have used spe13ial targets also
termed as mo13k targets A mo13k target (Fig 44) is an aluminum blo13k
with a base Three 13uts made on it help to fo13us the laser beam at the
desired position The 13enter of ea13h of the two laser spots should be lo13ated
at the interse13tion of the verti13al and horizontal marks and the pre13ision of
this alignment is less than 20 microm
Beside alignment purpose su13h targets are used to 13hara13terize the MAIN
and AUX spot diameters in the planes whi13h 13orrespond to the lo13ation of
the two Parylene foils Hen13e the laser beam fo13ussed on these targets
generates a 13rater (Chaulagain 2015) The detailed analysis of the shape
and size of these 13raters provides a quantitative information about the ho-
mogeneity of the in13ident beam and the size of the fo13al spot A detailed
dis13ussion about the fo13alisation of the laser with a massive target will be
presented in the se13tion 432
422 Gaseous Targets
The 13ore of the gaseous targets (Stehleacute et al 2014a) 13onsists of a 13hannel of
a parallel pipe shape having the dimension of 09 times 06 times 4 mm pla13ed at
the top of an aluminum stru13ture The pipe in whi13h the sho13ks propagate
is dened by two 500 microm thi13k verti13al lateral sides made with 13oated fused
sili13a (SiO2) windows (09 times 4 mm) as shown in the Fig 45 Su13h windows
are suitable for the visible interferometry diagnosti13 whi13h uses an auxiliary
Q-swit13hed NdYLF green Evolution laser (see Se13tion 441)
The 13hannel is 13losed at the top with a window made of a 100 nm thi13k
Si3N4 membrane whi13h is itself supported is by a 200 microm thi13k sili13on frame
(26 times 4 mm) made by SILSON 13ompany Su13h membrane is transparent
to XUV radiation whi13h is ne13essary for the XUV spe13tros13opi13 diagnosti13
used in the experiment The transmission of the aforementioned material is
obtained from the 13enter for X-ray opti13s (CXRO) database
1
and is plotted
between 0 and 40 nm in Fig 48
The pistons 13losing the tube from opposite sides and whi13h will laun13h
the two sho13ks are made of Parylene-N (11 microm) 13oated by Au (06 microm)
(SCITECH 13ompany) They are glued on 01 mm thi13k Ni13kel disks with
external diameter 5 mm and internal diameter 1 mm This disk helps in
assembling the targets It also 13ontributes together with a larger V-shield
(se13tion 49) to prevent hard X-ray emitted at the laser impa13t on the foil
to rea13h the gas in the tube The Parylene layer is fa13ing the laser and plays
the role of an ablator The gold layer whi13h is 13oated on the Parylene and
is lo13ated between the Parylene and the gas aims at blo13king the X-rays
generated by the laser plasma intera13tion to rea13h and preheat the gas in
1
httphenkelblgovopti13al_13onstants
52 CHAPTER 4 EXPERIMENTAL SETUP
(a)
(b)
Figure 45 Values on the gures are in mm (a) Verti13al 13ross se13tion view of
the gaseous target (b) Horizontal 13ross se13tion view
42 TARGETS 53
Figure 46 Pi13ture of a the gaseous target on its base
Figure 47 S13hemati13 of the gaseous target
54 CHAPTER 4 EXPERIMENTAL SETUP
Figure 48 Transmission of 100 nm thi13k Si3N4 membrane (CXRO database)
Figure 49 Snapshot of a target holder with one V-shield regarding the AUX
laser whi13h will pass through the hole in this shield marked in green in the Figure
The target whi13h fa13es the two diagnosti13s (tubes) and is lo13ated behind the
V-shield is thus not visible
42 TARGETS 55
the 13ell These various elements are manually glued on the base of the target
to built the sho13k 13hannel Ea13h target has been tested against leaks before
the experiment and before ea13h shot The axis of symmetry of the sho13k
waves is required to be lo13ated at 03 mm from the base and at 03 mm from
the two verti13al SiO2 windows S13hemati13s of verti13al and horizontal 13ross
se13tion views of the target are presented in Fig 45b The axis of symmetry
is visible in the verti13al view while the position of the laser fo13us point (`+
sign) in the horizontal view This is a13hieved during the alignment pro13edure
employing mo13k targets as dis13ussed in the se13tion 432 The base of the
target 13onsists of a mi13ro-ma13hined aluminum stru13ture A gas pipe at the
bottom of the stru13ture allows for in situ gas lling at a pressure of a fra13tion
of a bar
A total of 55 targets were manufa13tured for the experiment Among
them 31 gave valuable results
423 Target holder
We x the targets on a stainless holder and then pla13e the system into the
va13uum 13hamber The target holder has been designed and manufa13tured
at Observatoire de Paris It allows a very reprodu13ible positioning of the
target on the va13uum 13hamber breadboard The main element of the target
holder is a base where ea13h target is rst xed by s13rews and removed after
ea13h shot to be repla13ed by a new one It also in13ludes a diagnosti13s support
(Fig 46) whi13h is atta13hed on it from the top with s13rews One of these
diagnosti13s 13onsists in a fast XUV Si diode with a pinhole The diode is
atta13hed to the bla13k tube whi13h is visible on the left part the target holder
in Fig 47 This last diagnosti13 aimed for sho13k timing as the pinhole-diode
system allows to re13ord the sho13k passing at the imaged position of the tube
(Stehleacute et al 2012 Chaulagain 2015) Unfortunately this diagnosti13 did
not work well and will not be mentioned further in the thesis
The se13ond diagnosti13 is a visible time-and-spa13e-resolved visible spe13tro-
meter On the target holder this diagnosti13 13ontains a lens whi13h is lo13ated
inside the bla13k tube on the right part of Fig 49 The lens allows imaging
the sho13k on a linear bundle of 21 bers 13onne13ted to the visible spe13trometer
through a feedthrough (se13tion 443)
A snapshot of the target holder is shown in the gure 49 The red arrows
in the Fig 49 shows the distribution of the various diagnosti13s
In addition we used two 3 mm thi13k stainless steel V-shape shields on
both sides of the target whi13h were xed to the holder These V-shields have
a hole (sim25 mm) shaped entran13e to allow laser in13iden13e on the target (in
green on the Figure 49) The role of the V-shields is to prevent the target
from the extra laser or another kind of hard X-ray plasma self-emission linked
to the ablation pro13ess
56 CHAPTER 4 EXPERIMENTAL SETUP
424 Target lling
The gaseous targets were lled in situ at a fra13tion of 1 bar with a gas (viz
Xe Ar a mixture of Xe and He) whi13h thus provide the medium for the
sho13k propagation We 13hose an in-situ lling system (Fig 410) to redu13e
the dieren13e of pressures inside and outside of the tube This is 13riti13al as
a large pressure dieren13e between the target 13ell and the va13uum 13hamber
13an result in the breaking of the ultra-thin Si3N4 windows of the target
To 13ontrol the pressure we used a piezo-resistive pressure transdu13er
(Swagelok PTI-S-AC6-32) whi13h has 10 kΩ bridge resistan13e and works
with 14-30 V biasing voltage A sensitive Bourdan manometer and a gauge
allowed monitoring the pressure inside the target 13ell To read the pressure
remotely (from the 13ontrol room) until the last se13ond before the shot we
13onne13ted the system with a long 13able
The transdu13er has been operated with a +17 V DC biasing voltage The
pressure is read using an industrial pressure transdu13er from Swagelok (0 to
10 Volts -10 bar to 06 bar 05 pre13ision) and reading is made with a
voltmeter at better than 05 This system generally took approximately
one minute to stabilize A s13hemati13 of the aforementioned gas lling system
is presented in the Figure 410
The pro13edure of the target lling is presented as follows
Figure 410 In-situ system for target lling and pressure monitoring
The target is pumped during the 13hamber eva13uation
The target is isolated at the end of the pumping
The target is lled to the desired pressure
43 LASER FOCUSING 57
The target is isolated from the gas bottle to avoid any leakage in
13hamber
Then the pressure 13an be read from the 13ontrol room till the last se13ond
before the shot
The gases are taken from high grade bottles 99995 purity The Xe+He
mixture is 1007plusmn050 He molar fra13tion The lling pro13edure is pre-
venting the presen13e of impurities like air When 13losing the 13hamber and
during the primary pumping the lling 13ir13uit 13ommuni13ates with the 13ham-
ber thus the whole tubing and the target are eva13uated down to 001 mbar
After the turbo pump is a13tivated the 13onne13tion to 13hamber is 13losed and
the gas in inje13ted When the target pressure is rea13hed the 13ell is isolated
from the pressure regulator In 13ase there is an una1313eptable overpressure
it is possible to re13onne13t to 13hamber to eva13uate and to restart the lling
The pressure reading is possible in the 13ontrol room still at pre13ision better
than 1 and the value kept for the re13ords is the one just before the shot
whi13h happens normally 1-2 minutes after 13losing
43 Laser Fo13using
431 Fo13al Lenses and Phase Zone Plates
Two fo13al lenses are used to fo13us the laser beams on the target piston foil for
MAIN and AUX respe13tively The MAIN lens is lo13ated inside the va13uum
13hamber whereas the AUX lens is outside this 13hamber The spe13i13ations
of these lenses are given in the Table 42
laser Diameter (mm) fo13al length (mm) f-number
MAIN 340 564 at 3ω 19
AUX 150 1022 at ω 69
Table 42 Spe13i13ations of the MAIN and AUX fo13al lenses and the f- number
(fo13al lengthbeam diameter) is given (see table 41 for the beam diameters)
Whereas these lenses provide a smooth intensity at the fo13us this is not
su13ient to laun13h a planar sho13k wave To this purpose phase zone plates
(PZP) have been used between the output laser beam and the lens
A PZP 13omprises several phase zone elements whi13h divide the input
beam into several beamlets whi13h 13ombined with the main fo13al lens provi-
ding a uniform intensity distribution on the target with a spe13ied diameter
(Ross Pepler and Danson 1995 Bett et al 1995) The PZP are 13ommonly
used in EOS laser experiments whi13h require a uniform intensity distribution
on the target (Koenig et al 1994 Batani et al 2003)
Two PZP as dis13ussed below were used in the experiment
58 CHAPTER 4 EXPERIMENTAL SETUP
The rst one made by COLSICOAT was already used at PALS for
radiative sho13k experiments (Stehleacute et al 2010) This PZP used for
MAIN (438 nm) is made on a 13oated BK7 substrate of a diameter
of 310 mm and se13tion of 8 mm It is designed to provide a uniform
13ir13ular fo13al spot over a diameter of 05 mm
The se13ond plate made by SCITECH has been used for AUX (1315
nm) The plate is made on a 63 mm thi13k 13oated Viosil substrate
with a square 153 x 153 mm shape It is designed to provide a uniform
13ir13ular fo13al spot over a diameter of 0250 mm
Laser Spe13ied spot diameter (mm)
MAIN 05
AUX 025
Table 43 Spe13i13ations of the two Phase Zone Plates
In terms of energy distribution sim 80 of the laser energy is in the laser
spot (C Spindloe private 13omm) The energy of the laser whi13h is given
by the PALS installation should be 13orre13ted by this fa13tor of 80 For
AUX another 13orre13tion of sim 10 has to be added whi13h is linked to the
transmission of the entran13e window of the va13uum 13hamber As a result
for nominal energies of the MAIN and AUX lasers equal to 120 and 60 J (see
table 41) the uen13es on the target are equal to sim 48800 and 59500 J13m
2
respe13tively Taking into a1313ount the 03 ns laser duration of the lasers this
results in an intensity of sim 1-2 times 10
14W13m
2
432 Fo13using of the MAIN and AUX laser beams
The determinations of the fo13i of the two lenses were performed with a
Hartmann setup using a photographi13 paper and without any PZP plate
The determination of the size of the beam at the impa13t is important both to
know the initial 13onditions for the sho13k waves (laser irradian13e whi13h should
be few 10
14W13m
minus2and se13tion of the sho13k wave) The sele13tion of the
best spots size on the piston was done with mo13k targets and by translating
the two laser lenses The in13iden13e of a laser beam on a solid planar target
generates a 13rater in the foil and an ablated plume propagating ba13kward
at high temperature (Mahmood and Ur-Rehman 2009 Singh and Thakur
2007) The 13rater is the result of boiling and ablation of the material The
ablated plasma whi13h expands towards the in13ident laser beam emits in the
keV range
A standard pro13edure whi13h works well at low energy (sim 10 J) 13onsists
of the estimation of the spot size by measuring the 13rater on the bulk target
using a standard mi13ros13ope At higher energy the boiling ee13t makes
44 DIAGNOSTICS 59
this estimation impre13ise and we preferred to use a keV imaging diagnosti13s
(Chaulagain et al 2012) whi13h is based on a pinhole (25 microm for MAIN keV
and 20 microm for AUX) and a keV 13amera These two keV 13ameras have been
used during ea13h laser shot to image the impa13ts of the MAIN and AUX
laser beams on the two respe13tive pistons of the target Ea13h keV 13amera
is a CMOS dete13tor without any glass It is prote13ted by an Aluminium
lter (200 microm for MAIN and 42 microm for AUX) to blo13k the visible light and
to prote13t the ship from the debris The spe13i13ations of the 13ameras are
presented in the Table 44 and the keV 13amera system is shown in the Fig
411 The two PALS laser lenses were translated up to a13hieve the suitable
diameters on the target whi13h were nally set to 450 - 500 microm and 250 -
300 microm for the MAIN and AUX beams
The size of the impa13t was 13ontrolled ea13h day on mo13k targets before
using the laser beams for real shots on gaseous targets It is worth to pre13ise
that the keV diagnosti13 was still in pla13e for the shots on the gaseous targets
In that 13ase the beams impa13t Parylene-N instead of Aluminium The keV
signal is then weaker than for the mo13k aluminum targets
Figure 411 S13hemati13 of the keV 13amera
KeV 13amera KeV 13amera
MAIN laser AUX laser
Model UI-122xLE UI-164xLE
Resolution 752(H) x 480(V) 1280(H) x 1024(V)
Pixel size (microm x microm) 60 x 60 36 x 36
A13tive Area (mm x mm) 451 (H) x 288 (V) 46 (H) x 37 (V)
Magni13ation 107 068
Pinhole Material Ta Pt
Table 44 Spe13i13ations of the two uEye keV 13ameras (1stVsion 13ompany)
44 Diagnosti13s
Various kinds of diagnosti13s were implemented namely visible interfero-
metry XUV and visible spe13tros13opy For the same two keV 13ameras are
indu13ted to monitor the impa13t and two visible 13ameras for target align-
60 CHAPTER 4 EXPERIMENTAL SETUP
(a)
(b)
(13)
Figure 412 (a) Sign 13onvention for the respe13tive positions of the lens and the
target (b) keV image of a MAIN impa13t on a massive target (lens position +1500
microm) The spot size is 475plusmn25 microm Pixel size for MAIN keV 13amera is 56 microm (13)
keV image of an AUX impa13t (lens position -1500 microm) The spot size is sim 280plusmn20microm on target One pixel = 66 microm on AUXs keV 13amera
44 DIAGNOSTICS 61
ment In this se13tion I will des13ribe the setups of the visible interferometry
XUV spe13tros13opy and visible spe13tros13opy The analysis of the re13ords of
visible interferometry and XUV spe13tros13opy will be dis13ussed in subsequent
13hapters
441 Visible Interferometry
A Ma13h-Zehnder Interferometer (MZI) has been employed to perform visible
interferometry The re13orded interferometri13 images have been used to study
the sho13k speed and the ele13tron density during the sho13ks propagation The
experimental setup of this interferometer is presented below
Ma13h Zehnder Interferometer
We pla13ed the target in whi13h sho13k is produ13ed in one of the two arms
of the interferometer The probe laser wavelength (527 nm) is supposed to
be far away of any atomi13 absorption resonan13es in the plasma and thus the
13ontributions to the refra13tive index 13ome from the plasma free ele13trons and
not from bound ele13trons (Harilal and Tilla13k 2004)
Figure 413 First interferometri13 setup All the opti13al elements are 1 in13h For
simpli13ity the plasma slab is not reported in the probe beam
The rst experimental setup shown in Fig 413 was used until 04052015
62 CHAPTER 4 EXPERIMENTAL SETUP
In this setup
the 300 ns long probe laser of wavelength 527nm (beam diameter asymp 1
in13h energy on the target few tens of mi13rojoules) rst passes through
an alignment pinhole PH1
the non-polarizing beam splitter BS1 transmits 50 of the intensity
of the in13ident beam and the remaining 50 of intensity in the per-
pendi13ular dire13tion At this stage the two arms of the interferometer
namely the referen13e and probe beams are produ13ed
two half wave plates HWP1 and HWP2 are pla13ed respe13tively in
the path of the probe and referen13e beams They allow to play with
the polarization in order to optimize the fringes 13ontrast as it will be
explained below
the verti13ally polarized probe beam passes through the target (plasma
slab) Due to its polarization it is fully ree13ted by the polarizing 13ube
beam splitter PBS1 towards the polarizing 13ube beam splitter PBS2
the referen13e beam is ree13ted by the mirror M4 on to this 13ube splitter
PBS2
At the output of PBS2 the two beams overlap but they have ortho-
gonal polarizations and 13ant interfere They then travel through a
polarizing beam splitter PBS3 those axis makes an angle of 45
with
the 13ommon dire13tion of in13iden13e of the two beams This rotates
the polarizations of the referen13e (horizontally polarized) and probe
(verti13ally polarized) beams along the same axis
At this stage the beams share the same polarization and interfere
A last non-polarizing beamsplitter plate (90-10 ) BS2 is pla13ed after
PBS3 to enable the propagation of the two beams on the same axis as
the laser beam at the entran13e of the interferometer
The fringes are re13orded on a HAMAMATSU C7700 VIS Streak Camera
lo13ated after the last alignment pinhole PH4 To this purpose an imaging
setup makes the image of the target longitudinal axis (ie along the dire13tion
of the sho13k propagation) on the streak slit This is done with an a13hromati13
spheri13al doublet of fo13al length equal to 1000 mm The magni13ation is
measured using an AGAR grid to 226 whi13h means that 106 microm on target
are re13orded by one pixel (size of the pixel 24 microm) on the streak on both
dire13tions (Fig 413)
44 DIAGNOSTICS 63
Figure 414 Se13ond interferometri13 setup
Role of the polarizing devi13es
Let us suppose that the polarization state of the in13oming beam is unknown
The polarization of the probe beam is then xed to be verti13al by the pola-
rizer HWP1 Ree13ted by the polarizing 13ube PBS2 it remains verti13al at
the entran13e of PBS3 On this side the polarization of the referen13e beam
is set to be horizontal after the 13ube BS1 using the polariser HWP2 and is
transmitted as horizontal by PBS2 The rotation of PBS3 allows to proje13t
part of the two beams on a 13ommon polarization axis and to interfere
This setup presents the following interest the intensity of the laser beam
in the two arms of the interferometer 13an be modied (and thus the 13ontrast
of the fringes) 13an be optimized using the two polarizing plates HPW1 and
HPW2 However as the multimode laser is not fully 13oherent the dark
fringe (zero net intensity) has never been obtained
Moreover the setup is designed to also allow alignment and adjustment
of the imaging setup with a green HeNe laser whi13h shares the same path
as the probe beam up to PBS1
A more optimized setup was used after 05052015 (ie shot number
48033) where the mirror M1 in the referen13e beam was repla13ed by a
polarizing beam splitter PBS4 ree13ting the probe beam like the mirror M1
(Fig 414) This was done in order to redu13e the dieren13e between the
64 CHAPTER 4 EXPERIMENTAL SETUP
two paths l1 and l2 and then to optimize the fringe 13ontrast This was more
suitable due to the short 13oheren13e length (1- 2 13m) of the probe laser
Streak 13amera
Figure 415 S13hemati13 of the Streak 13amera (from Hamamatsu noti13e)
A streak 13amera is a setup whi13h allows measuring the temporal varia-
tion of a fast signal whi13h is re13orded on its slit As indi13ated in the noti13e)
of the Hamamatsu C770 13amera the light pulse rst passes through the slit
then it is fo13used on the photo13athode (size 7times17 mm) of the streak by an
input opti13s The visible photons are 13onverted into a number of ele13trons
whi13h is proportional to the intensity of the in13ident light These ele13trons
are a1313elerated and 13ondu13ted towards the phosphor s13reen when a high-
speed voltage whi13h is syn13hronized to the in13ident light is applied As a
result the ele13trons are swept at high speed from the top to the bottom
after whi13h they are bombarded against the phosphor s13reen of the streak
tube and 13onverted through an output opti13s to an opti13al image on a dual
mode 13ooled CCD 13amera with 512 times 512 pixels (Fig 415) In the PALS
experiment the sweeping times were kept either 50 or 200 ns The CCD
pixel size is 24 x 24 microm
The streak 13amera 13an be operated with time swap mode with the slit
of the 13amera almost 13losed The re13ord 13onsisted then in images (position
along the slit versus time) as dis13ussed above or in a stati13 mode (no time
swap) whatever the slit aperture In that 13ase the re13ord is a usual 2D
spatial image of the obje13t
During the experiment we have taken referen13e images (without any
fringe) for every target as shown in the Fig 417a for one target example
In order to re13ord su13h images we blo13ked the referen13e beam and let the
probe beam only to illuminate the target in a stati13 mode and with the slit
open Then we redu13ed the slit width to its nominal value (200 microm) This
enabled us to have the visualization of the portion of the target whi13h was
44 DIAGNOSTICS 65
Figure 416 Imaging setup of the Ma13h Zehnder Interferometer setup operating
in the longitudinal mode The 13hannel of the target is fully illuminated by the
probe beam In this mode the lens images the axis of symmetry of the two sho13k
waves along the slit of the streak 13amera
imaged in the slit on the target and also to lo13ate the position of the pistons
before the shot This image is termed as the referen13e target image Then
the system was pla13ed in the interferometri13 mode and the fringe pattern
was optimized to be perpendi13ular to the slit and thus in the longitudinal
mode perpendi13ular to the dire13tion of the sho13k propagation
The Fig 417b reports su13h a typi13al interferogram re13orded during our
experiment (13f Fig 413) The horizontal axis 13orresponds to the dire13tion
of the sho13k propagation (with a total s13ale of 54 mm on the target) and
the verti13al axis 13orresponds to time (s13aling 200 ns)
442 XUV Spe13tros13opy
The time and spa13e integrated XUV plasma emission are re13orded with a
Flat Field XUV spe13trometer (grazing in13iden13e XUV spe13trometer) using
a 13on13ave grating Typi13al ray tra13ing a grazing in13iden13e XUV spe13tro-
meter is shown in Fig 418 In pra13ti13al the entran13e slit was removed
as the plasma extension was small In the Fig 419 the drawing of the
XUV spe13trometer setup s13heme (red 13olor) is presented together with the
spheri13al 13hamber and the target The XUV spe13trometer is installed on
the top of the spheri13al 13hamber whi13h allows the XUV emission passing
through the Si3N4 membrane on the top of the target (see se13tion 422) to
be re13orded
A 13ooled Andor DX440 CCD (without glass prote13tion) is atta13hed to
the spe13trometer to re13ord the spe13trum of the XUV radiation between 12
66 CHAPTER 4 EXPERIMENTAL SETUP
(a)
(b)
Figure 417 (a) Referen13e 2D image of a target before the shot re13orded on the
Streak 13amera The positions of the two pistons 13losing the 4 mm long target are
lo13ated at 800 and 4800 microm The dark zones along them (between 800 to 1000 and
4600-4800 microm) 13orresponds to the glue This glue is then visible through verti13al
bla13k strips in the interferometri13 re13ord (b) Corresponding interferometri13 time-
spa13e re13ord
44 DIAGNOSTICS 67
Figure 418 Typi13al ray tra13ing in a grazing in13iden13e XUV spe13trograph The
dete13tor (GMCP or image plate) is installed tangentially to the Rowland 13ir13le
and 40 nm (31- 100 eV) Its 13hara13teristi13s are given in the Table 45 Two Al
lters of thi13kness 08 and 16 microm prote13t the grating and the CCD 13amera
from debris and visible light The XUV images re13orded on the CCD are
time (300 micros) and spa13e (sim 3 mm) integrated
Andor 13amera (DX440)
Pixel Size 135 x 135 micromArea 276 (H) x 69 (V) mm
A13tive pixels 2048 x 512
Table 45 Spe13i13ations of Andor CCD 13amera
The 13urved grating (Table 46) used to fo13us the XUV emission on to
the CCD 13hip has a variable growes spa13ing as shown in the bottom panel
of the Fig 419 A typi13al output re13ord is also presented for illustration
Grating
Type Dira13tion
Growes per mm 1200
Dimensions 30 x 50 x 10 mmRadius of 13urvature 5649 mm
Blazing angle 37 degree
Blazing wavelength 16 nm
Table 46 Spe13i13ations of the XUV grating
443 Visible spe13tros13opy
In addition to the interferometry and XUV spe13tros13opy we have also im-
plemented a time- and spa13e-resolved visible spe13tros13opi13 plasma emission
diagnosti13
68 CHAPTER 4 EXPERIMENTAL SETUP
Figure 419 (a) Setup of the XUV spe13trometer in the spheri13al 13hamber (b)
Typi13al re13ord between sim 10 and 40 nm is shown in the bottom panel
The 13ore of the diagnosti13s is a Czerny-Turner 300 mm Spe13trometer
2
lo13ated outside the va13uum 13hamber The light of the plasma is inje13ted on
the slit of this spe13trometer through a bers bundle This bundle 13onsists of
three sets of an aligned seven individual bers looking at dierent se13tions
of the target 13hannel through an imaging lens lo13ated in the bla13k tube of
Fig 49 The time resolution is performed through an iCCD 13amera lo13ated
at the exit port of the spe13trometer (Fig 420a)
A typi13al re13ord is reported in Fig 420b The results of this diagnosti13s
show several signatures whi13h dier along the tube Most of them are in
absorption and 13ould be linked to the heating of the target sili13a window
Their interpretation is not yet a13hieved and will not be presented in this
study
45 Summary
In this 13hapter I have presented the details of our experimental setup and
diagnosti13s The next 13hapter will be dedi13ated to the presentation of our
re13ords results and their analysis
2
see http wwwandorcomspectrographshamrock minus spectrographminus series
45 SUMMARY 69
(a)
(b)
Figure 420 (a) S13hemati13 of setup employed to re13ord the time and spa13e
visible plasma emission (b) A typi13al spe13trogram
70 CHAPTER 4 EXPERIMENTAL SETUP
Chapter 5
Results and interpretation
Contents
51 Visible interferometry 71
511 Longitudinal interferometry sho13k speed and
ele13tron density 74
512 Transverse interferometry lateral extension of
the sho13k 91
52 XUV Spe13tros13opy 92
53 Simulations based on experimental results 93
54 Summary 97
In the previous 13hapter I presented a brief des13ription of our experimen-
tal setup This setup in13ludes two main diagnosti13s namely visible interfe-
rometry and XUV spe13tros13opy The visible interferometry is an adequate
tool to probe the ele13tron density of the radiative pre13ursor up to the 13riti-
13al density (4 times 10
2113m
minus3at the wavelength 527 nm of the probing laser)
whereas the denser post-sho13k region is opaque to the visible light XUV
radiation is present in the whole sho13k stru13ture and thus the time- and
spa13e-integrated XUV spe13tros13opy may allow exploring the self-emission
13oming from the post-sho13k and pre13ursor regions
In this 13hapter
1
I present the results of the both diagnosti13s with a
parti13ular fo13us on the interferometri13 data whi13h I have extensively studied
with the help of a spe13i13 data analysis pro13edure that I have developed I
will then present the main results of the XUV spe13tros13opi13 diagnosti13s The
results will then be interpreted with the help of numeri13al simulations
51 Visible interferometry
In the visible interferometry the interferen13es between the main and refe-
ren13e beams overlapping on the dete13tor produ13e a pattern of fringes whi13h
follow the relative phase variations between the probe (whi13h passes through
the plasma) and the referen13e beams (see se13tion 441 of the previous 13hapter
for the experimental setup) The phase variation between two 13onse13utive
fringes of the unperturbed beams (ie without any plasma) is equal to 2π
1
Most of results presented in this 13hapter has been published in Singh et al (2017)
71
72 CHAPTER 5 RESULTS AND INTERPRETATION
Then in the presen13e of the plasma in the probe beam these fringes be-
13ome shifted due to the modi13ation of the refra13tive index More details
about the prin13iple of the interferometry and the Ma13h-Zehnder interfero-
metri13 setup may be found in the APPENDIX A and we re13all here only
the expression of the phase shift
∆φ asymp minus πd
λNclt Ne gt (51)
where λ = 527 nm is the wavelength of the probing laser Nc = 4 times 10
21
13m
minus3the 13riti13al density at this wavelength and lt Ne gt is the ele13tron
density averaged over the laser path d in the 13ell and is dened as
lt Ne gt=
int d
0
Ne(z t)dy
d(52)
In order to dedu13e the ele13tron density it is ne13essary to follow the
evolution of the positions of the maxima (respe13tively minima) of the fringes
versus time To this purpose I have developed a spe13i13 data treatment
pipeline in the intera13tive data language (IDL)
2
To improve the visibility of
the fringes obtained in the interferograms I pro13essed the images with the
Fast Fourier Transform (FFT) (Proakis 2001) This treatment transforms
the image from the spatial domain (ie our re13ords) into the frequen13y
domain It then allows to dene spe13i13 bands of frequen13y whi13h represent
noise for the image and to 13lean them The 13omplete pro13edure is des13ribed
below
First the FFT of the (512 times 512) interferometri13 image is derived In
the next step the low and high-frequen13y lterings within the FFT of the
image are done using low-pass Fminus and high-pass F+ Butterworth frequen13y
lters (Proakis 2001) as follows
Fminus(u) = 1[1 + C(uu0)2n] with C = 1 n = 1
F+(u) = 1[1 + C(u0u)2n] with C = 1 n = 1
(53)
where u represents the position in the FFT image and u0 is the nominal
lter 13uto frequen13y (represented as the width of the region in pixels) For
the low-pass Butterworth frequen13y lter u0 is 13onsidered to be 55 whereas
for high-pass Butterworth frequen13y lter it is taken to be 45 The values of
u0 are dened manually by a trial method with the aim to result in better
noise redu13tion
After this step the inverse FFT of the ltered FFT image is performed
whi13h results in a 13lean image in the spatial domain In Fig 51 the original
re13ord for the shot number 48055 (left) the FFT of this image (13enter)
and the nal frequen13y ltered image (right) are presented showing how the
2
httpwwwast13ama13uk~vasilyidlidl_introdu13tionpdf
51 VISIBLE INTERFEROMETRY 73
frequen13y ltering improves the fringes 13ontrast and thus fa13ilitates their
analysis
Figure 51 Original re13ord of shot 48055 (left) FFT of the original image
(13enter) and the frequen13y ltered image (right)
Next the 13ru13ial task is to identify and to follow the fringes with the
help of a pre13ise determination of the intensity (pixel 13ounts) maxima In
order to lo13ate these maxima I have adopted the following steps
Figure 52 A 13ropped se13tion of the re13ord from shot the 48055 The rst ve
positions have been sele13ted manually `+ signs (in red 13olor) on ea13h fringe On
this re13ord the distan13e between two unperturbed fringes is 13orrespond to 15 pixels
(ie 159 microm)
Firstly I have manually 13hosen a se13tion of a typi13al re13ord with the
aim to sele13t only the area asso13iated with the sho13k dynami13s
74 CHAPTER 5 RESULTS AND INTERPRETATION
Next ve representative points are sele13ted visually (`+ marks in red
13olor) on ea13h fringe 13overing the important lo13ations on the fringe
(13f Fig 52) They serve as the input to the spline interpolation for
estimating all intermediate pixels lo13ations along the fringe (see Fig
53a)
In order to obtain the lo13ations representing the fringe maxima the
spline interpolated lo13ation points are further rened by sele13ting the
pixel of maximum intensity within plusmn5 pixels of the respe13tive spline
lo13ations in the X-dire13tion (see Fig 53b)
The position of fringe maxima for the 13ropped image obtained in an
aforesaid manner is shifted so as to represent the same points however
on the full-image (see Fig 54)
The re13ords obtained from the experiments have been pro13essed in this
way to estimate the sho13k se13tion speed ele13tron temperature and density
511 Longitudinal interferometry sho13k speed and ele13tron
density
In the longitudinal interferometry we perform on the slit of the streak 13a-
mera the image of the axis of symmetry of the sho13k propagation along the
tube to analyse the sho13k propagation in this dire13tion
The interferometri13 images have been pro13essed as explained previously
to enhan13e the fringes 13ontrast The lo13ations of the maximum intensity in
ea13h fringe are then used to derive the sho13k speed and the average ele13tron
density as will be presented below
Sho13k speed dedu13ed from the last fringe method
An estimation of the sho13k speed 13an be obtained through the interferometri13
image In this se13tion I will present the method to derive sho13k speed
based on the last visible end points of the fringes also 13alled last fringe
method This determination provides rapidly a qualitative estimation of the
sho13k speed whi13h is based on the absorption behavior of the plasma (see
Equation A11 in appendix A) and not on the real position of the front
dis13ontinuity The front dis13ontinuity is not seen in the re13ords due to the
strong absorption
At the positions of the last visible end points of the fringes the ele13tron
density rea13hes the maximum value a1313essible to the diagnosti13 The frin-
ges are strongly bent and the absorption of the visible light be13omes also
important Therefore it is believed that the sho13k front is 13lose from this
lo13ation
The lo13ations of the fringe maxima have been already dedu13ed following
the analysis presented in previous se13tions In this regard the best visible end
51 VISIBLE INTERFEROMETRY 75
(a)
(b)
Figure 53 (a) Representative points of the fringes as derived from the spline
tting of the 5 manually sele13ted points (Fig 52 (II)) on ea13h fringe (b) Positions
of the fringes maxima along Y-axis for ea13h fringe derived by lo13ating the points
of maximum intensity in X-dire13tion of the previous points obtained by spline t
76 CHAPTER 5 RESULTS AND INTERPRETATION
Figure 54 Fringe maxima on the full image
Figure 55 Imaging setup of the Ma13h Zehnder Interferometer operating in the
longitudinal mode The lens images the axis of symmetry of the two sho13k waves
along the slit of the streak 13amera
51 VISIBLE INTERFEROMETRY 77
point of ea13h respe13tive fringe is 13onsidered for the average speed estimation
for the sho13k
At very initial time the in13rease in sho13k speed is non-linear This non-
linear trend may result in non-physi13al speed estimations Therefore I dis-
13arded the end points whi13h appear to be within the ve nanose13onds after
the sho13k laun13hing time (eg t = 145 ns on Fig 56)
However the above pro13edure may result in un13ertainties as the sele13ted
end points may have dierent intensity (13ounts) values In this regard I
have evaluated the un13ertainty in the determination of the average speeds
To this purpose I rst sele13ted the best visible end-point of a random fringe
Su13h an end point gives a referen13e intensity whi13h was then used to obtain
on the rest of fringes the end points having 13lose intensity This pro13edure
was performed thri13e in order to dedu13e three sets of su13h end-points (see
Fig 56) Ea13h of these three sets of points was then used to estimate the
speed through a linear-t method (lines in white red and green) It may be
noted that these sets of points and tted lines are very 13lose to ea13h other
Therefore it is tough to distinguish them on the Fig 56 These three speed
determinations enable us to derive an average value with an un13ertainty
whi13h is equal to the standard deviation of these three values
Figure 56 Interferometri13 image re13orded for the shot 48055 in Xe at 01 bar
The sho13k speeds for the sho13ks driven by MAIN (from left side) land AUX (from
right side) lasers are respe13tively equal to sim 54 and 23 kms The time of laser
arrival on the piston is at 146 ns The positions of the Au-Xe interfa13e on the
re13ord are respe13tively 950plusmn50 and 4950plusmn50 mi13rons
This estimation of the sho13k average speed is satisfa13tory for the re13ords
in whi13h the sho13k speed is almost 13onstant throughout its propagation Ho-
wever the variation with the time of the speed may be substantial for some
re13ords In su13h 13ases the un13ertainty in the average speed will be in prin-
13iple higher than the previous estimations Therefore I introdu13ed another
78 CHAPTER 5 RESULTS AND INTERPRETATION
method to determine the average velo13ity this velo13ity is then estimated
during the initial and in nal durations of the sho13k propagation by deriving
the slope of points on the rst middle and last few fringes respe13tively
The standard deviation estimated by this method is then termed as the
un13ertainty in the average speed of respe13tive re13ord
Using these two methods I have obtained two values of un13ertainties for
all the MAIN and AUX sho13ks observed in the experimental re13ords Finally
the largest un13ertainty value of the two is 13onsidered as the nal un13ertainty
in the estimation of the respe13tive speed and is shown by error bars in Fig
57a and 57b 13orresponding to the MAIN and AUX sho13ks respe13tively
The table 51 reports for dierent noble gases the values of the velo13ities
re13orded in several shots for the 13ounter-propagating sho13k waves In addi-
tion one re13ord (48131) 13orresponds to the 13ase of an isolated sho13k wave
laun13hed by the MAIN laser The values of the MAIN and AUX laser ener-
gies are also reported In some 13ases the glue on the window prevented the
probe laser to pass through the 13hannel Then it be13ame impossible to derive
any value for the velo13ity In su13h 13ases I put `NA for the 13orresponding
sho13k velo13ity
It may be noted that the sho13k speeds generated by the MAIN laser vary
in the range of 30-55 kms while for the sho13k originated from AUX laser
the speeds vary between 10 and 30 kms
Variations with laser energy and pressure the 13ase of Xenon
When the number of re13ords is su13ient to make a statisti13al analysis we
may analyse the variations of the sho13k speeds with the laser energy of
the MAIN and AUX lasers This is the 13ase for Xe or XeHe (90-10)
mixture and for gas pressures ranging between 01 and 02 bar At 01 bar
all the measurements 13on13ern Xenon (ρ= 54 times 10
minus4g13m
minus3) At 02 bar
the re13ords 13on13ern a mixture XeHe (90-10 ρ=10 times 10
minus3g13m
minus3)
ex13ept for one point at 121 J whi13h 13orrespond to the 13ase of pure Xe (ρ =
108 times 10
minus3g13m
minus3) We do believe that taking into a1313ount the pre13ision
of our re13ord the introdu13tion of tra13es of Helium do not ae13t mu13h the
dynami13s of the sho13k wave as the 13orresponding variation of mass density
is negligible
As expe13ted the sho13k speed in13reases with the laser energy and de13rea-
ses with the pressure (Fig 57a) A linear t gives the following dependen13e
of the speed in kms with the MAIN laser energy (in J) at 01 and 02 bar
respe13tively
3
v01bar = 1423 + 030E (54)
3
Two outlier points have been dis13arded from the set 13orresponding to 01 bar They
are shown for information on the Figure
51 VISIBLE INTERFEROMETRY 79
(a)
(b)
Figure 57 (a) MAIN sho13k speed (13al13ulated by the last fringe method) versus
the MAIN energy for Xenon or XeHe mixture at dierent pressures with the error
bars (b) AUX sho13k speed versus AUX energy also for Xenon or XeHe mixture
at dierent pressures
80 CHAPTER 5 RESULTS AND INTERPRETATION
v02bar = minus1405 + 047E (55)
The speeds generated by the AUX laser are lower than the previous ones
Although the re13ords are more sparse we note that the speeds at 01 bar
also in13rease with the laser energy However at 02 bar I 13ould not nd
similar trend due to the la13k of re13ords
Variations with the gas
As 13an be seen from the Table 51 we have also performed several shots in
Ar at 01 02 03 and 08 bar one shot in He at 05 and one in Kr at 02
bar
To 13ompare the velo13ities obtained for the dierent gases one 13an either
keep the pressure 13onstant (ie the number of atoms) or the mass density
For a given mass density and laser energy the velo13ity should be the same
in the adiabati13 limit However due to the radiation 13ooling and ionisation
this 13an not be the 13ase Nonetheless if we 13ompare the two re13ords of Ar
at 03 bar (ρ = 493 times 10
minus4g13m
minus3 E = 121 J) with Xe at 01 bar (54
times 10
minus4g13m
minus3 E = 133 J) we measure similar speeds (49 and 54 kms)
These two re13ords are thus interesting to 13ompare This will be done later
on in this 13hapter
More generally we note that the speed of Helium at 05 bar is higher by
sim 40 than for Ar at the same energy (see Fig 58) This is not surprising
for this lighter element (Table 52) However it is important to note that
the la13k of shots prevents us to make any pre13ise 13on13lusion
Ex13ept for this 13ase and for the unique re13ord of Ar at 02 bar whi13h
gives also a higher velo13ity the velo13ities of the other re13ords follow more or
less the variation with the energy than Xenon
To 13ompare the radiative ee13ts it seems justied to perform the 13om-
parison at a given sho13k speed independent of the laser energy and for mass
densities whi13h are 13lose to ea13h other In the following we shall thus 13om-
pare the results of Xe at 01 bar (54 kms 54times 10
minus4g13m
minus3) Ar at 03
bar (49 kms 49times 10
minus4g13m
minus3) and Kr at 02 bar (55 kms 68 times 10
minus4
g13m
minus3)
Line averaged ele13tron density lt Ne gt
I have estimated the ele13tron density lt Ne gt of the plasma averaged along
the path of the probe laser beam by analyzing the interferogram re13ords
assuming the same se13tion d of the plasma layer of 600 microm for both MAIN
and AUX side sho13ks
51 VISIBLE INTERFEROMETRY 81
Shot Gas Pressure E3ω MAIN sho13k Eω AUX sho13k
(bar) (J) speed (kms) (J) speed (kms)
at 298 K
48033 Air 03 124 54plusmn1 66 20plusmn248034 Air 03 131 52plusmn1 74 27plusmn248076 Ar 05 100 41plusmn1 66 18plusmn148077 Ar 01 115 42plusmn1 67 18plusmn148078 Ar 03 112 46plusmn3 65 25plusmn248079 Ar 03 121 49plusmn5 67 23plusmn348080 Ar 08 103 38plusmn2 62 21plusmn248081 Ar 08 113 38plusmn1 68 NA
48082 Ar 08 107 36plusmn1 65 NA
48141 Ar 02 111 63plusmn1 57 NA
48083 He 05 106 57plusmn3 63 NA
48146 Kr 02 125 55plusmn2 53 NA
48043 Xe 03 138 57plusmn1 72 25plusmn148051 Xe 01 123 58plusmn1 67 30plusmn148055 Xe 01 133 54plusmn3 68 22plusmn348057 Xe 01 127 53plusmn1 68 23plusmn148058 Xe 01 115 48plusmn3 63 18plusmn148059 Xe 01 116 50plusmn1 67 21plusmn148061 Xe 01 127 53plusmn1 67 17plusmn248065 Xe 01 122 52plusmn4 68 23plusmn348066 Xe 01 114 50plusmn3 67 15plusmn248067 Xe 05 115 39plusmn2 65 NA
48068 Xe 05 109 36plusmn2 65 12plusmn248070 Xe 05 109 33plusmn3 65 NA
48138 Xe 02 121 45plusmn5 0 0
48131 Xe +He 02 112 38plusmn1 0 0
48132 Xe +He 02 118 41plusmn4 56 18plusmn248133 Xe +He 02 112 41plusmn3 56 NA
48134 Xe +He 02 111 38plusmn1 60 NA
48136 Xe +He 02 115 39plusmn3 59 14plusmn148143 Xe +He 06 123 39plusmn4 63 18plusmn548144 Xe+He 02 133 45plusmn3 66 NA
Table 51 Sho13k speeds estimated from the `last fringe method `NA represents
the entries whi13h 13ould not be dedu13ed from the re13ord Further entries in the
bold font are dis13ussed in detail in this 13hapter
82 CHAPTER 5 RESULTS AND INTERPRETATION
Helium Argon Krypton Xenon
Atomi13 Number 2 18 36 54
Atomi13 Mass 4 3995 8380 13129
First Ionization Energy (eV) 246 157 14 12
Density (10
minus4g13m
minus3) at 01 bar 016 164 344 539
Table 52 Atomi13 data and density at 01 bar (at room temperature) for He Ar
Kr and Xe
Figure 58 MAIN sho13k speed (13al13ulated by the last fringe method) versus the
MAIN laser energy for Xe (at 01 bar only) Ar He and Kr at dierent pressures
with the error bars
51 VISIBLE INTERFEROMETRY 83
It should be noted that the beam se13tions dMAIN and dAUX of MAIN
and AUX lasers are approximately 600 and 300 microm respe13tively on the two
pistons and that the transverse horizontal se13tion of the sho13k tube is equal
to dtube = 600 microm Thus the value of lt Ne gt 13omputed for the sho13k wave
laun13hed by MAIN should be 13lose to the physi13al lo13al value Ne supposing
that the plasma is homogeneous along the transverse se13tion However
the value obtained for AUX is 13ertainly larger by a fa13tor whi13h 13an be
estimated at a rst step as the fra13tion of dtubedAUX giving a fa13tor of
about 2
As explained in the previous se13tion the pixels representing the positions
of the fringe maxima have been already determined with the best possible
pre13ision Therefore it is possible to 13al13ulate the relative phase shift (with
respe13t to zero time) variation along y-axis (time axis) at ea13h fringe maxima
Zero time is time of MAIN and AUX laser rival on the target With these
phase shifts estimated at ea13h pixel of ea13h fringe I then derived lt Ne gtusing Equation A22
Let us take the 13ase of three shots in Xenon for illustrating the method
whi13h will be followed (see the left panel of the Figure 59) The maximum
density is estimated to be sim11 times 1019cmminus3(13orresponding to phase shift sim
16) The number of subdivisions is taken to 5 bins with the following phase
(∆φ) ranges (in the units of 2π) and average ele13tron densities (ltNegt)
bin 1 ∆φ le 06 lt Ne gtle 39 10
1813m
minus3(white)
bin 2 06 - 08 39 - 57 10
1813m
minus3(red)
bin 3 08 - 11 57 - 75 10
1813m
minus3(blue)
bin 4 11 - 13 75 - 93 10
1813m
minus3(green)
bin 5 gt 13 gt 93 10
1813m
minus3(magenta)
All the re13ords shown in the Table 51 have been pro13essed using this
method The limit of dete13tion for the phase shift 13orresponds to 2 pixels
giving a threshold for the Ne measurement lt Nemin gt This threshold
diers from one re13ord to another due to the variation in the number of
fringes (and thus the distan13e between two unperturbed fringes)
In order to make a 13omparative investigation of the sho13ks propagation
and intera13tion in dierent gases as already mentioned previously I sele13ted
three 13ases (shot48055 shot48132 and shot48138) relative to Xe (or Xe-
He mixture) one 13ase relative to Ar (shot48079) and Kr (shot48146)
The 13orresponding experimental 13onditions are noted in bold in the Table
51
84 CHAPTER 5 RESULTS AND INTERPRETATION
lt Ne gt in Xenon
The Fig 59 reports the interferometri13 re13ords for the three Xenon 13ases
The top and the middle panels 13orrespond to the propagation of two 13ounter-
streaming radiative sho13k waves at 01 (shot48055) and 02 bar (48132)
For 13omparison one re13ord (48138 bottom panel) is dedi13ated to the pro-
pagation of single sho13k (MAIN) at 02 bar The Xe-He mixture (90 - 10
in numbers of atoms) is used for the investigation in the 13ase of 02 bar
pressure while it is only Xe in the 13ase of 01 bar As mentioned previously
at the pre13ision of our re13ords we 13onsider that this impurity 13on13entration
has a negligible ee13t on the sho13k speed and the pre13ursor ele13tron density
The limit of dete13tion of lt Ne gt over the se13tion of the tube (06 mm) is
13orresponds respe13tively to 9 times 10
17 6 times 10
17and 6 times 10
1713m
minus3for the
Figures 59(a) (b) and (13)
The variations of lt Ne gt with the distan13e along the sho13k tube (ie
parallel to the dire13tion of the sho13ks) are reported in the right panel of Fig
59 at 10 ns (in red) 20 ns (in blue) 30 ns (in green) and 40 ns (in magenta)
The intera13tion between the two pre13ursors is 13learly visible at 01 bar
(Fig 59(a)) at 10 ns the intera13tion of the 13ounter-propagating sho13ks
has either not yet started or is below the sensitivity of this diagnosti13 The
intera13tion o1313urs at later times with a typi13al signature whi13h is as follows
the slope of lt Ne gt is de13reasing from the left (MAIN pre13ursor) passes
through a minimum and in13reases at the right (AUX) The minimum itself
in13reases with time up to 7 times 10
1813m
minus3at 40 ns
At 02 bar we have not been able to re13ord the 13ollision in the temporal
range (50 ns) of the streak However the two re13ords (with MAIN only and
with the two sho13k waves) indi13ate a pre13ursor for MAIN The two gures
(Fig 59(b)) and (Fig 59(13)) show that the pre13ursor of the MAIN sho13k
wave is not inuen13ed by the presen13e of AUX sho13k wave up to 45 ns
There is no obvious indi13ation about a pre13ursor for AUX in the 13ase of
two 13ounter-propagating sho13k waves (Fig 59(b)) At this pressure and
13ompared with the previous 13ase at 01 bar the absen13e of pre13ursor for
AUX may be attributed to i) a low sho13k speed (18 kms) 13ombined with a
larger pressure (hen13e for a given gas the pre13ursor extension in13reases with
the speed and de13reases with pressure) ii) a too small longitudinal extension
of the eventual pre13ursor (see Fig 59(b)) 13ompared with the resolution of
20 mi13rons (2 pixels) Our 1D numeri13al simulations with Xenon opa13ity
multiplier times 20 (not presented here) indi13ate a small pre13ursor for AUX
sho13k At 15 ns its extension is 50 microm (900 microm for MAIN sho13k) with
a typi13al ele13tron density sim 35 times 10
1913m
minus3(23 times 10
1913m
minus3for MAIN
sho13k) whi13h does not agree with the re13ord At 42 ns the pre13ursor of MAIN
rea13hes the AUX sho13k front and the prole is similar to the prole at 20 ns
shown in Fig 313a at 01 bar with a plateau of almost 13onstant ele13tron
density between the two fronts This might be 13ompatible with small bending
51 VISIBLE INTERFEROMETRY 85
Figure 59 Left panel interferometri13 re13ords 48055 in Xe at 01 bar (a)
48132 in Xe+He at 02 bar (b) and 48138 in Xe+He at 02 bar (13) Right
panel ele13tron density at 10 20 30 and 40 ns versus distan13e for these re13ords
The positions of maxima have been identied on the re13ords in the left panel
The time t = 0 13orresponds to the time of laser arrival on the target and the
position x = 0 13orresponds to the interfa13e between the piston (Au layer) and the
gas Its determination is pre13ise within 100 mi13rons The distan13es between two
unperturbed fringes for re13ords 48055 48132 and 48138 are 159 244 and 244
microm respe13tively The lt Ne gt un13ertainty (plusmn 2 pixels) is indi13ated by the error bar
in the right panels It 13orresponds respe13tively to plusmn 9 times 10
17 plusmn 6 times 10
17and plusmn
6 times 10
1713m
minus3for the gures (a) (b) and (13) The limit of dete13tion (2 pixels) is
presented by a dotted line on ea13h gure
86 CHAPTER 5 RESULTS AND INTERPRETATION
of the 4
thfringe (from the right) between 45 and 50 ns As 1D simulations
are known to overestimate the pre13ursor ele13tron density 2D simulations are
ne13essary for a more pre13ise interpretation of the experimental result
lt Ne gt in Argon and in Krypton
As seen in the Fig 58 the variation with the laser energy of the sho13k speed
in Argon at 03 bar (in the red dashed 13ir13le) is in good agreement with the
13orresponding one of Xenon at 01 bar (blue linet) As these two 13ases have
a 13omparable mass density of sim 5 times 10
minus4g13m
3 this qualitative agreement
is expe13ted
Taking into a1313ount the larger pressure (and thus density) the slope of
the variation of the slower sho13k speeds in Ar at 08 bar (in a red dash-
dot 13ir13le) with the laser intensity is 13ompatible with the previous trends
However any 13on13lusion 13an not be derived for Argon at 02 and 05 bar
owing to the s13ar13ity of the re13ords
In Fig 510 the interferometri13 re13ord for Ar at 03 bar (shot 48079
49 times 10
minus4g13m
minus3) is presented The estimated speeds of MAIN and AUX
sho13ks dedu13ed from the last fringe method are respe13tively equal to 49 and
23 kms whi13h are 13lose to the speeds of 54 and 23 kms re13orded in Xe
at 01 bar (shot 48055 539 times 10
minus4g13m
minus3 Fig 59(a)) Contrarily to
the Xenon 13ase we do not see any signi13ant bending in the fringes The
maximum fringe shift is reported in red on the fringe 1 The 13orresponding
ele13tron density is equal to 38 times 10
1813m
minus3
Thus obviously the radiation ee13ts are less important for Argon than
for Xenon at the same density and speed Part of this dieren13e 13ould be
attributed to the huge dieren13e in the atomi13 numbers (40 and 131) of the
two gases whi13h leads to dierent peak temperatures for the ions as expe13ted
from equation 84 of 13hapter 2 However the post-sho13k temperatures dedu-
13ed from the jump relations for a real gas (Chapter 2) without radiation are
very 13lose (sim 25 eV) and thus this 13ould be not the relevant explanation
To 13larify this I performed HELIOSPROPACEOS simulations for two
13ounter-propagating sho13ks at 53 and 23 kms respe13tively in Ar and Xe
at the same initial mass density (54 times 10
minus4g13m
minus3) without any opa13ity
multiplier I have also reported for information the 13ase of Xenon with
an opa13ity multiplier equal to 20 The simulated temperature proles are
shown in the Fig 511 One notes immediately that the pre13ursor extension
is smaller for Ar than for Xe The peak temperatures of the MAIN sho13k
are respe13tively sim 21 eV (16 eV for opa13ity multiplier=20) for Xe and 21
eV for Ar In all the 13ases the post-sho13k temperatures are equal to sim 11
eV This indi13ates that the temperature is not the main explanation of the
dieren13es in the pre13ursor length
More interesting are the dieren13es in the Rosseland opa13ity (here per
unit of length) whi13h is reported in the Fig 511b We see that 13lose to the
51 VISIBLE INTERFEROMETRY 87
(a)
(b)
Figure 510 (a) Interferometri13 re13ord in Ar at 03 bar (48079) The estimated
speeds for MAIN and AUX are 49plusmn5 and 23plusmn3 kms (b) Same re13ord where the
fringes maxima are marked by points The bins denition is as fellows bin 0 Ne le11 times 10
1813m
minus3(white) bin 1 11 - 18 times 10
1813m
minus3(yellow) and bin 2 39 - 57
times 10
1813m
minus3(red) The time t = 0 13orresponds to the time of laser arrival on the
target and the position x = 0 13orresponds to the interfa13e between the piston (Au
layer) and the gas Its determination is pre13ise within 100 mi13rons
88 CHAPTER 5 RESULTS AND INTERPRETATION
(a)
(b)
Figure 511 Results for temperature (a) and Rosseland opa13ity (b) obtained from
HELIOS simulation at 10 ns for two 13ounter-propagating sho13ks at sim 50 and 18
kms for Xe (with opa13ity multiplier 1 and 20) Ar and Kr at 54 times 10
minus4g13m
minus3
initial mass density A 13omparison Ar Kr and Xe PROPACEOS opa13ity shown in
APPENDIX B
51 VISIBLE INTERFEROMETRY 89
front the Argon opa13ity is 13lose to 4 13m
minus1 whi13h means that the radiation
from the sho13k is absorbed within 25 mm This has to be 13ompared with the
Xenon 13ase (no multiplier) where it is equal to sim 03 13m
minus1 whi13h means
that the hot pre13ursor is almost transparent over the length of the tube
Thus the dierent behaviors of the opa13ity in the upstream gas explain the
quantitative dieren13e in the development of the pre13ursor The sharp peak
of the Xenon opa13ity with the multiplier of times 20 is due to the strong rise
of the opa13ity when the temperature de13reases below 5 eV whi13h marks the
end of the pre13ursor
I will present now the 13ase of two 13ounter-propagating sho13ks in Kr
(48146 in Fig 512) at 02 bar (69 times 10
minus4g13m
minus3) In this 13ase unfortu-
nately the AUX sho13k 13ould not be imaged 13ompletely due to the presen13e
of glue at the right edge of the target 13ell but the 13ollision was re13orded
at sim 40 ns The speed of the MAIN sho13k is estimated to be 55 kms
Thus despite a slightly higher initial mass density this sho13k wave may be
13ompared with the shot (48055) in Xe at 01 bar (54 times 10
minus4g13m
minus3 54
kms)
Despite the poor quality of the re13ord and the inadequate temporal range
(150 ns) sele13ted on the streak 13amera we note some bending in all the
fringes indi13ating the presen13e of pre13ursor The maximum fringe deviation
(063 of the distan13e between two fringes) is noted at the end of fringe 2
whi13h 13orresponds to the lt Ne gt value 45 times 10
1813m
minus3 The pre13ursor
seams to be more developed here than for the previous Ar 13ase
The result of the simulation in Xe and Kr with equal initial mass density
54 times 10
minus4g13m
minus3 and for the two respe13tive sho13k speeds of 50 and 20
kms is reported in the Figure 511 The peak temperatures of the MAIN
sho13k are respe13tively sim 21 eV (16 eV for opa13ity multiplier = 20) for Xe
and 22 eV for Kr The post-sho13k temperatures (sim 10-11 eV) are 13lose from
ea13h other
The pre13ursor extension in Krypton is smaller than in Xenon whi13h is lo-
gi13al if we 13onsider the opa13ity values (Fig 511b) However if we take into
a1313ount the multiplying fa13tor for the Xenon opa13ity the Krypton pre13ursor
is now more extended than for xenon whi13h is in 13ontradi13tion with the ex-
periment This probably means that either the Kr opa13ity is under estimated
by PROPACEOS or that our multiplier for Xenon is overestimated
Synthesis
The previous experimental investigation 13onrms that the sho13k velo13ity in-
13reases with the laser energy and de13reases with the mass density Moreover
for our given sho13k velo13ity and mass density the extension of the radiative
pre13ursor in the experiment de13reases with the atomi13 number Among Ar
Kr and Xe this last one appears to be the most adapted for the investiga-
90 CHAPTER 5 RESULTS AND INTERPRETATION
(a)
(b)
Figure 512 (a) Raw interferometri13 re13ord in Kr at 02 bar (shot 48146) The
estimated speed for MAIN sho13k is 53plusmn2 kms The time t = 0 13orresponds to the
time of laser arrival on the target(b) Same re13ord shown in (a) maxima is marked
by 13olored points The bins denition is as follows bin 0 Ne le 11 times 10
1813m
minus3
(white) bin 1 11 - 18 times 10
1813m
minus3(yellow) and bin 2 39 - 57 times 10
1813m
minus3
(red) The time t = 0 13orresponds to the time of laser arrival on the target and
the position x = 0 13orresponds to the interfa13e between the piston (Au layer) and
the gas Its determination is pre13ise within 100 mi13rons
51 VISIBLE INTERFEROMETRY 91
tion of the radiative pre13ursor A similar behaviour is found also at ORION
with higher speeds (Clayson et al 2017)
512 Transverse interferometry lateral extension of the sho13k
The transverse interferometry is a dierent geometri13al imaging setup of the
Ma13h-Zehnder interferometer whi13h provides qualitative information about
the 13urvature lo13alisation and transverse extension of the radiative pre13ur-
sor and thus of the sho13k
In the transverse interferometry we perform the image of a transverse
se13tion (on the slit of the streak 13amera) of the tube whi13h is lo13ated at a
distan13e dslit equal to 3 mm from the initial position of the MAIN piston
As the setup is originally adapted for the longitudinal interferometry we use
a Dove prism between the va13uum 13hamber and the streak 13amera to rotate
the image by 90 degrees
Figure 513 S13hemati13s of the Ma13h Zehnder Interferometer setup to re13ord
transverse interferometri13 images The lens allows to make on the slit of the
13amera the image of a se13tion perpendi13ular to dire13tion of sho13k propagation
A transverse interferometri13 re13ord for the MAIN sho13k alone in Xe at
02 bar is reported in Fig514(a) The streak 13amera rst re13ords the unper-
turbed plasma on the se13tion dslit in the tube Then it re13ords su1313essively
the pre13ursor and the post-sho13k (whi13h is opaque to visible radiation and
appears as a dark zone in the re13ords) Taking into a1313ount the oset of 14
ns the time of sho13k arrival is re13orded at 72 ns after the time t0 of laser
arrival on the target and the sho13k speed is estimated to be sim 35 kms Due
to glue on one lateral window (on the right part of the gure) only 6 fringes
92 CHAPTER 5 RESULTS AND INTERPRETATION
Figure 514 Transverse interferometri13 images for (a) shot48111 (MAIN sho13k
only) (b) shot48130 (AUX sho13k only) The time is measured after an oset
equal to 14 and 23 ns respe13tively after the time of the laser arrival on the target
The position zero on the x-axis of ea13h image 13orresponds to the base of the target
are visible The lateral extension of the sho13k stru13ture at this time is deri-
ved to be sim 570plusmn30 microm in qualitative agreement with the spe13i13ations of
the MAIN phase plate and the shape of the pre13ursor is relatively at The
axis of symmetry of the sho13k is determined to be at sim 350 mi13rons from
the base of the target (ie 50 microm above the nominal value of 300 microm)
A re13ord for the AUX sho13k alone is shown in Fig 514(b) whi13h 13orre-
sponds to a gate opening of 50 ns The start time of the image has an oset
of +23 ns from t0 and the distan13e dslit is set to 700 mi13rons from the initial
position of the AUX piston The AUX sho13k duration extends from 30 ns
to at least 34 ns after t0 The sho13k speed is then estimated to be ranging
between 23 and 20 kms The shape of the pre13ursor is strongly bent and
we note a tiny shift of the se13ond and third fringes (starting from the left)
at sim 8 ns It may further be noted that the lateral spread of the opaque
sho13k is ranging between 275plusmn25 microm (whi13h is also in agreement with the
spe13i13ations of the AUX phase plate) and that the axis of symmetry of the
sho13k system is also lo13ated at about 350 microm from the bottom of the 13ell
52 XUV Spe13tros13opy
XUV emission originates from the 13omparatively hot portions of the plasma
In the present 13ase it 13orresponds to the post-sho13k and the pre13ursor re-
53 SIMULATIONS BASED ON EXPERIMENTAL RESULTS 93
gion whi13h is 13lose to the sho13k front The spe13trum re13orded by the XUV
spe13trometer enables us to analyze the spe13tros13opi13 signatures of the radi-
ative sho13k One of the aims of this diagnosti13s was to identify the sho13ks
13ollision whi13h leads to higher temperatures through 13omparing the spe13tra
obtained for the 13ases of single and 13ounter-propagating sho13k waves Howe-
ver only a few re13ords were possible to be obtained during the experiment
and unfortunately the 13omparison with the 13ase of an isolated sho13k wave
was not performed
Among the shots re13orded the XUV spe13trum of the shot 48143 is
presented herewith in detail This shot was performed for [Xe (90)+ He
(10) mixture at 06 bar with laser energies of 123 J for MAIN and 63 J
for AUX The interferometri13 re13ord of this shot is shown in Fig 515 The
MAIN sho13k speed has been estimated to be sim 39 plusmn 4 kms The estimated
AUX sho13k speed (18 plusmn 5 kms) is not pre13ise due to the presen13e of glue
on the right se13tion of the re13ord (note Fig 515) In this interferometri13
re13ord we have not been able to retrieve the 13ollision time However an
extrapolation of the slope 13orresponding to the speeds sim 39 kms (MAIN)
and sim 18 kms (AUX) enables us to approximately determine the 13ollision
time to be between 60 to 65 ns
The raw spe13trum (shown in Fig 516a) re13orded for the wavelength
range of 15-35 nm (35-82 eV) shows the `L edge of Aluminum at 17 nm
(34 nm in se13ond order) in the rst and se13ond (34 nm)orders These two
wavelengths will be used for the wavelength 13alibration The net spe13trum
13orre13tion for the transmission (Henke Gullikson and Davis 1993) of the
100 nm thi13k Si3N4 window (3 mm times 04 mm) is introdu13ed in Fig 516b
A remarkable feature is a strong absorption dip between 19 and 22 nm (56-65
eV) This absorption probably 13omes from the 13older layers (thi13kness 300
microm) between the sho13k heated plasma and the Si3N4 window Few lines
of Xe VII-VIII are identied through NIST database
4
as also Oxygen IV
and V lines Lyman lines of He II (from 1-2 to 1-7) are also present in the
spe13trum This information will be useful for the estimation of the ele13tron
temperature
53 Simulations based on experimental results
In this se13tion I will 13ompare the experimental sho13k 13hara13teristi13s with the
results of HELIOS simulations using the PROPACEOS equation of state and
opa13ity (limited to 1 group) As indi13ated in the 13hapter 3 this opa13ity has
been multiplied by 20 for the Xenon only As our interest is to understand the
sho13k stru13ture in Xenon and not the laser matter intera13tion on the piston
we performed several simulations with Xe gas for various sets of uen13es for
4
[httpphysi13snistgovPhysRefDataASDlines_formhtml
94 CHAPTER 5 RESULTS AND INTERPRETATION
Figure 515 Interferometri13 image for the shot48143 The time t = 0 13orre-
sponds to the time of laser arrival on the target and the position x = 0 13orresponds
to the interfa13e between the piston (Au layer) and the gas Its determination is
pre13ise within 100 mi13rons
53 SIMULATIONS BASED ON EXPERIMENTAL RESULTS 95
(a)
(b)
Figure 516 Raw (a) and 13orre13ted XUV spe13trum (b) for the shot 48143
96 CHAPTER 5 RESULTS AND INTERPRETATION
the MAIN and AUX beams in order to obtain the best agreement between
the simulated and measured sho13k speeds
To analyse the results from the shot 48055 (Fig 59(a)) we set the u-
en13es to 32000 amp 7500 J13m
2 This allows produ13ing the experimental sho13k
speeds 54 and 23 kms in Xenon at 01 bar for the MAIN and AUX beams
respe13tively The two sho13ks appear in Xenon at 2 and 3 ns respe13tively for
MAIN and AUX The merging of the two pre13ursors starts at sim 15 ns and
the sho13k 13ollision time o1313urs at 47 ns In Fig 517 we present the ele13tron
density proles from the simulation (dotted lines) and the experiment (solid
lines) at 10 20 30 and 40 ns
At 10 ns the two simulated pre13ursor extensions are 0165 and 0022 13m
for MAIN and AUX respe13tively The ele13tron density is larger by a fa13tor
of 4 than in the experiment The shapes of the pre13ursors are also very dif-
ferent However this 1D pi13ture supposes the plasma to be uniform within
the tube In reality in the transverse dire13tion Ne is de13reasing from the
13enter to the walls whi13h results in a lower estimation of the average value
(lt Ne gt) 13ompared to the value at the target 13enter and in a smoother pro-
le near 02 amp 035 13m at 10 ns It is also important to note that for AUX
sho13k the average lt Ne gt value underestimates the lo13al one by a fa13tor
of about 2 (as it is averaged over 06 mm instead of 03 mm) Moreover
our 1D simulation suers from an inexa13t opa13ity and 2D ee13ts are pro-
bably important espe13ially for AUX Thus we have here only a qualitative
interpretation of the experimental results
The intera13tion between the two HELIOS radiative pre13ursors starts be-
tween 10 and 20 ns like in the experiment However the shape as well as
absolute values of the simulated ele13tron density 13urves are not in agreement
with the experimental results and the intera13tion is stronger in the simulation
than in the experiment
In order to interpret the spe13tros13opi13 data presented in se13tion 52 we
performed another 1D simulation in Xenon at 06 bar and adapted the
uen13es to generate two 13ounter-propagating sho13ks with the speeds 36 and
18 kms 13lose to the experiment The time evolutions of the ele13tron density
mean 13harge and ele13tron temperature at 56 57 58 60 64 and 65 ns are
presented in Fig 518 The two sho13ks appear in Xenon at 2 and 3 ns
respe13tively for MAIN and AUX Con13erning AUX the 13ombination of a
small speed and a relative high pressure does not allow to develop a radiative
pre13ursor in agreement with the experimental results (Fig 515) whereas
the MAIN sho13k has a pre13ursor and its length is in13reasing with time
The post sho13k temperature of the MAIN is sim 21 eV and the ion 13harge
sim 9 At 57 ns the pre13ursor of MAIN rea13hes the AUX sho13k front This
time is out of our re13ord (see Fig 515) whi13h means that the intera13tion
ee13t is either absent or o1313urs at later times The stru13ture of the AUX
post sho13k is modied by the intera13tion with the MAIN pre13ursor (Fig
518b) The sho13k 13ollision o1313urs at 65 ns (Fig 518a) resulting in the
54 SUMMARY 97
Figure 517 Re13orded ele13tron density (shot 48055) together with the HELIOS
results (with Xenon opa13ity times 20) at dierent times in Xenon at 01 bar
development of two reserve sho13k waves At the 13ollision time the ele13tron
density mass density ele13tron temperature and ion 13harge rea13h respe13tively
sim 14 times 10
2113m
minus3 0034 g13m
minus3 26 eV and 10 Mean 13harge is varying
between 5-10 whi13h 13ompatible with the presen13e of lines of Xe VII-VIII in
the experimental re13ord shown in Fig 516b
Moreover in order to interpret XUV spe13tros13opi13 results shown in se13tion
52 Rodriguez performed qualitative preliminary 13omputations (as des13ribed
in Rodriacuteguez et al (2015)) of the XUV spe13tra emerging from a 600 microm thi13k
plasma with two representative values of the mass density ρ = 32 times 10
minus2
and 33 times 10
minus3g13m
3 They show that the lines of HeII 13an only be ob-
served at a temperature of sim 15 eV and for the lowest density ie in the
radiative pre13ursor
54 Summary
In this 13hapter I have presented an extensive data analysis of few representa-
tive interferometri13 and spe13tros13opi13 re13ords The average sho13k speed and
ele13tron density have been estimated from the interferograms The sho13k
speeds of the MAIN and AUX radiative sho13k waves vary between sim 30-55
and 10-30 kms respe13tively and the averaged pre13ursor ele13tron density
ranges between 10
17and 10
1913m
minus3during the sho13ks propagation
We have demonstrated the intera13tion ee13t between the two radiative
pre13ursors in the 13ase of Xe at 01 bar at 54 and 23 kms The intera13tion is
13learly 13hara13terized in the experiment by the enhan13ement of the ionisation
wave followed by the merging of the two radiative pre13ursors at 20 ns The
13ollision time is re13orded at 47 ns Su13h behavior is reprodu13ed by the
98 CHAPTER 5 RESULTS AND INTERPRETATION
(a)
(b)
(13)
Figure 518 Time evolution of the mass density (a) ele13tron temperature (b) and
mean 13harge (13) at 56 57 58 60 64 and 65 ns within the sho13k tube derived from
the HELIOS simulations (with Xenon opa13ity multiplier = 20) for two 13ounter
streaming sho13ks of sim 39 and 18 kms in Xenon at 06 bar
54 SUMMARY 99
simulation
We have investigated this intera13tion at a larger pressure 02 bar with
the following speeds sim 41 kms for the MAIN and sim 18 kms for the AUX
sho13k waves We do not re13ord any experimental signature of the radiative
pre13ursor for AUX Further we have not been able to 13at13h experimentally
the 13ollision time The re13orded pre13ursor of MAIN is not inuen13ed by AUX
wave up to 48 ns (Fig 59(b) and (13) in data analysis 13hapter) whi13h is the
limit of the re13ord On its side the 1D simulation predi13ts a tiny pre13ursor
for AUX and that both pre13ursors start to intera13t at 49 ns This plausible
sho13ks intera13tion o1313urring at times whi13h are outside of the re13ord 13an
not be 13onrmed by our experiment
The results of the transverse interferometry at 02 bar with speeds of sim40 and 20 kms indi13ate that the MAIN pre13ursor has a lateral extension of
sim 600 microm whereas it is 300 microm for AUX The pre13ursor of MAIN is almost
at with a probable small bending at the edges of the tube whereas the
AUX pre13ursor is more 13urved This means that the 2D ee13ts are more
important for AUX than for MAIN Still in the 13ase of Xenon we have
obtained information about the temperature and the mean 13harge of gas
through our time integrated XUV spe13tra (shot48143) in Xenon at 06
bar From these results we may 13on13lude that the mean ion 13harge is at
least equal to 6 and that the temperature has rea13hed 15 eV
Our simulations give a qualitative des13ription of the sho13ks intera13tion
when the laser uen13e is adjusted to give the 13orre13t sho13k velo13ities Howe-
ver it is now well known that 2D simulations (together with state of the art
opa13ities) t better with experiments (Gonzaacutelez Audit and Stehleacute 2009
Leygna13 et al 2006 Stehleacute et al 2010) For the same laser energy the 2D
simulations lead to a diminution of the sho13k speed 13ompared to 1D as also
to a diminution of the ele13tron density For instan13e in the 13ase of a sho13k
wave laun13hed by a laser beam at 1315 nm in Xenon at 03 bar at PALS and
with a laser uen13e of 85000 J13m
2 ARWEN 2D simulations give a sho13k
speed of 44 kms in agreement with the experimental one (Cotelo et al
2015) 1D simulation would require in this 13ase a uen13e of 30000 J13m
2to
a13hieve the same velo13ity
The spa13e and time integrated XUV re13ords at 06 bar for respe13tive
speeds whi13h are equal to sim 39 and 18 kms indi13ate that the temperature
of the sho13k has rea13hed values up to 15 eV and that the Xenon mean ion
13harge has also rea13hed values of 6 - 7 whereas 1D simulations predi13t ele13tron
temperature 10-30 eV and ion 13harge 5-10 (Fig 51813) A more detailed
study based on 2D simulation and radiative transfer post-pro13essing will be
ne13essary to rene the analysis
We have investigated the 13ase of other noble gases (Ar Kr) and we have
observed that for a given laser energy the sho13k velo13ity de13reases with the
mass density This study has 13onrmed that for a given density and sho13k
velo13ity the radiative ee13ts in13rease with the atomi13 number For Kr we
100 CHAPTER 5 RESULTS AND INTERPRETATION
noti13ed a tiny pre13ursor without any intera13tion To get su13h ee13t it will
be ne13essary to in13rease the sho13k velo13ity and thus the laser energy
This has been done at the ORION laser fa13ility in UK where the 13ol-
lision of two 13ounter-propagating sho13ks at equal speed sim 80 kms with
laser uen13e sim 6 times 10
14W13m
2has been performed (Clayson et al 2017
Suzuki-Vidal et al 2016) for dierent noble gases and for pressures 13ompri-
sed between 01 and 1 bar A number of diagnosti13s setup X-ray ba13klig-
hting opti13al self-emission streak imaging and interferometry (multi-frame
and streak imaging) were used to study both the post-sho13k and the radia-
tive pre13ursor Although I have not parti13ipated in the experiments I have
performed 1D simulations to interpret ORION experimental results This
work is not presented in this thesis
Chapter 6
Optimization of an
ele13tromagneti13 generator for
strong sho13ks in low pressure
gas
Contents
61 Prin13iples of operation of a high 13urrent generator 102
62 Prin13iples of the run-down phase in a PFD 103
63 Proposed design for the plasma gun 104
64 Dynami13 13ir13uit modelling 106
65 3-D MHD simulations using GORGON 13ode 111
66 Measurements 116
67 Summary 118
The previous 13hapters were dedi13ated to the study of laser generated
radiative sho13ks With irradian13e more than 10
14W13m
2 we were able to
study sho13k waves propagating at velo13ity up to 50 kms in noble gases with
an initial mass density 13omprised between 5 times 10
minus4- 3 times 10
minus3g13m
minus3
Complementary to laser experiments 13ompa13t pulsed power generators may
drive an ele13tromagneti13 13oaxial plasma gun to 13reate astrophysi13al relevant
sho13ks in lower pressure noble gases (Kondo et al 2006) with a high avai-
lability and a rather modest 13apital 13ost The ele13tromagneti13ally driven
sho13k waves may have larger s13ales than those by laser thus they 13an be
observed rather easily (Kondo et al 2008)
Su13h ele13tri13al pulsed power devi13es may then i) 13reate a high-voltage
breakdown through a gas or more easily at the surfa13e of a diele13tri13 ii)
produ13e relatively hot plasma by ohmi13 heating when the rising 13urrent in the
devi13e is passing through a portion of gas iii) a1313elerate plasma layers under
the magneti13 pressure asso13iated with the self-generated magneti13 eld when
high ele13tri13al 13urrent (say 10s of kA) is maintained for a short but su13ient
time (say 1 micros) Su13h ionization and a1313eleration are present in Z-pin13h
plasmas and in plasma fo13us devi13es (PFD) providing a 13lever geometry is
employed and the mass under 13onsideration (say 1 mg) is 13ompatible with
the stored energy (Martin Williams and Kristiansen 1996)
101
102 CHAPTER 6 ELECTROMAGNETICALLY LAUNCHED SHOCK
In this 13hapter we will see rst how a fast ele13tri13al 13ir13uit works then
we will dis13uss the prin13iples of a 13oaxial plasma gun and of PFD We will
see what is in favor of our obje13tive and what has to be avoided A geometry
13apable to rea13h our obje13tive will be proposed
The optimization of the 13oaxial gun (in term of plasma speed) will be
performed and some typi13al gures will be given in the 13ase of a generator
with a stored energy around 1 kJ and a plasma slab of 4-mm diameter ie
quite 10 times the transverse dimension of a laser driven RS The operating
13onditions are supersoni13 sho13ks up to 10-30 kms speed at stati13 pressures
of few mbar in heavy rare gases (Ar Xe) The results of this 0-D model will
be then 13ompared to those obtained with 3-D MHD simulations performed
with the 13ode GORGON (by Andrea Ciardi at LERMA) whi13h has been
used su1313essfully to des13ribe other pulse-power driven plasma experiments
(Chittenden et al 2004) as well as laboratory plasma astrophysi13s experi-
ments (Ciardi et al 2007) The diagnosti13s whi13h have been implemented
will be presented to illustrate the model as well as preliminary re13ords of the
plasma speed
61 Prin13iples of operation of a high 13urrent gene-
rator
Our aim being to a1313elerate a plasma slab using the magneti13 pressure it
is obviously needed to drive a high intensity 13urrent be13ause the magneti13
pressure is expressed by B
22micro0 in the region where exists an indu13tion B
The pressure in bars is simply 4B
2 with B in Teslas and in a 13ylindri13al
geometry 1 Tesla is the eld around a 50-kA 13urrent at a radius of 1 13m
Thus we should 13ount on roughly 100 kA delivered by the generator but
su13h a high 13urrent is delivered only in a pulsed mode by a laboratory s13ale
devi13e
There are numbers of te13hni13al solutions to a13hieve su13h ele13tri13al pa-
rameters one may use rather slow generators at moderate high voltage like
13apa13itors bank with the advantage of well know te13hniques but with limited
adjustments A1313ording to an abundant literature
1
a 13onvenient devi13e is a
medium-energy 13apa13itor bank feeding a low-indu13tan13e 13ir13uit An R-L-C
13ir13uit is a well-known 13ombination delivering a high peak 13urrent in the
pseudo-periodi13 mode For a 13apa13itor C initially 13harged under U0 key
performan13es are as follows
Current intensity I(t) is a damped sinusoid
Pseudo pulsation ω =radic[1(LC)minusR2(4L2)]
1
Institute for Plasma Fo13us Studies resour13e website httpplasmafo13usnet last
13onne13tion in 2016
62 PRINCIPLES OF THE RUN-DOWN PHASE IN A PFD 103
First 13urrent peak Ipeak = U0[radic(LC) + 08R]
Time of 13urrent peaking T4 = π(2ω)
Equivalent impedan13e Z =radic(LC)
For a safe handling in air a voltage not higher than 30 kV is re13ommen-
ded Taking a total 13apa13itan13e of 6 microF 2700 J are stored under 30 kV For
an indu13tan13e of 240 nH the expe13ted peak 13urrent might be 150 kA at 19
micros This set of values will be a guideline for the development presented in
further se13tions
62 Prin13iples of the run-down phase in a PFD
Our aim is thus to 13reate a fast moving plasma sheath with quite a one-
dimension geometry The issues are to initiate this plasma at the best then
to maintain even improve its stru13ture during the a1313eleration nally to
laun13h it
Instead of 13reating a gas breakdown in volume whi13h would lead ra-
pidly to 13on13entrate the 13urrent in an ar13 the idea for PFD operation is to
start from a surfa13e dis13harge (Lee 1969 Bernard 2002) In the 13lassi13al
13oaxial geometry of the Mather-type PFD the 13entral ele13trode (usually the
anode) is a 13ut metal 13ylinder and the outer one is a se13ond metal 13ylin-
der of the same length or better a squirrel 13age allowing many viewpoints
to the plasma Both ele13trodes are atta13hed to a diele13tri13 bottom plate
made of polymer or 13erami13 When the 13apa13itor bank is swit13hed on the
high voltage is applied to this 13m-size gap and a radial surfa13e dis13harge is
initiated at the interfa13e with the gas forming a quite uniform ring-shaped
layer of thi13kness around 1 mm An insulating sleeve is adjusted around
the 13entral ele13trode to for13e the plasma to ow rapidly upwards under the
magneti13 pressure in this region The result is an elongation the dis13harge
path preventing an ar13ing in the plasma layer
At later times as the 13urrent is growing the plasma layer is strongly
inuen13ed it is pushed upwards by the magneti13 pressure jtimesB it is heatedby the ohmi13 ee13t then its ionization degree in13reases it be13omes denser
as a per13entage (10-40) of the heavy parti13les from the swept volume are
a1313reted (Potter 1971) This ele13tromagneti13ally driven motion is 13alled the
rundown phase Due to the a1313retion it is des13ribed by a so-13alled snowplow
model whi13h will be dis13ussed later in detail In quite all the designs even
if very high speeds are reported (Lee 1969 Serban 1995) the plasma is bell
shaped whi13h does not fulll our requirements
At the end of the rundown phase whi13h mat13hes roughly with the 13urrent
peak the plasma stays 13onne13ting the upper ends of the ele13trodes and the
radial 13omponent of the magneti13 for13es grows rapidly Then the plasma is
104 CHAPTER 6 ELECTROMAGNETICALLY LAUNCHED SHOCK
strongly pushed to the axis giving the so-13alled Z-pin13h This stage made
the PFD popular be13ause the pin13hed plasma is a sour13e of fast ele13trons
fast ions hard radiation and possibly neutron beams it must be avoided
in our 13ase It was also quoted (Lee and Serban 1996) that optimal PFD
are mat13hing a universal fa13tor 13omprising anode radius peak 13urrent and
gas density meaning a robustness of the design when a geometry has been
13hosen
63 Proposed design for the plasma gun
The previous des13ription has shown the positive inuen13e of a surfa13e dis-
13harge the modi13ation of the shape by the sleeve and the issue of the
pin13hing ee13t A1313ordingly the following design is proposed following a
work by Kondo et al (Kondo et al 2006 2008)
To ensure a rapid dis13harge with the values given in the se13tion 61 a
highly 13oaxial 13ir13uit is 13hosen for all the large parts the energy bank the
13losing swit13h and all the 13onne13tions The initial phase is kept with two
13oaxial ele13trodes pressed on a at insulator Homogeneity of the plasma
sheath is expe13ted by 13hoosing a rather small radial gap of 25 mm The ee13t
of magneti13 for13es during the rundown phase must be enhan13ed espe13ially
due to the in13rease of the mass of the plasma as des13ribed above As the
13urrent 13urve is the rising part of a sinusoid a tri13k is used to in13rease
the lo13al magneti13 eld B around a 13ondu13tor varies as the inverse of the
distan13e to axis so the proposed shape for the anode is a 13oneHowever a
nal divergen13e must be avoided and that is obtained by a rounded tip In
order to keep 13onstant the plasma length along the 13urrent path the radial
gap is kept 13onstant so the outer ele13trode is a hollow 13one with the same
angle Above the 13one the a1313elerator is 13onne13ted to a 13ondu13ting tube
where the plasma 13an propagate freely in the ba13kground gas The distan13e
of the plasma sheath to the 13one tip is in13reasing rapidly and the main
13urrent whi13h is still high will pass preferably through the diuse plasma
remaining between the ele13trodes Thus there will be no magneti13 pressure
anymore a13ting on the sheath whi13h 13an propagate freely The resulting
devi13e is des13ribed in Fig 61a and the a13ting magneti13 for13es shown in Fig
61b
Te13hni13ally su13h a 13onguration is 13onvenient even at 15 kV be13ause the
sharp edge of the anode tou13hing the insulator forms a so-13alled triple point
where the ele13tri13 eld is enhan13ed at the surfa13e of the insulator whi13h
triggers e13iently a dis13harge in presen13e of gas at low pressure (01 10
mbar) as proposed by Kondo et al (Kondo et al 2006) Polya13etal is a
13onvenient material for diele13tri13 and me13hani13al performan13es
The dire13tion and the orientation of the magneti13 for13e are other points to
be dis13ussed The high-intensity 13urrent (up to sim 150 kA) generates a strong
63 PROPOSED DESIGN FOR THE PLASMA GUN 105
(a)
(b)
Figure 61 (a) Sket13h of the sho13k generator showing the pulsed ele13tri13al 13ir13uit
the set of 13oaxial 13oni13al ele13trodes with a 13onstant radial gap and the plasti13
insulator featured in grey on whi13h a planar surfa13e dis13harge is initiated The
installation of three opti13al bers allows looking radially at the plasma moving in
the sho13k tube (b) S13hemati13s of the plasma dynami13s inside the 13oaxial gun
in fast-pulse mode the ele13tri13al 13urrent ows in the super13ial layers of the two
13oaxial 13oni13al ele13trodes and through an annular plasma layer The magneti13
pressure Pmag pushes the dis13harge axially
106 CHAPTER 6 ELECTROMAGNETICALLY LAUNCHED SHOCK
Figure 62 Exploded view of the plasma gun
azimuthal magneti13 eld between the ele13trodes thus the magneti13 pressure
jtimesB whi13h a1313elerates the annular plasma sheath stays perpendi13ular to the
sheath lo13ally (Fig 61b) The more planar is the initial sheath the more
axial will be the magneti13 pressure Finally if the roles of the ele13trodes are
ex13hanged the dire13tion of the 13urrent will be inverted and the same for B
so the pressure will stay oriented in order to expel the plasma Compared
with the Mather-type plasma fo13us (Potter 1971 Zambra et al 2009) the
plasma sheath 13onsidered here is quite planar 13lose to the insulating surfa13e
and later is expe13ted to stay planar and perpendi13ular to the axis As in a
PFD the thi13kness of the plasma sheath in13reases gradually (Zambra et al
2009) but it is assumed to stay in the mm-range We employ 13oaxial 13oni13al
brass ele13trodes ea13h 42-mm high with a 13onstant gap of 25 mm At the
bottom level internal and external radii of ele13trodes are 125 mm and 15
mm respe13tively This small-size dis13harge 13ell on top of the generator will
be modelled in the following
64 Dynami13 13ir13uit modelling
We optimize the ele13tri13 generator for various gases namely Ar and Xe
with the motivation to produ13e plasma sho13ks with speeds sim 1 - 30 kms
ie Ma13h numbers up to 200 For that we design the ele13trodes and set
64 DYNAMIC CIRCUIT MODELLING 107
the ele13tri13al 13ir13uit parameters to produ13e 1-micros pulses in the gas 13hamber
The geometry of the ele13trodes is as mentioned above The half angle of the
13oni13al ele13trode is 13hosen to be α =15deg and thus the height of the 13one is h
sim 42 mm
Eleven (N = 11) 13apa13itors ea13h with 13apa13itan13e C0 of 06 microF are13onne13ted in parallel giving an equivalent 13apa13itan13e of the bank C = 66
microF 13harging voltage U0 of the bank is set to 15 kV Thus the available
ele13tri13al energy is 750 J The 13ir13uit resistan13e is used to be 1 m Ω and
damping resistan13e is set to zero whi13h yields a total resistan13e (R = Rcir+
Rdamp N ) of 1 mΩ and a pseudo-periodi13 regime is expe13ted
Based on the above geometri13al and ele13tri13al parameters we have esti-
mated the time evolution of various parameters of the radial dis13harge layer
namely speed mass a1313eleration et13 at a dierent position along the axis
from z = 0 to h For this 13al13ulation we need to solve a set of three equa-
tions whi13h give the main 13urrent (- dqdt) supposed to ow only through
the plasma layer the mass a1313retion rate (dMdt) and the velo13ity v of
the plasma These equations together with initial 13onditions are explained
below The ele13tri13al equation is written 13lassi13ally
Ld2qdt2 +Rdqdt+ qC = 0 (61)
When the R L and C parameters are not time-dependent the 13ir13uit
equation 61 has an analyti13al solution a damped sinusoid for the 13urrent I
= -dqdt and that was used as a test of the iterative solving pro13ess and to
13onrm the parameters of the external 13ir13uit with a short 13ir13uit load The
indu13tan13e of the external 13ir13uit L0 is 13omputed from the ringing frequen13y
in short 13ir13uit (66 nH) whi13h results in a 13ir13uit impedan13e Z0 =
radic
L0Cto be 01 Ω In the plasma gun mode the sheath 13arrying the 13urrent (see
Fig 61b) is a1313elerated by the magneti13 pressure and it behaves as the de-
formable part of the 13ir13uit then L is time-dependent and a numeri13al solver
is ne13essary for equation 61 We used a lumped-parameter model whi13h has
proven to des13ribe similar 13ir13uits su1313essfully (Potter 1971 Zambra et al
2009) Gonzalez (Gonzalez et al 2004) showed that the mass and momen-
tum equations for a 13urrent sheath in the shape of an annular piston moving
forward in the axial dire13tion 13an be given by
dMdt = ερ0π(R2e minusR2
i)v (62)
where ε is a1313retion fa13tor and that the transformation of magneti13 to
kineti13 energy is expressed by
d(Mv)dt = lI22 (63)
where l is the linear indu13tan13e of the gun and 13an be expressed as l =
(micro04π) ln (ReRi) the radii being estimated where the plasma sheath is
13urrently lo13ated The equations 62 and 63 have been introdu13ed in the
108 CHAPTER 6 ELECTROMAGNETICALLY LAUNCHED SHOCK
(a)
(b)
(13)
Figure 63 Time dependen13e of the main parameters for Ar gas at 1 mbar a1313or-
ding to the 13ir13uit model (a) 13urrent (kA) and voltage (V) (b) a1313reted mass
(kg) and rate of a1313retion (kgs) (13) speed (ms) and kineti13 energy (J)
64 DYNAMIC CIRCUIT MODELLING 109
Figure 64 Plot of the dierent output parameters (normalized with respe13t to
the peak values) obtained at the top of the inner ele13trode For ea13h group upper
panel represents the variation for Ar gas whereas lower panel is for Xe gas Proles
of output parameters are given (a) with 13one angle (b) with a1313retion fa13tor (13)
with gas pressure and (d) with damping resistan13e
110 CHAPTER 6 ELECTROMAGNETICALLY LAUNCHED SHOCK
time-dependent iterative solving of equation 61 Therefore from equations
62 and 63 we get
dvdt =[ lI
2
2 minus ερ0π(R2e minusR2
i)v2]
M(64)
From equation 64 we may note that initial value of mass M(t0) 13annot
be set to zero as it would lead to a nonphysi13al divergen13e of the a1313eleration
In this regard we have 13onsidered an initial thi13kness of the 13urrent sheath
of 1 mm and integrated the gas density at rest leading to M(t0) equal to
35 times 10
minus10kg for Ar and 13 times 10
minus9kg for Xe A higher limit has to be
put too M(t0) must be small 13ompared with the mass at the end of the
laun13hing phase that is the mass 13ontained in the gap times the a1313retion
fa13tor The above value for M(t0) follows that 13riterion and within these
limits the 13hoi13e of initial mass is not inuen13ing the results
Parameter Referen13e Value Variation
Half angle of the 13one 15
13
14
15
16
middot middot middot 24A1313retion fa13tor ǫ 04 01 02 03 middot middot middot 09
Pressure P 100 Pa 10 20 50 100 200 500 1000
Damping resistan13e Rdamp 0 ohm 0 01 02
Table 61 Input parameters their standard values and range of variation
As the motion is mainly along the z-axis the speed v is the derivative
of the plasma position v = dzdt We solved these equations by a time-
dependent method where the initial 13ondition is that at the onset time t0
the 13harge q is set to CU0 Further the 13urrent i = -dqdt is taken to be zero
at t0 Thus using equation 61 the initial rate of 13hange of the 13urrent is
d
2qdt
2= - qL0C Then all the rst time-derivatives are updated at the rst
time step t0+∆t with ∆t=043 ns using equations 61-64 The quantities
like q M v z are updated at the next step then the iteration is pursued
Initially position a1313eleration and velo13ity of the sheath are set to be zero
as well as the rate of 13hange of the mass
As we initiate the ele13tri13al 13urrent into the gas inside the 13hamber the
plasma sheath is a1313elerated with respe13t to its initial mass as seen above
At later times the mass a1313retion is masking the inuen13e of this arbitrary
value We have 13al13ulated the plasma evolution in the dis13harge for dierent
sets of initial 13onditions whi13h are reported in Table 61
Fig 63 shows the behavior for the standard values reported in Table 61
in the 13ase of Argon at 1 mbar The 13orresponding values of 13harge mass
and velo13ity will be hereafter 13alled the referen13e values In Fig 63a the
variation of 13urrent and voltage in the rst half-period is 13lassi13al showing
that the pulse generator departs slightly from a pure RLC 13ir13uit Fig
63b represents the a1313reted mass whi13h in13reases in time as well whereas
65 3-D MHD SIMULATIONS USING GORGON CODE 111
the rate of mass a1313retion 13ulminates with a 13ompetition of speed growth
and de13rease of the swept volume per time step be13ause it depends on the
squared radii In Fig 6313 plotted are the speed and kineti13 energy of the
plasma sheath First the speed of the plasma is in13reasing slowly with time
quite linearly as the mass and the kineti13 energy is roughly a third power
of the time
That leads to 13hoose a design where the a1313eleration is imposed for a
su13iently long time However the estimated speed of plasma is sim 140 kms
whi13h is a very optimisti13 value The reason may 13ome from two reasons
(i) a part of the 13urrent ows through the large sho13ked volume redu13ing
the magneti13 pressure a13ting on the plasma sheath (ii) energy transfer from
magneti13 to thermal energy and ionization is redu13ing the kineti13 one The
observation of a very bright emission from the plasma will be a 13lear eviden13e
of this redistribution of the energy
Further we made a 13omparative study of the parameters indi13ated in
Table 61 and applied to two gases namely Ar and Xe In order to 13he13k the
ee13ts of the initial 13onditions (13one angle a1313retion fa13tor pressure and
damping resistan13e) we have performed a systemati13 study by varying only
one parameter and keeping the others 13onstant and equal to the referen13e
values given by the se13ond 13olumn of Table 61 The tested range is listed
in the third 13olumn of Table 61 Similarly for ea13h set the variation of the
plasma 13onditions at the top of the muzzle is normalized to the referen13e
value Fig 64 summarizes the variation of the sele13ted dierent output
parameters the time tm to rea13h the muzzle (z = h) the ele13tri13al 13urrent
and the axial speed at muzzle the laun13hed mass M the kineti13 energy and
the average speed at the top of the ele13trode 13al13ulated from the initial
point v=h( tm - t0)
Table 62 summarizes the respe13tive ratios for Ar and Xe at the top of
the 13oaxial ele13trodes As the values of interest are obtained at 13onstant
stored energy in the 13apa13itors there is a benet to work with a lighter gas
like Argon However the results obtained using this 13ir13uit model show that
the referen13e values are ensuring a satisfa13tory behavior for both gases
Speed Kineti13 energy Time to laun13h Mass
222 151 66 31
Table 62 Laun13hing performan13e ArXe
65 3-D MHD simulations using GORGON 13ode
Our simplied model (13f se13tion 64) is 13onvenient to perform qui13kly some
parametri13 tests in the a1313eleration phase and to help designing the devi13e
but it is not 13apable to treat the sho13k dynami13s in detail nor to give valuable
112 CHAPTER 6 ELECTROMAGNETICALLY LAUNCHED SHOCK
information on the plasma parameters in the sho13k region So rened nume-
ri13al simulations of the experiments were performed in 13ollaboration with A
Ciardi (Ciardi 2014) using the GORGON 13ode (Chittenden et al 2004 Ci-
ardi et al 2007 Suzuki-Vidal et al 2014) GORGON is an expli13it parallel
13ode designed to solve the resistive MHD equations on a three-dimensional
(3-D) Cartesian grid employing a Van Leer type algorithm The 13ode treats
the plasma as a single uid but solves separately the energy equations for
ions and ele13trons allowing dierent temperatures for the two spe13ies Both
thermal 13ondu13tion and resistive diusion are treated using Braginskii-like
transport 13oe13ients
Variables At rest Sho13k Post-sho13k
Ar density ρ (g13m
3) 18E-6 25E-5 10E-5
Ele13tron density Ne (13m3) 10 E+15 20E+18 15E+18
Ele13tron Temperature Te (eV) 003 (1lowast) 10 15-18
Ion density ni (13m3) 30E+16 35E+17 2-3E+7
Ion Temperature Ti 0 50 15-20
Average Speed ltVgt kms 0 50 20-40
ltzgt 02 6 7-9
Table 63 Values of variables in various 13onditions at rest (with seed ele13trons)
inside the sho13k and inside post-sho13k region A star lowast denotes the insulator surfa13e
This 13ode des13ribes the 13reation of the surfa13e dis13harge then its a1313ele-
ration by the magneti13 pressure It provides the mapping of all the plasma
parameters and lo13al B-eld in the laun13hing 13one as well as in the free ight
region The 3-D simulation was performed for Argon gas at 1 mbar only the
dis13harge 13urrent law being taken from experiment The Table 63 presents
the ranges of all the variables in the regions of interest
As an example Fig 65 shows an enlarged mapping of log (ne) at two
times in the laun13hing phase then when the sho13k has deeply penetrated
in the sho13k tube Taking the zero time as a referen13e the averaged speed
zt is 40 kms in this simulation When the two snapshots are 13onsidered
the estimate of the instantaneous speed zt is higher at 56 kms The
time history of plasma merging and early free ight into the sho13k tube
is presented in Fig 66 During the a1313eleration (inside the double 13one)
the plasma is well lo13alized on a planar annular sheath whi13h is 13arrying
the 13urrent as assumed in the lumped parameter model We see that the
merging is well a13hieved at the muzzle shortly before 1250 ns When the
sho13k has penetrated inside the tube (1500 ns) the 13omputed B-eld is
negligible at this lo13ation and the propagation is quite free as expe13ted for
the 13hosen design of the devi13e However a high 13urrent is still passing
through the gas at that time a su13ient ele13tron density remaining in the
13oni13al gap for that
65 3-D MHD SIMULATIONS USING GORGON CODE 113
Figure 65 Example of mapping of the ele13tron density from a 3-D MHD simu-
lation (ba13kground gas Argon at 1 mbar) when the plasma sheath is (left) at
the exit of the plasma gun (right) in free ight 13onditions (log s13ales for the false
13olors)
114 CHAPTER 6 ELECTROMAGNETICALLY LAUNCHED SHOCK
Figure 66 Time history of plasma merging and early free ight through the
mapping of the ele13tron density ele13tron temperature and average ioni13 13harge as
given by a 3-D MHD simulation (ba13kground gas Argon at 1 mbar) (s13ales for the
false 13olors)
65 3-D MHD SIMULATIONS USING GORGON CODE 115
Figure 67 Axial proles values taken in the sho13k tube along a line slightly o-
axis at a distan13e of 1 mm for mass density ion temperature ion density average
ion 13harge magneti13 eld ele13tron temperature ele13tron density and average speed
at 1500 ns (ba13kground gas Argon at 1 mbar) A maping of |B| is given with a
dashed line indi13ating sho13k front position as well
116 CHAPTER 6 ELECTROMAGNETICALLY LAUNCHED SHOCK
To 13onrm the steep stru13ture of the sho13k Fig 67 presents the axial
proles of ele13tron temperature Te number density Ne mass density ρ and
average ion 13harge ltZgt at 1500 ns inside the tube with an enlarged axial
position s13ale Steep fronts are observed for mass density ele13tron density
and temperature as in the earliest simulations (Potter 1971 Kondo et al
2008) The sho13k region is at 10 eV and sim2times1018 ele13trons13m
minus3 the post-
sho13k region is hotter Argon ionization stages of 5 (sho13k) to 9 (post-sho13k)
are obtained fore13asting an emission spe13trum in the UV or harder A wider
view of the plasma is given by other quantities at the sho13k front and behind
ion temperature average speed and lo13al magneti13 eld in Fig 67 We get
the 13onrmation of the extin13tion of B after the rst 5 mm inside the sho13k
tube as well as the sho13k speed at 50 kms 13oherent with the 13onstant speed
of 56 kms mentioned just above At present these parameters are guiding
the 13hoi13e of diagnosti13s like ultra-fast interferometry or UV-spe13tros13opy
66 Measurements
The aim of these tests was to show the 13onsisten13e of the above model by me-
asuring the plasma speed 13lose to the plasma gun muzzle The parameters of
the devi13e are the referen13e ones as dened in Table 61 A rather simple and
noise-free method for diagnosing a plasma gun during the rundown phase
is to re13ord the light emitted by the plasma sheath with a spatial resolution
13ompatible with the plasma stru13ture If we 13onsider a line of sight whi13h
is radial or tangential ie perpendi13ular to the dire13tion of propagation
one expe13ts to see a sudden rise when the plasma enters the dete13tion vo-
lume possibly a plateau when the plasma travels inside the volume then a
slow de13rease when the hotter plasma leaves the volume but when the sho13k
remnants are still present So the expe13ted signal is a triangular asymme-
tri13 pulse (Serban and Lee 1995) or some kind of double exponential pulse
(Stehleacute et al 2012) A peak indi13ates the time of ight for the given posi-
tion allowing to 13ompute a mean speed providing the laun13hing time (t=0)
and all positions are known An instant speed is a1313essible providing two
13lose dete13tors are 13onsidered That has been proposed and tested su1313ess-
fully in the 13ase of radiative sho13ks driven by one PALS laser beam (Stehleacute
et al 2012) the dete13tors being 13ollimated and ltered sili13on diodes For
13ooler plasmas the visible emission is more relevant Inside a squirrel 13age
plasma fo13us Serban has installed a set of opti13al ber looking tangentially
at the drift zone (Serban and Lee 1995) Despite a 13oni13al eld of view the
results were 13on13lusive at speeds as high as 100 kms in D2 An improved
version with 13ollimated opti13al bers (a1313eptan13e angle 2deg) was developed
too (Serban and Lee 1995) A non intrusive method was developed (Veloso
et al 2012ba) by imaging the plasma on a set of opti13al bers allowing to
play with the magni13ation
66 MEASUREMENTS 117
(a)
(b)
Figure 68 Experimental results (a) time history of the passive opti13al re13ords
13ompared to the main 13urrent for Argon gas 13lose to the referen13e pressure (b)
time-integrated signal vs Ar pressure for the dierent bers (13) averaged speed vs
Ar ba13king pressure
118 CHAPTER 6 ELECTROMAGNETICALLY LAUNCHED SHOCK
In our 13ase (see Fig 61a) three 1-mm PMMA (Poly(methyl metha13ry-
late)) opti13al bers are installed radially along a transparent se13tion of the
sho13k tube at z sim 625 mm 70 mm and 775 mm and fast DET10A dete13tors
re13ord the transient presen13e of the plasma sheath in the respe13tive lines of
sight The re13orded spe13tral range is limited to the whole visible in a preli-
minary stage a band-pass ltering might give information on the presen13e of
spe13i13 ioni13 spe13ies 13hara13teristi13 of a plasma temperature level however
the amount of light was not su13ient to do so Fig 68a presents the time
evolution of the ele13tri13al 13urrent with a period of 4 micros and the delayed
side-on observational data of the moving plasma inside the sho13k tube The
behavior of the ele13tri13al 13ir13uit is following well the expe13ted ringing of a
weakly damped RLC 13ir13uit the zero-time being xed at the onset of the
13urrent On opti13al 13hannels a steep rise and a peak are the signature of a
mm-thi13k fast-sho13k stru13ture (Serban and Lee 1995) 13onsistent with early
numeri13al simulations (Potter 1971 Tou Lee and Kwek 1989 Kwek Tou
and Lee 1990 Veloso et al 2012b) On ea13h ber signal se13ondary peaks
namely the one delayed by 4-5 micros are attributed to 13urrent restrikes at the
pseudoperiod To avoid the mixing of sho13ks further experiments should be
13ondu13ted in an aperiodi13 regime with a heavy-duty damping resistor added
in series to ea13h 13apa13itor Rdamp sim NZ
On Fig 68b and 13 are presented the exploitation of opti13al re13ords with
the Ar ba13kground pressure this parameter varying in the widest range (b)
the integrated emission whi13h presents a maximum between low pressure side
when the temperature is high but the number of emitters is low and the high
pressure side when the temperature is mu13h lower (13) the averaged speed
ziti for the rst peak on ea13h ber The relative intensities re13orded in the
three lo13ation s are not signi13ant be13ause of a strong 13opper deposition
13loser to the muzzle whi13h ae13ted the dete13tion The long-life noise on
the opti13al signals was making the lo13al speed 13al13ulated as (zi-zj)(ti-tj) a
non-reliable output One 13an note Ma13h numbers in the range 20 60 with
a pressure dependen13e similar to previous work (Kondo et al 2006) The
reported observations are 13onsistent with a high-speed millimetri13 planar
plasma travelling in the sho13k tube The speed is lower than dedu13ed from
the simplied model possibly due to the la13k of dissipation terms whi13h
is overestimating the transfer to kineti13 energy At the 13ontrary a realisti13
simulation as the one performed using GORGON is in better agreement
67 Summary
For this part of the work the obje13tive was to show that in 13omplement of
laser experiments a 13ompa13t pulsed power generator might drive astrophy-
si13ally relevant sho13ks in low pressure noble gases with a higher availability
and at a rather modest 13apital 13ost The longitudinal a1313eleration of a
67 SUMMARY 119
plasma sheath in a low ba13king gas pressure has been demonstrated in 13o-
axial plasma guns as early as in the 70s even leading to speeds of 100 kms
in light gas as H2 whi13h were measured and modeled A 13hoi13e was made
to revisit this type of devi13e At present the ele13tromagneti13 13oaxial plasma
guns are quite only 13onsidered as the initiators of a strong radial plasma
implosion plasma fo13us devi13es (PFD) due to a good ability to produ13e a
pin13h plasma and emission of energeti13 parti13les and hard X-ray For that
reason it was ne13essary to adapt the prin13iple of a PFD with two major
13onstraints in13rease the 13onversion of stored energy into the kineti13 energy
of a plasma sheath prevent totally any pin13hing
That obje13tive was rea13hed by dening the ele13tri13al and geometri13al
parameters of a low-indu13tan13e and 13ompa13t pulsed power generator An
optimization pro13ess has been established to mat13h a13hievable ele13tri13al pa-
rameters of the 13ir13uit (13apa13itan13e indu13tan13e peaking 13urrent intensity)
with a plasma motion in the a1313elerating tube over few 13m leading to a nal
speed of 10-30 kms A model was built up to des13ribe the dynami13s of an
RLC 13ir13uit with a varying element 13omposed of the moving plasma sheath
a1313elerated by the self generated magneti13 pressure The 13hange in plasma
position along the axis is in13reasing the indu13tan13e and the mass a1313retion
is in13reasing the plasma mass It was shown that a 13hange in the ele13trode
geometry namely a 30deg 13oni13al shape was in13reasing the energy 13onversion
and preventing any fo13using The further step in 13ontrast with PFD is the
merging of the plasma ring into a rather planar one whi13h is inje13ted in a
drift tube
Considering Ar or Xe at gas pressure in the range 01 10 mbar and
a highly 13oaxial 1-kJ devi13e the simple model was su13ient to predi13t
performan13es agreeing with the only work quoted in the literature Basi13
observations of the moving plasma using side-on dete13tion of the visible
emission give features 13oherent with the model Ma13h numbers from 20
to 60 are obtained tting with the obje13tive of strong sho13k formation A
rened 3-D MHD numeri13al simulation indi13ates very promising features of
the sho13k in view of exploring mm-size sho13ks in a regime 13omplementary
to laser driven ones and on a table-top devi13e allowing a higher repetition
rate
120 CHAPTER 6 ELECTROMAGNETICALLY LAUNCHED SHOCK
Chapter 7
Con13lusions and perspe13tives
71 Con13lusions
Although ubiquitous in the astrophysi13al environments radiative sho13ks are
13omplex phenomena whi13h still require an important eort to understand
them Beside observations whi13h suer from a la13k of angular resolution
laboratory experiments provide today an interesting approa13h to improve
our present knowledge In this 13ontext my thesis work is dedi13ated to the
experimental and numeri13al study of the 13hara13teristi13s of two 13ounter pro-
pagating radiative sho13ks propagating at dierent velo13ities (20-50 kms)
Most of the work presented here is the out13ome of the rst-of-its-kind ex-
perimental 13ampaign held in year 2015 at Prague and Laser Asterix system
(PALS) laser fa13ility
In this manus13ript I presented the setup for the aforesaid sho13k expe-
riment Following this the data analysis results interpretation as well as
relevant numeri13al simulations 13on13erning various diagnosti13s viz visible in-
terferometry and XUV spe13tros13opy have been presented Although a large
fra13tion of the experimental re13ords were performed for Xenon at various
pressures I have also studied the sho13k 13hara13teristi13s of dierent gases like
Ar Kr and He Our results are 13omplementary to those obtained at ORION
laser fa13ility also in 2015 relative to the 13ollision of identi13al radiative sho13k
waves at higher velo13ities (80 kms) In parallel in this thesis I have also
worked on the optimization of a setup dedi13ated to ele13tromagneti13ally ge-
nerated strong sho13ks at lower velo13ities In the following I briey present
a brief summary of the main results of my thesis work
In the third 13hapter I have presented a series of 1D hydrodynami13 simu-
lations (realized with the 13ommer13ial 13ode HELIOS) on isolated and 13ounter-
propagating sho13k waves at equal and dierent velo13ities The simulations
13onrm that a single radiative sho13k propagating in Xenon gas at 01 bar
is 13hara13terized by an extended pre13ursor and a large 13ompression of 30 in
the post sho13k At 50 kms the temperature evolution with the distan13e
shows identi13al post-sho13k and pre-sho13k values This indi13ates that the
numeri13al sho13k is of super13riti13al nature However I nd that there is an
important un13ertainty in the Xenon opa13ity whi13h makes the renement of
the simulation unne13essary espe13ially in terms of group numbers Next I
have investigated the intera13tion two 13ounter propagating sho13ks for iden-
ti13al (50-50 kms) and non identi13al (50-20 kms) sho13k speeds For the
121
122 CHAPTER 7 CONCLUSION
13ase of identi13al speeds the pre13ursors merge together at around 10 ns The
intera13tion is then 13hara13terized by a regular in13rease of the ele13tron density
and the temperature with the time The sho13ks 13ollision at 38 ns leads to a
jump in the ele13tron density (6 times 10
2113m
minus3) and temperature (39 eV) On
the other hand for the 13ase of non-identi13al sho13k speeds (13ase representing
our experiments) the pre13ursors intera13tion starts later than in the former
13ase The sho13ks 13ollision o1313urs at 49 ns and it is 13hara13terized by a sudden
in13rease of the ele13tron density also by an order of magnitude (4 times 10
21
13m
minus3) whereas the temperature in13reases up to 29 eV
Although the data analysis of all the experimental re13ords obtained du-
ring the PALS experimental 13ampaign has been 13arried out only a few repre-
sentative interferometri13 and spe13tros13opi13 re13ords for Xe gas are dis13ussed
in details in this thesis The interferometri13 re13ords allowed me to estimate
the average sho13k speed and time variation of the ele13tron density during the
sho13k propagation The sho13k speeds of the MAIN and AUX radiative sho13k
waves are found to be ranging between sim 30-55 and 12-25 kms respe13tively
whereas the averaged pre13ursor ele13tron density is varying between 10
17and
10
1913m
minus3during the sho13ks propagation
From the interferometri13 re13ords I have investigated the ee13t of the
intera13tion between the two radiative pre13ursors for the sho13ks propagating
in Xenon at 01 bar with respe13tive speeds of 50plusmn3 and 23plusmn3 kms The
intera13tion starts at 20 ns and is followed by the merging of the two ra-
diative pre13ursors This pre13ursor intera13tion is 13hara13terized through the
enhan13ement of the ionisation wave The sho13ks 13ollision is re13orded at 50
ns On the 13ontrary the same intera13tion behavior is not seen at 02 bar
(sho13k speeds sim 38plusmn4 and 18plusmn2 kms for the MAIN and AUX sho13k waves
respe13tively) Moreover in this 13ase there is no signature of a radiative pre-
13ursor for the AUX sho13k The pre13ursors intera13tion if any should then
13ould o1313ur at times whi13h are outside of the re13ord and 13an not be 13on-
rmed by the experiment This indi13ates that for a given gas and sho13k
speed the radiative ee13ts de13reases with the initial mass density
The investigation of the lateral extension of the sho13k has been made
through the analysis of transverse interferograms At 02 bar the MAIN
sho13k with a speed sim 40 kms has a lateral pre13ursor extension of sim570plusmn30microm whereas it is 275plusmn25 microm for AUX sho13k whi13h is propagating
with a speed of 20 kms The MAIN sho13k pre13ursor is almost of a at
spatial prole whereas the AUX pre13ursor is more 13urved suggesting that
the 2D ee13ts are mu13h more important for AUX than for MAIN One of
the explanations is that the spot size of AUX laser on the target is smaller
than the target width whereas the MAIN laser has a spot size whi13h is equal
to the target width
In order to interpret the experimental results with appropriate 1D simu-
lations I have optimized the lasers uen13es to numeri13ally obtain a sho13k
speed equal to that re13orded in the experiment In the 13orresponding simu-
72 PERSPECTIVES 123
lations for Xenon at 01 bar I note the intera13tion of the two pre13ursors
However I do not nd a good quantitative agreement for the ele13tron den-
sity Beside the question of the impre13ise opa13ities this disagreement might
also be attributed to 2D ee13ts (Gonzaacutelez Audit and Stehleacute 2009 Leygna13
et al 2006 Stehleacute et al 2010)(Cotelo et al 2015)
The analysis of spa13e- and time-integrated XUV re13ord at 06 bar is
presented for two 13ounter propagating sho13ks of speeds sim 36plusmn4 and 18plusmn5kms for MAIN and AUX respe13tively The presen13e of HeII Balmer lines
and Xenon lines tends to indi13ate a temperature of the sho13k of about 15
eV and a Xenon mean ion 13harge around 6 - 7 The 1D simulations predi13t
ele13tron temperature in the range of 10-30 eV while ion 13harge to be ranging
between 5 and 10
In addition to this study whi13h was fo13used on the Xe 13ase I have
investigated the sho13k 13hara13teristi13s for Ar Kr at 13lose mass density (sim 6
times 10
minus4g13m3) whi13h 13orrespond to 03 bar and Kr gas 02 bar Almost no
pre13ursor is found to be present in 13ase of Ar whereas a tiny pre13ursor have
been observed for Kr This indi13ates that for a given density and sho13k
speed the radiative ee13ts in13rease with the atomi13 number
In parallel I have worked on the optimization of the design of an experi-
mental setup where the sho13k is generated ele13tromagneti13ally This setup
allows studying sho13k sim 30 kms in noble gas at sim1 mbar The parameters
of a low-indu13tan13e and 13ompa13t pulsed power devi13e have been optimized
with a simple model in view of building-up and a1313elerating a plasma sheath
in Ar and Xe at gas pressure in the range 01-10 mbar Simple observations
of the moving plasma using side-on dete13tion of the visible emission give
features 13oherent with the model Sho13ks 13orresponding to ma13h numbers
ranging from 20 to 60 are obtained
72 Perspe13tives
The experimental results presented in this thesis 13orrespond to the 13ase of
two 13ounter-propagating radiative sho13ks We have been able to study the
time variations of the pre13ursor intera13tion and the ee13t of gas pressure
with 1D time dependent interferometri13 re13ords obtained with a streak 13a-
mera Together with the help of XUV spe13tros13opy we 13ould estimate the
sho13k speed ele13tron density and sho13k temperature However no informa-
tion is provided about the lateral extension of the pre13ursor and we were
not able to re13ord XUV data for all the shots Therefore in order to make
these results more 13omprehensive we have designed and parti13ipated in a
follow-up experimental 13ampaign in September 2016 at PALS This time we
fo13ussed on single sho13ks in 13lose experimental 13onditions but with a wider
range in terms of speeds (45-100 kms) We repla13ed the time dependent
streak 13amera interferometri13 study by an instantaneous 2D interferometri13
124 CHAPTER 7 CONCLUSION
image (GOI) obtained with a short pulse sub pi13ose13ond laser and 13ameras
XUV spe13tra were also re13orded The re13ords were performed at three dif-
ferent times of the sho13k propagation allowing to dedu13e the sho13k speed
and the time evolution of the pre13ursor morphology This follow up experi-
ment will allow to 13omplete the analysis of the laser generated sho13k waves
presented in this thesis
The preliminary analysis I have performed on these new re13ords is very
promising An example of these interferometri13 re13ords is given in the Figure
71a for the shot 30364 in Xe+He at 06 bar The high speed of the sho13k is
related to the energy of the laser whi13h is here 170 J whereas it was less than
120 J in the previous experiment The 13omparison with the interferometri13
re13ord (not shown) before the shot allows to dedu13e the ele13tron density
whi13h rea13hes 18 times 10
1913m
3 The high quality re13ord shows 13learly 2D
ee13ts for this sho13k wave
(a)
(b)
Figure 71 Interferometri13 re13ord obtained at 546 ns for a single radiative sho13k
of speed sim 100 kms propagating in Xe+He gas at 06 bar driven by laser at 348
nm with energy 170 J The dierent 13olors in (b) 13orrespond to yellow le Ne le35 times 10
1713m
minus3 13yan 37 - 74 times 10
1713m
minus3 blue74 - 15 times 10
1813m
minus3 green
15 - 18 times 10
1813m
minus3 magenta 18 - 22 times 10
1813m
minus3 red 22 - 26 times 10
1813m
minus3
orange26 times 10
1813m
minus3- 18 times 10
1913m
minus3
This analysis is now ongoing and the results of this experimental 13am-
72 PERSPECTIVES 125
paign will be 13ompared with that obtained from the analysis of re13ords from
2015 presented in this manus13ript This instantaneous imaging interferome-
tri13 diagnosti13 does not follow the sho13k 13ontinuously with the time but it
gives a pre13ise map of the ele13tron density whi13h 13an be inverted through
Abel inversion to obtain the lo13al estimation of the ele13tron density
In parallel in the ele13tromagneti13ally laun13hed sho13ks we have upgra-
ded our table top setup to a13hieve higher speeds For this we have used 12
13apa13itors ea13h of 1 microF at the pla13e of the previous 11 13apa13itors ea13h of
06 microF We have implemented new diagnosti13s namely a visible interferome-
ter working in the Gated Opti13al Imaging mode and time and spa13e visible
spe13tros13opy similar to the one des13ribed in the 13hapter 4 For the visible in-
terferometry we have Ma13h Zehnder Interferometer setup and we will image
the sho13k at dierent times with the help of a SL300 pi13ose13ond high energy
NdYAG laser (λ = 532 nm and energy = 024 J) The experiment will soon
be performed on this new setup and the analysis of su13h re13ords will also be
undertaken in future allowing a 13omparison of ele13tromagneti13ally laun13hed
sho13ks with the laser-driven sho13k whi13h should make the bridge between
thin and opti13ally thi13ker sho13ks
126 CHAPTER 7 CONCLUSION
Chapter 8
Thesis summary
Radiative sho13ks (RS) are present in various astrophysi13al environments es-
pe13ially in stars They are then present in all the supersoni13 a1313retioneje13tion
pro13esses during the stellar evolution from the early phase of the proto-
stellar 13ollapse up to the magnetosphere a1313retion on to the already formed
T- Tauri stars up to the late stage of the supernovae sho13ks
Radiative sho13ks are strong sho13ks (ie Ma13h number Mgtgt1) whi13hrea13h high temperatures and thus are the sour13e of intense radiation Alt-
hough the observation of (ie spatially resolved ) signatures of su13h sho13ks
in the astrophysi13al environments is very di13ult an alternate approa13h na-
med laboratory plasma astrophysi13s provides a powerful tool to study them
on the Earth
These waves have been experimentally studied sin13e more than a de13ade
mostly on large-s13ale laser fa13ilities in noble gases and with dierent targets
geometries The ablation pro13ess generated by the intera13tion between the
laser and a foil indu13es a sho13k wave in the foil whi13h then propagates
in the gas at a speed of several tens of kms Numeri13al simulations 13an
des13ribe these sho13ks with an improving pre13ision With laser intensities on
target 13omprised between 1014 and 1015 W13m
2 these experiments allowed
to re13ord sho13k speeds ranging between 40 and 150 kms
All previous experimental studies have been fo13used on the 13ase of single
radiative sho13ks However in astrophysi13al 13onditions the radiative sho13k
in general intera13ts with a denser medium leading to the development of
ree13ted and transmitted sho13ks A few representative examples of su13h
phenomena are the intera13tion of supernovae remnants with dense mole13ular
13louds the a1313retion sho13ks on the photosphere of T-Tauri stars and the bow
sho13ks at the head of fast stellar jets The 13ollision (or the intera13tion) of
two radiative sho13k waves is obviously a rare astrophysi13al event and the
template 13ase of the supernova remnants DEM L316 (see Fig 1 of Williams
et al (1997)) is still the subje13t of debates Williams et al (2005) Toledo-
Roy et al (2009) Velarde et al (2006) In this regard the development of
dedi13ated laboratory experiments to the study of propagation and intera13tion
of 13ounter-propagating sho13k waves is important as a tool to 13hara13terize
su13h events through their spe13i13 signatures
Complementary to laser experiments 13ompa13t pulsed power generators
may drive an ele13tromagneti13 13oaxial plasma gun to 13reate astrophysi13al
relevant sho13ks in lower pressure noble gases with a high availability and a
127
128 CHAPTER 8 THESIS SUMMARY
rather modest 13apital 13ost The ele13tromagneti13ally driven sho13k waves may
have larger s13ales than those by laser Thus they 13an be analyzed rather
easily with the adequate diagnosti13s (Kondo et al 2008)
In this thesis I am presenting radiative sho13ks in the 13ontext of labo-
ratory astrophysi13s using two experimental approa13hes laser-driven sho13ks
and ele13tromagneti13ally driven sho13ks
First 13hapter of my thesis is Radiative sho13k waves in whi13h the physi13s
of radiative sho13ks have been studded Radiative sho13k waves are hypersoni13
sho13k waves whi13h are heated to high temperature and as a 13onsequen13e
be13ome the sour13e of intense radiation This radiation in turn modies
the dynami13s and stru13ture of the sho13k itself whi13h makes its stru13ture
more 13ompli13ated Radiative sho13k waves o1313ur in several astrophysi13al 13ir-
13umstan13es su13h as in the a1313retion sho13k of protostellar formation (Stahler
Palla and Salpeter 1986) the supernova explosion and the intera13tion of
their remnants with the dense interstellar medium (Chevalier 1977) the bow
sho13ks at the head of stellar jets (Hartigan et al 2001)
Radiative sho13k waves 13an now be studied in the laboratory whi13h allows
to 13ompare the data with models existing in the literature and to 13he13k the
ability of the numeri13al 13odes to reprodu13e them (see for instan13e Bouquet
et al (2004) Leygna13 et al (2006)) The experimental study of radiative
sho13k 13hara13teristi13s is 13ru13ial in understanding these various energeti13 pro-
13esses o1313urring in the astrophysi13al plasmas
The jump 13onditions (Rankine-Hugoniot relations) for the simple 13ase
of an ideal gas whi13h rely on the values of the thermodynami13al quantities
on both sides of the dis13ontinuity Let us 13onsider a one-dimensional sho13k
propagating in a gas at rest with the speed us In the frame asso13iated with
the sho13k front the pre-sho13k (upstream) uid velo13ity is then u1 = - us
while behind the sho13k (post-sho13k or downstream region) this velo13ity is
u2 as shown in the Fig 21 Considering the 13ase of strong sho13ks where
M may be 13onsidered to be ≫1 jump 13onditions for ideal gas are
ρ2ρ1
=(γ + 1)
(γ minus 1)(81)
T2
T1=
2M2γ(γ minus 1)
(γ + 1)2(82)
kBT2 =2(γ minus 1)
(γ + 1)2mus
2(83)
For mono-atomi13 gases γ = 53 the 13ompression ratio is estimated to
be 4 and the post-sho13k temperature is
kBT2 =3
16mu1
2 =3
16mPAu1
2(84)
129
This temperature is thus proportional to the square of the sho13k velo13ity
and to the atomi13 mass In 13ontrast to the dis13ussion on ideal gas in the
13ase of a real gas is a little more 13ompli13ate It is to note that in this 13ase
a part of the kineti13 energy is used to ex13ite and ionize the post-sho13k gas
As a result its temperature is lower than that for the 13ase of ideal gas
In addition the pressure is also modied due to ionisation (Mi13haut et al
2004) In13luding the ee13t of ionization with the average 13harge z in the
medium for the 13ase real gas jump relations for real gas are
ρ2u2 = ρ1u1 (85)
ρ2u22 + ρ2
kBT2
m(1 + z2) = ρ1u
21 + ρ1
kBT1
m(1 + z1) (86)
ρ2u2
[5
2
(1 + z2)
mkBT2 + ǫexc2 +
u222
]
= ρ1u1
[5
2
(1 + z1)
mkBT1 + ǫexc1 +
u212
]
(87)
In the previous both 13ases the ee13ts of radiation heating and 13ooling
have been negle13ted However the radiation emitted in a strong sho13k will
ae13t its stru13ture These ee13ts 13an be determined using mass momentum
and energy 13onservation equations whi13h now have to in13lude the 13ontribu-
tions of the radiative ux energy and pressure This 13ase is more 13omplex
now Taking into a1313ount radiative ee13ts the equations of radiative hydro-
dynami13s in the 13o-moving frame of the sho13k (Mihalas and Mihalas 1999
Drake 2006) 13an be written as
partρ
partt= minusnabla(ρu) (88)
ρ
(
partu
partt+ unablau
)
= minusnabla(P + Prad) (89)
part
partt
(
ρu2
2+ ρǫ+ Erad
)
+nabla
[
ρu
(
ǫ+u2
2+
P
ρ
)
+ (Erad + Prad)u
]
= minusnablaFrad
(810)
The 13ontributions linked to Prad and Erad are ee13tive only at very high
temperature (ie velo13ity) However for the radiative sho13ks of interest in
our experiments the 13ontribution of the radiative ux Frad is non-negligible
13ompared to ρu3 These sho13ks are thus in the radiation ux dominated
regime Most of the experimental sho13k waves are in this regime These
hydrodynami13 equations are strongly non-linear
Depending on the opa13ity the radiation emitted from the sho13k may be
absorbed by the pre-sho13k region indu13ing its pre-heating Far away from
the dis13ontinuity the stru13ture of the upstream medium is determined by
the absorption On its side the stru13ture of the transition layer of the down-
stream medium 13lose to the dis13ontinuity depends on the balan13e between
130 CHAPTER 8 THESIS SUMMARY
the emission (13ooling) and the absorption (heating) Thus the full stru13ture
of the sho13k mainly depends on the variation of the opa13ity χ (13m
2g) The
opti13al depth (τ(s)) measured from the position of the jump is then the
relevant parameter for the absorption It is given by
τ(s) =
int s
sjump
χ(sprime)ρ(sprime) dsprime (811)
where sprime 13hara13terizes the path of the radiation As we are interested in
the mean frequen13y averaged radiative ux the relevant opa13ity here will be
the averaged Rosseland opa13ity whi13h will be dened in due 13ourse
If τ gtgt 1 the regime is referred as opti13ally thi13k while in the opposite
13ase (τ lt 1) it is opti13ally thin Flux dominated radiative sho13ks have
been 13lassied depending on the opti13al depth of upstream and downstream
regimes
We are more interested in the kind of sho13k have a radiative pre13ursor It
is possible to produ13e a radiative sho13k at the high energy laser fa13ilities with
laser intensity of sim 10
14W13m
2 A short summary of number of experiment
performed at high power laser fa13ilities have been listed in table 21 Other
than laser driven radiative sho13k it is possible to laun13h 13omparatively slower
radiative sho13ks (sim 10-30 kms) using ele13tromagneti13 generators
The 13hapter two is dedi13ated to the numeri13al analysis of radiative sho13k
waves and of the main physi13al pro13esses whi13h play a role in these 13omplex
pro13esses Radiative sho13ks are present in various astrophysi13al pro13esses
implying extreme 13onditions Laboratory experiments then allow investiga-
ting the underlying physi13al pro13esses whi13h take pla13e in these 13onditions
and whi13h are not observable from the Earth due to a la13k of spatial resolu-
tion However experiments are not always straightforward to interpret and
the help of numeri13al simulations be13omes pre13ious
Dierent numeri13al approa13hes are used for the simulation of laboratory
plasmas Some of them use xed grids Other introdu13e the possibility of
renement (on the y) in some meshes (Adaptative Mesh Renement) to
maintain a good resolution in the interesting parts of the plasma Another
approa13h is based on meshes following the uid (Lagrangian des13ription)
whi13h means that the meshes move with the uid and no matter may go
through a mesh to another The form of the equations dier in this 13ase (see
for instan13e Orban et al (2013))
HELIOS is su13h a one-dimensional Lagrangian radiation hydrodynami13
13ommer13ial 13ode (Ma13Farlane Golovkin and Woodru 2006) and I have
used it to simulate our experiment of intera13ting sho13k waves This 13ode
is easy to handle and has the presently interesting 13apability to be able to
simulate the 13ase of two sho13k waves propagating in opposite dire13tions and
laun13hed by two dierent lasers HELIOS may des13ribe non-LTE plasmas as
it in13ludes an in-line 13ollisional radiative (HELIOS-CR) model for 13omputing
131
non-LTE atomi13 level populations at ea13h time step of the hydrodynami13s
simulation However I used HELIOS in the LTE approximation whi13h is
appropriate to our experimental 13onditions (Rodriguez et al 2011) as will
be dis13ussed in the next se13tion
The 13ode solves the uid equations of motion using the pressure 13ontri-
butions of ele13trons ions and radiation It may des13ribe the ele13trons and
ions as two intera13ting uids at respe13tive temperatures Te and Ti Thermal
13ondu13tion is des13ribed within a ux-limited ele13tron 13ondu13tion model ba-
sed on the Spitzer 13ondu13tivity and the laser energy deposition is 13omputed
with an inverse Bremsstrahlung model
The radiation emission and absorption terms are introdu13ed in the equa-
tion of energy for the ele13trons and in the radiation transport equations One
of the methods whi13h are proposed is the ux-limited multi-group radia-
tion diusion model where the radiative ux is proportional to the gradient
of the radiative energy and is inversely proportional to the Rosseland opa-
13ity The expression is pondered by a ux-limited diusion 13oe13ient in
order to obtain the good opti13ally thin limit This 13oe13ient follows the
Larsen expression des13ribed in Olson Auer and Hall (2000) The se13ond
method is a (time independent) multi-angle short-13hara13teristi13s s13heme In
our 13ase where radiation and hydrodynami13s are strongly 13oupled and thus
the radiation eld varies rapidly we used the diusion model together with
LTE Plan13k and Rosseland multi-groups opa13ities
In addition our version of HELIOS uses the PROPACEOS (Ma13Farlane
Golovkin and Woodru 2006) Equation of State (EOS) and multi-group
opa13ity tables whi13h are generated by the 13ode
1
and it is not possible
to in13lude any other opa13ity or EOS table The knowledge of the thermo-
dynami13al 13onditions is required for an adequate simulation of the plasma
des13ription In our experiment the mass density ranges between sim 10
minus4and
10
minus1g13m
3 while the temperature values are 13omprised between and sim 0 to
50 eV The pre13ise 13al13ulations performed by Rodriguez et al (Fig2 of Rodri-
guez et al (2011)) for Xenon indi13ate that the thermodynami13al 13onditions
of our radiative sho13k experiments 13orrespond to the LTE regime
The equation of state (EOS) denes the dependen13e of the pressure
ionisation internal energy with the mass density and temperature Several
models do exist in the literature
Our version of HELIOS for LTE plasmas may handle polytropi13 or PRO-
PACEOS equation of state Unfortunately there are not many details about
this last method The authors of the 13ode mention that it uses a QEOS-type
model (More et al 1988) in the strong 13oupling regime (ie at relatively
high densities and low temperatures) and an isolated atom model whi13h
uses a detailed atomi13 stru13ture modeling in the weak 13oupling region The
properties between the weak and strong 13oupling regimes are said to be obtai-
1
see http wwwprism minus cscomSoftwarePROPACEOS
132 CHAPTER 8 THESIS SUMMARY
ned by interpolating in a thermodynami13ally 13onsistent manner This EOS
in13ludes the 13ontributions from the translation of ions and ele13trons ele13-
tron degenera13y atom ionization and ex13itation and Coulomb intera13tions
(Debye - Hu13kel 13orre13tion)
I have performed HELIOS 1D simulations for dierent 13ases for single
and 13ounter propagating sho13ks in 13hapter 2 of my thesis Radiative sho13k
waves propagating in Xenon at 01 bar with a velo13ity of 50 kms are 13ha-
ra13terized by the development of an extended radiative pre13ursor The huge
13ompression of 38 in the post-sho13k is a 13onsequen13e of both the sho13k and
the gas ionisation (fa13tor of 10) as also from the radiative 13ooling The post-
sho13k and pre-sho13k temperature on both side of the peak are identi13al
indi13ating that the sho13k is super13riti13al
The spe13ta13ular role of the opa13ity has been highlighted Due to the
un13ertainty in the opa13ity used for Xenon in our simulations I de13ided that
at this stage it was not ne13essary to rene the simulation in terms of group
numbers within the 1D des13ription of the sho13k wave
Our numeri13al study gives the main 13hara13teristi13s of the intera13tion of
two 13ounter-propagating sho13k waves with speeds equal to 50-50 kms and
50-20 kms The 13ase of identi13al speeds is simpler due to the symmetry of
the problem However whatever the speeds the most important signature
of the intera13tion is the merging of the pre13ursor at 8 ns for 50-50 kms
and at 15 ns for 50-20 kms This merging is followed by a regular in13rease
with time of the ele13tron density and the temperature The 13ollision time
is 13hara13terized by a sudden in13rease of the ele13tron density by an order of
magnitude rea13hing 66 times 10
21and 3 times 10
2113m
minus3respe13tively whereas the
temperature in13reases up to 39 and 28 eV
When a short pulse of a high-power laser beam is fo13ussed on a thin foil
it transfers a huge amount of energy to the foil material Almost instantane-
ously the laser absorption heats it up to the ablation A hot and low density
13orona is generated ba13kwards by this pro13ess Beyond the ablation surfa13e
a sho13k is formed in the foil as a 13onsequen13e of the momentum 13onservation
(ro13ket ee13t) The sho13k moves inward in the foil Our 13ase of the experi-
mental setup as the foil 13loses a tube lled with gas the sho13k propagates
then in the gas where it is studied through various kinds of diagnosti13s The
foil then a13ts as a piston
We performed the experiments at Prague Asterix Laser System (PALS)
in the Spring 2015 (ve weeks 20th April - 22th May) The obje13tive of the
experimental 13ampaign was to study the evolution of two 13ounter streaming
radiative sho13ks driven by two 13ounter-propagating lasers with an irradian13e
in the range of 10
14W13m
minus2 and therefore to 13hara13terize their intera13tion
and 13ollision In the 13hapter third 13hapter I have rst presented a brief
des13ription of the PALS laser fa13ility and of the laser beams used in our
experiment This followed by a presentation of the targets design and an
overview of the general setup and of our main diagnosti13s
133
The Prague Asterix Laser System (PALS) is a laser fa13ility based on
an Infrared high-power iodine laser system (Asterix IV) (Jungwirth et al
2001) Using dierent amplifying stages the laser fa13ility is able to deliver
energy up to 1 kJ in 03 ns at the fundamental wavelength 1315 nm The
output laser beam 13an be further subdivided in few auxiliary beams All
auxiliary beams may be frequen13y doubled (λ = 657 nm) or tripled (438
nm) The PALS laser fa13ility is 13apable of ring up to two high energy laser
shots per hour Compared with solid states lasers this gas laser is known to
deliver a quite homogenous beam intensity without hot spots
For our experiments we used two independent laser beams to drive the
two 13ounter streaming sho13ks in the gaseous target For this purpose the
fundamental PALS laser beam has been subdivided after the fourth amplier
(V4) into two beams with respe13tive energy fra13tions of 60 and 40 (see
Fig 42) The most energeti13 of these laser beams is then inje13ted through
the fth (V5) amplier and its frequen13y are then tripled This beam at 438
nm will be hereafter termed termed as the MAIN laser beam
The se13ond of the two previous laser beams propagates then without any
modi13ation It will be hereafter 13alled the AUX laser beam (1315 nm)
The s13heme of the energy distribution is shown in the Fig 42 and the
13hara13teristi13s of MAIN and AUX laser are presented in Table 41
For the experiments PALS oers two va13uum 13hambers with a spheri13al
and 13ylindri13al geometry respe13tively The spheri13al 13hamber assembly used
in our experiments is shown in Fig 43 This 13hamber having a diameter
of 100 13m is 13losed by two entran13e 13ir13ular windows of respe13tive diameters
80 13m and 50 13m to allow the MAIN and AUX beams to penetrate in the
13hamber and to rea13h the target Ea13h of these windows is outtted with a
hinged door Several 13ir13ular ports of various diameters also allow for the
dierent diagnosti13s The target holder and some diagnosti13s are mounted
on an opti13al ben13h whi13h is de13oupled from the 13hamber walls
Our targets (see Fig 45) s13hemati13ally 13onsist in small tubes of 4 mm
length 13losed on both sides by two spe13i13 foils of thi13kness 11 microm on whi13h
the laser beams are fo13used (one laser per foil) with an irradian13e of about
10
1413m
minus3 The tube is lled with gas in whi13h the radiative sho13k propa-
gates with a velo13ity of 30 - 60 kms The two foils 13losing the target insure
the 13onversion through ablation and sho13k generation of the laser energy
into me13hani13al energy
In our experiments we used spe13i13 targets for alignment and 13hara13te-
rization of the beam size These targets were manufa13tured at the Pole
instrumental of Observatoire de Paris
The gaseous targets were lled in situ at a fra13tion of 1 bar with a gas
(viz Xe Ar a mixture of Xe and He) whi13h thus provide the medium for the
sho13k propagation We 13hose an in-situ lling system (Fig 410) to redu13e
the dieren13e of pressures inside and outside of the tube This is 13riti13al as
a large pressure dieren13e between the target 13ell and the va13uum 13hamber
134 CHAPTER 8 THESIS SUMMARY
13an result in the breaking of the ultra-thin SiN windows of the target
First of all we performed fo13alization of MAIN and AUX laser beam
on target by using Al-massive targets The two PALS laser lenses were
translated up to a13hieve the suitable diameters on the target whi13h were
nally set to 450 - 500 microm and 250 - 300 microm for the MAIN and AUX beams
The size of the impa13t was 13ontrolled ea13h day on mo13k targets before using
the laser beams for real shots on gaseous targets It is worth to pre13ise that
the keV diagnosti13 was still in pla13e for the shots on the gaseous targets
In that 13ase the beams impa13t Parylene-N instead of Aluminium The keV
signal is then weaker than for the mo13k aluminum targets
Various kinds of diagnosti13s were implemented namely visible interfero-
metry XUV and visible spe13tros13opy For the same two keV 13ameras are
indu13ted to monitor the impa13t and two visible 13ameras for target alignment
For visible interferometory we have taken referen13e images (without any
fringe) for every target as shown in the Fig 417a for one target example
In order to re13ord su13h images we blo13ked the referen13e beam and let the
probe beam only to illuminate the target in a stati13 mode and with the slit
open Then we redu13ed the slit width to its nominal value (200 microm) This
enabled us to have the visualization of the portion of the target whi13h was
imaged in the slit on the target and also to lo13ate the position of the pistons
before the shot This image is termed as the referen13e target image Then
the system was pla13ed in the interferometri13 mode and the fringe pattern
was optimized to be perpendi13ular to the slit and thus in the longitudinal
mode perpendi13ular to the dire13tion of the sho13k propagation The Fig
417b reports su13h a typi13al interferogram re13orded during our experiment
(13f Fig 413) The horizontal axis 13orresponds to the dire13tion of the sho13k
propagation (with a total s13ale of 54 mm on the target) and the verti13al
axis 13orresponds to time (s13aling 200 ns)
The time and spa13e integrated XUV plasma emission are re13orded with
a Flat Field XUV spe13trometer using a 13on13ave grating In the Fig 419
the drawing of the XUV spe13trometer setup s13heme (red 13olor) is presented
together with the spheri13al 13hamber and the target The XUV spe13trometer
is installed on the top of the spheri13al 13hamber whi13h allows the XUV
emission passing through the Si3N4 membrane on the top of the target (see
se13tion 422) to be re13orded
In addition to the interferometry and XUV spe13tros13opy we have also im-
plemented a time- and spa13e-resolved visible spe13tros13opi13 (see Fig 420a)plasma
emission diagnosti13
In 13hapter 5 I present the results of the two diagnosti13s with a parti13u-
lar fo13us on the interferometri13 data whi13h I have extensively studied with
the help of a spe13i13 data analysis pro13edure that I have developed The
visible interferometry is an adequate tool to probe the ele13tron density of
the radiative pre13ursor up to the 13riti13al density (4 times 10
2113m
minus3at the wa-
velength 527 nm of the probing laser) whereas the denser post-sho13k region
135
is opaque to the visible light XUV radiation is present in the whole sho13k
stru13ture and thus the time- and spa13e-integrated XUV spe13tros13opy may
allow exploring the self-emission 13oming from the post-sho13k and pre13ursor
regions
In the visible interferometry the interferen13es between the main and
referen13e beams overlapping on the dete13tor produ13e a pattern of fringes
whi13h follow the relative phase variations between the probe (whi13h passes
through the plasma) and the referen13e beams (see se13tion 441 of the previ-
ous 13hapter for the experimental setup) The phase variation between two
13onse13utive fringes of the unperturbed beams (ie without any plasma) is
equal to 2π Then in the presen13e of the plasma in the probe beam these
fringes be13ome shifted due to the modi13ation of the refra13tive index More
details about the prin13iple of the interferometry and the Ma13h-Zehnder in-
terferometri13 setup may be found in the APPENDIX A and we re13all here
only the expression of the phase shift
∆φ asymp minus πd
λNclt Ne gt (812)
where λ = 527 nm is the wavelength of the probing laser Nc = 4 times 10
21
13m
minus3the 13riti13al density at this wavelength and lt Ne gt is the ele13tron
density averaged over the laser path d in the 13ell and is dened as
lt Ne gt=
int d
0
Ne(z t)dy
d(813)
In this 13hapter I have presented an extensive data analysis of few re-
presentative interferometri13 and spe13tros13opi13 re13ords I am presenting here
summary of results obtained bellow
The average sho13k speed and ele13tron density have been estimated from
the interferograms The sho13k speeds of the MAIN and AUX radiative sho13k
waves vary between sim 30-55 and 10-30 kms respe13tively and the averaged
pre13ursor ele13tron density ranges between 10
17and 10
1913m
minus3during the
sho13ks propagation
We have demonstrated the intera13tion ee13t between the two radiative
pre13ursors in the 13ase of Xe at 01 bar at 54 and 23 kms The intera13tion is
13learly 13hara13terized in the experiment by the enhan13ement of the ionisation
wave followed by the merging of the two radiative pre13ursors at 20 ns The
13ollision time is re13orded at 47 ns Su13h behavior is reprodu13ed by the
simulation
We have investigated this intera13tion at a larger pressure 02 bar with
the following speeds sim 41 kms for the MAIN and sim 18 kms for the AUX
sho13k waves We do not re13ord any experimental signature of the radiative
pre13ursor for AUX Further we have not been able to 13at13h experimentally
the 13ollision time The re13orded pre13ursor of MAIN is not inuen13ed by AUX
wave up to 48 ns (Fig 59(b) and (13) in data analysis 13hapter) whi13h is the
136 CHAPTER 8 THESIS SUMMARY
limit of the re13ord On its side the 1D simulation predi13ts a tiny pre13ursor
for AUX and that both pre13ursors start to intera13t at 49 ns This plausible
sho13ks intera13tion o1313urring at times whi13h are outside of the re13ord 13an
not be 13onrmed by our experiment
The results of the transverse interferometry at 02 bar with speeds of sim40 and 20 kms indi13ate that the MAIN pre13ursor has a lateral extension of
sim 600 microm whereas it is 300 microm for AUX The pre13ursor of MAIN is almost
at with a probable small bending at the edges of the tube whereas the
AUX pre13ursor is more 13urved This means that the 2D ee13ts are more
important for AUX than for MAIN Still in the 13ase of Xenon we have
obtained information about the temperature and the mean 13harge of gas
through our time integrated XUV spe13tra (shot48143) in Xenon at 06
bar From these results we may 13on13lude that the mean ion 13harge is at
least equal to 6 and that the temperature has rea13hed 15 eV
Our simulations give a qualitative des13ription of the sho13ks intera13tion
when the laser uen13e is adjusted to give the 13orre13t sho13k velo13ities Howe-
ver it is now well known that 2D simulations (together with state of the art
opa13ities) t better with experiments (Gonzaacutelez Audit and Stehleacute 2009
Leygna13 et al 2006 Stehleacute et al 2010) For the same laser energy the 2D
simulations lead to a diminution of the sho13k speed 13ompared to 1D as also
to a diminution of the ele13tron density For instan13e in the 13ase of a sho13k
wave laun13hed by a laser beam at 1315 nm in Xenon at 03 bar at PALS and
with a laser uen13e of 85000 J13m
2 ARWEN 2D simulations give a sho13k
speed of 44 kms in agreement with the experimental one (Cotelo et al
2015) 1D simulation would require in this 13ase a uen13e of 30000 J13m
2to
a13hieve the same velo13ity
The spa13e and time integrated XUV re13ords at 06 bar for respe13tive
speeds whi13h are equal to sim 39 and 18 kms indi13ate that the temperature
of the sho13k has rea13hed values up to 15 eV and that the Xenon mean ion
13harge has also rea13hed values of 6 - 7 whereas 1D simulations predi13t ele13tron
temperature 10-30 eV and ion 13harge 5-10 (Fig 51813) A more detailed
study based on 2D simulation and radiative transfer post-pro13essing will be
ne13essary to rene the analysis
We have investigated the 13ase of other noble gases (Ar Kr) and we have
observed that for a given laser energy the sho13k velo13ity de13reases with the
mass density This study has 13onrmed that for a given density and sho13k
velo13ity the radiative ee13ts in13rease with the atomi13 number For Kr we
noti13ed a tiny pre13ursor without any intera13tion To get su13h ee13t it will
be ne13essary to in13rease the sho13k velo13ity and thus the laser energy
This has been done at the ORION laser fa13ility in UK where the 13ol-
lision of two 13ounter-propagating sho13ks at equal speed sim 80 kms with
laser uen13e sim 6 times 10
14W13m
2has been performed (Clayson et al 2016
Suzuki-Vidal et al 2016) for dierent noble gases and for pressures 13ompri-
sed between 01 and 1 bar A number of diagnosti13s setup X-ray ba13klig-
137
hting opti13al self-emission streak imaging and interferometry (multi-frame
and streak imaging) were used to study both the post-sho13k and the radia-
tive pre13ursor Although I have not parti13ipated in the experiments I have
performed 1D simulations to interpret ORION experimental results This
work is not presented in this thesis
The previous 13hapters were dedi13ated to the study of laser generated
radiative sho13ks With irradian13e more than 10
14W13m
2 we were able to
study sho13k waves propagating at velo13ity up to 50 kms in noble gases with
an initial mass density 13omprised between 5 times 10
minus4- 3 times 10
minus3g13m
minus3
Complementary to laser experiments 13ompa13t pulsed power generators may
drive an ele13tromagneti13 13oaxial plasma gun to 13reate astrophysi13al relevant
sho13ks in lower pressure noble gases (Kondo et al 2006) with a high avai-
lability and a rather modest 13apital 13ost The ele13tromagneti13ally driven
sho13k waves may have larger s13ales than those by laser thus they 13an be
observed rather easily (Kondo et al 2008)
Su13h ele13tri13al pulsed power devi13es may then i) 13reate a high-voltage
breakdown through a gas or more easily at the surfa13e of a diele13tri13 ii)
produ13e relatively hot plasma by ohmi13 heating when the rising 13urrent in the
devi13e is passing through a portion of gas iii) a1313elerate plasma layers under
the magneti13 pressure asso13iated with the self-generated magneti13 eld when
high ele13tri13al 13urrent (say 10s of kA) is maintained for a short but su13ient
time (say 1 micros) Su13h ionization and a1313eleration are present in Z-pin13h
plasmas and in plasma fo13us devi13es (PFD) providing a 13lever geometry is
employed and the mass under 13onsideration (say 1 mg) is 13ompatible with
the stored energy (Martin Williams and Kristiansen 1996)
In this 13hapter we will see rst how a fast ele13tri13al 13ir13uit works then
we will dis13uss the prin13iples of a 13oaxial plasma gun and of PFD We will
see what is in favor of our obje13tive and what has to be avoided A geometry
13apable to rea13h our obje13tive will be proposed
The optimization of the 13oaxial gun (in term of plasma speed) will be
performed and some typi13al gures will be given in the 13ase of a generator
with a stored energy around 1 kJ and a plasma slab of 4-mm diameter ie
quite 10 times the transverse dimension of a laser driven RS The operating
13onditions are supersoni13 sho13ks up to 10-30 kms speed at stati13 pressures
of few mbar in heavy rare gases (Ar Xe) The results of this 0-D model will
be then 13ompared to those obtained with 3-D MHD simulations performed
with the 13ode GORGON (by Andrea Ciardi at LERMA) whi13h has been
used su1313essfully to des13ribe other pulse-power driven plasma experiments
(Chittenden et al 2004) as well as laboratory plasma astrophysi13s experi-
ments (Ciardi et al 2007) The diagnosti13s whi13h have been implemented
will be presented to illustrate the model as well as preliminary re13ords of the
plasma speed
Our aim being to a1313elerate a plasma slab using the magneti13 pressure
it is obviously needed to drive a high intensity 13urrent be13ause the magneti13
138 CHAPTER 8 THESIS SUMMARY
pressure is expressed by B
22micro0 in the region where exists an indu13tion B
The pressure in bars is simply 4B
2 with B in Teslas and in a 13ylindri13al
geometry 1 Tesla is the eld around a 50-kA 13urrent at a radius of 1 13m
Thus we should 13ount on roughly 100 kA delivered by the generator but
su13h a high 13urrent is delivered only in a pulsed mode by a laboratory s13ale
devi13e
There are numbers of te13hni13al solutions to a13hieve su13h ele13tri13al pa-
rameters one may use rather slow generators at moderate high voltage like
13apa13itors bank with the advantage of well know te13hniques but with limited
adjustments A1313ording to an abundant literature
1
a 13onvenient devi13e is a
medium-energy 13apa13itor bank feeding a low-indu13tan13e 13ir13uit An R-L-C
13ir13uit is a well-known 13ombination delivering a high peak 13urrent in the
pseudo-periodi13 mode For a 13apa13itor C initially 13harged under U0 key
performan13es are as follows
Current intensity I(t) is a damped sinusoid
Pseudo pulsation ω =radic[1(LC)minusR2(4L2)]
First 13urrent peak Ipeak = U0[radic(LC) + 08R]
Time of 13urrent peaking T4 = π(2ω)
Equivalent impedan13e Z =radic(LC)
For a safe handling in air a voltage not higher than 30 kV is re13ommen-
ded Taking a total 13apa13itan13e of 6 microF 2700 J are stored under 30 kV For
an indu13tan13e of 240 nH the expe13ted peak 13urrent might be 150 kA at 19
micros This set of values will be a guideline for the development presented in
further se13tions
For this part of the work the obje13tive was to show that in 13omple-
ment of laser experiments a 13ompa13t pulsed power generator might drive
astrophysi13ally relevant sho13ks in low pressure noble gases with a higher
availability and at a rather modest 13apital 13ost The longitudinal a1313elera-
tion of a plasma sheath in a low ba13king gas pressure has been demonstrated
in 13oaxial plasma guns as early as in the 70s even leading to speeds of 100
kms in light gas as H2 whi13h were measured and modeled A 13hoi13e was
made to revisit this type of devi13e At present the ele13tromagneti13 13oaxial
plasma guns are quite only 13onsidered as the initiators of a strong radial
plasma implosion plasma fo13us devi13es (PFD) due to a good ability to pro-
du13e a pin13h plasma and emission of energeti13 parti13les and hard X-ray For
that reason it was ne13essary to adapt the prin13iple of a PFD with two major
13onstraints in13rease the 13onversion of stored energy into the kineti13 energy
of a plasma sheath prevent totally any pin13hing
1
Institute for Plasma Fo13us Studies resour13e website httpplasmafo13usnet last
13onne13tion in 2016
139
That obje13tive was rea13hed by dening the ele13tri13al and geometri13al
parameters of a low-indu13tan13e and 13ompa13t pulsed power generator An
optimization pro13ess has been established to mat13h a13hievable ele13tri13al pa-
rameters of the 13ir13uit (13apa13itan13e indu13tan13e peaking 13urrent intensity)
with a plasma motion in the a1313elerating tube over few 13m leading to a nal
speed of 10-30 kms A model was built up to des13ribe the dynami13s of an
RLC 13ir13uit with a varying element 13omposed of the moving plasma sheath
a1313elerated by the self generated magneti13 pressure The 13hange in plasma
position along the axis is in13reasing the indu13tan13e and the mass a1313retion
is in13reasing the plasma mass It was shown that a 13hange in the ele13trode
geometry namely a 30deg 13oni13al shape was in13reasing the energy 13onversion
and preventing any fo13using The further step in 13ontrast with PFD is the
merging of the plasma ring into a rather planar one whi13h is inje13ted in a
drift tube
Considering Ar or Xe at gas pressure in the range 01 10 mbar and
a highly 13oaxial 1-kJ devi13e the simple model was su13ient to predi13t
performan13es agreeing with the only work quoted in the literature Basi13
observations of the moving plasma using side-on dete13tion of the visible
emission give features 13oherent with the model Ma13h numbers from 20
to 60 are obtained tting with the obje13tive of strong sho13k formation A
rened 3-D MHD numeri13al simulation indi13ates very promising features of
the sho13k in view of exploring mm-size sho13ks in a regime 13omplementary
to laser driven ones and on a table-top devi13e allowing a higher repetition
rate
140 CHAPTER 8 THESIS SUMMARY
Appendi13es
141
Appendix A
Visible Interferometry
Laser interferometry is used in plasmas to dedu13e the ele13tron density through
the variation of the refra13tive index 13aused by the ionisation of the matter
In this appendix rstly the experimental setup of Ma13h Zehnder interfero-
metry as well as the underlying physi13al prin13iples are presented
A01 Refra13tive index of a plasma
In order to derive the refra13tive index of a gas one starts with the set of
Maxwells equations for a mono13hromati13 plane wave propagating in the
plasma (see for instan13e equation 411 of referen13e (Hut13hinson 2002))
Free ele13trons and ions are supposed to be distributed uniformly and the net
total 13harge density is equal to zero
For harmoni13 waves propagating in the dire13tion of +x and with the
pulsation ω one obvious solution is given by
E(x t) = E0ei(kxminusωt)
(A1)
where E0 and k are 13omplex quantities
The relation between k and ω is given by the dispersion relation
k2 =εmicroω2
c2(1 + i
4πσ
ωε) (A2)
where micro is the ele13tri13 permitivity and ε is permeability
The free ele13trons are a1313elerated by the ele13tri13 eld and therefore equa-
tion their motion is given as
medv
dt= minuseE0e
minusiωt(A3)
The obvious solution for the ele13tron velo13ity v is
v = minusie
meωE (A4)
and thus the ele13trons 13arry a 13urrent with a density J
J = Neev = iNee
2
meωE (A5)
where Ne is the ele13tron density Due to the larger mass of the ions the
13orresponding 13urrent is negligible
143
144 APPENDIX A VISIBLE INTERFEROMETRY
The previous equation allows to derive the plasma 13ondu13tivity σ whi13his equal to iNee
2ωme Putting this value in the equation (A2) and assu-
ming a thin medium where ε asymp 1 and micro = 1 one obtains nally
k2 =ω2
c2(1minus
ω2p
ω2) (A6)
where ωp is the plasma frequen13y dened as
ω2p =
4πNee2
me(A7)
Equation A6 shows that a ele13tromagneti13 wave with a pulsation larger than
ωp 13ant propagate in the plasma This allows to derive a 13riti13al ele13tron
density Nc in 13m
minus3as
Nc =meω
2
4πe2=
4πmec2
e2λ2(A8)
For instan13e for a radiation of 1 microm wavelength this upper limit is equal
to Nc = 446times 1021cmminus3
Therefore the index of refra13tion is given by the expression
n =
radic
1minusω2p
ω2=
radic
1minus Ne
Nc(A9)
Finally one obtains
E(x t) = E0ei(kxminusωt) with k = n
ω
c(A10)
As a 13onsequen13e of the variation of the refra13tive index in the dierent
plasma layers the light is refra13ted Indeed the Snell laws stipulate that if a
beam is in13ident on a plane interfa13e between layers (of refra13tive index n1
and n2) it undergoes a bending from the angle of in13iden13e The in13ident
beam the refra13ted beam and the normal to the interfa13e lie in the same
plane and one has n1 sinθ1 = n2 sinθ2 where θ1 is the angle of in13iden13e
and θ2 is the angle of refra13tion Thus a mono13hromati13 beam propagating
in a plasma with in13reasing density be13omes more and more dee13ted
A02 Absorption of the laser beam
Beside refra13tion the light is absorbed by the plasma and the absorption
also in13reases rapidly near the 13riti13al density In the absen13e of any ato-
mi13 bound-bound or bound-free transition this absorption is due to the
inverse Bremsstrahlung pro13ess (ion-ele13tron 13ollision (Pfalzner 2006)) by
145
the plasma free ele13trons The absorption 13oe13ient κ (in 13m
minus1) is then
given by (NRL formulary)
κ(Ne Tev) = 317 10minus7LnΛ lt Z gt
(
Ne
ω
)2 1
T32ev (1minusNeNc)12
(A11)
where Tev is the temperature in eV Ne is the ele13tron density in 13m
minus3
and LnΛ is the Coulomb Logarithm dened as Λ = Max (2 Min(XY))
with (NRL formulary)
X = 23minus log
(
lt Z gtN
12e
T32ev
)
and Y = 24minus log
(
N12e
Tev
)
(A12)
The transmission T (d) through a layer of thi13kness d is given by
T (d) = exp(minusint d
0κ(l)dl) (A13)
For instan13e for λ = 527 nm taking Ne= 101913mminus3 ltZgt= 10 T = 10 eV
whi13h are typi13al values for the post-sho13k in our experiments in Xenon and
supposing a homogenous plasma thi13kness d = 600 microm the transmission is
about 09 whereas at 5times 1019cmminus3 it falls to 01
A03 Prin13iple of interferometry
In the opti13al interferometry te13hnique (Ovsyannikov and Zhukov 2000) the
interferen13e of two plane waves is performed by addition of the two 13orre-
sponding 13oherent ele13tri13 elds The analyzed medium (here the plasma)
in whi13h one of the beam is propagating indu13es a deshaping between the
beams whi13h ae13ts the resulting intensity depending if the elds interfere
in phase or out of phase ie 13onstru13tively or destru13tively
Two te13hniques are used the wave front and the amplitude division
The wave front division is obtained by using two portions (Fresnel mir-
rors Youngs double slit Lloyds mirror prisms et13) of the original wave
front whi13h are then superimposed to produ13e interferen13e
In the amplitude division the two beams are separated by division of
the amplitude over the same se13tion of the wave front Mi13helson and Ma13h
Zehnder interferometers et13 are the typi13al examples of this te13hnique of
amplitude division
The prin13iple of interferen13e of two 13oherent mono13hromati13 traveling
waves 1 and 2 with ele13tri13 elds E1 and E2 propagating in the same
dire13tion and polarized in the same plane 13an be understood as follows for
a given time t at the point r where these waves superpose the amplitude of
the ele13tri13 eld is the ve13tor addition of the two 13orresponding elementary
amplitudes ie
146 APPENDIX A VISIBLE INTERFEROMETRY
E(r t) = E1(r t) + E2(r t)
= a1exp(ikr + ωt+ iφ1) + a2exp(ikr + ωt+ iφ2)
where φ1 and φ2 are the phases of the two waves at r The total intensity
I(r t) = |E1 + E2|2 (A14)
One gets
I = I1 + I2 + 2radic
(I1I2)cos∆φ (A15)
where I1 = a12and I2 = a2
2are the intensities at the point of interferen13e
due to the two waves a13ting independently and ∆φ = φ1minusφ2 represents the
phase dieren13e between the two waves In the 13ase of equal intensities I1one gets
I = 4I1cos2(∆φ2) (A16)
and the interferometri13 pattern 13onsists of dark (∆φ = π + 2nπ) and
bright patterns (∆φ = 2nπ)In general the two beams are only partially 13oherent and the minimum
of the fringe intensity is not equal to zero The fringe 13ontrast
FC =Imax minus Imin
Imax + Imin(A17)
is then a measure of the interferen13e quality FC is maximum and equal to
1 in the 13ase of fully 13oherent beams of the same intensity I1 = I2
A04 Ma13h Zehnder Interferometer
The Ma13h Zehnder Interferometer is a devi13e 13ommonly and e13iently used
to estimate the 13hange in refra13tive index by the plasma Su13h 13hange is in
most the 13ases related to variations in the ele13tron density We employed
this interferometri13 setup in our experiment
The prin13iple of this interferometer is based on the division of the in13ident
light beam into two beams with the help of a beam-splitter whi13h 13an be
a 13oated glass plate or a 13ube After this division one of the beams (probe
beam) is allowed to pass through the medium of interest (a plasma in our
13ase) and then made to interfere with the other unae13ted part of the beam
(referen13e beam) with the help of se13ond beam-splitter as may be noted
from Fig A1
In order to understand the prin13iple of interferen13e and its use in esti-
mating plasma properties let us 13onsider l1 and l2 whi13h are the total path
lengths (in13luding the path in the opti13al elements) for the light travelling
from the sour13e to the dete13tor for the upper and lower paths (13f Figure A1)
respe13tively Suppose for the moment that the sample is removed from beam
147
Figure A1 S13hemati13s of Ma13h Zehnder interferometer with two mirrors two
50 beam splitters and the dete13tor A plasma slab is pla13ed in one of the two
arms of the interferometer
1 (see Figure A1) the two opti13al paths of the probe and referen13e beams
dier in terms of phases Simplify to supposing that the beams propagate
in air (n=1) these phases are then equal to the 2πl1λ and 2πl2λThus a1313ounting for the phase dieren13e δ between the two beams tra-
velled up to dete13tor A 13an be given as
2πl1 minus l2
λ= δ (A18)
If the two beams are perfe13tly parallel at the position of the dete13tor the
interferometri13 gure presents a at intensity whi13h passes through minima
and maxima when one length of the 2 arms is slightly 13hanged Usually one
introdu13es a small tilt between the two dire13tions of propagations whi13h
leads to the apparition of parallel linear dark and bright fringes
In order to obtain interferen13e patterns it is important to make the paths
l1 and l2 as 13lose as possible 13ompared to the 13oheren13e length of the laser
For the multimode Evolution 15 laser used in the PALS experiment this
length is approximately 1 13m (value 13ommuni13ated by the 13onstru13tor)
If we introdu13e the sample in the probe beam 1 this sample will introdu13e
an additional phase denoted by δsample and the net phase shift is then δ +δsample
The prin13iple of the dedu13tion of the ele13tron density with the Ma13h
Zehnder interferometer is to measure the refra13tive index of the test plasma
slab whi13h is pla13ed in the probe beam We assume for the moment that
148 APPENDIX A VISIBLE INTERFEROMETRY
this plasma slab is a phase obje13t for the in13oming beam whi13h means that
it does not signi13antly ae13t the intensity of the probe beam while it only
13hanges the phase of the wave [5]The additive phase dieren13e in the probe arm introdu13ed by the plasma
13an be dened as
φ =
int d
0kplasmadl =
int d
0nω
cdl (A19)
where n is the lo13al refra13tive index at position l of the plasma slab and
the integral is performed other the slab thi13kness d Therefore the phase
dieren13e introdu13ed by the plasma relative to the propagation into va13uum
of the referen13e beam is equal to
∆φ =
int d
0(kplasma minus k0)dl (A20)
where k0 = ωc = 2πλ is the wave ve13tor of the beam propagating in
va13uum
As long as Ne lt Nc the beam propagates in the plasma the plasma
refra13tive index is given by equation A9 and one has
∆φ =ω
c
int d
0
[
(
(1minus Ne
Nc
)12
minus 1]
dl (A21)
At very small ele13tron density (Ne ≪ Nc) the phase shift 13an be written as
∆φ asymp minus ω
2cNc
int d
0Nedl = minus πd
λNclt Ne gt (A22)
where
lt Ne gt=
int d
0
Ne(z t)dy
d(A23)
is the ele13tron density averaged over the path dIn pra13ti13e a referen13e re13ord is rst taken without plasma To dedu13e
the density one has to analyze how the fringes depart from their referen13e
positions
The spa13e between two fringes (inter-fringe) 13orresponds to a phase va-
riation of 2 π A shift of the fringe by 2 π from its referen13e position (where
Ne = 0) 13orresponds to an ele13tron density variation equal to lt Ne gt = 2
λNcd For λ = 527 nm and supposing an homogeneous plasma of thi13kness
600 microm this gives lt Ne gt= 7 times 10
1813m
minus3
Appendix B
Opa13ities and mean 13harge
The variations in logarithmi13 s13ale with the temperature of the PROPA-
CEOS Rosseland and Plan13k opa13ities (se13tion of 13hapter 5 see for intense
Fig 511)as also of the mean 13harge of the plasma are reported for two
densities 16 times 10
minus3g13m
3and 51 times 10
minus4g13m
3for three noble gases Ar
Kr and Xe in Fig B1 and B2
ρ = 51 times 10
minus4g13m
313orrespond to typi13al density of the pre13ursor
For this density Xenon and Krypton have quite similar variations for the
Rosseland opa13ity with a bump of 7000 13m
2g at 3 eV for Xe and 12000
13m
2g at 4 eV for Krypton At 10 eV the two opa13ities de13rease respe13tively
to 600 and 900 13m
2g In 13omparison with these two 13ases the opa13ity of
Argon is broader and more regular Its opa13ity peaks at 6 eV to 13000 13m
2g
At 10 eV it have de13reased to 7000 13m
2g whi13h is higher than for the two
previous gases
The mean 13harge variations in13rease regularly with the temperature At
10 eV these mean 13harges are respe13tively equal to 65 55 and 5 for Xe Kr
and Ar
149
150 APPENDIX B OPACITIES AND MEAN CHARGE
(a)
(b)
Figure B1 Rosseland and Plan13k opa13ity for three gases Xe Kr and Ar at mass
densities 51 times 10
minus4g13m
3(a) and 16 times 10
minus3g13m
3(b)
151
(a)
(b)
Figure B2 Mean 13harge for three gases Xe Kr and Ar at mass densities 16 times10
minus3g13m
3(a) and 51 times 10
minus4g13m
3(b)
152 APPENDIX B OPACITIES AND MEAN CHARGE
Referen13es
Batani D Strati F Telaro B Loumlwer T Hall T Benuzzi-Mounaix A
Koenig M 2003 Produ13tion of high quality sho13ks for equation of state
experiments European Physi13al Journal D 23 99 DOI ADS
Bernard A 2002 Plasma fo13us et appli13ations Le Vide 57(306)
Bett TH Jinks P Stevenson RM Danson CN Pepler DA Ross
IN 1995 Binary phase zone-plate arrays for laser-beam spatial-intensity
distribution 13onversion Applied Opti13s 34 4025 DOI ADS
Bouquet S Teyssier R Chieze JP 2000 Analyti13al study and stru13-
ture of a stationary radiative sho13k The Astrophysi13al Journal Supplement
Series 127(2) 245
Bouquet S Stehleacute C Koenig M Chiegraveze J Benuzzi-Mounaix A Ba-
tani D Leygna13 S Fleury X Merdji H Mi13haut C et al 2004
Observation of laser driven super13riti13al radiative sho13k pre13ursors Phy-
si13al review letters 92 225001
Bozier J Thiell G Le Breton J Azra S De13roisette M S13hirmann
D 1986 Experimental observation of a radiative wave generated in xenon
by a laser-driven super13riti13al sho13k Physi13al review letters 57(11) 1304
Burhop E Spitzer L 1957 Physi13s of fully ionized gases inters13ien13e
tra13ts on physi13s and astronomy no 3 JSTOR
Chaulagain U 2015 Radiative sho13ks expriments modellin and links to
Astrophysi13s PhD thesis Pierre and Marie Curie University
Chaulagain U Stehleacute C de Saacute L Larour J Auvray P Kozlova M
Krus M Dostal J Propupek J Suzuki-Vidal F et al 2012 Labo-
ratory experiments of radiative sho13ks in the 13ontext of stellar a1313retion
In SF2A-2012 Pro13eedings of the Annual meeting of the Fren13h So13iety
of Astronomy and Astrophysi13s 1 305
Chaulagain U Stehleacute C Larour J Kozlovaacute M Suzuki-Vidal F Bar-
roso P Cotelo M Velarde P Rodriguez R Gil JM Ciardi A
A13ef O Nejdl J de Saacute L Singh RL Ibgui L Champion N 2015
Stru13ture of a laser-driven radiative sho13k High Energy Density Physi13s
17 106 DOI ADS
Chevalier RA 1977 The intera13tion of supernovae with the interstellar
medium Annual review of astronomy and astrophysi13s 15 175 DOI ADS
153
154 REFERENCES
Chittenden JP Lebedev SV Jennings CA Bland SN Ciardi A
2004 X-ray generation me13hanisms in three-dimensional simulations of
wire array Z-pin13hesthe evolution of magneti13 tower jets in the laboratory
Plasma Physi13s and Controlled Fusion 46 B457 DOI ADS
Ciardi A 2014 private 13ommuni13ation
Ciardi A Lebedev SV Frank A Bla13kman EG Chittenden JP
Jennings CJ Ampleford DJ Bland SN Bott SC Rapley J Hall
GN Suzuki-Vidal FA Maro1313hino A Lery T Stehle C 2007 The
evolution of magneti13 tower jets in the laboratory Physi13s of Plasmas
14(5) 056501 DOI ADS
Clayson T Suzuki-Vidal F Lebedev SV Swadling GF Burdiak GC
Patankar S Smith RA Foster J Skidmore J Gumbrell E Gra-
ham P Danson C Stehleacute C Singh RL Chaulagain U Larour
J Kozlova M Spindloe C 2016 Counter-propagating radiative sho13k
experiments on the Orion laser fa13ility In APS Meeting Abstra13ts DOI
ADS
Clayson T Suzuki-Vidal F Lebedev S Swadling G Stehleacute C Bur-
diak G Foster J Skidmore J Graham P Gumbrell E et al 2017
Counter-propagating radiative sho13k experiments on the orion laser and
the formation of radiative pre13ursors High Energy Density Physi13s 23
60
Cotelo M Velarde P de la Varga AG Portillo D Stehleacute C Chaul-
again U Kozlova M Larour J Suzuki-Vidal F 2015 Simulation
of radiative sho13k waves in Xe of last PALS experiments High Energy
Density Physi13s 17 68 DOI ADS
de la Varga AG Velarde P Cotelo M de Gaufridy F
Zeitoun P 2011 Radiative properties for warm and hot
dense matter High Energy Density Physi13s 7(3) 163 DOI
httpwwws13ien13edire13t13oms13ien13earti13lepiiS1574181811000498
Diziegravere A Mi13haut C Koenig M Gregory C Ravasio A Sakawa
Y Kuramitsu Y Morita T Ide T Tanji H et al 2011 Highly
radiative sho13k experiments driven by gekko xii Astrophysi13s and Spa13e
S13ien13e 336(1) 213
Dobbs-Dixon I Cumming A Lin DNC 2010 Radiative Hydrodyna-
mi13 Simulations of HD209458b Temporal Variability The Astrophysi13al
Journal 710 1395 DOI ADS
Doss FW Robey HF Drake RP Kuranz CC 2009 Wall sho13ks in
high-energy-density sho13k tube experiments Physi13s of Plasmas 16(11)
112705
REFERENCES 155
Drake RP 2006 High-energy-density physi13s Fundamentals inertial fu-
sion and experimental astrophysi13s Sho13k wave and high pressure phe-
nomena Springer Berlin Heidelberg New York ISBN 3-540-29314-0
httpopa13inriafrre13ord=b1128433
Drake R Doss F M13Clarren R Adams M Amato N Bingham D
Chou C DiStefano C Fidkowski K Fryxell B et al 2011 Radiative
ee13ts in radiative sho13ks in sho13k tubes High Energy Density Physi13s
7(3) 130
Ensman L 1994 Test problems for radiation and radiation-hydrodynami13s
13odes The Astrophysi13al Journal 424 275 DOI ADS
Ensman L Burrows A 1992 Sho13k breakout in sn 1987a The Astrophy-
si13al Journal 393 742
Fadeyev YA Gillet D 2001 The stru13ture of radiative sho13k waves
III The model grid for partially ionized hydrogen gas Astronomy and
Astrophysi13s 368 901 DOI ADS
Fleury X Bouquet S Stehleacute C Koenig M Batani D Benuzzi-
Mounaix A Chieze J-P Grandjouan N Grenier J Hall T et al
2002 A laser experiment for studying radiative sho13ks in astrophysi13s
Laser and Parti13le Beams 20(02) 263
Fryxell B Olson K Ri13ker P Timmes FX Zingale M Lamb DQ
Ma13Nei13e P Rosner R Truran JW Tufo H 2000 FLASH An
Adaptive Mesh Hydrodynami13s Code for Modeling Astrophysi13al Ther-
monu13lear Flasheshera13les a three-dimensional radiation hydrodynami13s
13ode Astrophysi13al Journal Supplement 131 273 DOI ADS
Gonzalez JH Brollo FR Clausse A 2009 Modeling of the Dynami13
Plasma Pin13h in Plasma Fo13us Dis13harges Based in Von Karman Approx-
imations IEEE Transa13tions on Plasma S13ien13e 37 2178 DOI ADS
Gonzalez JH Clausse A Bruzzone H Florido PCMZ 2004 A Lum-
ped Parameter Model of Plasma Fo13us IEEE Transa13tions on Plasma
S13ien13e 32 1383 DOI ADS
Gonzaacutelez M Audit E Huynh P 2007 HERACLES a three-dimensional
radiation hydrodynami13s 13ode Astronomy and Astrophysi13s 464 429
DOI ADS
Gonzaacutelez M Audit E Huynh P 2007 Hera13les a three-dimensional
radiation hydrodynami13s 13ode Astronomy and Astrophysi13s 464(2) 429
Gonzaacutelez M Audit E Stehleacute C 2009 2D numeri13al study of the ra-
diation inuen13e on sho13k stru13ture relevant to laboratory astrophysi13s
Astronomy and Astrophysi13s 497 27 DOI ADS
156 REFERENCES
Gonzaacutelez M Stehleacute C Rus B Busquet M Thais F A13ef O Barroso
P Bar-Shalom A Bauduin D Kozlova M Lery T Madouri A To-
mas M Polan J 2006 Astrophysi13al radiative sho13ks From modeling
to laboratory experiments Laser and Parti13le Beams 24 535540 DOI
ADS
Gonzalez M StehlE C Audit E Busquet M Rus B Thais F A13ef
O Barroso P Bar-Shalom A Bauduin D et al 2006 Astrophysi13al
radiative sho13ks From modeling to laboratory experiments Laser and
Parti13le Beams-Pulse Power and High Energy Densities 24(4) 535
Gonzaacutelez M Vaytet N Commerccedilon B Masson J 2015 Multigroup
radiation hydrodynami13s with ux-limited diusion and adaptive mesh
renement Astronomy and Astrophysi13s 578 A12 DOI ADS
Griem HR 2005 Prin13iples of Plasma Spe13tros13opy 386 ADS
Gu MF 2008 The exible atomi13 13ode Canadian Journal of Physi13s
86(5) 675
Harilal SS Tilla13k MS 2004 Laser plasma
density measurements using interferometry
httplinkapsorgdoi101103PhysRevE64066412
Hartigan P 1989 The visibility of the Ma13h disk and the bow sho13k of a
stellar jet Astrophysi13al Journal 339 987 DOI ADS
Hartigan P 2003 Sho13k Waves in Outows from Young Stars Astrophysi13s
and Spa13e S13ien13e 287 111 DOI ADS
Hartigan P Morse JA Reipurth B Heath13ote S Bally J 2001 Pro-
per Motions of the HH 111 Jet Observed with the Hubble Spa13e Teles13ope
Astrophysi13al Journal Letters 559 L157 DOI ADS
Henke BL Gullikson E Davis JC 1993 X-ray intera13tions Photoab-
sorption s13attering transmission and ree13tion atlt igt eltigt= 50-30000
evlt igt zltigt= 1-92 Atomi13 data and nu13lear data tables 54(2) 181
Hugoniot PH 1887 Meacutemoire sur la propagation du mouvement dans les
13orps et ples speacute13ialement dans les gaz parfaits Journal de matheacutematiques
pures et appliqueacutees 3 477
Hugoniot PH 1889 Meacutemoire sur la propagation des mouvements dans les
13orps et speacute13ialement dans les gaz parfaits (deuxiegraveme partie) Journal de
lEacute13ole Polyte13hnique 58 1
Hut13hinson IH 2002 Prin13iples of Plasma Diagnosti13s 458 ADS
REFERENCES 157
Jungwirth K Cejnarova A Juha L Kralikova B Krasa J Krousky
E Krupi13kova P Laska L Masek K Mo13ek T et al 2001 The
prague asterix laser system Physi13s of Plasmas 8 2495
Keiter P Drake R Perry T Robey H Remington B Iglesias C
Walla13e R Knauer J 2002 Observation of a hydrodynami13ally driven
radiative-pre13ursor sho13k Physi13al review letters 89(16) 165003
Koenig M Faral B Boudenne JM Batani D Benuzzi A Bossi S
1994 Opti13al smoothing te13hniques for sho13k wave generation in laser-
produ13ed plasmas Physi13al Review E 50 R3314 DOI ADS
Kondo K Nakajima M Kawamura T Horioka K 2006 Compa13t pulse
power devi13e for generation of one-dimensional strong sho13k waves Review
of S13ienti13 Instruments 77(3) 036104 DOI ADS
Kondo K Nakajima M Kawamura T Horioka K 2008 Relaxation
layer in ele13tro-magneti13ally driven strong sho13ks Journal of Physi13s Con-
feren13e Series 112(4) 042028 DOI ADS
Kondo K Nakajima M Kawamura T Horioka K 2009a Atomi13 pro-
13ess in ele13tro-magneti13ally driven strong sho13k wave Nu13lear Instruments
and Methods in Physi13s Resear13h A 606 223 DOI ADS
Kondo K Nakajima M Kawamura T Horioka K 2009b Radiation ef-
fe13t and relaxation layer in ele13tro-magneti13ally driven strong sho13k waves
In APS Meeting Abstra13ts ADS
Kunze H-J 2009 Springer series on atomi13 opti13al and
plasma physi13s 125 Springer Berlin Heidelberg New York
httpwwwspringer13omseries411
Kwek K Tou T Lee S 1990 Current sheath stru13ture of the plasma
fo13us in the run-down phase IEEE Transa13tions on Plasma S13ien13e 18(5)
826
Larour J Singh RL Stehleacute C Ciardi A Chaulagain U Suzuki-Vidal
F 2015 Optimization of an ele13tromagneti13 generator for strong sho13ks
in low pressure gas High Energy Density Physi13s 17 129 DOI ADS
Lee S 1969 Transverse ionizing sho13k waves in a planar ele13tromagneti13
sho13k tube PhD thesis Australian National University
Lee S Serban A 1996 Dimensions and lifetime of the plasma fo13us pin13h
IEEE Transa13tions on plasma s13ien13e 24(3) 1101
Leygna13 S Boireau L Mi13haut C Lanz T Stehleacute C Clique C
Bouquet S 2006 Modeling multidimensional ee13ts in the propagation
of radiative sho13ks Physi13s of Plasmas (1994-present) 13(11) 113301
158 REFERENCES
Ma13Farlane JJ Golovkin IE Woodru PR 2006 HELIOS-CR A 1-
D radiation-magnetohydrodynami13s 13ode with inline atomi13 kineti13s mo-
deling Journal of Quantitative Spe13tros13opy and Radiative Transfer 99
381 DOI ADS
Ma13quorn Rankine WJ 1870 On the Thermodynami13 Theory of Waves of
Finite Longitudinal Disturban13e Philosophi13al Transa13tions of the Royal
So13iety of London Series I 160 277 ADS
Mahmood S Ur-Rehman H 2009 Ele13trostati13 solitons in unmagnetized
hot ele13tron-positron-ion plasmas Physi13s Letters A 373 2255 DOI ADS
Martin TH Williams M Kristiansen M 1996 J13 martin on pulsed
power 3 Springer S13ien13e amp Business Media
M13Clarren RG Drake RP Morel JE Holloway JP 2010 Theory of
radiative sho13ks in the mixed opti13ally thi13k-thin 13ase Physi13s of Plasmas
17(9) 093301
Mi13haut C Stehleacute C Leygna13 S Lanz T Boireau L 2004 Jump
13onditions in hypersoni13 sho13ks Quantitative ee13ts of ioni13 ex13itation
and radiation European Physi13al Journal D 28 381 DOI ADS
Mihalas D 1978 Stellar atmospheres 2nd edition ADS
Mihalas D Mihalas BW 1984 Foundations of radiation hydrodynami13s
ADS
Mihalas D Mihalas BW 1999 Foundations of radiation hydrodynami13s
More R Warren K Young D Zimmerman G 1988 A new quotidian
equation of state (qeos) for hot dense matter Physi13s of Fluids (1958-
1988) 31(10) 3059
Ogando F Velarde P 2001 Development of a radiation transport uid
dynami13 13ode under amr s13heme Journal of Quantitative Spe13tros13opy and
Radiative Transfer 71(2) 541
Olson GL Auer LH Hall ML 2000 Diusion P1 and other approx-
imate forms of radiation transport Journal of Quantitative Spe13tros13opy
amp Radiative Transfer 64 619 DOI ADS
Orban C Fatenejad M Chawla S Wilks SC Lamb DQ 2013 A
Radiation-Hydrodynami13s Code Comparison for Laser-Produ13ed Plasmas
FLASH versus HYDRA and the Results of Validation Experiments ArXiv
e-prints ADS
REFERENCES 159
Orlando S Bonito R Argiro C Reale F Peres G Mi13eli M Mat-
sakos T Stehleacute C Ibgui L de Saacute L Chiegraveze J-P Lanz T 2013
Radiative a1313retion sho13ks along nonuniform stellar magneti13 elds in 13las-
si13al T Tauri stars Astronomy amp Astrophysi13s
Ovsyannikov AA Zhukov MF 2000 Plasma diagnos-
ti13s Cambridge International S13ien13e Publishing Cambridge
httppubli13eblib13om13hoi13epubli13fullre13ordaspxp=258189
Pfalzner S 2006 An introdu13tion to inertial 13onnement fusion 1st
edn Series in Plasma Physi13s Taylor amp Fran13is Taylor amp Fran13is
httpgenlibruse13bookindexphpmd5=EACF38292FD79113F78BB532AB5301B8
Potter DE 1971 Numeri13al Studies of the Plasma Fo13us Physi13s of Fluids
14 1911 DOI ADS
Proakis JG 2001 Digital signal pro13essins Wiley Online Library
Raga A Mellema G Arthur S Binette L Ferruit P Steen W 1999
3d transfer of the diuse ionizing radiation in ism ows and the preioni-
zation of a herbig-haro working surfa13e Revista Mexi13ana de Astronomia
y Astrosi13a 35 123
Reighard AB 2007 Collapsing radiative sho13k experiments on the omega
laser PhD thesis The University of Mi13higan
Reighard A Drake R Dannenberg K Kremer D Grosskopf M Har-
ding E Leibrandt D Glendinning S Perry T Remington B et al
2006 Observation of 13ollapsing radiative sho13ks in laboratory experiments
Physi13s of Plasmas (1994-present) 13(8) 082901
Rodriguez R Gil JM Florido R Rubiano JG Mendoza MA Mar-
tel P Minguez E Symes DR Hohenberger M Smith RA 2011
Determination of the average ionization and thermodynami13 regimes of
xenon plasmas with an appli13ation to the 13hara13terization of blast waves
laun13hed in xenon 13lusters High Energy Density Physi13s 7(2) 71 DOI
httpwwws13ien13edire13t13oms13ien13earti13lepiiS157418181100022X
Rodriacuteguez R Espinosa G Gil JM Stehleacute C Suzuki-Vidal F Rubi-
ano JG Martel P Miacutenguez E 2015 Mi13ros13opi13 properties of xenon
plasmas for density and temperature regimes of laboratory astrophysi13s
experiments on radiative sho13ks Physi13al Review E 91(5) 053106 DOI
ADS
Ross IN Pepler DA Danson CN 1995 Binary phase zone plate de-
signs using 13al13ulations of far-eld distributions Opti13s Communi13ations
116 55 DOI ADS
160 REFERENCES
Seaton MJ 1987 Atomi13 data for opa13ity 13al13ulations I General des13rip-
tion Journal of Physi13s B Atomi13 and Mole13ular Physi13s 20(23) 6363
Serban A 1995 Anode geometry and fo13us 13hara13teristi13s PhD thesis
Nanyang Te13hnology University httphdlhandlenet104971568
Serban A Lee S 1995 A simple opti13al ber axial speed dete13tor Review
of S13ienti13 Instruments 66 4958 DOI ADS
Serban A Lee S 1995 A simple opti13al ber axial speed dete13tor Review
of s13ienti13 instruments 66(10) 4958
Singh JP Thakur SN (eds) 2007 Laser-indu13ed breakdown
spe13tros13opy Elsevier s13ien13edire13t
Singh R Stehleacute C Suzuki-Vidal F Kozlova M Larour J Chaulagain
U Clayson T Rodriguez R Gil J Nejdl J et al 2017 Experimental
study of the intera13tion of two laser-driven radiative sho13ks at the pals
laser High Energy Density Physi13s 23 20
Stahler SW Palla F Salpeter EE 1986 Primordial stellar evolution -
The protostar phase The Astrophysi13al Journal 302 590 DOI ADS
Stehleacute C Gonzaacutelez M Kozlova M Rus B Mo13ek T A13ef O Colom-
bier JP Lanz T Champion N Jakub13zak K et al 2010 Experi-
mental study of radiative sho13ks at pals fa13ility Laser and parti13le beams
28(02) 253
Stehleacute C Kozlovaacute M Larour J Nejdl J Champion N Barroso P
Suzuki-Vidal F A13ef O Delattre P-A Dostaacutel J et al 2012 New
probing te13hniques of radiative sho13ks Opti13s Communi13ations 285(1)
64
Stehleacute C Lefegravevre R Chaulagain U Champion N Barroso P Reix
F Jagourel P Larour J Melt13hakov E Mer13ier R et al 2014a
Criti13al 13omponents for xuv probing of laser driven sho13ks In X-Ray
Lasers 2012 Springer 239
Stehleacute C Kozlovaacute M Larour J Nejdl J Suzuki-Vidal F Cohen M
Chaulagain UP Champion N Barroso P A13ef O et al 2014b
Reply on the 13omment of the paper new probing te13hniques of radiative
sho13ks Opti13s Communi13ations 318 226
Suzuki-Vidal F Lebedev S Pi13kworth L Swadling G Burdiak G
Skidmore J Hall G Bennett M Bland S Chittenden J et al
2014 Sho13k formation in 13ounter-streaming jets on the magpie pulsed-
power generator In APS Meeting Abstra13ts 1 5005
REFERENCES 161
Suzuki-Vidal F Clayson T Swadling GF Patankar S Burdiak GC
Lebedev SV Smith RA Stehle C Chaulagain U Singh RL La-
rour J Kozlova M Spindloe C Foster J Skidmore J Gumbrell E
Graham P Danson C 2016 Counter-streaming radiative sho13k experi-
ments on the Orion laser In APS Meeting Abstra13ts ADS
Toledo-Roy JC Velaacutezquez PF de Colle F Gonzaacutelez RF Reynoso
EM Kurtz SE Reyes-Iturbide J 2009 Numeri13al model for the SNR
DEM L316 simulated X-ray emission Monthly Noti13es of the RAS 395
351 DOI ADS
Tou T Lee S Kwek K 1989 Nonperturbing plasma-fo13us measurements
in the run-down phase IEEE Transa13tions on Plasma S13ien13e 17(2) 311
Vaytet NMH Audit E Dubro13a B Delahaye F 2011 A numeri13al
model for multigroup radiation hydrodynami13s Journal of Quantitiative
Spe13tros13opy and Radiative Transfer 112 1323 DOI ADS
Vaytet N Chabrier G Audit E Commerccedilon B Masson J Ferguson
J Delahaye F 2013 Simulations of protostellar 13ollapse using mul-
tigroup radiation hydrodynami13s ii the se13ond 13ollapse arXiv preprint
arXiv13071010
Vaytet N Gonzaacutelez M Audit E Chabrier G 2013 The inuen13e
of frequen13y-dependent radiative transfer on the stru13tures of radiative
sho13ks Journal of Quantitiative Spe13tros13opy and Radiative Transfer 125
105 DOI ADS
Velarde P Gar13iacutea-Senz D Bravo E Ogando F Relantildeo A Gar13iacutea C
Oliva E 2006 Intera13tion of supernova remnants From the 13ir13umstellar
medium to the terrestrial laboratory Physi13s of Plasmas 13(9) 092901
DOI ADS
Veloso F Moreno J Tarifentildeo-Saldivia A Pavez C Zambra M Soto
L 2012a Non-intrusive plasma diagnosti13s for measuring sheath kine-
mati13s in plasma fo13us dis13harges Measurement S13ien13e and Te13hnology
23(8) 087002
Veloso F Tarifentildeo-Saldivia A Pavez C Moreno J Zambra M Soto
L 2012b Plasma sheath kinemati13s and some impli13ations on the mo-
deling of very low energy plasma fo13us devi13es Plasma Physi13s and Con-
trolled Fusion 54(9) 095007
Vis13o A Drake R Glenzer S Doumlppner T Gregori G Froula D
Grosskopf M 2012 Measurement of radiative sho13k properties by x-ray
thomson s13attering Physi13al review letters 108(14) 145001
162 REFERENCES
Williams RM Chu Y-H Di13kel JR Beyer R Petre R Smith RC
Milne DK 1997 Supernova Remnants in the Magellani13 Clouds I The
Colliding Remnants DEM L316 Astrophysi13al Journal 480 618 ADS
Williams RM Chu Y-H Di13kel JR Gruendl RA Seward FD
Guerrero MA Hobbs G 2005 Supernova Remnants in the Magella-
ni13 Clouds V The Complex Interior Stru13ture of the N206 Supernova
Remnant Astrophysi13al Journal 628 704 DOI ADS
Zambra M Kalise D Fernandez J Hernandez E Pas-
ten D Munoz V 2009 Current sheet thi13kness in the
plasma fo13us snowplow model J Plasma Fusion 8(879-882)
httpspeopleri13amoeawa13atdkalisefilesplasmapdf
Zeldovi13h IB Zeldovi13h YB Raizer YP 2002 Physi13s of sho13k waves
and high-temperature hydrodynami13 phenomena Courier Dover Publi13ati-
ons London New York Sydney
REFERENCES 163
Cho13s forts et radiatifs dinteacuterecirct pour les
environnements stellaires eacutetude expeacuterimentale et
appro13he numeacuterique
RESUME
Les 13ho13s forts sont preacutesents dans des pheacutenomegravenes astrophysiques varieacutes
De tels 13ho13s sont fortement inuen13eacutes par le rayonnement par son 13ouplage
ave13 lhydrodynamique Par suite leur topologie et leur dynamique sont
assez 13omplexes Geacuteneacuterer de tels 13ho13s hypersoniques en laboratoire dans
des 13onditions 13ontrleacutees est ainsi un outil pertinent pour eacutetudier linuen13e
du rayonnement et pour 13omparer aux reacutesultats des simulations numeacuteriques
Ces 13ho13s sont geacuteneacutereacutes par des lasers intenses et par des moyens eacutele13tromag-
neacutetiques La premiegravere partie du texte est 13onsa13reacute agrave leacutetude numeacuterique et
expeacuterimentale de lintera13tion de deux 13ho13s induits par laser se propageant
en sens 13ontraires Les expeacuterien13es ont eacuteteacute meneacutees sur linstallation laser kJ
PALS qui permet de former deux 13ho13s ave13 des vitesses propres dieacuteren-
tes (sim 30-55 et 10-25 kms respe13tivement) dans des gaz rares agrave pression
faible (moins de 1 bar) Des diagnosti13s ont eacuteteacute installeacutes interfeacuteromeacutetrie
visible spe13tros13opie visible agrave reacutesolution spatiale et temporelle spe13tros13o-
pie XUV inteacutegreacutee en temps Nos expeacuterien13es montrent une forte intera13tion
entre les deux preacute13urseurs radiatifs Les paramegravetres physiques du plasma
ont eacuteteacute deacuteduits de 13es diagnosti13s et 13ompareacutes aux reacutesultats de simulations
monodimensionnelles La se13onde partie est 13onsa13reacutee agrave la 13on13eption dune
expeacuterien13e ougrave le 13ho13 est geacuteneacutereacute de faccedilon eacutele13tromagneacutetique Loptimisation
de 13e geacuteneacuterateur est preacutesenteacutee ainsi que lenvironnement expeacuterimental per-
mettant deacutetudier des 13ho13s jusquagrave 30 kms dans des gaz rares peu denses
(1 mbar)
Mots 13leacutes 13ho13s radiatifs plasmas laser hydrodynamique spe13tros13opie
visible et XUV simulation numeacuterique astrophysique de laboratoire
physique agrave haute densiteacute deacutenergie
Reacutesumeacute long
En introdu13tion (13hapitre 1) les 13ho13s radiatifs (CR) sont preacutesenteacutes
dans de nombreux environnements astrophysiques en parti13ulier dans les
eacutetoiles Ce sont des 13ho13s forts ave13 un nombre de Ma13h eacuteleveacute (M1) et une
tregraves haute tempeacuterature 13e qui induit un rayonnement intense Alors que
lobservation de 13ette signature ave13 une reacutesolution spatiale est tregraves di13ile
en astrophysique une appro13he innovante 13elle dite des plasmas astrophy-
siques de laboratoire fournit un puissant moyen deacutetude des CR sur Terre
Depuis plus dune deacute13ennie 13es 13ho13s sont eacutetudieacutes en laboratoire prin13ipale-
ment sur de grandes installations laser dans les gaz rares et sous dieacuterentes
geacuteomeacutetries Lintera13tion entre un laser et une feuille min13e produit une
forte ablation et par eet fuseacutee a1313eacutelegravere la feuille qui se propage dans le
gaz agrave plusieurs dizaines de kms Pour des irradiations laser entre 10
14and
10
15W13m
2 on enregistre des vitesses de 13ho13s entre 40 et 150 kms En
parallegravele une des13ription ne est permise par des simulations numeacuteriques
Au deacutebut de 13e travail les expeacuterien13es eacutetaient 13on13entreacutees sur des CR
simples alors que la situation astrophysique est 13elle dun CR interagissant
ave13 un milieu plus dense donnant des 13ho13s reacuteeacute13his et transmis Des
exemples repreacutesentatifs de 13es pheacutenomegravenes sont la propagation de restes de
supernovae dans des nuages moleacute13ulaires denses les 13ho13s da1313reacutetion sur
la photosphegravere des eacutetoiles T-Tauri ou en13ore les 13ho13s deacutetrave agrave lavant
des jets stellaires La 13ollision (ou lintera13tion) de deux CR est bien sucircr un
eacutevegravenement rare en astrophysique mais un exemple est fourni par lintera13tion
de deacutebris de la supernova DEM L316 (13f Fig 1 de Williams et al (1997))
mecircme si lhypothegravese est 13ontesteacutee (Velarde et al (2006))
Il y a don13 une forte motivation pour reacutealiser une expeacuterien13e de labora-
toire pour eacutetudier la propagation et lintera13tion de deux 13ho13s fa13e agrave fa13e
et interpreacuteter les signatures observeacutees Cette appro13he 13onstitue une grande
part de mon travail En 13ompleacutement aux expeacuterien13es laser jai eacutetudieacute un
geacuteneacuterateur 13ompa13t de puissan13e pulseacutee alimentant un 13anon agrave plasma 13oax-
ial pour 13reacuteer des 13ho13s dinteacuterecirct astrophysique dans les gaz agrave basse pression
Les 13ho13s ainsi 13reacuteeacutes sont de plus grande taille et peuvent ecirctre eacutetudieacutes plus
fa13ilement (Kondo et al 2008) Pour 13ette thegravese je me suis atta13heacutee agrave preacute-
senter les 13ho13s radiatifs dans le 13ontexte de lastrophysique de laboratoire
en suivant 13es deux appro13hes
Le deuxiegraveme 13hapitre traite de la physique des CR Ils 13omposent
une 13lasse de 13ho13s supersoniques qui sont 13haueacutes agrave haute tempeacuterature
et sont par 13onseacutequent sour13es dun rayonnement intense En retour 13e
rayonnement modie la dynamique et la stru13ture du 13ho13 13e qui 13omplexie
la des13ription Les CR se ren13ontrent dans des situations astrophysiques
13omme les 13ho13s da1313reacutetion lors de la formation des protoeacutetoiles (Stahler
Palla et Salpeter 1986) lexplosion de supernovae et lintera13tion de leur
restes ave13 le milieu interstellaire dense (Chevalier 1977) ou en13ore les 13ho13s
deacutetrave en tecircte des jets stellaires (Hartigan et al 2001) Les CR sont
maintenant eacutetudieacutes en laboratoire 13e qui permet de 13omparer les reacutesultats
aux modegraveles de la litteacuterature et de veacuterier la possibiliteacute de les retrouver ave13
les simulations numeacuteriques disponibles (Bouquet et al (2004) Leygna13 et
al (2006)) Leacutetude expeacuterimentale des CR est 13ru13iale notamment pour
13omprendre les pro13essus eacutenergeacutetiques au sein des plasmas astrophysiques
Dans le 13as des gaz parfaits le saut au niveau dun 13ho13 est deacute13rit par les
eacutequations de Rankine-Hugoniot qui relient les quantiteacutes thermodynamiques
de part et dautre de la dis13ontinuiteacute La solution monodimensionnelle dun
13ho13 se propageant agrave la vitesse us dans un gaz au repos est usuellement
deacute13rite dans le repegravere lieacute au 13ho13 et on distingue la reacutegion amont (ou preacute-
13ho13) agrave la vitesse u1= - us et en arriegravere du 13ho13 (reacutegion post-13ho13 ou aval)
la vitesse est u2 selon la Fig 21 Pour les 13ho13s forts ougrave le nombre de Ma13h
M est tregraves grand devant 1 les sauts des quantiteacutes thermodynamiques dun
gaz parfait sont donneacutes par les eacuteq 21 agrave 24 Pour un gaz monoatomique
le rapport de 13ompression est de 4 et la tempeacuterature de la zone post-13ho13
est proportionnelle au 13arreacute de la vitesse du 13ho13 et agrave la masse atomique
La situation est plus 13omplexe pour un gaz reacuteel 13ar une part de leacutenergie
13ineacutetique sert agrave ex13iter et agrave ioniser le gaz du post 13ho13 La tempeacuterature
attendue est don13 infeacuterieure agrave 13elle du 13as ideacuteal en outre la pression est
modieacutee par lionisation (Mi13haut et al 2004) Un nouveau jeu deacutequations
in13orpore la 13harge ee13tive Z des ions (eacuteq 25 agrave 28) Ces modegraveles ignorent
le 13hauage et le refroidissement radiatifs alors que le rayonnement eacutemis
ae13te la stru13ture dun 13ho13 fort Cette des13ription demande dajouter aux
eacutequations de 13onservation de la masse quantiteacute de mouvement et eacutenergie les
13ontributions radiatives (ux eacutenergie et pression Le 13as est plus 13omplexe
et les eacutequations dhydrodynamique radiative ont eacuteteacute exprimeacutees dans le repegravere
du 13ho13 (Mihalas and Mihalas 1999 Drake 2006) selon les eacuteq 29 agrave 211
Les 13ontributions de la pression radiative et du 13hamp de rayonnement
sont importantes agrave tregraves haute tempeacuterature (vitesse) mais deacutejagrave pour nos 13ho13s
le ux radiatif est dominant Selon lopa13iteacute le rayonnement du 13ho13 peut
ecirctre absorbeacute par la region preacute-13ho13 induisant un 13hauage loin en avant de
la dis13ontinuiteacute Egalement la zone 13hoqueacutee pro13he de la dis13ontinuiteacute est
ae13teacutee par la 13ompeacutetition entre leacutemission (refroidissement) et labsorption
(13hauage) Ainsi leacutepaisseur optique lo13ale (eacuteq 211) devient le paramegravetre
pertinent Comme on raisonne sur le ux radiatif moyenneacute en freacutequen13e
13est lopa13iteacute de Rosseland qui est utiliseacutee et on dieacuteren13ie les reacutegimes op-
tiquement eacutepais et optiquement min13e 13e qui permet de 13lasser les 13ho13s
radiatifs
Notre inteacuterecirct sest 13on13entreacute sur les 13ho13s preacutesentant un preacute13urseur radi-
atif En laboratoire une intensiteacute laser au dessus de 10
14W13m
2le permet
et les installations sont rappeleacutees dans le tableau 21 Il est aussi possible de
lan13er des 13ho13s moins rapides don13 peu radiatifs (sim 10-30 kms) ave13 un
piston eacutele13tromagneacutetique
Le 13hapitre 3 est 13onsa13reacute agrave lanalyse numeacuterique des CR en deacute13rivant
les pro13essus physiques jouant un rle dans 13es systegravemes physiques De mecircme
que les observations les expeacuterien13es en laboratoire ne sont pas dire13tement
interpreacutetables et la simulation numeacuterique devient un outil preacute13ieux Diverses
appro13hes numeacuteriques sont utiliseacutees pour simuler les plasmas de laboratoire
On 13onsidegravere soit des grilles xes soit des grilles adaptables (Adaptative
Mesh Renement) pour maintenir une reacutesolution susante sur 13haque zone
du plasma Une autre appro13he utilise des grilles qui suivent le uide dans
son mouvement (modegravele Lagrangien) don13 sans passage de matiegravere dune
13ellule agrave la voisine Les eacutequations sont alors dieacuterentes (Orban et al 2013)
Pour deacute13rire les 13ollisions de 13ho13s jai utiliseacute le 13ode 13ommer13ial HE-
LIOS un 13ode Lagrangien mono-dimensionnel pouvant geacuterer le rayonnement
et lhydrodynamique (Ma13Farlane Golovkin et Woodru 2006) Assez fa13ile
agrave prendre en main il a le gros avantage de pouvoir simuler notre 13ongura-
tion deux 13ho13s de vitesses opposeacutees lan13eacutes par deux lasers HELIOS deacute13rit
les plasmas hors deacutequilibre thermodynamique (hors ETL) gracirc13e agrave un mo-
degravele 13ollisionnel-radiatif (HELIOS-CR) de 13al13ul des populations atomiques
agrave 13haque pas de simulation hydrodynamique De fait jai utiliseacute HELIOS
dans une approximation ETL qui 13orrespond aux 13onditions expeacuterimentales
(Rodriguez et al 2011) et 13ela est dis13uteacute en deacutetails
Le 13ode reacutesout les eacutequations uides en introduisant les pressions des eacutele13-
trons des ions et du rayonnement Ele13trons et ions sont deacute13rits par deux
uides en intera13tion de tempeacuteratures respe13tives Te et Ti La 13ondu13tion
thermique est geacutereacutee par 13elle des eacutele13trons agrave partir de la 13ondu13tiviteacute de
Spitzer et le deacutept deacutenergie laser par un modegravele de Bremsstrahlung inverse
Leacutemission et labsorption du rayonnement sont introduites dans leacutequation
deacutenergie des eacutele13trons et dans 13elles du transport radiatif Une meacutethode
utilise un modegravele de diusion du rayonnement multi-groupes et agrave ux li-
miteacute ougrave le ux radiatif est proportionnel au gradient de leacutenergie radiative
et inversement proportionnel agrave lopa13iteacute de Rosseland ave13 une pondeacuteration
pour obtenir une bonne des13ription du 13as optiquement min13e selon Olson
Auer et Hall (2000) Une se13onde meacutethode utilise un s13heacutema deacutependant du
temps agrave 13ourte distan13e et multi-angles Dans notre 13as ougrave rayonnement
et hydrodynamique sont fortement 13oupleacutes et aussi en variation rapide on
utilise le modegravele de diusion et les opa13iteacutes ETL multi-groupes de Plan13k et
Rosseland
En outre notre version dHELIOS utilise PROPACEOS (Ma13Farlane
Golovkin et Woodru 2006) une table deacutequation deacutetat et dopa13iteacute multi-
groupes geacuteneacutereacutee par le 13ode sans possibiliteacute dintroduire une autre table La
13onnaissan13e des 13onditions thermodynamiques est neacute13essaire pour la simu-
lation du plasma dans nos 13as masse volumique entre 10
minus4et 10
minus1g13m
3
tempeacuterature entre 0 et 50 eV Une eacutevaluation preacute13ise par Rodriguez et al
(13f Fig 2 de Rodriguez et al 2011) pour le Xeacutenon indique que les 13onditi-
ons thermodynamiques de nos CR 13orrespondent au reacutegime ETL Leacutequation
deacutetat relie pression ionisation et eacutenergie interne agrave la masse volumique et agrave
la tempeacuterature Notre version dHELIOS pour plasma ETL utilise PROPA-
CEOS qui est peu do13umenteacutee elle utilise le modegravele de More et al (1988)
en reacutegime de 13ouplage fort (haute densiteacute et basse tempeacuterature) et un modegravele
datome isoleacute pour un 13ouplage faible Des interpolations sont in13luses pour
passer au 13ouplage fort (httpwwwprism-13s13omSoftwarePROPACEOS)
Jai reacutealiseacute des simulations HELIOS 1D pour divers 13as de CR 13ho13 seul
ou 13ho13s en 13ollision qui sont deacute13rites dans 13e 13hapitre 3 Les CR se propa-
geant dans le Xe agrave 01 bar et agrave 50 kms sont 13ara13teacuteriseacutes par un preacute13urseur
radiatif eacutetendu Une 13ompression eacuteleveacutee (38) dans le post-13ho13 est attribueacutee
agrave lionisation du gaz et au refroidissement radiatif Les tempeacuteratures du
post-13ho13 et du preacute-13ho13 de part et dautre du pi13 sont identiques signe
dun 13ho13 super13ritique Le rle spe13ta13ulaire de lopa13iteacute a eacuteteacute remarqueacute
mais agrave 13onsideacuterer lin13ertitude sur lopa13iteacute du Xeacutenon et le traitement 1D
jai deacute13ideacute de ne pas 13her13her agrave raner la simulation en termes de nombre
de groupes
Leacutetude numeacuterique permet de retrouver les 13ara13teacuteristiques de lintera13tion
de deux 13ho13s opposeacutes ave13 des vitesses respe13tives 50-50 kms et 50-20 kms
Dans tous les 13as la prin13ipale signature de lintera13tion (don13 avant la 13ol-
lision) est le re13ouvrement des preacute13urseurs agrave t0+8 ns pour 50-50 kms et
agrave t0+15 ns pour 50-20 kms Ensuite on assiste agrave une remonteacutee reacuteguliegravere
de la densiteacute et de la tempeacuterature eacutele13troniques Linstant de la 13ollision
voit une augmentation soudaine de la densiteacute eacutele13tronique (par un ordre de
grandeur) qui atteint 66 times 10
21et 3 times 10
2113m
minus3respe13tivement alors que
la tempeacuterature monte agrave 39 et 28 eV
Dans le quatriegraveme 13hapitre je preacutesente linstallation laser PALS puis
la 13ellule dintera13tion son implantation et les diagnosti13s asso13ieacutes Rappe-
lons le prin13ipe expeacuterimental Quand une impulsion bregraveve de lumiegravere laser
est fo13aliseacutee sur une feuille min13e une part importante deacutenergie est trans-
feacutereacutee et le 13hauage du solide 13onduit agrave son ablation Un plasma 13oronal
13haud et peu dense est eacuteje13teacute vers larriegravere et un 13ho13 vers lavant se forme
dans la feuille par eet fuseacutee Comme 13ette feuille limite le tube le 13ho13
se transmet au gaz la feuille agissant 13omme un piston Des expeacuterien13es
ont eacuteteacute meneacutees sur le Prague Asterix Laser System (PALS) au printemps
2015 (5 semaines 20 avril - 22 mai) ave13 pour obje13tif la premiegravere eacutetude
de lintera13tion puis de la 13ollision de deux CR sous la13tion de deux lasers
au niveau de 10
14W13m
2 Le Prague Asterix Laser System (PALS) repose
sur un laser agrave iode infrarouge (Asterix IV Jungwirth et al 2001) Apregraves
ampli13ation il deacutelivre jusquagrave 1 kJ en 03 ns agrave la longueur donde fonda-
mentale de 1315 nm Des fais13eaux auxiliaires sont disponibles agrave freacutequen13e
doubleacutee (λ = 657 nm) ou tripleacutee (438nm) PALS deacutelivre deux tirs par heure
agrave haute eacutenergie et dans de bonnes 13onditions dhomogeacuteneacuteiteacute du fais13eau
Pour nos expeacuterien13es nous utilisons deux fais13eaux Le fais13eau fondamental
est diviseacute apregraves le 4egraveme ampli13ateur en deux fais13eaux deacutenergie 60 et
40 (voir Fig 42) Le plus puissant est inje13teacute dans le 5
meampli13ateur et
sa freacutequen13e est tripleacutee Ce fais13eau agrave 438 nm est nommeacute MAIN Le se13ond
fais13eau est utiliseacute sans modi13ation don13 agrave 1315 nm et est nommeacute AUX
Le s13heacutema de distribution est en Fig 42 et les proprieacuteteacutes de MAIN and
AUX laser sont rappeleacutees Tdans le ableau 41
PALS propose deux 13hambres agrave vide spheacuterique et 13ylindrique respe13ti-
vement nous avons utiliseacute la 13hambre spheacuterique (Fig 43) Dun diamegravetre
de 100 13m elle est a1313essible aux fais13eaux MAIN et AUX par deux hublots
de 80 13m et 50 13m Des portes sont ameacutenageacutees ainsi que de nombreux ports
A linteacuterieur la 13ellule et des diagnosti13s sont monteacutes sur une table optique
deacute13oupleacutee meacute13aniquement de la 13hambre Les 13ibles (Fig 45) 13omprennent
un petit tube long de 4 mm fermeacute agrave ses deux extreacutemiteacutes par des feuilles min-
13es de parylegravene-N doreacute de 11 microm sur lesquelles les deux lasers sont fo13aliseacutes
au niveau de 10
14W13m
minus2 Le tube est rempli du gaz dans lequel le 13ho13
se propagera agrave une vitesse attendue entre 30 et 60 kms Les feuilles min13es
assurent la 13onversion de leacutenergie laser en eacutenergie 13ineacutetique via lablation et
la geacuteneacuteration de 13ho13 Des 13ellules speacute13iques en dural massif permettent
lalignement et la mesure du diamegravetre du fais13eau Toutes les 13ellules sont
reacutealiseacutees par le Ple instrumental de lObservatoire de Paris
Les 13ellules agrave gaz sont remplies in situ agrave une fra13tion de bar ave13 Xe
Ar Xe90He10 et He permettant ainsi de bien 13onnaicirctre le milieu ougrave se
propagent les 13ho13s Le systegraveme de remplissage (Fig 410) permet de limiter
leacute13art de pression sur les fenecirctres de la 13ellule et permet de suivre la pression
jusquau moment du tir Cest 13ritique pour ne pas 13asser la fenecirctre ultra
ne de Si3N4 pour la spe13tros13opie XUV et pour geacuterer une fuite eacuteventuelle
Le reacuteglage de la position et de la fo13alisation permet dobtenir des ta13hes
laser 13entreacutees de diamegravetres 450 - 500 microm et 250 - 300 microm pour MAIN et
AUX respe13tivement Ce 13ontrle est reacutepeacuteteacute 13haque jour Les impa13ts sont
toujours suivis par des 13ameacuteras X au keV mecircme si limpa13t sur la feuille de
parylegravene-N donne un signal X faible 13ompareacute agrave 13elui du tir sur Al
Les diagnosti13s du plasma sont prin13ipalement linterfeacuteromeacutetrie visible et
la spe13tros13opie XUV et visible Lalignement est suivi par des 13ameacuteras dans
le visible Pour linterfeacuteromeacutetrie visible preacute13iseacutement on doit prendre une
image de reacutefeacuteren13e (sans franges) de 13haque 13ible (Fig 417a) Le fais13eau de
reacutefeacuteren13e de linterfeacuteromegravetre Ma13h-Zehnder est bloqueacute et le fais13eau sonde est
utiliseacute seul La fente est ensuite reacuteduite agrave 200 microm pour ne sonder que la zone
13entrale du tube mais seacutetendant le long de laxe jusquaux pistons pour avoir
une reacutefeacuteren13e des positions initiales Ensuite en mode interfeacuterogramme on
13reacutee un systegraveme de franges perpendi13ulaires agrave la fente don13 perpendi13ulaires
agrave la dire13tion de propagation du 13ho13 (13f Fig 417b et Fig 413) Le tube
de 13ho13 est imageacute sur la 13ameacutera ave13 un grandissement susant (54 mm) et
il est possible dassurer un balayage eacutele13tronique de la fente sur le deacutete13teur
jusquagrave 200 ns Un interfeacuterogramme dit streak permet ainsi de suivre limpa13t
des 13ho13s sur les franges en fon13tion du temps
Leacutemission XUV du plasma est enregistreacutee ave13 une inteacutegration spatiale
et temporelle sur un spe13trographe agrave 13hamp plan muni dun reacuteseau 13on13ave
Il est installeacute dans la partie haute de la 13hambre (13f Fig 418) et enregistre
leacutemission XUV traversant la fenecirctre de Si3N4 Une spe13tros13opie visible
reacutesolue en temps et en position longitudinale a eacuteteacute installeacutee (Fig 419a)
mais na pas en13ore pu ecirctre exploiteacutee 13omplegravetement
Le 13hapitre 5 preacutesente les reacutesultats des deux diagnosti13s surtout linterfeacute-
romeacutetrie qui a eacuteteacute 13omplegravetement analyseacutee Jai deacuteveloppeacute une pro13eacutedure
de deacutepouillement ave13 des outils numeacuteriques ad ho13 de 13al13ul et danalyse
dimage Certes linterfeacuteromeacutetrie ne permet pas de sonder des plasmas plus
denses que la densiteacute 13ritique imposeacutee par le laser (4 times 10
2113m
minus3agrave 627 nm)
et on 13onstate que le post-13ho13 est opaque Au 13ontraire le rayonnement
XUV est preacutesent dans toute la stru13ture et la spe13tros13opie XUV inteacutegreacutee
en temps et en espa13e permet dexplorer agrave la fois le post-13ho13 et le preacute13ur-
seur Lanalyse des interfeacuterogrammes streak en visible part dune eacutevaluation
de leet de lindi13e de reacutefra13tion des eacutele13trons deacutetailleacutee dans lappendi13e A
mais qui se heurte assez vite agrave une perte de 13ontraste et agrave une disparition
des franges bien en dessous de la valeur de la densiteacute 13ritique deacutejagrave menti-
onneacutee Les deacutephasages a1313essibles ave13 preacute13ision ne deacutepassent guegravere 2 ou 3
fois 2π La re13her13he de maxima des franges est semi automatiseacutee et permet
darriver agrave une densiteacute eacutele13tronique moyenneacutee sur les 600 microm de leacutepaisseur
du tube de 13ho13 Le prol non plan du 13ho13 13onduit agrave penser que la moyenne
sous-estime dun fa13teur 2 voire plus la densiteacute dans le preacute13urseur La dis-
parition des franges est attribueacutee agrave la preacutesen13e du front de 13ho13 Par suite
les interfeacuterogrammes permettent de mesurer des vitesses de 13ho13s de 30-55
et 10-30 kms pour MAIN et AUX respe13tivement ainsi que des densiteacutes
eacutele13tronique dans les preacute13urseurs entre 10
17et 10
1913m
minus3
Nous avons ainsi deacutemontreacute lintera13tion entre deux preacute13urseurs radiatifs
dans Xe at 01 bar pour deux 13ho13s agrave 54 et 23 kms Lintera13tion est
13lairement 13ara13teacuteriseacutee par laugmentation de londe dionisation puis par la
superposition des deux preacute13urseurs agrave t0+20ns La 13ollision est enregistreacutee agrave
t0+47 ns un reacutesultat retrouveacute par la simulation
Nous avons reacutepeacuteteacute mais agrave plus haute pression 02 bar et trouveacute des
vitesses de sim 41 kms pour MAIN et sim 18 kms pour AUX Cependant
nous navons pas de signature du preacute13urseur radiatif pour AUX et le temps de
13ollision na pas eacuteteacute a1313essible agrave lenregistrement Le preacute13urseur 13teacute MAIN
nest pas inuen13eacute par AUX jusquagrave la limite de t0+48 ns (Fig 59(b) et (13))
La simulation preacutedit un tregraves faible preacute13urseur 13teacute AUX et une intera13tion
des preacute13urseurs agrave t0+49 ns
Linterfeacuteromeacutetrie transverse agrave 02 bar ave13 des vitesses sim 40 et 20 kms
indique que le preacute13urseur 13teacute MAIN a une extension lateacuterale de sim 600 microm13ontre 300 microm pour AUX Le preacute13urseur de MAIN est leacutegegraverement ae13teacute
pregraves des parois alors que 13elui de AUX est fortement 13ourbeacute Les eets 2D
sont don13 pronon13eacutes pour AUX et faibles pour MAIN En outre pour le
Xeacutenon nous disposons de la tempeacuterature et de la 13harge moyenne par le
spe13tre XUV inteacutegreacute (tir48143 Xeacutenon 06 bar) On peut en 13on13lure que
la 13harge moyenne est au moins eacutegale agrave 6 et quon a atteint une tempeacuterature
de 15 eV
Nos simulations donnent une des13ription qualitative des CR en inte-
ra13tion agrave la reacuteserve pregraves que nous introduisions dans HELIOS une uen13e
laser ajusteacutee pour donner la bonne vitesse Cependant il est maintenant
admis que les simulations 2D (ave13 les opa13iteacutes 13orre13tes) deacute13rivent bien les
expeacuterien13es (Gonzaacutelez Audit et Stehleacute 2009 Leygna13 et al 2006 Stehleacute
et al 2010) A la mecircme eacutenergie laser le 13al13ul 2D 13onduit agrave diminuer la
vitesse du 13ho13 13ompare au 13al13ul 1D et aussi agrave une baisse de la densiteacute
eacutele13tronique Par exemple pour un 13ho13 lan13eacute agrave PALS par un laser agrave 1315
nm dans le Xeacutenon agrave 03 bar ave13 une uen13e de 85000 J13m
2 ARWEN 2D
donne une vitesse de 44 kms en a1313ord ave13 lexpeacuterien13e (Cotelo et al
2015) La simulation 1D demanderait 30000 J13m
2pour obtenir la mecircme
vitesse
De mecircme la spe13tros13opie XUV inteacutegreacutee agrave 06 bar pour des vitesses reacuteel-
les de sim 39 et 18 kms indique que la tempeacuterature a atteint 15 eV et que
la 13harge moyenne a atteint 6 ou 7 alors que la simulation 1D preacutedit 10-30
eV et 5-10 respe13tivement (Fig 518) On peut en 13on13lure quune eacutetude
deacutetailleacutee baseacutee sur des simulations 2D et un post-traitement du transfert de
rayonnement sont neacute13essaires pour raner lanalyse Pour les autres gaz
rares (Ar Kr) on a observeacute que agrave eacutenergie laser donneacutee la vitesse de CR est
une fon13tion deacute13roissante de la masse volumique Ce13i 13onrme que pour
une densiteacute et une vitesse les eets radiatifs augmentent ave13 le numeacutero
atomique Pour le Krypton on a observeacute un petit preacute13urseur mais au13une
intera13tion Celle-13i ne serait possible quen augmentant nettement la vi-
tesse don13 leacutenergie du laser Ce13i a eacuteteacute rendu possible sur linstallation
laser Orion (AWE Aldermaston G-B) ougrave la 13ollision de deux CR deacutegale
vitesse a eacuteteacute obtenue agrave sim 80 kms pour une uen13e laser sim 6 times 10
14W13m
2
(Clayson et al 2016 Suzuki-Vidal et al 2016) pour une large gamme de
gaz rares et des pressions entre 01 et 1 bar Pour eacutetudier agrave la fois les 13ho13s
et les preacute13urseurs radiatifs de nombreux diagnosti13s eacutetaient installeacutes radi-
ographie X imagerie optique en 13ameacutera agrave balayage de fente interfeacuteromeacutetrie
multi-vues et agrave balayage de fente Bien que je naie pas pu parti13iper aux
13ampagnes jai fait des simulations 1D pour interpreacuteter les reacutesultats Cet
aspe13t nest pas deacute13rit dans 13ette thegravese mais a eacuteteacute publieacute tregraves reacute13emment
(Clayson et al 2017)
Apregraves des 13hapitres 13onsa13reacutes aux 13ho13s radiatifs geacuteneacutereacutes par laser le
13hapitre 6 deacute13rit un moyen alternatif passant par la voie eacutele13trique agrave haute
puissan13e Les 13hires de la voie laser sont tregraves eacuteleveacutes en uen13e plus de
10
14W13m
2 mais repreacutesentent des eacutenergies modeacutereacutees dans limpulsion (120
et 60 J pour MAIN et AUX au PALS) Or un systegraveme eacutele13trique devant
deacutelivrer 100 J reste modeste pour sa partie sto13kage Les vitesses de 50
kms seront probablement di13iles agrave atteindre mais on peut tabler sur un
reacutegime 13ompleacutementaire des 13ho13s radiatifs ave13 plus de 10 kms si le ren-
dement de 13onversion en eacutenergie 13ineacutetique reste 13orre13t Dougrave un neacute13essaire
travail doptimisation Des vitesses jusquagrave 100 kms avaient eacuteteacute mesureacutees
tregraves tt dans des 13anons agrave plasma dhydrogegravene agrave basse pression (Lee 1969)
mais 13e13i a eacuteteacute exploiteacute ensuite pour dautres naliteacutes La litteacuterature ne
mentionne quun 13as de 13anon eacutele13tromagneacutetique 13oaxial deacutedieacute aux 13ho13s
dinteacuterecirct astrophysique (Kondo et al 2006) Il est signaleacute que le lan13eur
eacutele13tromagneacutetique produit des 13ho13s plans de grande dimension transverse
(1 ordre de grandeur par rapport agrave la voie laser) et don13 plus fa13iles agrave in-
vestiguer (Kondo et al 2008) Le prin13ipe de fon13tionnement est le suivant
i) 13reacuteation dun 13laquage haute tension initial sur une surfa13e dieacutele13trique
fa13e agrave du gaz ii) 13hauage ohmique du plasma par le 13ourant en 13roissan13e
rapide iii) a1313eacuteleacuteration de la 13ou13he de plasma sous la pression magneacuteti-
que auto-geacuteneacutereacutee par le 13ourant (dizaines de kA) iv) maintien (1 micros) de la
pression magneacutetique pour 13ommuniquer une impulsion importante Des tra-
vaux sur le Z-pin13h ou le plasma fo13us permettent de postuler des geacuteomeacutetries
13onvenables pour a1313eacuteleacuterer une masse de lordre de 10
minus510
minus4g
Apregraves avoir eacutetudieacute le fon13tionnement dun 13ir13uit eacutele13trique rapide jai
listeacute les points forts et les points faibles en fon13tion des obje13tifs et jai pro-
poseacute une geacuteomeacutetrie 13oaxiale 13ompa13te Loptimisation du 13anon agrave plasma a
eacuteteacute faite en terme de vitesse maximale agrave la sortie ave13 13omme seules 13ontrain-
tes une eacutenergie sto13keacutee de 1 kJ une dimension transverse de 4 mm et une
dimension axiale dun ordre plus grande Lobje13tif eacutetait le 13ho13 fortement
supersonique 10-30 kms agrave quelques mbar dAr ou de Xe Un modegravele simple
(0D) a eacuteteacute 13ompareacute agrave un modegravele 3D MHD par le 13ode GORGON qui deacute13rit
bien les expeacuterien13es de plasmas astrophysique par puissan13e pulseacutee (Ciardi
et al 2007)
La pression magneacutetique sexprimant par B
22micro0 un 13ourant supeacuterieur agrave
100 kA apparaicirct neacute13essaire et atteignable sur un geacuteneacuterateur 1 kJ - 1 micros 13om-
pa13t de type R-L-C Ensuite il a fallu monter un modegravele de 13ir13uit deacuteformable
13omportant une se13tion formeacutee de la lame de plasma en mouvement et qui
prenne en 13ompte la1313reacutetion du gaz dans un modegravele dit snowplow (Potter
1971) Des paramegravetres geacuteomeacutetriques 13omme la forme et les dimensions du
13anon ont pu ecirctre optimiseacutes pour donner une grande vitesse de sortie du
plasma pour Ar et Xe et dans la gamme 01 - 10 mbar Un 13anon a eacuteteacute 13on-
struit et 13onne13teacute agrave un geacuteneacuterateur eacutele13trique pulseacute existant Des observations
du mouvement du plasma par la dynamique du rayonnement radial donnent
des vitesses 13oheacuterentes ave13 le modegravele ave13 la simulation 3D MHD et des
nombres de Ma13h eacuteleveacutes de 20 agrave 60 Ces travaux ont soutenu la 13on13eption
dun autre geacuteneacuterateur pour obtenir des performan13es plus eacuteleveacutees en vitesse
de 13ho13 et en taux de reacutepeacutetition
Contents
List of Figures
List of Tables
1 Introdu13tion 1
11 General Context 1
12 Outline of Chapters 2
13 My 13ontribution 3
2 Radiative sho13k waves 5
21 Rankine-Hugoniot(R-H) relations 6
211 Jump relations for a real gas 9
212 Ee13t of radiation on sho13k waves 11
213 Radiative hydrodynami13s equations 13
22 Radiative sho13k regimes 15
221 Typi13al radiative sho13k waves with a pre13ursor 16
23 Radiative sho13k waves experiments 16
231 Laser-driven sho13k experiments 18
232 Ele13tromagneti13ally laun13hed sho13k 21
3 1D Simulations 23
31 HELIOS 23
32 LTE approximation 24
321 Mean opa13ity 25
322 Equation of State 27
33 Single radiative sho13k waves 27
34 Intera13ting radiative sho13k waves 40
35 Summary 44
4 Experimental Setup 47
41 Prague Asterix Laser System (PALS) fa13ility 48
42 Targets 49
421 Massive Targets 51
422 Gaseous Targets 51
423 Target holder 55
CONTENTS
424 Target lling 56
43 Laser Fo13using 57
431 Fo13al Lenses and Phase Zone Plates 57
432 Fo13using of the MAIN and AUX laser beams 58
44 Diagnosti13s 59
441 Visible interferometry 61
442 XUV spe13tros13opy 65
443 Visible spe13tros13opy 67
45 Summary 68
5 Results and Interpretation 71
51 Visible interferometry 71
511 Longitudinal interferometry sho13k speed and ele13tron
density 74
512 Transverse interferometry lateral extension of the sho13k 91
52 XUV Spe13tros13opy 92
53 Simulations based on experimental results 93
54 Summary 97
6 Ele13tromagneti13ally laun13hed sho13k 101
61 Prin13iples of operation of a high 13urrent generator 102
62 Prin13iples of the run-down phase in a PFD 103
63 Proposed design for the plasma gun 104
64 Dynami13 13ir13uit modelling 106
65 3-D MHD simulations using GORGON 13ode 111
66 Measurements 116
67 Summary 118
7 Con13lusion 121
71 Con13lusions 121
72 Perspe13tives 123
8 Thesis summary 127
Appendi13es 141
A Visible Interferometry 143
A01 Refra13tive index of a plasma 143
A02 Absorption of the laser beam 144
A03 Prin13iple of interferometry 145
A04 Ma13h Zehnder Interferometer 146
B Opa13ities and mean 13harge 149
Referen13es 152
List of Figures
21 S13hemati13 diagram for a 1-D sho13k moving in the x dire13tion
The speed of the sho13k front is us In the frame of referen13e of
the sho13k front the pre-sho13k region has a relative speed u1=
-us density ρ1 total pressure P1 temperature T1 whereas these
quantities are respe13tively u2 ρ2 P2 and T2 for the post-sho13k
region 6
22 Proles of temperature (Te = Ti = T ) and mass density (ρ) at 10ns for a sho13k propagating in Xenon at 01 bar (ρ1= 57 times 10
minus4
g13m
minus3) with a speed of sim 45 kms It is to note that the given
pressure value are at 298 K The simulation is performed with the
HELIOS 13ode ex13luding the radiation ee13t and for a polytropi13
gas (γ=53) The initial temperature of Xenon is set to 1 eV
and with a negligible (10
minus5WmK) thermal 13ondu13tivity The
Ma13h number is then equal to 40 For this simulation the sho13k
is laun13hed by a 100 microm thi13k Aluminium piston moving at the
velo13ity of 33 kms The position x=0 13orresponds to the position
of the XeAl interfa13e at time t=0 8
23 Ele13tron temperature (Te) ion temperature (Ti) and mass density
(ρ) proles at 10 ns derived from 1-D simulation for the 13ase of
ionisation (no radiation) The sho13k propagates in Xe gas at 01
bar (ρ1= 57 times 10
minus4g13m
minus3by sim 45 kms) It is to note that the
given pressure value is at 298 K The simulation is performed with
the HELIOS (Te 6= Ti 13ase) using the PROPACEOS equation
of state without any radiation and with a negligible thermal
13ondu13tivity (10
minus05WmK) The initial temperature in Xenon
is set to 1 eV and thus the Ma13h number is equal to 40 For this
simulation the sho13k is laun13hed by a 100 microm thi13k Aluminium
piston moving at the velo13ity of 38 kms The position x=0
13orresponds to the position of the XeAl interfa13e at time t=0 10
LIST OF FIGURES
24 Figure 14 amp 17 of Mi13haut et al (2004) (a) Compression ρ2ρ1(full 13ir13les thin line) kT2 in eV (inverted full triangles thin line)
in Argon (a) Xenon (b) for initial 13onditions kT1( = 10 eV for
Ar and = 01 eV for Xe) ρ1 = 5 times 10minus4 gcm3 versus sho13k
speed in kms with (full markers full lines) and without (empty
markers dashed lines) radiation The ionization stage lt z gt2 is
plotted with the diamond symbol 12
25 Simulated ele13tron temperature (Te) ion temperature (Ti) and
mass density (ρ) proles at 10 ns of a radiative sho13k propagatingwith a 15 kms (a sub-13riti13al) and 48 kms (b super13riti13al)
The sho13k is propagating in Xenon at 01 bar (ρ = 54 times 10
minus4
g13m
minus3 at 298 K) The initial temperature in Xenon is set to
1eV The two Ma13h numbers are respe13tively equal to 13 and 43
The radiation and ionisation ee13ts are in13luded in the simulation
whi13h is performed for two dierent temperatures (Te 6= Ti) and a
negligible thermal 13ondu13tivity (10
minus05WmK) For this simula-
tion the sho13k is laun13hed by a 100 microm thi13k Aluminium piston
moving at the speed of 12 kms and 45 kms for 13ase (a) and (b)
respe13tively The position x=0 13orresponds to the position of the
XeAl interfa13e at time t=0 17
31 PROPACEOS Mono13hromati13 opa13ity versus the photon energy
in eV of Xenon at 10 eV and at two ele13tron densities respe13tively
equal to 1018 and 1020 13m
minus3 26
32 Mass density and temperature (a) ele13tron density and mean
13harge (b) at 10 ns for a radiative sho13k of speed sim 47 kms
in Xenon at 01 bar The verti13al dotted bla13k lines show the
position of the interfa13e between piston and ba13king Xenon gas
Zero at x-axis is the interfa13e of piston and Xenon at time zero
Spitzer thermal 13ondu13tivity has been used in the simulation 29
33 Mean Rosseland opa13ity (in 13m
2g
minus1) and opa13ity (13m
minus1) at 10
ns for a radiative sho13k of speed sim 47 kms in Xenon at 01
bar The verti13al dotted bla13k lines show the position of the
interfa13e between piston and ba13king Xenon gas Zero at x-axis
is the interfa13e of piston and Xenon at time zero Spitzer thermal
13ondu13tivity has been used in the simulation 30
34 (a) Mass density and temperature (b) ele13tron density and
mean 13harge at 10 ns for a radiative sho13k of speed sim 47 kms
in Xenon at 01 bar The verti13al dotted bla13k lines show the
position of the interfa13e between piston and ba13king Xenon gas
Zero at x-axis is the interfa13e of piston and Xenon at time zero
A negligible 13onstant thermal 13ondu13tivity (10
minus12WmK) have
been used in the simulation for Xenon 31
LIST OF FIGURES
35 (a) Ele13tron density and ele13tron temperature (b) Mass density
and mean 13harge at 10 ns for a radiative sho13k of speed sim 48
kms in Xenon at 01 bar for the 13ase when the ele13tron and ion
temperature are dierent The verti13al dotted bla13k lines show
the position of the interfa13e between piston and ba13king Xenon
gas Zero at x-axis 13orresponds to the position of the interfa13e
between the piston and Xenon at time zero The Spitzer thermal
13ondu13tivity has been used in this simulation 32
36 (a) Mass density and ele13tron temperature (b) ele13tron density
and mean 13harge at 10 ns for a radiative sho13k of speed sim 45
kms in Xenon at 01 bar for the 13ase when the ele13tron and ion
temperature are dierent The verti13al dotted bla13k lines show
the position of the interfa13e between piston and ba13king Xenon
gas Zero at x-axis 13orresponds to the position of the interfa13e
between the piston and Xenon at time zero In this simulation
value of thermal 13ondu13tivity (10
minus12WmK) is negligible for
Xenon The peak value of ion temperature is found to be 405 eV
whereas theoreti13ally this value is expe13ted to be sim 600 eV It is
possible to a13hieve the expe13ted value by in13reasing the resolution
of the simulation 34
37 Ele13tron temperature proles for various numbers of frequen13y
groups N = 1 20 50 60 70 90 and 100 35
38 (a) Ele13tron temperature (a) mean 13harge (b) and mass density
(13) for four (1 10 30 and 40) multipliers of the Xenon opa13ity at
10 ns 36
39 Plan13k and Rosseland mono13hromati13 opa13ity proles Te for ρ=15 times 10
minus3from PROPACEOS (I) BIGBART (II) and Rodri-
guez et al (III) 38
310 Mono13hromati13 Xenon opa13ity versus the photon energy in eV
at T = 15 eV and for ρ = 15 times 10
minus3g13m
minus3for the PROPA-
CEOS (a) and Rodriacuteguez et al (2015) (Fig 16 of Rodriacuteguez et al
(2015)) (b) models 39
311 Ele13tron density Ne (a) and ele13tron temperature Te (b) versus
axial position (along a 04 13m long sho13k tube) at 3 10 20 30
and 38 ns from HELIOS simulations (with opa13ity times 20) for the
13ases of single sho13k of sim 50 kms (dotted line) and two identi-
13al 13ounter-propagating sho13ks of sim 50 kms (solid lines) The
verti13al dotted lines show the position of the interfa13e between
piston and ba13king Xenon gas 41
LIST OF FIGURES
312 Variations ofNe (a) and Te (b) versus axial position for the 13ase of
two identi13al 13ounter-propagating sho13ks (of speeds sim 50 kms)
at 3 10 20 30 35 and 40 ns as derived from HELIOS simulations
For these simulations we have negle13ted the ee13t of radiation
by keeping the Xe opa13ity equal to zero The verti13al dotted lines
show the position of the interfa13e between the piston and ba13king
Xenon gas 43
313 Variations of Ne (a) and Te (b) with axial position for the 13ase
of two non-identi13al 13ounter-propagating sho13ks (of speeds sim 50
amp 20 kms) and two single sho13ks (dotted lines) of speeds sim 50
amp 20 kms respe13tively at 3 10 30 35 and 46 ns as derived
from HELIOS simulations The verti13al dotted lines show the
position of the interfa13e between piston and ba13king Xenon gas
(with opa13ity times 20) 45
41 a Spatial prole of the PALS Laser beam b Intensity prole of
laser pulse I(t) with time 48
42 Energy distribution s13heme for MAIN and AUX laser beams 49
43 Snapshot of the spheri13al 13hamber showing the horizontal bred-
board and on the right the fo13using MAIN lens 50
44 Dimension drawing (a) and snapshot (b) of Aluminum massive
(mo13k) target 50
45 Values on the gures are in mm (a) Verti13al 13ross se13tion view
of the gaseous target (b) Horizontal 13ross se13tion view 52
46 Pi13ture of a the gaseous target on its base 53
47 S13hemati13 of the gaseous target 53
48 Transmission of 100 nm thi13k Si3N4 membrane (CXRO database) 54
49 Snapshot of a target holder with one V-shield regarding the AUX
laser whi13h will pass through the hole in this shield marked in
green in the Figure The target whi13h fa13es the two diagnosti13s
(tubes) and is lo13ated behind the V-shield is thus not visible 54
410 In-situ system for target lling and pressure monitoring 56
411 S13hemati13 of the keV 13amera 59
412 (a) Sign 13onvention for the respe13tive positions of the lens and
the target (b) keV image of a MAIN impa13t on a massive target
(lens position +1500 microm) The spot size is 475plusmn25 microm Pixel
size for MAIN keV 13amera is 56 microm (13) keV image of an AUX
impa13t (lens position -1500 microm) The spot size is sim 280plusmn20 micromon target One pixel = 66 microm on AUXs keV 13amera 60
413 First interferometri13 setup All the opti13al elements are 1 in13h
For simpli13ity the plasma slab is not reported in the probe beam 61
414 Se13ond interferometri13 setup 63
415 S13hemati13 of the Streak 13amera (from Hamamatsu noti13e) 64
LIST OF FIGURES
416 Imaging setup of the Ma13h Zehnder Interferometer setup opera-
ting in the longitudinal mode The 13hannel of the target is fully
illuminated by the probe beam In this mode the lens images
the axis of symmetry of the two sho13k waves along the slit of the
streak 13amera 65
417 (a) Referen13e 2D image of a target before the shot re13orded on
the Streak 13amera The positions of the two pistons 13losing the 4
mm long target are lo13ated at 800 and 4800 microm The dark zones
along them (between 800 to 1000 and 4600-4800 microm) 13orresponds
to the glue This glue is then visible through verti13al bla13k strips
in the interferometri13 re13ord (b) Corresponding interferometri13
time-spa13e re13ord 66
418 Typi13al ray tra13ing in a grazing in13iden13e XUV spe13trograph
The dete13tor (GMCP or image plate) is installed tangentially to
the Rowland 13ir13le 67
419 (a) Setup of the XUV spe13trometer in the spheri13al 13hamber (b)
Typi13al re13ord between sim 10 and 40 nm is shown in the bottom
panel 68
420 (a) S13hemati13 of setup employed to re13ord the time and spa13e
visible plasma emission (b) A typi13al spe13trogram 69
51 Original re13ord of shot 48055 (left) FFT of the original image
(13enter) and the frequen13y ltered image (right) 73
52 A 13ropped se13tion of the re13ord from shot the 48055 The
rst ve positions have been sele13ted manually `+ signs (in red
13olor) on ea13h fringe On this re13ord the distan13e between two
unperturbed fringes is 13orrespond to 15 pixels (ie 159 microm) 73
53 (a) Representative points of the fringes as derived from the spline
tting of the 5 manually sele13ted points (Fig 52 (II)) on ea13h
fringe (b) Positions of the fringes maxima along Y-axis for ea13h
fringe derived by lo13ating the points of maximum intensity in
X-dire13tion of the previous points obtained by spline t 75
54 Fringe maxima on the full image 76
55 Imaging setup of the Ma13h Zehnder Interferometer operating in
the longitudinal mode The lens images the axis of symmetry of
the two sho13k waves along the slit of the streak 13amera 76
56 Interferometri13 image re13orded for the shot 48055 in Xe at 01
bar The sho13k speeds for the sho13ks driven by MAIN (from left
side) land AUX (from right side) lasers are respe13tively equal to
sim 54 and 23 kms The time of laser arrival on the piston is at
146 ns The positions of the Au-Xe interfa13e on the re13ord are
respe13tively 950plusmn50 and 4950plusmn50 mi13rons 77
LIST OF FIGURES
57 (a) MAIN sho13k speed (13al13ulated by the last fringe method)
versus the MAIN energy for Xenon or XeHe mixture at dierent
pressures with the error bars (b) AUX sho13k speed versus AUX
energy also for Xenon or XeHe mixture at dierent pressures 79
58 MAIN sho13k speed (13al13ulated by the last fringe method) versus
the MAIN laser energy for Xe (at 01 bar only) Ar He and Kr
at dierent pressures with the error bars 82
59 Left panel interferometri13 re13ords 48055 in Xe at 01 bar (a)
48132 in Xe+He at 02 bar (b) and 48138 in Xe+He at 02 bar
(13) Right panel ele13tron density at 10 20 30 and 40 ns versus
distan13e for these re13ords The positions of maxima have been
identied on the re13ords in the left panel The time t = 0 13orre-
sponds to the time of laser arrival on the target and the position
x = 0 13orresponds to the interfa13e between the piston (Au layer)
and the gas Its determination is pre13ise within 100 mi13rons The
distan13es between two unperturbed fringes for re13ords 48055
48132 and 48138 are 159 244 and 244 microm respe13tively The
lt Ne gt un13ertainty (plusmn 2 pixels) is indi13ated by the error bar in
the right panels It 13orresponds respe13tively to plusmn 9 times 10
17 plusmn 6
times 10
17and plusmn 6 times 10
1713m
minus3for the gures (a) (b) and (13) The
limit of dete13tion (2 pixels) is presented by a dotted line on ea13h
gure 85
510 (a) Interferometri13 re13ord in Ar at 03 bar (48079) The esti-
mated speeds for MAIN and AUX are 49plusmn5 and 23plusmn3 kms (b)
Same re13ord where the fringes maxima are marked by points
The bins denition is as fellows bin 0 Ne le 11 times 10
1813m
minus3
(white) bin 1 11 - 18 times 10
1813m
minus3(yellow) and bin 2 39 -
57 times 10
1813m
minus3(red) The time t = 0 13orresponds to the time
of laser arrival on the target and the position x = 0 13orresponds
to the interfa13e between the piston (Au layer) and the gas Its
determination is pre13ise within 100 mi13rons 87
511 Results for temperature (a) and Rosseland opa13ity (b) obtained
from HELIOS simulation at 10 ns for two 13ounter-propagating
sho13ks at sim 50 and 18 kms for Xe (with opa13ity multiplier 1
and 20) Ar and Kr at 54 times 10
minus4g13m
minus3initial mass density
A 13omparison Ar Kr and Xe PROPACEOS opa13ity shown in
APPENDIX B 88
LIST OF FIGURES
512 (a) Raw interferometri13 re13ord in Kr at 02 bar (shot 48146)
The estimated speed for MAIN sho13k is 53plusmn2 kms The time
t = 0 13orresponds to the time of laser arrival on the target(b)
Same re13ord shown in (a) maxima is marked by 13olored points
The bins denition is as follows bin 0 Ne le 11 times 10
1813m
minus3
(white) bin 1 11 - 18 times 10
1813m
minus3(yellow) and bin 2 39 -
57 times 10
1813m
minus3(red) The time t = 0 13orresponds to the time
of laser arrival on the target and the position x = 0 13orresponds
to the interfa13e between the piston (Au layer) and the gas Its
determination is pre13ise within 100 mi13rons 90
513 S13hemati13s of the Ma13h Zehnder Interferometer setup to re13ord
transverse interferometri13 images The lens allows to make on
the slit of the 13amera the image of a se13tion perpendi13ular to
dire13tion of sho13k propagation 91
514 Transverse interferometri13 images for (a) shot48111 (MAIN
sho13k only) (b) shot48130 (AUX sho13k only) The time is
measured after an oset equal to 14 and 23 ns respe13tively after
the time of the laser arrival on the target The position zero on
the x-axis of ea13h image 13orresponds to the base of the target 92
515 Interferometri13 image for the shot48143 The time t = 0 13orre-
sponds to the time of laser arrival on the target and the position
x = 0 13orresponds to the interfa13e between the piston (Au layer)
and the gas Its determination is pre13ise within 100 mi13rons 94
516 Raw (a) and 13orre13ted XUV spe13trum (b) for the shot 48143 95
517 Re13orded ele13tron density (shot 48055) together with the HE-
LIOS results (with Xenon opa13ity times 20) at dierent times in
Xenon at 01 bar 97
518 Time evolution of the mass density (a) ele13tron temperature (b)
and mean 13harge (13) at 56 57 58 60 64 and 65 ns within the
sho13k tube derived from the HELIOS simulations (with Xenon
opa13ity multiplier = 20) for two 13ounter streaming sho13ks of sim39 and 18 kms in Xenon at 06 bar 98
61 (a) Sket13h of the sho13k generator showing the pulsed ele13tri13al
13ir13uit the set of 13oaxial 13oni13al ele13trodes with a 13onstant radial
gap and the plasti13 insulator featured in grey on whi13h a planar
surfa13e dis13harge is initiated The installation of three opti13al
bers allows looking radially at the plasma moving in the sho13k
tube (b) S13hemati13s of the plasma dynami13s inside the 13oaxial
gun in fast-pulse mode the ele13tri13al 13urrent ows in the super-
13ial layers of the two 13oaxial 13oni13al ele13trodes and through an
annular plasma layer The magneti13 pressure Pmag pushes the
dis13harge axially 105
62 Exploded view of the plasma gun 106
LIST OF FIGURES
63 Time dependen13e of the main parameters for Ar gas at 1 mbar
a1313ording to the 13ir13uit model (a) 13urrent (kA) and voltage
(V) (b) a1313reted mass (kg) and rate of a1313retion (kgs) (13)
speed (ms) and kineti13 energy (J) 108
64 Plot of the dierent output parameters (normalized with respe13t
to the peak values) obtained at the top of the inner ele13trode
For ea13h group upper panel represents the variation for Ar gas
whereas lower panel is for Xe gas Proles of output parameters
are given (a) with 13one angle (b) with a1313retion fa13tor (13) with
gas pressure and (d) with damping resistan13e 109
65 Example of mapping of the ele13tron density from a 3-D MHD
simulation (ba13kground gas Argon at 1 mbar) when the plasma
sheath is (left) at the exit of the plasma gun (right) in free
ight 13onditions (log s13ales for the false 13olors) 113
66 Time history of plasma merging and early free ight through
the mapping of the ele13tron density ele13tron temperature and
average ioni13 13harge as given by a 3-D MHD simulation (ba13k-
ground gas Argon at 1 mbar) (s13ales for the false 13olors) 114
67 Axial proles values taken in the sho13k tube along a line slightly
o-axis at a distan13e of 1 mm for mass density ion temperature
ion density average ion 13harge magneti13 eld ele13tron tempera-
ture ele13tron density and average speed at 1500 ns (ba13kground
gas Argon at 1 mbar) A maping of |B| is given with a dashed
line indi13ating sho13k front position as well 115
68 Experimental results (a) time history of the passive opti13al
re13ords 13ompared to the main 13urrent for Argon gas 13lose to the
referen13e pressure (b) time-integrated signal vs Ar pressure for
the dierent bers (13) averaged speed vs Ar ba13king pressure 117
71 Interferometri13 re13ord obtained at 546 ns for a single radiative
sho13k of speed sim 100 kms propagating in Xe+He gas at 06 bar
driven by laser at 348 nm with energy 170 J The dierent 13olors
in (b) 13orrespond to yellow le Ne le 35 times 10
1713m
minus3 13yan 37
- 74 times 10
1713m
minus3 blue74 - 15 times 10
1813m
minus3 green 15 - 18 times
10
1813m
minus3 magenta 18 - 22 times 10
1813m
minus3 red 22 - 26 times 10
18
13m
minus3 orange26 times 10
1813m
minus3- 18 times 10
1913m
minus3 124
A1 S13hemati13s of Ma13h Zehnder interferometer with two mirrors
two 50 beam splitters and the dete13tor A plasma slab is pla13ed
in one of the two arms of the interferometer 147
B1 Rosseland and Plan13k opa13ity for three gases Xe Kr and Ar at
mass densities 51 times 10
minus4g13m
3(a) and 16 times 10
minus3g13m
3(b) 150
LIST OF FIGURES
B2 Mean 13harge for three gases Xe Kr and Ar at mass densities 16
times 10
minus3g13m
3(a) and 51 times 10
minus4g13m
3(b) 151
List of Tables
21 Summary of the radiative sho13ks experiments and out13omes in
dierent laser installations All the experiments are performed in
Xe ex13ept Keiter et al (2002) and Vis13o et al (2012) In these
two experiments the rst uses SiO
lowast
2 and the se13ond uses Ar
lowastlowastas
a sho13ked medium 20
41 Nominal 13hara13teristi13s of MAIN and AUX laser beams 49
42 Spe13i13ations of the MAIN and AUX fo13al lenses and the f- num-
ber (fo13al lengthbeam diameter) is given (see table 41 for the
beam diameters) 57
43 Spe13i13ations of the two Phase Zone Plates 58
44 Spe13i13ations of the two uEye keV 13ameras (1stVsion 13ompany) 59
45 Spe13i13ations of Andor CCD 13amera 67
46 Spe13i13ations of the XUV grating 67
51 Sho13k speeds estimated from the `last fringe method `NA re-
presents the entries whi13h 13ould not be dedu13ed from the re13ord
Further entries in the bold font are dis13ussed in detail in this
13hapter 81
52 Atomi13 data and density at 01 bar (at room temperature) for
He Ar Kr and Xe 82
61 Input parameters their standard values and range of variation 110
62 Laun13hing performan13e ArXe 111
63 Values of variables in various 13onditions at rest (with seed ele13-
trons) inside the sho13k and inside post-sho13k region A star lowastdenotes the insulator surfa13e 112
Chapter 1
Introdu13tion
11 General Context
Radiative sho13ks (RS) are present in various astrophysi13al environments es-
pe13ially in stars They are then present in all the supersoni13 a1313retioneje13tion
pro13esses during the stellar evolution from the early phase of the proto-
stellar 13ollapse up to the magnetosphere a1313retion on to the already formed
T- Tauri stars up to the late stage of the supernovae sho13ks
Radiative sho13ks are strong sho13ks (ie Ma13h number Mgtgt1) whi13hrea13h high temperatures and thus are the sour13e of intense radiation Alt-
hough the observation of (ie spatially resolved ) signatures of su13h sho13ks
in the astrophysi13al environments is very di13ult an alternate approa13h na-
med laboratory plasma astrophysi13s provides a powerful tool to study them
on the Earth
These waves have been experimentally studied sin13e more than a de13ade
mostly on large-s13ale laser fa13ilities in noble gases and with dierent targets
geometries The ablation pro13ess generated by the intera13tion between the
laser and a foil indu13es a sho13k wave in the foil whi13h then propagates
in the gas at a speed of several tens of kms Numeri13al simulations 13an
des13ribe these sho13ks with an improving pre13ision With laser intensities on
target 13omprised between 1014 and 1015 W13m
2 these experiments allowed
to re13ord sho13k speeds ranging between 40 and 150 kms
All previous experimental studies have been fo13used on the 13ase of single
radiative sho13ks However in astrophysi13al 13onditions the radiative sho13k
in general intera13ts with a denser medium leading to the development of
ree13ted and transmitted sho13ks A few representative examples of su13h
phenomena are the intera13tion of supernovae remnants with dense mole13ular
13louds the a1313retion sho13ks on the photosphere of T-Tauri stars and the bow
sho13ks at the head of fast stellar jets The 13ollision (or the intera13tion) of
two radiative sho13k waves is obviously a rare astrophysi13al event and the
template 13ase of the supernova remnants DEM L316 (see Fig 1 of Williams
et al (1997)) is still the subje13t of debates (Velarde et al 2006) In this
regard the development of dedi13ated laboratory experiments to the study of
propagation and intera13tion of 13ounter-propagating sho13k waves is important
as a tool to 13hara13terize su13h events through their spe13i13 signatures
Complementary to laser experiments 13ompa13t pulsed power generators
may drive an ele13tromagneti13 13oaxial plasma gun to 13reate astrophysi13al
1
2 CHAPTER 1 INTRODUCTION
relevant sho13ks in lower pressure noble gases with a high availability and a
rather modest 13apital 13ost The ele13tromagneti13ally driven sho13k waves may
have larger s13ales than those by laser Thus they 13an be analyzed rather
easily with the adequate diagnosti13s (Kondo et al 2008)
In this thesis I am presenting radiative sho13ks in the 13ontext of labo-
ratory astrophysi13s using two experimental approa13hes laser-driven sho13ks
and ele13tromagneti13ally driven sho13ks
12 Outline of Chapters
In the rst 13hapter the physi13s of radiative sho13ks wave will be summarized
I will dis13uss the Rankine -Hugoniot (RH) for the perfe13t gas and genera-
lized RH equations for ionized gas Further the ee13ts of radiation on the
sho13k equations will be introdu13ed and I will review the dierent regimes of
the radiative sho13ks presenting a radiative pre13ursor Then I will present
a short histori13al review of laser and ele13tromagneti13ally laboratory experi-
ments on this topi13 Finally I will highlight the obje13tives of my work in
this framework
The 13hapter two is dedi13ated to the numeri13al analysis of radiative sho13k
waves and of the main physi13al pro13esses whi13h play a role in these 13omplex
pro13esses The simulations will be performed in 1D using the 13ommer13ial ra-
diation hydrodynami13 HELIOS 13ode (Ma13Farlane Golovkin and Woodru
2006) After a des13ription of the 13ode and his possibilities several simu-
lations will be performed to analyze the ee13ts of the two uids (ele13trons
and ions) approa13h 13ompared to the one uid 13ase (identi13al temperature
for both) of the thermal 13ondu13tivity and the multi-groups opa13ity des13rip-
tion Then a brief overview of single and two 13ounter-propagating radiative
sho13ks will be presented
Chapter three will start with a short overview of the Prague Asterix Laser
System fa13ility (PALS) It will be followed by the experimental setup in13lu-
ding the two high energy beams and targets des13ription These paragraphs
will be followed by the detail of the diagnosti13s
The fourth 13hapter will be dedi13ated to the data analysis the results
and the interpretation of the experimental re13ords I will rst des13ribe the
method I followed for the analysis of interferometri13 re13ords Subsequently
the results obtained from a few representative re13ords for Xe Ar and Kr
will be dis13ussed in details Complementary 1D simulations performed with
initial 13onditions similar to that in the experiment will be shown and help to
interpret the experimental results Finally the results of XUV spe13tros13opy
diagnosti13s will be presented
In the fth 13hapter I will present the 13alibration of a table top ele13tro-
magneti13 plasma generator whi13h is able to laun13h supersoni13 sho13ks up to
sim 40 kms speed at stati13 pressures of few mbar The results of this 13ali-
13 MY CONTRIBUTION 3
bration will also be presented and a 13omparison with 3-D MHD simulations
will be performed The diagnosti13s whi13h have been implemented will be
presented to illustrate the model as well as preliminary re13ords of the plasma
speed
The nal 13hapter of the thesis is dedi13ated to the summary of this work
together with perspe13tives for the follow-up of these experiments
13 My 13ontribution
I joined PhD on November 12 2013 under the supervision of Dr Jean
Larour and Dr Chantal Stehleacute My PhD thesis is 13omprised of two parts
the rst part of the work is to study the laser-driven sho13ks The se13ond
part of my thesis is to study the ele13tromagneti13ally laun13hed strong sho13ks
Con13erning the laser-driven sho13ks I have started to work on the analysis
and interpretation of XUV images re13orded previously during an experimen-
tal 13ampaign 13ondu13ted at the Prague Asterix Laser System (PALS) fa13ility
in 2011 This enabled me to estimate the lo13al opa13ity by employing Abel
inversion Although this work not presented in this thesis it shed light on
the basi13 understanding of radiative sho13k as seen in the experiments as well
as the knowledge of experimental setups Later in April - May 2015 I have
parti13ipated in a ve weeks experimental 13ampaign on PALS laser fa13ility
in Prague whi13h was dedi13ated to the study of the spe13tros13opi13 and in-
terferometri13 signatures of laser-produ13ed sho13ks Espe13ially the experiment
was designed to study the physi13s during the 13ollision of 13ounter-propagating
sho13ks In the preparation of this 13ampaign I be13ame a13quainted with the
instruments and the targets to be used as well as prepared a master re-
port 13ontaining all the 13ru13ial aspe13ts of the experiment to be referred by
all the parti13ipants during the experiments During the 13ampaign we have
performed around 55 shots 13orresponding to dierent gases (Xenon Argon
Helium and Krypton) at dierent pressure and used three types of diagnos-
ti13s ie visible interferometry visible spe13tros13opy and XUV spe13tros13opy
In this experiment parti13ularly I was in 13harge of the target sele13tion its
installation in the experimental 13hamber gas lling preparing logbook re-
13ording of the visible spe13tros13opy data from the 13ontrol room qui13k data
analysis for the spot size sho13k speed from interferometry data et13 In
addition I have worked on spe13tros13opi13 and interferometri13 data analysis
re13orded during the experiment Next in order to interpret the experimental
results I have performed several 1D hydrodynami13 simulations with varying
input 13onditions In parallel I have also performed simulations for another
experiment performed by my 13ollaborators at ORION laser fa13ility in the
United Kingdom However I have not presented this work in this thesis
Based on the results obtained from my analysis of re13ords obtained from
the PALS experiments in 2015 grabbing the opportunity we have planned
4 CHAPTER 1 INTRODUCTION
and parti13ipated to a 13omplementary experimental 13ampaign at PALS in
September 2016 The analysis of data re13orded during experiments is still
under progress and forms the future s13ope of the thesis
The se13ond part of my thesis has been 13omprised of the study of sho13k
laun13hed through the ele13tromagneti13 generator setup In this I have wor-
ked on the optimization of the design of the ele13tri13 generator in view of the
required sho13k parameters Moreover I have derived various sho13k 13hara13-
teristi13s viz radial dis13harge layer namely speed mass a1313eleration et13
at the dierent positions of its propagation In present a upgraded ele13tri13
generator with new diagnosti13s is ready to perform the new experiments
Chapter 2
Radiative sho13k waves
Contents
21 Rankine-Hugoniot(R-H) relations 6
211 Jump relations for a real gas 9
212 Ee13t of radiation on sho13k waves 11
213 Radiative hydrodynami13s equations 13
22 Radiative sho13k regimes 15
221 Typi13al radiative sho13k waves with a pre13ursor 16
23 Radiative sho13k waves experiments 16
231 Laser-driven sho13k experiments 18
232 Ele13tromagneti13ally laun13hed sho13k 21
Small-amplitude disturban13es in a medium often take the form of linear
waves On the 13ontrary strong disturban13es generally produ13ed by pro13esses
su13h as explosions lightening supersoni13 movements of bodies powerful
ele13tri13 dis13harges et13 where the uid properties 13hange rapidly generate
nonlinear waves like sho13k waves As a 13onsequen13e of sho13k waves the ow
moves faster than the sound speed A sho13k is 13hara13terized by a sudden
in13rease in the density and velo13ity of the medium The Ma13h number (M)
is 13ommonly used to 13hara13terize the strength of the sho13k This number
dened as the ratio of the sho13k speed to the speed of sound in the upstream
medium in whi13h the sho13k propagates
Radiative sho13k waves are hypersoni13 sho13k waves whi13h are heated to
high temperature and as a 13onsequen13e be13ome the sour13e of intense radi-
ation This radiation in turn modies the dynami13s and stru13ture of the
sho13k itself whi13h makes its stru13ture more 13ompli13ated Radiative sho13k
waves o1313ur in several astrophysi13al 13ir13umstan13es su13h as in the a1313retion
sho13k of protostellar formation (Stahler Palla and Salpeter 1986) the su-
pernova explosion and the intera13tion of their remnants with the dense in-
terstellar medium (Chevalier 1977) the bow sho13ks at the head of stellar
jets (Hartigan et al 2001)
Radiative sho13k waves 13an now be studied in the laboratory whi13h allows
to 13ompare the data with models existing in the literature and to 13he13k the
ability of the numeri13al 13odes to reprodu13e them (see for instan13e Bouquet
et al (2004) Leygna13 et al (2006)) The experimental study of radiative
5
6 CHAPTER 2 RADIATIVE SHOCK WAVES
sho13k 13hara13teristi13s is 13ru13ial in understanding these various energeti13 pro-
13esses o1313urring in the astrophysi13al plasmas In this 13hapter I will rst
present a short summary of the jump 13onditions (Rankine-Hugoniot rela-
tions) for the simple 13ase of an ideal gas whi13h rely on the values of the
thermodynami13al quantities on both sides of the dis13ontinuity Next I will
present the role played by the ionisation and ex13itation of the gas and then
the role played by the radiation energy and pressure The des13ription of the
dierent regimes of radiative sho13ks whi13h depend on the opa13ity of medium
and 13an be 13hara13terized by the presen13e or absen13e of a radiative pre13ursor
will follow Further I will give a brief overview of the literature fo13using on
the generation of radiative sho13ks in the laboratory and I will present the
motivations and obje13tives of my work
21 Rankine-Hugoniot(R-H) relations
Let us 13onsider a one-dimensional sho13k propagating in a gas at rest with
the speed us In the frame asso13iated with the sho13k front the pre-sho13k
(upstream) uid velo13ity is then u1 = - us while behind the sho13k (post-
sho13k or downstream region) this velo13ity is u2 as shown in the Fig 21
Figure 21 S13hemati13 diagram for a 1-D sho13k moving in the x dire13tion The
speed of the sho13k front is us In the frame of referen13e of the sho13k front the pre-
sho13k region has a relative speed u1= -us density ρ1 total pressure P1 temperature
T1 whereas these quantities are respe13tively u2 ρ2 P2 and T2 for the post-sho13k
region
Assuming that a stationary state is rea13hed the 13onditions whi13h deter-
mine the post-sho13k thermodynami13al quantities are the well known Rankine-
21 RANKINE-HUGONIOT(R-H) RELATIONS 7
Hugoniot equations (Ma13quorn Rankine 1870 Hugoniot 1887 1889) De-
du13ed from the Euler equations these 13onservation equations for the mass
momentum and energy are given by
ρ2u2 = ρ1u1 (21)
ρ2u22 + P2 = ρ1u
21 + P1 (22)
u2(ε2 +P2
ρ2) +
1
2u32 = u1(ε1 +
P1
ρ1) +
1
2u31 (23)
where ε1 and ε2 are the spe13i13 internal energies (energy per unit mass)
in the two (upstream and downstream) regions
For an ideal gas of atomi13 mass m =mPA (mP proton mass A atomi13
number) and having an adiabati13 index γ the speed of sound is given by
Cs =
radic
γP
ρ=
radic
γkBT
m(24)
and the Ma13h number (M) of the sho13k is then
M =u1Cs1
(25)
Inserting these the aforementioned form of M and Cs in the equations
21 22 and 23 we 13an derive the 13ompression pressure and temperature
ratios for the system as follows
ρ2ρ1
=u1u2
=M2(γ + 1)
2 +M2(γ minus 1)(26)
P2
P1=
2M2γ minus (γ minus 1)
(γ + 1)(27)
T2
T1=
P2
P1
ρ1ρ2
=(2M2γ minus (γ minus 1))(2 +M2(γ minus 1)
M2(γ + 1)2) (28)
Considering the 13ase of strong sho13ks where M may be 13onsidered to be
≫1 these equations be13ome
ρ2ρ1
=(γ + 1)
(γ minus 1)(29)
T2
T1=
2M2γ(γ minus 1)
(γ + 1)2(210)
kBT2 =2(γ minus 1)
(γ + 1)2mus
2(211)
8 CHAPTER 2 RADIATIVE SHOCK WAVES
For mono-atomi13 gases γ = 53 the 13ompression ratio is estimated to
be 4 and the post-sho13k temperature is
kBT2 =3
16mu1
2 =3
16mPAu1
2(212)
Temperature is thus proportional to the square of the sho13k velo13ity and
to the atomi13 mass Therefore to a13hieve a high temperature in the radiative
sho13k experiment we often use high atomi13 number and mass gases For Xe-
non (A=13129) the post-sho13k temperature is then equal to T2 = 0257u21(eV kms) whi13h for instan13e gives sim 520 eV for a sho13k propagating at
45 kms (see Fig 22)
Figure 22 Proles of temperature (Te = Ti = T ) and mass density (ρ) at 10ns for a sho13k propagating in Xenon at 01 bar (ρ1= 57 times 10
minus4g13m
minus3) with
a speed of sim 45 kms It is to note that the given pressure value are at 298 K
The simulation is performed with the HELIOS 13ode ex13luding the radiation ee13t
and for a polytropi13 gas (γ=53) The initial temperature of Xenon is set to 1
eV and with a negligible (10
minus5WmK) thermal 13ondu13tivity The Ma13h number
is then equal to 40 For this simulation the sho13k is laun13hed by a 100 microm thi13k
Aluminium piston moving at the velo13ity of 33 kms The position x=0 13orresponds
to the position of the XeAl interfa13e at time t=0
21 RANKINE-HUGONIOT(R-H) RELATIONS 9
211 Jump relations for a real gas
In 13ontrast to the dis13ussion on ideal gas presented in the previous se13tion
the 13ase of a real gas is a little more 13ompli13ated It is to note that in this
13ase a part of the kineti13 energy is used to ex13ite and ionize the post-sho13k
gas As a result its temperature is lower than that for the 13ase of ideal gas
In addition the pressure is also modied due to ionisation (Mi13haut et al
2004) as dis13ussed below
Let us now dene j as the ionization stage of the atom (varying between
0 and z) and i to be the atomi13 state in this ionisation stage We shall
denote by Pj amp Pji the fra13tional ionization of the ion and the population
of the atomi13 state i in the jth ionization state respe13tively Then the mean
ionization stage (average 13harge) per atom 13an be written as
z =
zsum
j=0
jPj (213)
and the ex13itation energy (in13luding ionization) per atom and per unit of
mass is
ǫexc =
sumzj=0
sum
i Pji Eji
mpA(214)
where Eji is the energy of the ionisation stage j in the atomi13 stage iThus the total pressure of gas 13an be written as
P = Pi + Pe (215)
where Pi and Pe are the ioni13 and ele13troni13 pressures 13ontributions
Using the 13harge neutrality 13ondition Ne = z Ni this gives
P = Ni(1 + z) kBT = ρkBT
m(1 + z) (216)
The thermal pressure and enthalpy of the real gas are given by
Pth = ρ(1 + z)
mkBT (217)
h =5
2
(1 + z)
mkBT + ǫexc (218)
Next the speed of sound in13luding the ee13t of ionization with the
average 13harge z in the medium is now modied by ionisation
Cs ≃radic
5
3
γ(z + 1)kBT
m(219)
Thus the 13ontinuity relations in the frame moving with the sho13k front
be13ome
ρ2u2 = ρ1u1 (220)
10 CHAPTER 2 RADIATIVE SHOCK WAVES
ρ2u22 + ρ2
kBT2
m(1 + z2) = ρ1u
21 + ρ1
kBT1
m(1 + z1) (221)
ρ2u2
[5
2
(1 + z2)
mkBT2 + ǫexc2 +
u222
]
= ρ1u1
[5
2
(1 + z1)
mkBT1 + ǫexc1 +
u212
]
(222)
Figure 23 Ele13tron temperature (Te) ion temperature (Ti) and mass density
(ρ) proles at 10 ns derived from 1-D simulation for the 13ase of ionisation (no
radiation) The sho13k propagates in Xe gas at 01 bar (ρ1= 57 times 10
minus4g13m
minus3by
sim 45 kms) It is to note that the given pressure value is at 298 K The simulation
is performed with the HELIOS (Te 6= Ti 13ase) using the PROPACEOS equation
of state without any radiation and with a negligible thermal 13ondu13tivity (10
minus05
WmK) The initial temperature in Xenon is set to 1 eV and thus the Ma13h
number is equal to 40 For this simulation the sho13k is laun13hed by a 100 microm thi13k
Aluminium piston moving at the velo13ity of 38 kms The position x=0 13orresponds
to the position of the XeAl interfa13e at time t=0
Similar to the Fig 22 however for the 13ase of a real gas the Fig
23 shows the variation the ele13tron and ion temperatures as well as the
mass density with the distan13e for a sho13k propagating in Xenon at sim 45
kms Now the post-sho13k ele13tron temperature peaks at 22 eV whi13h is
mu13h smaller than the maximum temperature of the ions (510 eV) The
13ompression is now equal to 10 instead of 4 as in the previous 13ase The ions
21 RANKINE-HUGONIOT(R-H) RELATIONS 11
are heated rst in the sho13k through ion-ion 13ollisions This ion temperature
de13reases and the ele13tron temperature in13reases as a 13onsequen13e of the
ele13tron-ion 13ollisions The two temperatures thus be13ome equal Sin13e the
momentum 13hange of the ions require only a few number of kineti13 13ollisions
the thi13kness of the abrupt transition between the pre-sho13k and the post-
sho13k is of the order of a mean free path of the ions
It should be noted that for sho13ks propagating with very high speed
the post-sho13k temperature is so high that the 13ompressed gas is fully ioni-
zed Therefore the ee13ts of ionisation and ex13itation be13ome negligible and
hen13e the 13ompression ratio rea13hes 4 as in the 13ase of the adiabati13 sho13k
212 Ee13t of radiation on sho13k waves
In the previous se13tion the ee13ts of radiation heating and 13ooling have
been negle13ted However the radiation emitted in a strong sho13k will ae13t
its stru13ture These ee13ts 13an be determined using mass momentum and
energy 13onservation equations whi13h now have to in13lude the 13ontributi-
ons of the radiative ux energy and pressure To this purpose and with
some simpli13ations we 13an use the same approa13h than in se13tion 21 still
assuming that the sho13k is stationary
Jump relations with radiation
Let us 13onsider a stationary sho13k propagating in an atomi13 gas far away from
the dis13ontinuity The medium is then opti13ally thi13k and we 13an negle13t
the 13ontribution of the radiative ux Let us also assume that the medium
is at LTE and that the radiation pressure and energy may be des13ribed as
follows
Prad =1
3Erad =
4
3cσT 4 =
1
3aradT
4(223)
where arad is the radiation density 13onstant and T is the sho13k radiation
temperature whi13h we shall suppose to be equal to the ele13tron temperature
Following Mi13haut et al (2004) the new jump 13onditions a13ross the sho13k
dis13ontinuity be13ome
ρ2u2 = ρ1u1 (224)
ρ2u22+ρ2
kBT2
m(1+z2)+
1
3aradT
42 = ρ1u
21+ρ1
kBT1
m(1+z1)+
1
3aradT
41 (225)
ρ2u2
(
5
2
(1 + z2)
mkBT2 + ǫexc2 +
u222
)
+4
3aradT
42 u2 =
ρ1u1
(
5
2
(1 + z1)
mkBT1 + ǫexc1 +
u212
)
+4
3aradT
41 u1 (226)
12 CHAPTER 2 RADIATIVE SHOCK WAVES
(a)
(b)
Figure 24 Figure 14 amp 17 of Mi13haut et al (2004) (a) Compression ρ2ρ1 (full13ir13les thin line) kT2 in eV (inverted full triangles thin line) in Argon (a) Xenon
(b) for initial 13onditions kT1( = 10 eV for Ar and = 01 eV for Xe) ρ1 = 5 times10minus4 gcm3
versus sho13k speed in kms with (full markers full lines) and without
(empty markers dashed lines) radiation The ionization stage lt z gt2 is plotted
with the diamond symbol
21 RANKINE-HUGONIOT(R-H) RELATIONS 13
In the radiative regime the 13ompression ratio ρ2ρ1 approa13hes to the
value of 7 at very high velo13ities (several hundred of kms) whereas it is equal
to 4 when the ee13ts of radiation energy and pressure have been negle13ted
This limit is the same for the 13ases of adiabati13 (Bouquet Teyssier and
Chieze 2000) as well as the real gas (Mi13haut et al 2004) This is due to
the fa13t that at very high velo13ities the medium be13omes fully ionized and
behaves as a perfe13t gas On the 13ontrary in the 13omparatively low velo13ity
regime the ee13t of ionisation and ex13itation dominates the radiation ee13ts
This is illustrated in Fig 24 taken from Mi13haut et al (2004) whi13h reports
the variations of the Ar and Xe 13ompression ratio versus the sho13k speed for
an initial mass density of 5 times 10
minus4g13m
3 At 50 kms the 13ompression
ratio and the temperature are respe13tively equal to (5 32eV) and (11 30
eV) for Argon and Xenon
The 13ompression ratio shows bumps for high velo13ities These bumps
may be attributed to the fa13t that as the temperature of the post-sho13k
in13reases with the velo13ity the atom ionises more and more Thus the ex-
ternal shells in its ioni13 atomi13 stru13ture are su1313essively opened indu13ing
a strong variation in the ex13itation energy
The post-sho13k temperature is very high for the very high-speed sho13ks
In su13h 13ases the 13ompressed gas is fully ionized and the ee13ts of ionisation
and ex13itation remain negligible like in the adiabati13 13ase It should be noted
that below 100 - 200 kms the ee13ts of radiation pressure remain negligible
Thus very energeti13 lasers like LMJ or NIF are required to rea13h a domain
where this ee13t 13an be observed
Thus whereas for the range of velo13ities of 20 - 50 kms 13overed by my
study the ee13ts of radiative energy and pressure 13an be negle13ted this is
not the 13ase for the radiative ux whi13h has been negle13ted in the previous
se13tions and whi13h may be13ome 13omparable to the material energy ux Its
ee13t on the system must be taken into a1313ount as will be shown below
213 Radiative hydrodynami13s equations
To in13lude the 13ontributions of radiation it is 13ompulsory to introdu13e the
relevant terms in the uid equations The propagation of the radiation in the
matter is governed by the pro13esses of absorption and emission of photons
whi13h respe13tively heats and 13ools the medium As the mean free path of
photons is always higher than the mean free path of parti13les the sho13k
stru13ture is more inuen13ed by the radiative transfer than the vis13osity of
the uid
The propagation of radiation in the matter is des13ribed by the radiative
transfer equation
(
1
c
part
partt+
part
parts
)
I(rn ν t) = η(r ν)minus χ(r ν) I(rn ν t) (227)
14 CHAPTER 2 RADIATIVE SHOCK WAVES
where χ(ρ ν T ) and η(ρ ν T ) are respe13tively the mono13hromati13 opa-
13ity and emissivity of the gas
The spe13i13 intensity I(rn ν t) (erg cmminus2 sminus1Hzminus1 srminus1) is the energy
radiated per units of surfa13e area time frequen13y and solid angle in the
dire13tion n
The radiative energy density Erad ux Frad and pressure Prad are re-
spe13tively dened as the zero rst and se13ond moments of the spe13i13 in-
tensity versus the angle after integration over the photon frequen13y ν
Erad =1
c
int ∮
I(rn ν t)dΩdν (228)
Frad =
int ∮
I (rn ν t) cos θ dΩdν (229)
Prad =1
c
int ∮
I (rn ν t) cos2 θ dΩdν (230)
(231)
Taking into a1313ount radiative ee13ts the equations of radiative hydro-
dynami13s in the 13o-moving frame of the sho13k (Mihalas and Mihalas 1999
Drake 2006) 13an be written as
partρ
partt= minusnabla(ρu) (232)
ρ
(
partu
partt+ unablau
)
= minusnabla(P + Prad) (233)
part
partt
(
ρu2
2+ ρǫ+ Erad
)
+nabla
[
ρu
(
ǫ+u2
2+
P
ρ
)
+ (Erad + Prad)u
]
= minusnablaFrad
(234)
As mentioned previously the 13ontributions linked to Prad and Erad are
ee13tive only at very high temperature (ie velo13ity) However for the
radiative sho13ks of interest in this work the 13ontribution of the radiative
ux Frad is non-negligible 13ompared to ρu3 These sho13ks are thus in the
radiation ux dominated regime
Most of the experimental sho13k waves are in this regime These hyd-
rodynami13 equations are strongly non-linear In the present work the one-
dimensional radiative-hydrodynami13 13ode `HELIOS (Ma13Farlane Golovkin
and Woodru 2006) has been employed to derive the physi13al parameters
of the sho13k in various 13onditions as presented in the following se13tion HE-
LIOS will be dis13ussed in detail in the next 13hapter
22 RADIATIVE SHOCK REGIMES 15
22 Radiative sho13k regimes
Radiative sho13ks are strong sho13ks (Ma13h number Mgtgt1) whi13h attain
high temperatures and thus are the sour13e of intense radiation (Mihalas
and Mihalas 1984 Zeldovi13h Zeldovi13h and Raizer 2002 Drake 2006)
Depending on the opa13ity the radiation emitted from the sho13k may be ab-
sorbed by the pre-sho13k region indu13ing its pre-heating Far away from the
dis13ontinuity the stru13ture of the upstream medium is determined by the
absorption On its side the stru13ture of the transition layer of the down-
stream medium 13lose to the dis13ontinuity depends on the balan13e between
the emission (13ooling) and the absorption (heating) Thus the full stru13ture
of the sho13k mainly depends on the variation of the opa13ity χ (13m
2g) The
opti13al depth (τ(s)) measured from the position of the jump is then the
relevant parameter for the absorption It is given by
τ(s) =
int s
sjump
χ(sprime)ρ(sprime) dsprime (235)
where sprime 13hara13terizes the path of the radiation As we are interested in
the mean frequen13y averaged radiative ux the relevant opa13ity here will be
the averaged Rosseland opa13ity whi13h will be dened in due 13ourse
If τ gtgt 1 the regime is referred as opti13ally thi13k while in the opposite
13ase (τ lt 1) it is opti13ally thin Flux dominated radiative sho13ks have
been 13lassied depending on the opti13al depth of upstream and downstream
regimes
An attempt of radiative sho13k 13lassi13ation has been performed by Drake
(Drake (2006)) a1313ording to the opti13al the thi13kness of the upstream and
downstream regimes resulting in the denition of four dierent regimes
namely a thin-thi13k thin-thin thi13k-thi13k thi13k-thin radiative sho13ks
Following this author a thin-thi13k type of radiative sho13k probably exists
only in theory In the 13ase of a thin-thin radiative sho13ks both upstream
and downstream regimes are opti13ally thin and the sho13k radiates away
its energy in both dire13tions This regime of radiative sho13k is present in
various astrophysi13al 13ontexts for instan13e in the internal sho13ks of stellar
jets (Hartigan 2003) It is possible to generate these type of sho13ks in the
laboratory in very low-density gases In thi13k-thi13k radiative sho13ks both
upstream and downstream zones are opti13ally thi13k This regime o1313urs
in the stellar interiors The radiation 13oming from the hot downstream
region heats the 13ooler upstream material and forms a pre13ursor The fourth
regime is the thi13k-thin radiative sho13k whi13h is 13hara13terized by an opti13ally
thi13k downstream and an opti13ally thin upstream region The radiation
from the heated downstream region passes ahead of the sho13k and 13reates a
13ooling layer in the downstream before rea13hing the nal state There is no
absorption of the radiation in the upstream region and therefore 13ontrary to
16 CHAPTER 2 RADIATIVE SHOCK WAVES
the thi13k-thi13k 13ase there is no radiative pre13ursor Sho13ks generated in the
laser experiments are usually between the thi13k-thi13k and thin-thin regimes
221 Typi13al radiative sho13k waves with a pre13ursor
The radiation from the post-sho13ked region just behind the sho13k passes
ahead of the sho13k and generates the pre13ursor However it also leads to a
radiative 13ooling layer downstream that extension depends on the opa13ity
A typi13al stru13ture of su13h sho13ks is presented in M13Clarren et al (2010)
In these 13ases the 13ompression in the 13ooling layer behind the sho13k is
mu13h higher than that in the hydrodynami13 13ase This kind of sho13k is
most 13ommonly generated in the laboratory experiments (Bozier et al 1986
Keiter et al 2002 Reighard et al 2006 Fleury et al 2002) In astrophysi13s
these sho13ks are present for instan13e in supernovae when the blast wave
emerges from the stellar interior (Ensman and Burrows 1992)
A detailed 13lassi13ation of the sho13ks presenting a radiative pre13ursor
is reported in several referen13es (Zeldovi13h Zeldovi13h and Raizer 2002
Ensman 1994 Vaytet et al 2013)
In the 13ase of sub-13riti13al sho13k the temperature just ahead of the sho13k
front is smaller than the post-sho13k temperature Su13h sho13ks are relatively
weak and hen13e the radiative ux transfer from the post-sho13k to the pre-
sho13k is small A piston moving with a 13onstant speed of 12 kms in Xenon
gas at 01 bar is able to generate su13h a sub-13riti13al radiative sho13k The
resulting mass density and ele13tron temperature proles have been presented
at 10 ns in the Fig 25a
The other regime of radiative sho13k is known as super13riti13al in whi13h
the pre-sho13k and the nal post-sho13k temperatures are the same The mass
density and ele13tron temperature proles of super13riti13al sho13k driven by a
13onstant moving piston with speed 45 kms in Xenon at 01 bar are presented
in Fig 25b
The speeds a13hieved in the PALS experiment range between sim 15 - 55
kms for pressure 13omprised between 01- 06 Our simulations indi13ate
that depending on the speeds the regimes 13over either the sub13riti13al or
super13riti13al 13ases
23 Radiative sho13k waves experiments
Radiative sho13k waves have been studied experimentally for more than a
de13ade on high-energy lasers fast magneti13 pin13h ma13hines and pulsed po-
wer fa13ilities Su13h fa13ilities are able to a13hieve strong sho13k 13onditions
(high-temperature high-pressure) over a very small volume and time Va-
rious diagnosti13s are able to re13ord the plasma 13onditions whi13h are then
13ompared with the results of numeri13al simulations The results of these
experiments are also employed to validate numeri13al 13odes some of them
23 RADIATIVE SHOCK WAVES EXPERIMENTS 17
(a)
(b)
Figure 25 Simulated ele13tron temperature (Te) ion temperature (Ti) and mass
density (ρ) proles at 10 ns of a radiative sho13k propagating with a 15 kms (a sub-
13riti13al) and 48 kms (b super13riti13al) The sho13k is propagating in Xenon at 01
bar (ρ = 54 times 10
minus4g13m
minus3 at 298 K) The initial temperature in Xenon is set to
1eV The two Ma13h numbers are respe13tively equal to 13 and 43 The radiation and
ionisation ee13ts are in13luded in the simulation whi13h is performed for two dierent
temperatures (Te 6= Ti) and a negligible thermal 13ondu13tivity (10
minus05WmK) For
this simulation the sho13k is laun13hed by a 100 microm thi13k Aluminium piston moving
at the speed of 12 kms and 45 kms for 13ase (a) and (b) respe13tively The position
x=0 13orresponds to the position of the XeAl interfa13e at time t=0
18 CHAPTER 2 RADIATIVE SHOCK WAVES
being relevant to the astrophysi13s like FLASH (Fryxell et al 2000) and
HERACLES (Gonzaacutelez Audit and Huynh 2007)
231 Laser-driven sho13k experiments
Most of laboratory experiments on radiative sho13ks have been performed
on large-s13ale laser fa13ilities (Bouquet et al 2004 Chaulagain et al 2015
Gonzaacutelez et al 2006 Reighard et al 2006 Stehleacute et al 2010 Doss et al
2009 Drake et al 2011 Diziegravere et al 2011 Stehleacute et al 2012) in noble
gases at pressures between 01 and 1 bar With laser intensities on the
target 13omprised between 1014 and 1015 W13m
2 these experiments allowed
to re13ord sho13k speeds ranging between 40 and 150 kms
Bozier et al (1986) for the rst time experimentally eviden13ed the ge-
neration of a radiative wave propagating ahead of a super13riti13al sho13k in
xenon (pressure up to 6 bars) at the CEAs Limeil laser fa13ilities (irradian13e
of 5times 10
14W13m
minus2) In this experiment the laser beam (wavelength of 106
microm duration 1 ns) of 40 J energy was fo13used on an aluminum foil a13ting
as a piston A sho13k of speed sim 50 kms was produ13ed in the gas A visible
streak 13amera was used to re13ord the pre13ursor emissivity over 10 ns
Later Keiter et al (2002) performed a similar experiment at the OMEGA
laser fa13ility (25 kJ 1ns fo13al spot diameter 600 microm) still in a planar
geometry The sho13k propagated here in a low-density SiO2 aerogel foam
(ρ sim 96 mg13m
3) In this experiment X-ray absorption spe13tros13opi13 data
were re13orded to qualitatively predi13t the temperature of the pre13ursor
In the year 2000 a new radiative sho13k experiment was performed in
Xenon at 02 bar at the LULI laser fa13ility (sim 100 J λ = 053 microm 4-
6times 10
13W13m
minus2) (Fleury et al (2002) Bouquet et al (2004)) This expe-
riment used a three-layered piston 13omposed of a thin polystyrene a13ting
as ablator (2 microm thi13k) a titanium foil (3microm) a13ting as x-ray shield and a
layer of a polyethylene foam (25 microm) as an a1313elerator Further a VISAR
(velo13ity interferometer system for any ree13tor) and a Ma13h - Zehnder in-
terferometer were used as a diagnosti13s to measure the front sho13k speed and
the pre13ursor ele13tron density Two streak 13ameras re13orded the sho13k pro-
pagation in longitudinal and transverse dire13tions This was the rst attempt
to estimate ele13tron density within the pre13ursor Their measurements indi-
13ate an ele13tron density ranging between 1018minus1020cm3 The experimental
results were in qualitative agreements with 1D numeri13al MULTI and FCI
simulations However the speed of the pre13ursor in these 1D simulations
was too large This was then explained later when multidimensional ee13ts
were in13luded in the simulations (Leygna13 et al 2006 Gonzaacutelez Audit and
Stehleacute 2009)
Complementary experiments were then performed by Gonzaacutelez et al
(2006) at the PALS laser fa13ility (60 J 035 ns 438 nm spot diameter 600microm
sim 7times 10
13W13m
minus2) in Xe gas The goal was to study the sho13ks over longer
23 RADIATIVE SHOCK WAVES EXPERIMENTS 19
times and to inspe13t the ee13ts of the walls albedo A doubled layered pis-
ton made of 10microm polystyrene and 05 microm gold was used The pre13ursor
edge was re13orded by shadowgraphy using a visible streak 13amera The ex-
perimental results were 13ompared with 2D numeri13al simulations performed
with HERACLES (Gonzaacutelez Audit and Huynh 2007) This 13omparison
allowed quantifying the albedo of the walls It also showed that the radia-
tive losses at the walls lead to a small 13urvature of the ionization front and
to a redu13tion of its longitudinal extension (Leygna13 et al 2006 Gonzaacutelez
Audit and Stehleacute 2009)
In parallel another experiment was performed at the OMEGA laser fa-
13ility (035microm spot sim 700 - 800microm sim 2times 10
15W13m
minus2) by Reighardt et
al (Reighard et al (2006) Reighard (2007)) The sho13k was imaged by an
X-ray ba13k-lighter at dierent times whi13h enabled to inspe13t for the rst
time the shape of the sho13k front At higher speeds (sim 100 kms) X-ray
radiography pointed out a 13ollapse of the post-sho13k (Reighard et al 2006)
whi13h was attributed to the radiation losses trough the sho13k front For
these high-speed 13onditions the wall heating leads to the development of se-
13ondary wall sho13ks intera13ting with the primary sho13k (Doss et al 2009)
Su13h wall sho13ks have not yet been observed at lower speeds
Subsequently an experiment by Stehleacute et al (2010) was performed at
PALS laser fa13ility with the similar laser parameters as taken in Gonzalez
et al (2006) In this experiment a Ma13h-Zehnder interferometer and a
streak 13amera were used to re13ord the sho13k propagation over 50 ns Time-
integrated XUV spe13tros13opi13 analysis between 16 and 22 nm of sho13k was
performed The experiment showed that of a quasi-stationary regime was
rea13hed after 20 ns
The rst experimental study in Argon gas was reported in 2012 by Vis13o
et al (2012) at OMEGA fa13ility (035microm spotsim 600 microm sim 7times 10
14W13m
minus2)
The ele13tron temperatures in the radiative pre13ursor and sho13k-front were
measured to be 34 eV and 60 eV respe13tively using X-ray Thomson s13atte-
ring diagnosti13
In addition a new experiment at PALS (Jungwirth et al 2001) was
presented by Chaulagain et al (2015) using for the rst time a Zn x-ray
laser for XUV imaging at 21 nm This diagnosti13 allowed imaging both the
pre13ursor and post-sho13k stru13ture of radiative sho13ks in Xe gas at 03 bar
In this experiment where the sho13k wave did not ll the tube 2D ee13ts
be13ame pronoun13ed The experimental results were su1313essfully 13ompared
with the results of 2D ARWEN simulations (Cotelo et al 2015)
A summary of these dierent experiments and their prin13ipal diagnosti13s
is presented in table 21
20
CHAPTER2RADIATIVESHOCKWAVES
Experiment Laser fa13ility I (times 1013) λ amp τ ρ0 u Prin13ipal diagnosti13s
W13m
minus2 microm ns mg 13m
minus3
km s
minus1
Bozier et al (1986) CEAs Limeil 20-50 0351 1 10 -30 50-80 Visible streak Camera
Fleury et al (2002) LULI 5 053 05 1 60-65 Visible streak Camera
Visible interferometry
VISAR
Keiter et al (2002) OMEGA 35- 85 0351 1 5 -15
lowast
100 x-ray spe13tros13opy
Gonzaacutelez et al (2006) PALS 10-15 0438 035 1 65 Visible streak 13amera
Visible interferometry
Reighard (2007) OMEGA 70-100 035 1 6 100-140 x-ray ba13k-lighting
VISAR
Stehleacute et al (2010) PALS 10-20 1315 035 1 60 Visible Interferometry
XUV spe13trometer
Stehleacute et al (2012) PALS 10-20 1315 035 15 50-55 XUV imaging
XUV fast Si diodes
Vis13o et al (2012) OMEGA 70 035 1 196
lowastlowast
100-150 x-ray Thomson S13atte-
ring
Chaulagain et al (2015) PALS 10-20 1315 035 15 50 XUV fast Si diodes
Table 21 Summary of the radiative sho13ks experiments and out13omes in dierent laser installations All the experiments are performed
in Xe ex13ept Keiter et al (2002) and Vis13o et al (2012) In these two experiments the rst uses SiO
lowast
2 and the se13ond uses Ar
lowastlowast
as a
sho13ked medium
23 RADIATIVE SHOCK WAVES EXPERIMENTS 21
Laser driven radiative sho13ks Goal of my thesis
All these experimental studies have been fo13used on the 13ase of single ra-
diative sho13ks propagating in a tube However in astrophysi13al 13onditions
radiative sho13ks often intera13t with a denser medium leading to the deve-
lopment of ree13ted and transmitted sho13ks A few representative examples
of su13h phenomena are the intera13tion of supernovae remnants with dense
mole13ular 13louds (Gonzaacutelez Audit and Stehleacute 2009) the a1313retion sho13ks
on the photosphere of T-Tauri stars (Orlando et al 2013) and the bow
sho13ks at the head of stellar jets (Hartigan 1989 Raga et al 1999) The
13ollision (or the intera13tion) of two radiative sho13k waves is obviously a rare
astrophysi13al event and the template 13ase of supernova remnant DEM L316
(see Fig 1 of Williams et al (1997)) is still the subje13t of debates (Williams
et al 2005 Toledo-Roy et al 2009 Velarde et al 2006) as the observation
of these two dierent sho13ks 13an be also interpreted as the superposition of
two blast waves in the eld of view of the teles13ope In this regard the de-
velopment of dedi13ated laboratory experiments to the study of propagation
and intera13tion of 13ounter-propagating sho13k waves is important as a tool to
13hara13terize su13h events through their spe13i13 signatures
In this thesis I will present the results of experiments performed at the
Prague Asterix Laser System (PALS) fa13ility (Jungwirth et al (2001)) on
the study of the intera13tion of two radiative sho13k waves The experimental
obje13tives were
To generate two sho13ks of dierent speeds propagating in opposite
dire13tions in the tube lled with Xenon gas at low pressure (lt 1 bar)
with speeds 13omprised between 12 and 55 kms
To re13ord the ele13tron density by using Ma13h-Zehnder Interferome-
ter over a duration 13omprised between 50 and 200 ns This allowed
studying the ee13t of pre13ursor intera13tion on to the other one
To determine the temperature and the ion 13harge of gas using a time
and spa13e integrated XUV emission spe13tros13opy
The experimental results were interpreted with the help of one-dimensional
simulations using the Lagrangian radiation hydrodynami13 13ode HELIOS
232 Ele13tromagneti13ally laun13hed sho13k by a 13ompa13t pul-
sed power devi13e
As dis13ussed in the previous se13tion most of the plasma laboratory astrop-
hysi13s studies are mainly performed on large-s13ale laser fa13ilities addressing
pure hydrodynami13 radiative sho13ks at very high speed (50 - 150 kms)
and moderate pressure (01 - 1 bar) Complementary to laser experiments
22 CHAPTER 2 RADIATIVE SHOCK WAVES
dis13harge produ13ed plasma has also been a well-established method for ge-
nerating the strong sho13ks (Kondo et al 2008 2009ab Larour et al 2015)
Su13h devi13e presents a high exibility and a larger repetition rate whi13h al-
lows a deeper understanding of sho13k physi13s in 13onditions of astrophysi13al
interest for instan13e in the 13ontext of stellar jets
In prin13iple a 13oaxial ele13trode pair lled with gas followed by an intense
ele13tri13al surfa13e dis13harge generates a thin layer of plasma (plasma sheath)
at the bottom of the ele13trode Due to the high rising 13urrent in the plasma
layer a magneti13 pressure is produ13ed whi13h a1313elerates the plasma layer
also termed as plasma fo13us (Gonzalez Brollo and Clausse 2009) During
this motion the plasma sheath a1313retes a noti13eable part of the ba13kground
gas Experiments performed by Serban and Lee (1995) in the light gasses
H2 D2 have shown fast axial sho13k waves (100 kms) but the plasma sheath
was annular
Later on 13ompa13t pulsed power ele13tri13 generators were able to laun13h
astrophysi13ally relevant strong sho13ks in low-pressure noble gases (Kondo
et al 2006) These authors reported sho13k of speed 45 kms using 13oni13al
13oaxial ele13trodes and Xenon gas (006 - 11 mbar) The ion - ele13tron energy
transfer and ionization relaxation pro13ess as also the radiation ee13t were
investigated by Kondo et al (2008 2009ab)
My work on ele13tromagneti13ally laun13hed sho13k experiment
My work fo13using on the ele13tromagneti13ally laun13hed strong sho13ks rele-
vant for astrophysi13s aimed to design and to optimize an ele13tromagneti13
generator to be able to produ13e strong sho13ks in noble gasses at low pres-
sures of few mbar The ele13tromagneti13 generator is a powerful a1313elerating
devi13e whi13h eje13ts a quasi-planar plasma sheath out of a set of 13oaxial 13o-
ni13al ele13trodes where a pulsed 100-kA 13urrent is passing
Due to the ele13tri13al dis13harge in the gas a hot and fast moving plasma
is generated whi13h travels along the ele13trodes The high-intensity 13urrents
produ13e a magneti13 eld of several Teslas whi13h a13ts as a piston a1313elerating
an annular plasma sheath and driving a strong sho13k towards the top of the
13one I have employed a simple model to optimize the operation parame-
ters With su13h optimization our ele13tromagneti13 generator should produ13e
strong sho13ks over few nanose13onds
Our preliminary experiments show that the generator is 13apable of laun-
13hing supersoni13 sho13ks in Argon in the form of a thin plasma layer with
the speed of sim 1 - 30 kms Three-dimensional MHD simulation performed
by A Ciardi is 13onsistent with the rst observations This is presented in
detail in the 13hapter 6 of the thesis
Chapter 3
1D Simulations
Contents
31 HELIOS 23
32 LTE approximation 24
321 Mean opa13ity 25
322 Equation of State 27
33 Single radiative sho13k waves 27
34 Intera13ting radiative sho13k waves 40
35 Summary 44
Radiative sho13ks are present in various astrophysi13al pro13esses implying
extreme 13onditions Laboratory experiments then allow investigating the un-
derlying physi13al pro13esses whi13h take pla13e in these 13onditions and whi13h
are not observable from the Earth due to a la13k of spatial resolution of the te-
les13opes instruments However experiments are not always straightforward
to interpret and the help of numeri13al simulations be13omes pre13ious
In this 13hapter I will present with the help of 1D radiative hydrodynami13
simulations a brief overview of the physi13s of isolated radiative sho13ks and
of two intera13ting radiative 13ounter-propagating sho13k waves
31 HELIOS
Dierent numeri13al approa13hes are used for the simulation of laboratory
plasmas Some of them use xed grids Other introdu13e the possibility of
renement (on the y) in some meshes (Adaptative Mesh Renement) to
maintain a good resolution in the interesting parts of the plasma Another
approa13h is based on meshes following the uid (Lagrangian des13ription)
whi13h means that the meshes move with the uid and no matter may go
through a mesh to another The form of the equations dier in this 13ase (see
for instan13e Orban et al (2013))
HELIOS is su13h a one-dimensional Lagrangian radiation hydrodynami13
13ommer13ial 13ode (Ma13Farlane Golovkin and Woodru 2006) and I have
used it to simulate our experiment of intera13ting sho13k waves This 13ode
is easy to handle and has the presently interesting 13apability to be able to
simulate the 13ase of two sho13k waves propagating in opposite dire13tions and
laun13hed by two dierent lasers HELIOS may des13ribe non-LTE plasmas as
23
24 CHAPTER 3 1D SIMULATIONS
it in13ludes an in-line 13ollisional radiative (HELIOS-CR) model for 13omputing
non-LTE atomi13 level populations at ea13h time step of the hydrodynami13s
simulation However I used HELIOS in the LTE approximation whi13h is
appropriate to our experimental 13onditions (Rodriguez et al 2011) as will
be dis13ussed in the next se13tion
The 13ode solves the uid equations of motion using the pressure 13ontri-
butions of ele13trons ions and radiation It may des13ribe the ele13trons and
ions as two intera13ting uids at respe13tive temperatures Te and Ti Ther-
mal 13ondu13tion is des13ribed within a ux-limited ele13tron 13ondu13tion model
based on the Spitzer 13ondu13tivity (Burhop and Spitzer 1957) and the laser
energy deposition is 13omputed with an inverse Bremsstrahlung model ()
The radiation emission and absorption terms are introdu13ed in the equa-
tion of energy for the ele13trons and in the radiation transport equations
One of the methods whi13h are proposed is the ux-limited multi-group
radiation diusion model (Gonzaacutelez et al 2015) where the radiative ux is
proportional to the gradient of the radiative energy and is inversely propor-
tional to the Rosseland opa13ity (Dobbs-Dixon Cumming and Lin 2010)
The expression is pondered by a ux-limited diusion 13oe13ient in order
to obtain the good opti13ally thin limit This 13oe13ient follows the Larsen
expression des13ribed in Olson Auer and Hall (2000) The se13ond method is
a (time independent) multi-angle short-13hara13teristi13s s13heme In our 13ase
where radiation and hydrodynami13s are strongly 13oupled and thus the ra-
diation eld varies rapidly we used the diusion model together with LTE
Plan13k and Rosseland multi-groups opa13ities
In addition our version of HELIOS uses the PROPACEOS (Ma13Farlane
Golovkin and Woodru 2006) Equation of State (EOS) and multi-group
opa13ity tables whi13h are generated by the 13ode
1
and it is not possible to
in13lude any other opa13ity or EOS table
32 Lo13al Thermodynami13 Equilibrium
The knowledge of the thermodynami13al 13onditions is required for an ade-
quate simulation of the plasma des13ription
The Lo13al Thermodynami13 Equilibrium (LTE) regime is valid at large
ele13tron densities In that 13ase the 13ollisions between the ele13trons and the
ions and atoms are so frequent that they maintain at ea13h time a steady-
state populations of ions Moreover within a given ioni13 stage the popu-
lations of the dierent energy levels follow the Boltzmann equilibrium and
the populations of the dierent ions the Saha-Boltzmann equation (Fadeyev
and Gillet 2001)
On the 13ontrary in the 13ase of Non Lo13al Thermodynami13 Equilibrium
(NLTE) regime one has to take into a1313ount the ex13itationdeex13itation
1
see http wwwprismminus cscomSoftwarePROPACEOS
32 LTE APPROXIMATION 25
indu13ed by the ele13tron 13ollisions and various radiative pro13esses This leads
to a set of dierential time dependent equations (rate equations) whi13h has
to be solved through and adequate Collisional Radiative model (CR) (Kunze
2009)
Therefore it is 13ru13ial to determine if the regime is LTE or NLTE A
13riterium is given by Griem (2005)
In our experiment the mass density ranges between sim 10
minus4and 10
minus1
g13m
3 while the temperature values are 13omprised between andsim 0 to 50 eV
The pre13ise 13al13ulations performed by Rodriguez et al (Fig2 of Rodriguez
et al (2011)) for Xenon indi13ate that the thermodynami13al 13onditions of
our radiative sho13k experiments 13orrespond to the LTE regime
321 Mean opa13ity
The opa13ity is a fundamental ingredient of the radiative transfer equations
whi13h provide the spe13i13 intensity and its moments (radiative ux energy
and pressure) These last ones enter in the uid equations (see se13tion 213
in 13hapter 2) The 13oupling of the uid equations with the radiative transfer
equation is demanding in terms of 13omputational resour13es and most of the
13odes use simplied radiation transport equations whi13h require the know-
ledge of adequate frequen13y averaged opa13ity 13oe13ients The form of the
average of the opa13ity is not obvious as the average has to be done on the
radiative ux or energy whi13h are unknown before the 13omputation Then
the 13ommonly used opa13ities are the Plan13k and Rosseland mean opa13ities
(Mihalas 1978 Seaton 1987)
The Rosseland mean opa13ity (κR in 13m
minus1and χR = κRρ in 13m
2g) isderived by 13onsidering harmoni13 weighting over the temperature derivative
of the Plan13k fun13tion It gives the 13orre13t radiative ux in the opti13ally
thi13k diusion limit The average is performed over the total mono13hromati13
opa13ity κtotν = κabsν +κscatν (in 13m
minus1) where in the present 13ase the s13attering
13ontribution 13omes from the ele13tron Thomson s13attering
1
κR=
int
infin
01
κtotν
dBν
dT dνint
infin
0dBν
dT dν(31)
The Plan13k mean opa13ity (κP in 13m
minus1and χP = κP ρ in 13m
2g) employs
the normalized Plan13k bla13kbody energy density distribution as a weighting
fa13tor It will give the 13orre13t value for the integrated thermal LTE emission
of an opti13ally thin plasma and is expressed as
κP =
int
infin
0 κabsν Bνdνint
infin
0 Bνdν(32)
In the multi-groups des13ription the radiation transport takes into a13-
13ount the propagation of radiation in N dierent wavelength bands (νk minusνk+1 k = 1 N) Hen13e for instan13e the medium may be transparent for
26 CHAPTER 3 1D SIMULATIONS
some wavelengths and opaque for other ones The radiation transport equa-
tions split into N 13ontributions ea13h of them being asso13iated to adequate
opa13ity 13oe13ients The expression of the 13oe13ients is similar to the pre-
vious grey 13ase (N =1) ex13ept that the integrals are performed between
the boundary frequen13ies of the group
In HELIOS the mono13hromati13 opa13ities κν are 13omputed for the die-
rent radiation groups using the ATBASE atomi13 suite of 13odes Energy levels
and other atomi13 data relevant for bound-bound and bound-free transitions
are 13omputed within a 13onguration intera13tion model using Hartree-Fo13k
wave-fun13tions (Ma13Farlane Golovkin and Woodru 2006)
Dividing the opa13ity by the mass density redu13es the variation of this
quantity with ρ The literature then often uses the mass absorption 13oe-
13ient (also termed as opa13ity) whi13h is dened as
χν = κνρ (33)
Figure 31 PROPACEOS Mono13hromati13 opa13ity versus the photon energy in
eV of Xenon at 10 eV and at two ele13tron densities respe13tively equal to 1018 and
1020 13mminus3
An example of the variations of the PROPACEOS opa13ity χν versus the
frequen13y in eV is given in Fig 310a for Xenon at 10 eV and two ele13tron
densities respe13tively equal to 1018 and 1020 13m
minus3 whi13h do 13orrespond to
typi13al 13onditions of the pre13ursor and post-sho13k of our laser generated
radiative sho13ks
33 SINGLE RADIATIVE SHOCK WAVES 27
322 Equation of State
The equation of state (EOS) denes the dependen13e of the pressure ionisa-
tion internal energy with the mass density and temperature Several models
do exist in the literature
Our version of HELIOS for LTE plasmas may handle polytropi13 or PRO-
PACEOS equation of state Unfortunately there are not many details about
this last method The authors of the 13ode mention that it uses a QEOS-type
model (More et al 1988) in the strong 13oupling regime (ie at relatively
high densities and low temperatures) and an isolated atom model whi13h
uses a detailed atomi13 stru13ture modeling in the weak 13oupling region The
properties between the weak and strong 13oupling regimes are said to be obtai-
ned by interpolating in a thermodynami13ally 13onsistent manner This EOS
in13ludes the 13ontributions from the translation of ions and ele13trons ele13-
tron degenera13y atom ionization and ex13itation and Coulomb intera13tions
(Debye - Hu13kel 13orre13tion)
Comparisons on spe13i13 13ases have been performed by the authors of the
13ode with Los Alamos SESAME data for CH Al and Au showing a good
agreement when using these two equations of state
In this 13hapter HELIOS 13ode will then be employed to simulate the
detailed 13hara13teristi13s of single and 13ounter-propagating radiative sho13ks
driven by one or two lasers in the 13onditions of our experiments (Chapter 5)
The target 13ell (4-mm length) is lled with Xenon gas at 01 bar This valueof pressure is representative and 13orrespond to standard ambient tempera-
ture ie 298 K and not the initial temperature 13onsidered in the simulations
This terminology is maintained throughout the thesis Two lasers are inte-
ra13ting at both ends with two gold-13oated `CH foils 11microm CH and 06 micromAu) ea13h of them resulting in a piston of total thi13kness 116 microm 13losing the
13ell
33 Single radiative sho13k waves
Typi13al simulations of the radiative sho13ks espe13ially multidimensional ones
whi13h are time-13onsuming use a frequen13y independent (grey or one group)
opa13ities However then the opa13ity of the gas under investigation presents
strong variations with the frequen13y a multi-group approa13h may be suitable
(Vaytet et al 2011 2013) When the number N of groups tends to innity
ea13h group opa13ity κkR and κkP tends to the lo13al mono13hromati13 opa13ity κν whi13h solves the question of the adequate opa13ity average
In the following I will investigate the inuen13e of the opa13ity and the
number of groups on the stru13ture of a sho13k wave propagating at sim 50 kms
in Xenon at 01 bar To this purpose I have performed HELIOS simulations
for a laser of uen13e 28000 Jcm2whi13h is able to drive a sho13k with the
relevant velo13ity In addition I have also analyzed the ee13ts of taking into
28 CHAPTER 3 1D SIMULATIONS
a1313ount the dierent ion and ele13tron temperatures
Referen13e simulation (1 group Te = Ti = T )
A result of the simulation is reported at 10 ns in the Fig 32 whi13h reports
the variations of ele13tron density (Ne) temperature (T ) mean 13harge (Z)and mass density (ρ) The initial 13onditions are 45 times 10
minus4g13m
3and T =
1 eV
The sho13k front is lo13ated at sim 0047 13m (speed of 47 kms) At this
position the temperature peaks at 22 eV The bla13k verti13al dotted line shows
the position of the interfa13e between the piston and the ba13king Xenon gas
This interfa13e is lo13ated at sim 20 microm from to sho13k front The post-sho13k gas
mass density pressure ele13tron density and ion 13harge are respe13tively equal
to 002 g13m
3 11times 10
4bars 52times 10
2013m
3and 56 The high 13ompression
is due to ionisation and radiation 13ooling (13hapter 2) Its temperature of 12
eV is the same than in the pre13ursor where it is 13onstant over the 2 mm of the
simulation The pre13ursor ele13tron density and mean 13harge are respe13tively
equal to 18 times 10
1913m
minus3and 7 The pre13ursor is not 13ompressed
The variations of the Rosseland opa13ities (13m
2g
minus1and 13m
minus1) with the
distan13e are reported in the Fig 33 The Rosseland opa13ity is low in the
pre13ursor Hen13e 600 13m
2g
minus113orresponds to 05 13m
minus1 whi13h means that
an opti13al depth of 1 is rea13hed at 2 13m from the sho13k front This explains
the at prole of the pre13ursor With a typi13al value in the post-sho13k of
2577 in 13m
2g
minus1 this opa13ity is now 53 13m
minus1 and an opti13al depth of 1 is
here rea13hed at 200 microm from the sho13k front (whi13h is mu13h smaller than
the post sho13k extension)
This simulation was performed using the Spitzer thermal 13ondu13tivity
To 13he13k the 13ondu13tivity ee13t I performed another simulation with a
negligible thermal 13ondu13tivity (ie 10
minus12WmK) for the Xenon layers
only The results are reported in the Fig 34 The interfa13e between the
piston and the ba13king Xenon gas is still 13lose to the sho13k front Ex13ept
for the peak of temperature whi13h is higher (sim 29 eV instead of 22 eV) and
thiner the sho13k stru13ture is similar to the previous 13ase
Two uids with dierent temperatures Te and Ti (1 group)
In the previous se13tion HELIOS simulations have been 13arried out for the
13ase of equal ion and ele13tron temperatures I will now investigate the ee13t
of su13h approximation on the simulated results employing 1) Spitzer and 2)
negligible 13onstant thermal 13ondu13tivity in the Xenon layers It is important
to note that this two uids approximation 13an not be restri13ted to the Xenon
layers only and thus it is valid from the piston (CHAu) up to the Xenon
gas Thus the dynami13s and 13onditions of the piston are also modied whi13h
may slightly modify the sho13k velo13ity
33 SINGLE RADIATIVE SHOCK WAVES 29
(a)
(b)
Figure 32 Mass density and temperature (a) ele13tron density and mean 13harge
(b) at 10 ns for a radiative sho13k of speed sim 47 kms in Xenon at 01 bar The
verti13al dotted bla13k lines show the position of the interfa13e between piston and
ba13king Xenon gas Zero at x-axis is the interfa13e of piston and Xenon at time zero
Spitzer thermal 13ondu13tivity has been used in the simulation
30 CHAPTER 3 1D SIMULATIONS
Figure 33 Mean Rosseland opa13ity (in 13m
2g
minus1) and opa13ity (13m
minus1) at 10 ns
for a radiative sho13k of speed sim 47 kms in Xenon at 01 bar The verti13al dotted
bla13k lines show the position of the interfa13e between piston and ba13king Xenon gas
Zero at x-axis is the interfa13e of piston and Xenon at time zero Spitzer thermal
13ondu13tivity has been used in the simulation
33 SINGLE RADIATIVE SHOCK WAVES 31
(a)
(b)
Figure 34 (a) Mass density and temperature (b) ele13tron density and mean
13harge at 10 ns for a radiative sho13k of speed sim 47 kms in Xenon at 01 bar The
verti13al dotted bla13k lines show the position of the interfa13e between piston and
ba13king Xenon gas Zero at x-axis is the interfa13e of piston and Xenon at time zero
A negligible 13onstant thermal 13ondu13tivity (10
minus12WmK) have been used in the
simulation for Xenon
32 CHAPTER 3 1D SIMULATIONS
(a)
(b)
Figure 35 (a) Ele13tron density and ele13tron temperature (b) Mass density and
mean 13harge at 10 ns for a radiative sho13k of speed sim 48 kms in Xenon at 01
bar for the 13ase when the ele13tron and ion temperature are dierent The verti13al
dotted bla13k lines show the position of the interfa13e between piston and ba13king
Xenon gas Zero at x-axis 13orresponds to the position of the interfa13e between the
piston and Xenon at time zero The Spitzer thermal 13ondu13tivity has been used in
this simulation
33 SINGLE RADIATIVE SHOCK WAVES 33
The Fig 35a reports the sho13k stru13ture at 10 ns for the 13ase of Spitzer
thermal 13ondu13tivity The sho13k is lo13ated at 0048 13m The ion temperature
peaks at 53 eV and the ele13tron temperature peaks at the same value of 20
eV than previously Ex13ept for the ion temperature the sho13k stru13ture is
not signi13antly 13hanged 13ompared to the 13ase of equal temperatures
Let us now inspe13t the role played by the thermal 13ondu13tivity (see Fig
36) In the 13ase of a negligible thermal 13ondu13tivity of (10
minus12WmK) for
the Xenon layers only the ele13tron temperature (and as a 13onsequen13e the
mean 13harge) peaks at higher value 25 eV (instead of 20 eV with the Spitzer
13ondu13tivity) and the ion temperature rea13hes to 405 eV (instead of 53 eV)
The rest of the sho13k is not ae13ted by this 13hange in the 13ondu13tivity This
13onrms that the width and height of the ion peak temperature are a dire13t
13onsequen13e of the thermal 13ondu13tivity
Several groups for the opa13ity one temperature
As mentioned earlier the number of frequen13y groups inuen13es the dyn-
ami13s and morphology of radiative sho13ks A set of simulations has been
performed 13onsidering (only for Xenon) dierent numbers (N) of frequen13y
group varying between 1 and 100 for Te = Ti The distribution in energy
of the groups is logarithmi13 and the grid is set up with approximately 85
of the groups having photon energies between 01 eV and 3 KeV while the
remaining 15 lie between 3 KeV and 1 MeV
The Figures 37 (a) and (b) show the variations of the ele13tron tempe-
rature with the number of groups (from 1 to 100) for the pre13ursor (a) and
the front (b) In13reasing the number of groups ae13ts mostly the pre13ursor
A similar behavior is found in Vaytet et al (2013)
It should be noted that this multigroup opa13ity treatment is applied
to all the layers (ex13ept gold only one group opa13ity is provided by the
13ompany) in the simulation This 13ould ae13t slightly the dynami13s of the
piston Con13erning the pre13ursor at this time of the simulation and taking
an a1313ountable variation in temperature of 10 the 13onvergen13e is a13hieved
after 20 groups
Variation with the opa13ity
In order to estimate the ee13t of the opa13ity I have performed several simu-
lations by in13reasing the Plan13k and Rosseland opa13ities in Xenon (Stehleacute
et al 2014b) using a 13ommon multiplier ranging between 1 to 40 Su13h
pro13edure is often used to redu13e the gap between the available and more
sophisti13ated opa13ities To ex13lude the impa13ts of the radiation 13oming from
the post-sho13k to the piston layers the opa13ity of the piston (CH and Au)
layers have been set to zero
I present in Fig 38 the results (proles of temperature mass density and
34 CHAPTER 3 1D SIMULATIONS
(a)
(b)
Figure 36 (a) Mass density and ele13tron temperature (b) ele13tron density and
mean 13harge at 10 ns for a radiative sho13k of speed sim 45 kms in Xenon at 01 bar
for the 13ase when the ele13tron and ion temperature are dierent The verti13al dotted
bla13k lines show the position of the interfa13e between piston and ba13king Xenon
gas Zero at x-axis 13orresponds to the position of the interfa13e between the piston
and Xenon at time zero In this simulation value of thermal 13ondu13tivity (10
minus12
WmK) is negligible for Xenon The peak value of ion temperature is found to be
405 eV whereas theoreti13ally this value is expe13ted to be sim 600 eV It is possible
to a13hieve the expe13ted value by in13reasing the resolution of the simulation
33 SINGLE RADIATIVE SHOCK WAVES 35
Figure 37 Ele13tron temperature proles for various numbers of frequen13y groups
N = 1 20 50 60 70 90 and 100
mean 13harge) of four 13ases 13orresponding to an opa13ity multiplier equal to
1 10 30 and 40 The ele13tron and ion temperatures are taken to be equal
In13reasing the opa13ity in13reases the absorption in the pre13ursor and thus
de13reases its length as the photons emitted by the sho13k are more rapidly
absorbed by the 13old pre-sho13k Taking as a referen13e the previous value of
the Rosseland opa13ity of 600 13m
2g
minus1and a multipli13ative fa13tor of 40 the
opti13al depth of 1 will now be rea13hed at 005 13m (instead of 2 13m without
any multiplying fa13tor)
The temperature peak value at the sho13k front also de13reases I do believe
that this 13an be attributed to the in13rease of the radiative 13ooling of the peak
of temperature with the multiplying fa13tor
I adopted here an oversimplied method in the sele13tion of the groups
boundaries A more realisti13 one would be to take into a1313ount the variation
of the opa13ity in the dierent layers with the wavelength In the 13ase of
xenon for instan13e it would be then more adapted to rene the des13ription
of the opa13ity between 5 and 100 eV (see Figure 310a)
Opa13ities 13omparison
The previous study illustrates the role played by the opa13ity for the radi-
ative sho13ks waves with a pre13ursor It seams then logi13al to 13ompare the
36 CHAPTER 3 1D SIMULATIONS
(a)
(b)
(13)
Figure 38 (a) Ele13tron temperature (a) mean 13harge (b) and mass density (13)
for four (1 10 30 and 40) multipliers of the Xenon opa13ity at 10 ns
33 SINGLE RADIATIVE SHOCK WAVES 37
PROPACEOS opa13ity whi13h is used by HELIOS with more sophisti13ated
13al13ulations
The Figures 39a and 39b report the variations of the Plan13k and Ros-
seland grey opa13ities 13omputed for the density of the pre13ursor ρ = 15 times10
minus3g13m
3by two dierent 13odes PROPACEOS (Ma13Farlane Golovkin
and Woodru 2006) and BiGBART (de la Varga et al 2011 Ogando and
Velarde 2001) PROPACEOS opa13ities and EOS used in HELIOS simula-
tions 13omputes frequen13y-dependent opa13ities along with spe13i13 internal
energies and pressures The atomi13 13ode BiGBART able to 13al13ulate two
type of opa13ities dependent on atomi13 stru13ture and frequen13y dependent
The opa13ities 13omputed by Rodriacuteguez et al (2015) are also reported for 13om-
parison The two last 13al13ulations use the FAC (Flexible Atomi13 Code) (Gu
2008) for the 13al13ulation of the atomi13 data
We see that for a temperature equal to 10 eV the Rosseland opa13ity is
equal respe13tively to 800 15000 and 8000 g13m
2 whereas the Plan13k opa13ity
is equal to 18000 65000 370000 g13m
2 This means that the PROPACEOS
opa13ity whi13h is used in HELIOS is smaller by more than one order of
magnitude than the two last opa13ities As it is not possible to 13hange the
opa13ity model within HELIOS we then de13ided to use the PROPACEOS
opa13ity with a multipli13ative fa13tor whi13h we have set equal to 20
The dis13repan13ies in the average opa13ity are a 13onsequen13e of important
dieren13es in the mono13hromati13 opa13ities This is illustrated in the Fig
310 whi13h 13ompares the mono13hromati13 opa13ities given by PROPACEOS
and Rodriacuteguez et al (2015) (see Fig 16 of Rodriacuteguez et al (2015)) at 15
eV and 15 times 10
minus3g13m
minus3) This gure indi13ates a noti13eable dieren13e
between 5 and 150 eV with more bb transitions for the last method This
explains the dieren13es observed in the 13orresponding averages
Synthesis
These dierent simulations for a sho13k propagating at sim 50 kms in Xenon
at 01 bar show that the simulations performed with only one temperature
are adequate for the des13ription of the sho13k
The opa13ity strongly ae13ts the pre13ursor extension whi13h de13reases from
more than 4 mm for 1 group to 3 mm then the number of groups is larger
than 20 The 13onvergen13e versus the number of groups is rea13hed after 20
groups
To t with more sophisti13ated opa13ities and as it was impossible to
in13lude them in HELIOS I de13ided to multiply the opa13ities of Xenon by a
fa13tor of 20 This redu13es strongly the pre13ursor extension up to sim 1 mm
at 10 ns Taking into a1313ount the present huge un13ertainty in the opa13ity
I 13onsidered that it was not ne13essary to use multi-group opa13ities This
also means that our 1D simulations will be used - only - for a qualitative
interpretation of the experimental results A quantitative 13omparison should
38 CHAPTER 3 1D SIMULATIONS
(a)
(b)
Figure 39 Plan13k and Rosseland mono13hromati13 opa13ity proles Te for ρ =15
times 10
minus3from PROPACEOS (I) BIGBART (II) and Rodriguez et al (III)
33 SINGLE RADIATIVE SHOCK WAVES 39
(a)
(b)
Figure 310 Mono13hromati13 Xenon opa13ity versus the photon energy in eV at T
= 15 eV and for ρ = 15 times 10
minus3g13m
minus3for the PROPACEOS (a) and Rodriacuteguez
et al (2015) (Fig 16 of Rodriacuteguez et al (2015)) (b) models
40 CHAPTER 3 1D SIMULATIONS
be in the future based on adequate opa13ities and multi-groups modeling
34 Intera13ting radiative sho13k waves
In order to pre13ise what is expe13ted from the experimental study I will now
investigate the 13hara13teristi13 parameters of two 13ounter-streaming sho13ks
propagating in Xenon at 01 bar (ρ = 54 times 10
minus4g13m
minus3) with equal and
dierent velo13ities To highlight the ee13t of the intera13tion I will 13ompare
the results with those obtained with 1D simulations All the simulations will
be performed with HELIOS For the Xenon opa13ity a multiplier times 20 will
be used The number of groups will be set to be 1 and the ele13tron and ion
temperatures will be supposed to be equal
Three representative sets will be 13onsidered
Case(I) two identi13al radiative sho13ks (2RS) at sim 50 kms propaga-
ting in opposite dire13tions (ie starting from the left and right end
respe13tively)
Case(II) same 13onditions but without any 13oupling with radiation
Case(III) two 13ounter-propagating radiative sho13ks propagating with
dierent speeds sim 50 kms from the left side of the 13ell and sim 20
kms from the right side
To a13hieve the aforementioned speeds on the left and right sides the
uen13es of the two laser beams have been adjusted The pulse duration is
set to 03 ns (peak at 015 ns) to reprodu13e the experimental 13onditions
detailed later in the manus13ript
Case(I) 13ounter propagating sho13ks with the same speed
This a13ademi13al 13ase is fully symmetri13al and it is equivalent to the 13ase of
one RS with a fully ree13tive boundary (for hydrodynami13s and radiation)
in the middle of the tube The Fig 311 shows the variations of the ele13tron
density (Ne) and temperature (Te) in the Xenon layers The two sho13ks
appear in Xenon at sim 2 ns and the 13ollision o1313urs at sim 38 ns At 3 ns the
pre13ursor extension is sim 008 13m whereas the post-sho13k ele13tron density
and ele13tron temperature are 78 times 10
2013m
minus3and 16 eV respe13tively The
length of pre13ursor in13reases rapidly with time and the two pre13ursors merge
suddenly at sim 8 ns After this time the merging ee13t in13reases signi13antly
It is 13hara13terized by a at 13ommon pre13ursor those ele13tron density and
temperature are in13reasing with time At the time of the 13ollision (sim 38
ns) the post-sho13k mass and ele13tron density in13rease from 0011 to 014
g13m
minus3and 67 times 10
20to 66 times 10
2113m
minus3 whereas the temperature rises
up to 39 eV The 13ollision leads to the development of two reverse sho13k
34 INTERACTING RADIATIVE SHOCK WAVES 41
(a)
(b)
Figure 311 Ele13tron density Ne (a) and ele13tron temperature Te (b) versus axial
position (along a 04 13m long sho13k tube) at 3 10 20 30 and 38 ns from HELIOS
simulations (with opa13ity times 20) for the 13ases of single sho13k of sim 50 kms (dotted
line) and two identi13al 13ounter-propagating sho13ks of sim 50 kms (solid lines) The
verti13al dotted lines show the position of the interfa13e between piston and ba13king
Xenon gas
42 CHAPTER 3 1D SIMULATIONS
waves propagating ba13k with a speed of sim 15 kms rst in Xenon and later
on in the dierent piston layers (not shown in the Figure) These reverse
sho13ks lead to a dense plasma (Ne gt 10
2113m
minus3) whi13h is not a1313essible to
the experimental diagnosti13s and will not be detailed here
To investigate the ee13ts of the intera13tion I have reported in Fig 311
the 13ase of 1RS moving from the left to the right dire13tion in the 13ell (dotted
lines) The wave propagates identi13ally to 2RS until 10 ns After this time
the proles of the temperature and ele13tron density dier strongly from the
previous 13ase and their values are lower than for the 2RS The post-sho13k
extension is slightly smaller than for the 2RS and the sho13k speed is thus
slightly smaller too This last ee13t is due to the fa13t that for the two inte-
ra13ting waves the pre13ursor is at a higher temperature than for the isolated
sho13k and thus the sho13k propagates in a warmer medium then modifying
the sho13k 13onditions (sound speed opa13ity et13 ) A similar ee13t in the post
- sho13k extension may be seen for an isolated sho13k wave propagating in a
warmer pre-sho13k gas
Case (II) Identi13al 13ounter-propagating sho13ks for an ideal gas
In order to highlight the ee13t of the radiation another simulation has been
performed with the same set of parameters as in the 13ase I however putting
the Xenon opa13ity equal to zero I still use here the PROPACEOS equation
of state The result of the simulation is presented in Fig312 The 13ollision
time is now 40 ns instead of 38 ns (thus the sho13k speed is slightly smaller)
The post-sho13k is no more 13ompressed by radiation 13ooling Its 13ompression
at 10 ns is 10 instead of 35 This 13ompression is due to the vis13ous sho13k
(fa13tor 4) followed by the ionisationex13itation of the gasThere is obviously
no radiative pre13ursor Moreover there are no dieren13es in the Ne and Tproles of the single sho13k and that of the two 13ounter-propagating sho13ks
before the 13ollision time
Case (III) Two sho13ks at dierent speeds
The spatial and temporal variations of Ne and Te are plotted at times 3
10 20 30 38 and 49 ns in Fig 313a and 313b The left and the right
sho13ks appear in Xenon at sim 2 and 3 ns respe13tively Later at 10 ns
the two pre13ursor extensions are respe13tively equal to 018 and 0034 13m
The merging of the two pre13ursors starts at sim 15 ns As expe13ted from the
values of sho13k speeds the 13ollision time o1313urs at 49 ns whi13h is delayed
in 13omparison to the 13ase (I)
In 2RS the two radiative pre13ursors merging results in a plateau for the
ele13tron density and the temperature The temperature at 13ollision time is
now 28 eV instead of 39 eV in the 13ase I and the ele13tron density rea13hes
up to 31 times 10
2113m
minus3instead of 66 times 10
2113m
minus3
34 INTERACTING RADIATIVE SHOCK WAVES 43
(a)
(b)
Figure 312 Variations of Ne (a) and Te (b) versus axial position for the 13ase
of two identi13al 13ounter-propagating sho13ks (of speeds sim 50 kms) at 3 10 20
30 35 and 40 ns as derived from HELIOS simulations For these simulations we
have negle13ted the ee13t of radiation by keeping the Xe opa13ity equal to zero
The verti13al dotted lines show the position of the interfa13e between the piston and
ba13king Xenon gas
44 CHAPTER 3 1D SIMULATIONS
It may be noted that up to this 13ollision time the post-sho13k density
13onditions and speed are identi13al for the 13ase of an isolated RS propagating
from the left (1RSL) at 50 kms and this present left sho13k This reveals
that in the 13ase of 2RS there is no noti13eable ee13t of the right sho13k with
speed 20 kms on the left post-sho13k of 50 kms
On the 13ontrary we note a dieren13e in extension of the right post sho13k
from the 2RS 13ompared with the 13ase of an isolated sho13k (1RSR) propa-
gating at 20 kms from the right whi13h indi13ates that the left sho13k of the
2RS has an inuen13e on the right post-sho13k (Fig 313a)
35 Summary
Radiative sho13k waves propagating in Xenon at 01 bar with a velo13ity of 50
kms are 13hara13terized by the development of an extended radiative pre13ur-
sor The huge 13ompression of 38 in the post-sho13k is a 13onsequen13e of both
the sho13k and the gas ionisation (fa13tor of 10) as also from the radiative
13ooling The post-sho13k and pre-sho13k temperature on both side of the peak
are identi13al indi13ating that the sho13k is super13riti13al
The spe13ta13ular role of the opa13ity has been highlighted Due to the
un13ertainty in the opa13ity used for Xenon in our simulations I de13ided that
at this stage it was not ne13essary to rene the simulation in terms of group
numbers within the 1D des13ription of the sho13k wave
Our numeri13al study gives the main 13hara13teristi13s of the intera13tion of
two 13ounter-propagating sho13k waves with speeds equal to 50-50 kms and
50-20 kms The 13ase of identi13al speeds is simpler due to the symmetry of
the problem However whatever the speeds the most important signature
of the intera13tion is the merging of the pre13ursor at 8 ns for 50-50 kms
and at 15 ns for 50-20 kms This merging is followed by a regular in13rease
with time of the ele13tron density and the temperature The 13ollision time
is 13hara13terized by a sudden in13rease of the ele13tron density by an order of
magnitude rea13hing 66 times 10
21and 3 times 10
2113m
minus3respe13tively whereas the
temperature in13reases up to 39 and 28 eV
35 SUMMARY 45
(a)
(b)
Figure 313 Variations of Ne (a) and Te (b) with axial position for the 13ase of
two non-identi13al 13ounter-propagating sho13ks (of speeds sim 50 amp 20 kms) and two
single sho13ks (dotted lines) of speeds sim 50 amp 20 kms respe13tively at 3 10 30 35
and 46 ns as derived from HELIOS simulations The verti13al dotted lines show the
position of the interfa13e between piston and ba13king Xenon gas (with opa13ity times20)
46 CHAPTER 3 1D SIMULATIONS
Chapter 4
Laser-driven radiative sho13k
Experimental Setup
Contents
41 Prague Asterix Laser System (PALS) fa13ility 48
42 Targets 49
421 Massive Targets 51
422 Gaseous Targets 51
423 Target holder 55
424 Target lling 56
43 Laser Fo13using 57
431 Fo13al Lenses and Phase Zone Plates 57
432 Fo13using of the MAIN and AUX laser beams 58
44 Diagnosti13s 59
441 Visible interferometry 61
442 XUV spe13tros13opy 65
443 Visible spe13tros13opy 67
45 Summary 68
When a short pulse of a high-power laser beam is fo13ussed on a thin foil
it transfers a huge amount of energy to the foil material Almost instantane-
ously the laser absorption heats it up to the ablation A hot and low density
13orona is generated ba13kwards by this pro13ess Beyond the ablation surfa13e
a sho13k is formed in the foil as a 13onsequen13e of the momentum 13onservation
(ro13ket ee13t) The sho13k moves inward in the foil Our 13ase of the experi-
mental setup as the foil 13loses a tube lled with gas the sho13k propagates
then in the gas where it is studied through various kinds of diagnosti13s The
foil then a13ts as a piston
We performed the experiments at the Prague Asterix Laser System (PALS)
in the Spring 2015 (ve weeks 20th April - 22th May) The obje13tive of the
experimental 13ampaign was to study the evolution of two 13ounter streaming
radiative sho13ks driven by two 13ounter-propagating lasers with an irradian13e
in the range of 10
14W13m
minus2 and therefore to 13hara13terize their intera13tion
and 13ollision In this 13hapter I will rst present a brief des13ription of the
PALS laser fa13ility and of the laser beams used in our experiment This will
47
48 CHAPTER 4 EXPERIMENTAL SETUP
be followed by a presentation of the targets design and an overview of the
general setup and of our main diagnosti13s
41 Prague Asterix Laser System (PALS) fa13ility
The Prague Asterix Laser System (PALS) is a laser fa13ility based on an
Infrared high-power iodine laser system (Asterix IV) (Jungwirth et al 2001)
Using dierent amplifying stages the laser fa13ility is able to deliver energy
up to 1 kJ in 03 ns at the fundamental wavelength 1315 nm The output
laser beam 13an be further subdivided in few auxiliary beams All auxiliary
beams may be frequen13y doubled (λ = 657 nm) or tripled (438 nm) The
PALS laser fa13ility is 13apable of ring up to two high energy laser shots per
hour Compared with solid state lasers this gas laser is known to deliver a
quite homogenous beam intensity without hot spots A spatial prole of the
PALS laser beam is shown in the Fig 41a (re13orded by PALS team during
the experimental 13ampaign) while its pulse 13hara13teristi13s are plotted with
time in the Fig 41b
(a)
(b)
Figure 41 a Spatial prole of the PALS Laser beam b Intensity prole of
laser pulse I(t) with time
For our experiments we used two independent laser beams to drive the
two 13ounter streaming sho13ks in the gaseous target For this purpose the
fundamental PALS laser beam has been subdivided after the fourth amplier
(V4) into two beams with respe13tive energy fra13tions of 60 and 40 (see
Fig 42) The most energeti13 of these laser beams is then inje13ted through
the fth (V5) amplier and its frequen13y are then tripled This beam at 438
nm will be hereafter termed termed as the MAIN laser beam
The se13ond of the two previous laser beams propagates then without any
modi13ation It will be hereafter 13alled the AUX laser beam (1315 nm)
The s13heme of the energy distribution is shown in the Fig 42 and the
13hara13teristi13s of MAIN and AUX laser are presented in Table 41
For the experiments PALS oers two va13uum 13hambers with a spheri13al
42 TARGETS 49
Figure 42 Energy distribution s13heme for MAIN and AUX laser beams
LASER MAIN AUX
Typi13al laser energy (J) 120 60
Beam diameter (mm) 290 148
Wavelength (nm) 1315 438
Pulse duration (ps) 350 350
Table 41 Nominal 13hara13teristi13s of MAIN and AUX laser beams
and 13ylindri13al geometry respe13tively The spheri13al 13hamber assembly used
in our experiments is shown in Fig 43 This 13hamber having a diameter
of 100 13m is 13losed by two entran13e 13ir13ular windows of respe13tive diameters
80 13m and 50 13m to allow the MAIN and AUX beams to penetrate in the
13hamber and to rea13h the target Ea13h of these windows is outtted with a
hinged door Several 13ir13ular ports of various diameters also allow for the
dierent diagnosti13s The target holder and some diagnosti13s are mounted
on an opti13al ben13h whi13h is de13oupled from the 13hamber walls
42 Targets
Our targets s13hemati13ally 13onsist in small tubes of 4 mm length 13losed on
both sides by two spe13i13 foils of thi13kness 11 microm on whi13h the laser beams
are fo13used (one laser per foil) with an irradian13e of about 10
1413m
minus3 The
tube is lled with gas in whi13h the radiative sho13k propagates with a velo-
13ity of 30 - 60 kms The two foils 13losing the target insure the 13onversion
through ablation and sho13k generation of the laser energy into me13hani13al
energy
In our experiments we used spe13i13 targets for alignment and 13hara13te-
rization of the beam size These targets were manufa13tured at the Pole
instrumental of Observatoire de Paris
50 CHAPTER 4 EXPERIMENTAL SETUP
Figure 43 Snapshot of the spheri13al 13hamber showing the horizontal bredboard
and on the right the fo13using MAIN lens
(a) (b)
Figure 44 Dimension drawing (a) and snapshot (b) of Aluminum massive
(mo13k) target
42 TARGETS 51
421 Massive Targets
For the laser alignment on the target we have used spe13ial targets also
termed as mo13k targets A mo13k target (Fig 44) is an aluminum blo13k
with a base Three 13uts made on it help to fo13us the laser beam at the
desired position The 13enter of ea13h of the two laser spots should be lo13ated
at the interse13tion of the verti13al and horizontal marks and the pre13ision of
this alignment is less than 20 microm
Beside alignment purpose su13h targets are used to 13hara13terize the MAIN
and AUX spot diameters in the planes whi13h 13orrespond to the lo13ation of
the two Parylene foils Hen13e the laser beam fo13ussed on these targets
generates a 13rater (Chaulagain 2015) The detailed analysis of the shape
and size of these 13raters provides a quantitative information about the ho-
mogeneity of the in13ident beam and the size of the fo13al spot A detailed
dis13ussion about the fo13alisation of the laser with a massive target will be
presented in the se13tion 432
422 Gaseous Targets
The 13ore of the gaseous targets (Stehleacute et al 2014a) 13onsists of a 13hannel of
a parallel pipe shape having the dimension of 09 times 06 times 4 mm pla13ed at
the top of an aluminum stru13ture The pipe in whi13h the sho13ks propagate
is dened by two 500 microm thi13k verti13al lateral sides made with 13oated fused
sili13a (SiO2) windows (09 times 4 mm) as shown in the Fig 45 Su13h windows
are suitable for the visible interferometry diagnosti13 whi13h uses an auxiliary
Q-swit13hed NdYLF green Evolution laser (see Se13tion 441)
The 13hannel is 13losed at the top with a window made of a 100 nm thi13k
Si3N4 membrane whi13h is itself supported is by a 200 microm thi13k sili13on frame
(26 times 4 mm) made by SILSON 13ompany Su13h membrane is transparent
to XUV radiation whi13h is ne13essary for the XUV spe13tros13opi13 diagnosti13
used in the experiment The transmission of the aforementioned material is
obtained from the 13enter for X-ray opti13s (CXRO) database
1
and is plotted
between 0 and 40 nm in Fig 48
The pistons 13losing the tube from opposite sides and whi13h will laun13h
the two sho13ks are made of Parylene-N (11 microm) 13oated by Au (06 microm)
(SCITECH 13ompany) They are glued on 01 mm thi13k Ni13kel disks with
external diameter 5 mm and internal diameter 1 mm This disk helps in
assembling the targets It also 13ontributes together with a larger V-shield
(se13tion 49) to prevent hard X-ray emitted at the laser impa13t on the foil
to rea13h the gas in the tube The Parylene layer is fa13ing the laser and plays
the role of an ablator The gold layer whi13h is 13oated on the Parylene and
is lo13ated between the Parylene and the gas aims at blo13king the X-rays
generated by the laser plasma intera13tion to rea13h and preheat the gas in
1
httphenkelblgovopti13al_13onstants
52 CHAPTER 4 EXPERIMENTAL SETUP
(a)
(b)
Figure 45 Values on the gures are in mm (a) Verti13al 13ross se13tion view of
the gaseous target (b) Horizontal 13ross se13tion view
42 TARGETS 53
Figure 46 Pi13ture of a the gaseous target on its base
Figure 47 S13hemati13 of the gaseous target
54 CHAPTER 4 EXPERIMENTAL SETUP
Figure 48 Transmission of 100 nm thi13k Si3N4 membrane (CXRO database)
Figure 49 Snapshot of a target holder with one V-shield regarding the AUX
laser whi13h will pass through the hole in this shield marked in green in the Figure
The target whi13h fa13es the two diagnosti13s (tubes) and is lo13ated behind the
V-shield is thus not visible
42 TARGETS 55
the 13ell These various elements are manually glued on the base of the target
to built the sho13k 13hannel Ea13h target has been tested against leaks before
the experiment and before ea13h shot The axis of symmetry of the sho13k
waves is required to be lo13ated at 03 mm from the base and at 03 mm from
the two verti13al SiO2 windows S13hemati13s of verti13al and horizontal 13ross
se13tion views of the target are presented in Fig 45b The axis of symmetry
is visible in the verti13al view while the position of the laser fo13us point (`+
sign) in the horizontal view This is a13hieved during the alignment pro13edure
employing mo13k targets as dis13ussed in the se13tion 432 The base of the
target 13onsists of a mi13ro-ma13hined aluminum stru13ture A gas pipe at the
bottom of the stru13ture allows for in situ gas lling at a pressure of a fra13tion
of a bar
A total of 55 targets were manufa13tured for the experiment Among
them 31 gave valuable results
423 Target holder
We x the targets on a stainless holder and then pla13e the system into the
va13uum 13hamber The target holder has been designed and manufa13tured
at Observatoire de Paris It allows a very reprodu13ible positioning of the
target on the va13uum 13hamber breadboard The main element of the target
holder is a base where ea13h target is rst xed by s13rews and removed after
ea13h shot to be repla13ed by a new one It also in13ludes a diagnosti13s support
(Fig 46) whi13h is atta13hed on it from the top with s13rews One of these
diagnosti13s 13onsists in a fast XUV Si diode with a pinhole The diode is
atta13hed to the bla13k tube whi13h is visible on the left part the target holder
in Fig 47 This last diagnosti13 aimed for sho13k timing as the pinhole-diode
system allows to re13ord the sho13k passing at the imaged position of the tube
(Stehleacute et al 2012 Chaulagain 2015) Unfortunately this diagnosti13 did
not work well and will not be mentioned further in the thesis
The se13ond diagnosti13 is a visible time-and-spa13e-resolved visible spe13tro-
meter On the target holder this diagnosti13 13ontains a lens whi13h is lo13ated
inside the bla13k tube on the right part of Fig 49 The lens allows imaging
the sho13k on a linear bundle of 21 bers 13onne13ted to the visible spe13trometer
through a feedthrough (se13tion 443)
A snapshot of the target holder is shown in the gure 49 The red arrows
in the Fig 49 shows the distribution of the various diagnosti13s
In addition we used two 3 mm thi13k stainless steel V-shape shields on
both sides of the target whi13h were xed to the holder These V-shields have
a hole (sim25 mm) shaped entran13e to allow laser in13iden13e on the target (in
green on the Figure 49) The role of the V-shields is to prevent the target
from the extra laser or another kind of hard X-ray plasma self-emission linked
to the ablation pro13ess
56 CHAPTER 4 EXPERIMENTAL SETUP
424 Target lling
The gaseous targets were lled in situ at a fra13tion of 1 bar with a gas (viz
Xe Ar a mixture of Xe and He) whi13h thus provide the medium for the
sho13k propagation We 13hose an in-situ lling system (Fig 410) to redu13e
the dieren13e of pressures inside and outside of the tube This is 13riti13al as
a large pressure dieren13e between the target 13ell and the va13uum 13hamber
13an result in the breaking of the ultra-thin Si3N4 windows of the target
To 13ontrol the pressure we used a piezo-resistive pressure transdu13er
(Swagelok PTI-S-AC6-32) whi13h has 10 kΩ bridge resistan13e and works
with 14-30 V biasing voltage A sensitive Bourdan manometer and a gauge
allowed monitoring the pressure inside the target 13ell To read the pressure
remotely (from the 13ontrol room) until the last se13ond before the shot we
13onne13ted the system with a long 13able
The transdu13er has been operated with a +17 V DC biasing voltage The
pressure is read using an industrial pressure transdu13er from Swagelok (0 to
10 Volts -10 bar to 06 bar 05 pre13ision) and reading is made with a
voltmeter at better than 05 This system generally took approximately
one minute to stabilize A s13hemati13 of the aforementioned gas lling system
is presented in the Figure 410
The pro13edure of the target lling is presented as follows
Figure 410 In-situ system for target lling and pressure monitoring
The target is pumped during the 13hamber eva13uation
The target is isolated at the end of the pumping
The target is lled to the desired pressure
43 LASER FOCUSING 57
The target is isolated from the gas bottle to avoid any leakage in
13hamber
Then the pressure 13an be read from the 13ontrol room till the last se13ond
before the shot
The gases are taken from high grade bottles 99995 purity The Xe+He
mixture is 1007plusmn050 He molar fra13tion The lling pro13edure is pre-
venting the presen13e of impurities like air When 13losing the 13hamber and
during the primary pumping the lling 13ir13uit 13ommuni13ates with the 13ham-
ber thus the whole tubing and the target are eva13uated down to 001 mbar
After the turbo pump is a13tivated the 13onne13tion to 13hamber is 13losed and
the gas in inje13ted When the target pressure is rea13hed the 13ell is isolated
from the pressure regulator In 13ase there is an una1313eptable overpressure
it is possible to re13onne13t to 13hamber to eva13uate and to restart the lling
The pressure reading is possible in the 13ontrol room still at pre13ision better
than 1 and the value kept for the re13ords is the one just before the shot
whi13h happens normally 1-2 minutes after 13losing
43 Laser Fo13using
431 Fo13al Lenses and Phase Zone Plates
Two fo13al lenses are used to fo13us the laser beams on the target piston foil for
MAIN and AUX respe13tively The MAIN lens is lo13ated inside the va13uum
13hamber whereas the AUX lens is outside this 13hamber The spe13i13ations
of these lenses are given in the Table 42
laser Diameter (mm) fo13al length (mm) f-number
MAIN 340 564 at 3ω 19
AUX 150 1022 at ω 69
Table 42 Spe13i13ations of the MAIN and AUX fo13al lenses and the f- number
(fo13al lengthbeam diameter) is given (see table 41 for the beam diameters)
Whereas these lenses provide a smooth intensity at the fo13us this is not
su13ient to laun13h a planar sho13k wave To this purpose phase zone plates
(PZP) have been used between the output laser beam and the lens
A PZP 13omprises several phase zone elements whi13h divide the input
beam into several beamlets whi13h 13ombined with the main fo13al lens provi-
ding a uniform intensity distribution on the target with a spe13ied diameter
(Ross Pepler and Danson 1995 Bett et al 1995) The PZP are 13ommonly
used in EOS laser experiments whi13h require a uniform intensity distribution
on the target (Koenig et al 1994 Batani et al 2003)
Two PZP as dis13ussed below were used in the experiment
58 CHAPTER 4 EXPERIMENTAL SETUP
The rst one made by COLSICOAT was already used at PALS for
radiative sho13k experiments (Stehleacute et al 2010) This PZP used for
MAIN (438 nm) is made on a 13oated BK7 substrate of a diameter
of 310 mm and se13tion of 8 mm It is designed to provide a uniform
13ir13ular fo13al spot over a diameter of 05 mm
The se13ond plate made by SCITECH has been used for AUX (1315
nm) The plate is made on a 63 mm thi13k 13oated Viosil substrate
with a square 153 x 153 mm shape It is designed to provide a uniform
13ir13ular fo13al spot over a diameter of 0250 mm
Laser Spe13ied spot diameter (mm)
MAIN 05
AUX 025
Table 43 Spe13i13ations of the two Phase Zone Plates
In terms of energy distribution sim 80 of the laser energy is in the laser
spot (C Spindloe private 13omm) The energy of the laser whi13h is given
by the PALS installation should be 13orre13ted by this fa13tor of 80 For
AUX another 13orre13tion of sim 10 has to be added whi13h is linked to the
transmission of the entran13e window of the va13uum 13hamber As a result
for nominal energies of the MAIN and AUX lasers equal to 120 and 60 J (see
table 41) the uen13es on the target are equal to sim 48800 and 59500 J13m
2
respe13tively Taking into a1313ount the 03 ns laser duration of the lasers this
results in an intensity of sim 1-2 times 10
14W13m
2
432 Fo13using of the MAIN and AUX laser beams
The determinations of the fo13i of the two lenses were performed with a
Hartmann setup using a photographi13 paper and without any PZP plate
The determination of the size of the beam at the impa13t is important both to
know the initial 13onditions for the sho13k waves (laser irradian13e whi13h should
be few 10
14W13m
minus2and se13tion of the sho13k wave) The sele13tion of the
best spots size on the piston was done with mo13k targets and by translating
the two laser lenses The in13iden13e of a laser beam on a solid planar target
generates a 13rater in the foil and an ablated plume propagating ba13kward
at high temperature (Mahmood and Ur-Rehman 2009 Singh and Thakur
2007) The 13rater is the result of boiling and ablation of the material The
ablated plasma whi13h expands towards the in13ident laser beam emits in the
keV range
A standard pro13edure whi13h works well at low energy (sim 10 J) 13onsists
of the estimation of the spot size by measuring the 13rater on the bulk target
using a standard mi13ros13ope At higher energy the boiling ee13t makes
44 DIAGNOSTICS 59
this estimation impre13ise and we preferred to use a keV imaging diagnosti13s
(Chaulagain et al 2012) whi13h is based on a pinhole (25 microm for MAIN keV
and 20 microm for AUX) and a keV 13amera These two keV 13ameras have been
used during ea13h laser shot to image the impa13ts of the MAIN and AUX
laser beams on the two respe13tive pistons of the target Ea13h keV 13amera
is a CMOS dete13tor without any glass It is prote13ted by an Aluminium
lter (200 microm for MAIN and 42 microm for AUX) to blo13k the visible light and
to prote13t the ship from the debris The spe13i13ations of the 13ameras are
presented in the Table 44 and the keV 13amera system is shown in the Fig
411 The two PALS laser lenses were translated up to a13hieve the suitable
diameters on the target whi13h were nally set to 450 - 500 microm and 250 -
300 microm for the MAIN and AUX beams
The size of the impa13t was 13ontrolled ea13h day on mo13k targets before
using the laser beams for real shots on gaseous targets It is worth to pre13ise
that the keV diagnosti13 was still in pla13e for the shots on the gaseous targets
In that 13ase the beams impa13t Parylene-N instead of Aluminium The keV
signal is then weaker than for the mo13k aluminum targets
Figure 411 S13hemati13 of the keV 13amera
KeV 13amera KeV 13amera
MAIN laser AUX laser
Model UI-122xLE UI-164xLE
Resolution 752(H) x 480(V) 1280(H) x 1024(V)
Pixel size (microm x microm) 60 x 60 36 x 36
A13tive Area (mm x mm) 451 (H) x 288 (V) 46 (H) x 37 (V)
Magni13ation 107 068
Pinhole Material Ta Pt
Table 44 Spe13i13ations of the two uEye keV 13ameras (1stVsion 13ompany)
44 Diagnosti13s
Various kinds of diagnosti13s were implemented namely visible interfero-
metry XUV and visible spe13tros13opy For the same two keV 13ameras are
indu13ted to monitor the impa13t and two visible 13ameras for target align-
60 CHAPTER 4 EXPERIMENTAL SETUP
(a)
(b)
(13)
Figure 412 (a) Sign 13onvention for the respe13tive positions of the lens and the
target (b) keV image of a MAIN impa13t on a massive target (lens position +1500
microm) The spot size is 475plusmn25 microm Pixel size for MAIN keV 13amera is 56 microm (13)
keV image of an AUX impa13t (lens position -1500 microm) The spot size is sim 280plusmn20microm on target One pixel = 66 microm on AUXs keV 13amera
44 DIAGNOSTICS 61
ment In this se13tion I will des13ribe the setups of the visible interferometry
XUV spe13tros13opy and visible spe13tros13opy The analysis of the re13ords of
visible interferometry and XUV spe13tros13opy will be dis13ussed in subsequent
13hapters
441 Visible Interferometry
A Ma13h-Zehnder Interferometer (MZI) has been employed to perform visible
interferometry The re13orded interferometri13 images have been used to study
the sho13k speed and the ele13tron density during the sho13ks propagation The
experimental setup of this interferometer is presented below
Ma13h Zehnder Interferometer
We pla13ed the target in whi13h sho13k is produ13ed in one of the two arms
of the interferometer The probe laser wavelength (527 nm) is supposed to
be far away of any atomi13 absorption resonan13es in the plasma and thus the
13ontributions to the refra13tive index 13ome from the plasma free ele13trons and
not from bound ele13trons (Harilal and Tilla13k 2004)
Figure 413 First interferometri13 setup All the opti13al elements are 1 in13h For
simpli13ity the plasma slab is not reported in the probe beam
The rst experimental setup shown in Fig 413 was used until 04052015
62 CHAPTER 4 EXPERIMENTAL SETUP
In this setup
the 300 ns long probe laser of wavelength 527nm (beam diameter asymp 1
in13h energy on the target few tens of mi13rojoules) rst passes through
an alignment pinhole PH1
the non-polarizing beam splitter BS1 transmits 50 of the intensity
of the in13ident beam and the remaining 50 of intensity in the per-
pendi13ular dire13tion At this stage the two arms of the interferometer
namely the referen13e and probe beams are produ13ed
two half wave plates HWP1 and HWP2 are pla13ed respe13tively in
the path of the probe and referen13e beams They allow to play with
the polarization in order to optimize the fringes 13ontrast as it will be
explained below
the verti13ally polarized probe beam passes through the target (plasma
slab) Due to its polarization it is fully ree13ted by the polarizing 13ube
beam splitter PBS1 towards the polarizing 13ube beam splitter PBS2
the referen13e beam is ree13ted by the mirror M4 on to this 13ube splitter
PBS2
At the output of PBS2 the two beams overlap but they have ortho-
gonal polarizations and 13ant interfere They then travel through a
polarizing beam splitter PBS3 those axis makes an angle of 45
with
the 13ommon dire13tion of in13iden13e of the two beams This rotates
the polarizations of the referen13e (horizontally polarized) and probe
(verti13ally polarized) beams along the same axis
At this stage the beams share the same polarization and interfere
A last non-polarizing beamsplitter plate (90-10 ) BS2 is pla13ed after
PBS3 to enable the propagation of the two beams on the same axis as
the laser beam at the entran13e of the interferometer
The fringes are re13orded on a HAMAMATSU C7700 VIS Streak Camera
lo13ated after the last alignment pinhole PH4 To this purpose an imaging
setup makes the image of the target longitudinal axis (ie along the dire13tion
of the sho13k propagation) on the streak slit This is done with an a13hromati13
spheri13al doublet of fo13al length equal to 1000 mm The magni13ation is
measured using an AGAR grid to 226 whi13h means that 106 microm on target
are re13orded by one pixel (size of the pixel 24 microm) on the streak on both
dire13tions (Fig 413)
44 DIAGNOSTICS 63
Figure 414 Se13ond interferometri13 setup
Role of the polarizing devi13es
Let us suppose that the polarization state of the in13oming beam is unknown
The polarization of the probe beam is then xed to be verti13al by the pola-
rizer HWP1 Ree13ted by the polarizing 13ube PBS2 it remains verti13al at
the entran13e of PBS3 On this side the polarization of the referen13e beam
is set to be horizontal after the 13ube BS1 using the polariser HWP2 and is
transmitted as horizontal by PBS2 The rotation of PBS3 allows to proje13t
part of the two beams on a 13ommon polarization axis and to interfere
This setup presents the following interest the intensity of the laser beam
in the two arms of the interferometer 13an be modied (and thus the 13ontrast
of the fringes) 13an be optimized using the two polarizing plates HPW1 and
HPW2 However as the multimode laser is not fully 13oherent the dark
fringe (zero net intensity) has never been obtained
Moreover the setup is designed to also allow alignment and adjustment
of the imaging setup with a green HeNe laser whi13h shares the same path
as the probe beam up to PBS1
A more optimized setup was used after 05052015 (ie shot number
48033) where the mirror M1 in the referen13e beam was repla13ed by a
polarizing beam splitter PBS4 ree13ting the probe beam like the mirror M1
(Fig 414) This was done in order to redu13e the dieren13e between the
64 CHAPTER 4 EXPERIMENTAL SETUP
two paths l1 and l2 and then to optimize the fringe 13ontrast This was more
suitable due to the short 13oheren13e length (1- 2 13m) of the probe laser
Streak 13amera
Figure 415 S13hemati13 of the Streak 13amera (from Hamamatsu noti13e)
A streak 13amera is a setup whi13h allows measuring the temporal varia-
tion of a fast signal whi13h is re13orded on its slit As indi13ated in the noti13e)
of the Hamamatsu C770 13amera the light pulse rst passes through the slit
then it is fo13used on the photo13athode (size 7times17 mm) of the streak by an
input opti13s The visible photons are 13onverted into a number of ele13trons
whi13h is proportional to the intensity of the in13ident light These ele13trons
are a1313elerated and 13ondu13ted towards the phosphor s13reen when a high-
speed voltage whi13h is syn13hronized to the in13ident light is applied As a
result the ele13trons are swept at high speed from the top to the bottom
after whi13h they are bombarded against the phosphor s13reen of the streak
tube and 13onverted through an output opti13s to an opti13al image on a dual
mode 13ooled CCD 13amera with 512 times 512 pixels (Fig 415) In the PALS
experiment the sweeping times were kept either 50 or 200 ns The CCD
pixel size is 24 x 24 microm
The streak 13amera 13an be operated with time swap mode with the slit
of the 13amera almost 13losed The re13ord 13onsisted then in images (position
along the slit versus time) as dis13ussed above or in a stati13 mode (no time
swap) whatever the slit aperture In that 13ase the re13ord is a usual 2D
spatial image of the obje13t
During the experiment we have taken referen13e images (without any
fringe) for every target as shown in the Fig 417a for one target example
In order to re13ord su13h images we blo13ked the referen13e beam and let the
probe beam only to illuminate the target in a stati13 mode and with the slit
open Then we redu13ed the slit width to its nominal value (200 microm) This
enabled us to have the visualization of the portion of the target whi13h was
44 DIAGNOSTICS 65
Figure 416 Imaging setup of the Ma13h Zehnder Interferometer setup operating
in the longitudinal mode The 13hannel of the target is fully illuminated by the
probe beam In this mode the lens images the axis of symmetry of the two sho13k
waves along the slit of the streak 13amera
imaged in the slit on the target and also to lo13ate the position of the pistons
before the shot This image is termed as the referen13e target image Then
the system was pla13ed in the interferometri13 mode and the fringe pattern
was optimized to be perpendi13ular to the slit and thus in the longitudinal
mode perpendi13ular to the dire13tion of the sho13k propagation
The Fig 417b reports su13h a typi13al interferogram re13orded during our
experiment (13f Fig 413) The horizontal axis 13orresponds to the dire13tion
of the sho13k propagation (with a total s13ale of 54 mm on the target) and
the verti13al axis 13orresponds to time (s13aling 200 ns)
442 XUV Spe13tros13opy
The time and spa13e integrated XUV plasma emission are re13orded with a
Flat Field XUV spe13trometer (grazing in13iden13e XUV spe13trometer) using
a 13on13ave grating Typi13al ray tra13ing a grazing in13iden13e XUV spe13tro-
meter is shown in Fig 418 In pra13ti13al the entran13e slit was removed
as the plasma extension was small In the Fig 419 the drawing of the
XUV spe13trometer setup s13heme (red 13olor) is presented together with the
spheri13al 13hamber and the target The XUV spe13trometer is installed on
the top of the spheri13al 13hamber whi13h allows the XUV emission passing
through the Si3N4 membrane on the top of the target (see se13tion 422) to
be re13orded
A 13ooled Andor DX440 CCD (without glass prote13tion) is atta13hed to
the spe13trometer to re13ord the spe13trum of the XUV radiation between 12
66 CHAPTER 4 EXPERIMENTAL SETUP
(a)
(b)
Figure 417 (a) Referen13e 2D image of a target before the shot re13orded on the
Streak 13amera The positions of the two pistons 13losing the 4 mm long target are
lo13ated at 800 and 4800 microm The dark zones along them (between 800 to 1000 and
4600-4800 microm) 13orresponds to the glue This glue is then visible through verti13al
bla13k strips in the interferometri13 re13ord (b) Corresponding interferometri13 time-
spa13e re13ord
44 DIAGNOSTICS 67
Figure 418 Typi13al ray tra13ing in a grazing in13iden13e XUV spe13trograph The
dete13tor (GMCP or image plate) is installed tangentially to the Rowland 13ir13le
and 40 nm (31- 100 eV) Its 13hara13teristi13s are given in the Table 45 Two Al
lters of thi13kness 08 and 16 microm prote13t the grating and the CCD 13amera
from debris and visible light The XUV images re13orded on the CCD are
time (300 micros) and spa13e (sim 3 mm) integrated
Andor 13amera (DX440)
Pixel Size 135 x 135 micromArea 276 (H) x 69 (V) mm
A13tive pixels 2048 x 512
Table 45 Spe13i13ations of Andor CCD 13amera
The 13urved grating (Table 46) used to fo13us the XUV emission on to
the CCD 13hip has a variable growes spa13ing as shown in the bottom panel
of the Fig 419 A typi13al output re13ord is also presented for illustration
Grating
Type Dira13tion
Growes per mm 1200
Dimensions 30 x 50 x 10 mmRadius of 13urvature 5649 mm
Blazing angle 37 degree
Blazing wavelength 16 nm
Table 46 Spe13i13ations of the XUV grating
443 Visible spe13tros13opy
In addition to the interferometry and XUV spe13tros13opy we have also im-
plemented a time- and spa13e-resolved visible spe13tros13opi13 plasma emission
diagnosti13
68 CHAPTER 4 EXPERIMENTAL SETUP
Figure 419 (a) Setup of the XUV spe13trometer in the spheri13al 13hamber (b)
Typi13al re13ord between sim 10 and 40 nm is shown in the bottom panel
The 13ore of the diagnosti13s is a Czerny-Turner 300 mm Spe13trometer
2
lo13ated outside the va13uum 13hamber The light of the plasma is inje13ted on
the slit of this spe13trometer through a bers bundle This bundle 13onsists of
three sets of an aligned seven individual bers looking at dierent se13tions
of the target 13hannel through an imaging lens lo13ated in the bla13k tube of
Fig 49 The time resolution is performed through an iCCD 13amera lo13ated
at the exit port of the spe13trometer (Fig 420a)
A typi13al re13ord is reported in Fig 420b The results of this diagnosti13s
show several signatures whi13h dier along the tube Most of them are in
absorption and 13ould be linked to the heating of the target sili13a window
Their interpretation is not yet a13hieved and will not be presented in this
study
45 Summary
In this 13hapter I have presented the details of our experimental setup and
diagnosti13s The next 13hapter will be dedi13ated to the presentation of our
re13ords results and their analysis
2
see http wwwandorcomspectrographshamrock minus spectrographminus series
45 SUMMARY 69
(a)
(b)
Figure 420 (a) S13hemati13 of setup employed to re13ord the time and spa13e
visible plasma emission (b) A typi13al spe13trogram
70 CHAPTER 4 EXPERIMENTAL SETUP
Chapter 5
Results and interpretation
Contents
51 Visible interferometry 71
511 Longitudinal interferometry sho13k speed and
ele13tron density 74
512 Transverse interferometry lateral extension of
the sho13k 91
52 XUV Spe13tros13opy 92
53 Simulations based on experimental results 93
54 Summary 97
In the previous 13hapter I presented a brief des13ription of our experimen-
tal setup This setup in13ludes two main diagnosti13s namely visible interfe-
rometry and XUV spe13tros13opy The visible interferometry is an adequate
tool to probe the ele13tron density of the radiative pre13ursor up to the 13riti-
13al density (4 times 10
2113m
minus3at the wavelength 527 nm of the probing laser)
whereas the denser post-sho13k region is opaque to the visible light XUV
radiation is present in the whole sho13k stru13ture and thus the time- and
spa13e-integrated XUV spe13tros13opy may allow exploring the self-emission
13oming from the post-sho13k and pre13ursor regions
In this 13hapter
1
I present the results of the both diagnosti13s with a
parti13ular fo13us on the interferometri13 data whi13h I have extensively studied
with the help of a spe13i13 data analysis pro13edure that I have developed I
will then present the main results of the XUV spe13tros13opi13 diagnosti13s The
results will then be interpreted with the help of numeri13al simulations
51 Visible interferometry
In the visible interferometry the interferen13es between the main and refe-
ren13e beams overlapping on the dete13tor produ13e a pattern of fringes whi13h
follow the relative phase variations between the probe (whi13h passes through
the plasma) and the referen13e beams (see se13tion 441 of the previous 13hapter
for the experimental setup) The phase variation between two 13onse13utive
fringes of the unperturbed beams (ie without any plasma) is equal to 2π
1
Most of results presented in this 13hapter has been published in Singh et al (2017)
71
72 CHAPTER 5 RESULTS AND INTERPRETATION
Then in the presen13e of the plasma in the probe beam these fringes be-
13ome shifted due to the modi13ation of the refra13tive index More details
about the prin13iple of the interferometry and the Ma13h-Zehnder interfero-
metri13 setup may be found in the APPENDIX A and we re13all here only
the expression of the phase shift
∆φ asymp minus πd
λNclt Ne gt (51)
where λ = 527 nm is the wavelength of the probing laser Nc = 4 times 10
21
13m
minus3the 13riti13al density at this wavelength and lt Ne gt is the ele13tron
density averaged over the laser path d in the 13ell and is dened as
lt Ne gt=
int d
0
Ne(z t)dy
d(52)
In order to dedu13e the ele13tron density it is ne13essary to follow the
evolution of the positions of the maxima (respe13tively minima) of the fringes
versus time To this purpose I have developed a spe13i13 data treatment
pipeline in the intera13tive data language (IDL)
2
To improve the visibility of
the fringes obtained in the interferograms I pro13essed the images with the
Fast Fourier Transform (FFT) (Proakis 2001) This treatment transforms
the image from the spatial domain (ie our re13ords) into the frequen13y
domain It then allows to dene spe13i13 bands of frequen13y whi13h represent
noise for the image and to 13lean them The 13omplete pro13edure is des13ribed
below
First the FFT of the (512 times 512) interferometri13 image is derived In
the next step the low and high-frequen13y lterings within the FFT of the
image are done using low-pass Fminus and high-pass F+ Butterworth frequen13y
lters (Proakis 2001) as follows
Fminus(u) = 1[1 + C(uu0)2n] with C = 1 n = 1
F+(u) = 1[1 + C(u0u)2n] with C = 1 n = 1
(53)
where u represents the position in the FFT image and u0 is the nominal
lter 13uto frequen13y (represented as the width of the region in pixels) For
the low-pass Butterworth frequen13y lter u0 is 13onsidered to be 55 whereas
for high-pass Butterworth frequen13y lter it is taken to be 45 The values of
u0 are dened manually by a trial method with the aim to result in better
noise redu13tion
After this step the inverse FFT of the ltered FFT image is performed
whi13h results in a 13lean image in the spatial domain In Fig 51 the original
re13ord for the shot number 48055 (left) the FFT of this image (13enter)
and the nal frequen13y ltered image (right) are presented showing how the
2
httpwwwast13ama13uk~vasilyidlidl_introdu13tionpdf
51 VISIBLE INTERFEROMETRY 73
frequen13y ltering improves the fringes 13ontrast and thus fa13ilitates their
analysis
Figure 51 Original re13ord of shot 48055 (left) FFT of the original image
(13enter) and the frequen13y ltered image (right)
Next the 13ru13ial task is to identify and to follow the fringes with the
help of a pre13ise determination of the intensity (pixel 13ounts) maxima In
order to lo13ate these maxima I have adopted the following steps
Figure 52 A 13ropped se13tion of the re13ord from shot the 48055 The rst ve
positions have been sele13ted manually `+ signs (in red 13olor) on ea13h fringe On
this re13ord the distan13e between two unperturbed fringes is 13orrespond to 15 pixels
(ie 159 microm)
Firstly I have manually 13hosen a se13tion of a typi13al re13ord with the
aim to sele13t only the area asso13iated with the sho13k dynami13s
74 CHAPTER 5 RESULTS AND INTERPRETATION
Next ve representative points are sele13ted visually (`+ marks in red
13olor) on ea13h fringe 13overing the important lo13ations on the fringe
(13f Fig 52) They serve as the input to the spline interpolation for
estimating all intermediate pixels lo13ations along the fringe (see Fig
53a)
In order to obtain the lo13ations representing the fringe maxima the
spline interpolated lo13ation points are further rened by sele13ting the
pixel of maximum intensity within plusmn5 pixels of the respe13tive spline
lo13ations in the X-dire13tion (see Fig 53b)
The position of fringe maxima for the 13ropped image obtained in an
aforesaid manner is shifted so as to represent the same points however
on the full-image (see Fig 54)
The re13ords obtained from the experiments have been pro13essed in this
way to estimate the sho13k se13tion speed ele13tron temperature and density
511 Longitudinal interferometry sho13k speed and ele13tron
density
In the longitudinal interferometry we perform on the slit of the streak 13a-
mera the image of the axis of symmetry of the sho13k propagation along the
tube to analyse the sho13k propagation in this dire13tion
The interferometri13 images have been pro13essed as explained previously
to enhan13e the fringes 13ontrast The lo13ations of the maximum intensity in
ea13h fringe are then used to derive the sho13k speed and the average ele13tron
density as will be presented below
Sho13k speed dedu13ed from the last fringe method
An estimation of the sho13k speed 13an be obtained through the interferometri13
image In this se13tion I will present the method to derive sho13k speed
based on the last visible end points of the fringes also 13alled last fringe
method This determination provides rapidly a qualitative estimation of the
sho13k speed whi13h is based on the absorption behavior of the plasma (see
Equation A11 in appendix A) and not on the real position of the front
dis13ontinuity The front dis13ontinuity is not seen in the re13ords due to the
strong absorption
At the positions of the last visible end points of the fringes the ele13tron
density rea13hes the maximum value a1313essible to the diagnosti13 The frin-
ges are strongly bent and the absorption of the visible light be13omes also
important Therefore it is believed that the sho13k front is 13lose from this
lo13ation
The lo13ations of the fringe maxima have been already dedu13ed following
the analysis presented in previous se13tions In this regard the best visible end
51 VISIBLE INTERFEROMETRY 75
(a)
(b)
Figure 53 (a) Representative points of the fringes as derived from the spline
tting of the 5 manually sele13ted points (Fig 52 (II)) on ea13h fringe (b) Positions
of the fringes maxima along Y-axis for ea13h fringe derived by lo13ating the points
of maximum intensity in X-dire13tion of the previous points obtained by spline t
76 CHAPTER 5 RESULTS AND INTERPRETATION
Figure 54 Fringe maxima on the full image
Figure 55 Imaging setup of the Ma13h Zehnder Interferometer operating in the
longitudinal mode The lens images the axis of symmetry of the two sho13k waves
along the slit of the streak 13amera
51 VISIBLE INTERFEROMETRY 77
point of ea13h respe13tive fringe is 13onsidered for the average speed estimation
for the sho13k
At very initial time the in13rease in sho13k speed is non-linear This non-
linear trend may result in non-physi13al speed estimations Therefore I dis-
13arded the end points whi13h appear to be within the ve nanose13onds after
the sho13k laun13hing time (eg t = 145 ns on Fig 56)
However the above pro13edure may result in un13ertainties as the sele13ted
end points may have dierent intensity (13ounts) values In this regard I
have evaluated the un13ertainty in the determination of the average speeds
To this purpose I rst sele13ted the best visible end-point of a random fringe
Su13h an end point gives a referen13e intensity whi13h was then used to obtain
on the rest of fringes the end points having 13lose intensity This pro13edure
was performed thri13e in order to dedu13e three sets of su13h end-points (see
Fig 56) Ea13h of these three sets of points was then used to estimate the
speed through a linear-t method (lines in white red and green) It may be
noted that these sets of points and tted lines are very 13lose to ea13h other
Therefore it is tough to distinguish them on the Fig 56 These three speed
determinations enable us to derive an average value with an un13ertainty
whi13h is equal to the standard deviation of these three values
Figure 56 Interferometri13 image re13orded for the shot 48055 in Xe at 01 bar
The sho13k speeds for the sho13ks driven by MAIN (from left side) land AUX (from
right side) lasers are respe13tively equal to sim 54 and 23 kms The time of laser
arrival on the piston is at 146 ns The positions of the Au-Xe interfa13e on the
re13ord are respe13tively 950plusmn50 and 4950plusmn50 mi13rons
This estimation of the sho13k average speed is satisfa13tory for the re13ords
in whi13h the sho13k speed is almost 13onstant throughout its propagation Ho-
wever the variation with the time of the speed may be substantial for some
re13ords In su13h 13ases the un13ertainty in the average speed will be in prin-
13iple higher than the previous estimations Therefore I introdu13ed another
78 CHAPTER 5 RESULTS AND INTERPRETATION
method to determine the average velo13ity this velo13ity is then estimated
during the initial and in nal durations of the sho13k propagation by deriving
the slope of points on the rst middle and last few fringes respe13tively
The standard deviation estimated by this method is then termed as the
un13ertainty in the average speed of respe13tive re13ord
Using these two methods I have obtained two values of un13ertainties for
all the MAIN and AUX sho13ks observed in the experimental re13ords Finally
the largest un13ertainty value of the two is 13onsidered as the nal un13ertainty
in the estimation of the respe13tive speed and is shown by error bars in Fig
57a and 57b 13orresponding to the MAIN and AUX sho13ks respe13tively
The table 51 reports for dierent noble gases the values of the velo13ities
re13orded in several shots for the 13ounter-propagating sho13k waves In addi-
tion one re13ord (48131) 13orresponds to the 13ase of an isolated sho13k wave
laun13hed by the MAIN laser The values of the MAIN and AUX laser ener-
gies are also reported In some 13ases the glue on the window prevented the
probe laser to pass through the 13hannel Then it be13ame impossible to derive
any value for the velo13ity In su13h 13ases I put `NA for the 13orresponding
sho13k velo13ity
It may be noted that the sho13k speeds generated by the MAIN laser vary
in the range of 30-55 kms while for the sho13k originated from AUX laser
the speeds vary between 10 and 30 kms
Variations with laser energy and pressure the 13ase of Xenon
When the number of re13ords is su13ient to make a statisti13al analysis we
may analyse the variations of the sho13k speeds with the laser energy of
the MAIN and AUX lasers This is the 13ase for Xe or XeHe (90-10)
mixture and for gas pressures ranging between 01 and 02 bar At 01 bar
all the measurements 13on13ern Xenon (ρ= 54 times 10
minus4g13m
minus3) At 02 bar
the re13ords 13on13ern a mixture XeHe (90-10 ρ=10 times 10
minus3g13m
minus3)
ex13ept for one point at 121 J whi13h 13orrespond to the 13ase of pure Xe (ρ =
108 times 10
minus3g13m
minus3) We do believe that taking into a1313ount the pre13ision
of our re13ord the introdu13tion of tra13es of Helium do not ae13t mu13h the
dynami13s of the sho13k wave as the 13orresponding variation of mass density
is negligible
As expe13ted the sho13k speed in13reases with the laser energy and de13rea-
ses with the pressure (Fig 57a) A linear t gives the following dependen13e
of the speed in kms with the MAIN laser energy (in J) at 01 and 02 bar
respe13tively
3
v01bar = 1423 + 030E (54)
3
Two outlier points have been dis13arded from the set 13orresponding to 01 bar They
are shown for information on the Figure
51 VISIBLE INTERFEROMETRY 79
(a)
(b)
Figure 57 (a) MAIN sho13k speed (13al13ulated by the last fringe method) versus
the MAIN energy for Xenon or XeHe mixture at dierent pressures with the error
bars (b) AUX sho13k speed versus AUX energy also for Xenon or XeHe mixture
at dierent pressures
80 CHAPTER 5 RESULTS AND INTERPRETATION
v02bar = minus1405 + 047E (55)
The speeds generated by the AUX laser are lower than the previous ones
Although the re13ords are more sparse we note that the speeds at 01 bar
also in13rease with the laser energy However at 02 bar I 13ould not nd
similar trend due to the la13k of re13ords
Variations with the gas
As 13an be seen from the Table 51 we have also performed several shots in
Ar at 01 02 03 and 08 bar one shot in He at 05 and one in Kr at 02
bar
To 13ompare the velo13ities obtained for the dierent gases one 13an either
keep the pressure 13onstant (ie the number of atoms) or the mass density
For a given mass density and laser energy the velo13ity should be the same
in the adiabati13 limit However due to the radiation 13ooling and ionisation
this 13an not be the 13ase Nonetheless if we 13ompare the two re13ords of Ar
at 03 bar (ρ = 493 times 10
minus4g13m
minus3 E = 121 J) with Xe at 01 bar (54
times 10
minus4g13m
minus3 E = 133 J) we measure similar speeds (49 and 54 kms)
These two re13ords are thus interesting to 13ompare This will be done later
on in this 13hapter
More generally we note that the speed of Helium at 05 bar is higher by
sim 40 than for Ar at the same energy (see Fig 58) This is not surprising
for this lighter element (Table 52) However it is important to note that
the la13k of shots prevents us to make any pre13ise 13on13lusion
Ex13ept for this 13ase and for the unique re13ord of Ar at 02 bar whi13h
gives also a higher velo13ity the velo13ities of the other re13ords follow more or
less the variation with the energy than Xenon
To 13ompare the radiative ee13ts it seems justied to perform the 13om-
parison at a given sho13k speed independent of the laser energy and for mass
densities whi13h are 13lose to ea13h other In the following we shall thus 13om-
pare the results of Xe at 01 bar (54 kms 54times 10
minus4g13m
minus3) Ar at 03
bar (49 kms 49times 10
minus4g13m
minus3) and Kr at 02 bar (55 kms 68 times 10
minus4
g13m
minus3)
Line averaged ele13tron density lt Ne gt
I have estimated the ele13tron density lt Ne gt of the plasma averaged along
the path of the probe laser beam by analyzing the interferogram re13ords
assuming the same se13tion d of the plasma layer of 600 microm for both MAIN
and AUX side sho13ks
51 VISIBLE INTERFEROMETRY 81
Shot Gas Pressure E3ω MAIN sho13k Eω AUX sho13k
(bar) (J) speed (kms) (J) speed (kms)
at 298 K
48033 Air 03 124 54plusmn1 66 20plusmn248034 Air 03 131 52plusmn1 74 27plusmn248076 Ar 05 100 41plusmn1 66 18plusmn148077 Ar 01 115 42plusmn1 67 18plusmn148078 Ar 03 112 46plusmn3 65 25plusmn248079 Ar 03 121 49plusmn5 67 23plusmn348080 Ar 08 103 38plusmn2 62 21plusmn248081 Ar 08 113 38plusmn1 68 NA
48082 Ar 08 107 36plusmn1 65 NA
48141 Ar 02 111 63plusmn1 57 NA
48083 He 05 106 57plusmn3 63 NA
48146 Kr 02 125 55plusmn2 53 NA
48043 Xe 03 138 57plusmn1 72 25plusmn148051 Xe 01 123 58plusmn1 67 30plusmn148055 Xe 01 133 54plusmn3 68 22plusmn348057 Xe 01 127 53plusmn1 68 23plusmn148058 Xe 01 115 48plusmn3 63 18plusmn148059 Xe 01 116 50plusmn1 67 21plusmn148061 Xe 01 127 53plusmn1 67 17plusmn248065 Xe 01 122 52plusmn4 68 23plusmn348066 Xe 01 114 50plusmn3 67 15plusmn248067 Xe 05 115 39plusmn2 65 NA
48068 Xe 05 109 36plusmn2 65 12plusmn248070 Xe 05 109 33plusmn3 65 NA
48138 Xe 02 121 45plusmn5 0 0
48131 Xe +He 02 112 38plusmn1 0 0
48132 Xe +He 02 118 41plusmn4 56 18plusmn248133 Xe +He 02 112 41plusmn3 56 NA
48134 Xe +He 02 111 38plusmn1 60 NA
48136 Xe +He 02 115 39plusmn3 59 14plusmn148143 Xe +He 06 123 39plusmn4 63 18plusmn548144 Xe+He 02 133 45plusmn3 66 NA
Table 51 Sho13k speeds estimated from the `last fringe method `NA represents
the entries whi13h 13ould not be dedu13ed from the re13ord Further entries in the
bold font are dis13ussed in detail in this 13hapter
82 CHAPTER 5 RESULTS AND INTERPRETATION
Helium Argon Krypton Xenon
Atomi13 Number 2 18 36 54
Atomi13 Mass 4 3995 8380 13129
First Ionization Energy (eV) 246 157 14 12
Density (10
minus4g13m
minus3) at 01 bar 016 164 344 539
Table 52 Atomi13 data and density at 01 bar (at room temperature) for He Ar
Kr and Xe
Figure 58 MAIN sho13k speed (13al13ulated by the last fringe method) versus the
MAIN laser energy for Xe (at 01 bar only) Ar He and Kr at dierent pressures
with the error bars
51 VISIBLE INTERFEROMETRY 83
It should be noted that the beam se13tions dMAIN and dAUX of MAIN
and AUX lasers are approximately 600 and 300 microm respe13tively on the two
pistons and that the transverse horizontal se13tion of the sho13k tube is equal
to dtube = 600 microm Thus the value of lt Ne gt 13omputed for the sho13k wave
laun13hed by MAIN should be 13lose to the physi13al lo13al value Ne supposing
that the plasma is homogeneous along the transverse se13tion However
the value obtained for AUX is 13ertainly larger by a fa13tor whi13h 13an be
estimated at a rst step as the fra13tion of dtubedAUX giving a fa13tor of
about 2
As explained in the previous se13tion the pixels representing the positions
of the fringe maxima have been already determined with the best possible
pre13ision Therefore it is possible to 13al13ulate the relative phase shift (with
respe13t to zero time) variation along y-axis (time axis) at ea13h fringe maxima
Zero time is time of MAIN and AUX laser rival on the target With these
phase shifts estimated at ea13h pixel of ea13h fringe I then derived lt Ne gtusing Equation A22
Let us take the 13ase of three shots in Xenon for illustrating the method
whi13h will be followed (see the left panel of the Figure 59) The maximum
density is estimated to be sim11 times 1019cmminus3(13orresponding to phase shift sim
16) The number of subdivisions is taken to 5 bins with the following phase
(∆φ) ranges (in the units of 2π) and average ele13tron densities (ltNegt)
bin 1 ∆φ le 06 lt Ne gtle 39 10
1813m
minus3(white)
bin 2 06 - 08 39 - 57 10
1813m
minus3(red)
bin 3 08 - 11 57 - 75 10
1813m
minus3(blue)
bin 4 11 - 13 75 - 93 10
1813m
minus3(green)
bin 5 gt 13 gt 93 10
1813m
minus3(magenta)
All the re13ords shown in the Table 51 have been pro13essed using this
method The limit of dete13tion for the phase shift 13orresponds to 2 pixels
giving a threshold for the Ne measurement lt Nemin gt This threshold
diers from one re13ord to another due to the variation in the number of
fringes (and thus the distan13e between two unperturbed fringes)
In order to make a 13omparative investigation of the sho13ks propagation
and intera13tion in dierent gases as already mentioned previously I sele13ted
three 13ases (shot48055 shot48132 and shot48138) relative to Xe (or Xe-
He mixture) one 13ase relative to Ar (shot48079) and Kr (shot48146)
The 13orresponding experimental 13onditions are noted in bold in the Table
51
84 CHAPTER 5 RESULTS AND INTERPRETATION
lt Ne gt in Xenon
The Fig 59 reports the interferometri13 re13ords for the three Xenon 13ases
The top and the middle panels 13orrespond to the propagation of two 13ounter-
streaming radiative sho13k waves at 01 (shot48055) and 02 bar (48132)
For 13omparison one re13ord (48138 bottom panel) is dedi13ated to the pro-
pagation of single sho13k (MAIN) at 02 bar The Xe-He mixture (90 - 10
in numbers of atoms) is used for the investigation in the 13ase of 02 bar
pressure while it is only Xe in the 13ase of 01 bar As mentioned previously
at the pre13ision of our re13ords we 13onsider that this impurity 13on13entration
has a negligible ee13t on the sho13k speed and the pre13ursor ele13tron density
The limit of dete13tion of lt Ne gt over the se13tion of the tube (06 mm) is
13orresponds respe13tively to 9 times 10
17 6 times 10
17and 6 times 10
1713m
minus3for the
Figures 59(a) (b) and (13)
The variations of lt Ne gt with the distan13e along the sho13k tube (ie
parallel to the dire13tion of the sho13ks) are reported in the right panel of Fig
59 at 10 ns (in red) 20 ns (in blue) 30 ns (in green) and 40 ns (in magenta)
The intera13tion between the two pre13ursors is 13learly visible at 01 bar
(Fig 59(a)) at 10 ns the intera13tion of the 13ounter-propagating sho13ks
has either not yet started or is below the sensitivity of this diagnosti13 The
intera13tion o1313urs at later times with a typi13al signature whi13h is as follows
the slope of lt Ne gt is de13reasing from the left (MAIN pre13ursor) passes
through a minimum and in13reases at the right (AUX) The minimum itself
in13reases with time up to 7 times 10
1813m
minus3at 40 ns
At 02 bar we have not been able to re13ord the 13ollision in the temporal
range (50 ns) of the streak However the two re13ords (with MAIN only and
with the two sho13k waves) indi13ate a pre13ursor for MAIN The two gures
(Fig 59(b)) and (Fig 59(13)) show that the pre13ursor of the MAIN sho13k
wave is not inuen13ed by the presen13e of AUX sho13k wave up to 45 ns
There is no obvious indi13ation about a pre13ursor for AUX in the 13ase of
two 13ounter-propagating sho13k waves (Fig 59(b)) At this pressure and
13ompared with the previous 13ase at 01 bar the absen13e of pre13ursor for
AUX may be attributed to i) a low sho13k speed (18 kms) 13ombined with a
larger pressure (hen13e for a given gas the pre13ursor extension in13reases with
the speed and de13reases with pressure) ii) a too small longitudinal extension
of the eventual pre13ursor (see Fig 59(b)) 13ompared with the resolution of
20 mi13rons (2 pixels) Our 1D numeri13al simulations with Xenon opa13ity
multiplier times 20 (not presented here) indi13ate a small pre13ursor for AUX
sho13k At 15 ns its extension is 50 microm (900 microm for MAIN sho13k) with
a typi13al ele13tron density sim 35 times 10
1913m
minus3(23 times 10
1913m
minus3for MAIN
sho13k) whi13h does not agree with the re13ord At 42 ns the pre13ursor of MAIN
rea13hes the AUX sho13k front and the prole is similar to the prole at 20 ns
shown in Fig 313a at 01 bar with a plateau of almost 13onstant ele13tron
density between the two fronts This might be 13ompatible with small bending
51 VISIBLE INTERFEROMETRY 85
Figure 59 Left panel interferometri13 re13ords 48055 in Xe at 01 bar (a)
48132 in Xe+He at 02 bar (b) and 48138 in Xe+He at 02 bar (13) Right
panel ele13tron density at 10 20 30 and 40 ns versus distan13e for these re13ords
The positions of maxima have been identied on the re13ords in the left panel
The time t = 0 13orresponds to the time of laser arrival on the target and the
position x = 0 13orresponds to the interfa13e between the piston (Au layer) and the
gas Its determination is pre13ise within 100 mi13rons The distan13es between two
unperturbed fringes for re13ords 48055 48132 and 48138 are 159 244 and 244
microm respe13tively The lt Ne gt un13ertainty (plusmn 2 pixels) is indi13ated by the error bar
in the right panels It 13orresponds respe13tively to plusmn 9 times 10
17 plusmn 6 times 10
17and plusmn
6 times 10
1713m
minus3for the gures (a) (b) and (13) The limit of dete13tion (2 pixels) is
presented by a dotted line on ea13h gure
86 CHAPTER 5 RESULTS AND INTERPRETATION
of the 4
thfringe (from the right) between 45 and 50 ns As 1D simulations
are known to overestimate the pre13ursor ele13tron density 2D simulations are
ne13essary for a more pre13ise interpretation of the experimental result
lt Ne gt in Argon and in Krypton
As seen in the Fig 58 the variation with the laser energy of the sho13k speed
in Argon at 03 bar (in the red dashed 13ir13le) is in good agreement with the
13orresponding one of Xenon at 01 bar (blue linet) As these two 13ases have
a 13omparable mass density of sim 5 times 10
minus4g13m
3 this qualitative agreement
is expe13ted
Taking into a1313ount the larger pressure (and thus density) the slope of
the variation of the slower sho13k speeds in Ar at 08 bar (in a red dash-
dot 13ir13le) with the laser intensity is 13ompatible with the previous trends
However any 13on13lusion 13an not be derived for Argon at 02 and 05 bar
owing to the s13ar13ity of the re13ords
In Fig 510 the interferometri13 re13ord for Ar at 03 bar (shot 48079
49 times 10
minus4g13m
minus3) is presented The estimated speeds of MAIN and AUX
sho13ks dedu13ed from the last fringe method are respe13tively equal to 49 and
23 kms whi13h are 13lose to the speeds of 54 and 23 kms re13orded in Xe
at 01 bar (shot 48055 539 times 10
minus4g13m
minus3 Fig 59(a)) Contrarily to
the Xenon 13ase we do not see any signi13ant bending in the fringes The
maximum fringe shift is reported in red on the fringe 1 The 13orresponding
ele13tron density is equal to 38 times 10
1813m
minus3
Thus obviously the radiation ee13ts are less important for Argon than
for Xenon at the same density and speed Part of this dieren13e 13ould be
attributed to the huge dieren13e in the atomi13 numbers (40 and 131) of the
two gases whi13h leads to dierent peak temperatures for the ions as expe13ted
from equation 84 of 13hapter 2 However the post-sho13k temperatures dedu-
13ed from the jump relations for a real gas (Chapter 2) without radiation are
very 13lose (sim 25 eV) and thus this 13ould be not the relevant explanation
To 13larify this I performed HELIOSPROPACEOS simulations for two
13ounter-propagating sho13ks at 53 and 23 kms respe13tively in Ar and Xe
at the same initial mass density (54 times 10
minus4g13m
minus3) without any opa13ity
multiplier I have also reported for information the 13ase of Xenon with
an opa13ity multiplier equal to 20 The simulated temperature proles are
shown in the Fig 511 One notes immediately that the pre13ursor extension
is smaller for Ar than for Xe The peak temperatures of the MAIN sho13k
are respe13tively sim 21 eV (16 eV for opa13ity multiplier=20) for Xe and 21
eV for Ar In all the 13ases the post-sho13k temperatures are equal to sim 11
eV This indi13ates that the temperature is not the main explanation of the
dieren13es in the pre13ursor length
More interesting are the dieren13es in the Rosseland opa13ity (here per
unit of length) whi13h is reported in the Fig 511b We see that 13lose to the
51 VISIBLE INTERFEROMETRY 87
(a)
(b)
Figure 510 (a) Interferometri13 re13ord in Ar at 03 bar (48079) The estimated
speeds for MAIN and AUX are 49plusmn5 and 23plusmn3 kms (b) Same re13ord where the
fringes maxima are marked by points The bins denition is as fellows bin 0 Ne le11 times 10
1813m
minus3(white) bin 1 11 - 18 times 10
1813m
minus3(yellow) and bin 2 39 - 57
times 10
1813m
minus3(red) The time t = 0 13orresponds to the time of laser arrival on the
target and the position x = 0 13orresponds to the interfa13e between the piston (Au
layer) and the gas Its determination is pre13ise within 100 mi13rons
88 CHAPTER 5 RESULTS AND INTERPRETATION
(a)
(b)
Figure 511 Results for temperature (a) and Rosseland opa13ity (b) obtained from
HELIOS simulation at 10 ns for two 13ounter-propagating sho13ks at sim 50 and 18
kms for Xe (with opa13ity multiplier 1 and 20) Ar and Kr at 54 times 10
minus4g13m
minus3
initial mass density A 13omparison Ar Kr and Xe PROPACEOS opa13ity shown in
APPENDIX B
51 VISIBLE INTERFEROMETRY 89
front the Argon opa13ity is 13lose to 4 13m
minus1 whi13h means that the radiation
from the sho13k is absorbed within 25 mm This has to be 13ompared with the
Xenon 13ase (no multiplier) where it is equal to sim 03 13m
minus1 whi13h means
that the hot pre13ursor is almost transparent over the length of the tube
Thus the dierent behaviors of the opa13ity in the upstream gas explain the
quantitative dieren13e in the development of the pre13ursor The sharp peak
of the Xenon opa13ity with the multiplier of times 20 is due to the strong rise
of the opa13ity when the temperature de13reases below 5 eV whi13h marks the
end of the pre13ursor
I will present now the 13ase of two 13ounter-propagating sho13ks in Kr
(48146 in Fig 512) at 02 bar (69 times 10
minus4g13m
minus3) In this 13ase unfortu-
nately the AUX sho13k 13ould not be imaged 13ompletely due to the presen13e
of glue at the right edge of the target 13ell but the 13ollision was re13orded
at sim 40 ns The speed of the MAIN sho13k is estimated to be 55 kms
Thus despite a slightly higher initial mass density this sho13k wave may be
13ompared with the shot (48055) in Xe at 01 bar (54 times 10
minus4g13m
minus3 54
kms)
Despite the poor quality of the re13ord and the inadequate temporal range
(150 ns) sele13ted on the streak 13amera we note some bending in all the
fringes indi13ating the presen13e of pre13ursor The maximum fringe deviation
(063 of the distan13e between two fringes) is noted at the end of fringe 2
whi13h 13orresponds to the lt Ne gt value 45 times 10
1813m
minus3 The pre13ursor
seams to be more developed here than for the previous Ar 13ase
The result of the simulation in Xe and Kr with equal initial mass density
54 times 10
minus4g13m
minus3 and for the two respe13tive sho13k speeds of 50 and 20
kms is reported in the Figure 511 The peak temperatures of the MAIN
sho13k are respe13tively sim 21 eV (16 eV for opa13ity multiplier = 20) for Xe
and 22 eV for Kr The post-sho13k temperatures (sim 10-11 eV) are 13lose from
ea13h other
The pre13ursor extension in Krypton is smaller than in Xenon whi13h is lo-
gi13al if we 13onsider the opa13ity values (Fig 511b) However if we take into
a1313ount the multiplying fa13tor for the Xenon opa13ity the Krypton pre13ursor
is now more extended than for xenon whi13h is in 13ontradi13tion with the ex-
periment This probably means that either the Kr opa13ity is under estimated
by PROPACEOS or that our multiplier for Xenon is overestimated
Synthesis
The previous experimental investigation 13onrms that the sho13k velo13ity in-
13reases with the laser energy and de13reases with the mass density Moreover
for our given sho13k velo13ity and mass density the extension of the radiative
pre13ursor in the experiment de13reases with the atomi13 number Among Ar
Kr and Xe this last one appears to be the most adapted for the investiga-
90 CHAPTER 5 RESULTS AND INTERPRETATION
(a)
(b)
Figure 512 (a) Raw interferometri13 re13ord in Kr at 02 bar (shot 48146) The
estimated speed for MAIN sho13k is 53plusmn2 kms The time t = 0 13orresponds to the
time of laser arrival on the target(b) Same re13ord shown in (a) maxima is marked
by 13olored points The bins denition is as follows bin 0 Ne le 11 times 10
1813m
minus3
(white) bin 1 11 - 18 times 10
1813m
minus3(yellow) and bin 2 39 - 57 times 10
1813m
minus3
(red) The time t = 0 13orresponds to the time of laser arrival on the target and
the position x = 0 13orresponds to the interfa13e between the piston (Au layer) and
the gas Its determination is pre13ise within 100 mi13rons
51 VISIBLE INTERFEROMETRY 91
tion of the radiative pre13ursor A similar behaviour is found also at ORION
with higher speeds (Clayson et al 2017)
512 Transverse interferometry lateral extension of the sho13k
The transverse interferometry is a dierent geometri13al imaging setup of the
Ma13h-Zehnder interferometer whi13h provides qualitative information about
the 13urvature lo13alisation and transverse extension of the radiative pre13ur-
sor and thus of the sho13k
In the transverse interferometry we perform the image of a transverse
se13tion (on the slit of the streak 13amera) of the tube whi13h is lo13ated at a
distan13e dslit equal to 3 mm from the initial position of the MAIN piston
As the setup is originally adapted for the longitudinal interferometry we use
a Dove prism between the va13uum 13hamber and the streak 13amera to rotate
the image by 90 degrees
Figure 513 S13hemati13s of the Ma13h Zehnder Interferometer setup to re13ord
transverse interferometri13 images The lens allows to make on the slit of the
13amera the image of a se13tion perpendi13ular to dire13tion of sho13k propagation
A transverse interferometri13 re13ord for the MAIN sho13k alone in Xe at
02 bar is reported in Fig514(a) The streak 13amera rst re13ords the unper-
turbed plasma on the se13tion dslit in the tube Then it re13ords su1313essively
the pre13ursor and the post-sho13k (whi13h is opaque to visible radiation and
appears as a dark zone in the re13ords) Taking into a1313ount the oset of 14
ns the time of sho13k arrival is re13orded at 72 ns after the time t0 of laser
arrival on the target and the sho13k speed is estimated to be sim 35 kms Due
to glue on one lateral window (on the right part of the gure) only 6 fringes
92 CHAPTER 5 RESULTS AND INTERPRETATION
Figure 514 Transverse interferometri13 images for (a) shot48111 (MAIN sho13k
only) (b) shot48130 (AUX sho13k only) The time is measured after an oset
equal to 14 and 23 ns respe13tively after the time of the laser arrival on the target
The position zero on the x-axis of ea13h image 13orresponds to the base of the target
are visible The lateral extension of the sho13k stru13ture at this time is deri-
ved to be sim 570plusmn30 microm in qualitative agreement with the spe13i13ations of
the MAIN phase plate and the shape of the pre13ursor is relatively at The
axis of symmetry of the sho13k is determined to be at sim 350 mi13rons from
the base of the target (ie 50 microm above the nominal value of 300 microm)
A re13ord for the AUX sho13k alone is shown in Fig 514(b) whi13h 13orre-
sponds to a gate opening of 50 ns The start time of the image has an oset
of +23 ns from t0 and the distan13e dslit is set to 700 mi13rons from the initial
position of the AUX piston The AUX sho13k duration extends from 30 ns
to at least 34 ns after t0 The sho13k speed is then estimated to be ranging
between 23 and 20 kms The shape of the pre13ursor is strongly bent and
we note a tiny shift of the se13ond and third fringes (starting from the left)
at sim 8 ns It may further be noted that the lateral spread of the opaque
sho13k is ranging between 275plusmn25 microm (whi13h is also in agreement with the
spe13i13ations of the AUX phase plate) and that the axis of symmetry of the
sho13k system is also lo13ated at about 350 microm from the bottom of the 13ell
52 XUV Spe13tros13opy
XUV emission originates from the 13omparatively hot portions of the plasma
In the present 13ase it 13orresponds to the post-sho13k and the pre13ursor re-
53 SIMULATIONS BASED ON EXPERIMENTAL RESULTS 93
gion whi13h is 13lose to the sho13k front The spe13trum re13orded by the XUV
spe13trometer enables us to analyze the spe13tros13opi13 signatures of the radi-
ative sho13k One of the aims of this diagnosti13s was to identify the sho13ks
13ollision whi13h leads to higher temperatures through 13omparing the spe13tra
obtained for the 13ases of single and 13ounter-propagating sho13k waves Howe-
ver only a few re13ords were possible to be obtained during the experiment
and unfortunately the 13omparison with the 13ase of an isolated sho13k wave
was not performed
Among the shots re13orded the XUV spe13trum of the shot 48143 is
presented herewith in detail This shot was performed for [Xe (90)+ He
(10) mixture at 06 bar with laser energies of 123 J for MAIN and 63 J
for AUX The interferometri13 re13ord of this shot is shown in Fig 515 The
MAIN sho13k speed has been estimated to be sim 39 plusmn 4 kms The estimated
AUX sho13k speed (18 plusmn 5 kms) is not pre13ise due to the presen13e of glue
on the right se13tion of the re13ord (note Fig 515) In this interferometri13
re13ord we have not been able to retrieve the 13ollision time However an
extrapolation of the slope 13orresponding to the speeds sim 39 kms (MAIN)
and sim 18 kms (AUX) enables us to approximately determine the 13ollision
time to be between 60 to 65 ns
The raw spe13trum (shown in Fig 516a) re13orded for the wavelength
range of 15-35 nm (35-82 eV) shows the `L edge of Aluminum at 17 nm
(34 nm in se13ond order) in the rst and se13ond (34 nm)orders These two
wavelengths will be used for the wavelength 13alibration The net spe13trum
13orre13tion for the transmission (Henke Gullikson and Davis 1993) of the
100 nm thi13k Si3N4 window (3 mm times 04 mm) is introdu13ed in Fig 516b
A remarkable feature is a strong absorption dip between 19 and 22 nm (56-65
eV) This absorption probably 13omes from the 13older layers (thi13kness 300
microm) between the sho13k heated plasma and the Si3N4 window Few lines
of Xe VII-VIII are identied through NIST database
4
as also Oxygen IV
and V lines Lyman lines of He II (from 1-2 to 1-7) are also present in the
spe13trum This information will be useful for the estimation of the ele13tron
temperature
53 Simulations based on experimental results
In this se13tion I will 13ompare the experimental sho13k 13hara13teristi13s with the
results of HELIOS simulations using the PROPACEOS equation of state and
opa13ity (limited to 1 group) As indi13ated in the 13hapter 3 this opa13ity has
been multiplied by 20 for the Xenon only As our interest is to understand the
sho13k stru13ture in Xenon and not the laser matter intera13tion on the piston
we performed several simulations with Xe gas for various sets of uen13es for
4
[httpphysi13snistgovPhysRefDataASDlines_formhtml
94 CHAPTER 5 RESULTS AND INTERPRETATION
Figure 515 Interferometri13 image for the shot48143 The time t = 0 13orre-
sponds to the time of laser arrival on the target and the position x = 0 13orresponds
to the interfa13e between the piston (Au layer) and the gas Its determination is
pre13ise within 100 mi13rons
53 SIMULATIONS BASED ON EXPERIMENTAL RESULTS 95
(a)
(b)
Figure 516 Raw (a) and 13orre13ted XUV spe13trum (b) for the shot 48143
96 CHAPTER 5 RESULTS AND INTERPRETATION
the MAIN and AUX beams in order to obtain the best agreement between
the simulated and measured sho13k speeds
To analyse the results from the shot 48055 (Fig 59(a)) we set the u-
en13es to 32000 amp 7500 J13m
2 This allows produ13ing the experimental sho13k
speeds 54 and 23 kms in Xenon at 01 bar for the MAIN and AUX beams
respe13tively The two sho13ks appear in Xenon at 2 and 3 ns respe13tively for
MAIN and AUX The merging of the two pre13ursors starts at sim 15 ns and
the sho13k 13ollision time o1313urs at 47 ns In Fig 517 we present the ele13tron
density proles from the simulation (dotted lines) and the experiment (solid
lines) at 10 20 30 and 40 ns
At 10 ns the two simulated pre13ursor extensions are 0165 and 0022 13m
for MAIN and AUX respe13tively The ele13tron density is larger by a fa13tor
of 4 than in the experiment The shapes of the pre13ursors are also very dif-
ferent However this 1D pi13ture supposes the plasma to be uniform within
the tube In reality in the transverse dire13tion Ne is de13reasing from the
13enter to the walls whi13h results in a lower estimation of the average value
(lt Ne gt) 13ompared to the value at the target 13enter and in a smoother pro-
le near 02 amp 035 13m at 10 ns It is also important to note that for AUX
sho13k the average lt Ne gt value underestimates the lo13al one by a fa13tor
of about 2 (as it is averaged over 06 mm instead of 03 mm) Moreover
our 1D simulation suers from an inexa13t opa13ity and 2D ee13ts are pro-
bably important espe13ially for AUX Thus we have here only a qualitative
interpretation of the experimental results
The intera13tion between the two HELIOS radiative pre13ursors starts be-
tween 10 and 20 ns like in the experiment However the shape as well as
absolute values of the simulated ele13tron density 13urves are not in agreement
with the experimental results and the intera13tion is stronger in the simulation
than in the experiment
In order to interpret the spe13tros13opi13 data presented in se13tion 52 we
performed another 1D simulation in Xenon at 06 bar and adapted the
uen13es to generate two 13ounter-propagating sho13ks with the speeds 36 and
18 kms 13lose to the experiment The time evolutions of the ele13tron density
mean 13harge and ele13tron temperature at 56 57 58 60 64 and 65 ns are
presented in Fig 518 The two sho13ks appear in Xenon at 2 and 3 ns
respe13tively for MAIN and AUX Con13erning AUX the 13ombination of a
small speed and a relative high pressure does not allow to develop a radiative
pre13ursor in agreement with the experimental results (Fig 515) whereas
the MAIN sho13k has a pre13ursor and its length is in13reasing with time
The post sho13k temperature of the MAIN is sim 21 eV and the ion 13harge
sim 9 At 57 ns the pre13ursor of MAIN rea13hes the AUX sho13k front This
time is out of our re13ord (see Fig 515) whi13h means that the intera13tion
ee13t is either absent or o1313urs at later times The stru13ture of the AUX
post sho13k is modied by the intera13tion with the MAIN pre13ursor (Fig
518b) The sho13k 13ollision o1313urs at 65 ns (Fig 518a) resulting in the
54 SUMMARY 97
Figure 517 Re13orded ele13tron density (shot 48055) together with the HELIOS
results (with Xenon opa13ity times 20) at dierent times in Xenon at 01 bar
development of two reserve sho13k waves At the 13ollision time the ele13tron
density mass density ele13tron temperature and ion 13harge rea13h respe13tively
sim 14 times 10
2113m
minus3 0034 g13m
minus3 26 eV and 10 Mean 13harge is varying
between 5-10 whi13h 13ompatible with the presen13e of lines of Xe VII-VIII in
the experimental re13ord shown in Fig 516b
Moreover in order to interpret XUV spe13tros13opi13 results shown in se13tion
52 Rodriguez performed qualitative preliminary 13omputations (as des13ribed
in Rodriacuteguez et al (2015)) of the XUV spe13tra emerging from a 600 microm thi13k
plasma with two representative values of the mass density ρ = 32 times 10
minus2
and 33 times 10
minus3g13m
3 They show that the lines of HeII 13an only be ob-
served at a temperature of sim 15 eV and for the lowest density ie in the
radiative pre13ursor
54 Summary
In this 13hapter I have presented an extensive data analysis of few representa-
tive interferometri13 and spe13tros13opi13 re13ords The average sho13k speed and
ele13tron density have been estimated from the interferograms The sho13k
speeds of the MAIN and AUX radiative sho13k waves vary between sim 30-55
and 10-30 kms respe13tively and the averaged pre13ursor ele13tron density
ranges between 10
17and 10
1913m
minus3during the sho13ks propagation
We have demonstrated the intera13tion ee13t between the two radiative
pre13ursors in the 13ase of Xe at 01 bar at 54 and 23 kms The intera13tion is
13learly 13hara13terized in the experiment by the enhan13ement of the ionisation
wave followed by the merging of the two radiative pre13ursors at 20 ns The
13ollision time is re13orded at 47 ns Su13h behavior is reprodu13ed by the
98 CHAPTER 5 RESULTS AND INTERPRETATION
(a)
(b)
(13)
Figure 518 Time evolution of the mass density (a) ele13tron temperature (b) and
mean 13harge (13) at 56 57 58 60 64 and 65 ns within the sho13k tube derived from
the HELIOS simulations (with Xenon opa13ity multiplier = 20) for two 13ounter
streaming sho13ks of sim 39 and 18 kms in Xenon at 06 bar
54 SUMMARY 99
simulation
We have investigated this intera13tion at a larger pressure 02 bar with
the following speeds sim 41 kms for the MAIN and sim 18 kms for the AUX
sho13k waves We do not re13ord any experimental signature of the radiative
pre13ursor for AUX Further we have not been able to 13at13h experimentally
the 13ollision time The re13orded pre13ursor of MAIN is not inuen13ed by AUX
wave up to 48 ns (Fig 59(b) and (13) in data analysis 13hapter) whi13h is the
limit of the re13ord On its side the 1D simulation predi13ts a tiny pre13ursor
for AUX and that both pre13ursors start to intera13t at 49 ns This plausible
sho13ks intera13tion o1313urring at times whi13h are outside of the re13ord 13an
not be 13onrmed by our experiment
The results of the transverse interferometry at 02 bar with speeds of sim40 and 20 kms indi13ate that the MAIN pre13ursor has a lateral extension of
sim 600 microm whereas it is 300 microm for AUX The pre13ursor of MAIN is almost
at with a probable small bending at the edges of the tube whereas the
AUX pre13ursor is more 13urved This means that the 2D ee13ts are more
important for AUX than for MAIN Still in the 13ase of Xenon we have
obtained information about the temperature and the mean 13harge of gas
through our time integrated XUV spe13tra (shot48143) in Xenon at 06
bar From these results we may 13on13lude that the mean ion 13harge is at
least equal to 6 and that the temperature has rea13hed 15 eV
Our simulations give a qualitative des13ription of the sho13ks intera13tion
when the laser uen13e is adjusted to give the 13orre13t sho13k velo13ities Howe-
ver it is now well known that 2D simulations (together with state of the art
opa13ities) t better with experiments (Gonzaacutelez Audit and Stehleacute 2009
Leygna13 et al 2006 Stehleacute et al 2010) For the same laser energy the 2D
simulations lead to a diminution of the sho13k speed 13ompared to 1D as also
to a diminution of the ele13tron density For instan13e in the 13ase of a sho13k
wave laun13hed by a laser beam at 1315 nm in Xenon at 03 bar at PALS and
with a laser uen13e of 85000 J13m
2 ARWEN 2D simulations give a sho13k
speed of 44 kms in agreement with the experimental one (Cotelo et al
2015) 1D simulation would require in this 13ase a uen13e of 30000 J13m
2to
a13hieve the same velo13ity
The spa13e and time integrated XUV re13ords at 06 bar for respe13tive
speeds whi13h are equal to sim 39 and 18 kms indi13ate that the temperature
of the sho13k has rea13hed values up to 15 eV and that the Xenon mean ion
13harge has also rea13hed values of 6 - 7 whereas 1D simulations predi13t ele13tron
temperature 10-30 eV and ion 13harge 5-10 (Fig 51813) A more detailed
study based on 2D simulation and radiative transfer post-pro13essing will be
ne13essary to rene the analysis
We have investigated the 13ase of other noble gases (Ar Kr) and we have
observed that for a given laser energy the sho13k velo13ity de13reases with the
mass density This study has 13onrmed that for a given density and sho13k
velo13ity the radiative ee13ts in13rease with the atomi13 number For Kr we
100 CHAPTER 5 RESULTS AND INTERPRETATION
noti13ed a tiny pre13ursor without any intera13tion To get su13h ee13t it will
be ne13essary to in13rease the sho13k velo13ity and thus the laser energy
This has been done at the ORION laser fa13ility in UK where the 13ol-
lision of two 13ounter-propagating sho13ks at equal speed sim 80 kms with
laser uen13e sim 6 times 10
14W13m
2has been performed (Clayson et al 2017
Suzuki-Vidal et al 2016) for dierent noble gases and for pressures 13ompri-
sed between 01 and 1 bar A number of diagnosti13s setup X-ray ba13klig-
hting opti13al self-emission streak imaging and interferometry (multi-frame
and streak imaging) were used to study both the post-sho13k and the radia-
tive pre13ursor Although I have not parti13ipated in the experiments I have
performed 1D simulations to interpret ORION experimental results This
work is not presented in this thesis
Chapter 6
Optimization of an
ele13tromagneti13 generator for
strong sho13ks in low pressure
gas
Contents
61 Prin13iples of operation of a high 13urrent generator 102
62 Prin13iples of the run-down phase in a PFD 103
63 Proposed design for the plasma gun 104
64 Dynami13 13ir13uit modelling 106
65 3-D MHD simulations using GORGON 13ode 111
66 Measurements 116
67 Summary 118
The previous 13hapters were dedi13ated to the study of laser generated
radiative sho13ks With irradian13e more than 10
14W13m
2 we were able to
study sho13k waves propagating at velo13ity up to 50 kms in noble gases with
an initial mass density 13omprised between 5 times 10
minus4- 3 times 10
minus3g13m
minus3
Complementary to laser experiments 13ompa13t pulsed power generators may
drive an ele13tromagneti13 13oaxial plasma gun to 13reate astrophysi13al relevant
sho13ks in lower pressure noble gases (Kondo et al 2006) with a high avai-
lability and a rather modest 13apital 13ost The ele13tromagneti13ally driven
sho13k waves may have larger s13ales than those by laser thus they 13an be
observed rather easily (Kondo et al 2008)
Su13h ele13tri13al pulsed power devi13es may then i) 13reate a high-voltage
breakdown through a gas or more easily at the surfa13e of a diele13tri13 ii)
produ13e relatively hot plasma by ohmi13 heating when the rising 13urrent in the
devi13e is passing through a portion of gas iii) a1313elerate plasma layers under
the magneti13 pressure asso13iated with the self-generated magneti13 eld when
high ele13tri13al 13urrent (say 10s of kA) is maintained for a short but su13ient
time (say 1 micros) Su13h ionization and a1313eleration are present in Z-pin13h
plasmas and in plasma fo13us devi13es (PFD) providing a 13lever geometry is
employed and the mass under 13onsideration (say 1 mg) is 13ompatible with
the stored energy (Martin Williams and Kristiansen 1996)
101
102 CHAPTER 6 ELECTROMAGNETICALLY LAUNCHED SHOCK
In this 13hapter we will see rst how a fast ele13tri13al 13ir13uit works then
we will dis13uss the prin13iples of a 13oaxial plasma gun and of PFD We will
see what is in favor of our obje13tive and what has to be avoided A geometry
13apable to rea13h our obje13tive will be proposed
The optimization of the 13oaxial gun (in term of plasma speed) will be
performed and some typi13al gures will be given in the 13ase of a generator
with a stored energy around 1 kJ and a plasma slab of 4-mm diameter ie
quite 10 times the transverse dimension of a laser driven RS The operating
13onditions are supersoni13 sho13ks up to 10-30 kms speed at stati13 pressures
of few mbar in heavy rare gases (Ar Xe) The results of this 0-D model will
be then 13ompared to those obtained with 3-D MHD simulations performed
with the 13ode GORGON (by Andrea Ciardi at LERMA) whi13h has been
used su1313essfully to des13ribe other pulse-power driven plasma experiments
(Chittenden et al 2004) as well as laboratory plasma astrophysi13s experi-
ments (Ciardi et al 2007) The diagnosti13s whi13h have been implemented
will be presented to illustrate the model as well as preliminary re13ords of the
plasma speed
61 Prin13iples of operation of a high 13urrent gene-
rator
Our aim being to a1313elerate a plasma slab using the magneti13 pressure it
is obviously needed to drive a high intensity 13urrent be13ause the magneti13
pressure is expressed by B
22micro0 in the region where exists an indu13tion B
The pressure in bars is simply 4B
2 with B in Teslas and in a 13ylindri13al
geometry 1 Tesla is the eld around a 50-kA 13urrent at a radius of 1 13m
Thus we should 13ount on roughly 100 kA delivered by the generator but
su13h a high 13urrent is delivered only in a pulsed mode by a laboratory s13ale
devi13e
There are numbers of te13hni13al solutions to a13hieve su13h ele13tri13al pa-
rameters one may use rather slow generators at moderate high voltage like
13apa13itors bank with the advantage of well know te13hniques but with limited
adjustments A1313ording to an abundant literature
1
a 13onvenient devi13e is a
medium-energy 13apa13itor bank feeding a low-indu13tan13e 13ir13uit An R-L-C
13ir13uit is a well-known 13ombination delivering a high peak 13urrent in the
pseudo-periodi13 mode For a 13apa13itor C initially 13harged under U0 key
performan13es are as follows
Current intensity I(t) is a damped sinusoid
Pseudo pulsation ω =radic[1(LC)minusR2(4L2)]
1
Institute for Plasma Fo13us Studies resour13e website httpplasmafo13usnet last
13onne13tion in 2016
62 PRINCIPLES OF THE RUN-DOWN PHASE IN A PFD 103
First 13urrent peak Ipeak = U0[radic(LC) + 08R]
Time of 13urrent peaking T4 = π(2ω)
Equivalent impedan13e Z =radic(LC)
For a safe handling in air a voltage not higher than 30 kV is re13ommen-
ded Taking a total 13apa13itan13e of 6 microF 2700 J are stored under 30 kV For
an indu13tan13e of 240 nH the expe13ted peak 13urrent might be 150 kA at 19
micros This set of values will be a guideline for the development presented in
further se13tions
62 Prin13iples of the run-down phase in a PFD
Our aim is thus to 13reate a fast moving plasma sheath with quite a one-
dimension geometry The issues are to initiate this plasma at the best then
to maintain even improve its stru13ture during the a1313eleration nally to
laun13h it
Instead of 13reating a gas breakdown in volume whi13h would lead ra-
pidly to 13on13entrate the 13urrent in an ar13 the idea for PFD operation is to
start from a surfa13e dis13harge (Lee 1969 Bernard 2002) In the 13lassi13al
13oaxial geometry of the Mather-type PFD the 13entral ele13trode (usually the
anode) is a 13ut metal 13ylinder and the outer one is a se13ond metal 13ylin-
der of the same length or better a squirrel 13age allowing many viewpoints
to the plasma Both ele13trodes are atta13hed to a diele13tri13 bottom plate
made of polymer or 13erami13 When the 13apa13itor bank is swit13hed on the
high voltage is applied to this 13m-size gap and a radial surfa13e dis13harge is
initiated at the interfa13e with the gas forming a quite uniform ring-shaped
layer of thi13kness around 1 mm An insulating sleeve is adjusted around
the 13entral ele13trode to for13e the plasma to ow rapidly upwards under the
magneti13 pressure in this region The result is an elongation the dis13harge
path preventing an ar13ing in the plasma layer
At later times as the 13urrent is growing the plasma layer is strongly
inuen13ed it is pushed upwards by the magneti13 pressure jtimesB it is heatedby the ohmi13 ee13t then its ionization degree in13reases it be13omes denser
as a per13entage (10-40) of the heavy parti13les from the swept volume are
a1313reted (Potter 1971) This ele13tromagneti13ally driven motion is 13alled the
rundown phase Due to the a1313retion it is des13ribed by a so-13alled snowplow
model whi13h will be dis13ussed later in detail In quite all the designs even
if very high speeds are reported (Lee 1969 Serban 1995) the plasma is bell
shaped whi13h does not fulll our requirements
At the end of the rundown phase whi13h mat13hes roughly with the 13urrent
peak the plasma stays 13onne13ting the upper ends of the ele13trodes and the
radial 13omponent of the magneti13 for13es grows rapidly Then the plasma is
104 CHAPTER 6 ELECTROMAGNETICALLY LAUNCHED SHOCK
strongly pushed to the axis giving the so-13alled Z-pin13h This stage made
the PFD popular be13ause the pin13hed plasma is a sour13e of fast ele13trons
fast ions hard radiation and possibly neutron beams it must be avoided
in our 13ase It was also quoted (Lee and Serban 1996) that optimal PFD
are mat13hing a universal fa13tor 13omprising anode radius peak 13urrent and
gas density meaning a robustness of the design when a geometry has been
13hosen
63 Proposed design for the plasma gun
The previous des13ription has shown the positive inuen13e of a surfa13e dis-
13harge the modi13ation of the shape by the sleeve and the issue of the
pin13hing ee13t A1313ordingly the following design is proposed following a
work by Kondo et al (Kondo et al 2006 2008)
To ensure a rapid dis13harge with the values given in the se13tion 61 a
highly 13oaxial 13ir13uit is 13hosen for all the large parts the energy bank the
13losing swit13h and all the 13onne13tions The initial phase is kept with two
13oaxial ele13trodes pressed on a at insulator Homogeneity of the plasma
sheath is expe13ted by 13hoosing a rather small radial gap of 25 mm The ee13t
of magneti13 for13es during the rundown phase must be enhan13ed espe13ially
due to the in13rease of the mass of the plasma as des13ribed above As the
13urrent 13urve is the rising part of a sinusoid a tri13k is used to in13rease
the lo13al magneti13 eld B around a 13ondu13tor varies as the inverse of the
distan13e to axis so the proposed shape for the anode is a 13oneHowever a
nal divergen13e must be avoided and that is obtained by a rounded tip In
order to keep 13onstant the plasma length along the 13urrent path the radial
gap is kept 13onstant so the outer ele13trode is a hollow 13one with the same
angle Above the 13one the a1313elerator is 13onne13ted to a 13ondu13ting tube
where the plasma 13an propagate freely in the ba13kground gas The distan13e
of the plasma sheath to the 13one tip is in13reasing rapidly and the main
13urrent whi13h is still high will pass preferably through the diuse plasma
remaining between the ele13trodes Thus there will be no magneti13 pressure
anymore a13ting on the sheath whi13h 13an propagate freely The resulting
devi13e is des13ribed in Fig 61a and the a13ting magneti13 for13es shown in Fig
61b
Te13hni13ally su13h a 13onguration is 13onvenient even at 15 kV be13ause the
sharp edge of the anode tou13hing the insulator forms a so-13alled triple point
where the ele13tri13 eld is enhan13ed at the surfa13e of the insulator whi13h
triggers e13iently a dis13harge in presen13e of gas at low pressure (01 10
mbar) as proposed by Kondo et al (Kondo et al 2006) Polya13etal is a
13onvenient material for diele13tri13 and me13hani13al performan13es
The dire13tion and the orientation of the magneti13 for13e are other points to
be dis13ussed The high-intensity 13urrent (up to sim 150 kA) generates a strong
63 PROPOSED DESIGN FOR THE PLASMA GUN 105
(a)
(b)
Figure 61 (a) Sket13h of the sho13k generator showing the pulsed ele13tri13al 13ir13uit
the set of 13oaxial 13oni13al ele13trodes with a 13onstant radial gap and the plasti13
insulator featured in grey on whi13h a planar surfa13e dis13harge is initiated The
installation of three opti13al bers allows looking radially at the plasma moving in
the sho13k tube (b) S13hemati13s of the plasma dynami13s inside the 13oaxial gun
in fast-pulse mode the ele13tri13al 13urrent ows in the super13ial layers of the two
13oaxial 13oni13al ele13trodes and through an annular plasma layer The magneti13
pressure Pmag pushes the dis13harge axially
106 CHAPTER 6 ELECTROMAGNETICALLY LAUNCHED SHOCK
Figure 62 Exploded view of the plasma gun
azimuthal magneti13 eld between the ele13trodes thus the magneti13 pressure
jtimesB whi13h a1313elerates the annular plasma sheath stays perpendi13ular to the
sheath lo13ally (Fig 61b) The more planar is the initial sheath the more
axial will be the magneti13 pressure Finally if the roles of the ele13trodes are
ex13hanged the dire13tion of the 13urrent will be inverted and the same for B
so the pressure will stay oriented in order to expel the plasma Compared
with the Mather-type plasma fo13us (Potter 1971 Zambra et al 2009) the
plasma sheath 13onsidered here is quite planar 13lose to the insulating surfa13e
and later is expe13ted to stay planar and perpendi13ular to the axis As in a
PFD the thi13kness of the plasma sheath in13reases gradually (Zambra et al
2009) but it is assumed to stay in the mm-range We employ 13oaxial 13oni13al
brass ele13trodes ea13h 42-mm high with a 13onstant gap of 25 mm At the
bottom level internal and external radii of ele13trodes are 125 mm and 15
mm respe13tively This small-size dis13harge 13ell on top of the generator will
be modelled in the following
64 Dynami13 13ir13uit modelling
We optimize the ele13tri13 generator for various gases namely Ar and Xe
with the motivation to produ13e plasma sho13ks with speeds sim 1 - 30 kms
ie Ma13h numbers up to 200 For that we design the ele13trodes and set
64 DYNAMIC CIRCUIT MODELLING 107
the ele13tri13al 13ir13uit parameters to produ13e 1-micros pulses in the gas 13hamber
The geometry of the ele13trodes is as mentioned above The half angle of the
13oni13al ele13trode is 13hosen to be α =15deg and thus the height of the 13one is h
sim 42 mm
Eleven (N = 11) 13apa13itors ea13h with 13apa13itan13e C0 of 06 microF are13onne13ted in parallel giving an equivalent 13apa13itan13e of the bank C = 66
microF 13harging voltage U0 of the bank is set to 15 kV Thus the available
ele13tri13al energy is 750 J The 13ir13uit resistan13e is used to be 1 m Ω and
damping resistan13e is set to zero whi13h yields a total resistan13e (R = Rcir+
Rdamp N ) of 1 mΩ and a pseudo-periodi13 regime is expe13ted
Based on the above geometri13al and ele13tri13al parameters we have esti-
mated the time evolution of various parameters of the radial dis13harge layer
namely speed mass a1313eleration et13 at a dierent position along the axis
from z = 0 to h For this 13al13ulation we need to solve a set of three equa-
tions whi13h give the main 13urrent (- dqdt) supposed to ow only through
the plasma layer the mass a1313retion rate (dMdt) and the velo13ity v of
the plasma These equations together with initial 13onditions are explained
below The ele13tri13al equation is written 13lassi13ally
Ld2qdt2 +Rdqdt+ qC = 0 (61)
When the R L and C parameters are not time-dependent the 13ir13uit
equation 61 has an analyti13al solution a damped sinusoid for the 13urrent I
= -dqdt and that was used as a test of the iterative solving pro13ess and to
13onrm the parameters of the external 13ir13uit with a short 13ir13uit load The
indu13tan13e of the external 13ir13uit L0 is 13omputed from the ringing frequen13y
in short 13ir13uit (66 nH) whi13h results in a 13ir13uit impedan13e Z0 =
radic
L0Cto be 01 Ω In the plasma gun mode the sheath 13arrying the 13urrent (see
Fig 61b) is a1313elerated by the magneti13 pressure and it behaves as the de-
formable part of the 13ir13uit then L is time-dependent and a numeri13al solver
is ne13essary for equation 61 We used a lumped-parameter model whi13h has
proven to des13ribe similar 13ir13uits su1313essfully (Potter 1971 Zambra et al
2009) Gonzalez (Gonzalez et al 2004) showed that the mass and momen-
tum equations for a 13urrent sheath in the shape of an annular piston moving
forward in the axial dire13tion 13an be given by
dMdt = ερ0π(R2e minusR2
i)v (62)
where ε is a1313retion fa13tor and that the transformation of magneti13 to
kineti13 energy is expressed by
d(Mv)dt = lI22 (63)
where l is the linear indu13tan13e of the gun and 13an be expressed as l =
(micro04π) ln (ReRi) the radii being estimated where the plasma sheath is
13urrently lo13ated The equations 62 and 63 have been introdu13ed in the
108 CHAPTER 6 ELECTROMAGNETICALLY LAUNCHED SHOCK
(a)
(b)
(13)
Figure 63 Time dependen13e of the main parameters for Ar gas at 1 mbar a1313or-
ding to the 13ir13uit model (a) 13urrent (kA) and voltage (V) (b) a1313reted mass
(kg) and rate of a1313retion (kgs) (13) speed (ms) and kineti13 energy (J)
64 DYNAMIC CIRCUIT MODELLING 109
Figure 64 Plot of the dierent output parameters (normalized with respe13t to
the peak values) obtained at the top of the inner ele13trode For ea13h group upper
panel represents the variation for Ar gas whereas lower panel is for Xe gas Proles
of output parameters are given (a) with 13one angle (b) with a1313retion fa13tor (13)
with gas pressure and (d) with damping resistan13e
110 CHAPTER 6 ELECTROMAGNETICALLY LAUNCHED SHOCK
time-dependent iterative solving of equation 61 Therefore from equations
62 and 63 we get
dvdt =[ lI
2
2 minus ερ0π(R2e minusR2
i)v2]
M(64)
From equation 64 we may note that initial value of mass M(t0) 13annot
be set to zero as it would lead to a nonphysi13al divergen13e of the a1313eleration
In this regard we have 13onsidered an initial thi13kness of the 13urrent sheath
of 1 mm and integrated the gas density at rest leading to M(t0) equal to
35 times 10
minus10kg for Ar and 13 times 10
minus9kg for Xe A higher limit has to be
put too M(t0) must be small 13ompared with the mass at the end of the
laun13hing phase that is the mass 13ontained in the gap times the a1313retion
fa13tor The above value for M(t0) follows that 13riterion and within these
limits the 13hoi13e of initial mass is not inuen13ing the results
Parameter Referen13e Value Variation
Half angle of the 13one 15
13
14
15
16
middot middot middot 24A1313retion fa13tor ǫ 04 01 02 03 middot middot middot 09
Pressure P 100 Pa 10 20 50 100 200 500 1000
Damping resistan13e Rdamp 0 ohm 0 01 02
Table 61 Input parameters their standard values and range of variation
As the motion is mainly along the z-axis the speed v is the derivative
of the plasma position v = dzdt We solved these equations by a time-
dependent method where the initial 13ondition is that at the onset time t0
the 13harge q is set to CU0 Further the 13urrent i = -dqdt is taken to be zero
at t0 Thus using equation 61 the initial rate of 13hange of the 13urrent is
d
2qdt
2= - qL0C Then all the rst time-derivatives are updated at the rst
time step t0+∆t with ∆t=043 ns using equations 61-64 The quantities
like q M v z are updated at the next step then the iteration is pursued
Initially position a1313eleration and velo13ity of the sheath are set to be zero
as well as the rate of 13hange of the mass
As we initiate the ele13tri13al 13urrent into the gas inside the 13hamber the
plasma sheath is a1313elerated with respe13t to its initial mass as seen above
At later times the mass a1313retion is masking the inuen13e of this arbitrary
value We have 13al13ulated the plasma evolution in the dis13harge for dierent
sets of initial 13onditions whi13h are reported in Table 61
Fig 63 shows the behavior for the standard values reported in Table 61
in the 13ase of Argon at 1 mbar The 13orresponding values of 13harge mass
and velo13ity will be hereafter 13alled the referen13e values In Fig 63a the
variation of 13urrent and voltage in the rst half-period is 13lassi13al showing
that the pulse generator departs slightly from a pure RLC 13ir13uit Fig
63b represents the a1313reted mass whi13h in13reases in time as well whereas
65 3-D MHD SIMULATIONS USING GORGON CODE 111
the rate of mass a1313retion 13ulminates with a 13ompetition of speed growth
and de13rease of the swept volume per time step be13ause it depends on the
squared radii In Fig 6313 plotted are the speed and kineti13 energy of the
plasma sheath First the speed of the plasma is in13reasing slowly with time
quite linearly as the mass and the kineti13 energy is roughly a third power
of the time
That leads to 13hoose a design where the a1313eleration is imposed for a
su13iently long time However the estimated speed of plasma is sim 140 kms
whi13h is a very optimisti13 value The reason may 13ome from two reasons
(i) a part of the 13urrent ows through the large sho13ked volume redu13ing
the magneti13 pressure a13ting on the plasma sheath (ii) energy transfer from
magneti13 to thermal energy and ionization is redu13ing the kineti13 one The
observation of a very bright emission from the plasma will be a 13lear eviden13e
of this redistribution of the energy
Further we made a 13omparative study of the parameters indi13ated in
Table 61 and applied to two gases namely Ar and Xe In order to 13he13k the
ee13ts of the initial 13onditions (13one angle a1313retion fa13tor pressure and
damping resistan13e) we have performed a systemati13 study by varying only
one parameter and keeping the others 13onstant and equal to the referen13e
values given by the se13ond 13olumn of Table 61 The tested range is listed
in the third 13olumn of Table 61 Similarly for ea13h set the variation of the
plasma 13onditions at the top of the muzzle is normalized to the referen13e
value Fig 64 summarizes the variation of the sele13ted dierent output
parameters the time tm to rea13h the muzzle (z = h) the ele13tri13al 13urrent
and the axial speed at muzzle the laun13hed mass M the kineti13 energy and
the average speed at the top of the ele13trode 13al13ulated from the initial
point v=h( tm - t0)
Table 62 summarizes the respe13tive ratios for Ar and Xe at the top of
the 13oaxial ele13trodes As the values of interest are obtained at 13onstant
stored energy in the 13apa13itors there is a benet to work with a lighter gas
like Argon However the results obtained using this 13ir13uit model show that
the referen13e values are ensuring a satisfa13tory behavior for both gases
Speed Kineti13 energy Time to laun13h Mass
222 151 66 31
Table 62 Laun13hing performan13e ArXe
65 3-D MHD simulations using GORGON 13ode
Our simplied model (13f se13tion 64) is 13onvenient to perform qui13kly some
parametri13 tests in the a1313eleration phase and to help designing the devi13e
but it is not 13apable to treat the sho13k dynami13s in detail nor to give valuable
112 CHAPTER 6 ELECTROMAGNETICALLY LAUNCHED SHOCK
information on the plasma parameters in the sho13k region So rened nume-
ri13al simulations of the experiments were performed in 13ollaboration with A
Ciardi (Ciardi 2014) using the GORGON 13ode (Chittenden et al 2004 Ci-
ardi et al 2007 Suzuki-Vidal et al 2014) GORGON is an expli13it parallel
13ode designed to solve the resistive MHD equations on a three-dimensional
(3-D) Cartesian grid employing a Van Leer type algorithm The 13ode treats
the plasma as a single uid but solves separately the energy equations for
ions and ele13trons allowing dierent temperatures for the two spe13ies Both
thermal 13ondu13tion and resistive diusion are treated using Braginskii-like
transport 13oe13ients
Variables At rest Sho13k Post-sho13k
Ar density ρ (g13m
3) 18E-6 25E-5 10E-5
Ele13tron density Ne (13m3) 10 E+15 20E+18 15E+18
Ele13tron Temperature Te (eV) 003 (1lowast) 10 15-18
Ion density ni (13m3) 30E+16 35E+17 2-3E+7
Ion Temperature Ti 0 50 15-20
Average Speed ltVgt kms 0 50 20-40
ltzgt 02 6 7-9
Table 63 Values of variables in various 13onditions at rest (with seed ele13trons)
inside the sho13k and inside post-sho13k region A star lowast denotes the insulator surfa13e
This 13ode des13ribes the 13reation of the surfa13e dis13harge then its a1313ele-
ration by the magneti13 pressure It provides the mapping of all the plasma
parameters and lo13al B-eld in the laun13hing 13one as well as in the free ight
region The 3-D simulation was performed for Argon gas at 1 mbar only the
dis13harge 13urrent law being taken from experiment The Table 63 presents
the ranges of all the variables in the regions of interest
As an example Fig 65 shows an enlarged mapping of log (ne) at two
times in the laun13hing phase then when the sho13k has deeply penetrated
in the sho13k tube Taking the zero time as a referen13e the averaged speed
zt is 40 kms in this simulation When the two snapshots are 13onsidered
the estimate of the instantaneous speed zt is higher at 56 kms The
time history of plasma merging and early free ight into the sho13k tube
is presented in Fig 66 During the a1313eleration (inside the double 13one)
the plasma is well lo13alized on a planar annular sheath whi13h is 13arrying
the 13urrent as assumed in the lumped parameter model We see that the
merging is well a13hieved at the muzzle shortly before 1250 ns When the
sho13k has penetrated inside the tube (1500 ns) the 13omputed B-eld is
negligible at this lo13ation and the propagation is quite free as expe13ted for
the 13hosen design of the devi13e However a high 13urrent is still passing
through the gas at that time a su13ient ele13tron density remaining in the
13oni13al gap for that
65 3-D MHD SIMULATIONS USING GORGON CODE 113
Figure 65 Example of mapping of the ele13tron density from a 3-D MHD simu-
lation (ba13kground gas Argon at 1 mbar) when the plasma sheath is (left) at
the exit of the plasma gun (right) in free ight 13onditions (log s13ales for the false
13olors)
114 CHAPTER 6 ELECTROMAGNETICALLY LAUNCHED SHOCK
Figure 66 Time history of plasma merging and early free ight through the
mapping of the ele13tron density ele13tron temperature and average ioni13 13harge as
given by a 3-D MHD simulation (ba13kground gas Argon at 1 mbar) (s13ales for the
false 13olors)
65 3-D MHD SIMULATIONS USING GORGON CODE 115
Figure 67 Axial proles values taken in the sho13k tube along a line slightly o-
axis at a distan13e of 1 mm for mass density ion temperature ion density average
ion 13harge magneti13 eld ele13tron temperature ele13tron density and average speed
at 1500 ns (ba13kground gas Argon at 1 mbar) A maping of |B| is given with a
dashed line indi13ating sho13k front position as well
116 CHAPTER 6 ELECTROMAGNETICALLY LAUNCHED SHOCK
To 13onrm the steep stru13ture of the sho13k Fig 67 presents the axial
proles of ele13tron temperature Te number density Ne mass density ρ and
average ion 13harge ltZgt at 1500 ns inside the tube with an enlarged axial
position s13ale Steep fronts are observed for mass density ele13tron density
and temperature as in the earliest simulations (Potter 1971 Kondo et al
2008) The sho13k region is at 10 eV and sim2times1018 ele13trons13m
minus3 the post-
sho13k region is hotter Argon ionization stages of 5 (sho13k) to 9 (post-sho13k)
are obtained fore13asting an emission spe13trum in the UV or harder A wider
view of the plasma is given by other quantities at the sho13k front and behind
ion temperature average speed and lo13al magneti13 eld in Fig 67 We get
the 13onrmation of the extin13tion of B after the rst 5 mm inside the sho13k
tube as well as the sho13k speed at 50 kms 13oherent with the 13onstant speed
of 56 kms mentioned just above At present these parameters are guiding
the 13hoi13e of diagnosti13s like ultra-fast interferometry or UV-spe13tros13opy
66 Measurements
The aim of these tests was to show the 13onsisten13e of the above model by me-
asuring the plasma speed 13lose to the plasma gun muzzle The parameters of
the devi13e are the referen13e ones as dened in Table 61 A rather simple and
noise-free method for diagnosing a plasma gun during the rundown phase
is to re13ord the light emitted by the plasma sheath with a spatial resolution
13ompatible with the plasma stru13ture If we 13onsider a line of sight whi13h
is radial or tangential ie perpendi13ular to the dire13tion of propagation
one expe13ts to see a sudden rise when the plasma enters the dete13tion vo-
lume possibly a plateau when the plasma travels inside the volume then a
slow de13rease when the hotter plasma leaves the volume but when the sho13k
remnants are still present So the expe13ted signal is a triangular asymme-
tri13 pulse (Serban and Lee 1995) or some kind of double exponential pulse
(Stehleacute et al 2012) A peak indi13ates the time of ight for the given posi-
tion allowing to 13ompute a mean speed providing the laun13hing time (t=0)
and all positions are known An instant speed is a1313essible providing two
13lose dete13tors are 13onsidered That has been proposed and tested su1313ess-
fully in the 13ase of radiative sho13ks driven by one PALS laser beam (Stehleacute
et al 2012) the dete13tors being 13ollimated and ltered sili13on diodes For
13ooler plasmas the visible emission is more relevant Inside a squirrel 13age
plasma fo13us Serban has installed a set of opti13al ber looking tangentially
at the drift zone (Serban and Lee 1995) Despite a 13oni13al eld of view the
results were 13on13lusive at speeds as high as 100 kms in D2 An improved
version with 13ollimated opti13al bers (a1313eptan13e angle 2deg) was developed
too (Serban and Lee 1995) A non intrusive method was developed (Veloso
et al 2012ba) by imaging the plasma on a set of opti13al bers allowing to
play with the magni13ation
66 MEASUREMENTS 117
(a)
(b)
Figure 68 Experimental results (a) time history of the passive opti13al re13ords
13ompared to the main 13urrent for Argon gas 13lose to the referen13e pressure (b)
time-integrated signal vs Ar pressure for the dierent bers (13) averaged speed vs
Ar ba13king pressure
118 CHAPTER 6 ELECTROMAGNETICALLY LAUNCHED SHOCK
In our 13ase (see Fig 61a) three 1-mm PMMA (Poly(methyl metha13ry-
late)) opti13al bers are installed radially along a transparent se13tion of the
sho13k tube at z sim 625 mm 70 mm and 775 mm and fast DET10A dete13tors
re13ord the transient presen13e of the plasma sheath in the respe13tive lines of
sight The re13orded spe13tral range is limited to the whole visible in a preli-
minary stage a band-pass ltering might give information on the presen13e of
spe13i13 ioni13 spe13ies 13hara13teristi13 of a plasma temperature level however
the amount of light was not su13ient to do so Fig 68a presents the time
evolution of the ele13tri13al 13urrent with a period of 4 micros and the delayed
side-on observational data of the moving plasma inside the sho13k tube The
behavior of the ele13tri13al 13ir13uit is following well the expe13ted ringing of a
weakly damped RLC 13ir13uit the zero-time being xed at the onset of the
13urrent On opti13al 13hannels a steep rise and a peak are the signature of a
mm-thi13k fast-sho13k stru13ture (Serban and Lee 1995) 13onsistent with early
numeri13al simulations (Potter 1971 Tou Lee and Kwek 1989 Kwek Tou
and Lee 1990 Veloso et al 2012b) On ea13h ber signal se13ondary peaks
namely the one delayed by 4-5 micros are attributed to 13urrent restrikes at the
pseudoperiod To avoid the mixing of sho13ks further experiments should be
13ondu13ted in an aperiodi13 regime with a heavy-duty damping resistor added
in series to ea13h 13apa13itor Rdamp sim NZ
On Fig 68b and 13 are presented the exploitation of opti13al re13ords with
the Ar ba13kground pressure this parameter varying in the widest range (b)
the integrated emission whi13h presents a maximum between low pressure side
when the temperature is high but the number of emitters is low and the high
pressure side when the temperature is mu13h lower (13) the averaged speed
ziti for the rst peak on ea13h ber The relative intensities re13orded in the
three lo13ation s are not signi13ant be13ause of a strong 13opper deposition
13loser to the muzzle whi13h ae13ted the dete13tion The long-life noise on
the opti13al signals was making the lo13al speed 13al13ulated as (zi-zj)(ti-tj) a
non-reliable output One 13an note Ma13h numbers in the range 20 60 with
a pressure dependen13e similar to previous work (Kondo et al 2006) The
reported observations are 13onsistent with a high-speed millimetri13 planar
plasma travelling in the sho13k tube The speed is lower than dedu13ed from
the simplied model possibly due to the la13k of dissipation terms whi13h
is overestimating the transfer to kineti13 energy At the 13ontrary a realisti13
simulation as the one performed using GORGON is in better agreement
67 Summary
For this part of the work the obje13tive was to show that in 13omplement of
laser experiments a 13ompa13t pulsed power generator might drive astrophy-
si13ally relevant sho13ks in low pressure noble gases with a higher availability
and at a rather modest 13apital 13ost The longitudinal a1313eleration of a
67 SUMMARY 119
plasma sheath in a low ba13king gas pressure has been demonstrated in 13o-
axial plasma guns as early as in the 70s even leading to speeds of 100 kms
in light gas as H2 whi13h were measured and modeled A 13hoi13e was made
to revisit this type of devi13e At present the ele13tromagneti13 13oaxial plasma
guns are quite only 13onsidered as the initiators of a strong radial plasma
implosion plasma fo13us devi13es (PFD) due to a good ability to produ13e a
pin13h plasma and emission of energeti13 parti13les and hard X-ray For that
reason it was ne13essary to adapt the prin13iple of a PFD with two major
13onstraints in13rease the 13onversion of stored energy into the kineti13 energy
of a plasma sheath prevent totally any pin13hing
That obje13tive was rea13hed by dening the ele13tri13al and geometri13al
parameters of a low-indu13tan13e and 13ompa13t pulsed power generator An
optimization pro13ess has been established to mat13h a13hievable ele13tri13al pa-
rameters of the 13ir13uit (13apa13itan13e indu13tan13e peaking 13urrent intensity)
with a plasma motion in the a1313elerating tube over few 13m leading to a nal
speed of 10-30 kms A model was built up to des13ribe the dynami13s of an
RLC 13ir13uit with a varying element 13omposed of the moving plasma sheath
a1313elerated by the self generated magneti13 pressure The 13hange in plasma
position along the axis is in13reasing the indu13tan13e and the mass a1313retion
is in13reasing the plasma mass It was shown that a 13hange in the ele13trode
geometry namely a 30deg 13oni13al shape was in13reasing the energy 13onversion
and preventing any fo13using The further step in 13ontrast with PFD is the
merging of the plasma ring into a rather planar one whi13h is inje13ted in a
drift tube
Considering Ar or Xe at gas pressure in the range 01 10 mbar and
a highly 13oaxial 1-kJ devi13e the simple model was su13ient to predi13t
performan13es agreeing with the only work quoted in the literature Basi13
observations of the moving plasma using side-on dete13tion of the visible
emission give features 13oherent with the model Ma13h numbers from 20
to 60 are obtained tting with the obje13tive of strong sho13k formation A
rened 3-D MHD numeri13al simulation indi13ates very promising features of
the sho13k in view of exploring mm-size sho13ks in a regime 13omplementary
to laser driven ones and on a table-top devi13e allowing a higher repetition
rate
120 CHAPTER 6 ELECTROMAGNETICALLY LAUNCHED SHOCK
Chapter 7
Con13lusions and perspe13tives
71 Con13lusions
Although ubiquitous in the astrophysi13al environments radiative sho13ks are
13omplex phenomena whi13h still require an important eort to understand
them Beside observations whi13h suer from a la13k of angular resolution
laboratory experiments provide today an interesting approa13h to improve
our present knowledge In this 13ontext my thesis work is dedi13ated to the
experimental and numeri13al study of the 13hara13teristi13s of two 13ounter pro-
pagating radiative sho13ks propagating at dierent velo13ities (20-50 kms)
Most of the work presented here is the out13ome of the rst-of-its-kind ex-
perimental 13ampaign held in year 2015 at Prague and Laser Asterix system
(PALS) laser fa13ility
In this manus13ript I presented the setup for the aforesaid sho13k expe-
riment Following this the data analysis results interpretation as well as
relevant numeri13al simulations 13on13erning various diagnosti13s viz visible in-
terferometry and XUV spe13tros13opy have been presented Although a large
fra13tion of the experimental re13ords were performed for Xenon at various
pressures I have also studied the sho13k 13hara13teristi13s of dierent gases like
Ar Kr and He Our results are 13omplementary to those obtained at ORION
laser fa13ility also in 2015 relative to the 13ollision of identi13al radiative sho13k
waves at higher velo13ities (80 kms) In parallel in this thesis I have also
worked on the optimization of a setup dedi13ated to ele13tromagneti13ally ge-
nerated strong sho13ks at lower velo13ities In the following I briey present
a brief summary of the main results of my thesis work
In the third 13hapter I have presented a series of 1D hydrodynami13 simu-
lations (realized with the 13ommer13ial 13ode HELIOS) on isolated and 13ounter-
propagating sho13k waves at equal and dierent velo13ities The simulations
13onrm that a single radiative sho13k propagating in Xenon gas at 01 bar
is 13hara13terized by an extended pre13ursor and a large 13ompression of 30 in
the post sho13k At 50 kms the temperature evolution with the distan13e
shows identi13al post-sho13k and pre-sho13k values This indi13ates that the
numeri13al sho13k is of super13riti13al nature However I nd that there is an
important un13ertainty in the Xenon opa13ity whi13h makes the renement of
the simulation unne13essary espe13ially in terms of group numbers Next I
have investigated the intera13tion two 13ounter propagating sho13ks for iden-
ti13al (50-50 kms) and non identi13al (50-20 kms) sho13k speeds For the
121
122 CHAPTER 7 CONCLUSION
13ase of identi13al speeds the pre13ursors merge together at around 10 ns The
intera13tion is then 13hara13terized by a regular in13rease of the ele13tron density
and the temperature with the time The sho13ks 13ollision at 38 ns leads to a
jump in the ele13tron density (6 times 10
2113m
minus3) and temperature (39 eV) On
the other hand for the 13ase of non-identi13al sho13k speeds (13ase representing
our experiments) the pre13ursors intera13tion starts later than in the former
13ase The sho13ks 13ollision o1313urs at 49 ns and it is 13hara13terized by a sudden
in13rease of the ele13tron density also by an order of magnitude (4 times 10
21
13m
minus3) whereas the temperature in13reases up to 29 eV
Although the data analysis of all the experimental re13ords obtained du-
ring the PALS experimental 13ampaign has been 13arried out only a few repre-
sentative interferometri13 and spe13tros13opi13 re13ords for Xe gas are dis13ussed
in details in this thesis The interferometri13 re13ords allowed me to estimate
the average sho13k speed and time variation of the ele13tron density during the
sho13k propagation The sho13k speeds of the MAIN and AUX radiative sho13k
waves are found to be ranging between sim 30-55 and 12-25 kms respe13tively
whereas the averaged pre13ursor ele13tron density is varying between 10
17and
10
1913m
minus3during the sho13ks propagation
From the interferometri13 re13ords I have investigated the ee13t of the
intera13tion between the two radiative pre13ursors for the sho13ks propagating
in Xenon at 01 bar with respe13tive speeds of 50plusmn3 and 23plusmn3 kms The
intera13tion starts at 20 ns and is followed by the merging of the two ra-
diative pre13ursors This pre13ursor intera13tion is 13hara13terized through the
enhan13ement of the ionisation wave The sho13ks 13ollision is re13orded at 50
ns On the 13ontrary the same intera13tion behavior is not seen at 02 bar
(sho13k speeds sim 38plusmn4 and 18plusmn2 kms for the MAIN and AUX sho13k waves
respe13tively) Moreover in this 13ase there is no signature of a radiative pre-
13ursor for the AUX sho13k The pre13ursors intera13tion if any should then
13ould o1313ur at times whi13h are outside of the re13ord and 13an not be 13on-
rmed by the experiment This indi13ates that for a given gas and sho13k
speed the radiative ee13ts de13reases with the initial mass density
The investigation of the lateral extension of the sho13k has been made
through the analysis of transverse interferograms At 02 bar the MAIN
sho13k with a speed sim 40 kms has a lateral pre13ursor extension of sim570plusmn30microm whereas it is 275plusmn25 microm for AUX sho13k whi13h is propagating
with a speed of 20 kms The MAIN sho13k pre13ursor is almost of a at
spatial prole whereas the AUX pre13ursor is more 13urved suggesting that
the 2D ee13ts are mu13h more important for AUX than for MAIN One of
the explanations is that the spot size of AUX laser on the target is smaller
than the target width whereas the MAIN laser has a spot size whi13h is equal
to the target width
In order to interpret the experimental results with appropriate 1D simu-
lations I have optimized the lasers uen13es to numeri13ally obtain a sho13k
speed equal to that re13orded in the experiment In the 13orresponding simu-
72 PERSPECTIVES 123
lations for Xenon at 01 bar I note the intera13tion of the two pre13ursors
However I do not nd a good quantitative agreement for the ele13tron den-
sity Beside the question of the impre13ise opa13ities this disagreement might
also be attributed to 2D ee13ts (Gonzaacutelez Audit and Stehleacute 2009 Leygna13
et al 2006 Stehleacute et al 2010)(Cotelo et al 2015)
The analysis of spa13e- and time-integrated XUV re13ord at 06 bar is
presented for two 13ounter propagating sho13ks of speeds sim 36plusmn4 and 18plusmn5kms for MAIN and AUX respe13tively The presen13e of HeII Balmer lines
and Xenon lines tends to indi13ate a temperature of the sho13k of about 15
eV and a Xenon mean ion 13harge around 6 - 7 The 1D simulations predi13t
ele13tron temperature in the range of 10-30 eV while ion 13harge to be ranging
between 5 and 10
In addition to this study whi13h was fo13used on the Xe 13ase I have
investigated the sho13k 13hara13teristi13s for Ar Kr at 13lose mass density (sim 6
times 10
minus4g13m3) whi13h 13orrespond to 03 bar and Kr gas 02 bar Almost no
pre13ursor is found to be present in 13ase of Ar whereas a tiny pre13ursor have
been observed for Kr This indi13ates that for a given density and sho13k
speed the radiative ee13ts in13rease with the atomi13 number
In parallel I have worked on the optimization of the design of an experi-
mental setup where the sho13k is generated ele13tromagneti13ally This setup
allows studying sho13k sim 30 kms in noble gas at sim1 mbar The parameters
of a low-indu13tan13e and 13ompa13t pulsed power devi13e have been optimized
with a simple model in view of building-up and a1313elerating a plasma sheath
in Ar and Xe at gas pressure in the range 01-10 mbar Simple observations
of the moving plasma using side-on dete13tion of the visible emission give
features 13oherent with the model Sho13ks 13orresponding to ma13h numbers
ranging from 20 to 60 are obtained
72 Perspe13tives
The experimental results presented in this thesis 13orrespond to the 13ase of
two 13ounter-propagating radiative sho13ks We have been able to study the
time variations of the pre13ursor intera13tion and the ee13t of gas pressure
with 1D time dependent interferometri13 re13ords obtained with a streak 13a-
mera Together with the help of XUV spe13tros13opy we 13ould estimate the
sho13k speed ele13tron density and sho13k temperature However no informa-
tion is provided about the lateral extension of the pre13ursor and we were
not able to re13ord XUV data for all the shots Therefore in order to make
these results more 13omprehensive we have designed and parti13ipated in a
follow-up experimental 13ampaign in September 2016 at PALS This time we
fo13ussed on single sho13ks in 13lose experimental 13onditions but with a wider
range in terms of speeds (45-100 kms) We repla13ed the time dependent
streak 13amera interferometri13 study by an instantaneous 2D interferometri13
124 CHAPTER 7 CONCLUSION
image (GOI) obtained with a short pulse sub pi13ose13ond laser and 13ameras
XUV spe13tra were also re13orded The re13ords were performed at three dif-
ferent times of the sho13k propagation allowing to dedu13e the sho13k speed
and the time evolution of the pre13ursor morphology This follow up experi-
ment will allow to 13omplete the analysis of the laser generated sho13k waves
presented in this thesis
The preliminary analysis I have performed on these new re13ords is very
promising An example of these interferometri13 re13ords is given in the Figure
71a for the shot 30364 in Xe+He at 06 bar The high speed of the sho13k is
related to the energy of the laser whi13h is here 170 J whereas it was less than
120 J in the previous experiment The 13omparison with the interferometri13
re13ord (not shown) before the shot allows to dedu13e the ele13tron density
whi13h rea13hes 18 times 10
1913m
3 The high quality re13ord shows 13learly 2D
ee13ts for this sho13k wave
(a)
(b)
Figure 71 Interferometri13 re13ord obtained at 546 ns for a single radiative sho13k
of speed sim 100 kms propagating in Xe+He gas at 06 bar driven by laser at 348
nm with energy 170 J The dierent 13olors in (b) 13orrespond to yellow le Ne le35 times 10
1713m
minus3 13yan 37 - 74 times 10
1713m
minus3 blue74 - 15 times 10
1813m
minus3 green
15 - 18 times 10
1813m
minus3 magenta 18 - 22 times 10
1813m
minus3 red 22 - 26 times 10
1813m
minus3
orange26 times 10
1813m
minus3- 18 times 10
1913m
minus3
This analysis is now ongoing and the results of this experimental 13am-
72 PERSPECTIVES 125
paign will be 13ompared with that obtained from the analysis of re13ords from
2015 presented in this manus13ript This instantaneous imaging interferome-
tri13 diagnosti13 does not follow the sho13k 13ontinuously with the time but it
gives a pre13ise map of the ele13tron density whi13h 13an be inverted through
Abel inversion to obtain the lo13al estimation of the ele13tron density
In parallel in the ele13tromagneti13ally laun13hed sho13ks we have upgra-
ded our table top setup to a13hieve higher speeds For this we have used 12
13apa13itors ea13h of 1 microF at the pla13e of the previous 11 13apa13itors ea13h of
06 microF We have implemented new diagnosti13s namely a visible interferome-
ter working in the Gated Opti13al Imaging mode and time and spa13e visible
spe13tros13opy similar to the one des13ribed in the 13hapter 4 For the visible in-
terferometry we have Ma13h Zehnder Interferometer setup and we will image
the sho13k at dierent times with the help of a SL300 pi13ose13ond high energy
NdYAG laser (λ = 532 nm and energy = 024 J) The experiment will soon
be performed on this new setup and the analysis of su13h re13ords will also be
undertaken in future allowing a 13omparison of ele13tromagneti13ally laun13hed
sho13ks with the laser-driven sho13k whi13h should make the bridge between
thin and opti13ally thi13ker sho13ks
126 CHAPTER 7 CONCLUSION
Chapter 8
Thesis summary
Radiative sho13ks (RS) are present in various astrophysi13al environments es-
pe13ially in stars They are then present in all the supersoni13 a1313retioneje13tion
pro13esses during the stellar evolution from the early phase of the proto-
stellar 13ollapse up to the magnetosphere a1313retion on to the already formed
T- Tauri stars up to the late stage of the supernovae sho13ks
Radiative sho13ks are strong sho13ks (ie Ma13h number Mgtgt1) whi13hrea13h high temperatures and thus are the sour13e of intense radiation Alt-
hough the observation of (ie spatially resolved ) signatures of su13h sho13ks
in the astrophysi13al environments is very di13ult an alternate approa13h na-
med laboratory plasma astrophysi13s provides a powerful tool to study them
on the Earth
These waves have been experimentally studied sin13e more than a de13ade
mostly on large-s13ale laser fa13ilities in noble gases and with dierent targets
geometries The ablation pro13ess generated by the intera13tion between the
laser and a foil indu13es a sho13k wave in the foil whi13h then propagates
in the gas at a speed of several tens of kms Numeri13al simulations 13an
des13ribe these sho13ks with an improving pre13ision With laser intensities on
target 13omprised between 1014 and 1015 W13m
2 these experiments allowed
to re13ord sho13k speeds ranging between 40 and 150 kms
All previous experimental studies have been fo13used on the 13ase of single
radiative sho13ks However in astrophysi13al 13onditions the radiative sho13k
in general intera13ts with a denser medium leading to the development of
ree13ted and transmitted sho13ks A few representative examples of su13h
phenomena are the intera13tion of supernovae remnants with dense mole13ular
13louds the a1313retion sho13ks on the photosphere of T-Tauri stars and the bow
sho13ks at the head of fast stellar jets The 13ollision (or the intera13tion) of
two radiative sho13k waves is obviously a rare astrophysi13al event and the
template 13ase of the supernova remnants DEM L316 (see Fig 1 of Williams
et al (1997)) is still the subje13t of debates Williams et al (2005) Toledo-
Roy et al (2009) Velarde et al (2006) In this regard the development of
dedi13ated laboratory experiments to the study of propagation and intera13tion
of 13ounter-propagating sho13k waves is important as a tool to 13hara13terize
su13h events through their spe13i13 signatures
Complementary to laser experiments 13ompa13t pulsed power generators
may drive an ele13tromagneti13 13oaxial plasma gun to 13reate astrophysi13al
relevant sho13ks in lower pressure noble gases with a high availability and a
127
128 CHAPTER 8 THESIS SUMMARY
rather modest 13apital 13ost The ele13tromagneti13ally driven sho13k waves may
have larger s13ales than those by laser Thus they 13an be analyzed rather
easily with the adequate diagnosti13s (Kondo et al 2008)
In this thesis I am presenting radiative sho13ks in the 13ontext of labo-
ratory astrophysi13s using two experimental approa13hes laser-driven sho13ks
and ele13tromagneti13ally driven sho13ks
First 13hapter of my thesis is Radiative sho13k waves in whi13h the physi13s
of radiative sho13ks have been studded Radiative sho13k waves are hypersoni13
sho13k waves whi13h are heated to high temperature and as a 13onsequen13e
be13ome the sour13e of intense radiation This radiation in turn modies
the dynami13s and stru13ture of the sho13k itself whi13h makes its stru13ture
more 13ompli13ated Radiative sho13k waves o1313ur in several astrophysi13al 13ir-
13umstan13es su13h as in the a1313retion sho13k of protostellar formation (Stahler
Palla and Salpeter 1986) the supernova explosion and the intera13tion of
their remnants with the dense interstellar medium (Chevalier 1977) the bow
sho13ks at the head of stellar jets (Hartigan et al 2001)
Radiative sho13k waves 13an now be studied in the laboratory whi13h allows
to 13ompare the data with models existing in the literature and to 13he13k the
ability of the numeri13al 13odes to reprodu13e them (see for instan13e Bouquet
et al (2004) Leygna13 et al (2006)) The experimental study of radiative
sho13k 13hara13teristi13s is 13ru13ial in understanding these various energeti13 pro-
13esses o1313urring in the astrophysi13al plasmas
The jump 13onditions (Rankine-Hugoniot relations) for the simple 13ase
of an ideal gas whi13h rely on the values of the thermodynami13al quantities
on both sides of the dis13ontinuity Let us 13onsider a one-dimensional sho13k
propagating in a gas at rest with the speed us In the frame asso13iated with
the sho13k front the pre-sho13k (upstream) uid velo13ity is then u1 = - us
while behind the sho13k (post-sho13k or downstream region) this velo13ity is
u2 as shown in the Fig 21 Considering the 13ase of strong sho13ks where
M may be 13onsidered to be ≫1 jump 13onditions for ideal gas are
ρ2ρ1
=(γ + 1)
(γ minus 1)(81)
T2
T1=
2M2γ(γ minus 1)
(γ + 1)2(82)
kBT2 =2(γ minus 1)
(γ + 1)2mus
2(83)
For mono-atomi13 gases γ = 53 the 13ompression ratio is estimated to
be 4 and the post-sho13k temperature is
kBT2 =3
16mu1
2 =3
16mPAu1
2(84)
129
This temperature is thus proportional to the square of the sho13k velo13ity
and to the atomi13 mass In 13ontrast to the dis13ussion on ideal gas in the
13ase of a real gas is a little more 13ompli13ate It is to note that in this 13ase
a part of the kineti13 energy is used to ex13ite and ionize the post-sho13k gas
As a result its temperature is lower than that for the 13ase of ideal gas
In addition the pressure is also modied due to ionisation (Mi13haut et al
2004) In13luding the ee13t of ionization with the average 13harge z in the
medium for the 13ase real gas jump relations for real gas are
ρ2u2 = ρ1u1 (85)
ρ2u22 + ρ2
kBT2
m(1 + z2) = ρ1u
21 + ρ1
kBT1
m(1 + z1) (86)
ρ2u2
[5
2
(1 + z2)
mkBT2 + ǫexc2 +
u222
]
= ρ1u1
[5
2
(1 + z1)
mkBT1 + ǫexc1 +
u212
]
(87)
In the previous both 13ases the ee13ts of radiation heating and 13ooling
have been negle13ted However the radiation emitted in a strong sho13k will
ae13t its stru13ture These ee13ts 13an be determined using mass momentum
and energy 13onservation equations whi13h now have to in13lude the 13ontribu-
tions of the radiative ux energy and pressure This 13ase is more 13omplex
now Taking into a1313ount radiative ee13ts the equations of radiative hydro-
dynami13s in the 13o-moving frame of the sho13k (Mihalas and Mihalas 1999
Drake 2006) 13an be written as
partρ
partt= minusnabla(ρu) (88)
ρ
(
partu
partt+ unablau
)
= minusnabla(P + Prad) (89)
part
partt
(
ρu2
2+ ρǫ+ Erad
)
+nabla
[
ρu
(
ǫ+u2
2+
P
ρ
)
+ (Erad + Prad)u
]
= minusnablaFrad
(810)
The 13ontributions linked to Prad and Erad are ee13tive only at very high
temperature (ie velo13ity) However for the radiative sho13ks of interest in
our experiments the 13ontribution of the radiative ux Frad is non-negligible
13ompared to ρu3 These sho13ks are thus in the radiation ux dominated
regime Most of the experimental sho13k waves are in this regime These
hydrodynami13 equations are strongly non-linear
Depending on the opa13ity the radiation emitted from the sho13k may be
absorbed by the pre-sho13k region indu13ing its pre-heating Far away from
the dis13ontinuity the stru13ture of the upstream medium is determined by
the absorption On its side the stru13ture of the transition layer of the down-
stream medium 13lose to the dis13ontinuity depends on the balan13e between
130 CHAPTER 8 THESIS SUMMARY
the emission (13ooling) and the absorption (heating) Thus the full stru13ture
of the sho13k mainly depends on the variation of the opa13ity χ (13m
2g) The
opti13al depth (τ(s)) measured from the position of the jump is then the
relevant parameter for the absorption It is given by
τ(s) =
int s
sjump
χ(sprime)ρ(sprime) dsprime (811)
where sprime 13hara13terizes the path of the radiation As we are interested in
the mean frequen13y averaged radiative ux the relevant opa13ity here will be
the averaged Rosseland opa13ity whi13h will be dened in due 13ourse
If τ gtgt 1 the regime is referred as opti13ally thi13k while in the opposite
13ase (τ lt 1) it is opti13ally thin Flux dominated radiative sho13ks have
been 13lassied depending on the opti13al depth of upstream and downstream
regimes
We are more interested in the kind of sho13k have a radiative pre13ursor It
is possible to produ13e a radiative sho13k at the high energy laser fa13ilities with
laser intensity of sim 10
14W13m
2 A short summary of number of experiment
performed at high power laser fa13ilities have been listed in table 21 Other
than laser driven radiative sho13k it is possible to laun13h 13omparatively slower
radiative sho13ks (sim 10-30 kms) using ele13tromagneti13 generators
The 13hapter two is dedi13ated to the numeri13al analysis of radiative sho13k
waves and of the main physi13al pro13esses whi13h play a role in these 13omplex
pro13esses Radiative sho13ks are present in various astrophysi13al pro13esses
implying extreme 13onditions Laboratory experiments then allow investiga-
ting the underlying physi13al pro13esses whi13h take pla13e in these 13onditions
and whi13h are not observable from the Earth due to a la13k of spatial resolu-
tion However experiments are not always straightforward to interpret and
the help of numeri13al simulations be13omes pre13ious
Dierent numeri13al approa13hes are used for the simulation of laboratory
plasmas Some of them use xed grids Other introdu13e the possibility of
renement (on the y) in some meshes (Adaptative Mesh Renement) to
maintain a good resolution in the interesting parts of the plasma Another
approa13h is based on meshes following the uid (Lagrangian des13ription)
whi13h means that the meshes move with the uid and no matter may go
through a mesh to another The form of the equations dier in this 13ase (see
for instan13e Orban et al (2013))
HELIOS is su13h a one-dimensional Lagrangian radiation hydrodynami13
13ommer13ial 13ode (Ma13Farlane Golovkin and Woodru 2006) and I have
used it to simulate our experiment of intera13ting sho13k waves This 13ode
is easy to handle and has the presently interesting 13apability to be able to
simulate the 13ase of two sho13k waves propagating in opposite dire13tions and
laun13hed by two dierent lasers HELIOS may des13ribe non-LTE plasmas as
it in13ludes an in-line 13ollisional radiative (HELIOS-CR) model for 13omputing
131
non-LTE atomi13 level populations at ea13h time step of the hydrodynami13s
simulation However I used HELIOS in the LTE approximation whi13h is
appropriate to our experimental 13onditions (Rodriguez et al 2011) as will
be dis13ussed in the next se13tion
The 13ode solves the uid equations of motion using the pressure 13ontri-
butions of ele13trons ions and radiation It may des13ribe the ele13trons and
ions as two intera13ting uids at respe13tive temperatures Te and Ti Thermal
13ondu13tion is des13ribed within a ux-limited ele13tron 13ondu13tion model ba-
sed on the Spitzer 13ondu13tivity and the laser energy deposition is 13omputed
with an inverse Bremsstrahlung model
The radiation emission and absorption terms are introdu13ed in the equa-
tion of energy for the ele13trons and in the radiation transport equations One
of the methods whi13h are proposed is the ux-limited multi-group radia-
tion diusion model where the radiative ux is proportional to the gradient
of the radiative energy and is inversely proportional to the Rosseland opa-
13ity The expression is pondered by a ux-limited diusion 13oe13ient in
order to obtain the good opti13ally thin limit This 13oe13ient follows the
Larsen expression des13ribed in Olson Auer and Hall (2000) The se13ond
method is a (time independent) multi-angle short-13hara13teristi13s s13heme In
our 13ase where radiation and hydrodynami13s are strongly 13oupled and thus
the radiation eld varies rapidly we used the diusion model together with
LTE Plan13k and Rosseland multi-groups opa13ities
In addition our version of HELIOS uses the PROPACEOS (Ma13Farlane
Golovkin and Woodru 2006) Equation of State (EOS) and multi-group
opa13ity tables whi13h are generated by the 13ode
1
and it is not possible
to in13lude any other opa13ity or EOS table The knowledge of the thermo-
dynami13al 13onditions is required for an adequate simulation of the plasma
des13ription In our experiment the mass density ranges between sim 10
minus4and
10
minus1g13m
3 while the temperature values are 13omprised between and sim 0 to
50 eV The pre13ise 13al13ulations performed by Rodriguez et al (Fig2 of Rodri-
guez et al (2011)) for Xenon indi13ate that the thermodynami13al 13onditions
of our radiative sho13k experiments 13orrespond to the LTE regime
The equation of state (EOS) denes the dependen13e of the pressure
ionisation internal energy with the mass density and temperature Several
models do exist in the literature
Our version of HELIOS for LTE plasmas may handle polytropi13 or PRO-
PACEOS equation of state Unfortunately there are not many details about
this last method The authors of the 13ode mention that it uses a QEOS-type
model (More et al 1988) in the strong 13oupling regime (ie at relatively
high densities and low temperatures) and an isolated atom model whi13h
uses a detailed atomi13 stru13ture modeling in the weak 13oupling region The
properties between the weak and strong 13oupling regimes are said to be obtai-
1
see http wwwprism minus cscomSoftwarePROPACEOS
132 CHAPTER 8 THESIS SUMMARY
ned by interpolating in a thermodynami13ally 13onsistent manner This EOS
in13ludes the 13ontributions from the translation of ions and ele13trons ele13-
tron degenera13y atom ionization and ex13itation and Coulomb intera13tions
(Debye - Hu13kel 13orre13tion)
I have performed HELIOS 1D simulations for dierent 13ases for single
and 13ounter propagating sho13ks in 13hapter 2 of my thesis Radiative sho13k
waves propagating in Xenon at 01 bar with a velo13ity of 50 kms are 13ha-
ra13terized by the development of an extended radiative pre13ursor The huge
13ompression of 38 in the post-sho13k is a 13onsequen13e of both the sho13k and
the gas ionisation (fa13tor of 10) as also from the radiative 13ooling The post-
sho13k and pre-sho13k temperature on both side of the peak are identi13al
indi13ating that the sho13k is super13riti13al
The spe13ta13ular role of the opa13ity has been highlighted Due to the
un13ertainty in the opa13ity used for Xenon in our simulations I de13ided that
at this stage it was not ne13essary to rene the simulation in terms of group
numbers within the 1D des13ription of the sho13k wave
Our numeri13al study gives the main 13hara13teristi13s of the intera13tion of
two 13ounter-propagating sho13k waves with speeds equal to 50-50 kms and
50-20 kms The 13ase of identi13al speeds is simpler due to the symmetry of
the problem However whatever the speeds the most important signature
of the intera13tion is the merging of the pre13ursor at 8 ns for 50-50 kms
and at 15 ns for 50-20 kms This merging is followed by a regular in13rease
with time of the ele13tron density and the temperature The 13ollision time
is 13hara13terized by a sudden in13rease of the ele13tron density by an order of
magnitude rea13hing 66 times 10
21and 3 times 10
2113m
minus3respe13tively whereas the
temperature in13reases up to 39 and 28 eV
When a short pulse of a high-power laser beam is fo13ussed on a thin foil
it transfers a huge amount of energy to the foil material Almost instantane-
ously the laser absorption heats it up to the ablation A hot and low density
13orona is generated ba13kwards by this pro13ess Beyond the ablation surfa13e
a sho13k is formed in the foil as a 13onsequen13e of the momentum 13onservation
(ro13ket ee13t) The sho13k moves inward in the foil Our 13ase of the experi-
mental setup as the foil 13loses a tube lled with gas the sho13k propagates
then in the gas where it is studied through various kinds of diagnosti13s The
foil then a13ts as a piston
We performed the experiments at Prague Asterix Laser System (PALS)
in the Spring 2015 (ve weeks 20th April - 22th May) The obje13tive of the
experimental 13ampaign was to study the evolution of two 13ounter streaming
radiative sho13ks driven by two 13ounter-propagating lasers with an irradian13e
in the range of 10
14W13m
minus2 and therefore to 13hara13terize their intera13tion
and 13ollision In the 13hapter third 13hapter I have rst presented a brief
des13ription of the PALS laser fa13ility and of the laser beams used in our
experiment This followed by a presentation of the targets design and an
overview of the general setup and of our main diagnosti13s
133
The Prague Asterix Laser System (PALS) is a laser fa13ility based on
an Infrared high-power iodine laser system (Asterix IV) (Jungwirth et al
2001) Using dierent amplifying stages the laser fa13ility is able to deliver
energy up to 1 kJ in 03 ns at the fundamental wavelength 1315 nm The
output laser beam 13an be further subdivided in few auxiliary beams All
auxiliary beams may be frequen13y doubled (λ = 657 nm) or tripled (438
nm) The PALS laser fa13ility is 13apable of ring up to two high energy laser
shots per hour Compared with solid states lasers this gas laser is known to
deliver a quite homogenous beam intensity without hot spots
For our experiments we used two independent laser beams to drive the
two 13ounter streaming sho13ks in the gaseous target For this purpose the
fundamental PALS laser beam has been subdivided after the fourth amplier
(V4) into two beams with respe13tive energy fra13tions of 60 and 40 (see
Fig 42) The most energeti13 of these laser beams is then inje13ted through
the fth (V5) amplier and its frequen13y are then tripled This beam at 438
nm will be hereafter termed termed as the MAIN laser beam
The se13ond of the two previous laser beams propagates then without any
modi13ation It will be hereafter 13alled the AUX laser beam (1315 nm)
The s13heme of the energy distribution is shown in the Fig 42 and the
13hara13teristi13s of MAIN and AUX laser are presented in Table 41
For the experiments PALS oers two va13uum 13hambers with a spheri13al
and 13ylindri13al geometry respe13tively The spheri13al 13hamber assembly used
in our experiments is shown in Fig 43 This 13hamber having a diameter
of 100 13m is 13losed by two entran13e 13ir13ular windows of respe13tive diameters
80 13m and 50 13m to allow the MAIN and AUX beams to penetrate in the
13hamber and to rea13h the target Ea13h of these windows is outtted with a
hinged door Several 13ir13ular ports of various diameters also allow for the
dierent diagnosti13s The target holder and some diagnosti13s are mounted
on an opti13al ben13h whi13h is de13oupled from the 13hamber walls
Our targets (see Fig 45) s13hemati13ally 13onsist in small tubes of 4 mm
length 13losed on both sides by two spe13i13 foils of thi13kness 11 microm on whi13h
the laser beams are fo13used (one laser per foil) with an irradian13e of about
10
1413m
minus3 The tube is lled with gas in whi13h the radiative sho13k propa-
gates with a velo13ity of 30 - 60 kms The two foils 13losing the target insure
the 13onversion through ablation and sho13k generation of the laser energy
into me13hani13al energy
In our experiments we used spe13i13 targets for alignment and 13hara13te-
rization of the beam size These targets were manufa13tured at the Pole
instrumental of Observatoire de Paris
The gaseous targets were lled in situ at a fra13tion of 1 bar with a gas
(viz Xe Ar a mixture of Xe and He) whi13h thus provide the medium for the
sho13k propagation We 13hose an in-situ lling system (Fig 410) to redu13e
the dieren13e of pressures inside and outside of the tube This is 13riti13al as
a large pressure dieren13e between the target 13ell and the va13uum 13hamber
134 CHAPTER 8 THESIS SUMMARY
13an result in the breaking of the ultra-thin SiN windows of the target
First of all we performed fo13alization of MAIN and AUX laser beam
on target by using Al-massive targets The two PALS laser lenses were
translated up to a13hieve the suitable diameters on the target whi13h were
nally set to 450 - 500 microm and 250 - 300 microm for the MAIN and AUX beams
The size of the impa13t was 13ontrolled ea13h day on mo13k targets before using
the laser beams for real shots on gaseous targets It is worth to pre13ise that
the keV diagnosti13 was still in pla13e for the shots on the gaseous targets
In that 13ase the beams impa13t Parylene-N instead of Aluminium The keV
signal is then weaker than for the mo13k aluminum targets
Various kinds of diagnosti13s were implemented namely visible interfero-
metry XUV and visible spe13tros13opy For the same two keV 13ameras are
indu13ted to monitor the impa13t and two visible 13ameras for target alignment
For visible interferometory we have taken referen13e images (without any
fringe) for every target as shown in the Fig 417a for one target example
In order to re13ord su13h images we blo13ked the referen13e beam and let the
probe beam only to illuminate the target in a stati13 mode and with the slit
open Then we redu13ed the slit width to its nominal value (200 microm) This
enabled us to have the visualization of the portion of the target whi13h was
imaged in the slit on the target and also to lo13ate the position of the pistons
before the shot This image is termed as the referen13e target image Then
the system was pla13ed in the interferometri13 mode and the fringe pattern
was optimized to be perpendi13ular to the slit and thus in the longitudinal
mode perpendi13ular to the dire13tion of the sho13k propagation The Fig
417b reports su13h a typi13al interferogram re13orded during our experiment
(13f Fig 413) The horizontal axis 13orresponds to the dire13tion of the sho13k
propagation (with a total s13ale of 54 mm on the target) and the verti13al
axis 13orresponds to time (s13aling 200 ns)
The time and spa13e integrated XUV plasma emission are re13orded with
a Flat Field XUV spe13trometer using a 13on13ave grating In the Fig 419
the drawing of the XUV spe13trometer setup s13heme (red 13olor) is presented
together with the spheri13al 13hamber and the target The XUV spe13trometer
is installed on the top of the spheri13al 13hamber whi13h allows the XUV
emission passing through the Si3N4 membrane on the top of the target (see
se13tion 422) to be re13orded
In addition to the interferometry and XUV spe13tros13opy we have also im-
plemented a time- and spa13e-resolved visible spe13tros13opi13 (see Fig 420a)plasma
emission diagnosti13
In 13hapter 5 I present the results of the two diagnosti13s with a parti13u-
lar fo13us on the interferometri13 data whi13h I have extensively studied with
the help of a spe13i13 data analysis pro13edure that I have developed The
visible interferometry is an adequate tool to probe the ele13tron density of
the radiative pre13ursor up to the 13riti13al density (4 times 10
2113m
minus3at the wa-
velength 527 nm of the probing laser) whereas the denser post-sho13k region
135
is opaque to the visible light XUV radiation is present in the whole sho13k
stru13ture and thus the time- and spa13e-integrated XUV spe13tros13opy may
allow exploring the self-emission 13oming from the post-sho13k and pre13ursor
regions
In the visible interferometry the interferen13es between the main and
referen13e beams overlapping on the dete13tor produ13e a pattern of fringes
whi13h follow the relative phase variations between the probe (whi13h passes
through the plasma) and the referen13e beams (see se13tion 441 of the previ-
ous 13hapter for the experimental setup) The phase variation between two
13onse13utive fringes of the unperturbed beams (ie without any plasma) is
equal to 2π Then in the presen13e of the plasma in the probe beam these
fringes be13ome shifted due to the modi13ation of the refra13tive index More
details about the prin13iple of the interferometry and the Ma13h-Zehnder in-
terferometri13 setup may be found in the APPENDIX A and we re13all here
only the expression of the phase shift
∆φ asymp minus πd
λNclt Ne gt (812)
where λ = 527 nm is the wavelength of the probing laser Nc = 4 times 10
21
13m
minus3the 13riti13al density at this wavelength and lt Ne gt is the ele13tron
density averaged over the laser path d in the 13ell and is dened as
lt Ne gt=
int d
0
Ne(z t)dy
d(813)
In this 13hapter I have presented an extensive data analysis of few re-
presentative interferometri13 and spe13tros13opi13 re13ords I am presenting here
summary of results obtained bellow
The average sho13k speed and ele13tron density have been estimated from
the interferograms The sho13k speeds of the MAIN and AUX radiative sho13k
waves vary between sim 30-55 and 10-30 kms respe13tively and the averaged
pre13ursor ele13tron density ranges between 10
17and 10
1913m
minus3during the
sho13ks propagation
We have demonstrated the intera13tion ee13t between the two radiative
pre13ursors in the 13ase of Xe at 01 bar at 54 and 23 kms The intera13tion is
13learly 13hara13terized in the experiment by the enhan13ement of the ionisation
wave followed by the merging of the two radiative pre13ursors at 20 ns The
13ollision time is re13orded at 47 ns Su13h behavior is reprodu13ed by the
simulation
We have investigated this intera13tion at a larger pressure 02 bar with
the following speeds sim 41 kms for the MAIN and sim 18 kms for the AUX
sho13k waves We do not re13ord any experimental signature of the radiative
pre13ursor for AUX Further we have not been able to 13at13h experimentally
the 13ollision time The re13orded pre13ursor of MAIN is not inuen13ed by AUX
wave up to 48 ns (Fig 59(b) and (13) in data analysis 13hapter) whi13h is the
136 CHAPTER 8 THESIS SUMMARY
limit of the re13ord On its side the 1D simulation predi13ts a tiny pre13ursor
for AUX and that both pre13ursors start to intera13t at 49 ns This plausible
sho13ks intera13tion o1313urring at times whi13h are outside of the re13ord 13an
not be 13onrmed by our experiment
The results of the transverse interferometry at 02 bar with speeds of sim40 and 20 kms indi13ate that the MAIN pre13ursor has a lateral extension of
sim 600 microm whereas it is 300 microm for AUX The pre13ursor of MAIN is almost
at with a probable small bending at the edges of the tube whereas the
AUX pre13ursor is more 13urved This means that the 2D ee13ts are more
important for AUX than for MAIN Still in the 13ase of Xenon we have
obtained information about the temperature and the mean 13harge of gas
through our time integrated XUV spe13tra (shot48143) in Xenon at 06
bar From these results we may 13on13lude that the mean ion 13harge is at
least equal to 6 and that the temperature has rea13hed 15 eV
Our simulations give a qualitative des13ription of the sho13ks intera13tion
when the laser uen13e is adjusted to give the 13orre13t sho13k velo13ities Howe-
ver it is now well known that 2D simulations (together with state of the art
opa13ities) t better with experiments (Gonzaacutelez Audit and Stehleacute 2009
Leygna13 et al 2006 Stehleacute et al 2010) For the same laser energy the 2D
simulations lead to a diminution of the sho13k speed 13ompared to 1D as also
to a diminution of the ele13tron density For instan13e in the 13ase of a sho13k
wave laun13hed by a laser beam at 1315 nm in Xenon at 03 bar at PALS and
with a laser uen13e of 85000 J13m
2 ARWEN 2D simulations give a sho13k
speed of 44 kms in agreement with the experimental one (Cotelo et al
2015) 1D simulation would require in this 13ase a uen13e of 30000 J13m
2to
a13hieve the same velo13ity
The spa13e and time integrated XUV re13ords at 06 bar for respe13tive
speeds whi13h are equal to sim 39 and 18 kms indi13ate that the temperature
of the sho13k has rea13hed values up to 15 eV and that the Xenon mean ion
13harge has also rea13hed values of 6 - 7 whereas 1D simulations predi13t ele13tron
temperature 10-30 eV and ion 13harge 5-10 (Fig 51813) A more detailed
study based on 2D simulation and radiative transfer post-pro13essing will be
ne13essary to rene the analysis
We have investigated the 13ase of other noble gases (Ar Kr) and we have
observed that for a given laser energy the sho13k velo13ity de13reases with the
mass density This study has 13onrmed that for a given density and sho13k
velo13ity the radiative ee13ts in13rease with the atomi13 number For Kr we
noti13ed a tiny pre13ursor without any intera13tion To get su13h ee13t it will
be ne13essary to in13rease the sho13k velo13ity and thus the laser energy
This has been done at the ORION laser fa13ility in UK where the 13ol-
lision of two 13ounter-propagating sho13ks at equal speed sim 80 kms with
laser uen13e sim 6 times 10
14W13m
2has been performed (Clayson et al 2016
Suzuki-Vidal et al 2016) for dierent noble gases and for pressures 13ompri-
sed between 01 and 1 bar A number of diagnosti13s setup X-ray ba13klig-
137
hting opti13al self-emission streak imaging and interferometry (multi-frame
and streak imaging) were used to study both the post-sho13k and the radia-
tive pre13ursor Although I have not parti13ipated in the experiments I have
performed 1D simulations to interpret ORION experimental results This
work is not presented in this thesis
The previous 13hapters were dedi13ated to the study of laser generated
radiative sho13ks With irradian13e more than 10
14W13m
2 we were able to
study sho13k waves propagating at velo13ity up to 50 kms in noble gases with
an initial mass density 13omprised between 5 times 10
minus4- 3 times 10
minus3g13m
minus3
Complementary to laser experiments 13ompa13t pulsed power generators may
drive an ele13tromagneti13 13oaxial plasma gun to 13reate astrophysi13al relevant
sho13ks in lower pressure noble gases (Kondo et al 2006) with a high avai-
lability and a rather modest 13apital 13ost The ele13tromagneti13ally driven
sho13k waves may have larger s13ales than those by laser thus they 13an be
observed rather easily (Kondo et al 2008)
Su13h ele13tri13al pulsed power devi13es may then i) 13reate a high-voltage
breakdown through a gas or more easily at the surfa13e of a diele13tri13 ii)
produ13e relatively hot plasma by ohmi13 heating when the rising 13urrent in the
devi13e is passing through a portion of gas iii) a1313elerate plasma layers under
the magneti13 pressure asso13iated with the self-generated magneti13 eld when
high ele13tri13al 13urrent (say 10s of kA) is maintained for a short but su13ient
time (say 1 micros) Su13h ionization and a1313eleration are present in Z-pin13h
plasmas and in plasma fo13us devi13es (PFD) providing a 13lever geometry is
employed and the mass under 13onsideration (say 1 mg) is 13ompatible with
the stored energy (Martin Williams and Kristiansen 1996)
In this 13hapter we will see rst how a fast ele13tri13al 13ir13uit works then
we will dis13uss the prin13iples of a 13oaxial plasma gun and of PFD We will
see what is in favor of our obje13tive and what has to be avoided A geometry
13apable to rea13h our obje13tive will be proposed
The optimization of the 13oaxial gun (in term of plasma speed) will be
performed and some typi13al gures will be given in the 13ase of a generator
with a stored energy around 1 kJ and a plasma slab of 4-mm diameter ie
quite 10 times the transverse dimension of a laser driven RS The operating
13onditions are supersoni13 sho13ks up to 10-30 kms speed at stati13 pressures
of few mbar in heavy rare gases (Ar Xe) The results of this 0-D model will
be then 13ompared to those obtained with 3-D MHD simulations performed
with the 13ode GORGON (by Andrea Ciardi at LERMA) whi13h has been
used su1313essfully to des13ribe other pulse-power driven plasma experiments
(Chittenden et al 2004) as well as laboratory plasma astrophysi13s experi-
ments (Ciardi et al 2007) The diagnosti13s whi13h have been implemented
will be presented to illustrate the model as well as preliminary re13ords of the
plasma speed
Our aim being to a1313elerate a plasma slab using the magneti13 pressure
it is obviously needed to drive a high intensity 13urrent be13ause the magneti13
138 CHAPTER 8 THESIS SUMMARY
pressure is expressed by B
22micro0 in the region where exists an indu13tion B
The pressure in bars is simply 4B
2 with B in Teslas and in a 13ylindri13al
geometry 1 Tesla is the eld around a 50-kA 13urrent at a radius of 1 13m
Thus we should 13ount on roughly 100 kA delivered by the generator but
su13h a high 13urrent is delivered only in a pulsed mode by a laboratory s13ale
devi13e
There are numbers of te13hni13al solutions to a13hieve su13h ele13tri13al pa-
rameters one may use rather slow generators at moderate high voltage like
13apa13itors bank with the advantage of well know te13hniques but with limited
adjustments A1313ording to an abundant literature
1
a 13onvenient devi13e is a
medium-energy 13apa13itor bank feeding a low-indu13tan13e 13ir13uit An R-L-C
13ir13uit is a well-known 13ombination delivering a high peak 13urrent in the
pseudo-periodi13 mode For a 13apa13itor C initially 13harged under U0 key
performan13es are as follows
Current intensity I(t) is a damped sinusoid
Pseudo pulsation ω =radic[1(LC)minusR2(4L2)]
First 13urrent peak Ipeak = U0[radic(LC) + 08R]
Time of 13urrent peaking T4 = π(2ω)
Equivalent impedan13e Z =radic(LC)
For a safe handling in air a voltage not higher than 30 kV is re13ommen-
ded Taking a total 13apa13itan13e of 6 microF 2700 J are stored under 30 kV For
an indu13tan13e of 240 nH the expe13ted peak 13urrent might be 150 kA at 19
micros This set of values will be a guideline for the development presented in
further se13tions
For this part of the work the obje13tive was to show that in 13omple-
ment of laser experiments a 13ompa13t pulsed power generator might drive
astrophysi13ally relevant sho13ks in low pressure noble gases with a higher
availability and at a rather modest 13apital 13ost The longitudinal a1313elera-
tion of a plasma sheath in a low ba13king gas pressure has been demonstrated
in 13oaxial plasma guns as early as in the 70s even leading to speeds of 100
kms in light gas as H2 whi13h were measured and modeled A 13hoi13e was
made to revisit this type of devi13e At present the ele13tromagneti13 13oaxial
plasma guns are quite only 13onsidered as the initiators of a strong radial
plasma implosion plasma fo13us devi13es (PFD) due to a good ability to pro-
du13e a pin13h plasma and emission of energeti13 parti13les and hard X-ray For
that reason it was ne13essary to adapt the prin13iple of a PFD with two major
13onstraints in13rease the 13onversion of stored energy into the kineti13 energy
of a plasma sheath prevent totally any pin13hing
1
Institute for Plasma Fo13us Studies resour13e website httpplasmafo13usnet last
13onne13tion in 2016
139
That obje13tive was rea13hed by dening the ele13tri13al and geometri13al
parameters of a low-indu13tan13e and 13ompa13t pulsed power generator An
optimization pro13ess has been established to mat13h a13hievable ele13tri13al pa-
rameters of the 13ir13uit (13apa13itan13e indu13tan13e peaking 13urrent intensity)
with a plasma motion in the a1313elerating tube over few 13m leading to a nal
speed of 10-30 kms A model was built up to des13ribe the dynami13s of an
RLC 13ir13uit with a varying element 13omposed of the moving plasma sheath
a1313elerated by the self generated magneti13 pressure The 13hange in plasma
position along the axis is in13reasing the indu13tan13e and the mass a1313retion
is in13reasing the plasma mass It was shown that a 13hange in the ele13trode
geometry namely a 30deg 13oni13al shape was in13reasing the energy 13onversion
and preventing any fo13using The further step in 13ontrast with PFD is the
merging of the plasma ring into a rather planar one whi13h is inje13ted in a
drift tube
Considering Ar or Xe at gas pressure in the range 01 10 mbar and
a highly 13oaxial 1-kJ devi13e the simple model was su13ient to predi13t
performan13es agreeing with the only work quoted in the literature Basi13
observations of the moving plasma using side-on dete13tion of the visible
emission give features 13oherent with the model Ma13h numbers from 20
to 60 are obtained tting with the obje13tive of strong sho13k formation A
rened 3-D MHD numeri13al simulation indi13ates very promising features of
the sho13k in view of exploring mm-size sho13ks in a regime 13omplementary
to laser driven ones and on a table-top devi13e allowing a higher repetition
rate
140 CHAPTER 8 THESIS SUMMARY
Appendi13es
141
Appendix A
Visible Interferometry
Laser interferometry is used in plasmas to dedu13e the ele13tron density through
the variation of the refra13tive index 13aused by the ionisation of the matter
In this appendix rstly the experimental setup of Ma13h Zehnder interfero-
metry as well as the underlying physi13al prin13iples are presented
A01 Refra13tive index of a plasma
In order to derive the refra13tive index of a gas one starts with the set of
Maxwells equations for a mono13hromati13 plane wave propagating in the
plasma (see for instan13e equation 411 of referen13e (Hut13hinson 2002))
Free ele13trons and ions are supposed to be distributed uniformly and the net
total 13harge density is equal to zero
For harmoni13 waves propagating in the dire13tion of +x and with the
pulsation ω one obvious solution is given by
E(x t) = E0ei(kxminusωt)
(A1)
where E0 and k are 13omplex quantities
The relation between k and ω is given by the dispersion relation
k2 =εmicroω2
c2(1 + i
4πσ
ωε) (A2)
where micro is the ele13tri13 permitivity and ε is permeability
The free ele13trons are a1313elerated by the ele13tri13 eld and therefore equa-
tion their motion is given as
medv
dt= minuseE0e
minusiωt(A3)
The obvious solution for the ele13tron velo13ity v is
v = minusie
meωE (A4)
and thus the ele13trons 13arry a 13urrent with a density J
J = Neev = iNee
2
meωE (A5)
where Ne is the ele13tron density Due to the larger mass of the ions the
13orresponding 13urrent is negligible
143
144 APPENDIX A VISIBLE INTERFEROMETRY
The previous equation allows to derive the plasma 13ondu13tivity σ whi13his equal to iNee
2ωme Putting this value in the equation (A2) and assu-
ming a thin medium where ε asymp 1 and micro = 1 one obtains nally
k2 =ω2
c2(1minus
ω2p
ω2) (A6)
where ωp is the plasma frequen13y dened as
ω2p =
4πNee2
me(A7)
Equation A6 shows that a ele13tromagneti13 wave with a pulsation larger than
ωp 13ant propagate in the plasma This allows to derive a 13riti13al ele13tron
density Nc in 13m
minus3as
Nc =meω
2
4πe2=
4πmec2
e2λ2(A8)
For instan13e for a radiation of 1 microm wavelength this upper limit is equal
to Nc = 446times 1021cmminus3
Therefore the index of refra13tion is given by the expression
n =
radic
1minusω2p
ω2=
radic
1minus Ne
Nc(A9)
Finally one obtains
E(x t) = E0ei(kxminusωt) with k = n
ω
c(A10)
As a 13onsequen13e of the variation of the refra13tive index in the dierent
plasma layers the light is refra13ted Indeed the Snell laws stipulate that if a
beam is in13ident on a plane interfa13e between layers (of refra13tive index n1
and n2) it undergoes a bending from the angle of in13iden13e The in13ident
beam the refra13ted beam and the normal to the interfa13e lie in the same
plane and one has n1 sinθ1 = n2 sinθ2 where θ1 is the angle of in13iden13e
and θ2 is the angle of refra13tion Thus a mono13hromati13 beam propagating
in a plasma with in13reasing density be13omes more and more dee13ted
A02 Absorption of the laser beam
Beside refra13tion the light is absorbed by the plasma and the absorption
also in13reases rapidly near the 13riti13al density In the absen13e of any ato-
mi13 bound-bound or bound-free transition this absorption is due to the
inverse Bremsstrahlung pro13ess (ion-ele13tron 13ollision (Pfalzner 2006)) by
145
the plasma free ele13trons The absorption 13oe13ient κ (in 13m
minus1) is then
given by (NRL formulary)
κ(Ne Tev) = 317 10minus7LnΛ lt Z gt
(
Ne
ω
)2 1
T32ev (1minusNeNc)12
(A11)
where Tev is the temperature in eV Ne is the ele13tron density in 13m
minus3
and LnΛ is the Coulomb Logarithm dened as Λ = Max (2 Min(XY))
with (NRL formulary)
X = 23minus log
(
lt Z gtN
12e
T32ev
)
and Y = 24minus log
(
N12e
Tev
)
(A12)
The transmission T (d) through a layer of thi13kness d is given by
T (d) = exp(minusint d
0κ(l)dl) (A13)
For instan13e for λ = 527 nm taking Ne= 101913mminus3 ltZgt= 10 T = 10 eV
whi13h are typi13al values for the post-sho13k in our experiments in Xenon and
supposing a homogenous plasma thi13kness d = 600 microm the transmission is
about 09 whereas at 5times 1019cmminus3 it falls to 01
A03 Prin13iple of interferometry
In the opti13al interferometry te13hnique (Ovsyannikov and Zhukov 2000) the
interferen13e of two plane waves is performed by addition of the two 13orre-
sponding 13oherent ele13tri13 elds The analyzed medium (here the plasma)
in whi13h one of the beam is propagating indu13es a deshaping between the
beams whi13h ae13ts the resulting intensity depending if the elds interfere
in phase or out of phase ie 13onstru13tively or destru13tively
Two te13hniques are used the wave front and the amplitude division
The wave front division is obtained by using two portions (Fresnel mir-
rors Youngs double slit Lloyds mirror prisms et13) of the original wave
front whi13h are then superimposed to produ13e interferen13e
In the amplitude division the two beams are separated by division of
the amplitude over the same se13tion of the wave front Mi13helson and Ma13h
Zehnder interferometers et13 are the typi13al examples of this te13hnique of
amplitude division
The prin13iple of interferen13e of two 13oherent mono13hromati13 traveling
waves 1 and 2 with ele13tri13 elds E1 and E2 propagating in the same
dire13tion and polarized in the same plane 13an be understood as follows for
a given time t at the point r where these waves superpose the amplitude of
the ele13tri13 eld is the ve13tor addition of the two 13orresponding elementary
amplitudes ie
146 APPENDIX A VISIBLE INTERFEROMETRY
E(r t) = E1(r t) + E2(r t)
= a1exp(ikr + ωt+ iφ1) + a2exp(ikr + ωt+ iφ2)
where φ1 and φ2 are the phases of the two waves at r The total intensity
I(r t) = |E1 + E2|2 (A14)
One gets
I = I1 + I2 + 2radic
(I1I2)cos∆φ (A15)
where I1 = a12and I2 = a2
2are the intensities at the point of interferen13e
due to the two waves a13ting independently and ∆φ = φ1minusφ2 represents the
phase dieren13e between the two waves In the 13ase of equal intensities I1one gets
I = 4I1cos2(∆φ2) (A16)
and the interferometri13 pattern 13onsists of dark (∆φ = π + 2nπ) and
bright patterns (∆φ = 2nπ)In general the two beams are only partially 13oherent and the minimum
of the fringe intensity is not equal to zero The fringe 13ontrast
FC =Imax minus Imin
Imax + Imin(A17)
is then a measure of the interferen13e quality FC is maximum and equal to
1 in the 13ase of fully 13oherent beams of the same intensity I1 = I2
A04 Ma13h Zehnder Interferometer
The Ma13h Zehnder Interferometer is a devi13e 13ommonly and e13iently used
to estimate the 13hange in refra13tive index by the plasma Su13h 13hange is in
most the 13ases related to variations in the ele13tron density We employed
this interferometri13 setup in our experiment
The prin13iple of this interferometer is based on the division of the in13ident
light beam into two beams with the help of a beam-splitter whi13h 13an be
a 13oated glass plate or a 13ube After this division one of the beams (probe
beam) is allowed to pass through the medium of interest (a plasma in our
13ase) and then made to interfere with the other unae13ted part of the beam
(referen13e beam) with the help of se13ond beam-splitter as may be noted
from Fig A1
In order to understand the prin13iple of interferen13e and its use in esti-
mating plasma properties let us 13onsider l1 and l2 whi13h are the total path
lengths (in13luding the path in the opti13al elements) for the light travelling
from the sour13e to the dete13tor for the upper and lower paths (13f Figure A1)
respe13tively Suppose for the moment that the sample is removed from beam
147
Figure A1 S13hemati13s of Ma13h Zehnder interferometer with two mirrors two
50 beam splitters and the dete13tor A plasma slab is pla13ed in one of the two
arms of the interferometer
1 (see Figure A1) the two opti13al paths of the probe and referen13e beams
dier in terms of phases Simplify to supposing that the beams propagate
in air (n=1) these phases are then equal to the 2πl1λ and 2πl2λThus a1313ounting for the phase dieren13e δ between the two beams tra-
velled up to dete13tor A 13an be given as
2πl1 minus l2
λ= δ (A18)
If the two beams are perfe13tly parallel at the position of the dete13tor the
interferometri13 gure presents a at intensity whi13h passes through minima
and maxima when one length of the 2 arms is slightly 13hanged Usually one
introdu13es a small tilt between the two dire13tions of propagations whi13h
leads to the apparition of parallel linear dark and bright fringes
In order to obtain interferen13e patterns it is important to make the paths
l1 and l2 as 13lose as possible 13ompared to the 13oheren13e length of the laser
For the multimode Evolution 15 laser used in the PALS experiment this
length is approximately 1 13m (value 13ommuni13ated by the 13onstru13tor)
If we introdu13e the sample in the probe beam 1 this sample will introdu13e
an additional phase denoted by δsample and the net phase shift is then δ +δsample
The prin13iple of the dedu13tion of the ele13tron density with the Ma13h
Zehnder interferometer is to measure the refra13tive index of the test plasma
slab whi13h is pla13ed in the probe beam We assume for the moment that
148 APPENDIX A VISIBLE INTERFEROMETRY
this plasma slab is a phase obje13t for the in13oming beam whi13h means that
it does not signi13antly ae13t the intensity of the probe beam while it only
13hanges the phase of the wave [5]The additive phase dieren13e in the probe arm introdu13ed by the plasma
13an be dened as
φ =
int d
0kplasmadl =
int d
0nω
cdl (A19)
where n is the lo13al refra13tive index at position l of the plasma slab and
the integral is performed other the slab thi13kness d Therefore the phase
dieren13e introdu13ed by the plasma relative to the propagation into va13uum
of the referen13e beam is equal to
∆φ =
int d
0(kplasma minus k0)dl (A20)
where k0 = ωc = 2πλ is the wave ve13tor of the beam propagating in
va13uum
As long as Ne lt Nc the beam propagates in the plasma the plasma
refra13tive index is given by equation A9 and one has
∆φ =ω
c
int d
0
[
(
(1minus Ne
Nc
)12
minus 1]
dl (A21)
At very small ele13tron density (Ne ≪ Nc) the phase shift 13an be written as
∆φ asymp minus ω
2cNc
int d
0Nedl = minus πd
λNclt Ne gt (A22)
where
lt Ne gt=
int d
0
Ne(z t)dy
d(A23)
is the ele13tron density averaged over the path dIn pra13ti13e a referen13e re13ord is rst taken without plasma To dedu13e
the density one has to analyze how the fringes depart from their referen13e
positions
The spa13e between two fringes (inter-fringe) 13orresponds to a phase va-
riation of 2 π A shift of the fringe by 2 π from its referen13e position (where
Ne = 0) 13orresponds to an ele13tron density variation equal to lt Ne gt = 2
λNcd For λ = 527 nm and supposing an homogeneous plasma of thi13kness
600 microm this gives lt Ne gt= 7 times 10
1813m
minus3
Appendix B
Opa13ities and mean 13harge
The variations in logarithmi13 s13ale with the temperature of the PROPA-
CEOS Rosseland and Plan13k opa13ities (se13tion of 13hapter 5 see for intense
Fig 511)as also of the mean 13harge of the plasma are reported for two
densities 16 times 10
minus3g13m
3and 51 times 10
minus4g13m
3for three noble gases Ar
Kr and Xe in Fig B1 and B2
ρ = 51 times 10
minus4g13m
313orrespond to typi13al density of the pre13ursor
For this density Xenon and Krypton have quite similar variations for the
Rosseland opa13ity with a bump of 7000 13m
2g at 3 eV for Xe and 12000
13m
2g at 4 eV for Krypton At 10 eV the two opa13ities de13rease respe13tively
to 600 and 900 13m
2g In 13omparison with these two 13ases the opa13ity of
Argon is broader and more regular Its opa13ity peaks at 6 eV to 13000 13m
2g
At 10 eV it have de13reased to 7000 13m
2g whi13h is higher than for the two
previous gases
The mean 13harge variations in13rease regularly with the temperature At
10 eV these mean 13harges are respe13tively equal to 65 55 and 5 for Xe Kr
and Ar
149
150 APPENDIX B OPACITIES AND MEAN CHARGE
(a)
(b)
Figure B1 Rosseland and Plan13k opa13ity for three gases Xe Kr and Ar at mass
densities 51 times 10
minus4g13m
3(a) and 16 times 10
minus3g13m
3(b)
151
(a)
(b)
Figure B2 Mean 13harge for three gases Xe Kr and Ar at mass densities 16 times10
minus3g13m
3(a) and 51 times 10
minus4g13m
3(b)
152 APPENDIX B OPACITIES AND MEAN CHARGE
Referen13es
Batani D Strati F Telaro B Loumlwer T Hall T Benuzzi-Mounaix A
Koenig M 2003 Produ13tion of high quality sho13ks for equation of state
experiments European Physi13al Journal D 23 99 DOI ADS
Bernard A 2002 Plasma fo13us et appli13ations Le Vide 57(306)
Bett TH Jinks P Stevenson RM Danson CN Pepler DA Ross
IN 1995 Binary phase zone-plate arrays for laser-beam spatial-intensity
distribution 13onversion Applied Opti13s 34 4025 DOI ADS
Bouquet S Teyssier R Chieze JP 2000 Analyti13al study and stru13-
ture of a stationary radiative sho13k The Astrophysi13al Journal Supplement
Series 127(2) 245
Bouquet S Stehleacute C Koenig M Chiegraveze J Benuzzi-Mounaix A Ba-
tani D Leygna13 S Fleury X Merdji H Mi13haut C et al 2004
Observation of laser driven super13riti13al radiative sho13k pre13ursors Phy-
si13al review letters 92 225001
Bozier J Thiell G Le Breton J Azra S De13roisette M S13hirmann
D 1986 Experimental observation of a radiative wave generated in xenon
by a laser-driven super13riti13al sho13k Physi13al review letters 57(11) 1304
Burhop E Spitzer L 1957 Physi13s of fully ionized gases inters13ien13e
tra13ts on physi13s and astronomy no 3 JSTOR
Chaulagain U 2015 Radiative sho13ks expriments modellin and links to
Astrophysi13s PhD thesis Pierre and Marie Curie University
Chaulagain U Stehleacute C de Saacute L Larour J Auvray P Kozlova M
Krus M Dostal J Propupek J Suzuki-Vidal F et al 2012 Labo-
ratory experiments of radiative sho13ks in the 13ontext of stellar a1313retion
In SF2A-2012 Pro13eedings of the Annual meeting of the Fren13h So13iety
of Astronomy and Astrophysi13s 1 305
Chaulagain U Stehleacute C Larour J Kozlovaacute M Suzuki-Vidal F Bar-
roso P Cotelo M Velarde P Rodriguez R Gil JM Ciardi A
A13ef O Nejdl J de Saacute L Singh RL Ibgui L Champion N 2015
Stru13ture of a laser-driven radiative sho13k High Energy Density Physi13s
17 106 DOI ADS
Chevalier RA 1977 The intera13tion of supernovae with the interstellar
medium Annual review of astronomy and astrophysi13s 15 175 DOI ADS
153
154 REFERENCES
Chittenden JP Lebedev SV Jennings CA Bland SN Ciardi A
2004 X-ray generation me13hanisms in three-dimensional simulations of
wire array Z-pin13hesthe evolution of magneti13 tower jets in the laboratory
Plasma Physi13s and Controlled Fusion 46 B457 DOI ADS
Ciardi A 2014 private 13ommuni13ation
Ciardi A Lebedev SV Frank A Bla13kman EG Chittenden JP
Jennings CJ Ampleford DJ Bland SN Bott SC Rapley J Hall
GN Suzuki-Vidal FA Maro1313hino A Lery T Stehle C 2007 The
evolution of magneti13 tower jets in the laboratory Physi13s of Plasmas
14(5) 056501 DOI ADS
Clayson T Suzuki-Vidal F Lebedev SV Swadling GF Burdiak GC
Patankar S Smith RA Foster J Skidmore J Gumbrell E Gra-
ham P Danson C Stehleacute C Singh RL Chaulagain U Larour
J Kozlova M Spindloe C 2016 Counter-propagating radiative sho13k
experiments on the Orion laser fa13ility In APS Meeting Abstra13ts DOI
ADS
Clayson T Suzuki-Vidal F Lebedev S Swadling G Stehleacute C Bur-
diak G Foster J Skidmore J Graham P Gumbrell E et al 2017
Counter-propagating radiative sho13k experiments on the orion laser and
the formation of radiative pre13ursors High Energy Density Physi13s 23
60
Cotelo M Velarde P de la Varga AG Portillo D Stehleacute C Chaul-
again U Kozlova M Larour J Suzuki-Vidal F 2015 Simulation
of radiative sho13k waves in Xe of last PALS experiments High Energy
Density Physi13s 17 68 DOI ADS
de la Varga AG Velarde P Cotelo M de Gaufridy F
Zeitoun P 2011 Radiative properties for warm and hot
dense matter High Energy Density Physi13s 7(3) 163 DOI
httpwwws13ien13edire13t13oms13ien13earti13lepiiS1574181811000498
Diziegravere A Mi13haut C Koenig M Gregory C Ravasio A Sakawa
Y Kuramitsu Y Morita T Ide T Tanji H et al 2011 Highly
radiative sho13k experiments driven by gekko xii Astrophysi13s and Spa13e
S13ien13e 336(1) 213
Dobbs-Dixon I Cumming A Lin DNC 2010 Radiative Hydrodyna-
mi13 Simulations of HD209458b Temporal Variability The Astrophysi13al
Journal 710 1395 DOI ADS
Doss FW Robey HF Drake RP Kuranz CC 2009 Wall sho13ks in
high-energy-density sho13k tube experiments Physi13s of Plasmas 16(11)
112705
REFERENCES 155
Drake RP 2006 High-energy-density physi13s Fundamentals inertial fu-
sion and experimental astrophysi13s Sho13k wave and high pressure phe-
nomena Springer Berlin Heidelberg New York ISBN 3-540-29314-0
httpopa13inriafrre13ord=b1128433
Drake R Doss F M13Clarren R Adams M Amato N Bingham D
Chou C DiStefano C Fidkowski K Fryxell B et al 2011 Radiative
ee13ts in radiative sho13ks in sho13k tubes High Energy Density Physi13s
7(3) 130
Ensman L 1994 Test problems for radiation and radiation-hydrodynami13s
13odes The Astrophysi13al Journal 424 275 DOI ADS
Ensman L Burrows A 1992 Sho13k breakout in sn 1987a The Astrophy-
si13al Journal 393 742
Fadeyev YA Gillet D 2001 The stru13ture of radiative sho13k waves
III The model grid for partially ionized hydrogen gas Astronomy and
Astrophysi13s 368 901 DOI ADS
Fleury X Bouquet S Stehleacute C Koenig M Batani D Benuzzi-
Mounaix A Chieze J-P Grandjouan N Grenier J Hall T et al
2002 A laser experiment for studying radiative sho13ks in astrophysi13s
Laser and Parti13le Beams 20(02) 263
Fryxell B Olson K Ri13ker P Timmes FX Zingale M Lamb DQ
Ma13Nei13e P Rosner R Truran JW Tufo H 2000 FLASH An
Adaptive Mesh Hydrodynami13s Code for Modeling Astrophysi13al Ther-
monu13lear Flasheshera13les a three-dimensional radiation hydrodynami13s
13ode Astrophysi13al Journal Supplement 131 273 DOI ADS
Gonzalez JH Brollo FR Clausse A 2009 Modeling of the Dynami13
Plasma Pin13h in Plasma Fo13us Dis13harges Based in Von Karman Approx-
imations IEEE Transa13tions on Plasma S13ien13e 37 2178 DOI ADS
Gonzalez JH Clausse A Bruzzone H Florido PCMZ 2004 A Lum-
ped Parameter Model of Plasma Fo13us IEEE Transa13tions on Plasma
S13ien13e 32 1383 DOI ADS
Gonzaacutelez M Audit E Huynh P 2007 HERACLES a three-dimensional
radiation hydrodynami13s 13ode Astronomy and Astrophysi13s 464 429
DOI ADS
Gonzaacutelez M Audit E Huynh P 2007 Hera13les a three-dimensional
radiation hydrodynami13s 13ode Astronomy and Astrophysi13s 464(2) 429
Gonzaacutelez M Audit E Stehleacute C 2009 2D numeri13al study of the ra-
diation inuen13e on sho13k stru13ture relevant to laboratory astrophysi13s
Astronomy and Astrophysi13s 497 27 DOI ADS
156 REFERENCES
Gonzaacutelez M Stehleacute C Rus B Busquet M Thais F A13ef O Barroso
P Bar-Shalom A Bauduin D Kozlova M Lery T Madouri A To-
mas M Polan J 2006 Astrophysi13al radiative sho13ks From modeling
to laboratory experiments Laser and Parti13le Beams 24 535540 DOI
ADS
Gonzalez M StehlE C Audit E Busquet M Rus B Thais F A13ef
O Barroso P Bar-Shalom A Bauduin D et al 2006 Astrophysi13al
radiative sho13ks From modeling to laboratory experiments Laser and
Parti13le Beams-Pulse Power and High Energy Densities 24(4) 535
Gonzaacutelez M Vaytet N Commerccedilon B Masson J 2015 Multigroup
radiation hydrodynami13s with ux-limited diusion and adaptive mesh
renement Astronomy and Astrophysi13s 578 A12 DOI ADS
Griem HR 2005 Prin13iples of Plasma Spe13tros13opy 386 ADS
Gu MF 2008 The exible atomi13 13ode Canadian Journal of Physi13s
86(5) 675
Harilal SS Tilla13k MS 2004 Laser plasma
density measurements using interferometry
httplinkapsorgdoi101103PhysRevE64066412
Hartigan P 1989 The visibility of the Ma13h disk and the bow sho13k of a
stellar jet Astrophysi13al Journal 339 987 DOI ADS
Hartigan P 2003 Sho13k Waves in Outows from Young Stars Astrophysi13s
and Spa13e S13ien13e 287 111 DOI ADS
Hartigan P Morse JA Reipurth B Heath13ote S Bally J 2001 Pro-
per Motions of the HH 111 Jet Observed with the Hubble Spa13e Teles13ope
Astrophysi13al Journal Letters 559 L157 DOI ADS
Henke BL Gullikson E Davis JC 1993 X-ray intera13tions Photoab-
sorption s13attering transmission and ree13tion atlt igt eltigt= 50-30000
evlt igt zltigt= 1-92 Atomi13 data and nu13lear data tables 54(2) 181
Hugoniot PH 1887 Meacutemoire sur la propagation du mouvement dans les
13orps et ples speacute13ialement dans les gaz parfaits Journal de matheacutematiques
pures et appliqueacutees 3 477
Hugoniot PH 1889 Meacutemoire sur la propagation des mouvements dans les
13orps et speacute13ialement dans les gaz parfaits (deuxiegraveme partie) Journal de
lEacute13ole Polyte13hnique 58 1
Hut13hinson IH 2002 Prin13iples of Plasma Diagnosti13s 458 ADS
REFERENCES 157
Jungwirth K Cejnarova A Juha L Kralikova B Krasa J Krousky
E Krupi13kova P Laska L Masek K Mo13ek T et al 2001 The
prague asterix laser system Physi13s of Plasmas 8 2495
Keiter P Drake R Perry T Robey H Remington B Iglesias C
Walla13e R Knauer J 2002 Observation of a hydrodynami13ally driven
radiative-pre13ursor sho13k Physi13al review letters 89(16) 165003
Koenig M Faral B Boudenne JM Batani D Benuzzi A Bossi S
1994 Opti13al smoothing te13hniques for sho13k wave generation in laser-
produ13ed plasmas Physi13al Review E 50 R3314 DOI ADS
Kondo K Nakajima M Kawamura T Horioka K 2006 Compa13t pulse
power devi13e for generation of one-dimensional strong sho13k waves Review
of S13ienti13 Instruments 77(3) 036104 DOI ADS
Kondo K Nakajima M Kawamura T Horioka K 2008 Relaxation
layer in ele13tro-magneti13ally driven strong sho13ks Journal of Physi13s Con-
feren13e Series 112(4) 042028 DOI ADS
Kondo K Nakajima M Kawamura T Horioka K 2009a Atomi13 pro-
13ess in ele13tro-magneti13ally driven strong sho13k wave Nu13lear Instruments
and Methods in Physi13s Resear13h A 606 223 DOI ADS
Kondo K Nakajima M Kawamura T Horioka K 2009b Radiation ef-
fe13t and relaxation layer in ele13tro-magneti13ally driven strong sho13k waves
In APS Meeting Abstra13ts ADS
Kunze H-J 2009 Springer series on atomi13 opti13al and
plasma physi13s 125 Springer Berlin Heidelberg New York
httpwwwspringer13omseries411
Kwek K Tou T Lee S 1990 Current sheath stru13ture of the plasma
fo13us in the run-down phase IEEE Transa13tions on Plasma S13ien13e 18(5)
826
Larour J Singh RL Stehleacute C Ciardi A Chaulagain U Suzuki-Vidal
F 2015 Optimization of an ele13tromagneti13 generator for strong sho13ks
in low pressure gas High Energy Density Physi13s 17 129 DOI ADS
Lee S 1969 Transverse ionizing sho13k waves in a planar ele13tromagneti13
sho13k tube PhD thesis Australian National University
Lee S Serban A 1996 Dimensions and lifetime of the plasma fo13us pin13h
IEEE Transa13tions on plasma s13ien13e 24(3) 1101
Leygna13 S Boireau L Mi13haut C Lanz T Stehleacute C Clique C
Bouquet S 2006 Modeling multidimensional ee13ts in the propagation
of radiative sho13ks Physi13s of Plasmas (1994-present) 13(11) 113301
158 REFERENCES
Ma13Farlane JJ Golovkin IE Woodru PR 2006 HELIOS-CR A 1-
D radiation-magnetohydrodynami13s 13ode with inline atomi13 kineti13s mo-
deling Journal of Quantitative Spe13tros13opy and Radiative Transfer 99
381 DOI ADS
Ma13quorn Rankine WJ 1870 On the Thermodynami13 Theory of Waves of
Finite Longitudinal Disturban13e Philosophi13al Transa13tions of the Royal
So13iety of London Series I 160 277 ADS
Mahmood S Ur-Rehman H 2009 Ele13trostati13 solitons in unmagnetized
hot ele13tron-positron-ion plasmas Physi13s Letters A 373 2255 DOI ADS
Martin TH Williams M Kristiansen M 1996 J13 martin on pulsed
power 3 Springer S13ien13e amp Business Media
M13Clarren RG Drake RP Morel JE Holloway JP 2010 Theory of
radiative sho13ks in the mixed opti13ally thi13k-thin 13ase Physi13s of Plasmas
17(9) 093301
Mi13haut C Stehleacute C Leygna13 S Lanz T Boireau L 2004 Jump
13onditions in hypersoni13 sho13ks Quantitative ee13ts of ioni13 ex13itation
and radiation European Physi13al Journal D 28 381 DOI ADS
Mihalas D 1978 Stellar atmospheres 2nd edition ADS
Mihalas D Mihalas BW 1984 Foundations of radiation hydrodynami13s
ADS
Mihalas D Mihalas BW 1999 Foundations of radiation hydrodynami13s
More R Warren K Young D Zimmerman G 1988 A new quotidian
equation of state (qeos) for hot dense matter Physi13s of Fluids (1958-
1988) 31(10) 3059
Ogando F Velarde P 2001 Development of a radiation transport uid
dynami13 13ode under amr s13heme Journal of Quantitative Spe13tros13opy and
Radiative Transfer 71(2) 541
Olson GL Auer LH Hall ML 2000 Diusion P1 and other approx-
imate forms of radiation transport Journal of Quantitative Spe13tros13opy
amp Radiative Transfer 64 619 DOI ADS
Orban C Fatenejad M Chawla S Wilks SC Lamb DQ 2013 A
Radiation-Hydrodynami13s Code Comparison for Laser-Produ13ed Plasmas
FLASH versus HYDRA and the Results of Validation Experiments ArXiv
e-prints ADS
REFERENCES 159
Orlando S Bonito R Argiro C Reale F Peres G Mi13eli M Mat-
sakos T Stehleacute C Ibgui L de Saacute L Chiegraveze J-P Lanz T 2013
Radiative a1313retion sho13ks along nonuniform stellar magneti13 elds in 13las-
si13al T Tauri stars Astronomy amp Astrophysi13s
Ovsyannikov AA Zhukov MF 2000 Plasma diagnos-
ti13s Cambridge International S13ien13e Publishing Cambridge
httppubli13eblib13om13hoi13epubli13fullre13ordaspxp=258189
Pfalzner S 2006 An introdu13tion to inertial 13onnement fusion 1st
edn Series in Plasma Physi13s Taylor amp Fran13is Taylor amp Fran13is
httpgenlibruse13bookindexphpmd5=EACF38292FD79113F78BB532AB5301B8
Potter DE 1971 Numeri13al Studies of the Plasma Fo13us Physi13s of Fluids
14 1911 DOI ADS
Proakis JG 2001 Digital signal pro13essins Wiley Online Library
Raga A Mellema G Arthur S Binette L Ferruit P Steen W 1999
3d transfer of the diuse ionizing radiation in ism ows and the preioni-
zation of a herbig-haro working surfa13e Revista Mexi13ana de Astronomia
y Astrosi13a 35 123
Reighard AB 2007 Collapsing radiative sho13k experiments on the omega
laser PhD thesis The University of Mi13higan
Reighard A Drake R Dannenberg K Kremer D Grosskopf M Har-
ding E Leibrandt D Glendinning S Perry T Remington B et al
2006 Observation of 13ollapsing radiative sho13ks in laboratory experiments
Physi13s of Plasmas (1994-present) 13(8) 082901
Rodriguez R Gil JM Florido R Rubiano JG Mendoza MA Mar-
tel P Minguez E Symes DR Hohenberger M Smith RA 2011
Determination of the average ionization and thermodynami13 regimes of
xenon plasmas with an appli13ation to the 13hara13terization of blast waves
laun13hed in xenon 13lusters High Energy Density Physi13s 7(2) 71 DOI
httpwwws13ien13edire13t13oms13ien13earti13lepiiS157418181100022X
Rodriacuteguez R Espinosa G Gil JM Stehleacute C Suzuki-Vidal F Rubi-
ano JG Martel P Miacutenguez E 2015 Mi13ros13opi13 properties of xenon
plasmas for density and temperature regimes of laboratory astrophysi13s
experiments on radiative sho13ks Physi13al Review E 91(5) 053106 DOI
ADS
Ross IN Pepler DA Danson CN 1995 Binary phase zone plate de-
signs using 13al13ulations of far-eld distributions Opti13s Communi13ations
116 55 DOI ADS
160 REFERENCES
Seaton MJ 1987 Atomi13 data for opa13ity 13al13ulations I General des13rip-
tion Journal of Physi13s B Atomi13 and Mole13ular Physi13s 20(23) 6363
Serban A 1995 Anode geometry and fo13us 13hara13teristi13s PhD thesis
Nanyang Te13hnology University httphdlhandlenet104971568
Serban A Lee S 1995 A simple opti13al ber axial speed dete13tor Review
of S13ienti13 Instruments 66 4958 DOI ADS
Serban A Lee S 1995 A simple opti13al ber axial speed dete13tor Review
of s13ienti13 instruments 66(10) 4958
Singh JP Thakur SN (eds) 2007 Laser-indu13ed breakdown
spe13tros13opy Elsevier s13ien13edire13t
Singh R Stehleacute C Suzuki-Vidal F Kozlova M Larour J Chaulagain
U Clayson T Rodriguez R Gil J Nejdl J et al 2017 Experimental
study of the intera13tion of two laser-driven radiative sho13ks at the pals
laser High Energy Density Physi13s 23 20
Stahler SW Palla F Salpeter EE 1986 Primordial stellar evolution -
The protostar phase The Astrophysi13al Journal 302 590 DOI ADS
Stehleacute C Gonzaacutelez M Kozlova M Rus B Mo13ek T A13ef O Colom-
bier JP Lanz T Champion N Jakub13zak K et al 2010 Experi-
mental study of radiative sho13ks at pals fa13ility Laser and parti13le beams
28(02) 253
Stehleacute C Kozlovaacute M Larour J Nejdl J Champion N Barroso P
Suzuki-Vidal F A13ef O Delattre P-A Dostaacutel J et al 2012 New
probing te13hniques of radiative sho13ks Opti13s Communi13ations 285(1)
64
Stehleacute C Lefegravevre R Chaulagain U Champion N Barroso P Reix
F Jagourel P Larour J Melt13hakov E Mer13ier R et al 2014a
Criti13al 13omponents for xuv probing of laser driven sho13ks In X-Ray
Lasers 2012 Springer 239
Stehleacute C Kozlovaacute M Larour J Nejdl J Suzuki-Vidal F Cohen M
Chaulagain UP Champion N Barroso P A13ef O et al 2014b
Reply on the 13omment of the paper new probing te13hniques of radiative
sho13ks Opti13s Communi13ations 318 226
Suzuki-Vidal F Lebedev S Pi13kworth L Swadling G Burdiak G
Skidmore J Hall G Bennett M Bland S Chittenden J et al
2014 Sho13k formation in 13ounter-streaming jets on the magpie pulsed-
power generator In APS Meeting Abstra13ts 1 5005
REFERENCES 161
Suzuki-Vidal F Clayson T Swadling GF Patankar S Burdiak GC
Lebedev SV Smith RA Stehle C Chaulagain U Singh RL La-
rour J Kozlova M Spindloe C Foster J Skidmore J Gumbrell E
Graham P Danson C 2016 Counter-streaming radiative sho13k experi-
ments on the Orion laser In APS Meeting Abstra13ts ADS
Toledo-Roy JC Velaacutezquez PF de Colle F Gonzaacutelez RF Reynoso
EM Kurtz SE Reyes-Iturbide J 2009 Numeri13al model for the SNR
DEM L316 simulated X-ray emission Monthly Noti13es of the RAS 395
351 DOI ADS
Tou T Lee S Kwek K 1989 Nonperturbing plasma-fo13us measurements
in the run-down phase IEEE Transa13tions on Plasma S13ien13e 17(2) 311
Vaytet NMH Audit E Dubro13a B Delahaye F 2011 A numeri13al
model for multigroup radiation hydrodynami13s Journal of Quantitiative
Spe13tros13opy and Radiative Transfer 112 1323 DOI ADS
Vaytet N Chabrier G Audit E Commerccedilon B Masson J Ferguson
J Delahaye F 2013 Simulations of protostellar 13ollapse using mul-
tigroup radiation hydrodynami13s ii the se13ond 13ollapse arXiv preprint
arXiv13071010
Vaytet N Gonzaacutelez M Audit E Chabrier G 2013 The inuen13e
of frequen13y-dependent radiative transfer on the stru13tures of radiative
sho13ks Journal of Quantitiative Spe13tros13opy and Radiative Transfer 125
105 DOI ADS
Velarde P Gar13iacutea-Senz D Bravo E Ogando F Relantildeo A Gar13iacutea C
Oliva E 2006 Intera13tion of supernova remnants From the 13ir13umstellar
medium to the terrestrial laboratory Physi13s of Plasmas 13(9) 092901
DOI ADS
Veloso F Moreno J Tarifentildeo-Saldivia A Pavez C Zambra M Soto
L 2012a Non-intrusive plasma diagnosti13s for measuring sheath kine-
mati13s in plasma fo13us dis13harges Measurement S13ien13e and Te13hnology
23(8) 087002
Veloso F Tarifentildeo-Saldivia A Pavez C Moreno J Zambra M Soto
L 2012b Plasma sheath kinemati13s and some impli13ations on the mo-
deling of very low energy plasma fo13us devi13es Plasma Physi13s and Con-
trolled Fusion 54(9) 095007
Vis13o A Drake R Glenzer S Doumlppner T Gregori G Froula D
Grosskopf M 2012 Measurement of radiative sho13k properties by x-ray
thomson s13attering Physi13al review letters 108(14) 145001
162 REFERENCES
Williams RM Chu Y-H Di13kel JR Beyer R Petre R Smith RC
Milne DK 1997 Supernova Remnants in the Magellani13 Clouds I The
Colliding Remnants DEM L316 Astrophysi13al Journal 480 618 ADS
Williams RM Chu Y-H Di13kel JR Gruendl RA Seward FD
Guerrero MA Hobbs G 2005 Supernova Remnants in the Magella-
ni13 Clouds V The Complex Interior Stru13ture of the N206 Supernova
Remnant Astrophysi13al Journal 628 704 DOI ADS
Zambra M Kalise D Fernandez J Hernandez E Pas-
ten D Munoz V 2009 Current sheet thi13kness in the
plasma fo13us snowplow model J Plasma Fusion 8(879-882)
httpspeopleri13amoeawa13atdkalisefilesplasmapdf
Zeldovi13h IB Zeldovi13h YB Raizer YP 2002 Physi13s of sho13k waves
and high-temperature hydrodynami13 phenomena Courier Dover Publi13ati-
ons London New York Sydney
REFERENCES 163
