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Tampere University of Technology
Real-time measurements of dissipative solitons in a mode-locked fiber laser
CitationRyczkowski, P., Närhi, M., Billet, C., Merolla, J. -M., Genty, G., & Dudley, J. M. (2017). Real-time measurementsof dissipative solitons in a mode-locked fiber laser. arXiv eprint.
Year2017
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Published inarXiv eprint
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1
Real‐timefullfieldmeasurementsrevealtransientdissipative
solitondynamicsinamode‐lockedlaserP.Ryczkowski1†,M.Närhi1†,C.Billet2†,J.‐M.Merolla2,G.Genty1,J.M.Dudley2*
1 LaboratoryofPhotonics,TampereUniversityofTechnology,Tampere,Finland.
2 InstitutFEMTO‐ST,UMR6174CNRS‐UniversitéBourgogneFranche‐Comté,Besançon,France.
† Equal contributions
* Corresponding author [email protected]
Dissipativesolitonsareremarkablelocalizedstatesofaphysicalsystemthatarisefromthe
dynamicalbalancebetweennonlinearity,dispersionandenvironmentalenergyexchange.
Theyarethemostuniversalformofsolitonthatcanexistinnature,andareseeninfar‐
from‐equilibriumsystemsinmanyfieldsincludingchemistry,biology,andphysics.There
hasbeenparticularinterestinstudyingtheirpropertiesinmode‐lockedlasersproducing
ultrashort light pulses, but experiments have been limited by the lack of convenient
measurement techniques able to track the soliton evolution in real‐time.Here,we use
dispersiveFouriertransformandtimelensmeasurementstosimultaneouslymeasurereal‐
timespectralandtemporalevolutionofdissipativesolitonsinafiberlaserastheturn‐on
dynamicspassthroughatransientunstableregimewithcomplexbreak‐upandcollision
dynamics before stabilizing to a regularmode‐locked pulse train. Ourmeasurements
enable reconstruction of the soliton amplitude and phase and calculation of the
correspondingcomplex‐valuedeigenvalue spectrum toprovide furtherphysical insight.
These findingsaresignificant inshowinghowreal‐timemeasurementscanprovidenew
perspectivesintotheultrafasttransientdynamicsofcomplexsystems.
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INTRODUCTION
Whenoperatingintheirsteady‐stateregime,mode‐lockedlasersproducehighlyregular
stable pulse trains which have found widespread application in many fields of science (1‐3).
However,themode‐lockedlaserisalsowell‐knowntoexhibitarichvarietyofunstabledynamical
phenomenawhendetunedfromitssteady‐state,orasstablemode‐lockingbuildsupfromnoise
(4‐6).Thesetransient instabilitiesareasubjectofparticular interest inpassivelymode‐locked
fiberlasers,sincetheinterplayofthefibernonlinearityanddispersionwithcavitygainandloss
yieldsarichlandscapeofdissipativesolitondynamics(7‐10).Dissipativesolitonsareinfactthe
mostuniversalclassoflocalizedsolitonstateinphysics,existingforanextendedperiodoftimein
thepresenceofnonlinearityanddispersion,evenwhilepartsofthestructurecanexperiencegain
andloss.Incontrasttosolitonsinenergy‐conservingsystems,dissipativesolitonsexistinsystems
farfromequilibrium,andaredynamicalobjectsthatdisplayhighlynon‐trivialbehaviour(7).
Althoughthereisanextensivetheoreticalliteratureondissipativesolitonsinfiberlasers,
experimentalstudieshavebeenmorelimited,andoftenrestrictedtomeasurementsusingonly
fastphotodetectors(6,9,11‐13).Theseexperimentshavecertainlyprovidedinsightintounstable
modesoflaseroperation,yetthelimitedtemporalresolutionofthedetectorsusedhasgenerally
preventedanydetailedstudyoftheunderlyingsolitondynamics.
Recentyears,however,haveseendramaticadvancesintechniquescapableofreal‐time
measurementsofnon‐repetitiveopticalsignals(14‐25).Thefirstmethodtoseewidespreaduse
wasthedispersiveFouriertransform(DFT)techniqueforreal‐timespectralmeasurementsatthe
MHzrepetitionrate typicalofmode‐locked lasers (14). TheDFTwas firstused tostudy fiber
instabilitiessuchasroguewavedynamics,modulationinstabilityandsupercontinuumgeneration
(15‐18), andmore recently has been applied to study the spectral properties ofmode‐locked
lasers.Theabilitytoresolvethespectrumaftereachcavityroundtripinalaserhasrevealednovel
decoherenceandsolitonexplosiondynamicsina fiber laser(19,20),andwavelengthevolution
dynamics in a Kerr‐lens Ti:Sapphire laser (21). In fact, soliton dynamics in the Kerr‐lens
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Ti:SapphirelaserhavealsobeenstudiedusingamodifiedversionoftheDFTadaptedforreal‐time
spectralinterferometry(22).Inparallelwiththeseadvancesinreal‐timespectralmeasurements,
thedevelopmentofatemporalanalogofaspatialthinlenshasresultedinreal‐timepulseintensity
measurementsinwithsub‐picosecondresolution(23).Thetime‐lenshasbeenusedinstudiesof
incoherentsolitonpropagationinopticalturbulence(24),andstochasticbreatheremergencein
modulationinstability(25).
In this work, we combine, for the first time, real‐time spectral DFT and time lens
techniques to perform simultaneous measurements of the spectral and temporal profiles of
dissipative solitons evolving during the turn‐on phase of a mode‐locked fiber laser. Our
measurementsaremadewithsub‐nmandsub‐psresolution,providingaunifiedsnapshotofhow
thesolitonsevolvefromroundtriptoroundtrip,anddirectlyrevealingarangeofhighlycomplex
dynamicstypicalofdissipativesolitons.Inaddition,phaseretrievalallowsthereconstructionof
the full field (intensity and phase) of the measured pulses, allowing us to determine the
correspondingcomplexeigenvaluespectrumofthepulsesindifferentdynamicalregimes.These
results are significant in providing a unique picture of the internal evolution of fiber laser
dissipativesolitons,andweanticipatetheirapplicationintheoptimizationanddesignoflasers
withimprovedstabilitycharacteristics.Moregenerally,webelievethatourresultswillstimulate
the widespread use of simultaneous temporal and spectral characterization as a standard
techniqueforthestudyofultrafastcomplexopticalsystems.
RESULTS
Ourexperimentsstudiedthetransient turn‐ondynamicsofasaturableabsorbermode‐locked
fiberlaserconfigured(instableoperation)togeneratepulsesof~4.5psdurationat1545nmwith
20MHzrepetitionrate(i.e.cavityroundtriptimeof50ns).Thelasercavityhasnetanomalous
dispersion such that in steady‐state operation the circulating pulses are close to fundamental
solitons,andindeedfrequency‐resolvedopticalgatingmeasurementsconfirmedthatthestable
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outputpulseswerewell‐fittedbyahyperbolicsecantprofileandweretransformlimitedwitha
uniformtemporalphase(26).
Wecharacterizedtheturn‐ondynamicsusingbothdirectphotodiodedetectionaswellas
simultaneousDFTandtime‐lensmeasurements(seeMaterialsandMethodsforfulldetails).Fig.1
showsresultsusingdirectphotodiodedetection.Specifically,Fig.1(a)plotstheenvelopeofthe
pulses measured with reduced detection bandwidth to capture a time‐window of 130ms,
correspondingto2.6 million cavityroundtrips.The figureclearlyrevealsa~100ms transient
regime of Q‐switched mode‐locked operation before the appearance of a stable pulse train.
Figure1(b)showsaseparatemeasurementofaportionofthetransientregimeasindicated,but
hereusing30GHzdetectionbandwidthsothatitispossibletoresolveaseriesof~10μsbursts
separatedbya~70 μsperiod.TheexpandedviewinFig.1(b)showshowundereachburst,an
irregulartrainofmode‐lockedpulses(at50nsperiod)ispresent.Incontrastwiththisunstable
regime,Fig.1(c)showsmeasurementsinthestablemode‐lockedregimeusing30GHzdetection,
toillustratearegularpulsetrainofconstantintensity.Ofcourse,suchcomplextransientdynamics
havebeenseeninbothnumericalandexperimentalstudiesofarangeofotherpassively‐mode‐
lockedlasers(27‐34),butweincludethesemeasurementshereforcompleteness.Thekeypoint
isthat,evenwithadetectionbandwidthof30GHz,thetemporalwidthofthepulsesseeninthe
photodiode intensity record is~30ps,precludinganydetailedstudyof theunderlyingsoliton
dynamicsinthetransientregime.It isthis limitationthatweovercomewithoursimultaneous
real‐timeDFTandtime‐lensmeasurements.
Figure2showsthesetupused.Lightfromthefiberlaseroutputwassplitintotwopaths
andsenttothereal‐timeDFTandtime‐lensacquisitionarms.Realtimespectralmeasurements
usedastandardDFTsetup(18)withspectralresolutionof~0.3nm.Thetimelensset‐upwas
similar to that described inRef. (25) andwas capable of real‐timemeasurements of intensity
profilesupto60psdurationwithaneffectivetemporalresolutionof400fs.BothDFTandtime
lenssignalswererecordedusingahighstoragecapacitydigitaloscilloscope,withdataacquisition
triggeredbythetime‐lenssignaldetectedafterthelaserwasturnedon.Wecouldsimultaneously
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measurespectralandtemporalintensityprofilesoveramaximumof400cavityroundtrips(see
MaterialsandMethods).
We recorded multiple data sequences to examine the spectral and temporal soliton
dynamicsbothintheQ‐switchedmode‐lockedandthestable‐modelockingregime.Webeginin
Fig.3(a)byshowingresultsforthestablemodelockingregimewhereasexpectedthespectral(i)
and temporal (ii) intensity profiles are constant with roundtrip. The extracted spectral and
temporalwidths and the integratedpulse energy are shownas the right subfigures; the small
(noise)variationsinthesefiguresisoftheorderoftheexperimentalresolutionofthespectraland
temporalmeasurements.
Althoughtheabilitytomeasurebothtemporalandspectralintensityinrealtimeisitselfa
highly significant advance, we extend this technique further using Gerchberg‐Saxton phase‐
retrieval(35,36)torecoverthecorrespondingcomplexelectricfieldofeachpulse(seeMaterials
andMethods).TheseresultsareshowninFig.3(a‐iii)whereweplottheretrievedintensity(left
axis)andphase(rightaxis)revealingthesolitoncharacteristicsofnear‐uniformphase.Accessto
thefullcomplexelectricfieldallowsustocalculatetheassociatedwavelength‐timespectrogram,
and these results are shown in Fig.3(a‐iv), confirming the localized soliton‐like nature of the
pulses.Fig4showssimilar results,but fromadatasequence justprior to theregimeof stable
modelocking.Here,overarelativelysmallnumberofroundtrips,weseesignificantmodulationin
boththemeasuredspectralandtemporalamplitudes,aswellasenergyvariationof~30%.This
“breathing” of the intracavitypulses justprior to theonsetof stability is aknownpropertyof
dissipativesolitondynamicsinmode‐lockedlasers(21),butourresultsarethefirsttobeableto
experimentallycharacterizeitcompletelyinboththetimeandfrequencydomains.
Dissipative solitons candisplayamuch richer rangeof interactiondynamics (7‐9), and
Fig.4showsexamplesof thisbehavior in theevolutionofpulsesundera transientQ‐switched
burst. To capture different bursts at different timeswe used a variable hold off time for our
measurementwindowtoscanvariouspointsofthetransientregime.Infact,wefoundthatwhilst
thegeneraldynamicsweresimilarforallburstdurations(i.e.showingtheemergenceofmultiple
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pulsesfromaninitialnoisyfieldfollowedbysubsequentdecay)thedetailedevolutiondynamics
varied significantly for different bursts. Indeed, ourmeasurement set‐up allowed us to reveal
previously unobserved regimes of dissipative soliton propagation and interaction under the
transientenvelope.
Remarkingfirstlythatthespectralandtemporalevolutionwithroundtripisplottedfrom
toptobottom,twotypicalresultsareshowninFig.4(a)and4(b)toillustratethedifferenttypes
ofbehaviorobserved.Alsonotethatenergyisnotconstantinthetransientturn‐onregime(12).
We first discuss Fig.4(a)which shows an initial noisy field splitting into 3 distinct pulses (of
duration~5ps)thatpropagatecoherentlytogetherover~100roundtrips.Eachpulsedisplaysa
well‐localizedtemporalintensitypeak,andthethreepulsesaremutuallycoherentasisseenby
thefactthatthereisdistinctmodulationinthecorrespondingspectrum.Significantly,thepulse
separationdoesnotvaryover~100roundtripsasthepulsesevolvewithoutseemingtointeract
beforedecaying.Figure4(b)showsaqualitativelydifferentcase.Here, twopulsesemerge,but
rather than propagating without interacting, they undergo attraction and eventually collide,
displayingamorecomplexnonlinearphaseprofileatthispoint.
Althoughadetailedstudyofthesedynamicsisbeyondthescopeofthispaper,weremark
thatanalysisoftheseresults(andamoreextensivedatasetofover100similarcases)revealthat
whetherornottheemergentsolitonsevolvewithoutinteractingorcollidedependsonwhether
ornotthepulseshavethesamecentralfrequency.Thecentralfrequenciesofthepulsescanbe
clearly seenon thespectrogramplots, and inFig.4(a)weseehowthe three (non‐interacting)
pulseshaveslightlydifferentcentralfrequencieswhereasinFig.4(b)thetwopulsesthatcollide
have the same central frequency and similar amplitudes. These results are consistent with
previousstudiesofboundsolitondynamics(37‐39).Figure4(c)ontheotherhandshowsanother
exampleofburstevolutionwhereacombinationof theabovedynamicsareobservedwiththe
emergenceof3solitons.Here,thetwosolitonswiththesamecentralfrequencyattracteachother
and collideafter~100 roundtrips,whereas the remaining solitonwith loweramplitudeand a
slightlydifferentfrequencypropagateswithoutinteractionbeforeeventuallydecaying(40,41).
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Theabilitytocompletelyreconstructthecomplexelectricfieldfromthemeasuredpulse
intensityandphaseishighlysignificant,andindeedallowscalculationofderivedquantitiesthat
yieldadditionalandimportantphysical insights. Inparticular, it ispossibletoapplynumerical
techniquesfromscatteringtheory(42‐45)tocalculateanonlinearFouriertransformthatyields
spectraleigenvalueportraitsofthepulsestructureswhichcanthenbecomparedwiththeknown
signaturesof ideal solitons (42).Although theuseof suchscattering theory in fiberopticshas
generallybeenfromatheoreticalperspectivetoobtainanalyticsolutionstothegoverningpartial
differentialequations,therehasalsobeenrecentinterestinnumericalapproachestoyieldinsight
intoroguewavedynamics(46)andasanovelapproachtoovercomingchannellimitsinoptical
communications(47).Significantly,although the theoreticalanalysis isonlystrictlyvalid inan
integrablesystem,numericalcomputationoftheeigenvaluespectrumforthedissipativesoliton
systemconsideredhere(seeMaterialsandMethods)nonethelessyieldsresultsthathaveaclear
physicalinterpretation.Thisisbecauseweconsiderevolutionregionswherethepulseproperties
areprimarilydeterminedbynonlinearanddispersiveeffects(i.e.wecanneglectresonantgain
shaping) so the systemmaybe considered tobeonlyweaklynon‐integrable. In this case the
evolutionofpulsesthroughthedifferentdiscreteelementsofthecavitycanbeapproximatedby
anequivalentnonlinearSchrödingerequation(NLSE)withuniformdistributednonlinearityand
dispersion,andthenonlinearspectrumcanbeconsideredasyieldinglocaleigenvaluesthatcan
revealapproximatesolitoncontent(48).
TheresultsofthisanalysisareshowninFig.5wherethereddotsineachsubfigureshow
thecalculateddiscreteeigenspectrumforthenormalizedpulseintensityprofilesshownasinsets.
In this case,we associate the stable pulse regime show in Fig. 3(a)with solitons havingwith
eigenvaluesat=±0.5iinthecomplexplane.BasedonthemeasurementsinFig.3(a),wecanthen
normalizetheintensitiesofmeasuredpulsesinthetransientregimewithrespecttotheseideal
solitonsandperformdirect scattering analysis todetermine the correspondingmore complex
eigenvaluespectrum(seeMaterialsandMethods).
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WefirstconsidertheresultsinFig.5(a)and(b)whichcorrespondtosinglepulsesattwo
pointsofthebreathercycleshowninFig.3(b‐iii).Ineachcasethescatteringanalysisyieldsone
distincteigenvalue,andweseethatasthepulseintensityvariesbelowandabovetheunityvalue
ofanormalizedfundamentalsoliton(dashedblackline),theeigenvalueIm()alsovariesbelow
andabovetheidealsolitonvalueof±0.5i.Inotherwords,thevariationintheeigenvaluereflects
thebreathingofthepulsepropertiesinthisregime.
Figure5(c) and (d) considermore complex evolution. Figure5(c)plots the eigenvalue
spectrumforthedouble‐solitonpulseshowninFig.4(b‐iii)wherethedirectscatteringprocedure
yieldstwodiscreteeigenvalues,bothwithIm()~±0.5i.Figure5(d)showsresultsforthethree‐
solitoncaseinFig.4(a)andhereweseetwodiscreteeigenvalueswithIm(~±0.5iandathird
eigenvalueataslightlylowervalue.Inallcases,theclearseparationoftheeigenvaluesfromthe
realaxisindicatesthatthisparticulardynamicalregimewithinthelaserburstspriortosteady‐
statemodelockingcanbeconsideredascreatingdiscretepulsecomplexes,similartothoseseen
withmultipleboundsolitons(7‐9).Thisisaveryimportantphysicalinsightthatthenonlinear
Fouriertransformrevealsdirectly(48).
DISCUSSION
Dissipativenonlinearsystemscanbehighlycomplex,butitisclearthatrecentyearshave
seentremendousadvancesinbeingabletounderstandtheirbehaviorthroughnoveltheoretical
approachesandnumericalmodelling.Ofcourse,afullunderstandingofsuchcomplexdynamical
phenomena requires that theoretical and numerical results are carefully compared with
experiment,andtothisendthemeasurementtechniqueandphaseretrievalanalysisreportedin
thispapermakeasignificantcontribution inenablingcompletecharacterizationofpicosecond
dissipativesolitonsystems.
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Theresultsobtainedshowawiderangeofultrafasttemporalandspectraldynamicsneverbefore
seendirectly,andweanticipatethatourresultswillstimulatemanyfuturetheoreticalstudiesand
numericalmodellingthatcannowbecompareddirectlywithexperimentinawaythatwasnot
previouslypossible.
A further area of particular significance concerns our use of the nonlinear Fourier
transformtocalculatealocaleigenspectrumofthedissipativesolitonsinthelaser.Thecalculated
spectraclearlyshowthepresenceoflocalsolitoncontentinthecomplexpulseprofilesmeasured,
whichprovidesanewwindowintothephysicsoftheunderlyinglaserdynamics.Wealsobelieve
ourresultswillmotivateinterestinmuchbroaderapplicationsofthenonlinearFouriertransform
in all dissipative systems where field evolution through discrete elements can be modelled
approximately by uniformly distributed nonlinearity and dispersion (49,50). In optics, we
anticipateparticularinterestinreal‐timespectralandtemporalstudiesofnonlinearsingle‐pass
propagationdynamicsinopticalfiberssuchasroguewaveandmodulationinstabilitythatalso
displaycomplextransientnoisespikes.
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SUPPLEMENTARYMATERIAL
MATERIALSANDMETHODS
ExperimentalSetup
ThefiberlaserusedinourexperimentswasacommercialPritelFFL‐500modelusinga
linearFabry‐Perotcavityconfiguration,similartothatdescribedin(51)butwitha978nmpump
laserandEr:dopedfibergainmedium.Modelockingissustainedbya2mthickbulksaturable
absorber(InGaAsonInPsubstrate)contactbondedtooneofthecavitymirrors.Thesteady‐state
modelockingdynamicsaredominatedbysolitonpropagationeffectsbecauseofthenetanomalous
dispersioninthecavity,withthegenerationofhyperbolic‐secantlikepulsesofflatphaseinstable
operation.
WeimplementedtheDFTtechniqueusing850mofdispersioncompensatingfiber(DCF)
withgroupvelocitydispersioncoefficientof100ps/nm.kmanddispersionslope0.33ps/nm2.km
at 1550nm.We attenuated the input to the DCF in order to ensure linear propagation, and
confirmedthefidelityofthetime‐stretchingtechniquewhenthefiberlaserwasoperatedinthe
stablemode‐lockedregimebycomparingtheDFTspectrumwiththatmeasuredusinganoptical
spectrum analyser (Anritsu MS9710B). The real‐time DFT signal was measured by a 35GHz
photodiode (New Focus 1474A) connected to a 30GHz channel of a real‐time oscilloscope
(LeCroy845Zi‐A,80GS/s),resultinginaspectralresolutionof0.3nm.
The time lens measurements used a commercial Picoluz UTM-1500 system described
previously in (24) withatemporalmagnificationfactorof76.4.Totalaccumulateddispersionfor
theinputandoutputpropagationstepswas:D1=4.16ps/nmandD2=318ps/nmrespectively,
withmagnification|M|=D2/D1.Thetemporalquadraticphase(toreproducetheeffectofathin
lens)wasimposedthroughfourwavemixingfromapumppulse(100MHzMenloC‐FiberSync
andP100‐EDFA)withlinearchirpaccumulatedfrompropagationinapre‐chirpingfiberDp. The
imaging condition for magnification is 2/Dp = 1/D1 +1/D2 so that the dispersion for the pump is
around twice that of the signal input step. Thesignalatthetimelensoutputisrecordedbya
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13GHzphotodiode(Miteq135GE)connectedtothe30GHzchannelofthereal‐timeoscilloscope
at a sampling rate of 80Gs/s, resulting in an effective 400 fs temporal resolution over a
(demagnified) timewindowof 60ps.The temporalwindow isdeterminedby thepumppulse
durationusedinthetimelenstoimposetherequiredquadraticchirpviafour‐wavemixingina
highlynonlinearSiwaveguide(52).Instablemode‐lockedoperationitispossibletosynchronize
the20MHzlaserunderstudywiththe100MHzpumplaser,butthisisnotthecasewhenstudying
the transient dynamics as there is significant amplitude and phase noise that precludes the
detectionofawell‐defined20MHzharmonicforrepetition‐ratelocking.Thetimelensistherefore
operatedinasynchronousmodewithfree‐runningacquisitiontriggeredbythearrivalofthetime
lenssignal,althoughthislimitsthenumberofroundtripsthatcanbesimultaneouslymeasuredto
~400(asthereisawalk‐offbetweentheQ‐switchedmode‐lockedpulsesrelativetothetimelens
gate).ToobtainrepresentativedatasetsatdifferentpointsintheQ‐switchedmode‐lockedburst
(whichtypicallydevelopsover200‐300roundtrips)weperformedmultiplemeasurementswith
differentdelaysbetweentheswitch‐ontimeofthefiberlaserandthetimelenstrigger.Inthisway,
wewere able to characterize different phases of the pulse evolution in the transient regime.
Finally,wenotethatbecausetheopticalpathlengthsoftheDFTandtimelensstepsweredifferent,
thedelaybetween time‐lens andDFT recordswas calibrated (in a separatemeasurement)by
comparing thearrival timeson theoscilloscopeofa characteristic intensitypattern imprinted
ontoaCWlaser.Thisallowedustomatchwithoutambiguitythereal‐timetemporalandspectral
intensityprofilesofthedissipativesolitonsduringtheirtransientevolutionphase.
PhaseRetrieval
PhaseretrievalwasperformedwiththeGerchberg‐Saxtonalgorithm(35).Asidefromafewtrivial
ambiguities,thisalgorithmisknowntobeaccurateforpulseretrievalwhenintensityenvelopes
aremeasuredbothinthespectralandtimedomains.Thealgorithmconstructsaninitialguessfor
theprofileoftheelectricfieldusingthemeasuredtemporalintensityIM(t)andaninitialrandom
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phase rand(t): which is then Fourier transformed to the spectral
domaintoyieldacorrespondinginitialguessforthespectralamplitudeandphaseFT[ ]=
. The next step involves retaining the calculated spectral phase but replacing the
calculated spectral amplitude with the measured spectral amplitude i.e. . This
updated spectral profile is then transformed back into the time domainwherewe retain the
calculated temporal phase but again replace the calculated amplitude with that from
experiment . This procedure is iteratively repeated until the root mean square error
betweenthemeasurementsandretrievedintensityprofilesbecomessmallerthanachosenvalue
(3×10‐5inourcase).Convergencewasimprovedbyapplyingthemeasuredtemporalandspectral
constraintsonlywherethemeasuredintensitieswerewellabovethenoisefloor(‐20dBfromthe
maximum).Elsewheretheamplitudesweremultipliedwithasmallconstantvalueof0.001forcing
thembelowthenoise.Ifthealgorithmwasdetectedtostagnate(nochangeintheretrievalerror
beforereachingthedesiredvalue),asmalladditionalrandomphasecontributionwasadded.The
reliabilityofthealgorithmwastestedwithsimulatedpulseswithcomplexpropertiessimilarto
thoseseeninexperiments,andbyperformingmultipleretrievalsonthesameexperimentaldata,
whichallconvergedtothesameresults(withintheretrievalerror).
NonlinearFourierTransform
The nonlinear Fourier transform (also known as the direct scattering transform) is a
mathematicalprocedurethatidentifiesandquantifiessolitoncontentinagivenpulsestructure.
Significantly,whilstitsuseintheoreticalanalysistoobtainclosed‐formanalyticsolitonsolutions
is only strictly valid in an integrable system, computation of the eigenvalue spectrum can be
performednumericallyforanyarbitraryopticalfield.Ofcourse,thequestioninthiscaseishow
such results should be interpreted, but for a laser where the pulse properties are primarily
determined by nonlinear and dispersive effects, the nonlinear spectrum can be considered as
yieldinglocaleigenvaluesthatcanrevealapproximatesolitoncontent(48).Thisisbecauseifwe
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canneglectstrongpulsefilteringeffects(e.g.fromtheresonanttransitionorsaturableabsorber)
then the evolution through the different discrete elements of the cavity can be considered as
yielding an average pulse that is a solution to an equivalent NLSE with uniform distributed
nonlinearityanddispersion.Infactthisapproachisthebasisofthewell‐knownuseofthecomplex
GinzbergLandau(orHausMasterEquation)modelthathasprovenhighlysuccessfulinmodelling
mode‐lockedlasers(8,9,53).
UndertheseconditionsconsideringasystemdescribedbytheNLSEinnormalisedform:
(1)
Theassociatedscatteringproblemyieldsthefollowingsystem(42):
(2)
where v1 and v2 are the amplitudes of the waves scattered by the potential, and is the
correspondingcomplexeigenvalue.Forourresults,stablemodelockedoperationwasassumedto
correspondto(intracavity)solitonswith||=1,andthefieldprofilescorrespondingtotheother
pulsesanalysedwerenormalisedrelativetothisvalue.Standardnumericaltechniques(matrix
methods)wereusedtodeterminetheeigenvaluespectrum(48).
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2
10
2i
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ACKNOWLEDGEMENTS
Funding
ThisworkwassupportedbytheAgenceNationaledelaRechercheprojectLABEXACTIONANR11‐
LABX‐0001‐01,theRegionofFranche‐ComtéProjectCORPSandtheAcademyofFinland(Grants
267576and298463).
Contributions
Allauthorsparticipatedintheexperimentalworkanddataanalysisreported,andtothewriting
and reviewof the finalmanuscript.G.G. and J.M.D.planned the researchproject andprovided
overallsupervision.TheauthorsalsothankK.V.Reddyforprovidingtechnicaldetailsconcerning
thesolitonoperatingregimeofthePritellaserusedintheseexperiments.
Competingfinancialinterests
Theauthorsdeclarenocompetingfinancialinterests.
Correspondingauthors
Correspondenceto:JohnM.Dudley
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Figure1.Directphotodetectormeasurementof transient laserdynamics. (a)Results recorded
with reduced bandwidth detection of 20MHz illustrating the~100ms transient regime ofQ‐
switched mode‐locked operation before stable mode‐locking. The total time record shown
corresponds to2.6millionround trips. (b)Results fromseparatemeasurementsusing30GHz
bandwidthdetection,showinghowthelaseroutputduringtheQ‐switchedmode‐lockedregime
consistsoftransientburstsoftemporalwidth~10sseparatedby~70speriod.Theexpanded
viewshowshowunstablemode‐lockedpulsesat50nsperiodaregeneratedundereachburst.
(c)Resultsusing30GHzdetectionbutinthestablemode‐lockedregime,showingaregulartrain
ofpulseswithconstantamplitude.
21
Figure2.Setupusedforreal‐timecharacterization.Pulsesfroma20MHzpassivelymode‐locked
fiberlaseraresplitintotwopathsinafibercoupler(FC)andsenttoaDFTrealtimespectralsetup
andatime‐lens.TheDFTused850mofdispersioncompensatingfiber(DCF)toperformtimeto
wavelengthconversion,withdetectionusinga35GHzbandwidthphotodiode(PD2).Anoptical
spectrumanalyzerwasalsousedforcontrolmeasurementsofthespectrum.Thetimelensuses
twodispersivepropagationsteps(D1andD2),oneoneachsideofasiliconwaveguidestagethat
appliesaquadratictemporalphasethroughfourwavemixing(FWM)withlinearly‐chirpedpump
pulsesgeneratedfroma100MHzfemtosecondpulsefiberlaser(TLPump)afterstretchingina
dispersivestretching fiberDp.Thetime lenssignal isextracted fromtheFWMspectrumbyan
opticalfilterandwasmeasuredusinga13GHzbandwidthphotodiode(PD1).BothPD1andPD2
wereinputtoa30GHzchannelofthereal‐timeoscilloscope.ValuesfordispersionmodulesD1,D2
andDpinthetimelensaregiveninMaterialsandMethods.TheinputstoboththeDFTandtime
lenswereattenuated(Att.)toavoidnonlineareffectsandpolarizationcontrol(PC)wasneededto
ensureoptimalsignalfromthetime‐lens.
22
Figure3.Resultsover200roundtripsshowingreal‐timespectralandtemporalcharacterization
for(a)stableand(b)breathingmode‐lockingregimes.(i)Measuredspectralintensitywiththe
extractedspectralwidthshownintherightsubfigure.(ii)Measuredtemporalintensitywiththe
extractedtemporalwidthandenergyshownintherightsubfigures.(iii)Correspondingtemporal
intensityandphaseextractedusingtheGerchberg‐Saxtonalgorithmforpulsesatpointsindicated
by arrows. For each of the extracted pulses in (iii), the plots in (iv) show the calculated
wavelength‐timespectrogram.
23
Figure4.Resultsover200roundtripsshowingreal‐timespectralandtemporalcharacterization
for (a) a non‐interacting triplet of three solitons, (b) more complex break‐up and collision
dynamicsofasolitondoublepulseand(c)acombinationofasinglenon‐interactingsolitonanda
twopulsecollision.(i)Measuredspectral intensity.(ii)Measuredtemporal intensity.Theright
subfigure shows the integrated energy. (iii) Corresponding temporal intensity and phase
extractedusingtheGerchberg‐Saxtonalgorithmforpulsesatpointsindicatedbyarrows.Foreach
oftheextractedpulsesin(iii),theplotsin(iv)showthecalculatedwavelength‐timespectrogram.
24
Figure5.Resultsapplying thescattering transformto intensityandphasemeasurementsofa
selectionofmeasureddissipativesolitons.Ineachcasethereddotsshowthecalculateddiscrete
eigenspectrum corresponding to the normalized pulse intensity profiles shown as insets. The
resultsinFig5(a)and(b)correspondtothetwopointsinthedissipativesolitonbreathercycle
showninFig.3(b).Asthepulseintensityvariesbelowandabovetheunityvalueofanormalized
fundamentalsoliton,theretrievedeigenvaluesIm()variesbelowandabove±0.5i.(c)Thedouble
solitoncaseinFig.4(b)whereweseetwodiscreteeigenvalueswithIm()~±0.5i.(d)Thethree
solitoncaseinFig.4(a)whereweseethreediscreteeigenvalueswithIm()around±0.5i.(Note
that the intensity profiles plotted correspond to results shown in Fig. 3 and 4 but have been
smoothedforclaritywhenshownasinsets.)Thedashedlineintheinsetsshowstheintensityof
anidealsoliton.