Quasistatic magnetic and electric fields generated in intense laser plasma interactionBin Qiao, Shao-ping Zhu, C. Y. Zheng, and X. T. He

Citation: Physics of Plasmas (1994-present) 12, 053104 (2005); doi: 10.1063/1.1889090 View online: http://dx.doi.org/10.1063/1.1889090 View Table of Contents: http://scitation.aip.org/content/aip/journal/pop/12/5?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Fluid theory for quasistatic magnetic field generation in intense laser plasma interaction Phys. Plasmas 13, 053106 (2006); 10.1063/1.2200298 Plasma Expansion in Presence of Electric and Magnetic Fields AIP Conf. Proc. 762, 1295 (2005); 10.1063/1.1941712 Intense longitudinal electric fields generated from transverse electromagnetic waves Appl. Phys. Lett. 84, 3855 (2004); 10.1063/1.1748843 Magnetic field generation in a plasma in the presence of an ultrashort laser pulse Phys. Plasmas 8, 2918 (2001); 10.1063/1.1371770 Electric and magnetic field mapping Phys. Teach. 35, 136 (1997); 10.1119/1.2344621

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Quasistatic magnetic and electric fields generated in intense laserplasma interaction

Bin Qiaoa!

Graduate School of China Academy of Engineering Physics, P.O. Box 2101,Beijing 100088, People’s Republic of China

Shao-ping Zhu and C. Y. ZhengInstitute of Applied Physics and Computational Mathematics, P.O. Box 8009,Beijing 100088, People’s Republic of China

X. T. HeInstitute of Applied Physics and Computational Mathematics, P.O. Box 8009,Beijing 100088, People’s Republic of China and Department of Physics, Zhejiang University,Hangzhou 310027, People’s Republic of China

sReceived 30 November 2004; accepted 8 February 2005; published online 20 April 2005d

A self-consistent kinetic model based on relativistic Vlasov–Maxwell equations is presented for thegeneration of quasistatic spontaneous fields, i.e., both the quasistatic magneticsQSMd field and thequasistatic electricsQSEd field, in intense laser plasma interaction. For the circularly polarized laser,QSM field includes two parts, the axial partBz as well as the azimuthalBu; the QSE fieldEs,corresponding to the space-charge potential, forms a plasma density channel. For the linearlypolarized laser,Bz is absent. Equations forBz, Bu, and Es are uniformly derived from oneself-consistent model under the static-state approximation, which satisfies the conservation law ofcharge. The profile of the plasma density channel and the dependence of the peak QSM fields on thelaser intensity are discussed. The experiment and simulation results are explained by the model. Thepredicted QSM and QSE fields are also observed in the three-dimensional particle simulation.© 2005 American Institute of Physics. fDOI: 10.1063/1.1889090g

I. INTRODUCTION

Since the fast ignition scheme1 for inertial confinementfusion was put forward, much attention has been given to theinteraction of an intense short-pulse laser with relativisticplasma.2–4 Studies show that various nonlinear effects5–8 canlead to the generation of many nonlinear spontaneous fields.Among them, quasistatic spontaneous fields, including thequasistatic magneticsQSMd field Bs and quasistatic electricsQSEd field Es, have been considered to be most critical. Theformer has a very important influence on the collimation ofhot electrons that deposit energy at the edge of the high-compressed density fuel to form an ignition spark with a sizeof about 10mm. The latter affects the formation of theplasma density channel, through which an intense laser canpropagate a long distance. The study of this issue also haswide applications in analyzing the mechanism of particle ac-celeration and x-ray generation, as well as some phenomenaof astrophysics, such as the radiation of the pulsar.9,10 Someefforts11–17have been devoted to the study of the axial mag-netic fieldBz from a fluid scheme. However, these fluid mod-els are not perfect. So far, to our knowledge, no work gives acomplete QSM fieldBs, which should include both the axialpart Bz and the azimuthal partBu, in particular,Bu and Es

have not been given. Some of the present authors18–20 pro-posed a kinetic model forBz. However, unfortunately, exceptfor Bz, Bu andEs are also not given in Refs. 18–20, for therethe nonlinear currents are incomplete, dissatisfying the con-

servation law of charge, and the direction of the slow-time-scale wave-vectorq is improperly assumed to be parallel tothat of the laser wave vectork0. In principle, a self-consistentkinetic model satisfying the conservation law of charges]r /]td+ = ·j =0 sr and j are charge and current, respec-tivelyd should give a reasonable expectation for all quasi-static fields, including the completeBs sBz andBud andEs.

In this paper, we establish such a self-consistent kineticmodel to uniformly discuss the generation mechanism ofboth the complete QSM fieldsincluding Bz andBud and theQSE fieldEs from relativistic Vlasov–Maxwell equations. Itis found that the nonlinear beat interaction of laser fieldswith relativistic plasma causes a rotationalsazimuthald cur-rent in second order to generateBz, while the axial irrota-tional current driven by the ponderomotive force generatesBu. The ponderomotive force also expels electronssself-channelingd to form an electron density channel, and accord-ingly the QSE fieldEs is generated, corresponding to thespace-charge potential=fs. We deduce the equations forEs,Bz and Bu, among which the equations ofBu and Es areexplicitly given for the first time. For a given laser beam, wediscuss the approximate solutions, analyze the formation ofthe plasma density channel fromEs, and explain the experi-ment and simulation results forBz sRef. 21d andBu.

22,23 Wealso analyze the dependence of the peak magnetic field onthe laser intensity. In order to check our analytical results, werun a three-dimensionals3Dd particle-in-cell sPICd simula-tion to make further study and analysis.

In Sec. II, the kinetic model for the QSM and QSE fieldsadElectronic mail: binIqiao1979@yahoo.com

PHYSICS OF PLASMAS12, 053104s2005d

1070-664X/2005/12~5!/053104/12/$22.50 © 2005 American Institute of Physics12, 053104-1

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is given. The QSE fieldEs and the profile of the plasmadensity channel for a given laser beam are studied in Sec. III.The complete QSM fields, includingBz and Bu for a givenlaser beam, are given in Sec. IV. In Sec. V, we compare oursolutions with the results of experimental measurements.21,22

In Sec. VI, we explain the particle simulation results23 andmake further study using our 3D PIC simulation code“LARAD-P.” And, the conclusion is given in Sec. VII.

II. KINETIC MODEL FOR QSM AND QSE FIELDS

In this section, we present our kinetic model and giveequations for the QSM and QSE fields generated by laserfields interacting with an underdense plasma. Consider therelativistic Vlasov–Maxwell equations

F ]

]t+

cpa

p0a

·]

]x+ eaGsx,td ·

]

]paG fasx,pa,td = 0, s1d

= 3 E = −1

c

]B

]t, = 3 B =

1

c

]E

]t+

4p

cj , s2d

Gsx,td = Esx,td +pa 3 B

p0a

, s3d

j = oaE ea

cpa

p0a

fasx,pa,tdd3pa, s4d

where the indexa sa= i ,ed represents ions and electrons,respectively; p0a= gamac, pa=gamav, ga=Î1+pa

2 /ma2c2,

namely,p0a=Îpa2 +ma

2c2, are taken; and the other notationsare standard. We split the particle distribution functions upinto their background, slow-time-scale and fast-time-scaleparts,

fa = na0fa0 + fas + faf , s5d

wherena0 represents the initial plasma density and the iso-trope for fa0 with efa0d

3pa=1 is assumed. Also, we dividefields and currents into slow-time-scale and fast-time-scaleparts. Making Fourier transforms with respect to both spaceand time for Eqs.s1d–s4d and definingk=sk ,vd as wavevector and frequency of fast time scale,q=sq ,Vd as those ofslow time scale, we can obtain the slow-time-scale distribu-tion function in the Fourier representation from Eqs.s1d–s4das

fassq,pad =eana0

iSV −cq ·pa

p0aDo

s

Essses

ss8sqd ·]fa0

]pa

−ea

2na0

SV −cq ·pa

p0aDEssd

os1s2

ef8s1sk1d

·]

]pa

ef8s2sk2d ·

]fa0

]pa

Sv2 −ck2 ·pa

p0aD

3Efs1sk1dEf

s2sk2dd4sq − k1 − k2dd4k1d

4k2

s2pd4 , s6d

whereEfsiskidsi =1,2d is the amplitude of the fast-time-scale

electric field ssuch as laser electric field, Langmuir field,and wakefield in laser produced plasmad. In Eq. s6d, wekeep only the lowest nonlinear terms, and the symbolessd refers to a slow-time-scale motion caused by the differentbeat of two fast-time-scale motions. Also,Es

sssqd, Gssqd,es8

sssqd, and jssssqd are defined as18–20 Essqd=oss=t,lEs

ssess,Gssqd=oss=t,lEs

sse8ss, es8sssqd=f1−scq ·pa /Vp0adges

sssqd+sces

sssqd ·pa /Vp0adq, j ssqd=oss=l,t j ssssqdes

ss, where t indi-cates the transverse part andl the longitudinal part, respec-tively, perpendicular and parallel to the slow-time-scale wavevectorq.

We use the conventionEsss

*

sqd=−Essss−qd and es

ss*

sqd=−es

sss−qd, wherep notes the complex conjugate. Similarly,we have the same convention for the fast-time-scale field.Substituting Eq.s6d into Eq. s4d and ignoring the ion contri-bution to electric current due to much larger mass, we get theslow-time-scale electric currents and electric fields, respec-tively, as

jsss = − ine0e

2esss ·E cpe

p0eSV −cq ·pe

p0eDEs

sssqdesss1

·]fe0

]ped3pe + n0e

3 os1s2

Essd

Pesss1s2sq,k1,k2d

3Efs1sk1dEf

s2sk2dd4sq − k1 − k2dd4k1d

4k2

s2pd4 , s7d

UssssqdEs

sssqd = −i4pne0e

3

Vo

s1s2

Essd

Pesss1s2sq,k1,k2d

3Efs1sk1dEf

s2sk2dd4sq − k1 − k2dd4k1d

4k2

s2pd4 ,

s8d

where

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Ussssqd = 1 −

c2uqu2

V2 +c2sq ·es

ssd2

V2 + oa,ss8

vpa2 ma

Ves

ss ·

3E cpaesss8 ·

]fa0

]pa

p0aSV −cq ·pa

p0aDd3pa, s9d

Pesss1s2sq,k1,k2d =

1

2es

ss ·E 3 cpe

p0eSV −cq ·pe

p0eDef8

s1sk1d

·]

]pe

ef8s2sk2d ·

]fe0

]pe

Sv2 −ck2 ·pe

p0eD + s1 ↔ 2d4d3pe

s10d

are defined ands1↔2d represents the preceding term withindices 1 and 2 interchanged.Pe

sss1s2sq,k1,k2d is a nonlinearelectric current matrix.vpa is the plasma frequency definedby the rest massma asvpa=Î4pna0e

2/me. Here it should benoted that there are two important points of differences be-tween the kinetic and fluid theories. On the one hand, inkinetic theory, the slow-time-scale electric current is deter-mined only by the slow-time-scale distribution functionfes

as j s=−eescpe/p0edfesd3pe, which corresponds to the whole

of the three termsn0vs+nsvs+nfv f svs and v f are, respec-tively, the slow-time-scale and fast-time-scale velocitiesd influid theory. On the other hand, the integration offes overmomentum,ns=efesd

3pe, is the plasma density variation,which is usually not equal to zero iffesÞ0. Accordingly, theconclusion can be obtained that the QSM fieldBs is onlygenerated in the existence of plasma density variationns, forthe slow-time-scale currents are the basic source for the gen-eration ofBs. If ns=0, noBs will be generated. This conclu-sion is consistent with that of the usual fluid model. How-ever, in the fluid model,ns is usually inconsistently assumedas in Refs. 15 and 16, rather than self-consistently includedas in our model.

As mentioned above, the slow-time-scale motion iscaused by the different beat of two fast-time-scale motions,thus it is reasonable to assumeV=v1+v2, q=k1+k2, andv−ck ·pa /p0aÞ0. After tedious calculations, one can showthat the following relation holds for relativistic plasmasseethe Appendixd:

1

VPe

sss1s2sq,k1,k2d =1

v2Pe

s2sss1sk2,q,− k1d. s11d

As the conditionsvtiuqu!V!vteuqu, V!cuqu, and v@ uk uvte are usually satisfied under the static-state approxima-tion, using relations11d, we can obtain the approximate ex-pressions of Eqs.s9d and s10d as

Pets1s2sq,k1,k2d = es

t ·fq 3 b1se1 3 e2dg

me2v2

2v1

+ est ·

fe1sq · b2e2dgme

2v22v1

+psq · b3e2dsq ·e1d

16me2v2

2cuqu2+

pb4se1 ·e2d4me

2v22c

,

s12d

Pels1s2sq,k1,k2d =

V

me2v2

2c2uqub5se1 ·e2d, s13d

Ustsqd = −

c2uqu2

V2 −vpe

2

c2uqu2b6 −

vpi2

V2 b7 . −c2uqu2

V2 ,

s14d

Usl = 1 +

vpe2

c2uqu2b6 −

vpi2

V2 b7 .vpe

2

c2uqu2b6.

In Eqs. s12d–s14d, the coefficients related to therelativistic effect are defined asb1=me

2c2e1/p0e2 fs1/2d

−supeu2/3p0e2 dgfe0d

3pe, b2=me2c2esupeu2/3p0e

4 dfe0d3pe, b3

=me2c2esupeu /p0e

3 dfe0d3pe, b4=me

2c2efs1/p0eupeud−s3upeu /4p0e

3 dgfe0d3pe, b5=me

2c2efs1/2upeu2d+supeu2/3p0e4 d

−s1/2p0e2 dgfe0d

3pe, b6=mecefs1/p0ed+sp0e/ upeu2dgfe0d3pe,

and b7=micefs1/p0id−spi2/3up0iu3dgf i0d

3pi. The dependenceof the coefficientsb1–b5 on the electron temperature is plot-ted in Fig. 1sad, and b6, b7 in Fig. 1sbd. In Eqs. s14d, thecondition 1@ uqule@ svte

Îb6/c2dsV / uqud holds and the ioneffect is ignored. From Eq.s12d, we can see that the nonlin-ear current matrix calculated in Refs. 18–20 is incomplete,missing the latter three terms of Eq.s12d. We find there aretwo main reasons for this slip. One is that in Ref. 19 thedirection ofq is improperly taken for the direction ofk1 ork2 and accordinglyes

t ·k i =0si =1,2d is improperly taken. Theother is thates

t has been incorrectly regarded as a fixed di-rection perpendicular to the direction ofq, rather than aplane including two possible directions that are both perpen-dicular to q, and thus many terms of the currentsj s

t in thisplane have been incorrectly averaged to zero asj s

t is pro-jected on a line other than a plane, whenPe

ts1s2sq,k1,k2d iscalculated.

From Eqs. s12d–s14d and s7d–s10d, transforming intospace-time coordinates, after tedious calculation, we obtainthe slow-time-scale electric fields and the nonlinear currents,respectively, perpendicular and parallel toq as

¹2¹2Estsx,td = − i

vpe2 e

mev03c2

]

]t¹2 = 3 fb1E fsx,td 3 E f

*sx,tdg

− ivpe

2 e

mev03c2

]

]t¹2fE fsx,td = · b2E f

*sx,tdg

+pvpe

2 e

16mev02c3es

t ]

]t= · fb3E f

*sx,td = ·E fsx,tdg

+pvpe

2 e

4mev02c3es

t ]

]t¹2b4uE fsx,tdu2, s15d

053104-3 Quasistatic magnetic and electric fields… Phys. Plasmas 12, 053104 ~2005!

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¹2j stsx,td = − i

vpe2 e

4pmev03¹2 = 3 fb1E fsx,td 3 E f

*sx,tdg

− ivpe

2 e

4pmev03¹2fE fsx,td = · b2E f

*sx,tdg

+vpe

2 e

64mev02c

est = · fb3E f

*sx,td = ·E fsx,tdg

+vpe

2 e

16mev02c

est¹2b4uE fsx,tdu2, s16d

Esl = −

eb5

mev02b6

= uE fu2, s17d

j sl =

eb5

4pmev02b6

]

]t= uE fu2. s18d

It can be seen from Eqs.s15d and s17d that the transverseelectric fieldEs

t is slowly varying, and its effects on electronmotion and other nonlinear plasma dynamics are embodied

by the QSM fieldBs as =3Est =−1/cs]Bs/]td; the longitu-

dinal electric fieldEsl is just the QSE field, which corre-

sponds to space-charge potential asEsl =−=fs. It can also be

seen from Eq.s16d that the transverse currentsj st are com-

posed of four kinds of currents. The first is the nonlinearhelical srotationald current driven by different beats of fast-time-scale fieldsslaser fieldsd, which will generateBz. Thesecond is the nonlinear azimuthal current driven by interac-tions of fast-time-scale velocityv f and fast-time-scale elec-tron density perturbationnf, which corresponds to the currentnfv f in the fluid model and will generateBz as well. How-ever, in our kinetic model, the gradient of plasma tempera-ture is assumed to be small, whereTe represents an averagedtemperature of plasma electrons in the main region of laserplasma interaction, i.e.,=b=0; and for an intense laserpropagating in underdense plasma,= ·E f .0 can be approxi-mately satisfied becausenf is much smaller, which will beexplained in the following paragraph. Thus this current van-ishes. The third term always equals zero for= ·E f .0. Thefourth is the nonlinear irrotational current driven by the pon-deromotive force, which will generateBu. It should be notedthat these transverse currentsj s

t of Eq. s17d are the basicsources for generation of QSM fields as=3Bs=s4p /cdj s

t ,and the longitudinal currentsj s

l of Eq. s18d are to keep thecharge conservation ass]rs/]td+ = ·j s

l =0. In other words,our model satisfies the conservation law of charge throughthe longitudinal currents.

When the laser beam propagates in plasma, a longitudi-nal fast-time-scale electric fieldEz will be generated by theinteraction of laser fieldEL with the fast-time-scale plasmadensity variationnf as = ·E f =−4penf; and a plasma wavefield swakefieldd EW of frequencyvpe may also be generatedvia a self-modulation or stimulated Raman scattering insta-bility, which always exists behind the laser pulse. That is, thefast-time-scale electric fieldE f should includeEL, Ez, andEW. As mentioned above, we only keep the second-orderterms representing nonlinear interactions of laser fields andplasmas. The wakefieldEW, as a subfield to laser field,here can be ignored, because the amplitude of the wakefieldis much smaller than the laser field for underdense plasmasas uEwu,0.1−0.3smcvp/ed=0.1−0.3sn0/ncrd1/2uELu! uELu,which is shown in Ref. 24. For a laser beam withwave numberk0=2p /l0, assuming it propagates along thez-axis direction, under the Coulomb gauge, the electricfields are usually expressed asELsx ,td= 1

2E0sr ,u ,tdser

+ iaeudeiaueik0z−iv0t+c.c., wherea=0, ±1 for the linearly po-larized sLPd and the circularly polarizedsCPd laser, respec-tively. We assume the laser intensity is of Gaussian distribu-

tion, asI = I0e−r2/r0

2−sz−ctd2/L2andE0=ÎI, whereL is the pulse

longitudinal width andr0 the transverse radius. We canapproximately estimate the value ofEz through thefluid equations. From the continuity equations]nf /]td+ = ·sne0v fd=0 and the motion equation for plasmaelectrons quivering in the laser field]p f /]t=]gmv f /]t=−eEL fg=f1+sp f

2/me2c2dg1/2.f1+s1+a2dIlmm

2 /2.76g1/2g,we get nf =−sene0/v0

2med= ·sEL /gd. Then, fromPoisson’s equation, = ·E f = ik0Ez+=' ·EL=−4penf, wecan obtain Ez as Ez= i /k0f1−svpe

2 /v02gdg=' ·EL−si /k0d

FIG. 1. The dependence of the relativistic coefficientsbi si =1–7d on thetemperature of the electrons, wheresad plots of b1–b5 and sbd plots of b6

andb7.

053104-4 Qiao et al. Phys. Plasmas 12, 053104 ~2005!

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3svpe2 /v0

2dEL ·='s1/gd+c.c. After substituting the detailedGaussian intensity distribution into them, we getnf =sene0/2v0

2meds1/g3dE0eiaueik0z−iv0tsr / r0

2d+c.c. and Ez

=−si /2k0dsr / r02dE0e

iaueik0z−iv0t+si /2k0dsr / r02dsvpe

2 /v02ds1/g3d

3E0eiaueik0z−iv0t+c.c. For intense laser propagating in under-

dense plasma,g.1 andvpe2 /v0

2,1 should be taken, assum-ing the laser beam transverse radiusr0 to be about 2l andintensity to be I0=1019 W/cm2 sg=3d after focus, uEzu, sr /k0r0

2dE0, s1/4pdE0!E0 suEzu→0 when r →0d andunfu, sr /k0r0

2dsÎg2−1/g3dne0, s1/k0r0g2dne0, s1/36pdne0

!ne0 are satisfied. That is, for intense laser propagating inunderdense plasma,uEzu! uELu and unfu!ne0 are both satis-fied. Thus, it is reasonable to ignoreEz itself in the detailedcalculation and to approximately take the relation= ·E f .0.But note that the effect ofik0Ez is not ignored in our modelas the relation= ·E f .0 is used above, for the value ofik0Ez

is comparable with that of=' ·EL.Ignoring ]EL /]t and using the Ampere law=3Bs

=s4p /cdj st , we obtain the QSE fieldEs and the QSM fieldBs

as,

Es = −eb5

mev02b6

= uELu2, s19d

= 3 Bssx,td = − ivpe

2 e

mev03c

= 3 fb1ELsx,td 3 EL* sx,tdg

+pvpe

2 e

4mev02c2es

tb4uELsx,tdu2. s20d

Note thatEs and Bs in Eqs. s19d and s20d are quasistatic,assumingV! uquvte. That is, they do have important averageeffects stime-averaged over one laser periodT=2p /v0d onplasma dynamics, especially on electron motion, such as theponderomotive force, while the effects of all fast-time-scalefields are averaged to zero.

III. QUASISTATIC ELECTRIC FIELD AND PLASMADENSITY CHANNEL

In this section, we deduceEs and analyze the profile ofthe plasma density channel for the given focused laser beamas that in Sec. II. Assuming the laser beam propagates alongthe z-axis direction with wave numberk0=2p /l0, under theCoulomb gauge, the electric fields are usually expressed asELsx ,td= 1

2E0sr ,u ,tdser + iaeudeiaueik0z−iv0t+c.c, where a=0, ±1 for the linearly polarizedsLPd and the circularly po-larized sCPd laser, respectively. We assume laser intensity is

of Gaussian distribution asI = I0e−r2/r0

2−sz−ctd2/L2and E0=ÎI,

whereL is the pulse longitudinal width, andr0 the transverseradius. All these descriptions of the laser beam are the sameas those in Sec. II. Introducing the normalized transformt→ t /v0,x→cx /v0,E→Esmecv0/ed ,B→Bsmecv0/ed, wecan obtainEs, plasma density variationns through Poissonequation= ·Es

l =−4pens and plasma density divided by itsinitial value ne/ne0=sne0+nsd /ne0 as

Es =s1 + a2d

2

b5

b6I0e

−sz − td2/L2e−r2/r0

2S r

r02er +

z− t

L2 ezD , s21d

ns = − s1 + a2db5

b6I0e

−sz − td2/L2e−r2/r0

2 1

r02S1 −

r2

r02D , s22d

ne

ne0= 1 −

s1 + a2dne0

b5

b6I0e

−sz − td2/L2e−r2/r0

2 1

r02S1 −

r2

r02D . s23d

Equationss22d ands23d just describe the transverse profile ofplasma density channel, ignoring the longitudinal densityvariation due toL@ r0. It can be seen thatnene0 is smallerthan 1 in the region ofr , r0, that is, a density channel isformed; whenr * r0, then ne/ne0.1 is taken; and whenr→`, thenne/ne0=1 is satisfied.

We can see that the expression of the right-hand side ofEq. s19d or Eq. s21d is similar to that of the ponderomotiveforce except that the sign is opposite. This can be easilyunderstood, as it is just the ponderomotive force that expelssome electrons from the axis so that a space-charge potentialis formed to generateEs. This is also just the mechanism forboth the generation of QSE field and the formation of plasmadensity channel. In the meantime, it should be noted that thehot electrons with temperatureThe are mainly originatedfrom this part of electrons, which are accelerated by pon-deromotive force in the process of being expelled.

IV. QUASISTATIC MAGNETIC FIELD

In this section, we give the solutions of the QSM fieldsBz andBu for the given focused laser beam described in Secs.II and III. For real intense laser plasma interaction, in themain interacting region, some of electrons are warmed bylaser fields due to heat exchange, others are accelerated intothe relativistic regime by the ponderomotive force, as men-tioned at the end of Sec. III. It is reasonable to assume thatthere are two groups of electrons in the interacting region,thermal ones and hot ones. Certainly, there are still a largenumber of cold electrons in the noninteracting region. Butthey do not evidently affect the generation of the QSM fieldsand thus we ignore their contribution. If we take the param-eter j to represent the ratio of the number of thermal elec-trons to that of total electrons, we get the normalized equa-tions, respectively, ofBz andBu from Eq. s20d as follows:

Bz = −1

2ane0fjb1sTted + s1 − jdb1sThedgI0e

−r2/r02−sz − ctd2/L2

,

s24d

Bu =ps1 + a2d

64ne0fjb4sTted

+ s1 − jdb4sThedgI0e−sz − ctd2/L2

rr0

2

r2s1 − e−r2/r02d, s25d

whereTte is the averaged temperature for thermal electronsandThe is the averaged temperature for hot electrons. In Eq.s25d, we also assumeq makes an angleuq with the axisz andkez·es

tl=ksinuql=1/2.

053104-5 Quasistatic magnetic and electric fields… Phys. Plasmas 12, 053104 ~2005!

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From Eqs.s20d, s24d, ands25d, the mechanism for QSMfield generation can be obtained where the beat interactionsof CP laser fields drive a rotational azimuthal currentfthefirst term of Eq.s20dg to generateBz of Eq. s24d, in whichelectrons following the CP lasersa= ±1d move spirallyaround the laser propagation direction; the ponderomotiveforce drives a irrotational axial currentfthe second term ofEq. s20dg for the generation ofBu of Eq. s25d. Yet, for LPlaser fieldssa=0d, which are only along a fixed linearly po-larized direction, the rotational current should vanish andBz

is nothing, as seen in Eq.s24d.

V. COMPARING WITH EXPERIMENTAL RESULTS

Although the QSE fields can not be easily measured,there are some experiments involving the QSM fields. Re-cently, measurements of the QSM fields were reported byNajmudinet al.21 and Fuchset al.22 In Ref. 21, Najmudinetal. paid attention to the QSM fields parallel to the laser beampropagation direction, that is,Bz in the present paper. ButFuchset al.22 measured the magnetic fields perpendicular tothe laser beam propagation direction, that is,Bu in thepresent paper. We compare our theoretical expectation withthe experiments in this section.

Before the detailed comparisons, we have to give somereasonable estimations of three parametersj, Tte, andThe dueto their complexity. The values of these parameters may havesome little effects on the numerical value ofBz andBu, butdo not essentially affect the physical character ofBz andBu

and the comparisons between our model and the experi-ments, as well as the following numerical simulation. First,as mentioned above and in the end of Sec. III, if we approxi-mately consider that hot electrons are mainly produced bythe acceleration of ponderomotive force ignoring fast-varying density variation, we can get the approximatej fromEqs.s22d and s23d as

j .ne

ne0= 1 −

s1 + a2dne0

b5

b6I0e

−sz − td2/L2e−r2/r0

2 1

r02S1 −

r2

r02D .

s26d

Second, it is widely proved that the Wilk’s scaling law can beused to determine the averaged hot electron temperature asThe=511sg−1d [g=(1+fs1+a2dI0lmm

2 /231.3831018g)1/2].Lastly, the value estimation of the averaged thermal electrontemperatureTte is the most difficult. So far, to our knowl-edge, no paper has give the exact value ofTte. Through alarge number of numerical calculations and simulation, it isfound thatTte seems to be several tens of keV under theabove estimation ofj and The. Here we simply assume allfollowing Tte to be 75 keV. It is also found that ifj is chosento be smaller,Tte may be chosen to be lower; and in contrast,j greater,Tte higher.

A CP laser beam with “vacuum” intensityI0=6.731018 W/cm2 is used in the experiment given in Ref. 21,the main parameters are laser wavelengthl=1.05mm,vacuum laser transverse radiusr0=10l, laser pulse durationt=1 ps and plasma densityne0=2.831019 cm−3. As men-tioned in Ref. 21, the laser beam may be self-focused inplasma, the intensity may be enhanced, while the transverse

radius decreased in some regions. Here we assumeI0 can betwo times of vacuum laser intensity andr0 be decreased to5l. Figures 2sad and 2sbd showBz calculated by Eq.s24d forthe CP laser beam with wavelengthl=1.05, z−ct=0, andne0=2.831019 cm−3 for sad intensity I0=6.731018 W/cm2,transverse pulse radiusr0=10l and sbd I0=1.3431019 W/cm2, r0=5l, respectively. As mentioned above,The=511sg−1d andTte=75 kev are taken andj is chosen asEq. s26d. Other parameters accord with those given in Ref.21. It can be seenBz dominates in the region near the axisand the profile ofBz is consistent with Fig. 2 of Ref. 21. Thepeak magnetic fieldBz is about sad 2.13 megagausssMGdand sbd 3.07 MG at the laser axisr =0, both of which arevery close to the experiment result of 1.6–3.2 MG.

The experiment measurement onBu generated by a LPlaser is reported in Ref. 22. The main parameters are vacuumlaser intensity I0=4.731018 W/cm2, laser wavelengthl

FIG. 2. sColor onlined. Bz calculated by Eq.s24d for the CP laser beam withl=1.05mm and sad I0=6.731018 W/cm2, r0=10l; sbd I0=1.3431019 W/cm2, r0=5l, where z−ct=0, ne=2.831019 cm−3, Tte=75 keV,andTre=511sg−1d are adopted.

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=1.058mm, vacuum laser transverse radiusr0=4l, laserpulse duration t=600 fs, and plasma densityne0=231020 cm−3. Figures 3sad and 3sbd show Bu calculated byEq. s25d for the LP laser beam withl=1.05mm, wheresadI0=4.731018 W/cm2, r0=4l andsbd I0=9.431018 W/cm2,r0= 4

3l, as two times of vacuum laser intensity and one-thirdof the initial transverse radius due to self-focusing, andne0

=231020 cm−3 is adopted. The parameters used here are inaccordance with those in Ref. 22. The averaged thermal elec-tron temperatureTte is also assumed to be 75 keV,j is alsochosen as Eq.s26d and The is determined by Wilk’s scalinglaw The=511sg−1d, as well. It can be seen from Figs. 3sadand 3sbd that the peakBu is aboutsad 85 MG andsbd 59 MG,respectively, which is either very close to the results35–70 MG in Ref. 22. We can see thatBu is close to zeronear the axis, increases gradually withr until reaches itsmaximum value at aboutr =r0, and then decreases due toreturn currents.

Notice that the experiment results21,22both show that theQSM fields increase with laser intensity. But few models cangive the explanation of the dependence of the peak QSM

fields Bz andBu on the laser intensity. Sheng’s model12 sug-gests that the peakBz decreases as laser intensity increasesssee theoretical curve in Fig. 3 of Ref. 21d, which is in con-tradiction with the experimental result given in Ref. 21. Thereason is that in Sheng’s model12 all electrons are assumed tobe hotsrelativisticd accelerated by ponderomotive force, thatis, j=0. However in fact, as mentioned in our model, not allelectrons had been accelerated into the relativistic regime,there are some thermal electrons, that is,jÞ0. Figures 4sadand 4sbd plot the dependence ofsad Bz andsbd Bu on the laserintensityI0, wheresad r0=5l andsbd r0=4/3l are taken afterself-focusing, other parameters are consistent with the aboveexperiments given in Refs. 21 and 22, respectively. It can beseen from Fig. 4sad that Bz increases with laser intensityI0,which accords with the experimental results. It also shouldbe noted that whileI0 rises to high enough that all electronsat r =0 were expelled and accelerated to be hot, i.e.,ne→0and j→0 here, the peakBz then decreases ifI0 still in-creases, as seen in Fig. 4sad, because the peakBz is mainlydetermined by electrons at the axisr =0. However, the peakBu is determined mainly by electrons atr =r0, where all elec-

FIG. 3. Bu calculated by Eq.s25d for the LP laser beam withl=1.05mm,where sad I0=4.731018 W/cm2, r0=4l; sbd I0=9.431018 W/cm2, r0

=1.33l, wherene=231020 cm−3, Tte=75 keV,Tre=511sg−1d, z−ct=0 areadopted.

FIG. 4. The dependence of the peakBz sad andBu sbd on the laser intensityI0, where sad r0=5l and sbd r0=4/3l are taken after self-focusing, otherparameters are consistent with Figs. 2 and 3, respectively.

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trons are thermal andj=1 is always satisfied, soBu willincrease all along withI0, as seen in Fig. 4sbd. This explana-tion can be more explicitly seen from Figs. 5sad and 5sbd,which plots the corresponding plasma density divided by itsinitial value calculated by Eq.s23d, that is, just describing thetransverse profile of the corresponding plasma density chan-nel, under the same parameters as Figs. 2sbd and 3sbd, re-spectively. From Figs. 5sad and 5sbd we can evidently seethat electrons in the regionr , r0 are expelled by pondero-motive force so thatne/ne0,1, i.e., j,1; at the placer=r0, ne/ne0=1, i.e.,j=1; and the expelled electrons will de-posited at a cavity edge ofr * r0, wherene/ne0.1 should besatisfied; then whenr @ r0, ne/ne0=1 should be taken again.

VI. PARTICLE SIMULATION

In this section, we check our analytical model throughthe particle simulations. The values of parametersj, Tte, andThe are approximately estimated in the same way as Sec. V.First, we compare our theoretical expectation with the simu-lation of Pukhov and Meyer-ter-Vehn.23 A LP laser beamwith vacuum intensityI0=1.2431019 W/cm2 and vacuumtransverse radiusr0=6l was used in the simulation given inRef. 23. The main parameters are laser wavelengthl

=1 mm and plasma densityne0=0.36ncr. From Figs.s1d–s4dof Ref. 23, we can see the results thatsid the laser beam willbe self-focused when it enters plasmas, the intensity can beenhanced to be 1.2431019–6.031020 W/cm2, transverseradius can be collapsed to be 1−2l; sii d about 50–100 MGazimuthal QSM field is generated;siii d a density cavitation isformed and guides the laser beam. Using our model, we plotthe azimuthal QSM fieldBu calculated by Eq.s24d in Fig.6sad, whereI0=2.4831019 W/cm2 as two times of vacuumintensity, r0=1l after self-focusing andne0=0.36ncr aretaken. The parameters accord with that given in Ref. 23. Itcan be seen that the profile ofBu in Fig. 6sad is very similarto that of Fig. 2 in Ref. 23. The peakBu is 100 MG, also veryclose to the simulation results 50–100 MG. Figure 6sbdshows transversesY,Zd plots vp

2/v2=ne/ncr using Eq.s23d.The parameters areI0=4.531020 W/cm2, r0=1l, and ne0

=0.36ncr, which are consistent with that of Fig. 4 in Ref. 23.As can be seen from the figure, our model can give well the

FIG. 5. The plasma density divided by its initial value, that is, the transverseprofile of the corresponding plasma density channel, under the same param-eters as Figs. 2sbd and 3sbd, respectively.

FIG. 6. sColor onlined. sad Plots Bu calculated by Eq.s24d for LP laser,where I0=2.4831019 W/cm2 as two times of “vacuum” intensity,r0=1lafter self-focusing, andne0=0.36ncr are taken;sbd transversesY,Zd plotsvp

2/v2=ne/ncr using Eq. s23d, where I0=4.531020 W/cm2, r0=1l, andne0=0.36ncr are adopted.

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density cavitation, whose profile and value are both verysimilar to the simulation results given in Ref. 23.

Then, in order to further study the QSM and QSE fieldsfor the CP laser beam, we commence with the numericalsimulation using our 3D PIC code LARAD-P. The CP laserbeam sl=1.054mmd with an initial transverse Gaussianwidth d0=6l is incident from the leftz=0 in the intensityI =531018 W/cm2 to the plasmas. The initial plasma densityis ne=0.36ncr, wherencr is the nonrelativistic critical density;the rest mass ratio for ion and electron ismi /me=4500; Te

=16 keV; 33108 electrons and ions are uniformly distrib-uted in 16036603160s12.7l352.5l312.7ld spatialmeshes.

The numerical simulation results show that the laserbeam is strongly focused when it enters plasmas, the laserintensity increases fromI0=531018 W/cm2sg2−1=4d to I0

=1.8631019 W/cm2sg2−1=15d, while the transverse widthdecreases tod0<2l–3l, as seen in Figs. 7sad and 7sbd. Forthe CP laser, the longitudinal profiles of the axial magneticfield Bz and the azimuthal oneBx at t=48t are illustrated inFigs. 7scd and 7sdd. At t=48t, uBzu=20–45 MG is distributedin z=14l–30l, approximately uniform in the transverse di-rections with the transverse diameterd=1l and dominates inthe region near the axis;uBxu=70 MG is distributed inz=8l–28l and dominates at the outlying zone where thetransverse diameter isd0<2l–3l. Figures 7sed–7sgd, re-spectively, plot the profiles ofI0, Bz, and Bx, which trans-verse cuts atz=26.25l, t=48t. It can be seen that the shapesof the transverse profiles forBz and Bx are both consistentwith our theoretical expectationfjust asBz in Fig. 2 andBu in

Fig. 3 or Fig. 6sadg. And it can be seen that the peaksuBzu anduBxu are, respectively, about 40 MG and 70 MG, whereI0

=1019 W/cm2sg2−1=8d and d0<2l–3l are satisfied.Adopting the same parameters and using Eqs.s24d ands25d,the peakuBzu=24.6 MG anduBx= 1

Î2Buu=98.4 MG are esti-

mated by our model, which are both very close to the simu-lation results. Figure 8 plots the corresponding QSE fieldsand plasma density channel,sad and sbd, respectively, illus-trate the longitudinal profiles of the radial QSE fieldEsx

=ex·Es sEsrd and axial QSE fieldEsz=ez·Es at t=92t; scd–sed, respectively, plot the transverse profiles ofI0, Esx, andEsz transverse cutting atz=26.25l, t=92t; sfd and sgd, re-spectively, plot the longitudinal and transverse profiles ofplasma density channelne/ne0 cutting, respectively, assadand sdd. As seen from Figs. 8sad and 8sbd, uEsxu=30 CGSEsEsrd is distributed inz=10l–45l and dominates at the out-lying zone aboutd0=2−3l, uEszu=20 CGSE is distributed inz=8l–28l dominating in the region near the axis. Theseprofiles ofEsx andEszare both consistent with our theoreticalexpectationfcan be seen from Eq.s21dg, which can be moreexplicitly seen from the transverse profiles of them in Figs.8sdd and 8sed. From Figs. 8scd–8sed, we can also see that thepeak uEsxu and uEszu are, respectively, about 30 CGSE and20 CGSE for I0=1.2931019 W/cm2sg2−1=10.4d and d0

=2l. Adopting the same parameters and using Eq.s21d, thepeakuEsxu=

1Î2

Esr=29.3 CGSE anduEszu=14.7 CGSEsassum-ing laser beam longitudinal widthL=2r0=2l after focuseddare estimated by our model, which are both very close to thesimulation results. From Figs. 8sfd and 8sgd, we can see that

FIG. 7. sColor onlined. Instantaneous laser intensitysg2−1d longitudinal sz,yd for the CP laser cuts att=28t sad and t=48t sbd, respectively. Att=48tmagnetic fields longitudinalsz,yd for the CP laser cut,scd Bz andsdd Bx sazimuthald, respectively. The profiles ofsed I0, sfd Bz, andsgd Bx sazimuthald transversesy,xd cut atz=26.25l, t=48t. The units ofx,y,z arel.

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a plasma density channelfne/ne0=0.3 in Fig. 8sgdg can beformed and guide the laser propagation, which also basicallyaccords with our theoretical prediction.ne/ne0=0.4 is esti-mated using Eq.s22d with the same parameters as Fig. 8sgd.

VII. CONCLUSION

In conclusion, we have studied the generation of both theQSM field and the QSE field in intense laser plasma interac-tion analytically and numerically, which includes the axialQSM fieldBz as well as the azimuthalBu, and the QSE fieldEs. A self-consistent kinetic model satisfying the conserva-tion law of charge is set up, which can give a good uniformexplanation of the generation mechanism of all these fieldsand even the formation of the plasma density channel. Wecompare our theoretical expectations with the results of theexperiments21,22 and Pukhov’s simulation.23 Through ourown particle simulation, we further investigate and verify ourmodel. It is found thatBz as high as about 40 MG andBu

about 70 MG can both be generated by the CP laser in arelativistic plasma with laser intensityI0=531018 W/cm2;but for the LP laserBz is absent. In the meantime,Esr about30 CGSE andEsz about 20 CGSE can both be generated byLP or CP laser. It is also found that the peak magnetic fieldsdepend heavily on the laser intensity and the ratio of the

number of thermal electrons to that of total electrons. As,actually, not all electrons have been accelerated into the rela-tivistic regime by intense laser beam, the peak magneticfieldsBz andBu both increase with the laser intensity. For thepurposes of laser plasma acceleration and fast ignition, suchquasistatic fields are important because the fields will affectthe electron motion and the laser propagation, which are stillunclear and under further investigation.

ACKNOWLEDGMENTS

This work was supported by National Hi-Tech InertialConfinement Fusion Committee of China, National NaturalScience Foundation of China Under Grant No. 10135010,Special Funds for Major State Basic Research Projects inChina, National Basic Research Project “nonlinear Science”in China, and Science Foundation of CAEP.

APPENDIX: PROOF OF EQ. „11…

Let es8=es8sssqd, e18=ef8

s1sk1d, e28=ef8s2sk2d, es=es

sssqd, e1

=efs1sk1d, e2=ef

s2sk2d, then

FIG. 8. sColor onlined. The longitudinal profiles ofsad radial QSE fieldEsx=s1/Î2dEsr andsbd axial QSE fieldEsz cut att=92t. The transverse profiles ofI0

scd, Esx sdd, andEsz sed cut atz=26.25l, t=92t; the longitudinal and transverse profiles of plasma density channelne/ne0 cut att=92t, z=26.25l, respectively,of sfd and sgd.

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Pesss1s2sq,k1,k2d =

1

2E 3 ces ·pe

p0eSV −cq ·pe

p0eDe18

]

]pe

e28 ·]fe0

]pe

Sv2 −ck2 ·pe

p0eD + s1 ↔ 2d4d3pe. sA1d

We now prove that

1

VE 3 ces ·pe

p0eSV −cq ·pe

p0eDe18 ·

]

]pe

e28 ·]fe0

]pe

Sv2 −ck2 ·pe

p0eD + s1 ↔ 2d4d3pe −

1

v2E ce2 ·pe

p0eSv2 −ck2 ·pe

p0eD

33es8 ·]

]pe

e18 ·]fe0

]pe

Sv1 −ck1 ·pe

p0eD − e18 ·

]

]pe

es8 ·]fe0

]pe

SV −cq ·pe

p0eD4d3pe = 0. sA2d

EquationsA2d = E 5−1

V

e28 ·]fe0

]pe

Sv2 −ck2 · pe

p0eDe18 · 3 ces

SV −cq · pe

p0eDp0e

−cses · pedpe

SV −cq · pe

p0eDp0e

3

+cses · ped

p0eSV −cq · pe

p0eD2

3S cq

p0e−

csq · pedpe

p0e3 D4 −

1

V

e18 ·]fe0

]pe

Sv1 −ck1 · pe

p0eDe28 · 3 ces

SV −cq · pe

p0eDp0e

−cses · pedpe

SV −cq · pe

p0eDp0e

3

+cses · ped

p0eSV −cq · pe

p0eD2S cq

p0e−

csq · pedpe

p0e3 D4 +

1

v2

e18 ·]fe0

]pe

Sv1 −ck1 · pe

p0eDes8 · 3 ce2

Sv2 −ck2 · pe

p0eDp0e

−cse2 · pedpe

Sv2 −ck2 · pe

p0eDp0e

3

+cse2 · ped

p0eSv2 −ck2 · pe

p0eD2Sck2

p0e−

csk2 · pedpe

p0e3 D4

−1

v2

es8 ·]fe0

]pe

SV −cq · pe

p0eDe18 · 3 ce2

Sv2 −ck2 · pe

p0eDp0e

−cse2 · pedpe

Sv2 −ck2 · pe

p0eDp0e

3

+cse2 · ped

p0eSv2 −ck2 · pe

p0eD2Sck2

p0e

−csk2 · pedpe

p0e3 D46d3pe =

1

Vv1v2E c

p0e

]fe0

]pe· 5 e18fe28 · es8g − e28fe18 · es8g

SV −cq · pe

p0eD2Sv2 −

ck2 · pe

p0eD

+e18fe28 · es8g − es8fe18 · e28g

SV −cq · pe

p0eDSv2 −

ck2 · pe

p0eD26d3pe −

1

Vv1v2E c

p0e3

]fe0

]pe· 5e18fpe · e28gfpe · es8g − e28fpe · e18gfpe · es8g

SV −cq · pe

p0eD2Sv2 −

ck2 · pe

p0eD

+e18fpe · e28gfpe · es8g − es8fpe · e18gfpe · e28g

SV −cq · pe

p0eDSv2 −

ck2 · pe

p0eD2 6d3pesA3d. sA3d

After tedious calculation, we can further deduce Eq.sA2d from Eq. sA3d as

053104-11 Quasistatic magnetic and electric fields… Phys. Plasmas 12, 053104 ~2005!

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Eq. sA2d =1

V2v12v2

2 E 5 ]

]pe·Fce1 ·pe

p0ese2es − ese2d ·

]

]peG

+]

]pe3 1

SV −cq ·pe

p0eDSv2 −

ck2 ·pe

p0eD

c3se1 ·pedse2 ·pedses ·pedp0e

3 sk2q − qk2d ·]

]pe4+

]

]pe·3 V −

cq ·pe

p0e

v2 −ck2 ·pe

p0e

ce2 ·pe

p0e3 sese1 − e1esd ·

]

]pe4 +]

]pe·3v2 −

ck2 ·pe

p0e

V −cq ·pe

p0e

ce2 ·pe

p0e3 se1e2 − e2e1d ·

]

]pe4+

]

]pe·3 1

V −cq ·pe

p0e

c2ses ·pedse2 ·pedp0e

2 se2k1 − k1e2d ·]

]pe4 +]

]pe·3 1

V −cq ·pe

p0e

c2ses ·pedse2 ·pedp0e

2 se1k2

− k2e1d ·]

]pe4 +]

]pe3 1

v2 −ck2 ·pe

p0e

c2se1 ·pedse2 ·pedp0e

2 sesk1 − k1esd ·]

]pe4+

]

]pe3 1

v2 −ck2 ·pe

p0e

c2ses ·pedse2 ·pedp0e

2 sesq − qesd ·]

]pe46d3pe = 0. sA4d

Then Eq.s11d has been proved. It should be noted that theisotropy of distribution functionfe0 has been used in theabove proof.

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