PHYSICS OF PLASMAS VOLUME 6, NUMBER 8 AUGUST 1999

Plasma fluctuations driven by a randomized laser beamA. V. Brantov, V. Yu. Bychenkov, and V. T. TikhonchukP. N. Lebedev Physics Institute, Russian Academy of Sciences, Moscow 117924, Russia

W. RozmusTheoretical Physics Institute, Department of Physics, University of Alberta, Edmonton T6G 2J1,Alberta, Canada

V. K. SenechaCenter for Advanced Technology, Laser Programme, Indore-452 013, India

~Received 11 December 1998; accepted 27 April 1999!

Nonthermal fluctuations produced by a randomized laser beam in an underdense plasma have beeninvestigated in the regime of a nonlocal electron transport. A nonlocal closure to linearized fluidequations for fluctuating hydrodynamic variables properly accounts for inverse bremsstrahlungheating and ponderomotive force effects. It has been shown that a typical randomized laser beam asused in the inertial confinement fusion experiments can generate large amplitude densityfluctuations well above thermal levels. An expression for the Thomson scattering cross-section fromthese fluctuations has been derived and discussed. The vortical velocity and magnetic fieldfluctuations can also be enhanced with the vortical plasma velocity reaching values comparable tothe ion acoustic velocity. ©1999 American Institute of Physics.@S1070-664X~99!02008-X#

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I. INTRODUCTION

An efficient coupling of laser energy to fusion targeand their uniform compression in direct and indirect driinertial confinement fusion~ICF! experiments are of vitaimportance to achieve ignition conditions at a minimuvalue of driver energy. In the past decade several new lbeam smoothing techniques,1 namely, random phase plate~RPP!,2 smoothing by spectral dispersion~SSD!,3 and in-duced spatial incoherence~ISI!,4 have been used for controling natural intensity nonuniformities in laser beams. Thetechniques introduce randomness into the laser beam thrspatial and/or temporal incoherence to produce a smoothser intensity distribution in a focal spot region. This in tuimproves efficiency of laser energy coupling to the plasand uniformity of fuel compression in ICF targets. In all sustudies, a tacit assumption is made that smoothing proceoccur on a time scale shorter compared to the hydrodynatime scales. As a result they are expected to reduceRayleigh–Taylor instability.5,6 This has been confirmed iboth experimental and computational studies.7 In addition,the use of smoothing techniques has been effective inpressing parametric instabilities such as the stimulated Blouin scattering~SBS!,8 stimulated Raman scattering~SRS!,9

and two plasmon decay instability.10 An exact mechanismwhich is responsible for the reduction of parametric instabties could be related to different processes, including dirchanges in the threshold of scattering instabilities due tofluctuating laser field, an increased level of density andlocity fluctuations, and an additional nonlinearity in laseplasma coupling. It has already been demonstrated btheoretically11 and experimentally12 how the randomized laser beam reduces SBS reflectivity and the laser speself-focusing.13

In the investigation of a single hot spot temperature

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laxation it has been shown that under inhibited electron thmal transport conditions the relaxation takes significanlonger time as compared to classical collisional case andaccompanied by ion acoustic wave excitation which can lto large-amplitude density perturbations.14 For the ICF plas-mas the effect of inverse bremsstrahlung~IB! heating as wellas the ponderomotive force~PF! effect due to intense lasefield are equally important. Therefore, it is essential tocount for both of these effects in studies of temperaturedensity fluctuations in localized hot spots. The IB heatiand PF effects are, in essence, kinetic processes involperturbations of the distribution function which take placespatial scales comparable with the particle mean free pPlasma evolution under these conditions can be describethe nonlocal electron–ion transport theory, which has bdeveloped in Refs. 15–17 and recently it has been geneized to include IB and PF processes in Ref. 18.

We start our discussion of plasma fluctuations by intducing a set of nonlocal transport equations following ttheory18 for small amplitude potential perturbations. Theequations are generalized for the vortical component ofelectron flow velocity. We consider the randomized lasertensity as a driving force for plasma perturbations insteadthe spontaneous plasma fluctuations due to the pardiscreteness.19,20 The statistical properties of a randomizelaser beam are described in terms of a correlation functioa fashion similar to Refs. 21–23 which is closely relatedthe ISI optically smoothed laser beam. The theory of drivhydrodynamical fluctuations based on the system of nonlotransport equations is similar to Refs. 20 and 24.

The paper has been organized as follows: in Sec. IIpresent the main physics of laser driven fluctuations inplasma. In Sec. III we discuss the correlation functionlaser intensity that defines spectral density of nonther

2 © 1999 American Institute of Physics

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3003Phys. Plasmas, Vol. 6, No. 8, August 1999 Plasma fluctuations driven by a randomized laser beam

plasma fluctuations. In the next sections the detailed theof laser driven fluctuations is presented. It includes effectselectron–electron and electron–ion collisions, IB heatiand PF effects in the entire range of plasma collisionalitySec. IV we present closure relations for plasma hydronamic equations, while Sec. V describes expressionselectron fluxes and transport coefficients defined by IB aPF laser–plasma interactions. A new set of closed hydronamic equations is also derived in Sec. V including the eqtion for vortical component of the plasma velocity and seconsistent magnetic field. In Sec. VI we derive couplicoefficients relating IB and PF source terms to plasma pturbations. Section VII is devoted to calculations of exprsions for density, temperature, magnetic field, and vortvelocity fluctuations of a plasma using the spectral corretion function of the laser intensity introduced in Sec. III.Sec. VIII we discuss the Thomson scattering of a probe labeam from laser driven density fluctuations. Finally, Sec.contains conclusions of our studies.

II. ENHANCED FLUCTUATIONS IN LASER DRIVENPLASMAS

The simple physical model of nonthermal fluctuatioproduced by a randomized laser beam assumes plaquasineutrality, neglects all particle collisions and plasheating, and introduces a source term defined by the pderomotive force}¹I , where I is the laser intensity. Themodel describes linear response of density perturbationsdnto the ponderomotive force

]2dn

]t2 12ga

]dn

]t2cs

2Ddn52cs

2

2

ne

ncTeDI , ~1!

whereTe is the electron temperature,cs is the ion acousticvelocity, nc5v0

2me/4pe2 is the critical density for a laselight with the frequencyv0, andga is the ion acoustic damping. The dampingga is introduced in Eq.~1! as a constancoefficient; however, in more realistic models it is an integoperator in space and time. The ponderomotive force termthe right hand side of Eq.~1! is given by the collisionlessplasma expression.25,26

Consider a laser beam which is characterized by amogeneous and stationary statistically averaged inten^I &5I 0 and exhibits small scale fluctuations~hot spots! char-acterized by the spectral density^I 2&v,k . In response to theinhomogeneous~randomized! part of laser radiation Eq.~1!predicts density fluctuations which are described by the sptral density

K dn2

ne2 L

v,k

5U 1/2

~v/kcs!212igav/kcs

221U2 ^I 2&v,k

nc2Te

2. ~2!

In order to estimate the maximum amplitude of density flutuations~2! we first introduce a result of Sec. III, mainly thmaximum of intensity fluctuations, max^I 2&v,k;I 0

2, corre-sponds to the value ofv of the order of the inverse correlation time and tok of the order of the inverse hot spot size. Othe other hand, the plasma response to the ISI laser bdescribed by the resonant term in Eq.~2!, produces maxi-mum density fluctuations atv;kcs . By including the ion

ryf,

n-

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r--l-

er

maan-

ln

o-ity

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-

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acoustic resonance we can estimate frequency integrdensity fluctuations asdne

2&1/2/ne;(kcs /ga)1/2(I 0 /ncTe),where we have assumed a laser pump bandwidth to be laor comparable with the ion acoustic frequency,kcs . For anion acoustic damping defined by the electron Landau daing the level of fluctuations is ^dne

2&1/2/ne

;(mi /me)1/4(I 0 /ncTe). This is a rather large value, particu

larly when the smoothed laser beam displays intensity molations, which are comparable to mean laser intensity. Ttime needed to reach this level of fluctuations is on the orof ion acoustic damping time,;1/ga .

Our simple discussion does not take into account partcollisions which could be important for small scale lasfluctuations. Indeed, the typical sizea0 of a hot spot in therandomized laser beam is a few laser wavelengths. Thfore, resonantly excited ion acoustic perturbations withk;a0

21 correspond toklei where lei is the electron–ionmean free path, which is a relatively small number charterizing a weakly collisional plasma. For example, one cestimate that for 0.35mm laser light in a plasma withZ510, ne50.1nc and Te52 keV this parameter is'5 fora0'1 mm. The theory15,18 of a nonlocal electron transporpredicts dramatic changes in plasma response forklei&10as compared to collisionless results.

In this article we will generalize the simple theory of E~2! to account for plasma heating and particle collisionaleffects. Particle collisions have two important effects onfluctuation levels: they change ion acoustic damping inexpression~2! and the coupling coefficient12 in the numeratorof Eq. ~2! which corresponds to the usual ponderomotinteraction. In collisional plasmas a ponderomotive forterm differs from the standard expression~cf. Refs. 25 and26!. This has already been demonstrated in Ref. 18. IB hing as well as nonlocal heat transport also change the cpling coefficient. However, the present theory demonstrathat the form of density fluctuations is the same as~2! if onereplaces the coupling coefficient12 by the new Ak term~which is derived below! and the correct expressions for thion acoustic damping and dispersion are used in the denonator. Together with the potential perturbations the vortiones can also be excited by a randomized laser beam duthe nonpotential component of the electromagnetic strtensor. An important characteristic of vortical perturbatiois their nonzero amplitudes due to the plasma responsintensity fluctuations in the regime of a nonlocal transpoThese perturbations vanish in collisionless and strongly clisional limits.

III. LASER INTENSITY CORRELATION FUNCTION

We assume a translational invariance of a laser light iplasma, i.e., the scale length of an average laser intenvariation is much longer than the correlation length. Alsthe characteristic time of an average intensity variationlonger than the relaxation time of plasma modes, which isthe order of an ion acoustic damping time. Under these cditions the laser intensity correlation function satisfiesfollowing relation:22

^I ~ t,r !I ~ t8,r 8!&5uCEE* ~ t2t8,r2r 8!u2, ~3!

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3004 Phys. Plasmas, Vol. 6, No. 8, August 1999 Brantov et al.

where CEE* is the laser electric field correlation functionThe latter depends on the particular way of the laser figeneration and the choice of a smoothing method. As a tcal example of practical interest we consider the Gauscorrelation function

^I ~ t,r !I ~ t8,r 8!&5I 0

2

11~z2z8!2/LR2

3expF2~r'2r'8 !2

a02~11~z2z8!2/LR

2 !

2~ t2z/vg2t81z8/vg!2

t02 G , ~4!

which describes reasonably well statistical properties of lahot spots produced by RPP.22 Heret0 is the laser correlationtime ~it could be as small as a few psec!, a0 is the minimumhot spot radius~typically it is a few laser wavelengths ilength!, LR5k0a0

2 is the Rayleigh length or the longitudinacorrelation length~it varies from tens to hundreds of laswavelengths!, andvg5k0c2/v0;c is the laser wave groupvelocity, wherek0 is the wave number of a laser light.

Performing a Fourier transform of the laser intensity crelation function~3! we obtain a spectral function:

^I ~v,k!I ~v8,k8!&5~2p!4d~v1v8!d~k1k8!^I 2&v,k ,~5!

where d-functions reflect the stationary and homogeneonature of laser intensity fluctuations, i.e., the average laintensity,I 0, changes slow in space and/or in time. The nmalization of the spectral correlation function is standard

E dv

2p

d3k

~2p!3^I 2&v,k5I 0

2 .

The final expression for the spectral correlation functionour example reads

^I 2&v,k5p

k'a0I 0

2VHSt0expF21

4v2t0

221

4k'

2 a02

2S kz2v

vgD 2 LR

2

k'2 a0

2G , ~6!

where VHS52pa02LR is the characteristic single hot sp

volume. In the linear theory of plasma response to randoized laser beams laser light statistical properties directlyfine characteristic scales of driven plasma fluctuations:vt0

&2, k'a0&2, andkz /k'&a0 /LR!1. The intensity distri-bution ~6! approximates most closely an ISI opticalsmoothed laser beam.

IV. BASIC EQUATIONS

The starting point for a general theory of laser drivplasma fluctuations is a set of linearized quahydrodynamical equations18 for spatial Fourier componentof fluctuations of the following physical quantities: electroand ion densitiesdne anddni , electron and ion temperaturedTe anddTi , and electron and ion fluid velocitiesue andui :

di-n

er

-

ser-

-e-

-

]

]tdne1 ik•ue ne50 ,

]

]tdni1 ik•ui ni50 ,

]

]tui5

Ze

miE2 ikvTi

2 S dni

ni1

dTi

TiD2

4

3

k2vTi2

n ih iui1

1

nimiRie ,

~7!]

]tdTe1

2

3neik•qe1

2

3ik•ue Te

52

3ncnei

T I k11

3nc

]

]tI k ,

]

]tdTi52

2

3ik•ui Ti .

Notations and definitions are standard: neiT

54A2pe2ei2niL/3Te

3/2Ame denotes the electron–ion collision frequency,n i54Apei

4niL/3Ti3/2Ami is the ion–ion col-

lision frequency,vTi(e)5ATi (e) /mi (e) is the ion ~electron!thermal velocity,I k(t) is the spatial Fourier component othe laser intensity, andRie is the ion–electron friction forceWe assume electrical quasi-neutrality,ne5Zni , and Max-wellian distribution functions for electrons and ions in thbackground plasma. The latter assumption requires a rtively small average laser intensityZI0 /ncTe!1.

The electron and ion fluid velocities are related throuthe electric current,j5ene(ui2ue), and the electron heaflux qe . Closure relations forj and qe are discussed in thenext section. Nonlocal ion transport processes are resented by the ion viscosityh i which has been obtained in 2moment Grad approximation.17 It is an operator in a timedomain and its Fourier component reads

h i5in i~v11.46in i !

~v11.20in i !~v11.46in i !10.23n i2

. ~8!

This expression provides a good approximation for the atrary ratio between the frequencyv and the ion–ion collisionfrequencyn i . We neglected in Eqs.~7! an ion thermal con-ductivity, ion collisionless damping and the electron–ion eergy exchange by considering relatively fast processesv@$kvTi ,nei

T me /mi% and highZ plasma,Z@1.The fluid Eqs.~7! are supplemented by Maxwell’s equa

tions

ik•E54pe~Zdni2dne!, ]B/]t52 ic@k3E#,~9!

ic@k3B#54p j ,

where the displacement current is neglected. The lattevalid for slow electron motions with characteristic velocitiwhich are much smaller as compared to the velocity of lig

V. HYDRODYNAMIC EQUATIONS AND CLOSURERELATIONS

To obtain closure relations for nonlocal hydrodynamone has to express electron fluxes in terms of ‘‘effectforces:’’ electric fieldE, thermal pressure forceik(Tedne

1nedTe), temperature gradientikdTe , plasma~ion! flowvelocity ui , and laser intensity gradientikI k . The following

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3005Phys. Plasmas, Vol. 6, No. 8, August 1999 Plasma fluctuations driven by a randomized laser beam

closure relations have been derived in Ref. 18 for the potial parts of electron fluxes, which are parallel to the wavectork,

j i5sEi* 1a ikdTe* 1b jeneui i1 ikj j

ene

ncTevTeleiI k ,

qei52aTeEi* 2x ikdTe* 2bqneTeui i1 ikjq

ne

ncvTeleiI k ,

~10!

Reii52~12b j !eneEi* 1bqneikdTe* 2b rmeneneiT ui i

2 ikj r

ne

ncI k

wherelei5vTeneiT is the electron–ion mean free path. He

we have introduced a renormalized electric field and renmalized electron temperature perturbations

Ei* 5Ei1 ikTe

e S dne

ne1

dTe*

Te1

I k

2ncTeD , dTe* 5dTe2

I k

3nc.

~11!

A similar set of relations holds for vortical parts of electrofluxes. We have defined them following a procedurescribed in Refs. 15, 16, and 18 for a plasma without the lafield

j'5s'E'1b j'eneui'1 ie'~k•e! j j'

ene

ncTevTeleiI k ,

qe'52a'TeE'2bq'neTeui'

1 ie'~k•e! jq'

ne

ncvTeleiI k , ~12!

Rei'52~12b j'!eneE'2b r'meneneiT ui'

2 ie'~k•e! j r'

ne

ncI k ,

wheree is the polarization vector of a laser beam andperpendicular component of an arbitrary vectorA is definedas A'5@k3A#3k/k2, while the parallel component isAi5k (A•k)/k2.

Nonlocal,k-dependent coefficientss, a, x, b j ,q,r , j j ,q,r

define transport relations for the electrostatic perturbatiand s' , a' , b ( j ,q,r )' correspond to nonpotential, electromagnetic perturbations. They have been introduced in R15, 16, and 18. These transport coefficients describe elecal current (s), thermoelectric current (a), heat transport(x), ion convective fluxes (b), and laser induced plasmresponse (j). Their properties, asymptotics, and dependeon klei and Z are presented in Refs. 15, 16, and 18.particular, Figs. 1–3 in Ref. 15 shows, a, x, b j ,q,r , Figs.8–10 in Ref. 16 displays' , a' , b ( j ,q,r )' , and Figs. 1–3 inRef. 18 showj j ,q,r .

The new set of nonlocal transport coefficientsj ( j ,q,r )'

which describe contributions from the laser intensity pertbations to vortical parts of the electron fluxes reads

n-e

r-

-er

e

s

fs.ri-

e

-

j j'5~2p!3/2vTe

3

nek2lei

2 E0

`dvv

h1121

h11S 62

v2

vTe2 D F0 ,

jq'5~2p!3/2vTe

3

2nek2lei

2 E0

`dvv

h1121

h11S 62

v2

vTe2 D

3S 52v2

vTe2 D F0 , ~13!

j r'56p2vTe

6

nek2lei

2 E0

`dvv4

h1121

h11S 62

v2

vTe2 D F0

whereF0(v) is the electron Maxwellian distribution andh11

is the dimensionless effective collision frequency introducin Ref. 16. Coefficientsj ( j ,q,r )' have the following proper-ties. They reach constant values in the long wavelen~strongly collisional! limit, klei!1:

j j'5264

45p, jq'5

352

45p, j r'5

1

15. ~14!

The coefficientj j' changes a sign atklei.0.2 and has localmaximum atklei;1. The coefficientj r' is the most slowlyvarying one. In contrast toj j' andjq' it is practically con-stant in the regionklei&1 having a local maximum.0.1 atklei.0.4. All coefficients decrease asklei increases. In theshort wavelength~collisionless! limit klei@1 they have thefollowing asymptotic behavior:j j'}jq'}(klei)

22ln(klei)andj r'}(klei)

25/4. The vortical laser coupling coefficientj ( j ,q,r )' normalized by their long wavelength limit magntudes~14! are plotted in Fig. 1.

Fluid equations~7! and nonlocal transport relations~10!and~12! with thek-dependent nonlocal transport coefficiens, a, x, b j ,q,r , j j ,q,r , s' , a' , and b ( j ,q,r )' are fullyequivalent to the linearized kinetic description in a Maxweian plasma over the wide range of variations of the plascollisionality parameter,klei . For a plasma without electromagnetic fields,16 I 50, our model describes plasma evoltion in the entire range of particle collisionality 0,klei

,`. In the presence of a laser field,IÞ0, our theory is alsovalid for the arbitrary collisionality, but only as far as thevolution of potential perturbations is concerned.18 Quasi-

FIG. 1. Dependence of the nonlocal coefficientsj'[j ( j ,q,r )' ~curves 1, 2,and 3, respectively! on the collisionality parameterklei . All coefficients arenormalized to their long wavelength limits~14!. The inset showsj j' in theregion of its small and negative values.

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e

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nspo-the

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sat-

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andupat

toandport

icy

it,

3006 Phys. Plasmas, Vol. 6, No. 8, August 1999 Brantov et al.

hydrodynamical equations are not applicable for vortiplasma motion in the collisionless regime because they hbeen derived without taking into account second order lasplasma coupling terms}v0

22. This approximation neglectsmagnetization current27 which is a reasonable simplification for j ( j ,q)'@nei

T /v0 corresponding to klei!(v0 /kvTe)ln(klei).

Equations~7!, ~9!, ~10!, and~12! constitute a closed system for a given laser intensityI k . We will further simplifythe model by assuming a zero net longitudinal currentj i50 (dne5dni[dn, uei5ui i[ui). Eliminating an electricfield Ei* from the potential part of hydrodynamic equatio~7! we can rewrite them in the following form:

]

]tdn1 ikuine50 ,

]

]tui52 ik~cs

21vTi2 !S dn

ne1

dTe*

TeD 1 ikcs

2bdTe*

Te

2 ikvTi2 dTi

Ti2S 4k2vTi

2

3n ih i1nei

T bu

cs2

vTe2 D ui

2 ikcs2S 1

21juD I k

ncTe,

~15!

]

]tdTe* 52

2k2k

3nedTe* 2

2

3ikuiTe~12b!

12

3ncnei

T ~11jk2lei2 !I k ,

]

]tdTi52

2

3ikuiTi .

Herecs5AZTe /mi is the ion acoustic velocity and the folowing definitions of the transport coefficients have beused:18 k5x2a2Te /s, b5bq2eab j /s, bu5b r

1(3p/32)b j2s0 /s, j5jq1eaj j /s, and ju5j r

1(3p/32)b jj js0 /s.Nonpotential parts of hydrodynamical equations d

scribe perturbations of an ion vorticity,W5¹3u, and aquasi-static magnetic field,B. Neglecting small terms relateto the electron–ion energy exchange and ion–ion collisiowe find from~7!, ~9!, and~12! the following set of equationsfor the vortical perturbations:

S114ps'

k2c2

]

]tDB524pene

k2cvTelei~k•e! @k3e# j j'

I k

ncTe,

~16!

]

]tW52b j'

Ze

mic

]

]tB2~k•e! @k3e# j r'cs

2 I k

ncTe. ~17!

In the long wavelength limitklei!1, the right hand side oEq. ~16! corresponds to the well-known collisional curredriven by a high-frequency electromagnetic field.28

lver–

n

-

s,

VI. PLASMA RESPONSE TO A RANDOMIZED LASERBEAM

The linear fluid theory of Sec. V describes fluctuatioof hydrodynamical variables induced by the random comnent of a laser intensity. Following standard steps ofgeneral theory of plasma fluctuations~cf., e.g., Ref. 19! weapply a time Fourier transform to Eqs.~15!–~17!, solve thelinear system of algebraic equations and express densitylocity, temperature, vorticity, and magnetic field perturbtions in terms of the Fourier component of a laser intensWe also assume that the heat transport is a fast processisfying condition,v!kk2/ne . One finds the following rela-tions for potential perturbations :

ui~v,k!5v

knedn~v,k! , dn~v,k!5DN~v,k!

neI v,k

ncTe,

dTi~v,k!52Ti

3nedn~v,k! , ~18!

dTe~v,k!5@neiT ~11jk2lei

2 !2 iv~12b!DN~v,k!#neI v,k

nck2k

.

The functionDN(v,k),

DN~v,k!5Ak

~v/kcs!212igav/k2cs

22~vs /cs!2 , ~19!

describes spectral properties of longitudinal plasma pertutions ~denominator! and the efficiency of their coupling tolaser intensity fluctuations in terms of the coefficientAk .The denominator of the functionD describes ion acousticresonance, where the ion acoustic wave damping,ga , isgiven by

ga52k2vTi

2

3n iReh i1nei

Tcs

2

2vTe2

bu1cs2ne

~12b!2

2k. ~20!

This expression accounts for both electron collisionlesselectron and ion collisional effects. The ion acoustic grovelocity, vs5Acs

21vTi2 G i , depends on the ion specific he

ratio G i55/31(4v/3n i) Im h i . Finally, the coefficient

Ak51

21ju1

nevTelei

k~12b!S 1

k2lei2

1j D ~21!

gives the strength of the coupling of density fluctuationsthe laser intensity perturbation. It depends on IB heatingPF effects which are described using our nonlocal transtheory. Figure 2 shows variations of coupling coefficientAk

as a function of collisionality parameterklei . The coeffi-cientAk decreases asklei increases and grows with the ioncharge,Z. Deviations from the classical collisional theor~dashed-dotted lines in Fig. 2! start in the long wavelengthregime where the coupling coefficientAk depends on IBheating. Nonlocal effects are more pronounced in highZplasmas~compare curves forZ58 and 64 in Fig. 2!. This istypical behavior for all nonlocal transport coefficients.15,16,18

The coupling coefficientAk displays the followingasymptotic behavior. In the long wavelength lim

on

m

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th

l-fe

ll

ba

nsin

be

sma

c-ine.rst

lve

-mit

3007Phys. Plasmas, Vol. 6, No. 8, August 1999 Plasma fluctuations driven by a randomized laser beam

30Zk2lei2 ,1, where IB heating dominates, Ak

50.074/k2lei2 12.54Z. In the weakly collisional limit,klei

@Z21/2, the coupling coefficient reads

Ak'1210.88Z5/7~klei!

24/7. ~22!

For example, one can see thatAk'1 for klei;Z5/4. Forshorter wavelengths the IB contribution is small and the pderomotive term dominates, i.e.,Ak.1/2.

Relations for vortical spectral components follow froEqs.~16! and ~17!:

V5~k•e!@k3e#

k2

vpe2 vTe

2

neiT c2

I v,k

ncTeAB,k ,

AB,k5j j'

4p is'v/k2c221,

~23!

W5~k•e!@k3e#

k2

vpi2 vTe

2

neiT c2

I v,k

ncTeAW,k ,

AW,k5nei

T k2c2

v vpe2

j r'1 ib j'AB,k ,

wherevpe( i ) is the electron~ion! plasma frequency andV5eB/mec is the electron cyclotron frequency. Both the manetic field and the vortex flow are generated by the elecmagnetic stress tensor driven by laser radiation. The eciency of their generation strongly depends oncharacteristic frequency~laser pulse correlation time,t0

;v21) or more exactly on the parametervts

[(vTe /c)2(vpe /neiT )3(v/vpe) where ts is characteristic

electron skin time. Figure 3 shows plots ofuAB,ku anduAW,kuas functions of the collisionality parameter for different vaues ofvt andZ. Notice that efficiency of the generation omagnetic fluctuations achieves its maximum for the intermdiate collisionality,klei*1, while the efficiency of the vor-tical flow excitation increases as one moves into the cosionless limit.

VII. PLASMA CORRELATION FUNCTIONS

From the equations~18! and~23!, which relate velocity,density, temperature, magnetic field, and vorticity pertur

FIG. 2. Dependence of the coupling coefficientAk on the collisionalityparameterklei for Z58 and 64~solid lines!. Dashed-dotted lines correspond to the strongly collisional limit results.

-

--

fi-e

-

i-

-

tions to laser intensity, one finds the following expressiofor the spectral densities of plasma correlation functionsterms of laser intensity fluctuation spectral densities,^I 2&v,k~6!,

^u2&v,k5v2

k2 K dne2

ne2 L

v,k

,K dTi2

Te2 L

v,k

52

3 K dne2

ne2 L

v,k

,

K dne2

ne2 L

v,k

5uDN~v,k!u2 ^I 2&v,k

nc2Te

2, ~24!

K dTe2

Te2 L

v,k

5uneiT ~11jk2lei

2 !2 iv~12b!DN~v,k!u2

31

k2k2

^I 2&v,k

nc2Te

2.

The correlation functions for vortical components couldconveniently written as follows:

^V2&v,k,i j 5u~k•e!u2

k4@k3e# i@ke# j

3S vpe2 vTe

2

neiT c2 D 2

uAB,ku2 ^I 2&v,k

nc2Te

2,

~25!

^W2&v,k,i j 5u~k•e!u2

k4@k3e# i@ke# j

3S vpi2 vTe

2

neiT c2 D 2

uAW,ku2^I 2&v,k

nc2Te

2.

These expressions clearly show the dependence of placorrelation functions on the coupling coefficientsAk , AB,k ,andAW,k .

Due to properties of the laser intensity correlation funtion, ^I 2&v,k ~6!, the fluctuation spectra are very narrowthe z direction (kz!k') and have a simple Gaussian shapInvestigating general properties of the fluctuations we fiintegrate the spectra with respect tokz (*dkz/2p) and con-sider their properties in (k' ,v)-plane. Spectra of potentiaperturbations are azimuthally symmetric and they ha

FIG. 3. Dependence of the coupling coefficientsuAB,ku and uAW,ku on thecollisionality parameterklei . Solid and dashed lines correspond tovts

510 and 100, respectively. Gray lines indicate the strongly collisional liresults.

-

d

eroofh

-reityorol

onicm

ic

al-to

eec-

is

the

atef

er

serdedngdin-

tion

ion

serity

ruple

onron-a-

ic

les

3008 Phys. Plasmas, Vol. 6, No. 8, August 1999 Brantov et al.

maxima corresponding to ion acoustic resonances,v5va

5k'vs , provided the laser correlation time is short,t0va

!1. Taking the integral over the frequencies (*dv/2p) wefinally obtain the radialk-spectrum of the plasma fluctuations:

K dne2

ne2 L

k'

5Ak'

2 S I 0

ncTeD 2

p3/2a02 k'cst0

4na

3exp@2 14k'

2 ~a021vs

2t02!# , ~26!

where the functionAk'accounts for the nonlocal effects an

na5ga(k')/va is a dimensionless ion acoustic damping.The behavior of stationary density fluctuations~26! is

shown in Figs. 4 for different values of (a021vs

2t02)1/2/lei

andZ. As compared with the collisionless expression~2! theamplitude of fluctuations~26! is strongly enhanced due to ththermal effects. These involve collisions and include pcesses like IB heating, heat transport and redistributionkinetic energy between different groups of electrons. Tcoupling coefficientAk'

~21! is increased from the collisionless value1

2, which characterizes a purely pondermotivesponse. HereAk'

shows further increase as the collisionalparameterklei decreases and collisions start to play a mimportant role. In addition, for the intermediate range of clisionality, klei&1, and for the highZ plasma density fluc-tuations are about two orders of magnitude higher for a nlocal case as compared to predictions of the classtransport theory. However, for shorter wavelengths the aplitude of fluctuations declines sharply approaching classcollisionless values forklei;5 – 10. The value ofklei at

FIG. 4. Dependence of the induced density fluctuations^dne2/n2&k'

normal-

ized to (I 0a0 /ncTe)2t0nei

T on the radial collisionality parameterk'lei for aplasma withTi50.5Te , A/Z52. Panel~a!: Z564, (vs

2t021a0

2)/lei2 530

~curves 1! and (vs2t0

21a02)/lei

2 52.25 ~curves 2!. Panel ~b!: Z58, (vs2t0

2

1a02)/lei

2 50.5 ~curves 1! and (vs2t0

21a02)/lei

2 50.04 ~curves 2!. Dashedcurves correspond to the results in strongly collisional and collisionlimits.

-ae

-

e-

-al-

al

which the sharp fall starts is smaller for highZ values. In theintermediate collision range,klei;1, the stationary ampli-tude of density fluctuations stays approximately constantthough it may take a longer time for longer wavelengthattain this stationary level.

Equation~26! can be used to qualitatively describe thfeatures of laser-enhanced fluctuations. The intensity sptrum has a maximum at the characteristic scalek* ;2(a0

2

1cs2t0

2)21/2. For the parameters of practical interest thwave number corresponds to the collisional ionsk* cs&n i

and intermediate or weakly collisional electronsk* lei*1.This is exactly the parameter range where our theory ofnonlocal electron transport is in effect. BecauseAk'

is a slowvarying function in this parameter range, one can integr~26! overk' (*k'dk'/2p) and find the average amplitude oplasma density fluctuations:

K dne2

ne2 L 5

p

4naS I 0

ncTeD 2 a0

2cst0Ak'

2

~a021cs

2t02!3/2

, ~27!

where functions Ak*

and na are taken at k* .2(a02

1cs2t0

2)21/2.Typically a0;cst0 and Ak

*;1, then the amplitude of

density fluctuations ~27! can be estimated asdne /ne

;na21/2(I 0 /ncTe). If we consider the highest possible las

intensity for our theoryI 0 /ncTe;Z21 and the characteristicvalue of an ion acoustic damping coefficientna;(225)•1022, then one may expect a level of a few percent for lastimulated density perturbations. However, the time neeto reach this level is of the order of ion acoustic dampitime, ;1/ga , that is relatively a long time, about a hundrepicoseconds for typical plasma parameters. Fluctuationscrease linearly with time fort,1/ga . These laser driven ionacoustic waves propagate predominantly across the direcof laser light propagation.

The vortical fluctuations do not need so long a relaxattime. They are established in thet0 time scale and the flowsare also localized in the plane perpendicular to the labeam axis. Correspondingly, the magnetic field and vorticare directed along the laser beam axis and have a quadstructure in the azimuthal direction: ^V2&k'

5^V2&k'sin2 2w and^W2&k'

5^W2&k'sin2 2w. The structure

of the fluctuation spectra in the radial direction dependsthe relation between the correlation time and the electskin time,ts(k)54ps'(k)/k2c2. They can be obtained following the method applied before to longitudinal fluctutions:

^V2&k'5

p3/2j j'2 a0

2t0

8ts~k'! S I 0

ncTeD 2S vpe

2 vTe2

neiT c2 D 2

3F12ErfS t0

2ts~k'! D GexpS t02

4ts2~k'!

2a0

2k'2

4 D ,

~28!

where Erf(x) is the error function. The behavior of magnetfluctuations is shown in Fig. 5 for different values ofa0 /lei

andts /t0 wherets5vpe2 vTe

2 /c2(neiT )3;ts(lei

21).

s

tic

a

u

thins

ahed

ororl

e-ie

alafoom

tantseruc-

e

ies:sticres-

ser

sby

htthe

ofeenfer-

ron

la-isto

orobe

3009Phys. Plasmas, Vol. 6, No. 8, August 1999 Plasma fluctuations driven by a randomized laser beam

The total level~integrated overk') of the magnetic fluc-tuations forlei.a0 can be estimated for two characteriscasest0,ts and t0.ts . For broadband laser beams,t0

,ts , the amplitude of vortical perturbations is proportionto the correlation time:

V;neiT S vTet0

a0D I 0

ncTe.

For longer correlation times,t0.ts , the amplitude of mag-netic fluctuations does not depend ont0,

V;neiT S vpea0

c D 2 I 0

ncTe.

This expression can lead to values of the magnetic field sthat V*vTe /a0, i.e., an electron Larmor radius,vTe /V, issmaller than the characteristic size of the hot spot,a0. There-fore the magnetic field can have a siginifant effect ontransport in a plasma. The ion flow velocity increases learly with t0 due to the direct effect of acceleration of ionby electromagnetic pressure:

u'

cs;t0

cs

leiS a0

leiD 1/4 I 0

ncTe.

The latter estimate is valid if the correlation time is less thion collisional damping time. If this is not a case, then tsaturated level of plasma vortical velocity can be estimateone replacest0 by t i .

For the typical laser parameters witha0;3 mm andc/vpe;0.3mm, ts is about 10 psec which is comparablelarger than the laser correlation time. For relatively long crelation times,t0;lei /cs;100 psec, the velocity of vorticaflow might be comparable to the ion acoustic velocity.

VIII. THOMSON SCATTERING FROM ENHANCEDFLUCTUATIONS

Thomson scattering~TS! of a probe laser beam is a usful tool for diagnostic of plasma parameters and for studof basic physical processes in laser produced plasmas.29–32

An interpretation of TS experiments relies strongly onproper theoretical description of an electron density corretion function. In this paper we have developed a theoryenhanced plasma fluctuations which are driven by rand

FIG. 5. Dependence of the induced magnetic fluctuations^V2&k'normal-

ized by (I 0a0 /ncTe)2(vpe

2 vTe2 /nei

T c2)2 on the radial collisionalityk'lei forthe following parameters:t/t0510 ~solid lines!, ts /t05100 ~dashed lines!,anda0 /lei50.2 ~curves 1!, a0 /lei51 ~curves 2!.

l

ch

e-

n

if

-

s

-r-

ized laser beams. This enhancement is particularly imporin the regime of a nonlocal electron transport, where lainduced fluctuations can be much larger than thermal fltuations due to particle discreetness effects.20,24 A cross sec-tion for a TS of a probe laser light~with the wavelengthlp)from a plasma driven by a randomized laser beam~with thewavelengthl0) is determined by the Fourier transform of thelectron density correlation function~24!. A spectrum ofdensity fluctuations depends on three physical quantitspectrum of laser intensity perturbations, an ion acoudamping, and the coupling coefficient. The standard expsion for the TS differential cross section~per unit volume!reads~cf. Ref. 33!

ds5r 0

2

4p~11cos2u!^dne

2&Dv,qdo8 dv8, ~29!

where r 05e2/mec2 is the classical electron radius,q5kp8

2kp , Dv5vp82vp is a small frequency shift (Dv!v), uis the angle of scattering cosu5kp•kp8/kpkp8 and the frequen-cies and wave vectors marked by thep subscript with andwithout primes correspond to scattered and probing lalight, respectively.

Using density fluctuation spectrum~24! for an under-dense plasma (ne!nc) in the TS cross section~29! we ob-tain the following expression:

ds5r 02ne

2VHSS I 0

ncTeD 2S t0

4q'a0D

3q4cs

4Aq2~11cos2u!do8 dvp8

@~Dv!22~qcs!2#214ga

2~Dv!2

3expF2S Dvt0

2 D 2

2S a0q'

2 D 2

2S LR

a0q'D 2

3S Dv

c1qzD 2G , ~30!

where the following definitions have been used:q52(vp /c)sin(u/2), qz5q@cosa sin(u/2)2sina cosw3cos(u/2)#, q'5Aq22qz

2, a is the angle between vectork0 andkp , andw is the angle between the planes spannedvectorsk0 , kp andkp8 , kp .

In Fig. 6 we have plotted the spectrum of scattered ligfor a probing beam propagating in the same direction asmain, interaction laser beam (kpik0). This spectrum is com-pared to results of the electron collisionless theory~2!. Thiscomparison shows significant differences in the shapespectral lines and in the height of ion acoustic peaks betwresults of a nonlocal and collisionless theories. These difences become smaller for larger values ofkplei @compareFigs. 6~a! and 6~b!#. Note also that the amplitude of laseinduced fluctuations is well above the thermal fluctuatilevel.24

Equation ~30! predicts deviation from the azimuthasymmetry for the scattered light, if the probing light propgates obliquely (aÞ0) with respect to main laser beam. Thcharacteristic property of scattered light could be usedidentify laser driven fluctuations and it may be useful fplasma diagnostics. For example, for a Thomson pr

ca

ra

nsordanara

mn

meeW

ostefpll

orwsuc

e

edmam-of amlar-uc-

ofc-sticby

ngnsasercts.m-oflocaln-ing

e

tua-eirase-

3010 Phys. Plasmas, Vol. 6, No. 8, August 1999 Brantov et al.

propagating perpendicularly to the main laser beam the stering predominantly occurs forkp8'k0. This strong scatter-ing anisotropy is demonstrated in Fig. 7 which shows a dmatic decrease of the scattered light intensity whenwchanges from 90° to 80°.

IX. CONCLUSIONS

Enhanced ion acoustic fluctuations could modify traport processes, growth of paramertic instabilities and abstion of light in laser produced plasmas. We have discussethis article nonthermal fluctuations directly generated by rdomized ISI laser beams. The enhancement is particulstrong in the regime of weak collisionality and the nonloctransport. A plasma response to smoothed laser beamsbeen described in terms of linearized nonlocal hydrodynacal equations incorporating effects of laser IB heating athe ponderomotive force. Correlation functions of plasdensity, velocity, and other physical quantities have bdefined in terms of a spectral laser correlation function.have shown that longitudinal fluctuation spectra dependthree quantities: the spectrum of laser intensity, ion acoudamping, and plasma collisionality. In general thermalfects and collisions enhance the plasma response andduce higher level of fluctuations as compared to the cosionless part of the fluctuation spectrum (klei@1). Inaddition the correct application of the nonlocal transptheory in the regime of the intermediate collisionality shoup to two orders of magnitude higher values for density fltuations as compared to approximate predictions basedthe classical transport theory. We have also studied magn

FIG. 6. Frequency spectra of the Thomson scattering~dashed curves!ds/do8dvp8 normalized by (ner 0I 0 /ncTe)

2(VHSt0lei /a0) compared tospectra from the electron collisionless theory~solid curves! for kpik0 , u510°, l051.06mm and a gold plasma (Z564) with ne51020 cm23, Te

52Ti51 keV, t052 psec, anda053 mm. Panel~a! corresponds to theprobe wavelengthlp51.06mm and panel~b! corresponds tolp50.53mm.

t-

-

-p-in-lylhasi-danenic-ro-i-

t

-ontic

field and plasma vorticity fluctuations which are determinby the laser correlation function and the characteristic plasskin length. We have shown for the first time that randoized laser field can generate large amplitude fluctuationsquasistatic magnetic field. An identification of such randomagnetic fields may be possible by measurements of poization changes in a probing laser beam. Magnetic field fltuations at the levelV*vTe /a0 could result in the inhibitionof an electron heat transport from the localized regionhigh laser intensity and IB absorption. Vortical velocity flutuations with the amplitude comparable to the ion acouvelocity may be important for the scattering instabilitiesintroducing depolarization of the laser light.

A differential cross section for the Thomson scatteriof a probing laser light from nonthermal density fluctuatiois derived. This cross section is enhanced both due to lintensity fluctuations and nonlocal electron transport effeAn interesting feature of the scattering light is an axial asymetry in its angular distribution. A significant broadeningion acoustic resonances has been shown due to the nonelectron transport. The possibility of directly inferring ehanced density fluctuations from the Thomson scatterdata is demonstrated for realistic plasma parameters.

ACKNOWLEDGMENTS

Two of us, V.T.T. and W.R. would like to acknowledgimportant discussions with Denis Pesme, Stefan Hu¨ller andJason Myatt on the subject of enhanced ion-acoustic fluctions and their nonlinear evolution. We are aware about thstudies of the laser produced fluctuations. This work wpartly supported by the Russian Foundation for Basic R

FIG. 7. Frequency spectra of the Thomson scattering~dashed curves!ds/do8dvp8 normalized by (ner 0I 0 /ncTe)

2(VHSt0lei /a0) for kp'k0 , u510°, lp51.06mm and a gold plasma (Z564) with ne51020 cm23, Te

52Ti51 keV, t052 psec, anda053 mm. Dashed lines correspond tow590° and solid lines correspond tow580°.

dhid

a.

a,

. M

s-hy

et

hy

ne

lu

k,

v,

k,

s.

s,

-

s.

.

e-

J.ys.

. J.ev.

3011Phys. Plasmas, Vol. 6, No. 8, August 1999 Plasma fluctuations driven by a randomized laser beam

search~Grant No. 98-02-17385-a!, the Natural Sciences anEngineering Research Council of Canada, the INTAS witthe research program of the International Center for Funmental Physics in Moscow~Grant No. 96-0457!, and theDepartment of Science and Technology, New Delhi, Indi

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