Modeling of two-dimensional effects in hot spot relaxationin laser-produced plasmas

J.-L. Feugeas,1,a� Ph. Nicolaï,1 X. Ribeyre,1 G. Schurtz,1 V. Tikhonchuk,1 and M. Grech1,2

1Centre Lasers Intenses et Applications, Université Bordeaux 1–CNRS–CEA,33405 Talence Cedex, France2Max-Planck-Institut für Physik Komplexer Systeme, D-01187 Dresden, Germany

�Received 11 December 2007; accepted 14 April 2008; published online 5 June 2008�

Two-dimensional numerical simulations of plasma heating and temperature hot spots relaxation arepresented in the domain where the diffusive approximation for heat transport fails. Under relevantconditions for laser plasma interactions, the effects of the nonlocality of heat transport on the plasmaresponse are studied comparing the Spitzer–Härm model with several frequently used nonlocalmodels. The importance of using a high-order numerical scheme to correctly model nonlocal effectsis discussed. A significant increase of the temperature relaxation time due to nonlocal heat transportis observed, accompanied by enhanced density perturbations. Applications to plasma-inducedsmoothing of laser beams are considered. © 2008 American Institute of Physics.�DOI: 10.1063/1.2919791�

I. INTRODUCTION

Controlling the propagation of high-power laser beamsthrough large-scale underdense plasmas is a crucial point inthe context of inertial confinement fusion �ICF�. Controllingthe growth rate and saturation level of parametric instabili-ties such as filamentation1 and stimulated scattering2–4 hasbeen a challenging issue for several tens of years. In theindirect drive approach,5 the growth of plasma instabilitiesmay induce beam spreading, deflection, and energy lossesleading to a significantly deteriorated target performance. Inthe direct drive approach,6,7 inhomogeneities in the laserbeam intensity distribution may create pressure perturbationson the target surface and induce hydrodynamic instabilities.This will decrease the fuel compression8 and reduce the en-ergy gain.9–11

In order to control the laser energy distribution inplasma, the optical techniques of laser beam smoothing areimplemented. They consist in breaking the temporal12 andspatial13 coherence of the beam and creating a small-scaledynamic interference pattern. The resulting intensity distri-bution presents many nonstationary local maxima. These so-called speckles are randomly distributed in space and timebut have well-known, reproducible average properties.14,15

In the past two decades, it has been shown16,17 that thepropagation of such smoothed beams through the plasmamay lead to an enhanced laser incoherence. This phenom-enon, known as plasma-induced smoothing, follows from thestimulated18,19 and spontaneous20 scattering on self-induceddensity perturbations. Understanding of such phenomena re-quires correct modeling of the laser plasma coupling. Theinverse bremsstrahlung absorption is one of the mechanismsresponsible for this coupling. It may modify the temperaturedistribution and strongly enhance the hydrodynamic plasmaresponse.

For parameters of interest, the speckles have a width of

the order of electron-ion mean free path so that the electronheat transport is nonlocal.21–26 Previous studies have shown astrong effect of the electron transport model on the plasmaresponse.27,28 The nonlocality of heat transport modifies therelaxation of the hot spot temperature and enhances theplasma density perturbation as observed in Ref. 29. How-ever, these studies were based on the perturbation approachor on one-dimensional �1D� kinetic codes. They were re-stricted to relatively short time periods and small volumes.

In this paper, the plasma response to a localized laserheating is analyzed using a two-dimensional �2D� hydrody-namic code CHIC

30 with a 2D nonlocal transport modulebased on the Schurtz–Nicolaï–Busquet �SNB� nonlocalmodel.26,31

Although based on the kinetic equation, this model usesvarious assumptions in order to reduce the computationalcost and to allow its implementation in 2D code. It assumesa small departure of the distribution function from the Max-wellian function, and the high velocity approximation is usedto simplify the collision operator.26,31 These restrictive con-ditions reduce the limits of validity of the model, which nev-ertheless correctly describes the heat transport in the ICFcontext. This model is fully 2D and compatible with hydro-dynamic modules. It allows us to consider time intervals atthe nanosecond level and large plasma dimensions, whichcould exceed several millimeters. The SNB model has beencompared with the direct solution of the Fokker–Planckequation and a good agreement was found.26

The paper considers a sequence of test problems withincreasing difficulty. All along, each simulation is performedwith the classical Spitzer–Härm �SH� model,32 with or with-out limitation �model regularly used in large hydrodynamiccodes� and with the SNB model. Section II is devoted to ashort presentation of this SNB electron heat flux model. InSec. III, we start from the case discussed in Ref. 29, studyingthe relaxation time of a single 1D temperature hot spot. Re-sults obtained using the SNB model are compared with thea�Electronic mail: feugeas@celia.u-bordeaux1.fr.

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classical SH model, the flux limited SH model, and the non-local Brantov model.33 This latter is based on another ap-proach �solution to a linearized Fokker–Planck equation�yielding a significantly different domain of validity. A goodagreement is found under the conditions in which both mod-els are valid. Utilization of nonlocal models in a 2D geom-etry is a complicated problem, because numerical methods ofinterpolation of heat transport may introduce modificationsin the value and direction of the heat flux.

In Sec. IV, the numerical schemes of discretization arediscussed and the relaxation of a 2D cylindrical hot spot isconsidered. It is shown that using inappropriate numericalschemes may strongly modify the physical results and evencancel the expected effects. In Sec. V, more realistic 2D hotspot configurations are studied. The cases of a nonround hotspot and two neighboring hot spots are considered. It is dem-onstrated that the SNB heat flux model correctly captures thekinetic effects. The laser-plasma coupling due to the inversebremsstrahlung absorption is studied in 2D geometry in Sec.VI in the case of a spatially and temporally smoothed laserbeam. The SNB effects on the plasma response and beampropagation under conditions of interest for ICF are dis-cussed. Section VII presents the concluding remarks.

II. ELECTRON NONLOCAL HEAT FLUX MODEL

The temperature relaxation rate of a hot spot depends onthe heat flux, which in the classical strongly collisional casereads Q=−�SH�Te, where Te is the electron temperature and�SH= �128 /3�� nevT�ei��Z� is the SH heat conductivity.32

The electron density is ne, vT=�Te /me is the thermal veloc-ity, and Z is the ion charge number. The function ��Z�= �0.24+Z� / �2.4+Z� accounts for the electron-electron colli-sions. The electron-ion mean free path reads �ei

=3Te2 / �4�2�Znee

4��, where � is the Coulomb logarithmand e is the electron charge. The classical formula for Q failsif the characteristic scale length of temperature variation be-comes comparable to one-hundredth of the electron-ionmean free path. Luciani et al.22 proposed a convolution for-mulation to describe the electron nonlocal effects. Theynoted that the energy flux depends not only on the localthermodynamic conditions, but also on the conditions withinthe interval of a few hundred �ei from which the electronsmay come. The heat flux is therefore determined by the tem-perature profile enclosed in this interval and it is, in thissense, nonlocal. The heat flux was mathematically expressedas a 1D integral of the local SH flux multiplied by a kernelW�x ,x�� such as the heat flux at x is given by Q�x�=�QSH�x��W�x ,x��dx�. Several 1D kernels have been de-rived from the kinetic Fokker–Planck �FP�equation.21,22,25,33–35 In particular, for small temperaturevariations the kernel W depends only on the relative positionx−x�. Consequently, the nonlocal heat flux can be presented

in Fourier space in the form Q̂k=−ik�̂�k�Tk with the heatconductivity that depends on the wave number of tempera-ture perturbation. A simple interpolation formula for the non-local heat conductivity has been derived in Ref. 22, �̂�k�= �̂SH / �1+ �10�Zk�ei�0.9�.

The comparisons with kinetic models show that the in-tegral form of Q provides good results in 1D geometry evenif the temperature perturbations are not too small. However,its computational cost becomes unaffordable in 2D or 3Dgeometry. Moreover, the integral formula for Q in Ref. 22does not account appropriately for 2D effects where the heatflux vector is not necessarily parallel to the temperature gra-dient. Another nonlocal transport model especially adaptedto the two-dimensional geometry has been developed inRefs. 26 and 31. It is based on a multigroup approach andaccounts for the dependence of the electron-ion mean freepath on the electron energy. For numerical realization, thismodel is interpreted as a first-order angular momentum ofthe integral solution of a linear steady-state transport equa-tion with an exponential kernel. This formulation allows oneto use very efficient standard numerical 2D methods. Thenonlocal convolution kernel is considered as a Green func-tion of a simple transport operator. This latter satisfies a dif-fusion equation well adapted to implementation in hydrody-namic codes. In particular, this model has been implementedin the 2D Lagrangian radiation hydrodynamic code CHIC.30

III. SINGLE CYLINDRICAL HOT SPOT

A. Temperature relaxation

Senecha et al.29 have considered a sufficiently simpleand well-posed problem of temperature relaxation in a singlehot spot. It describes the physics clearly and allows us tocompare the SNB with another nonlocal model in the spe-cific hot spot configuration. First, following Ref. 29 we con-sider a given initial temperature profile. A cylindrical hotspot is supposed to have a Gaussian profile of the electrontemperature in the radial direction,

T�r,t = 0� = Te1 exp�− r2/R2� + Te0, �1�

where the background temperature Te0=0.7 keV, Te1

=0.12 Te0, and R=6.6 �m is the hot spot radius. In this pa-per, such a temperature perturbation is called a “hot spot” incontrast to a “speckle,” which is a laser intensity local maxi-mum �see Sec. VI�. The ion charge number is fixed, Z=5,and the electron density ne0=1021 cm−3. Under these condi-tions, the electron-ion mean free path is about 2.2 �m andthe hot spot radius R=3�ei. Two other characteristic valuesare the electron-ion collision time �ei=�ei /vT=0.2 ps and thecollision frequency �ei=�ei

−1.The 1D cylindrical domain of computation is 10�ei

10�ei with 100 cells. The time step is about 10 fs. Theboundary conditions are periodic. Second-order schemes areused for the hydrodynamic and the conduction models.

The radial distribution of temperature obtained using theSH flux and the SNB model are shown in Fig. 1�a� at t=�ei.This figure has to be compared to Fig. 1 of Ref. 29, where asimple analytical formula derived in the linear theory33 wasused. That gives an accurate result for a constant density andk�ei1. In order to reproduce results29 with the code CHIC,we first consider the temperature relaxation of a hot spot in aplasma treated as a perfect gas with a fixed ions.

A good agreement with Ref. 29 can be seen by compar-ing the time of temperature relaxation. The temperature is

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reduced by a factor 2, using the SH model, after about �0

=3ne0R2 /8�0=0.08 ps. In the nonlocal case, Ref. 29 found atime around 0.9 ps while our simulation gives 1 ps. It isinteresting to test whether a SH flux limitation could repro-duce the nonlocal result. Figure 1�b� presents temperaturedecrease versus time using the SNB model and the SH flux-limited model. The simulation with a flux limiter gives asatisfactory agreement with the SNB one.

The radial temperature profile is also shown in Fig. 1�a�for the simulation with the SH flux-limited model. Such anempirical model can partly reproduce the nonlocal behavior.

After one collision time t=�ei, a flux limiter f =7% re-produces the temperature maximum but the spatial shape isdifferent. The peak has a top hat profile. At t=3�ei, a limita-tion close to 10% agrees better with the SNB result. In con-clusion, flux limitation is a method that provides crude re-sults and is required to be time- and space-dependent, whichis never the case.

B. Plasma density response

The nonlinear plasma hydrodynamic response to a giventemperature perturbation was analyzed with the code CHIC todemonstrate the validity of a simplified equation for theplasma motion in Ref. 29. Figure 2 shows the temporal evo-lution of the spatial profile of the normalized density pertur-bation. After t=150�ei�30 ps, the maximum of densitydepletion in the hot spot center is about 0.55% in the SNBcase, whereas the depletion is much smaller in the SH case�Fig. 2�b��.

Figure 2�a� is in good agreement with Fig. 4 of Ref. 29.A deeper density cavity is predicted in the SNB case.

The temporal evolution of the density profiles in Figs.2�c� and 2�d� demonstrates that the use of flux limiters in theSH model allows us to obtain an appropriate density deple-tion but it modifies the profile shape. After 75�ei, the densityprofile obtained with the flux limitation of 10% �Fig. 2�c�� is

very different from the one observed in the SNB simulation�Fig. 2�a��. This can be explained by the fact that the plasmaresponse to a thermal input is determined by two effects.First, the temperature gradients initiate in a short time a den-sity modulation in the initially unperturbed plasma. Second,these density and temperature perturbations create a pressuregradient of density having an opposite effect.

The flux-limited SH model does not deplete the densityin the center of the hot spot, because the gradient of tempera-ture and consequently of pressure is null. This creates a strik-ingly different density profile. The density minimum in theflux-limited SH model is initially at the periphery of the hotspot. It arrives at the center only after 100 collision times, along time after the hot spot relaxation. The amplitude ofdensity perturbation depends on how long the temperaturepeak exists. The characteristic time of the SH diffusion�0.08 ps� is smaller than the electron-ion collision time�0.2 ps� and than the SNB one �1 ps�. Consequently, the den-sity perturbation in the nonlocal case is much larger. Thiswill be demonstrated in the last part of the paper, where theplasma response to realistic partially incoherent laser beamsis considered.

FIG. 1. �a� Radial distribution of the electron temperature ��= �T−Te0� /Te1� for a hot spot in a plasma with Z=5 and ne0=1021 cm−3 att=�ei. �b� Temperature maximum as a function of time. Solid lines, SNBmodel; dotted lines, the SH heat conductivity; dashed and dot-dashed lines,the SH model with 7 and 10% limitation, respectively. The simulation pa-rameters are given in the text.

FIG. 2. Radial distribution of the ion density at time �eit=0, 25, 50, 75, 100,125, 150. �a� is the SNB model; �c� is the SH model with a 10% flux limiter.Panels �b� and �d� show the evolution of the ion density at the hot spotcenter for cases �a� and �c�, respectively.

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IV. NUMERICAL EFFECTS ON HEAT CONDUCTIONMODELING

The discretization method and structure of the numericalgrid may affect the physical results especially in 2D geom-etry. To check the effects of numerical resolution, we con-sider the previous problem as a test case. As shown in Fig. 3,the 2D SNB or unlimited-SH model agree with the 1D re-sults on a regular mesh.

A. Discretization of the limited heat flux

An inappropriate discretization of the limited SH fluxcould lead to an anisotropic and thus unphysical hot spotrelaxation. Panels �b� and �c� in Fig. 3 shows the electrontemperature distribution obtained with different numericalschemes using the limited SH model. The conditions are thesame as in the case shown in Fig. 1.

The 2D domain of computation is 10�ei10�ei lengthmeshed with 100100 cells. The time step is about 10 fs.The boundary conditions are periodic. Second-order schemesare used for the hydrodynamic and the conduction models.

As is shown in the previous section, a SH limited fluxcould produce, at a given time, a correct temperature maxi-mum, even if the spatial shape is not correct. In the 2D case,the SH limited flux gives a square-shape relaxation, whichfollows the mesh �Fig. 3�c��. This effect is due to an inap-propriate discretization method of the flux.

Usually, in a two-dimensional code, the fluxes are com-puted at the faces of the mesh. In some cases, as in theproblem considered above, this induces preferential direc-tions and breaks down the cylindrical geometry of the prob-

lem. One way to avoid this deformation is to compute theflux and the limitation coefficient at cell centers. The cell-centered flux method allows, as shown in Fig. 3�b�, to pre-serve the symmetry.

B. Semi-implicit flux approximation

The SH heat flux is proportional to the gradient of tem-perature. An explicit time difference scheme induces a pro-hibitive short time step unaffordable for multinanosecondcalculations required for the ICF. Conversely, the amplifica-tion factor of an implicit time difference leads to a schemestable for any time step. The values of temperature at thespatial mesh are found from the solution of a sparse non-negative symmetric matrix in the SH case. An efficient andreliable numerical method and high-order schemes have beendeveloped for this case. Conversely, in the SNB case, thenumber of nonzero off-diagonal elements increases as theelectron-ion mean free path becomes larger. As it is not apriori known, the determination of each unknown element ofthe mesh depends on every other element of the mesh. Thesolution of the linear set of equations becomes much moreexpensive in terms of data storage and computing time. Acrude solution has been proposed in the past to partly cir-cumvent this problem.25 The idea is to force the diffusiveformulation calculating an effective thermal conductivity.36

This approach is in general ill-posed. It suffers from severedrawbacks, because an effective conductivity cannot be cal-culated in places where the temperature gradient is zero, norin places where the temperature gradient has the same sign asthe SNB heat flux. Therefore, the fast electron preheatingand counter streaming fluxes cannot be predicted by thisapproach.25,26

The SNB nonlocal model allows us to circumvent thisdifficulty because the stability criterion for an explicit timedifference scheme is much less restrictive compared to theSH case. The introduction of electric fields into the modelinduces a linear stability condition. This condition hasproven to be difficult to fulfill rigorously in practical multi-group calculations, where the temperature over- �or under-�shoots are sometimes observed. Solutions that lay some-where in between a full implicit or explicit resolution aresuggested by a SNB flux form that appears as the sum of alocal contribution and a correction for nonlocality,26

QSNB =2�me

3�

0

�

�f1m − �ei

v � � f0�v5dv = QLOC − QNL, �2�

where �eiv =v4me / �4�nee

4�� is the mean free path of theelectron. Here, f1

m is the anisotropic part of the distributionfunction in the local case and �� f0 is the nonisotropic partof the distribution function, which induces the SNB correc-tion on the heat flux. Equation �2� allows a different numeri-cal treatment for the “local” part of the flux �QLOC� and itsnonlocal correction �QNL�. The SNB correction is computedexplicitly from the temperature profile of the previous timestep, whereas the local component is implicitly time-differenced. This choice leads to the following finite differ-ence energy equation:

FIG. 3. Temperature contour lines ��= �T−Te0� /Te1� at t=�ei; �a� the SNBmodel; �b� the SH model with a 10% flux limiter; a cell-centered approxi-mation; �c� the same as �b� but with the edge approximation; �d� the SHmodel. The simulation parameters are given in the text.

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CvTn+1 − Tn

�t+ � · QLOC

n+1 = � · QNLn , �3�

where the upper subscript n stands for the time n� t. Thisimplicit-explicit scheme allows us to retain the main featuresof the nonlocal model, and the implicit part of it is efficientin damping eventual overshoots caused by the explicit calcu-lation.

C. High-order scheme module

In a realistic configuration, where the laser specklesswitch on and off, a hot spot sequence gives rise to a com-plicated density distribution consisting of many compres-sions and depressions. In a Lagrangian code, the mesh fol-lows the matter and then can be distorted.

This high-order spatial numerical scheme has beenimplemented for the heat transport description in the codeCHIC. First, the diffusion problem is solved for each energygroup.26 After that, the solution is used to modify the localSH flux and to build the SNB flux. This flux is then used tosolve the equation for the temperature. For each diffusionproblem in each group, the divergence operator is discretizedwith the same space scheme as the main diffusion operator.The divergence operators presented in the model are ex-pressed as matricial diffusion operators. Every diffusion op-erator is written as the divergence of a flux � ·Q. The equa-tion is resolved by a finite volumes scheme in each cell ofthe mesh. The application of the theorem of the divergenceinduces already the choice of a scheme using only the directneighbors. We do not apply the theorem of divergence to theoperator of diffusion. This opens the possibility of usingnaturally all the neighbors.

To analyze the effect of mesh on the temperature relax-ation, we rerun the same problem of a hot spot relaxation aswas presented above in Sec. III A, but on a strongly de-formed grid. The domain of computation is 10�ei10�ei

with 6060 cells and periodic boundary conditions. Asecond-order scheme is used for the hydrodynamics. Thisso-called Kershaw grid,37 shown in Fig. 4�a�, is currentlyused to test the diffusion solvers. Figure 4 shows the effect ofa five-point low-order scheme. The original spherical shapeis distorted and the mesh structure clearly influences the dif-fusion. The maximum temperature evolution with time is notreproduced, and the numerical diffusion exceeds the differ-ence between the SH and SNB models and shadows the ex-pected effects. This scheme, still used for its speed, cannot beapplied in our case. As is demonstrated in Fig. 4, the highKershaw scheme37 correctly simulates the temperature relax-ation and discriminates between the physical effects. Thisscheme retains an isotropic shape of the heat wave front. Inwhat follows, we use this high-order scheme.

V. REALISTIC HOT SPOT CONFIGURATIONS

In the ICF context, a realistic energy distribution in thelaser beam involves several speckles. Interference betweenthem produces intensity distributions with complicatedspeckle shapes. In what follows, several basic profiles areconsidered.

FIG. 4. Temperature profiles and contours �in keV� in a round hot spot att=�ei /5: �b� SH model and �c� SNB models. Transverse sections are pre-sented for the five points diffusion scheme. The dashed line demonstratesthe superposition of the high-order scheme and reference case on an unde-formed mesh shown in Fig. 3; �a� the Kershaw mesh.

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A. Elliptic hot spot

1. Temperature relaxation

Characteristic temperature profiles at the time t=�ei forthe case of an elliptic hot spot with the aspect ratio 1:2 areshown in Fig. 5. The domain of computation is 20 �m10 �m with 200100 cells. The time step is about 10 fs.The boundary conditions are periodic. Second-order schemesare used for the hydrodynamic and the conduction models.

As we have already noticed in the 1D case for the pa-rameter chosen, the SH model �Fig. 5�a�� gives a very fast

diffusion of the temperature: the initial form disappears after0.2 ps and the hot spot relaxes to a round shape. The SHmodel with a 10% flux limiter �Fig. 5�a�� gives a slowerdiffusion. A numerical 2D effect is observed, however. Thediscretization on cells edges induces a dependence on themesh �Fig. 5�b�� as it was noticed in the previous analysis�Fig. 4�. The flux-limited SH model gives, following the nu-merical scheme, a rectangular form of hot spot if attention isnot paid to the approximation. The approximation of the fluxlimitation in the center of the cells �Fig. 5�c�� effectivelyimproves the hot spot shape, but the radial profile shape re-mains different. The top hat profile already observed in the1D case indicates that the limiter value should be a functionof coordinate and of time. This analysis confirms the conclu-sions of 1D simulations. The SNB model preserves the pro-portion of the initial shape �Fig. 5�d��. It does not present anyspace or time singularity and it is a starting point of a hydro-dynamic process.

2. Hydrodynamic response

The hydrodynamic response to the thermal solicitation isvery dependent on the model as it was already observed inthe 1D case. The SH model diffuses the temperature so fastthat it does not produce any noticeable density perturbation�Fig. 6�a��, although it reproduces the initial elliptic shape ofthe temperature distribution. The response to the flux-limitedSH model depends strongly on the geometrical consider-ations and it is different along the two axes. The cell centeredapproximation decreases but does not completely removethis defect �Figs. 6�b� and 6�c��. In addition, the flattenedtemperature profile in the hot spot center does not create thedensity depletion. Figure 6�d� presents the response in theSNB model. The strong gradient along the y axis induces adepletion in the middle of the hot spot and generates twodivergent compression waves. This configuration inducescrossed gradients of density and temperature and it can gen-erate vortical flows and magnetic fields.38 However, theseeffects are not accounted for in the present simulations.

B. Double maximum hot spot

A hot spot with two maxima separated by a distance offive electron mean free paths formed by two neighboringspeckles is shown in Fig. 7 along with the SNB and SH heatfluxes.

The domain of computation is 100 �m50 �m meshedby 200100. The time step is about 10 fs. The boundaryconditions are periodic. Second-order schemes are used forthe hydrodynamic and the conduction models. Other param-eter are the same as those described in Sec. III A.

The SH flux follows the temperature gradient while thisis not the case for the SNB flux. It presents regions where theflux sign changes and does not follow the gradient of tem-perature. This behavior comes from the competition of twoopposite temperature gradients as a distance of a fewelectron-ion mean free paths.26 The SH fluxes are largeenough to eliminate two maxima and reduce the hot spot to around shape. The temperature profile quickly disappears,while it is preserved using the SNB model.

FIG. 5. Contour lines of temperature �keV� for an elliptic hot spot at t=�ei; �a� the SH heat conductivity; �b� the SH heat conductivity with a 10%limitation; the edge approximation; �c� is the same model as �b� but with acell-centered approximation; �d� the SNB model. The simulation parametersare given in the text.

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VI. PLASMA RESPONSE TO A SPECKLE PATTERN

Optical smoothing techniques used on high-power laserfacilities reduce the laser temporal and spatial coherence pro-viding a highly inhomogeneous and nonstationary laser in-tensity distribution. The speckle pattern has been simulatedfor the LIL conditions.16 The details of modeling of this ran-dom intensity distribution are discussed in the Appendix .

This section considers the effect of the laser beam speck-led structure on the density and the temperature distributions.The interaction parameters are chosen to be characteristic ofcoronal plasmas on current laser facilities.16 The plasma iscompletely ionized with the charge state Z=5; the electrondensity and temperature are ne0=3.31019 cm−3 and Te0

=500 eV, respectively.The laser parameters are those of the LMJ laser facility41

with the wavelength �0�0.351 �m, the speckle radius d�3.1 �m, the coherence time �c�2.2 ps, and the averageintensity 1015 W /cm2.

Under such conditions, the electron-ion collision time isestimated to be �ei�0.9 ps, the SH characteristic time oftemperature relaxation is about 0.4 ps, which is shorter than�c, whereas the SNB one is almost 5 ps, which is larger than�c. The hydrodynamic characteristic time is d /cs�25 ps.

The inverse bremsstrahlung absorption is considered asthe mechanism responsible for the coupling between the la-ser and the plasma. The ponderomotive force is not ac-counted for in the present simulation. The conservation ofmomentum equation allows us to evaluate the static plasmaresponse to the ponderomotive force,

�ne

ne

PF�

1

2

I

cncTe� 2 % .

The speckle size d is �3 times the electron mean free path;the heating effect is �10 times stronger than the effect of theponderomotive force. Under these conditions, the pondero-motive effect is weaker than the bremsstrahlung absorption.

The domain of computation is 60 �m60 �m meshedby 128128. The time step is about 10 fs. The boundary

FIG. 6. Density contour lines driven by the initial elliptic temperature dis-tribution ��ne−ne0� / �ne

max−ne0��; �a� the SH heat conductivity; �b� the SHmodel with a 10% limitation; the edge approximation scheme; �c� the sameas �b� but with a cell-centered approximation; �d� the SNB model. Corre-sponding temperature profiles are shown in Fig. 5. The simulation param-eters are given in the text.

FIG. 7. Distribution of the temperature �dashed line� and the heat fluxes forthe SH model �bold line� and the SNB model �thin line�. The simulationparameters are given in the text. The initial temperature is given in the inset.

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conditions are periodic. Second-order schemes are used forthe hydrodynamic and the conduction models.

Figures 8 and 9 present the electron temperature anddensity distributions obtained from CHIC simulations after 1and 20 ps, respectively. Panels �a� and �b� show results ob-tained with the SNB model, panels �c� and �d� are resultsfrom the SH model, and panels �e� and �f� are results fromthe SH model with a 7% limiter.

After one collision time �Fig. 8�, the SH model alreadydiffused the temperature �Fig. 8�c�� and the hydrodynamicresponse is weak �Fig. 8�d��. For the limited SH and SNBmodels, the temperature distribution �Figs. 8�a� and 8�e�� ismuch more inhomogeneous leading to a strong electron pres-sure and to enhanced density perturbations �Figs. 8�b� and8�f��.

After 20 collision times �Fig. 9�, the differences betweenthe SH model and the limited SH and SNB models becomeeven more apparent. For the SH model �Figs. 9�c� and 9�d��,the temperature has been completely smoothed and the den-sity perturbations are very small. Conversely, for the limitedSH and SNB models, the electron temperature and densitydistribution are even more inhomogeneous �Figs. 9�a� and9�b��. Whereas no difference between limited SH and SNBmodels can be seen after 1 ps �Fig. 8�, the density perturba-tions are stronger in the SNB model at 20 ps.

In order to better understand the differences between the

models, more quantitative diagnostics have been developed.Figure 10�a� presents the temporal evolution of the

plasma temperature spatially averaged over many speckles.As one can expect, all three models give almost the sameresults. The global plasma heating is not modified by asmall-scale heat transport.

Figures 10�b� and 10�c� show the temporal evolution ofthe normalized density fluctuation level �n�t�= ��ne�t ,r�−ne0�2 /ne0

2 1/2 and the average temperature pertur-bation level �Te�t�= ��Te�t ,r�−Te�t��2 1/2, respectively.

Figure 10�b� shows that all models lead to a strong in-crease of the temperature perturbation for the very first mo-ment of the interaction. The heat transport still has no time tomodify the temperature distribution and no difference ob-served between the models. Then the SH model, where theestimated relaxation time is about 0.4 ps, quickly diffuses theelectron temperature. After 5 ps, the temperature perturba-tions do not exceed a few eV. The density fluctuation levelstays to a very low level �less than 0.2%, as observed in Fig.10�b��.

On the contrary, for the limited SH and SNB simula-tions, the temperature perturbations remain at a higher level.In the limited SH case, temperature perturbations up to30 eV are observed after almost 10 ps. Then they decreasegradually in time to the level of the order of a few eV �asalready observed in Fig. 9�. The density perturbations arisein the place of temperature maxima in the first 20 ps, whenthe electron pressure is the highest. Then, these density per-turbations propagate freely in the plasma. The density fluc-

FIG. 8. �Color online� Temperature �left� and density �right� distributionsobtained in CHIC simulations with the SNB model �a�,�b�; the SH model�c�,�d�; and the limited SH model �e�,�f� at the time t=�ei. The laser andplasma parameters are given in the text.

FIG. 9. �Color online� Same as Fig. 8 but for t=20�ei.

062701-8 Feugeas et al. Phys. Plasmas 15, 062701 �2008�

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tuations stay almost at a constant level of the order of 5%.The SNB model exhibits a different temporal evolution

of the temperature perturbations after 10 ps. The temperatureperturbations are maintained at an almost constant level�25 eV during the simulation time. The plasma response tothis nonstationary temperature distribution occurs on thecharacteristic hydrodynamic time scale, d /cs�25 ps. Thedensity fluctuations are growing to a quite high level, up to12.5%. This is almost twice the level obtained in the limited

SH model and 25 times higher than the one obtained in theSH model.

VII. CONCLUSION

The processes of plasma heating and temperature relax-ation in a hot spot have been studied in a 2D geometry usinghydrodynamic simulations with nonlocal heat transport usingSH, limited SH, and SNB models.

Simulations of a simple hot spot in a cylindrical geom-etry demonstrated good agreement of the SNB model withanother one used in Ref. 29. The flux-limited SH model canreproduce the characteristic time of density perturbations,but the temperature profile is not correct and the hydrody-namic response is two to three times weaker for the set ofparameters used in present simulations.

An appropriate choice of the numerical discretizationscheme is important for the correct description of the heattransport with the SNB model. A high-order numericalscheme is required to account for the nonlocal effects.

Multispeckle simulations have been performed underconditions typical for present-day high-energy laser systems,where the inverse bremsstrahlung absorption is the dominantmechanism of laser plasma coupling. The ordering of thetemperature relaxation and the laser coherence times is ofgreat importance for the level of the plasma density fluctua-tions excited by the nonstationary speckle pattern. The non-local SNB model of heat transport predicts the density per-turbation at the level �10%. The laser beam propagationthrough it might then be strongly modified. It may experi-ence a multiple scattering and the forward stimulated Bril-louin scattering might be enhanced.20 The associated loss ofcoherence, due to the plasma-induced smoothing, may leadto angular spraying of the laser light. An accurate descriptionof the nonlocal heat transport is therefore of great importanceto correct the laser-plasma interaction in the coronal plasmaof ICF targets.

ACKNOWLEDGMENTS

The authors are grateful to the Aquitaine region for itsfinancial support. Fruitful discussions with P. Charrier, S.Weber, J. Breil, and P. H. Maire are acknowledged.

APPENDIX: NONSTATIONARY SPECKLEPATTERN GENERATION

Generation of a nonstationary speckle pattern of the laserbeam is accomplished in two steps.39 First, a random set ofspatial speckle distributions is created by using a spatial filterin the wave-vector space. This provides a spatially incoher-ent laser beam with a very large temporal spectrum �����t−1, where �t is the time step�. The second step consists inlimiting the temporal spectrum by use of a spectral filter inthe frequency space. The resulting speckle pattern has de-sired coherence time and length. This technique has been

implemented in the code CHIC30 and provides a speckle pat-

tern with a good accuracy for the current laser facilities.

FIG. 10. Temporal evolution of the plasma average temperature Te �keV��a�, the normalized density perturbation level �n �b�, and the temperaturefluctuation level �T �keV� �c� with the SH model �dotted line�, limited SHmodels �7% dashed line and 10% dot-dashed line�, and the SNB model�solid line�. The simulation parameters are given in the text.

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FIG. 11. �Color online� Formation of the speckle pattern: �a� generation of a static speckle pattern randomly distributed for each pixel. By using a spectralfilter �b� in the Fourier space, we generate a pattern corresponding to the speckle size �c�. In the time domain, generation of a new random speckle pattern ateach time step �e� and a spectral filter in frequency domain allow us to choose the lifetime of the speckle �f�. Panel �g� shows a spacetime speckle pattern.Panels �d� and �h� present the space and time correlation functions; the first zero gives the size d and the lifetime �c of the speckle, respectively.

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1. Static speckle pattern

A laser field E�x ,y� with a random real and imaginarypart is generated following a Gaussian statistics,40

E�x,y� = Randr�x,y� + i Randi�x,y� , �A1�

where Randr�x ,y� and Randi�x ,y� are randomly distributedreal numbers in the mesh in �x ,y� space. They follow a nor-mal low, i.e., �E�x ,y� =0 and �Randi =0 with�Re�E�x ,y��2 = �Im�E�x ,y��2 =�2. We define a specklewith a Gaussian statistics as �I =�2, where �I = �EE* is themean intensity. Then the speckle pattern contrast C=�2 / �I is 100%. At this stage, the beam correlation length is veryshort, defined by the mesh; see Fig. 11�a�.

To control the speckle width, we use a filter Fs�kx ,ky� inthe Fourier space. To obtain the same statistical properties asthe LIL beam,41 we use a door function ��kx ,ky�, which isequal to 1 for ��kx � , �ky � ���D / ��0f�; see Fig. 11�b� �D is thequad beam width, �0 is the laser wavelength, and f is thefocal length of the lens�. In the LIL conditions, one obtainskx ,ky �1.0 �m−1 �D=0.9 m, f =8 m, and �0=0.351 �m�.

Then using the inverse Fourier transform �Fig. 11�c��,one obtains the final speckle pattern,

Ef�x,y� = Fx,y−1�Fs�kx,ky�F�E�x,y��� , �A2�

where Fx,y and Fx,y−1 are the spatial Fourier and inverse Fou-

rier transforms, respectively.The randomized laser intensity distribution is character-

ized by the intensity correlation function,42

�s�x,y� = �I�X + x,Y + y�I�X,Y� . �A3�

The speckle radius is defined as the first zero of thecorrelation function �Fig. 11�d��. It is d�3.1 �m, which isconsistent with the theoretical value d=�0f /D.

2. Dynamic speckle pattern

The procedure described above generates one specklepattern. One can also consider a temporal evolution by intro-ducing the speckles lifetime. First we generate a randomlyindependent speckle pattern at each time step �Fig. 11�e��.The lifetime of a speckle is then equal to the time step. Thena spectral filter in the frequency domain �Fig 11�f�� is used,

E�t� = ei��t�,Ft���� = FE�t� . �A4�

For the case of a phase modulation ��t�=2� sin��mt�, � isthe modulation depth and �m is the modulation frequency.For the LIL beams, �=15 �at �0=0.351 �m� and �m

=14 GHz.An inverse Fourier transform is computed to obtain the

final spacetime speckle pattern, EF�x ,y , t�,

EF�x,y,t� = Ft−1�Ft���Ft�E�x,y,t��� , �A5�

where Ft and Ft−1 are the time Fourier and inverse Fourier

transform, respectively. Figure 11�g� shows the final space-time intensity. The temporal correlation function �t�t�= �I�T+ t� I�T� characterizes the speckle lifetime, i.e., the co-herence time �c �Fig. 11�h��. We define the �c as the positionof the first zero of the autocorrelation function �t�t�. For the

LIL case, �c=2.2 ps, which agrees with the theoretical value�c�1 / �2��m�. Moreover, to validate the calculations, weevaluated the instantaneous contrast to assure that it is near100%.

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