plasma induced laser beam smoothing below the ﬁlamentation ...skupin/article/pop_13_093104.pdf ·...

PHYSICS OF PLASMAS 13, 093104 �2006�

D

Plasma induced laser beam smoothing below the filamentation thresholdM. Grecha�

Centre Lasers Intenses et Applications, UMR 5107, Université Bordeaux 1, 351 cours de la Libération,33405 Talence, France and Département de Physique Théorique et Appliquée, CEA/DAM-Ile-de-France,91680 Bruyéres le Chatel, France

V. T. TikhonchukCentre Lasers Intenses et Applications, UMR 5107, Université Bordeaux 1, 351 cours de la Libération,33405 Talence, France

G. RiazueloDépartement de Physique Théorique et Appliquée, CEA/DAM-Ile-de-France, 91680 Bruyères le Chatel,France

S. WeberCentre Lasers Intenses et Applications, UMR 5107, Université Bordeaux 1, 351 cours de la Libération,33405 Talence, France and Centre de Physique Théorique, UMR 7644, Ecole Polytechnique,91128 Palaiseau, France

�Received 20 April 2006; accepted 25 July 2006; published online 21 September 2006�

This paper deals with a statistical approach for description of the laser field interaction withunderdense plasmas and modification of the laser beam temporal coherence during its propagationthrough a plasma at power well below the filamentation threshold. The main properties of theplasma density perturbations driven by a randomized laser beam are derived from a stochastic waveequation. The laser spectral and angular broadening is shown to occur on a distance that dependsessentially on the ratio of the average power in a speckle to the critical power for filamentation. Thecoherence time of the transmitted light is reduced to the plasma acoustic time of response to thelaser. It is typically a few picoseconds. Dedicated diagnostics have been developed for theinteraction code PARAX in order to analyze the laser and plasma statistical properties. The effect ofthe plasma length on the transmitted light coherence is found to be in good agreement withtheoretical predictions. Forward stimulated Brillouin scattering is shown to play a key role in thelaser coherence loss in this low-intensity regime. The limitations of the analytical model arediscussed in terms of the deviation of the electric field distribution from the Gaussian statistics andcreation of density-electric field correlations. This regime of laser induced incoherence is especiallyinteresting in that the associated angular broadening is not as deleterious as observed for higherintensities. Moreover, beam smoothing can be achieved in low-density plasmas where energy lossesdue to absorption and backscattering are not too important. © 2006 American Institute of Physics.�DOI: 10.1063/1.2337791�

I. INTRODUCTION

A detailed understanding of the characteristics of laser-plasma interaction is essential for inertial confinementfusion1 �ICF�. The control of laser beam coherence is neces-sary to achieve a homogeneous ablation and a symmetriccompression of the target. For this, optical smoothing tech-niques have been developed in the past decades. They consistin breaking the spatial and temporal coherence of the laserpulse. Spatial smoothing is achieved by using random phaseplates2 �RPP� to redistribute the energy among a larger num-ber of small hot spots, the so-called speckles. The obtainedintensity distribution is highly inhomogeneous but has well-known, reproducible average properties. The temporal coher-ence also needs to be broken so that the time-integrated in-tensity distribution is smoothed. This is achieved by thephase modulation and dispersion of the laser beam by

a�
Electronic mail: [email protected]
1070-664X/2006/13�9�/093104/19/$23.00 13, 09310

ownloaded 07 Oct 2008 to 193.174.246.167. Redistribution subject to

smoothing by spectral dispersion.3 The efficiency of smooth-ing techniques in reducing parametric instabilities,4,5

self-focusing,4,6,7 and Rayleigh-Taylor hydrodynamicinstability8–11 has been proved. However, such techniquesare expensive and face severe technological constraints.

In the last few years, it has been shown that the interac-tion of a spatially incoherent laser beam with an underdenseplasma enhances spatio-temporal smoothing. This is an at-tractive possibility for an efficient control of the laser energydeposition. At sufficiently high intensities, the interplay be-tween filament instability and forward stimulated Brillouinscattering �FSBS� has been identified as the dominantmechanism for plasma induced smoothing. This has beenconfirmed in theory,12,14 numerical calculations13,14,16 andexperiments.15,17,18 However, this regime requires relativelyhigh laser intensities and is accompanied by undesirable ef-fects as enhanced backscattering and important angularspreading associated with the filamentation instability.

At lower intensities, below the critical power for fila-
mentation, the propagation through a very low density
© 2006 American Institute of Physics4-1

AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp

http://dx.doi.org/10.1063/1.2337791

http://dx.doi.org/10.1063/1.2337791

093104-2 Grech et al. Phys. Plasmas 13, 093104 �2006�

D

plasma, typically a few percent of the critical density, canalso induce smoothing.19,20 This regime could be interestingfor ICF applications since it is not associated with parasiteeffects of beam spreading and backscattering parametric in-stabilities. Moreover, the standard ICF laser pulse containstwo parts, a low-intensity plateau followed by a higher-intensity peak. Parameters used in this paper correspond tothe plateau regime. It has been suggested that the observedtemporal coherence loss is due to multiple scattering of thetransmitted light on self-induced density fluctuations. How-ever, so far this effect has not well been understood. Thiswork attempts a theoretical analysis of this phenomenonwithin a statistical approach and its numerical simulation us-ing the three-dimensional interaction code PARAX.21

Smoothing techniques introduce into the laser beam ran-dom spatial and/or temporal perturbations so that statisticalmethods are necessary to describe the electric field22,23 andthe plasma perturbations.26 The laser electric field coherenceproperties and their modification during the propagationthrough a plasma are described by the correlation functions.The plasma density fluctuations are driven by a randomizedlaser beam. They consist of static and propagating large scaleion acoustic waves. Their interaction with a laser beam isresponsible for the induced smoothing.

The paper is organized as follows. Section II introducesthe statistical framework to describe the electric field corre-lation function. The main properties of the electric fieldcorrelation function in vacuum are addressed. The equationthat governs the modification of this function through arandom medium is derived. The effect of given densityfluctuations is considered in terms of the modification ofthe laser coherence along the propagation direction. SectionIII considers the plasma density fluctuations driven by lasers.Using a simple wave model, the dynamics of plasma pertur-bations due to the ponderomotive force is studied analyti-cally in two transverse dimensions for regular and random-ized laser beams. Their spectra and level are derived.The effects on the density fluctuation level of electron-ioncollisions and non-local transport are addressed within thestatistical approach and compared to two-dimensional nu-merical simulations from PARAX. Section IV considersthe system of coupled equations for the electric field anddensity correlation functions. The angular and spectralbroadening of the transmitted light is derived within a per-turbation approach. The general properties of the transmittedlight are discussed. In Sec. V, three-dimensional simulationsof the laser smoothing below the filamentation threshold areperformed. Specific diagnostics have been used to character-ize the modification of the light coherence along the propa-gation direction. Numerical results are compared to analyti-cal results from the statistical model. The importance ofFSBS from an ensemble of speckle in the induced coherenceloss is shown and the main assumptions of the model arediscussed. Finally, Sec. VI contains the conclusions of the
present study.

II. STATISTICAL DESCRIPTION OF A RANDOMLASER BEAM

A. Field correlation function

This paper considers the laser electromagnetic field inthe envelope approximation assuming that the complex am-plitude E�r ,z , t� varies slowly over the laser wavelength andperiod. Moreover, it is assumed that E is a stochastic quantitythat can be characterized by its statistical average �E� and thecorrelation function22

�EE*�R,2�,T,2�,z�

= �E�R + �,T + �,z�E*�R − �,T − �,z�� , �1�

where R, z, and T are the macroscopic coordinates and timeand � and � describe the correlations in the plane perpen-dicular to the beam propagation axis z and in time.

The spatial correlations can be introduced in the laserbeam before it enters the plasma with the RPP technique.Temporal and spatial behaviors are then decoupled and�EE*�R ,� ,T ,� ,z�=CEE*�R ,��F�T ,��, where CEE* is the spa-tial correlation function and F describes the temporal evolu-tion. The purpose of this paper is to describe how temporalcorrelations are affected during the laser propagation throughthe plasma, assuming that spatial correlations are known atthe plasma entrance, z=0. These initial correlations are as-sumed to follow Gaussian statistics with zero mean value.

B. Propagation equation for the electric fieldcorrelation function

The propagation of a laser beam through a plasma withelectron density n0 is described by the scalar paraxial waveequation:

�2ik0�

�z+ ��

2 �E�r,t� =�p0

2

c2 �n�r,t�E�r,t� , �2�

where �n is the density perturbation, �p0= �e2n0 /me�0�1/2 isthe plasma frequency, k0= ��0 /c��1−n0 /nc�1/2 is the laserwave number, nc=�0�0

2me /e2 is the critical density, and c thelight velocity in vacuum. This paraxial description accountsfor small-angle refraction and laser beam scattering in thenear-forward direction.

It is widely used to describe the propagation of regularlaser beams.7,21 It also applies to smoothed beams but itsusage becomes more complicated. Indeed, solving this equa-tion for a randomized beam involves disparate scales in timeand space, and therefore requires powerful computationaltools. We will present the results of numerical simulationsusing the code PARAX later in this article. However, from atheoretical point of view, it is more efficient to deal withaverage quantities.

A propagation equation for the electric field correlationfunction �1� is derived from the paraxial equation in Appen-dix A.

For this, a diffusion approximation is introduced that as-sumes the electric field distribution taken in plane z1 to beindependent on density fluctuations taken in plane z2�z1. Itis justified if the longitudinal coherence length of density
fluctuations, LC, is much shorter than the distance �C �see

093104-3 Plasma induced laser beam smoothing¼ Phys. Plasmas 13, 093104 �2006�

D

Eq. �9� below� over which the nonlinearity is significant. Itshould be noted that this diffusion approximation, discussedin Ref. 27, is different from the one used in transport theory.It considers the multiple scattering of the laser light on den-sity fluctuations as a random process, leading to an equationfor the electric field correlation function:

��z −i

k0�R · ��EE* = − ��zln ��z��EE*, �3�

where

��z� =exp− ik0

2

n0

nc�

0

z ��n�R +�

2,T +

�

2,z��

− �n�R −�

2,T −

�

2,z�� dz�� . �4�

The diffusion approximation assumes that the laser beamcrosses many density fluctuations along its propagation, i.e.,z�LC, so that the integral �0

z�n�z�dz should follow a Gauss-ian distribution, according to the central limit theorem. Equa-tion �3� then takes the following form:

� �

�z−

i

k0�R · ��EE*

=�2

4LCk0

2�n0

nc�2

�DN��,T,�� − DN�0,T,0��EE*, �5�

where the plasma density correlation function DN is given as

DN�2�,T,2�� = ��n�R + �,T + �,z��n�R − �,T − �,z�� .

�6�

If the beam is wide enough so that the density fluctuationspropagating with the acoustic velocity cs have no time totraverse it �T2csL0� and assuming that they are homoge-neously distributed, one can neglect macroscopic effects.Then, DN depends only on the relative position �. The situ-ation is different for time. Since nonstationary processes arestudied, it is important to account for the variable T.

The same considerations can be applied to the equationfor the field-field correlation function �EE:

� �

�z−

i

2k0��R

2 + ��2��EE

=�2

4LCk0

2�n0

nc�2

�DN��,T,�� + DN�0,T,0��EE. �7�

For the RPP-created laser beams, �EE is negligible in com-parison with �EE*. Equations �5� and �7� show that this prop-erty is conserved during the beam propagation.

Let us assume that the density correlation function fol-lows the Gaussian distribution:

DN��,T,�,z� = �n02�z�exp�− �2/2�M

2 − �2/2�M2 � . �8�

A part of the laser light is scattered on these perturbations.Its coherence time is strongly reduced to the order of �M,while the coherence radius varies much less provided that�M ��0. The scattered light intensity increases a distance es-
timated from Eq. �5�:

�c−1 � k0

2LC�n0/nc�2�n02. �9�

This corresponds to the medium length needed for the laserbeam smoothing. For larger distances, the laser beam char-acteristic spatial correlation length �c�z� and the coherencetime �c�z� vary with z as follows:

�c�z� = �0�1 +z

�c

�02

�M2 �−1/2

, �c�z� = �M��c

z. �10�

The coherence radius of the light is slightly reduced alongthe propagation because of the multiple scattering, whichreduces the coherence time. Figure 1 presents the solution ofEq. �5� for the electric field spectrum as a function of thepropagation distance. The spectral broadening of the laserbeam is symmetric, which is a characteristic feature of the

FIG. 1. �a� Variation of the spectrum of the transmitted light with the propa-gation length z �dotted line z=0, dashed line z=1.2�c, and solid linez=3�c�. Inset: Fraction of the energy outside of the monochromatic peakversus the propagation length. �b� Corresponding temporal correlation func-tion for T�� dotted line z=0, dashed line z=1.2�c, and solid linez=3�c�.

multiple scattering process on independently created density



D

fluctuations. The effect of the induced smoothing can be de-scribed in terms of the increase of the fraction of scatteredlight with frequency outside of the monochromatic, incidentpeak; i.e., with �� 1/T:

FS =

��1/T

d��E��2

� d��E��2. �11�

The evolution of FS along the z axis can be approximated bythe curve presented in the inset of Fig. 1:

FS�z� � 0.82�1 − exp�− z/�c�� . �12�

It is suitable for z /�c2, and it will be compared with theresults of numerical simulations in Sec. V.

III. PLASMA DENSITY FLUCTUATIONS DRIVENBY A LASER

A. Enhanced fluctuations in laser driven plasmas

Plasma fluctuations responsible for laser multiple scat-tering could be created by the laser beam itself due to ther-mal effects and/or the ponderomotive force. Small-amplitudedensity perturbations can be described within a linear acous-tic wave model. Assuming plasma quasineutrality and ne-glecting plasma heating, the linear response in a transverseplane to a laser hot spot with the intensity I�r , t� is describedby the wave equation

��t2 + 2�d�t − vs

2��2 ��n�r,t� =

cs2

2ncTec��

2 AI�r,t� , �13�

where vs2=cs

2+3vTi2 , cs

2=ZTe /mi is the ion acoustic wave ve-locity, and vTi

2 =Ti /mi is the ion thermal velocity, which de-pends only on the ion temperature Ti and mass mi. �d is the

ion acoustic wave damping rate and A is an operator16,28 thataccounts for the thermal and collisional effects on the exci-tation of density fluctuations. It is important if the specklewidth is larger or comparable to the electron-ion mean freepath ei. Its Fourier spectrum is detailed in Appendix B.

In most situations of practical interest, Ti�Te, so thatvs�cs. The characteristic time for ion acoustic wave damp-ing is much larger that the ion acoustic time, and often, thanthe duration of the simulation. Moreover, the spatio-temporallaser beam smoothing appears as the consequence of the in-teraction of speckles with density fluctuations driven by theirneighbors. Therefore, it would be efficient provided the ionacoustic wave is not damped before it interacts with manyspeckles. For these reasons, the ion acoustic damping is ne-glected in what follows.

In general, the density perturbation created by the lasercontains several components. Quasistationary density deple-tions are embedded in the location of stationary intensitymaxima. They are created during the characteristic acoustictime td��0 /cs. Density bumps move away from the intensitypeaks with the velocity cs. Finally, in the case of a nonsta-tionary illumination, density depressions are released whenthe laser pressure is not strong enough to maintain the de-
pression. They also propagate with the velocity cs. The

propagating perturbations are the ion acoustic waves trans-ferring the perturbations across the laser beam and leading toits eventual temporal smoothing.

In the case of a spatially smoothed beam, the intensitydistribution consists of many randomly distributed spikes. Itcan be characterized by its correlation function CI��=�I�R+ 1

2��I�R− 12��. Two methods can be used to describe it: ei-

ther the density correlation function is deduced directly byaveraging the solutions of Eq. �13�, or a stochastic equationfor the correlation function DN is derived from Eq. �13�.

Let us consider the solution in two transverse dimen-sions �2TD� of Eq. �13� in the case of a ponderomotive cou-pling and assuming that the laser source is instantaneouslyturned on at T=0. By taking the average of the products ofthe solutions of Eq. �13�, one then finds the following ex-pression for the density correlation function

DN�k,T,�� =CI�k�T

2�ncTec�2G1 sinc�2�T�

+ �±

�G2 sinc�2�� ± csk�T�

+ G3 sinc��2� ± csk�T�� , �14�

where CI�k� is the Fourier transform of the intensity corre-lation function and sinc�x�=sin�x� /x. The first term in thecurly brackets reads

G1�T,k� = 1 + 12 cos�2cskT� . �15�

It results from two contributions: on one hand, the correla-tions between stationary depressions, and on the other hand,the correlations between bumps propagating in the same di-rection. The second term,

G2�T,k� = 14 , �16�

denotes the correlations between bumps and corresponds tothe acoustic free mode of the plasma ��= ±csk�. Its ampli-tude is a constant. The third term reads

G3�T,k� = − 2 cos�cskT� . �17�

It describes the correlations between propagating bumps andstationary depressions. The particularity of this componentis that its frequency �= ±csk /2 is half the ion-acousticfrequency.

This correlation function describes the density perturba-tions driven by a step-like laser. It can be generalized to apulse with a characteristic rise-time tm. A similar treatmentshows that the bump amplitude decreases as 1/ tm, whereasits width increases linearly with tm.

One can also obtain an evolution equation for the densitycorrelation function by taking the average of the product oftwo Eqs. �13�. In the case in which the damping is small, one
finds


D

� 1

16�T

4 +1

2��2 + cs

2k2��T2��2 − cs

2k2�2 DN =cs

4k4

�ncTec�2Ak2�I,

�18�

where �I is the spatio-temporal intensity correlation functionin the Fourier space. This approach allows one to consider anarbitrary spatially and/or temporally smoothed beam pro-vided its intensity correlation function is known. For a spa-tially smoothed beam, Eq. �14� is the solution to Eq. �18�.For a beam with a coherence time �c��0 /csT, the in-duced density correlation function reads

DN�T,k,�� = −��ccs

3k3Ak2CI�k�

8�ncTec�2�

�exp�−�c

2�2

4��

±�±�

sin2�� ± csk�T�� ± csk�2 .

�19�

For such a short correlation time, stationary depressions donot exist and one can see that the excited resonance�= ±csk /2 disappears.

Finally, as the macroscopic time T is much longer thanthe characteristic correlation time, Eq. �18� can be reducedto a second order in time equation by assuming��2−cs

2k2 � ��2:

��T2 +

��2 − cs2k2�2

�2 DN =cs

2k2

�ncTec�2Ak2�I. �20�

This equation can also be derived directly from Eq. �13� inthe envelope approximation.

B. Properties of density fluctuations

The density correlation function defined by Eq. �18� con-tains all the general properties of the density fluctuationsdriven by a randomized laser beam. It depends on three vari-ables: T, �, and k, and it is difficult to calculate it numeri-cally and to visualize it. The reduced correlation functionsare easier to manipulate and also contain important informa-tions. Their properties are discussed below.

1. Time-averaged k-spectra

By averaging Eq. �14� over time T and integrating over�, one finds the k-spectrum of RPP-driven fluctuations:

��n2�k�� =CI�k�

8�ncTec�2 �3 + sinc�2cskT� − 4 sinc�cskT�� ,

�21�

while, from Eq. �19�, one finds the k-spectrum of fluctuationsdriven by a spatially and temporally smoothed laser beam:

��n2�k�� =�cs

2k2�cT

16�ncTec�2 CI�k�exp�−cs

2�c2k2

4� . �22�

Figure 2�a� shows normalized k-spectra of density fluctua-tions driven by a RPP beam for time moments T=10�0 /cs

and 50�0 /cs. Figure 2�b� corresponds to the case of a tempo-


rally incoherent beam laser with the coherence time�c=0.1�0 /cs and �c=�0 /cs. In both panels, the normalizedGaussian intensity k-spectrum is shown. In the RPP case, themaximum wavelength is given by the laser spatial propertiesand the minimum wavelength depends on time. The spec-trum then spreads from k��csT�−1 to k��0

−1. For densityperturbations driven by a spatially and temporally incoherentbeam, the density fluctuations have a narrow k-spectrum withwidth of the order of �0

−1. The spectrum has a maximum forkM =2��0

2+cs2�c

2�−1/2 and no small-wavelength perturbationsare excited. A similar behavior has been observed by Brantovet al.26

2. Time-averaged frequency spectra

In the same way, integration over wave numbers k andaveraging over T allows one to access the �-spectra. For

FIG. 2. �a� Wave-number spectra of the density fluctuations driven by a RPPbeam for T=10�0 /cs �dashed curve� and T=50�0 /cs �solid curve�. Normal-ized to ��0I /ncTec�2. �b� Wave-number spectra of fluctuations driven by atemporally and spatially incoherent beam with �c=0.1�0 /cs �dashed line�,and �c=�0 /cs �solid curve�. Normalized to ��0I /ncTec�2�Tcs /�0�. The nor-malized intensity k-spectrum is also shown on both panels �dotted curves�.

RPP-driven fluctuations in 2TD, one finds



D

��n2�� =�I�

4�ncTec�2�3�� +�0

2

4cs2 ��exp�−

�02�2

4cs2 � .

�23�

For a spatially and temporally smoothed laser beam, one hasin the 2TD case:

��n2�� =3/2�cT�0

2�I�32�ncTec�2cs

2 ��3exp�− ��02

cs2 + �c

2�2�2

4 .

�24�

Figure 3 presents the corresponding spectra driven by a RPPand a temporally incoherent laser beam. In the RPP case, thespectrum of density fluctuations is very large �0��cs /�0� and it contains a static component related to thestationary density depressions. For a temporally incoherentbeam, the spectrum is much narrower with a maximum at

� 2 2 2 −1/2

FIG. 3. Frequency spectra of ponderomotively driven density fluctuationsfor a RPP beam �a� and a temporally smoothed laser beam �b� in 2TD. Thespectra are normalized to �0 /cs��I� /ncTec�2 and to ��cTcs /�0��I� /ncTec�2,respectively.

�M = 6��0 /cs +�c� .


3. Level of density fluctuations

The level of density fluctuations can be derived by inte-grating the density correlation function over k and �. For aRPP laser beam, in the 2TD ponderomotive case, the squareof the density perturbation level reads

��n2�T�� =�I�2csT

4�0�ncTec�2 �h�2csT/�0� − 2h�csT/�0�� , �25�

where h�x�=�i erf�ix�exp�−x2��2�0xdtet2−x2

. For x�1,this function behaves as h�x��1/x. Therefore, the level ofdensity fluctuations saturates for time larger than �0 /cs. For aspatially and temporally incoherent beam with the coherencetime shorter than the acoustic time, the square of the fluctua-tion level can be estimated from Eq. �19�:

��n2�T�� =��I�2

2�ncTec�2

�cTcs2�0

2

��02 + �c

2cs2�2 . �26�

In this case, the fluctuation level does not saturate, and for atime longer than �0 /cs it increases as the square root of thetime. The non-stationary speckles continuously excite den-sity fluctuations, therefore their level gradually increases.

These analytical expressions have been compared to the2TD PARAX simulations of the plasma response to a randomlaser beam. The basic characteristics of the code are pre-sented in Appendix B. The results for the RPP case areshown in Fig. 4. A beam with the average intensity2�1013 W/cm2 was focused in a helium plasma with thedensity n0=0.05nc and Te=500 eV. The calculated k- and�-spectra are in a good agreement with theory, as well as theaverage level of fluctuations.

The temporal growth of the density fluctuations drivenby spatially and temporally incoherent laser beam is pre-sented in Fig. 5. The beam coherence time �c=4.1 ps isshorter than the plasma acoustic time �a�8 ps. Its averageintensity is 3.7�1013 W/cm2. The characteristic increase ofthe fluctuation level as �T is observed provided that the char-acteristic correlation time is shorter than the acoustic time.

The thermal effects related to the inverse bremsstrahlungabsorption and plasma heating have also been accounted forin some simulations. According to Eq. �13�, the density per-turbations were enhanced by a factor

�T � �1 + 3Z5/7� ei/�0�−4/7. �27�

This estimate is in agreement with the simulation resultsshown in Fig. 6.

IV. LASER BEAM PROPAGATION IN SELF-INDUCEDDENSITY FLUCTUATIONS

A. Hypothesis of the Gaussian statistics

This section presents the analysis of the set of coupledEqs. �5� and �18�, which provide a self-consistent descriptionfor the modification of the laser coherence due to self-exciteddensity perturbations. The coupling of these two equationsrequires a relation between the intensity and the electric fieldcorrelation functions. This is possible provided the electricfield follows the Gaussian statistics. The intensity correlation
function, which is fourth order in the field, can then be writ-

the level �c� of density fluctuations driven by a RPP laser beam.


Downloaded 07 Oct 2008 to 193.174.246.167. Redistribution subject to

ten as a bilinear combination of the electric field correlationfunctions:22 �I�� ,T ,��= �I�2+ ��EE�2+ ��EE*�2. The contribu-tion of �EE is negligible, as discussed in Sec. II, and themean intensity �I� does not participate in the excitation ofdensity perturbations. Thus, only the term with �EE* contrib-utes to the interaction.

The Gaussian hypothesis is justified for a random laserbeam propagating in vacuum because of the large number ofspeckles and due to the central limit theorem. The statisticscould be affected by nonlinear effects during the propagationthrough a plasma. However, for powers below the filamen-tation threshold, one can assume that there is no physicalmechanism that can modify the statistics on distances shorterthan �c. This assumption is compared to numerical simula-tions and it is discussed in the following section.

The system of Eqs. �5� and �20� defines the characteristiclevel of density fluctuations �n��T�I� /ncTec discussed inSec. III B 3. Correspondingly, the characteristic length �c �9�becomes inversely proportional to the square of the laserintensity. It is convenient to express it in terms of the ratio ofthe average power in a speckle: Psp=�0

2�I� to the criticalpower for the speckle self-focusing:12

Pc = �8c2/�02�ncTec�1 − n0/nc�1/2nc/n0. �28�

In the case of laser driven density perturbations, the expres-sion of �c becomes

�c =0.066LR

�T2�1 − n0/nc�P2

. �29�

The ratio P= Psp / Pc is the key scaling parameter: for a givenratio Psp / Pc, the laser-plasma interaction has similar behav-

FIG. 5. Comparison of theoretical �dashed line� and numerical �solid line�level of the density fluctuations driven by a temporally smoothed beam with�ccs /�0�0.5.

FIG. 4. Comparison of theoretical and numerical spectra in k �a�, � �b�, and
ior. This will be demonstrated in the following section.


D

B. Perturbation approach

The characteristic features of the system of Eqs. �5� and�18� can be seen within the perturbation approach. For this,the electric field and density correlation functions are split ina sum of decoupled components �EE*

0 and DN0 , and perturbed

components �EE*1 and DN

1 . Here, �EE*0 is the field correlation

function of the incident laser beam and DN0 =0 if the sponta-

neous �thermal� plasma density fluctuations are neglected. Atfirst order, the system for the perturbed components reads

�z�EE*1 = �c

−1DN1 � �EE*

0 , �30�

��2�T2 + ��2 − k2�2�DN

1 = cs4k4Ak

2�EE*0 � �EE*

0* , �31�

where �EE* is normalized to the average intensity �I� and DN

to �I� / �cncTe� and � denotes the convolution products overwave numbers and frequencies. These equations describe theprocess of multiple scattering. In contrast to parametric cou-pling, the effect of the scattered field on the density pertur-bation is ignored here. Equation �31� has been studied in Sec.III and its solution is given by Eq. �14�. The solution to Eq.�30� for the perturbed component of the electric field corre-lation function then reads

�EE*1 �T,k,��

= 2�02 z

�c�

3��exp�−

�02k2

6�

− 2�0

cscos�2�T��0k�2

�0

cs�� +

1

4

�0

cs��0k��0

cs�� , �32�

2

FIG. 6. Effect of plasma heating on the density fluctuation level for Z=2�solid curve, circle�, Z=3.5 �dashed line, square�, and Z=4.57 �dot-dashedline, triangle�. Lines show the theoretical prediction of Eq. �27�; points,PARAX simulations. The results from PARAX simulations are obtained by cal-culating the ratio of the density fluctuation level in the case in which thermaleffects are accounted for over the one obtained in ponderomotive calcula-tions. This ratio contains oscillations on the acoustic time scale, which leadsto the measured uncertainties.

where for the 2TD case, ��= �� I0��exp�−� /2


−3�2 /4� and I0 is the modified Bessel function of the firstkind.

Each component of the density correlation functiongives rise to a branch of scattered light. The static densitydepressions induce a reduction of the transverse coherence ofthe transmitted light without frequency broadening. The lasttwo terms describe the scattering from moving density per-turbations. They are responsible for the reduction of spatialcoherence and frequency broadening. The induced compo-nent in the transmitted spectrum �second term� would bedifficult to observe as it requires a resolution time shorterthan the acoustic time �0 /cs. The last term in Eq. �32� ac-counts for the scattering on the density perturbations thatpropagate transversally to the beam with cs. It dominates thefrequency broadening effect.

This solution is real, which corresponds to a symmetricbroadening of the transmitted light in k and �. Furthermore,the function ��0k��0� /cs� has a maximum for k�� /cs. Thatmeans that the spatial and temporal dependencies are mixed,which is the requirement for any smoothing technique.

The time-averaged k-spectrum is obtained by integratingthe field correlation function over �. As �2�−1�d��= 1

3 exp�−�2 /6�, the transmitted light coherence width is re-duced from �0 to �0 /�3. This result follows also from Eq.�10� by taking for the density transverse correlation length�M =�0 /�2, which corresponds to the speckle characteristicwidth. Therefore, the angular broadening of the transmittedlight is enhanced by a factor of �3.

The spectral properties of the transmitted light followfrom Eq. �32� by integration over k. As �2�−2�d�2��= 1 � 2 �� exp�−�2 /4�, the spectral broadening of the light issymmetric. The coherence time �0 / �cs

�2� agrees with theestimate of Eq. �10� by taking for the density correlation time�M =�M /cs. That is, the characteristic coherence time of thetransmitted light is given by the density response time.

V. NUMERICAL SIMULATIONS OF LASER BEAMSMOOTHING

A. Incident light and plasma properties

Three-dimensional simulations of the interaction wereperformed to verify the theoretical model and to study theeffect of plasma induced smoothing at low laser intensitiesfor the experimentally relevant conditions. In addition tostandard PARAX diagnostics,21 several new ones have beendeveloped, inspired by Schmitt and Afeyan’s work.13

The laser and plasma parameters are characteristic forpresent day experiments.16,17,19,20 The Gaussian laser beamwith the wavelength 0=1.053 �m was focused in the centerof a 2 mm long helium plasma through a RPP with squareelements, which provides a focal spot containing half theenergy in a 230 �m large square. The average intensity was�I��3�1013 W/cm2. The speckles have a Gaussian shapewith width �0=4.3 �m. The laser power brought by an av-erage speckle was therefore Psp=�0

2�I��17 MW. Theplasma density was varied from 1% to 5% of the criticaldensity, the electron temperature was Te=500 eV, and theion temperature 50 eV. The ion acoustic velocity was
cs=0.17 �m/ps, which leads to the plasma response time


D

�0 /cs�25 ps. The critical power for self-focusing Pc wasdecreased from 1600 to 330 MW as the density was in-

creased. Therefore, the ratio P= Psp / Pc was increased from1% to 5%. The characteristic length for plasma inducedsmoothing �c Eq. �29� was decreased from 78 to about1 mm.

The electric field correlation function of the incidentfield is shown in Fig. 7�a�. Its real part is in agreement withthe analytical predictions while the imaginary part accountsfor a small asymmetry of the speckle pattern and is negli-gible. It has also been verified that the function �EE�� iszero at the plasma entrance. The shape of the time correla-tion function is characteristic for a temporally coherent laserbeam: it is real and depends on the macroscopic time T, asshown in Fig. 7�b�. The time-averaged spatio-temporal spec-trum �E�k ,��2 is presented in Fig. 7�c�. The spatial and tem-poral components are decoupled: the spatial component islarge with the width ��0

−1, whereas the temporal componentis very narrow with the width given by 1/T.

B. Numerical simulations of the beam temporalsmoothing

1. Properties of the transmitted light

A visual evidence of the temporal beam smoothing in aplasma is presented in Fig. 8. It shows the temporal evolu-tion of the intensity of the transmitted light I�x , t� at thecentral section y=0. In panel a �n0 /nc=1% �, the specklepattern has no temporal modifications. The distance�c=78 mm in this case is too large and the plasma does notaffect the laser beam. In panel �b� �n0 /nc=3% �, the intensitydistribution presents some temporal fluctuations on timescale of about 50–100 ps. Finally, in panel �c�, the intensitydistribution is strongly modified with time. Here, the charac-teristic distance �c�2.5 mm is comparable to the plasmalength and the smoothing effect is rather efficient. The con-trast of the time-integrated intensity distribution

Cint = �I��−1��I2�� − �I��

2 �33�

presents a quantitative characteristic of the smoothing effect.

Here I is the time-averaged intensity and the spatial averagein Eq. �33� is taken over the beam cross section at a givendistance z. It is known that, for a temporally coherent beamwith a Gaussian statistics, this contrast is equal to 1 andtherefore its reduction characterizes the smoothing effect.

Figure 9 shows the contrast evolution along the propa-gation direction for different conditions of interaction. In or-der to reduce the data volume, the average was calculated inone transverse direction �x ,y=0� instead of in the wholetransverse plane. The gray curve corresponds to the low-density case in which the contrast stays almost constant. Thedashed line corresponds a 3% of nc plasma density, wherethe contrast is reduced to about 70%. For the case of theplasma density of n0=0.05nc, the contrast is strongly reducedto less than 50% after the distance of 2 mm. Thermal effectsenhance even more the smoothing efficiency. The character-istic length is therefore reduced by a factor �T

2 �2.2 and the
contrast is reduced to about 38%. At the end of the simula-

FIG. 7. Coherence properties of the incident light: �a� real �solid curve� andimaginary �dashed curve� parts of the spatial correlation function of the laserelectric field, �b� coherence time of the laser electric field as a function ofthe macroscopic time T, and �c� spatio-temporal spectrum of the incident
light.


D

tion box the contrast reduction is saturated. This is due to thediffraction of the beam, as discussed in the following para-graph.

2. Modification of the temporal correlation function

Figure 10 presents the real and imaginary parts of thetime correlation function CEE*�T ,�� for the case in which�c�2.5 mm. The real part is strongly modified during thepropagation. Whereas the function contains only the macro-

FIG. 8. �left� Time resolved intensity distribution in a transverse directionafter propagating through 2 mm plasma with the density n0 /nc=1% �a�, 3%�b�, and 5% �c�. �right� Comparison between time integrated intensity invacuum �dashed curve� and after propagation through the plasma �solidcurve�.

scopic scale T at z=0, a smaller scale appears for larger


values of z. It characterizes the coherence time, which isshown in Fig. 11. It is reduced from 250 ps at the beginningof the interaction to about 20 ps after a few hundred �m ofinteraction, which is in good agreement with the analyticalprediction �0 / �cs

�2��18 ps. The coherence time does notevolve so much in time, but it is reduced as the interactionlength increases. This reduction is explained by the reductionof the speckle radius with z due to the multiple scattering. Tounderline this effect, the time integrated speckle radius iscalculated as function of z, and the corresponding time�c�z� / �cs

�2� is shown in Fig. 11.The calculated real part of the temporal correlation be-

haves according to theoretical predictions shown in Fig. 1�b�.The only difference is due to the fact that the average inten-sity varies with z because of the diffraction effect that modi-fies the value for �=0. Otherwise, in both figures the corre-lation function consists of two parts: the homogeneous partdecreases with z, while the perturbation with the coherencetime about �0 /cs /�2 increases on the characteristic length�c.

Moreover, the temporal correlation function contains agrowing imaginary part that becomes comparable to the realpart at the distance of 800 �m. This effect is not describedby the current model of multiple scattering presented in Sec.IV, where the correlation function remains real along thepropagation axis. Although the imaginary part of the corre-lation function has no effect on the coherence time, it impliesthat the real and imaginary parts of the electric field, ER andEI are correlated. Indeed, the imaginary part of the temporalcorrelation function reads

Im �EE*�T,2�� = �ER�T + ��EI�T − �� − �ER�T − ��EI�T + �� .

�34�

It is not zero if the real and imaginary parts of the electricfield have a different behavior on the time scale ��0 / �cs

�2�,

FIG. 9. Evolution of the time integrated contrast with the propagationlength. The plasma density n0 /nc=0.01 �gray curve�, n0 /nc=0.03 �dashedcurve�, and n0 /nc=0.05 in the ponderomotive case �dot-dashed curve� andwith the thermal effects �solid curve�.

where the imaginary part of �EE* has an extremum. Further-



D

more, as the temporal electric field correlation function con-tains an imaginary part, the spectral density of energy,

��E��2� = T−1� dT�EE*�T,�� , �35�

is no longer symmetric. Whereas the model of multiple scat-tering leads to a symmetric spreading of the transmitted lightspectrum, numerical simulations suggest an asymmetricspreading of the frequency spectrum as z increases.

FIG. 11. Evolution along the propagation direction of the coherence timemeasured from the numerical 2TD electric field correlation function forT�30 ps �circles�, T�130 ps �squares�, and T�230 ps �diamonds�. Thegray line shows the theoretical coherence time calculated from the measuredcoherence width of the electric field ��0 /cs

�2�. This corresponds to the case
in which �c�2.5 mm.

3. Spectral properties of the transmitted light

The evolution of the laser spectral energy density alongthe propagation direction is shown in Fig. 12 in the case inwhich �c�2.5 mm �panel �a�, ponderomotive case� and�c�1.1 mm �panel �b�, accounting for thermal effects�. It iscalculated from the temporal evolution of the electric field ina transverse direction. One observes that, as z increases, theintensity of stray light is reduced. The scattered componentgrows with z and a spectral broadening of the order of0.25 ps−1�2cs /�0 as well as a redshift ��0.1 ps−1 arepresent. This redshift is characteristic of FSBS and has al-ready been observed in numerical simulations at higher pow-ers of the order of Pc.

13,14 It has also been observed in thenumerical simulations at powers below the filamentationthreshold by Michel.29 Moreover, the amplitude of thestraight light ��=0� decreases more quickly if thermal ef-fects are accounted for. This shows the importance of sucheffects on the density fluctuation level and their effects onthe modification of the laser coherence properties.

In the case of ponderomotive coupling, the transversallyresolved frequency spectrum of the light is shown in Fig. 13.The incident light �panel �a�� contains in the x direction twovery different scales. The macroscopic one is the beam en-velope �400 �m, the microscopic one is the speckle width,which is a few micrometers. The spectrum has a narrowdistribution in frequencies with the width �1/T. After800 �m propagation, a spreading and redshift appear at thecenter of the beam. It means that the light at the center of thefocal spot is already partially smoothed, whereas, for largervalues of x, the laser light coherence properties are not yetmodified. As z increases one observes an increase of theintensity of the scattered light and the spectral broadeningextends to the beam edges. This is due to the macroscopic

FIG. 10. Real �top� and imaginary �bottom� parts ofthe temporal electric field correlation function forT=33 ps �a�,�b�; and T=130 ps �c�,�d�. For z=0 �m�gray curve�, z=800 �m �dashed curve�, z=1200 �m�dot-dashed curve�, and z=2000 �m �solid curve�.

properties of the intensity distribution provided by the RPP.



D

The intensity at the center of the beam is higher, thus, thelight at the center of the focal spot needs a shorter lengththan at edges to be smoothed.

The light that is scattered in the middle of the beam hasboth, a reduced temporal coherence and a reduced coherencewidth. It diverges faster than the straight light, and, after acertain distance, enhances the density fluctuations at thebeam edges and thus increases the smoothing efficiency.

The �-k spectrum of the light is presented in Fig. 14 forthe same conditions of interaction. It shows how spatio-temporal correlations are created as the beam propagatesdeeper into the plasma. After 400 �m �panel �a��, the laserlight contains a quasisymmetric frequency spread compo-nent. However, the part of the energy outside of the centralpeak, i.e., �� T−1, is relatively small. It increases for largervalues of z where the laser light undergoes both a stronger

FIG. 12. Frequency spectrum of the laser electric field in the ponderomotivecase �a� and accounting for thermal effects �b�, for z=0 �gray curve�,800 �m �dashed curve�, 1600 �m �dot-dashed curve�, and 2000 �m �solidcurve�. The corresponding smoothing length is about 2.5 mm in the pon-deromotive case �a� and 1.1 mm with thermal effects �b�.

frequency broadening and a redshift. The scattered waves are


FIG. 13. Light spectrum resolved in the transverse direction for z=0 �a�,800 �m �b�, and 2000 �m �c�. This corresponds to the case in which� �1.1 mm.
c

ture. This corresponds to the case in which �c�1.1 mm.



filling the ion acoustic cone as it is suggested by the statis-tical model presented in Sec. IV. A similar behavior has al-ready been reported by Schmitt and Afeyan13 but for muchhigher laser intensities where the average power in a specklewas of the order of the self-focusing threshold. It was ex-plained as a combined effect of FSBS and filamentation. Inpresent conditions, the average power in a speckle is wellbelow the self-focusing threshold. Consequently, the angularspread does not increase �cf. panel �c��, while the �-spectrumis significantly broadened. The speckle width calculated fromthe spatial electric field correlation function decreases from4.3 �m at the entrance to 3 �m at the exit. That correspondsto increase of the beam aperture angle from 1.2° to 2.5° atdistance of 2 mm. This is ten times smaller than the apertureangle reported in Ref. 13.

C. Efficiency of the plasma induced smoothingand the role of FSBS

The theory of multiple scattering predicts that thesmoothing length �29� depends essentially on the ratio

P= Psp / Pc. It has been verified in PARAX simulations that for

the same ratio P�0.05 but for different n0 and �I� thesmoothing length and the fraction of scattered light FS isalmost the same if the thermal effects are switched off. Thisis illustrated in Fig. 15 for the case in which �c�2.5 mmwith n0 /nc=0.05 and �I��3�1013 W/cm2 �circles�, n0 /nc

=0.025 and �I��6�1013 W/cm2 �diamonds� and n0 /nc

=0.01 and �I��1.5�1014 W/cm2�squares�. It has been veri-fied that the thermal effects on the plasma induced smooth-ing can be described by the factor �T �27�. Indeed, one cansee in Fig. 15 that the evolution of FS is almost the same byaccounting for the thermal effects �up triangles� than by mul-tiplying the intensity by a factor �T in the ponderomotivecase �down triangles�. The corresponding smoothing length

FIG. 15. Evolution along the propagation axis of the part of the energyoutside of the monochromatic peak for different smoothing lengths:�c�2.5 mm �circles, diamonds, squares� and �c�1.1 mm �up and downtriangles�.

FIG. 14. �-k spectrum of the laser electric field for z=400 �m �a�,z=1200 �m �b�, and z=2000 �m �c�. Dashed lines show the acoustic dis-persion �= ±csk. Dotted lines in panel �c� indicate the incident beam aper-

is �c�1.1 mm.



D

Considering in more detail the case in whichn0=0.05nc and accounting for thermal effects, one can definethree characteristic regions of interaction. In the first, i.e.,500 �m, the first speckles induce some density fluctuationsand the light is scattered on them. The temporal coherence isreduced but the fraction of scattered light is small. The den-sity fluctuation level is constant in time as it can be seen inFig. 16�b� and in Fig. 17. In the second region, typicallyfrom 500 to 1500 �m, the scattered intensity increases with zand attains the saturation level of 50%–60%. The perturba-tion approach that has been developed in Sec. V does nothold here. Indeed the scattered light with a reduced coher-ence plays an important role in the excitation of density per-turbations. They are continuously excited by the fluctuatingponderomotive force and increase in time from �0.8% att�30 ps to 1.5% at t�260 ps as �t �cf. Fig. 17�. This in-duces a further reduction of the characteristic length �c andtherefore increases the induced smoothing efficiency. In thethird region the diffraction of the beam takes over and itreduces the average laser intensity �panel �a��. Density fluc-tuations are no longer excited �panel �b��, and the smoothingefficiency saturates �panel �c��.

The enhanced scattering in the second region is associ-ated with spectral broadening and redshift of the laser light.These features are characteristic for FSBS.13,14 In contrast tothe case considered by Schmitt and Afeyan13 and Maximovet al.,14 in which it was seeded by the filament instability, thepresent study shows that initial RPP-driven fluctuations andassociated frequency broadening of the laser light in the firstregion serve as a strong seed for the FSBS. As a conse-quence, the laser beam does not suffer filamentation and suchundesirable effects as enhanced backscattering and angularspreading.

Analytical predictions for FSBS were proposed in Ref.14 using the random phase approximation �RPA� technique.The convective amplification was derived for scattered lightwith large wave numbers, for the light scattered outside ofthe incident cone. This approach is not appropriate to de-scribe the current situation in which the k-spectrum is not somuch broadened. Indeed, it was underlined in Ref. 14 thatthe RPA does not hold in the incident cone where incidentand scattered light could be partially correlated.

Nevertheless, the characteristic gain length for FSBS,LFSBS, can be estimated from Ref. 14 by setting the correla-tion time to �0 /cs. One then obtains

LFSBS−1 � k0

n0

nc

�I�cncTe

, �36�

which, for our simulation parameters, is of the order of1 mm. In a more general way, one shows from Eqs. �29� and

�36� that �c /LFSBS� P−1, so that the multiple scattering onself-induced density fluctuations can not be observed withoutFSBS in the low-intensity regime. It just serves as a seed forit. As it is amplified over the whole interaction length, itquickly enters the nonlinear regime of saturation. The tem-poral evolution of the instability is mainly governed by the
temporal increase of the density perturbations driven by a

temporally incoherent beam, as described in Sec. II, whichleads to the reduction of the effective length for smoothing.

The multiple scattering as well as FSBS activitiesstrongly depend on the ion acoustic wave damping.30 In plas-mas with ZTe /Ti�3, where �d�0.3kcs, the density pertur-bations are damped before they interact with neighboringspeckles. As a result, numerical simulations show that mul-tiple scattering does not occur, FSBS has not been excitedand no induced incoherence is observed.

D. Limitations of the model of multiple scattering

The key role of FSBS observed in numerical simulationsprompts us to revisit the main assumptions that have beenused in the model presented in Sec. IV. In what follows, thedifferences between the interaction of laser beams with self-induced and already existing density perturbations are dis-cussed. A RPP laser beam was creating the density fluctua-tions during about 30 ps and then the coupling term in theacoustic wave equation was instantaneously turned off att=0. Thus, for t�0, the laser beam propagates through al-ready existing fluctuations without exciting new ones. Pro-vided that the acoustic damping is negligible, the densityfluctuation level stays constant for time shorter than L0 /cs

and the laser beam undergoes the multiple scattering on den-sity fluctuations with the level of about 0.6%.

The simulation results are presented in Fig. 18. One ob-serves that the redshift in the frequency spectrum has disap-peared �panel a�, the imaginary part of the electric field cor-relation function is negligible. The spectral broadening issymmetric and the efficiency of the smoothing has been re-duced by a factor of about 2 �panel �b��. The resulting evo-lution of FS is in good agreement with theoretical predictionof Eq. �12�. These results demonstrate that a continuous ex-citation of density fluctuations by FSBS enhances thesmoothing efficiency. This effect is not captured by the per-turbation approach of Eqs. �5� and �20�.

An important assumption of the model presented in Sec.IV is that the electric field distribution in the focal spot fol-lows the Gaussian statistics. This holds true if one considersthe propagation either in a vacuum or through medium witha slow nonlinearity, �0 /cs��c. Under such conditions,Garnier et al.24 proved that the Gaussian statistics is pre-served. In the contrary, in Ref. 25, the authors show that themain effect of the propagation through an instantaneouslyresponding medium, �0 /cs��c, is to destroy the Gaussianstatistics, leading to a contrast enhancement.

The departure from the Gaussian statistics was observedin numerical simulations from the beam instantaneous con-trast: C= �I�−1��I2�− �I�2�1/2. In the case in which the real andimaginary parts of the electric field are statistically indepen-dent, a Gaussian field distribution is characterized by theexponential intensity distribution: p�I�= �I�−1exp�−I / �I��, andthe corresponding instantaneous contrast is equal to 1. In theregion of the first 500 �m, in full PARAX simulations, thecontrast increases from 1 to about 1.5 �cf. Fig. 19�; that is,the Gaussian statistics is lost. Indeed, in the case of RPP laserbeams, the coherence time is given by the pulse duration,
typically a few hundred picoseconds. It is therefore much

thermal effects are accounted for.



longer than the response time of the medium, typically a fewpicoseconds, and the nonlinearity destroys the Gaussianstatistics.25 The Gaussian statistics of a transverse distribu-tion of hot spots holds if there is only one microscopic trans-verse scale. However, even below the filamentation thresh-old, the stationary speckles undergo a focusing in the densitydepression they dig. This modifies the field distribution andtwo different scales appear: the reduced size of the specklesand the average distance between them. This physicalmechanism is responsible for enhancement of the instanta-neous contrast and thus, for deviation from the Gaussian sta-tistics. It is not accounted for in the current model for mul-tiple scattering.

On the contrary, in the case in which the source term isswitched off in the acoustic equation, the contrast stays closeto 1 within 5%. The Gaussian properties are conserved. Thisis a clear demonstration of the effect of the instantaneousnonlinearity on breaking the Gaussian statistics.

After propagating on a distance of the order of one-tenth�c the temporal coherence of the beam is reduced to the timeof nonlinear response. Numerical simulations show that thecontrast returns back to 1 at distances larger than 1000 �m;that is, the temporal smoothing restores the Gaussian statis-tics. This case, in which �c is reduced to almost �0 /cs, isintermediary between the fast and slow nonlinearity case dis-cussed above. It is therefore difficult to make any theoreticalpredictions; however, numerical simulations suggest that theGaussian properties is restored.

As it has been discussed, the reason for the increase ofinstantaneous contrast is the speckle focusing in density de-pressions. This occurs provided that density depressions andintensity local maxima are correlated. These correlations areneglected in the construction of Eq. �5� that describes theelectric field correlation function. This is the major differ-

FIG. 17. Temporal evolution of the density fluctuation level for z=0 �graycurve�, 400 �m �dashed curve�, 1200 �m �dot-dashed curve�, and 2000 �m�solid curve�. The characteristic evolution as �T of fluctuations driven by atemporally incoherent beam is shown by the dotted line. The parameters arethe same as in Fig. 16.

FIG. 16. Evolution along the propagation axis of the laser average intensityin a circular zone containing 2/3 and 1/2 of the total energy �a�, of thedensity fluctuations level �b�, and of the part of the laser energy outside ofthe monochromatic peak �c�. Dashed curves: t�30 ps, dot-dashed curves:end of the simulation �t�260 ps�, solid curves: time averaged over thewhole simulation duration. This case corresponds to �c�1.1 mm in which

ence between the propagation through self-induced and al-



D

ready existing density fluctuations. A diagnostic for thesecorrelations has been proposed by Schmitt and Afeyan.13

Figure 20 illustrates the evolution of these correlations dur-ing the propagation through the plasma. In the first 500 �m,the correlation between density depressions and intensitymaxima increases. It is of the form: �n� I. It corresponds tothe region where the instantaneous contrast is increasing. Assoon as the induced smoothing becomes efficient, forz�1000 �m, the correlation between intensity and densityperturbations is then reduced and the instantaneous contrastdecreases to 1. Similar behavior has been observed at powersabove the self-focusing threshold in Ref. 13. This again con-firms that the effect of multiple scattering is rather universalas soon as the density perturbations attain a sufficiently high

FIG. 18. Effect of the multiple scattering decoupled from the FSBS: spec-trum of the transmitted light �a� and fraction of laser light outside of themonochromatic peak �b�. The parameters and notations are the same as inFig. 16.

level. In the case of decoupled density fluctuations, there is


no correlation between intensity and perturbations and theGaussian statistics is conserved along the whole interactionlength �cf. Fig. 19�.

VI. CONCLUSIONS

The problem of modification of coherence properties ofa partially incoherent beam during its propagation through arandomly fluctuating medium is considered in this paper. Us-ing a stochastic field equation, the effect of such densityfluctuations on the spectral �temporal� and angular �spatial�properties of a laser beam was addressed. It is found that,after having propagated through a certain distance �c, thetemporal and spatial coherences of the transmitted light arereduced. The length �c depends essentially on the amplitudeof the fluctuations and on their characteristic scales.

These density perturbations can be excited by an exter-nal source or by the laser beam itself. In the latter case, theyare driven by the ponderomotive force and the inhomoge-neous heating associated with the random distribution of hotspots. Their average properties are derived from the analysisof the correlation function. The thermal effects are shown toplay a non-negligible role in the excitation of density pertur-bations for conditions of recent experiments.15–20

The set of coupled equations for the electric field andplasma density correlation functions shows, within a pertur-bation approach, the characteristic features of the transmittedlight spectral and angular broadening. The multiple scatter-ing requires a certain distance to affect the laser coherence,which depends essentially on the speckle Rayleigh lengthand the average power in a speckle normalized to the criticalpower for filamentation. The analytical model shows a sym-metric frequency broadening of the transmitted light withoutstrong angular spreading.

Three-dimensional numerical simulations using the in-

FIG. 19. Evolution of the instantaneous contrast with the propagation lengthat t�30 ps in the case of propagation through self-induced �ponderomotivecase, dashed line and accounting for thermal effects, solid line�, and alreadyexisting fluctuations �gray line� density fluctuations. The departure from 1denotes a deviation from the Gaussian statistics.

teraction code PARAX have been performed and compared to



D

predictions of the statistical model. A spatially incoherent�RPP� beam with an average intensity well below the fila-mentation threshold was propagated through a low-densitymillimetric size plasma. Appropriate diagnostics have beenused to analyze the smoothing effect. The spectral broaden-ing distance is in good agreement with the model predic-tions. The coherence time of the transmitted light is reducedto the ion acoustic time ��0 /cs and angular beam divergenceis enhanced less than twice.

Numerical simulations show that the spectral broadeningof the scattered light is associated with a redshift, which wasnot predicted by the model, and which is characteristic forFSBS. Multiple scattering on RPP-driven density fluctua-tions occurs in the first speckle length and serves as a strongseed for FSBS that grows along the whole interaction length.The reduced coherence time of the scattered light is respon-sible for a continuous excitation of the density perturbationsthat increases as the square root of time. As a consequencethe smoothing effect is enhanced and its efficiency increases.

The redshift in the frequency spectrum of the transmittedlight is associated to deviation from the Gaussian statisticsfor the electric field distribution. This deviation is due to thefast nonlinearity of the plasma. The intensity-density corre-lations that occur when the RPP laser beam is not alreadysmoothed are not accounted for in the model and are respon-sible for the departure from the Gaussian statistics. When thetemporal coherence of the beam is broken, these correlationsare reduced and the Gaussian statistics is restored. This cor-responds to the case in which the coherence time of the laserlight is of the order of the nonlinearity time of response.

The use of low-intensity lasers smoothed by only theRPP enables to avoid deleterious effects of filamentation in-stability and to obtain an efficient smoothing in the sametime. The temporal coherence properties are modified while
the angular beam spreading stays almost the same. More-

over, RPP-driven density fluctuations serve as a much stron-ger seed for FSBS than thermal noise for backward scatter-ing instabilities31 that grow in the more intense hot-spots.32

That is, plasma induced laser beam smoothing can operateunder conditions where backscattering instabilities are notyet excited. This low-intensity regime of interaction appearsas an interesting possibility to control the laser coherenceproperties in the ICF context. Parameters chosen in this ar-ticle, I 0

2�0.3 MW, are characteristic for current laser facili-ties and for the plateau stage of standard ICF pulses.

ACKNOWLEDGMENTS

One of the authors �M.G.� would like to thank ClaudeGouédard for fruitful discussions on the statistical propertiesof partially incoherent laser beams. Usage of the computingcenter Centre de Calcul Recherche et Technologie �CCRT� ofthe CEA is acknowledged.

APPENDIX A: DERIVATION OF THE EQUATIONFOR THE ELECTRIC FIELD CORRELATION FUNCTION

Let us consider a correlation of electric fields at twopositions and at two instants �r1 , t1� and �r2 , t2� denoted as +and −, respectively. The equation for the product of twofields E+�z�E−

��z� follows from Eq. �2�:

�2ik0�z −�p0

2

c2 ��n+�z� − �n−�z�� E+�z�E−��z�

= ��+ − ��

− �E+�z�E−��z� . �A1�

By considering the right-hand side as a known function, the

FIG. 20. Spatial correlation of the intensity and theself-induced density fluctuations at t�260 ps for z=0�a�, 400 �m �b�, 1600 �m �c�, and 2000 �m �d�. Con-ditions are the same as in Fig. 16.

formal solution to Eq. �A1� reads



D

E+�z�E−��z� = E+�0�E−

��0�

�exp�− i�p0

2

2k0c2�0

z

��n+�z�� − �n−�z��dz��+

i

2k0�

0

z

exp�− i�p0

2

2k0c2�z�

z

��n+�z� − �n−�z��dz��

+ − ��− �E+�z��E−

��z��dz�. �A2�

It is now assumed that the longitudinal coherence length ofdensity fluctuations LC is much shorter than the propagationlength, z�LC and that the electric field in the z1 plane isindependent of density fluctuations in the plane z2�z1. Thus,according to the integration limits in Eq. �A2�, one can de-couple the electric field and density components when takingthe average:

�E+�z�E−��z�� = �E+�0�E−

��0��z��

+i

2k0�

0

z

dz�exp�− i�p0

2

2k0c2�z�

z

��n+�z�

− �n−�z��dz��+ − ��

− ��E+�z��E−��z�� ,

�A3�

where

��z� = exp�− i�p0

2

2k0c2�0

z

��n+�z�� − �n−�z��dz�� . �A4�

Provided the longitudinal length of density fluctuations ismuch shorter than the nonlinear distance �C, the densityfluctuation integral �0

z�n�z�dz behaves as a Brownian motion.That is, it has independent increments, so that

��z�� = ��z��exp�− i�p0

2

2k0c2�z�

z

�n�z�dz�� , �A5�

leading to

�E+�z�E−��z�� = �E+�0�E−

��0��z��

+i

2k0�

0

z

dz��z��z��

��+ − ��

− �

��E+�z��E−��z�� . �A6�

The differential form of the previous equation reads

��z −i

2k0��

+ − ��− ��EE* = �zln ��z��EE*, �A7�

where

��z� =exp− ik0

2

n0

nc�

0

z ��n�R +�

2,T +

�

2,z��

− �n�R −�

2,T −

�

2,z�� dz�� . �A8�

The propagation through density fluctuations is considered as
a diffusion process over many randomly distributed density

fluctuations, the diffusion approximation ensures, accordingto the central limit theorem, the integral �0

z�n�z�dz shouldfollow Gaussian statistics. As a consequence, the expressionof ln ��z� can be simplified to:

��z� = exp� �p04

8k02c4�

0

z

�DN�0,T = t1,0� + DN�0,T = t2,0�

− 2DN�r1 − r2,T,��dz�� . �A9�

Finally, using more relevant quantities R= �r1+r2� /2 and�=r1−r2, T= �t1+ t2� /2 and �= t1− t2 and writing ��

+ −��− as

2�R ·��, one obtains Eq. �5�.

APPENDIX B: THE INTERACTION CODE PARAX

The code PARAX simulates the propagation of an electro-magnetic wave through an underdense plasma within theparaxial approximation. The equation for the slowly varyingenvelope of the laser electric field is, in Fourier space for thetransverse coordinate:

��z +�

c2�IB +i

2k0k2 − i

�p02

2k0c2 ��n�k,z,t� − 1� E�k,z,t� = 0,

�B1�

where �IB=�ein0 /nc is the electromagnetic wave dampingdue to electron-ion collisions. In simulations presented here,the plasma density and charge state are low so that the ab-sorption by inverse bremsstrahlung is negligible. Equation�B1� generalizes Eq. �2� and is valid for LR�c�c.

The plasma density perturbations induced by the laserponderomotive force and thermal effects are treated withinan ion-acoustic wave model. In the absence of a transverseplasma flow, it reads

��t2 + 2�d�t + cs

2k2�ln��n�k,z,t��

=�0cs

2k2

2ncTeAk� dk�E�k�,z,t�E*�k� − k,z,t� . �B2�

�d is the ion acoustic wave damping and accounts for theLandau damping due to both electrons and ions and for theion-ion collision effects. The Ak operator is derived fromBrantov et al.26 and reads

Ak =1

2+

0.88Z5/7

��k� ei�4/7 +2.54Z

1 + 5.5��k� ei�2 . �B3�

This equation is similar to Eq. �13� except for the logarithmon the left-hand side, which allows to saturate the densitydepressions and to ensure the positivity of the density intransient regimes. As this paper is interested in the low-intensity regime, this precaution is not necessary.

1J. D. Lindl, P. Amendt, R. L. Berger, S. Gail Glendinning, S. H. Glenzer,S. W. Haan, R. L. Kauffman, O. L. Landen, and L. J. Suter, Phys. Plasmas11, 339 �2004�.

2Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka,and C. Yamanaka, Phys. Rev. Lett. 53, 1057 �1984�.

3S. Skupsky, R. W. Short, T. Kessler, R. S. Craxton, S. Letzring, and J. M.
Soures, J. Appl. Phys. 66, 3456 �1989�.


D

4C. Labaune, S. Baton, T. Jalinaud, H. A. Baldis, and D. Pesme, Phys.Fluids B 4, 2224 �1992�.

5P. N. Guzdar, C. S. Liu, and R. H. Lehmberg, Phys. Fluids B 5, 910�1993�.

6E. Lefebvre, R. L. Berger, A. B. Langdon, B. J. Macgowan, J. E. Rothen-berg, and E. A. Williams, Phys. Plasmas 5, 2701 �1998�.

7A. J. Schmitt, Phys. Fluids 31, 3079 �1988�.8R. H. Lehmberg and S. P. Obenschain, Opt. Commun. 46, 27 �1983�.9J. Grun, M. E. Emery, C. K. Manka, T. N. Lee, E. A. Mostovych, J.Stamper, S. Bodner, S. P. Obenschain, and B. H. Ripin, Phys. Rev. Lett.58, 2672 �1987�.

10M. Desselberger, O. Willy, M. Savage, and M. J. Lamb, Phys. Rev. Lett.65, 2997 �1990�.

11M. H. Emery, J. H. Gardner, R. H. Lehmberg, and S. P. Obenschain, Phys.Fluids B 3, 2640 �1991�.

12D. Pesme, S. Hüller, J. Myatt, C. Riconda, A. Maximov, V. T. Tikhonchuk,C. Labaune, J. Fuchs, S. Depierreux, and H. A. Baldis, Plasma Phys.Controlled Fusion 44, B53 �2002�.

13A. J. Schmitt and B. B. Afeyan, Phys. Plasmas 5, 503 �1998�.14A. V. Maximov, I. G. Ourdev, D. Pesme, W. Rozmus, V. T. Tikhonchuk,

and C. E. Capjack, Phys. Plasmas 8, 1319 �2001�.15J. D. Moody, B. J. MacGowan, S. H. Glenzer, R. K. Kirkwood, W. L.

Kruer, A. J. Schmitt, E. A. Williams, and G. F. Stone, Phys. Rev. Lett. 83,1783 �1999�.

16P. Michel, C. Labaune, S. Weber, V. T. Tikhonchuk, G. Bonnaud, G.Riazuelo, and F. Walraet, Phys. Plasmas 10, 3545 �2003�.

17P. Michel, C. Labaune, H. C. Bandulet, K. Lewis, S. Depierreux, S. Hulin,G. Bonnaud, V. T. Tikhonchuk, S. Weber, G. Riazuelo, H. A. Baldis, and
A. Michard, Phys. Rev. Lett. 92, 175001 �2004�.

18C. Labaune, H. C. Bandulet, S. Depierreux, K. Lewis, P. Michel, A.Michard, H. A. Baldis, S. Hulin, D. Pesme, S. Hüller, V. T. Tikhonchuk, C.Riconda, and S. Weber, Plasma Phys. Controlled Fusion 46, B301 �2004�.

19V. Malka, J. Faure, S. Hüller, V. T. Tikhonchuk, S. Weber, and F.Amiranoff, Phys. Rev. Lett. 90, 075002 �2003�.

20J. Fuchs, C. Labaune, H. C. Bandulet, P. Michel, S. Depierreux, and H. A.Baldis, Phys. Rev. Lett. 88, 195003 �2002�.

21G. Riazuelo and G. Bonnaud, Phys. Plasmas 7, 3841 �2000�.22H. A. Rose and D. F. Dubois, Phys. Fluids B 5, 590 �1992�.23J. Garnier, Phys. Plasmas 6, 1601 �1999�.24J. Garnier, J.-P. Ayanides, and O. Morice, J. Opt. Soc. Am. B 20, 1409

�2003�.25J. Garnier, L. Videau, C. Gouédard, and A. Migus, J. Opt. Soc. Am. B 15,

2773 �1998�.26A. V. Brantov, V. Yu. Bychenkov, V. T. Tikhonchuk, W. Rozmus, and V.

K. Senecha, Phys. Plasmas 6, 3002 �1999�.27V. I. Klyatskin, Stochastic Equations and Waves in Randomly Inhomoge-

neous Media �Nauka, Moscow, 1980�; Dynamics of Stochastic Systems�Elsevier, New York, 2005�.

28V. Yu. Bychenkov, W. Rozmus, A. V. Brantov, and V. T. Tikhonchuk,Phys. Plasmas 7, 1511 �2000�.

29P. Michel, Perte de cohérence d’un faisceau laser intense lors de sapropagation dans un plasma, Ph.D. thesis, Ecole Polytechnique �2003�.

30M. D. Tracy, E. A. Williams, K. G. Estabrook, J. S. De Groot, and S. M.Cameron, Phys. Fluids B 5, 1430 �1993�.

31R. L. Berger, E. A. Williams, and A. Simon, Phys. Fluids B 1, 414 �1989�.32V. T. Tikhonchuk, C. Labaune, and H. A. Baldis, Phys. Plasmas 3, 3777

�1996�.


plasma induced laser beam smoothing below the ﬁlamentation ...skupin/article/pop_13_093104.pdf ·...

Documents