J. Plasma Fusion Res. SERIES, Yol2 (1999) 19-24

Non-Linear Interaction Mechanisms of an Ultra-lntense Laser
Pulse with an Underdense Plasma
ADAM Jean C., HERON Anne, LAVAL Guy and MORA Patrick
Centre de Physique Thdorique, CNRS-Ecole Polytechnique, Palaiseau, France(Received: 26 January 1999 / Accepted: 11 August 1999)

Abstract
The interaction of an intense beam (1Ore < / < l020Wcm2; with moderately dense plasmas has been
investigated by using theoretical analysis as well as numerical simulations. The parametric instabilities ofa plane intense light wave have been theoretically studied in this strongly coupled case, with a particularemphasis on filamentation and the generalization of the TPD instability. Then these results are used to
discuss numerical simulations. A 2D PIC code is used to study the propagation of picosecond pulses. The
theoretical conclusions may be compared with present experimental results. These experiments show thatthe plasma strongly absorbs the intense radiation for densities where it would be expected to betransparent or reflective via Raman instabilities. Moreover, very high electron temperatures are achieved
and multi-Mev particles are generated. These behaviors result from the strong non-linear interaction of
the radiation with the plasma.

Keywords:
relativistic plasma, laser-plasma interaction, absorption, MeV electrons

1. lntroduction

assumption that the oscillating electrons can beconverted into a high energy particle beam which will
heat the dense core of a compressed pellet. Preliminary
experiments and numerical simulations show that theconversion does occur when the ultra-intense laser beaminteracts with a very overdense plasma.

The chirped pulse amplification of laser light has

opened a new field of investigation for plasma
physicists. Laser peak powers have reached up to one
petawatt . By focusing this light beam, intensities larger
than l020Wcm2 are now available. In this intensity
range, the electrons are brought to relativistic energies

However, several physics issues have to be
examined before being able to make a realistic

by the mere oscillation in the laser electric field.
Consequently, the physics of laser plasma interaction isstrongly modified. The electron mass depends on thelight intensity and the usual picture of wave propagationassessment of the method. Among them, the channelingdoes not apply any more to this relativistic plasma.

through the long coronal plasma without deviation,
filamentation and absorption. The current solutionof the ultra-intense pulse through the underdense coronais challenging. The short intense pulse has to propagate

A large number of ultra-high intensity lasers have

implies that a long intense prepulse will drill a channel
in the coronal plasma. Once this channel formed, thebeen built to generate high energy particles for several

purposes. Among them, the fast ignition of fusion
targets raised great hopes by lowering the laser requiredenergy to obtain the fuel burning and a thermonuclear

ultra-intense pulse will reach safely the dense core.
In order to define the desirable parameters of thegain. The fast ignition principle relies upon the@1999 by The Japan Society of Plasma
C orre s pondin g aut ho r' s e - mail : adam @ c p ht. p o ly t e c hniqueJr
t9

Science and Nuclear Fusion ResearchAdam J.C. er al., Non-Linear Interaction Mechanisms of an Ultra-Intense Laser Pulse with an Underdense Plasma

residual plasma inside the channel, it is necessary toknow how the ultra-intense pulse will interact with it.
This is the purpose of several experiments in which an
ultra-intense pulse interacts with a preformed
underdense or slightly overdense plasma. In thesebeen performed with 0.O5 < nln" < 0.4 and 1016 < 1, <experiments, anomalous absorption has been observed[] as well as a strong reduction of the transmission
through moderately underdense plasmas. For the sameand the reflection coefficients were obtained by

1020W/cm2 where f. is the maximum intensity of thepulse. In this range of intensities, the relativistic massfactor 7of an electron oscillating in the laser field varies
from 1.01 to 6.1 for a I pm radiation. The transmission

computing the outgoing flux of the Poynting vector

through the simulation box boundaries and theparameters, high energy electrons have been detected.

absorption coefficient was given by the overall energy

In this paper, the interaction of an ultra-intense I
pm laser pulse with moderately dense plasmas will be
investigated by using mainly 2D numerical particle

flux balance. One run lasts 15 hours on the 64
processors of a T3E.

Figures 1-a,b,c are plots of the transmission,

simulations. The transmission, reflection and absorptionof the pulse will be studied for densities in the range nl
n" = 0.05 - 0.4 and for intensities from 1016 to 1020W/
cm2. The results will then be discussed in the light oftheoretical calculations, mainly about the parametricinstabilities in the strongly relativistic regime. It will bes^o(a)r" o oE
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shown that the interaction is strongly non-linear andmay be understood by assuming that the laser plasmainteraction leads to a bulk electron heating to relativisticenergies. Calculations of parametric instabilities in such
hot plasma will be necessary in order to understand the

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numerical results.
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2. Numerical Simulations
In the numerical simulations, the electromagnetic
field is propagated by integrating the full Maxwell
equations. The electromagnetic wave is linearly

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polarized with the electric field in the plane ofincidence. In order to simulate a focused beam, the0.1ointensity profile is assumed to be gaussian with the same
5 ,l,e full width at half maximum (FWHM) for all cases,
where ,l,s is the vacuum wavelength. For the plasma, 2D
particle-in-cell simulations are used [2-3]. The

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simulation box initially contains a plasma slab, withsharp edges normal to the direction of incidence of thelaser. The slab width is 200 2.6 and in the transverse
direction, normal to the direction of incidence, thesimulation box is 73 .l,s wide. There are 74 million
particles and 8.4 million grid points. The boundary
conditions are periodic in the transverse direction. In

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order to avoid interaction between the simulation and itso(n /cmz)

periodic images, the particles are cooled as they crossthe transverse boundary. In front of the plasma slab, thelength of the vacuum region is 35 2eThe electron/ion
mass ratio is l/1836 and the initial temperature is I
KeV. The temporal pulse shape is approximately
gaussian with a full width at half maximum LT = l2OO
rrlq- 1 where @o= ftclLo. A series of simulations have

Fig. 1 a) Transmission coefficient as a function of flux

for three different densities of plasma ll) n/n. =0.1 ,* nln.=0.2,O nln.=9.4,Y nln"= 0.4and
fixed ions), b) Reflection coefficient as a functionof flux for the same parameters as figure 1a),c) Absorption coefficient as a function of flux forthe same parameters as figure 1a).
20Adam J.C. er a/., Non-Linear Interaction Mechanisms of an Ultra-Intense Laser Pulse with an Underdense plasmareflection and absorption coefficients as a function oflm

for three different density rutios nlnc. These results showthat the transmission initially decreases with 1.,
becoming very low at 1m - 10reWcm2 for nln. = 0.2 or
nln" = Q.4 and increasing again for 1, > l0reWcm2, toreach 30Vo for 1* = l02oWcm2.

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The rather large transmission at very highintensities could result from the action of the

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ponderomotive force which creates a hollow density
channel where the laser propagates without interaction
with the plasma. In order to check this interpretation,
simulations have been performed with fixed ions for 1.

1o-3-

n/ \"
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> l01eWcm2. It is seen, on figure l-a, that there is noo.1

significant modification of the transmission in this case.
Consequently, this simple interpretation does not seemto apply. Instead, it suggests that the low transmission
results from an interaction taking place in the front oft o.oothe pulse where the electrons have not yet been

do o.o4

evacuated.
0.02

The reflection coefficient reaches 0.25 for fluxes in
the l0r7 Wcm2 range and falls down for higher fluxes tonegligible values for 1, = l020Wcm2. The behavior of
the reflection coefficient for n/n" = 0.4 is enlightening:for 1, < l017Wcm2, the reflection coefficient is muchsmaller than for lower densities, as it is expected since
backward Raman instabilities are forbidden. At higher
intensities, relativistic effects allow again Raman
instabilities [4] and the reflection coefficient behaves inthe same way as for lower densities.

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The absorption coefficient brings out the
surprisingly strong energy transfer between theelectromagnetic wave and the plasma in the relativistic
regime 1, > l0l8Wcm2, whereas, at lower fluxes and ino.2

this density range, the absorption is very low, as
predicted from the linear propagation theory. The
anomalous absorption is not modified if the ions aren/noFig. 2 a) Transmission coefficient as a function ofdensity of plasma for two different fluxes

fixed, at least for 1, > l018Wcm2. It rules out the radialacceleration of ions by the ponderomotive potential as
the origin of the anomalous absorption. The main
purpose of this work is to try to understand thisrelativistic induced opacity.
Figures 2-a,b,c are plots of the transmission,reflection and absorption coefficients as a function of n/n" for three different values of 1, in the relativistic
regime. The very fast growth of the absorption
coefficient with density excludes a direct acceleration ofelectrons by the laser since it would lead to a linear
growth of the absorption coefficient with the density. It
implies that the heating process is very sensitive to the(! 3.1018Wcm'z, * 1020Wcm'?), b) Reflection
coefficient as a function of density of plasma forthe same parameters as figure 2a), c) Absorption
coefficient as a function of density of plasma forthe same parameters as figure 2a).

can be widely scattered if the plasma density is notcontrolled with a sufficient precision.
The importance of transverse effects is clear on
figure 3 where the distribution of the laser intensity has
been plotted at a given time. As already noticed [5], it isseen that strong self-focusing may occur andplasma density. It also shows that experimental results

filamentation takes place even for such narrow beams.

2lAdam J.C. et cl., Non-Linear Interaction Mechanisms of an Ultra-Intense Laser Pulse with an Underdense Plasma

These transverse processes have to be taken into accountin any explanation of the strong absorption since lD
simulations exhibit a much higher transparency [6].

F 44.

3. Parametric Instabilities
In the relativistic regime, parametric instabilities?0.

cannot be studied by using the resonant weak coupling
approximation. The laser wave propagation is non-linear

and simple solutions are available only for circularpolarization. In such a case and for a cold plasma, thestability of the solution with respect to longitudinal
perturbation has been thoroughly investigated [4]. In therelativistic regime and for 0.O5 < nln" < 0.4, the usual

forward and backward branches of the Raman
stimulated scattering merge into a mixed instability inwhich the Stokes and the anti-Stokes waves are strongly
coupled. The growth rate peaks for modes such that the
Stokes component is backscattered. Moreover thegrowth rate is very high as indicated on figure 4. In thisfigure, the maximum growth rate is plotted as a functionof the intensity as well as the corresponding plasma

s 40.
p0.

wave number /<, its frequency rat and its phase velocity
v6. The growth rates and frequency unit is the laser

pulsation ab, the wavenumber unit is the laser lightlna! {1{3wavenumber in vacuum fte and the phase velocity unit isthe velocity of light c. In these units, the frequency of
the Stokes component is l-ar and its wavenumber is 1-ft.
For intensities larger than l01e Wcm2, it is seen that ft =Fig. 3 lsovalues of the flux of the Poynting vector acrossthe y direction in dimensionless units lor nln. =0.2 and 1021&/cm2 at two different times a) 500fs,b) 840fs.

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Fig. 4 Solutions of the cold dispersion relation as a function of flux for three different densities of plasma (l) nln. = 9.1,
nln.= 0.2, A, nln. = 0.4), a) the maximum of the growth rate, b) the wave number corresponding to the
maximum groMh rate, c) the real part of the frequency corresponding to the maximum growth rate, d) the phasevelocity corresponding to the maximum growth rate.

*

22

Adam J.C. et a/., Non-Linear Interaction Mechanisms of an Ultra-Intense Laser Pulse with an Underdense Plasma

2 while a and vo become very small. These features arethose of the backward Raman instability in a very low
density plasma. It can be understood by noticing that therelativistic increase of the electron mass brings down thethe longitudinal perturbation, leading to an efficientlongitudinal heating of the electrons. This heating
should stabilize the backward modes with low phasevelocity. Such stabilization of the backward Raman
scattering has already been observed in experiments [7],where the bursting of the reflected light was interpreted

effective plasma frequency. However and on thecontrary, the growth rate is very high since it peaks at
0.4 for nln"- O.4 and reaches 0.26 for n/n" = 9.1. 11tit
values are larger than the effective plasma frequency a4l
7. It implies that the Raman instability is in the direct
Compton regime. It is also seen that the absorption
coefficient follows nicely the growth rate dependence on
the intensity. Consequently, it is natural to link theobserved large absorption to the parametric instability inthe relativistic regime.

as signs of the wavebreaking for the longitudinal
perturbation.

In order to settle this point in a more quantitative
way, the parametric instability dispersion equation has
been established for longitudinally heated plasmas [4]and solved numerically. The unperturbed electron
distribution function in momentum space is assumed tobe cold in the transverse direction. As regards thelongitudinal velocity dependence, the distribution

4. Electron Heating
If the reflectivity is now considered, the picture
does not fit so well the results, since the instability isfunction is the sum of a dirac function and of a gaussianwith a zero average momentum and a rms momentum d
in mc units, which is supposed to take into account theformation of hot tail. It has been found that a hot taildoes not modify the instability, even when it contains
20Vo of the particles. Consequently, only distributions
with 1007o hot particles will be considered here.

essentially a stimulated backward scattering whichshould generate a strong reflection. In order to explain
this discrepancy, the very fast growth of the instability
and the very low phase velocity of the plasma wave
have to be taken into account. These features show that,after a few femtoseconds, wavebreaking will occur for

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Figure 5 shows that for an intensity I = 3 x l0r8w- 0.2, the maximum growth rate is only

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Fig. 5 Solution of the warm dispersion relation as a function of flux for nln. = 0.2 and different values of the <thermal>

spread(46=0.,*6=1.,!6=2..O6=4.),a)themaximumofthegrowthrate,b)thewavenumber

corresponding to the maximum growth rate, c) the real part of the frequency corresponding to the maximum
growth rate, d) the phase velocity corresponding to the maximum growth rate.
z3

Adam J.C. et al., Non-Linear Interaction Mechanisms of an Ultra-Intense Laser Pulse with an Underdense Plasma

0.06 for 6= I and there is no more backscattering. For
larger 6, the growth rate decreases and it becomes
negligible for 6 > 3. It is noticed that, for 6 = I thephase velocity is 0.8 in c units, which implies that theallowing to reach very high average energies.

6. Conclusions
Numerical simulations have shown that moderately
dense plasmas absorb efficiently short pulses of laser
light in the relativistic regime. This anomalous
absorption seems to be linked to the excitation of strong
Raman-type parametric instabilities. These instabilities
lead to a bulk heating of the electrons which stabilizes
the backward scattering instability and leaves stimulated
Compton scattering instabilities to further heat thelongitudinal wave is strongly damped. In this case andfor hotter plasmas, the laser pulse is scattered andabsorbed by a strong kinetic stimulated Compton
scattering with a large frequency shift. At high intensity,
the backscattering is ruled out and the growth rate
becomes negligible for much higher values of 6. This isconfirmed by the PIC simulations which show that the
average electron energy, in the pulse, is 3 MeV for 1= 3
x 10r8Wcm2, while it peaks at 20 MeV for 1= 1020W

electrons to relativistic energies.

The intensity, the pulse duration and the plasma
density have been chosen in ranges corresponding to

cmz.

present experiments and fitting the current capacity of
the numerical simulations. It is not yet possible to draw
conclusions for the conditions of fast ignition. However,it indicates already that the main pulse may loose energy
if the preformed channel still contains a low density

5. Transverse Effects
Transverse effects have been studied theoreticallyin the cold plasma case [8]. It has been shown that the
instability features of the lD case extends in thetransverse direction, so that instabilities with large

plasma.

transverse mode numbers are expected to be excited at
the same level as the purely longitudinal modes.

References

However, the growth rate peaks always for purely
longitudinal wavenumbers. In the simulations, these[1] J.A. Cobble et al.,Phys. Plasmas 4, 3006 (1996).

[2] A.B. Langdon, B. Lasinski, in Methods ofComputational Physics (Academic Press, New
York, 1976) Vol.9, p.327.

oblique modes are found to have amplitudes lower thanthe parallel ones but they are responsible for theobserved transverse heating which is slightly smaller[3] J.C. Adam et al., J. of Comput. Phys. 47, 2 (1982).
[4] S. Guerin et al.,Phys. Plasmas 2,2807 (1995).

than in the longitudinal direction. The absorption ismuch larger in 2D simulations than in lD siniulations.

l5l A. Pukhov and J. Meyer-ter-Vehn, Phys. Rev. Lett.
76,3975 (1996).
[6] J.C. Adam et c/., Phys. Rev. Lett. 78,4765 (1997).
t7l C.A. Coverdale et al., Phys. Rev. Lett. 74, 4659
(1995); A. Modena et al., Nature,377, 606 (1995).
B.
t8l Quesnel et al.,Phys. Rev. Lett. 78,2132 (1997).

This increase cannot be explained only by the transverse
heating. The oblique modes could have also an efficienteffect on the longitudinal heating by destroying the
distribution function plateau which would be formed ina short time with purely lD longitudinal modes,
avoidine then a saturation of the heatine mechanism and24