# SRS-SBS competition and nonlinear laser energy absorption in a high
temperature plasma

S. A. Shekhanov<EMAIL_ADDRESS>ELI-Beamlines Center,
Institute of Physics, Czech Academy of Sciences, 25241 Dolní Břežany, Czech
Republic  Faculty of Nuclear Sciences and Physical Engineering, Czech
Technical University in Prague, CZ-115 19 Prague, Czech Republic  V. T.
Tikhonchuk  ELI-Beamlines Center, Institute of Physics, Czech Academy of
Sciences, 25241 Dolní Břežany, Czech Republic  Centre of Lasers Intenses and
Applications, University of Bordeaux, CNRS, CEA, 33405 Talence, France

###### Abstract

Stimulated Raman and Brillouin scattering of laser radiation in a plasma
corona are outstanding issues for the inertial confinement fusion. Stimulated
Raman scattering may produce absorption of a significant fraction of laser
energy near the plasma quarter critical density associated with plasma
cavitation and generation of hot electrons. By contrast, stimulated Brillouin
scattering operates in a lower density plasma and prevents the laser light
access to the absorption region. In the present paper, we report the results
of analysis of competition of these two parametric instabilities with a series
of one-dimensional kinetic simulations of laser-plasma interactions. By
controlling the Brillouin backscattering through variation such plasma
parameters as ion acoustic wave damping, divergence of the plasma expansion
velocity or the laser bandwidth, we demonstrate the possibility of controlling
the level of nonlinear laser absorption and scattering in a hot, weakly
collisional plasma.

##  I Introduction

An important topic in inertial confinement fusion are the effects caused by
various parametric instabilities generated by interaction of intense
electromagnetic wave with hot dense plasma. It is important to control the
interplay between scattering instabilities (stimulated Brillouin and Raman
scattering) and laser absorption. Efficient absorption of the laser spike in
the shock ignition scheme [1] is a serious and unresolved issue. Laser pulse
propagates through a long and hot plasma corona created by the preceding
compressing laser pulse, collisional absorption is very low in such plasmas
and collective absorption related to excitation of parametric instabilities
could be important. However, it is not known how much energy can be absorbed
and reflected and how absorbed energy is distributed between thermal and
suprathermal (hot) particles.

Studies of the laser-plasma interaction in the shock ignition conditions [2]
show that stimulated Raman scattering (SRS) – scattering of laser radiation on
electron plasma waves – is a dominant process resulting in producing a large
quantity of hot electrons. These electrons can be absorbed in cold fuel and
increase the shock strength [3, 4], but there is a risk that a fraction of
these electrons penetrates upstream the shock, preheats the plasma and reduces
the shock strength. The energy distribution of hot electrons is thus needs to
be controlled in the shock igniiton scenario.

Another issue is related to absorption efficiency of the laser spike.
Stimulated Brillouin scattering (SBS) – scattering of laser radiation on low
frequency ion acoustic waves – may prevent penetration of laser radiation into
a dense plasma where SRS is excited, and thus compromise the efficiency of
laser absorption. Large amounts of SBS were observed in experiments but do not
fully explained theoretically [5].

While multidimensional simulations are certainly closer to the reality, the
simulated volume and simulation time are limited, the spatial and temporal
resolution could be insufficient and parameter space cannot be studied in
detail. For these reasons, one-dimensional (1D) simulations are still of great
importance assuming that the dominant physical processes are correctly
described. This applies to the physics of laser plasma interaction in the
shock ignition conditions, where the main competitor of SRS – two plasmon
decay (TPD) [6] – is suppressed because of high plasma temperature and
interaction is dominated by SRS and SBS competition [7].

Simplified theoretical models considering each parametric instability
separately [8] are necessary for understanding the conditions where they can
be excited but such models are insufficient for predicting the effects taking
place in realistic conditions where several instabilities are interrelated and
competing for same laser source. Moreover, secondary effects such as density
profile modification and hot electron generation modify the conditions of
instability excitation and subsequent nonlinear effects. It was demonstrated
recently that two plasmon decay instability and stimulated Raman scattering
could produce anomalous absorption of the incident laser light [9, 10]. Full
kinetic simulations using particle-in-cell (PIC) codes are widely developed.
Starting from simplified 1D geometry [11, 12, 13], contemporary PIC codes
provide possibilities for extended 2D simulations [14, 7] and 3D simulations
are also accessible [15, 16, 17].

For these reasons we return in this paper to analysis of SBS-SRS competition
in 1D geometry. Compared to previous studies [13, 15, 18, 19], we focus our
attention on the aspects that were observed but not investigated in detail.
These are: (i) dynamics of SRS evolution leading to plasma cavitation and hot
electron generation; (ii) development of SBS in a lower density plasma and
(iii) controlling of SBS-SRS competition by changing the plasma expansion
velocity, ion species composition and the laser pulse bandwidth. The
cavitation process has been reported in the 1970s [11], but it role was
underestimated. Other publications has reported on cavitation due to SBS [12]
and SRS [13, 20], but its relation to laser energy absorption and hot electron
generation was not quantified. These studies show that cavitation is a
promising process for increasing of a nonlinear laser energy transfer to
electrons with energies in the range of $10-20$ keV, which are favorable for
shock ignition.

The paper is organized as follows. The theoretical background is recalled in
section II. Numerical model and results are presented in section III. We are
using the PIC code SMILEI [21] in the collisionless version. However, several
results, which take into account electron collisions, are also presented. They
are important for evaluation of the SRS process near the quarter critical
density where the scattered wave is excited near its turning point. The
results are discussed and concluded in section IV.

##  II Theoretical analysis

###  II.1 Temporal and spatial characteristics of SBS in a linear regime

Stimulated Brillouin scattering is a three-wave parametric instability
corresponding to a decomposition of the pump electromagnetic wave with a
frequency $\omega_{0}$ into a scattered electromagnetic wave and an ion
acoustic wave with a frequency $\omega_{ia}\ll\omega_{0}$. In the case of
backward scattering, the characteristic temporal growth rate of SBS reads [8,
22]:

$\gamma_{\rm
SBS}=\frac{1}{4}\omega_{pi}v_{os}\sqrt{\frac{k_{ia}}{\omega_{0}c_{s}}},$ (1)

where $\mathbf{k}_{ia}=\mathbf{k}_{0}-\mathbf{k}_{s}\approx 2\mathbf{k}_{0}$
is the ion acoustic wave vector, $\mathbf{k}_{0}$ and
$\mathbf{k}_{s}\approx-\mathbf{k}_{0}$ are the wave vectors of the pump and
scattered waves, $k_{0}=(\omega_{0}/c)\,(1-n_{e}/n_{c})^{1/2}$, $\omega_{pi}$
is the ion plasma frequency, $\omega_{ia}=k_{ia}c_{s}$ is ion acoustic wave
frequency, $c_{s}$ is the ion acoustic velocity, $c$ is the light speed,
$n_{e}$ is the local electron density, $n_{c}$ is the critical density for the
laser light and $v_{os}$ is the amplitude of the electron quiver velocity in
the laser field.

The temporal gain is reduced if the pump wave has the finite spectral
bandwidth $\Delta\omega_{0}\gtrsim\gamma_{\rm SBS}$ [23, 24, 25, 26]. However
even in the case $\Delta\omega_{0}\lesssim\gamma_{\rm SBS}$ one can expect a
decrease of temporal gain. The growth rate can be estimated with approximate
expression:

$\gamma_{\Delta\omega_{0}}\simeq\frac{\gamma_{\rm SBS}^{2}}{\sqrt{\gamma_{\rm
SBS}^{2}+\xi^{2}\Delta\omega_{0}^{2}}},$ (2)

where the numerical coefficient $\xi$ in the right hand side depends on the
pump wave power spectrum. The temporal gain is also affected by the ion
acoustic wave damping $\gamma_{ia}$, which is of particular importance in a
multi-ion species plasma [27, 28, 29, 30, 31].

SBS is a convective instability, the amplitude of scattered wave in an
inhomogeneous plasma is amplified spatially along its propagation direction
near the resonant point corresponding to the perfect wave matching. The
spatial growth rate for SBS reads [8, 22, 32]:

$G_{\rm SBS}=\frac{1}{4}\frac{k_{ia}^{2}\,v_{os}^{2}}{v_{s}\,\omega_{s}}\int
dx\,{\rm Im}\frac{\chi_{e}\,(1+\chi_{i})}{1+\chi_{e}+\chi_{i}},$ (3)

where the integral is taken over the resonance region, $v_{s}$ is the group
velocity of scattered wave and $\chi_{e,i}$ is the electron/ion susceptibility
corresponding to the ion acoustic wave.

Examples of the SBS temporal growth rate dependence on the plasma density for
the set of parameters considered in numerical simulations presented in the
next section are shown in figure 1a for a plasma with the electron density
increasing linearly with the coordinate $x$ according to the relation
$n_{e}/n_{c}=0.05+0.23\,x/L_{n}$. Here, $n_{c}=1.1\times
10^{21}\lambda_{0}^{-2}$ cm-3 is the electron critical density for the pump
wave, $\lambda_{0}=0.351\,\mu$m is the pump wavelength and
$L_{n}=300\,\lambda_{0}$ is the characteristic density scale length. The red,
green and blue lines show the SBS growth rate in a plasma with fully ionized
carbon and hydrogen ions of equal concentration,
$n_{C}=n_{H}=\frac{1}{7}n_{e}$. The green line shows the case where two ion
species are replaced by one species with an average mass
$A_{av}=m_{i,av}/m_{p}=6.5$ and charge $Z_{av}=3.5$, where $m_{p}$ is the
proton mass. There is a significant difference from the case where two ion
species are considered separately. There are two ion acoustic modes: the slow
mode is strongly damped due to the resonant interaction with ions and,
consequently, its growth rate is suppressed (blue line). By contrast, the fast
mode is less damped and it has a significantly larger growth rate (red line).

Figure 1: Dependence of the growth rate (a) of SBS on the resonance position
and spatial gain (b) on the coordinate for the case of a plasma with
inhomogeneous density profile with the density scale length
$L_{n}=300\,\lambda_{0}$ and velocity scale length $L_{u}=300\,\lambda_{0}$.
Here, the laser intensity is $6\times 10^{15}$ W/cm2, laser wavelength is
$0.351\,\mu$m, electron temperature is 3.4 keV and ion temperature is 1 keV.

The purple line represents the growth rate for a phase-modulated laser pulse
with a Lorentzian power spectrum with a correlation time
$\tau_{c}=2/\Delta\omega_{0}\sim 0.5$ ps in the case of a single average ion
species. The laser bandwidth $\Delta\omega_{0}\simeq 4$ ps-1 is comparable
with the growth rate $\gamma_{sbs}\sim 5-10$ ps-1 in the monochromatic pump
shown in figure 1a (green line) and, according to equation (2), the growth
rate has suppressed.

The golden line in figure 1b represents the spatial gain for an inhomogeneous
plasma with the ion flow velocity $u$ depending on coordinate. Here we
consider a single ion model and a linear flow velocity profile,
$u(x)=u_{s}\,(x/L_{u}-1)$ where $L_{u}=300\,\lambda_{0}$ is the velocity scale
length and $u_{s}=0.75\,\mu$m/ps is approximately twice the ion sound velocity
$c_{s}\simeq 0.47\,\mu$m/ps. In this case the frequency of ion acoustic wave
is Doppler-shifted, $\omega_{ia}=k_{ia}(c_{s}+u)$, and the resonance
condition, $\kappa(x)=k_{0}-k_{s}-k_{ia}\approx 0$, is fulfilled in a narrow
zone depending on the density, $L_{n}=n_{e}\,(dx/dn_{e})$, and velocity,
$L_{u}=u\,(dx/du)$, scale lengths. Consequently, amplification of SBS-driven
waves is suppressed if the plasma expansion is considered. This can be seen by
comparing the golden line in figure 1b with the green line corresponding to
zero flow velocity.

Ion acoustic wave damping has a smaller effect on the spatial SBS gain as
shown in figure 1b. There is practically no difference between the spatial
gains for slow and fast modes, but the gain for a plasma with a single average
ion species is higher. These examples show the possibilities of controlling
the SBS by laser bandwidth and ion species ratio.

###  II.2 Temporal and spatial characteristics of SRS in a linear regime

Stimulated Raman scattering is the parametric instability corresponding to a
decomposition of the pump wave into a scattered electromagnetic wave and an
electron plasma (Langmuir) wave. As the frequencies of plasma and scattered
waves are comparable, SRS can be excited only in a plasma with electron
density smaller than the quarter of critical density. This is an electronic
instability characterized by a high growth rate [8, 22] as shown in figure 2a

$\gamma_{\rm
SRS}=\frac{k_{p}v_{os}}{4}\sqrt{\frac{\omega_{pe}}{\omega_{0}-\omega_{pe}}},$
(4)

where $\mathbf{k}_{p}=\mathbf{k}_{0}-\mathbf{k}_{s}$ is the plasma wave
vector. Similarly to SBS, SRS in the backward direction is a convective
instability with the spatial gain

$G_{\rm
SRS}=\frac{\pi}{8}\frac{v_{os}^{2}}{c^{2}}\frac{k_{p}^{2}L_{n}}{k_{s}}.$ (5)

which is shown in figure 2b. There is also an absolute SRS instability [8, 33,
18], which develops near the quarter critical density corresponding to the
turning point of the scattered wave. This absolute instability has a rather
low threshold

$v_{os}/c\simeq(k_{0}L_{n})^{-2/3},$ (6)

and it often dominates the interaction. The phase-modulation of the pump pulse
with correlation time $\tau_{c}\sim 0.5$ ps does not affect on the SRS
instability development because the bandwidth $\Delta\omega_{0}$ is much
smaller than the SRS growth rate.

The SRS growth rate shown in figure 2a attends its maximum in a $\sim
10\,\mu$m wide region slightly below the quarter critical plasma density. In
the lower density part of plasma, SRS is suppressed due to strong Landau
damping because the characteristic parameter $k_{p}\lambda_{D}$ increases,
were $\lambda_{D}$ is the Debye length. In our simulations, we have fixed the
intensity at $6\times 10^{15}$ W/cm2 and density scale length at
$L_{n}=300\,\lambda_{0}$.

Figure 2: Dependence of the SRS temporal growth rate (a) and spatial gain (b)
on the coordinate of the resonance point for the case of a plasma with
inhomogeneous density profile. The density scale length
$L_{n}=300\,\lambda_{0}$ in case (a) and varies from 300 to $450\,\lambda_{0}$
in case (b). The laser intensity is $(2-6)\times 10^{15}$ W/cm2 in case (a)
and $6\times 10^{15}$ W/cm2 in case (b). Here the laser wavelength is
$0.351\,\mu$m, electron temperature is 3.4 keV and ion temperature is 1 keV.

The SRS instability could significantly contribute the laser absorption via
supra-thermal electrons with energies about ten times higher than the thermal
energy. They could lead to fuel preheat in the inertial confinement fusion, so
their energy distribution has to be controlled.

Collisional damping may be an important issue for the SRS-driven plasma waves.
The scattered light group velocity, $v_{g}\simeq
c\,(1-\omega_{pe}^{2}/\omega_{s}^{2})^{1/2}$, could be small near the
corresponding critical surface, and electron-ion collisions may contribute to
the plasma wave damping [13, 34]. The minimal scattered wave group velocity
near the turning point is defined by the limit of applicability of the WKB
approximation, that is, $k_{s}\gtrsim(L_{n}\,\lambda_{D}^{2})^{-1/3}$. The WKB
theory of SRS was revisited near these turning points in Refs. [8, 33].

Collisional damping rate of the scattered electromagnetic wave is proportional
to the electron-ion collision frequency evaluated at the quarter critical
density $\nu_{ei}^{*}$, which is $\sim 2.1$ ps-1 under the considered
conditions. This value is smaller than the SRS temporal growth rate but could
be comparable or larger than the collisionless damping rate of the
electromagnetic wave trapped in the cavity (10), which is considered in the
next section. Absorption of the scattered wave propagating from the quarter
critical zone $x_{1/4}$ to the plasma edge reads:

$f_{\rm
coll.abs}=1-\exp\Bigl{[}-\frac{\nu_{ei}^{*}}{c}\int_{0}^{x_{1/4}}\,\Bigl{(}\frac{\omega_{pe}}{\omega_{s}}\Bigr{)}^{4}\,\Bigl{(}1-\frac{\omega_{pe}^{2}}{\omega_{s}^{2}}\Bigr{)}^{-1/2}\,dx\Bigr{]}.$
(7)

For the chosen plasma parameters the value of absorption coefficient is
$f_{\rm col.abs}\approx 0.3$. Thus about 30% of the scattered wave energy
could be absorbed in plasma due to collisions.

###  II.3 Collisionless SRS absorption in a cavity

Numerical simulations described in the next section show that the absolute SRS
instability may lead to a significant absorption of laser energy in plasma. It
is related to formation of density cavities near the quarter critical density
where a part of energy of scattered wave is trapped and absorbed by the
electrons. As cavity expands under the action of pondermotive pressure of
trapped wave, a part of energy is also transferred to ions. Here, we estimate
the efficiency of absorption of electromagnetic waves trapped in a cavity.

After a fast transient process, the cavity can be considered as a deep
rectangular well of a width $\Delta x_{w}$, top density $n_{e,t}$ and bottom
density $n_{e,b}$ filled with a resonant electromagnetic wave. The frequency
of the trapped wave $\omega_{w}$ is defined by the dispersion equation:

$i\,\zeta(\omega_{w})=R(\omega_{w})=\sqrt{1-\omega_{p,b}^{2}/\omega_{w}^{2}}\cot\left[(\Delta
x_{w}/2c)\,\sqrt{\omega_{w}^{2}-\omega_{p,b}^{2}}\right],$ (8)

which is obtained by imposing the reflecting boundary conditions at the cavity
edges. Here, $\omega_{p,b}$ is the plasma frequency corresponding to the
bottom well density $n_{e,b}$, $\zeta(\omega)$ is the surface impedance
expressed through the Fried-Conte $Z$-function [35]:

$\zeta(\omega)=-\frac{2\sqrt{2}}{\pi}\,\frac{v_{Te}}{c}\int_{0}^{\infty}\frac{d\xi}{1-2\,\xi^{2}\,(v_{Te}/c)^{2}-2\,\xi^{3}\,(v_{Te}/c)^{2}\,(\omega_{p,t}/\omega)^{2}\,Z(\xi)},$
(9)

where $\omega_{p,t}$ is the plasma frequency corresponding to the top well
density $n_{e,t}$ and $v_{Te}$ is electron thermal velocity. The real and
imaginary parts of an impedance calculated numerically are shown in figure 3.

Figure 3: Real (a) and imaginary (b) parts of the surface impedance for a
representative set of parameters: $\Delta x_{w}/\lambda_{0}=7.7$,
$v_{Te}/c=0.082$, $\omega_{p,t}/\omega_{0}=0.49$ and
$\omega_{b,t}/\omega_{0}=0.255$.

Equation (8) has multiple solutions, the lowest frequency modes
$\omega_{w}\gtrsim\omega_{p,t}$ are of main interest. For a representative set
of parameters, $\Delta x_{w}/\lambda_{0}=7.7$, $v_{Te}/c=0.082$,
$\omega_{p,t}/\omega_{0}=0.49$ and $\omega_{b,t}/\omega_{0}=0.255$, the real
and imaginary parts of impedance are shown in figure 3. The top plasma
frequency corresponds to the quarter critical plasma density region where SRS
occurs. As the real part of impedance is much smaller than the imaginary part,
the real part of the lowest proper frequencies $\omega_{w1}/\omega_{0}=0.258$
and $\omega_{w2}/\omega_{0}=0.266$ can be found from equation (8) with only
imaginary part of $\zeta$ retained, $-{\rm
Im}\,\zeta(\omega_{w})=R(\omega_{w})$.

Then, by taking the imaginary part of equation (8) one can find expression for
the imaginary part of the proper frequency, ${\rm Im}\,\omega_{w}={\rm
Re}\,\zeta(\omega_{w})/R^{\prime}(\omega_{w})$. The values of damping for both
modes ${\rm Im}\,\omega_{w}/\omega_{0}\simeq-1\times 10^{-4}$ correspond to
the collisionless absorption of trapped modes due to the interaction with
plasma electrons in the skin layers from both sides of the well. The
absorption coefficient averaged by a laser period is defined as:

$f_{\rm cav.abs}=-4\pi\,{\rm Im}\,\omega_{w}/{\rm Re}\,\omega_{w}.$ (10)

The value of $f_{\rm cav.abs}$ is of the order of 0.1% for the chosen
parameters.

SRS provides a flexible way for controlling the nonlinear laser absorption.
The collisional and collisionless processes compete each other since the
damping rate of the trapped electromagnetic wave is compatible with the
collision absorption. We explore this feature numerically in the next section
III.

##  III Simulation model and results

###  III.1 Input parameters

In order to demonstrate the possibilities of controlling nonlinear laser
energy absorption, we consider the interaction of an intense laser pulse
incident normally on an expanding underdense inhomogeneous plasma made of a
plastic (CH) with parameters relevant to the shock ignition scenario [1, 2].
Plasma has a linear density profile, $n_{e}/n_{c}=0.05+0.23\,x/L_{n}$,
increasing from 0.05 to $0.28\,n_{c}$ over a length $L_{n}=286\,\lambda_{0}$
for the laser wavelength $\lambda_{0}=0.351\,\mu$m. The initial electron
temperature, $T_{e}=3.4$ keV, and ion temperature, $T_{i}=1.0$ keV, are
assumed to be homogeneous in space. These are representative values reported
in numerical simulations and experiments [2, 7]. The laser pulse intensity,
$I_{0}=\frac{1}{2}c\epsilon_{0}E_{0}^{2}=6\times 10^{15}$ W/cm2, is maintained
constant during the simulation time of 8 ps, that is, about $10^{4}$ laser
periods, after a linear ramp during first ten laser periods. This time is
sufficient for achieving a quasi-steady state in the simulation.

Simulations were performed with a massively parallel fully-relativistic
electromagnetic PIC code SMILEI [21] in a 1D geometry. The code uses smooth
high-order shape functions for particles in order to suppress numerical
heating and reduce the number of particles per cell. The number of particles
per cell in our simulation is 1500, which is sufficient for maintaining a low
noise level in a 1D simulation. The simulation cell size is
$dx=0.02\,\lambda_{0}$ and the computational time step is
$dt=0.02\,\lambda_{0}/c$.

The simulation box length is about $314\,\lambda_{0}$ with $14\,\lambda_{0}$
vacuum margins at the front and rear sides of the box in order to enable a
free plasma expansion. Boundary conditions are open for electromagnetic waves,
the particles reaching front and rear boundaries are re-injected with a
Maxwellian distribution corresponding to the initial temperature. These
boundary conditions, however, do not describe ejection of hot electrons into a
dense plasma as it happens in the experiment. Instead, we evaluate the
distribution of the energy flux carried by electrons and ions in plasma near
the right boundary.

In order to evaluate the SBS-SRS competition and its role in the efficiency of
laser energy absorption, we consider six representative cases:

  1. i)

the reference case of the interaction of a monochromatic laser pulse with a
collisionless plasma with zero fluid velocity and a single ion species having
effective charge $Z_{av}=3.5$ and mass $6.5\,m_{p}$ corresponding to the
equimolar mixture of hydrogen and carbon;

  2. ii)

the reference case is repeated with the electron-ion collisions switched on;

  3. iii)

the same as case i) but with expanding plasma with a fluid velocity linearly
increasing with the coordinate, $u(x)=u_{s}(x/L_{u}-1)$, with
$L_{u}=300\lambda_{0}$ and $u_{s}=.075\,\mu$m/ps;

  4. iv)

the case iii) of expanding plasma is repeated with the electron-ion collisions
switched on.

  5. v)

the same as case i) but with two ion species, fully ionized carbon and
hydrogen of equal concentrations;

  6. vi)

the same as the reference case i) but with a phase-modulated laser pulse with
a correlation time $\tau_{c}\sim 0.5$ ps.

The results are compared in terms of laser energy absorption in plasma,
reflection and transmission, absorbed laser energy partition between electrons
and ions, and energy partition between the bulk (thermal) and hot
(suprathermal) particles.

The phase-modulated laser case was implemented using a random phase jump
technique. The time of jump is fixed on 0.5 ps and the magnitude is random
within the $2\pi$ interval. In this case the phase undergoes a random walk
following the Poisson probability function. According to the theoretical
analysis presented in section II, the expected linear growth rate for SBS
instability shown in figure 1a is $\gamma_{\Delta\omega_{0}}\sim 5.4$ ps-1 for
the laser bandwidth $\Delta\omega_{0}\simeq 4$ ps-1, which corresponds to the
correlation time $\tau_{c}\sim 0.5$ ps. Thus, the laser pulse bandwidth is
comparable with the SBS growth rate and may affect its development.

###  III.2 Particle energy evolution

In absence of electron-ion collisions, which are of minor importance under the
conditions considered here, the energy transfer between the laser and the
particles is mediated by electrostatic fields. In the case of SRS, the laser
energy is transferred to electron plasma wave, which then transfers it to
electrons, in addition, in the case of cavitation, the scattered
electromagnetic wave is trapped and transfers a part of its energy to ions.
The energy transfer in the case of SBS is weaker because of smallness of the
ion acoustic wave frequency, but then, ion acoustic wave transfers its energy
directly to ions.

The temperature of species $j=e,i$ at the position $x$ is defined via the
distribution function $f_{j}$:

$T_{j}=\frac{2}{3n_{j}}\int
d\mathbf{p}\,\varepsilon_{j}f_{j}(\mathbf{p},x,t),$ (11)

where $\varepsilon_{j}=m_{j}c^{2}(\gamma_{j}-1)$ is the particle energy,
$\gamma_{j}$ is the relativistic factor, $m_{j}$ is the particle mass and
$n_{j}=\int d\mathbf{p}\,f_{j}$ is the density. Temporal evolution of the
effective temperature of electrons and ions in the simulations averaged over
the length of simulation box is shown in figure 4.

Figure 4: Temporal evolution of the space averaged temperature of electrons
(a) and ions (b) in the runs described in the legend. Parameters are given in
section III.1.

The particle energy starts evolving at $t=0.7$ ps when laser pulse enters the
plasma. The first fast increase of electron temperature, same in all
simulations in figure 4a, is related to the excitation of SRS near the quarter
critical density. After that, two different evolution paths are observed. In
the reference case of a single species plasma with zero velocity, the electron
temperature gradually decreases independently on the presence (red) or absence
(blue) of electron collisions. A comparison of the collisional and
collisionless runs confirms the minor role played by collisions in the
reference case. By contrast, the ion temperature shown in figure 4b increases
at a later time of $\sim 3.5$ ps when the SBS instability is excited.

In the cases i) and ii) of a plasma with zero velocity shown in figure 4 with
red (collisional case) and blue (collisionless case), SBS dominates the
interaction. After a short delay of $\sim 3$ ps a strong backscattering
prevents the laser radiation to penetrate to the quarter critical density
where SRS can be excited. Consequently, SRS is suppressed and electron
temperature decreases. A small peak in the electron temperature near $t=3.5$
ps is explained by the collisional absorption of SRS reflected light excited
near the quarter critical density. As shown in table 1, only about 2% of laser
energy is absorbed in the reference case i) and distributed between electrons
and ions. The level of SBS reflectivity is very high, about 93% at the late
stage of interaction ($t>5$ ps).

# | Simulation | Reflected | Transmitted | Absorbed
---|---|---|---|---
i) | Reference, collisionless plasma | 91% | 4% | 5%
ii) | Reference, collisional plasma | 88% | 10% | 2%
iii) | Expanding collisionless plasma | 36% | 37% | 27%
iv) | Expanding collisional plasma | 37% | 27% | 36%
v) | Two ion species, collisionless | 36% | 30% | 34%
vi) | Phase-modulated pump, collisionless | 81% | 16% | 3%
Table 1: Averaged values of reflected and transmitted light for the last 4 ps
of simulation time for the six considered cases.

In the cases iii) and iv) of expanding plasma shown in figure 4 with green
(collisional case) and gold (collisionless case) lines, electron and ion
temperatures are increasing to higher levels, thus demonstrating a much better
laser absorption. As discussed in section II, divergence of the plasma
velocity suppresses SBS as the amplification length of the scattered wave is
shorter. Consequently, laser absorption is dominated by SRS near the quarter
of critical density and plasma cavitation. Similar effect has been already
observed by Klimo et al. [13]. In our case about 9% of laser energy is
absorbed and deposited into electrons and ions. The electron energy rises to
4.5 keV during first $4-5$ ps and then remains approximately constant when the
interaction enters in a quasi-stationary regime.

The role of electron collisions is more important in the expanding plasma.
Switching the collisions on results in increase of the average electron
temperature from 4.5 to 4.7 keV (figure 4a), which corresponds to a difference
of more than 20% compared to the total temperature increment $\sim 1$ keV with
respect to the initial temperature of 3.5 keV. By contrast, collisions have an
opposite effect on the average ion temperature. It increases by 40% in the
collisionless case but only by 30% in the collisional case. Explanation of the
effect of collisions on particle heating is given in the next section.

In the case v) of two ion species shown in figure 4 with a purple line, the
increase of electron temperature is similar to in the case of expanding plasma
without collisions. The improved laser absorption in this case is also
explained by the SBS suppression, but it is related in this case to a stronger
ion acoustic damping on light (hydrogen) ions. The level of SBS reflectivity
in this case is 36%. Consequently, less laser radiation penetrates to the
quarter critical density and produces a stronger electron heating. The effect
of hydrogen ions on SRS saturation and hot electron production was reported in
several experiments [36, 37, 38].

The case vi) of a phase-modulated laser pulse shows a smaller increase of
electron and ion temperatures compared to the cases iii) and v). This particle
heating is also explained by a partial SBS suppression related to a stochastic
nature of laser driver. However, the considered pump correlation time of 0.5
ps is of the same order as the SBS growth time $1/\gamma_{\rm SBS}$ as
described in section II. Consequently, the finite bandwidth suppression is
less efficient, and the level of SBS reflectivity in this case is maintained
on a high level of 81%. The energy partition in table 1 shows that laser
energy absorption in plasma increases significantly if SBS is suppressed.

###  III.3 Laser backscattering

Both SBS and SRS contribute to the backscattered signal. Their contributions
can be distinguished in the spatio-temporal plots in figure 5. Excitation of
Raman scattered electromagnetic waves, shown in the top row plots, is
correlated with the excitation of large amplitude plasma waves shown in the
bottom row. Dynamics of the Raman scattered electromagnetic wave is rather
different in the three cases presented in this figure: in the left and right
columns corresponding to cases i) and vi), the SBS reflectivity is high, which
leads to the inhibition of SRS near the quarter critical density and to
suppression of the cavitation process. By contrast, in the central column
corresponding to case iii) SBS is suppressed resulting in a stronger SRS and
cavitation.

First short SRS burst comes from the region near the quarter critical density
at $t\sim 0.7$ ps. The SRS instability develops very fast, the Langmuir waves
are excited in a region near the point $x\simeq 100\,\mu$m, corresponding to
$n_{e}/n_{c}=0.23-0.25$, where temporal and spatial gains plotted in figure 2
are maximized. At this time, reflected SRS wave is strong enough to excite
secondary SRS near the plasma density of $n_{c}/16$ at $x\approx 12\,\mu$m
manifested by short-lived bursts of plasma waves. This, however, has a very
weak effect on the laser reflection and absorption. A steady excitation of SRS
is produced later in time $t>4$ ps in the cases iii) and vi), and it is
manifested by formation of cavities – narrow localized packets of
electromagnetic field trapped in deep density depressions. There is
practically no electrostatic field in the cavities because the electrons are
expelled. Formation of plasma cavities near the quarter and one-sixteenths of
the critical density due to Raman backscattering was reported by Klimo et al.
[13].

Figure 5: Spatial and temporal evolution of the backscattered light (top row)
and electrostatic field $E_{x}$ (bottom row) for the reference case (i, left
column), expanding collisionless plasma (iii, central column) and broadband
laser pulse (vi, right column).

In the case i) (left column in figure 5), the SRS instability is excited near
the quarter critical density only at the transient stage. It produces one or
two bursts of scattered light with frequency $\sim 0.5\,\omega_{0}$ as shown
in figure 6a, which could be partially absorbed via collisional damping while
propagating through the plasma. Later in time, for $t>4$ ps, the scattered
light originates from a plasma with density less than $0.1n_{c}$, and it is
due to SBS. Similar effect is observed in the case vi) of a broadband laser
pulse (right column in figure 5). However, here SBS is partially suppressed
and the absolute SRS excited at later time, $t>4$ ps, which is manifested in
formation of two cavities and stronger backscattering in figure 6.

Figure 6: Overall view (a) and a detailed structure (b) of the spectrum of the
light reflected from the plasma for the reference case (i, red), expanding
collisionless (iii, green) and broadband (vi, blue) cases.

By contrast, SBS is much stronger suppressed in the case iii) of expanding
plasma, which is shown in the central column in figure 5. Here, in addition to
the absolute SRS, excited near the quarter critical density and produced
multiple cavities, a convective SRS is excited in a lower density plasma. The
latter is manifested by excitation of Langmuir waves in a plasma with density
$\gtrsim 0.15\,n_{c}$ and a broad spectrum of backscattered waves in figure 6.
Strong SRS excitation in the case iii) is accompanied with electron
acceleration in the Langmuir waves. The electron energy flux shown in figure 7
(red) could be fitted with a Maxwellian two-temperature distribution:

$\frac{dF}{dp_{x}}=\sum_{j={\rm bulk,\,hot}}\frac{n_{j}\,v_{x}}{\sqrt{2\pi
m_{e}T_{e\,j}}}\Bigl{(}\frac{p_{x}^{2}}{2m_{e}}+T_{e\,j}\Bigr{)}\exp\Bigl{(}-\frac{p_{x}^{2}}{2m_{e}T_{e\,j}}\Bigr{)},$
(12)

with temperatures of bulk $T_{e\,\rm bulk}\simeq 3.5$ keV and hot $T_{e\,\rm
hot}\simeq 35$ keV components, respectively. The ratio of hot to bulk relative
concentrations $n_{j}$ is about 10.7%. As the bulk electrons are reflected
from the vacuum zone to the right of plasma edge at $x\simeq 120$, the
electron distribution function is approximately symmetric and only part
corresponding to $p_{x}>0.3\,m_{e}c$ is absorbed at the right boundary.

The interpretation of respective roles of SRS and SBS in reflection of the
incident laser light is corroborated by analysis of the spectrum of reflected
light in figure 6b. The peak near the laser frequency dominates the spectrum
in simulation cases i), vi). It is downshifted by $\Delta\omega_{a}\simeq
0.0028\,\omega_{0}$, which is consistent with the frequency of a driven ion
wave $\omega_{ia}=2k_{0}c_{s}$. Spectral downshift in the two species case v)
$\Delta\omega_{a}\simeq 0.0032\,\omega_{0}$ is consistent with excitation of
the fast wave, as discussed in section II.1

Figure 7: Distribution of the electron energy flux near the quarter critical
plasma ($n_{e}\sim 0.25\,n_{c}$) on the parallel momentum, $dF/dp_{x}$
averaged over the first (blue) and last 0.5 ps (red) for the case iii) of
expanding plasma. Black dashed line is an electron energy flux fitted with
Maxwellian function (12) for $p_{x}>0$ corresponding to the initial
temperature of 3.4 keV. All curves are normalized to the incident laser energy
flux $I_{0}$.

A particular feature of SRS spectrum shown in figure 6 is that it extends to
the frequencies smaller than $\omega_{0}/2$. It contains several spectral
components but the strongest one is downshifted by
$\Delta\omega_{s}=\omega_{s}-\omega_{0}/2\approx-0.01\omega_{0}$. This shift
was predicted theoretically in Ref. [33]. It is explained by thermal electron
motion, which leads to effective increase of the plasma wave frequency and to
a correspondent decrease of the scattered wave frequency. According to
equation (56) in Ref. [33], expression for the frequency shift of the
scattered wave reads:

$\frac{\Delta\omega_{s}}{\omega_{0}}=\frac{\omega_{s}-\omega_{0}/2}{\omega_{0}}\simeq-\frac{9T_{e}}{8m_{e}c^{2}}+\frac{1}{4k_{0}L_{n}(v_{os}/c)^{1/2}}.$
(13)

The first term in the right hand side is negative and it is on the order of
0.01 under our conditions, which agrees very well with the observed shift. The
second term is much smaller, it is on the order of 0.001. A similar negative
shift was observed in several other publications [39, 7], but its origin was
not explained.

Formation of cavities is related to the fact that the group velocity of the
scattered electromagnetic wave near the quarter critical density is close to
zero and thereby it stays a relatively long time in the resonance with the
pump wave [8, 33]. Consequently, a strong ponderomotive force produced by the
daughter electromagnetic and plasma waves induces a density depression where
they are self-trapped. The plasma wave disappears when electrons are expelled
from the cavity, while electromagnetic waves are weakly damped and persist for
a long time. Ions are accelerated under the ponderomotive pressure of trapped
light as discussed in section III.4. The cavity formation is manifested by a
strong reduction of backscattering level to $\sim 25$%.

The cavity formation is terminated when the ponderomotive pressure of trapped
waves is equalized with the thermal pressure of the ambient plasma. So, the
energy stored in the cavity can be estimated as $0.25n_{c}T_{e}\Delta x_{w}$,
which corresponds to about 1 kJ/cm2 for our conditions. The cavity life time
is defined by the damping of the trapped mode, which is about 1 ps according
to equation (10). Therefore, cavitation and subsequent collisionless
absorption of trapped electromagnetic waves could be an important mechanism of
laser energy absorption as one can see in table 1.

Collisional absorption of the scattered electromagnetic wave makes also a
notable contribution to the absorption process. This can be seen in figure 4
when comparing the cases with and without collisions. In the reference case,
there is no difference between the runs i) and ii): since the SRS is
suppressed by a strong SBS in a low density plasma. By contrast, there is a
notable electron and ion heating in the case of expanding plasma: in the run
iii) without collisions, both electrons and ions are heated due to SRS
excitation and energy absorption in cavities. In the run iv) with collisions,
there is less ion heating and more electron heating due to the collisional
absorption of the SRS scattered wave.

###  III.4 Ion heating and cavitation

Excitation of parametric instabilities leads also to the ion heating. It is
related to two processes: damping of ion acoustic waves excited by SBS in a
low density plasma, $n_{e}/n_{c}\lesssim 0.2$, and expansion of cavities
created at nonlinear stage of SRS evolution near the quarter critical density.
A comparison of the ion temperature evolution for different cases in figure 4b
shows that cavitation makes a strong contribution with ion temperature
increase by more than 30% during the simulation time.

Figure 8: Distribution of the ion energy flux near the quarter critical plasma
on the parallel momentum, $dF/dp_{x}$ averaged over the first (blue) and last
0.5 ps (red) for the case v) with two ion species: (a) hydrogen and (b)
carbon. Black dashed line is an ion energy flux fitted with Maxwellian
function (12) for $p_{x}>0$ corresponding to the initial temperature of 1 keV.
All curves are normalized to the incident laser energy flux $I_{0}$. Total ion
energy flux through the quarter critical plasma is about $1.6\times
10^{-3}I_{0}$ for hydrogen and $\sim 10^{-4}I_{0}$ for carbon.

A more detailed analysis of the ion dynamics is presented in figures 8 and 9
showing distribution of the ion energy flux on the momentum $p_{x}$ near the
quarter critical density. Figure 8 shows the case v) of two ion species.
According to [28, 29], for the temperature ratio, $T_{i}/T_{e}\simeq 0.3$, the
fast mode has a smaller damping increment, ${\rm Im}\,\omega_{ia}/{\rm
Re}\,\omega_{ia}\simeq 0.24$, and it is excited in the SBS process. The phase
velocity of this mode $\simeq 0.76\,\mu$m/ps is approximately two times the
hydrogen thermal velocity and it is approximately eight times the carbon
thermal velocity. Consequently, a comparison of the left and right panels in
figure 8 shows that hydrogen ions are gaining much more energy than carbon
ions from the ion acoustic waves.

SBS develops in the case near the quarter critical density. Indeed, the
distribution function of both ion species is modified essentially for positive
velocities and in the range larger than the ion acoustic velocity. The cutoff
of the proton distribution at $p_{x}\simeq 10\,m_{e}c$ corresponds to the
proton energy of 12 keV and velocity approximately three times the phase
velocity of the slow mode. Velocity of carbon ions is about 5 times smaller
than the hydrogen velocity and they are carrying an order of magnitude smaller
energy flux.

Figure 9: Distribution of the ion energy flux entering the quarter critical
plasma on the parallel momentum, $dF/dp_{x}$ averaged over the first (blue)
and last 0.5 ps (red) for broadband laser pulse vi) (a) and for expanding
plasma iii) (b). Black dashed line is an ion energy flux fitted with
Maxwellian function (12) for $p_{x}>0$ corresponding to the initial
temperature of 1 keV. All curves are normalized to the incident laser energy
flux $I_{0}$.

Figure 9 shows the ion energy flux for two other simulations with a single ion
species with an expanding plasma iii) and a broadband laser pulse vi). SBS is
suppressed in these two runs and modification of the ion distribution function
is completely different. The ions are accelerated symmetrically in the
positive and negative directions and their velocities are 1.5 times larger
than the ion acoustic velocity $c_{s}\simeq 0.5\,\mu$m/ps, which corresponds
to the ion momentum $\sim 20\,m_{e}c$. This symmetric ion acceleration is
explained by their expansion from the cavities produced at the nonlinear stage
of SRS evolution. This interpretation is confirmed by the reduction of
backward reflectivity and electron heating observed in these runs.

##  IV Conclusion

We discussed and compared several methods of controlling laser absorption in a
hot, weakly collisional plasma via SRS-SBS competition. The reference case of
a plasma with zero fluid velocity and a single average ion species is
characterized by a relatively week laser absorption and strong SBS
backscattering. This situation corresponds qualitatively to experiments with
single ion species plasmas [38] showing low level SRS saturation and small
number of hot electrons.

Such a situation is not typical for experiments with solid density targets
where underdense plasma is expanding with a supersonic velocity. It is
important to account for this factor in numerical simulations of laser plasma
interaction. Divergence of the plasma flow velocity leads to a strong
reduction of the SBS spatial gain and makes a dramatic effect on the SBS-SRS
competition. The SBS reflectivity decreases by a factor of three, thus
allowing laser to penetrate to the quarter critical density region and excite
absolute SRS instability. The leads to a significant nonlinear laser
absorption through the cavitation and collisional absorption of the scattered
electromagnetic wave, which could be as big as 30%. Hot electron temperature
observed in our simulations does not exceed 50 keV, which is compatible with
the shock ignition scenario of inertial fusion.

Two other methods of suppression of a strong SBS amplification are the use of
a multi-species plasma and/or a broadband laser pulse. By adding hydrogen ions
with the thermal velocity close to the phase velocity of ion acoustic waves
one can dramatically reduce the SBS gain and favor SRS excitation near the
quarter critical density. The SBS-SRS competition in this case is manifested
in a very different modification of the ion distribution function. Instead of
ion acceleration in the forward direction in large amplitude ion acoustic
waves, SRS and cavitation manifest themselves in a symmetric ion acceleration
in the plasma cavities.

A phase-modulated laser pulse can also suppress SBS and favor SRS, but the
considered case with a correlation time $\tau_{c}\sim 0.5$ ps is too large
thus leading to a relatively small reduction of reflectivity. It is desirable
to increase the laser bandwidth at least $2-3$ times for a more efficient SBS
suppression.

To summarize, we revised a set of flexible methods for controlling the SRS-SBS
competition in inertial fusion plasmas. Suppression of SBS reflectivity favors
absolute and convective SRS excitation, electron acceleration and cavitation.
The latter effect could make an important contribution to the laser absorption
by trapping the backscattered light.

## Acknowledgments

Fruitful discussions with S. Weber are gratefully acknowledged. This research
was partially supported by the Project LQ1606 with the financial support of
the Ministry of Education, Youth and Sports as part of targeted support from
the Czech National Programme of Sustainability II.

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