## 1. Introduction

In laser fusion, hundreds of laser beams propagate through underdense coronal (ablator material) plasmas surrounding the Deuterium-Tritium (fusion fuel) filled target. At least tens of targets would have to be irradiated in succession per second for laser fusion energy cycle economics to be viable with modest or high (50-100) gain targets. The propagation of multiple laser beams in this coronal plasma and the subsequent energy deposition must be well controlled to achieve thermonuclear ignition and significant gain [1]. Because of high laser intensities and long plasma scalelengths, the underdense plasma environment is invariably the scene of nonlinear coherent processes which are detrimental to laser fusion. This is because they can lead to backscattering losses, hot electron preheating and implosion non-uniformity. Considerable attention has been given to parametric instabilities or nonlinear optical processes in plasmas (for an entry level introduction, see [2]). The linear theory of such instabilities is well understood (see [3-6] for high frequency parametric instabilities involving electron plasma waves (EPW)) but the nonlinear kinetic theory is still rich with mysteries to be uncovered (for an introduction with some advanced elements see the recent text in [7]). This is because kinetic effects add new dimensions of velocity space dynamics, changing the distribution functions strongly (away from a dull Maxwellian) and making the resonant wave-wave interaction picture much more intricate via wave-particle and particle-particle interactions (see [8,9] for an older perspective, and [10] for a more in depth and modern one). In particular, trapping, untrapping and retrapping of (a sufficiently large number of) particles makes the *transient* behaviour well beyond the reach of nonlinear knob-kludged fluid models. Coherent, phase sensitive, nonlocal memory effects, disparate scales and bursty or intermittent structures, all make predictions difficult, toy models irrelevant, and useful simulations very challenging. In particular, the initiation of large backscattering, for instance, may depend on many processes that precede that growth in leading the distribution function away from a Maxwellian, which, had it not changed, would not have allowed backscattering growth at all.

In the present work, we apply an Eulerian Vlasov code for the numerical simulation of the one-dimensional (1D) relativistic Vlasov-Maxwell equations, to study the laser-plasma interaction process known as stimulated Raman scattering (SRS). The Eulerian Vlasov code we use was presented and applied in several references [11-16]. The numerical scheme applies a direct solution method of the Vlasov equation as a partial differential equation in phase-space without dimensional splitting. The numerical scheme is based on a 2D advection technique, of second-order accuracy in time-step. The distribution function is advanced in time by interpolating in 2D along the phase space characteristic using a tensor product of cubic *B*-spline. Interest in Eulerian grid-based solvers associated with the method of characteristics for the numerical simulation of the Vlasov equation comes from the very low noise levels inherent in these codes, which allow us to study accurately nonlinear physics in low density regions of phase-space (see also [17]), without being inundated by numerical artifacts.

Besides SRBS, and SRFS, which is the analogous Raman forward scattering instability, other high frequency instabilities may occur at least kinetically when modified distribution functions exist (see [5,6,18-20]). In the family of such structures, which are beyond the scope of fluid models, are stimulated scattering off Electron Acoustic Waves (EAW) and Kinetic Electrostatic Electron Nonlinear (KEEN) waves. These stimulated processes are therefore called SEAS and SKEENS. There are also Beam Acoustic Modes (BAM) having been identified as possibly being linked to SRBS nonlinear evolution and saturation (see for instance [21-23]). A different perspective has also been promulgated under the heading of transient enhanced instability levels attributed to rapidly changing distribution functions which diminish damping rates and thus allow larger levels of SRS than would be expected in models ignoring transient tracking of distribution functions. These come under the heading of inflationary models of SRS.

Of these, KEEN waves have the interesting feature that they do not require a pre-flattened (zero slope at the phase velocity of the wave) distribution function and are not steady-state, time-independent solutions. In other words, they are not BGK modes (see for instance [8,9]). On the other hand, EAWs and nonlinear EPWs are BGK modes. In contrast, KEEN waves involve multiple phase-locked harmonics which produce a steepened multi-mode electric field pattern that can throw particles a good distance ahead as untrapped particles which then may become retrapped and help maintain an overall wave amplitude that is not in strict local equilibrium with the plasma particle distribution. The slope of the averaged distribution function need not be zero anywhere (a necessity for EPW and EAW and BGK modes, or for stationarity) and there is no infinitesimal amplitude version of KEEN waves which are nonlinearly strongly modified phase space distribution function states. They were discovered by Afeyan *et al.* in 2002 while performing ponderomotively driven Vlasov-Poisson simulations, and while trying to explore the limits of resonance physics of EAWs. The latter were found to be of measure zero compared to KEEN waves. This was first published in the proceedings of the 2003 IFSA meeting [18].

Returning to the choice of a Vlasov code over a PIC code, say, we offer this argument. If some violent rapidly and strongly driven regime is adopted in a PIC simulation, say, solely to render the initial conditions of an underesolved model less troubling, then all such intricate physics (as found here) might go unnoticed. Or if all diagnostics that can be afforded after massively parallel and data distributed simulations only look at time averaged or time integrated quantities, again, the transient and exciting initiation processes might go unnoticed or obscured by much larger final state signatures. This is what we avoid here, and it appears the sequence of events in time that are revealed here have not been seen before by PIC or Vlasov code studies.

What causes the pre-distortion of distribution functions can easily be missed or miscalculated if coarse means of tracking the fine scale structures of phase space are adopted. This is an inherent risk in PIC codes. Typical practitioners tend to emphasize very rapidly imposed and large amplitude perturbations since otherwise they would risk drowning in slowly brewing, artificial-noise generated physics. Even with Vlasov codes, the important transient kinetic physics can be easily missed if the backscattered process is strongly promoted over all other processes by seeding it externally and artificially. Not allowing the plasma to develop its own response as it sees fit, when confronted with a high intensity laser and in the presence of thermal level plasma fluctuations, leads to blocking many phase space pathways of self-organization beyond just the backscattering channel. Instead of just stimulated Raman backscattering (SRBS), in a plasma at a given density and temperature (for sufficiently large wave vector-Debye length product values for stimulated Raman backscattering electron plasma waves, SRBS EPWs), one may also expect Raman forward scatter (SRFS), especially when the wavenumber of a small perturbation imposed in the plasma as an initial condition in the transverse field, corresponds to SRFS. We may then expect to see Kinetic Electrostatic Electron Nonlinear (KEEN) waves [18-20] driven by the pump beating with the backscattering portion of the imposed standing wave initial condition perturbation at the SRFS wavenumber. We observe that only after stimulated KEEN wave scattering, SKEENS, has caused sufficient KEEN wave growth, and the background distribution function sufficiently flattened, that Raman backscatter finally can develop in earnest. This is a novel scenario of SRBS initiation and entrenched entanglement with KEEN waves which coevolve even though eventually SRBS having the far superior growth rate outstrips the KEEN wave influence reaching even more nonlinear states later in time with positive slopes in the electron distribution function and the accompanying chaotic, bursty behaviour.

In this chapter we report new results that show that SRFS is first driven in such plasmas before SRBS can grow from small perturbations that do not directly seed it. We will show that SRFS will give rise to the excitation of KEEN waves due to the opposite direction wavenumber of the SRFS wave that was seeded by the initial scattered standing wave light field. This is then amplified by the pump through the SKEENS process. This then drives KEEN waves into their characteristic multiple-harmonic phase-locked structure and steepened electric field profiles which facilitate retrapping of particles that escape any given potential well as the overall electrostatic field adjusts to all these waves being driven and amplified. The back of the simulation box is where these processes coexist most markedly. In the middle of the simulation box, SRBS finally grows after KEEN waves reach that area and change the local distribution function by softening it. SRBS eventually swallows up the KEEN wave and dominates since its growth rate is far higher. This new scenario confirms that nonlinear trapping evolution of SRBS is not just a question of EPWs but also of KEEN waves and SKEENS and that SRFS wavenumber perturbations can initiate the latter if a standing wave already exists much before SRBS can occur. The later evolution of all these processes is very complicated still involving positive slope electron distribution functions which will then accelerate the self-destruction of these modes and render the picture even more transient, intermittent and chaotic. We stop the simulations short of that eventuality where even Brillouin scatter begins to occur and dynamics and predictions become more challenging to track requiring ion acoustic waves and fluid saturation of SRS as well via Langmuir decay instability, etc.

Nonlocal and collective kinetic effects are involved in this physics. A direct Vlasov solver, capable of resolving these kinetic processes, is used here to address some aspects of the scattering properties of SRS and SKEENS. The code evolves relativistically both electrons and ions. In Vlasov codes used to simulate these problems, noise and other numerical fluctuations are very low for the SRBS to grow from, therefore it is usual in several simulations to stimulate artificially the counter-propagating daughter light wave at a low level as an injected seed, in order to enhance the SRBS growth and to allow the saturation phase to be reached rapidly. This saturation results from the competition of non-linear effects which include frequency shift, pump depletion, damping reduction, trapped particle instability, spatiotemporal chaos, among others. A detailed study of the resulting distribution function obtained at saturation has been presented, for instance, in Strozzi *et al*, 2007, which showed that the stage following saturation involved the transformation of Raman Langmuir waves into a set of beam acoustic modes or BAM (see also [24,25]), and an EAW appears at this stage with a weak reflected light that phase-matches for scattering off this mode, a process called electron acoustic scatter (EAS). SEAS has been experimentally observed in [26,27], and has been reported in simulations of plasmas overdense to SRBS and at relativistic pump intensities [11,28], and has been also observed in underdense Vlasov simulations [29]. Distinguishing between BAMs and EAWs is discussed in the literature, see for instance [22,29]. We will avoid this discussion, because they play secondary roles in the results presented here. For the parameters we are using, which involves strongly damped electron plasma waves, our simulations are dominated by SRFS, SKEENS and SRBS, in that order. It is the purpose of the present work to study these three processes and their mutual interactions, SRFS, SKEENS and SRBS which arise during the SRS dynamics process, and which appear in the early stage which precedes the saturation of the SRBS. To avoid any interference from artificially distorted distribution functions or imposed seeding, we start the code from an initial Maxwellian distribution, and the system evolves under the influence of a pump light wave which provides fluctuations from which SRS develops, without any additional imposed initial perturbation except for a standing wave at the resonant wavenumber of SRFS. This then develops SRFS but also drives SKEENS at the backscattering portion beating with the pump electrostatic ponderomotive field which seeds a KEEN wave directly. We do not seed the counter-propagating daughter light wave to stimulate the growth of the SRBS. We identify in the early phase of the Raman interaction a backscattered light that phase-matches for scattering off a KEEN wave, and which precedes the growth and saturation of the SRBS process. These SKEENS events arise during the Raman physics, from the initial Maxwellian distribution. The signature of this KEEN wave is clearly identified in the electron distribution function phase-space, and the evolution of the system until the appearance of the growth and the saturation of the SRBS process will be followed. Possible effect of this SKEENS on the initiation and subsequent saturation of the SRBS process will be discussed.

## 2. The relevant equations of the Eulerian Vlasov code and the numerical scheme

We study this current problem by using an Eulerian Vlasov code for the numerical solution of the one-dimensional (1D) relativistic Vlasov-Maxwell equations. The relevant equations for the Eulerian Vlasov formulation are those previously presented in references [12-15] for instance. We present here these equations in order to fix the notation. Time *t* is normalized to the inverse plasma frequency *c* and to *c* is the velocity of light. We have the following Vlasov equations for the electrons and the ions distribution functions

where

(the upper sign in Eq.(1) is for the electron equation and the lower sign for the ion equation, and subscripts *e* or *i* denote electrons or ions, respectively). In our normalized units

and

The transverse electromagnetic fields

In our normalized units we have the following expressions for the normal current densities:

The longitudinal electric field is calculated from Ampère’s equation:

Test runs were made in which Poisson’s equation was used instead of Ampère’s equation to obtain the longitudinal electric field, with identical results.

The Eulerian Vlasov code we use to solve Eqs.(1-5) was recently presented and applied in [11-16] for instance. We outline the main steps for the numerical solution of Eq.(1), using an Eulerian scheme. Given *i* denotes the ion distribution function), we calculate the new value *j*_{x}, and *j*_{p} corresponding to the mesh points

We assume that at the time *x* is at the grid point *j*_{x}, and *j*_{p}. The following leapfrog scheme can be written for the solution of (6):

where

Put

Equations (7) and (8) can be rewritten as:

Which are implicit equations for

*k*=0. Usually two or three iterations are sufficient to get a good convergence. The shifted values in Eqs.(10,11) are calculated by a two-dimensional interpolation using a tensor product of cubic *B*-splines [30]. We now write that the distribution function is constant along the characteristics. Then

Again the shifted values in Eq.(13) are calculated with a two-dimensional interpolation using a tensor product of cubic *B*-splines. Details have been presented in [30]. These methods compared favourably with other Eulerian methods for the numerical solution of the Vlasov equation [31].

The numerical scheme to advance Eq.(1) from time

with

From Eq.(2) we also have

## 3. The relevant parameters

We use a fine resolution grid in phase-space, with *N* = 60000 grid points in space, and 512 grid points in momentum space for the electrons and 256 grid points in momentum space for the ions (extrema of the electron momentum are *n=*0.0825*n*_{cr}, where *n*_{cr} is the critical density. The electron temperature is *T*_{e} =2 keV. The ions have a temperature *T*_{i} = 0.5 keV. Ions are allowed to move, especially to adjust the sheath structure at the boundaries on both sides, but we noted at the end of the simulations beginning traces of stimulated Brillouin backscattering, which remained at a very weak level, and therefore ion dynamics can be ignored in the results we are presenting [This is not logical. Ion dynamics affects SRS saturation by creating IAW mediated LDI and similar instabilities. Given that EPW and KEEN waves live in these simulations, it is incorrect to assume a priori that IAWs could play no role. The only exception to this would be that you ran for a very short time in which case the results are not demonstrative of real situations which occur over 100s of ps at the very least at these intensities]. The initial flat profile of the uniform plasma with the density *n*_{e}*=n*_{i}*=*1 (normalized to *n*) extends over a length

A characteristic parameter of laser beams is the normalized vector potential or quiver momentum *I* is the laser intensity in W/cm^{2}, and *M*_{i}*/M*_{e}*=*1836. A forward propagating linearly polarized wave is injected in the domain at the left boundary at *x*=0 with

The frequency and wavenumber (

with *T*_{e} = 2 keV. The resulting roots are *et al*., 2009, for these parameters the SRBS plasma wave is heavily damped, and the damping of the SRFS plasma wave is negligible. The heavily damped regime with *et al.*, 2005). In our normalized units the Debye length *T*_{e} = 2 keV. We finally get

Equation (16) has the following roots:

The results in Eqs.(17-18) obey the dispersion relation for the electromagnetic wave:

## 4. Results

We follow the evolution of the system with a close look to the evolution in two regions of the domain, a first one at about a quarter of the length in the domain, and a second one closer to the center of the domain. We will point out important differences in the initial evolution of the spectra between these two regions depending on the level of the round-off errors which now act as a perturbation in the noiseless Vlasov code. So the initial evolution of the SRFS and SRBS is not uniform through the domain, and consequently this affect the initial evolution of the KEEN waves which, as we shall show, develops from the beginning of the Raman scattering.

### 4.1. Evolution of the system in the first quarter of the length of the plasma domain

For the parameters used in these simulations, the SRFS plasma mode with *x* between *t*= 351, 468 and 761 respectively. We see in these figures a modulation with wavelength *t*=351 and 468 respectively, in the domain

Then ^{nd} harmonic of the laser wave.

We note in Figs.(1c) a mode with a wavenumber 5.616. These results are confirmed in Figure (3), where we present the spatial Fourier modes at the time *t*=527 in the same domain *x*=280. At this stage at time *t*=351, it shows on a logarithmic scale the straight line of a Maxwellian. However, in Figure (2b) at time *t*=468, and in Figure (4,bottom right) at time *t*=761, the distribution function shows a slightly distorted (but not fully flattened) distribution function, which lower the damping rate and which can facilitate the excitation of the mode around *x*=280, is shown at time *t*=761 in Figure (4,bottom left). Figure (4,top right) shows a plot of the longitudinal electric field in

Figure (5) presents the spatial Fourier spectrum at time *t*=761 in

At this stage, we look in Figure (5b) to the wavenumber spectrum of the forward electromagnetic wave

In free space, the forward propagating wave *et al.*, 2004 and Afeyan *et al*., 2013a-f) when discussing above the spectrum. The frequency

We present in Figure (6a) the frequency spectrum of the forward propagating wave *x*=300 between *t*_{1}=664 and *t*_{2}=824. We identify the pump frequency *x*=300, during the same time. It shows the peaks with the forward wave at 3.495 (at much lower level than in Figure (6a)) and at 2.474 (at essentially the same level as the SRFS mode in Figure (6a)), and a small peak for the weakly growing heavily damped SRBS wave at

Figure (7,left) shows the frequency spectrum for the longitudinal electric field recorded at the position *x*=300, during the same time between *t*_{1}=664 and *t*_{2}=824 as in Figure (6), and Figure (7,right) shows the frequency spectrum at the same position, but between *t*_{1}=664 and *t*_{2}=984. The comparison emphasizes the growth of the modes present during this phase. We can identify the mode for the SRFS plasma wave

We have so far shown in this early stage, that the excitation of the KEEN wave and the forward scattering are dominating. We present in Figure (8) the contour plots of the distribution function for *t*=761, 937 and 966. We see at *t*=761 in Figure (8a) the same pattern as in Figure (4, top left). However, in Figures (8b,c) we see a mode coming from the right, and propagating to the left while growing, which is the SRBS plasma wave with *t*_{1}=664 to *t*_{2}=984, we see the mode at

In Figure (9a) we present in the wavenumber spectrum of the longitudinal electric field in the domain *t*=879 (to be compared with Figure (5a)). We note that the SRBS plasma peak at *t*=879. We see the now growing backscattered wave at *e*
*E*^{+} in Figure (9b) at *t*=1289, in the final stage close to saturation. Figures (10b) and (10c) shows the spatially averaged distribution function over one wavelength of the SRBS plasma wave,

There is also a local maximum on the bumpy plateau. We look in Figure (11a) to the wavenumber spectrum of the longitudinal electric field at *t*=1289, close to saturation, in the domain

In Figure (11c) we present the wavenumber spectrum for the backward wave *t*= 1289, in the domain

We present in Figure (12) the frequency spectra between *t*_{1}=1132 and *t*_{2}=1292, at the position *x*=300. The dominant peak in Figure (12a) is now for the SRBS plasma wave at

### 4.2. Evolution of the system around the center of the domain

We present in Figure (13) a plot of the longitudinal electric field at *t*=761 (at the time we present the spectrum in Figure (5), after about 39000 time steps). The maximum of the electric field in Figure (13) is between *x*=400 or *x*=500. We look in Figure (14) to the phase-space contour plot of the electron distribution function in *x*=400 or *x*=500 have reached a sufficiently high level to act as a perturbation, which stimulates the growth of the SRBS plasma wave, at the same time as the SKEENS we have identified and studied in section 4.1.

In Figure (15) we present in the domain *t*=820, the phase-space contour plots of the electron distribution function. We note the rapid growth of the SRBS plasma wave, in addition to the SKEENS. We observe in Figures (15a,b) the same pattern as in Figures.(8a,b). In Figure (15a), we see the dominant modes excited are the SRFS plasma wave with

So the pattern of excitation of the modes to the right of the domain, for instance in *t*
*t*=820, during the period of the growth, is given in Figure (16). It shows similar peaks as in Figure (5a), with the exception that now the SBRS plasma wave at

Figure (17) shows the longitudinal electric field profile at *t*=1289. We finally present in Figures (18,19) the wavenumber and frequency spectra at *t*=1289, at the end of the simulation, in the domain

In Figure (18b,c) we present the wavenumber spectrum of the longitudinal electric field, the forward wave *t*= 1289, in the domain

To identify the frequency spectra at the end of the simulation, we present in Figure (19) the frequency spectra between *t*_{1}=1113 and *t*_{2}=1273, at the position *x*=450. We identify in Fig.(19a) the local peak at *T*_{e}=2keV. With

## 5. Conclusion

In laser fusion, the coupling and propagation of the laser beams in the plasma surrounding the pellet can be the scene of nonlinear processes such as parametric instabilities, which must be well understood and controlled to keep them at low levels, since they are detrimental to laser fusion because they can lead to losses of energy and illumination uniformity. Recent publications [22,23,34-40] have identified the need for a deeper understanding of laser-plasma interactions, and the importance of a kinetic treatment of the plasma, particularly in the regimes currently being approached by the new generation of lasers, and for the treatment of modes such as KEEN waves [18-20], even newer horizons are opened up. The old picture of EPWs and their evolution is now replaced by a much richer scenario of multiple harmonic waves transiently trapping, untrapping and retrapping paricle distributions that maintain the wave on average but without the need for flat distribution functions as in the canonical BGK mode setting of lore.

We showed for the first time in this study that a seemless transition occurs from Raman forward scatter, to the standing wave excited KEEN wave very near the backscattering plasma wave so that the distribution function is strongly modified by the KEEN wave before the EPW can be excited in SRBS. For the parameters we have investigated, the SRBS process is preceded by KEEN waves and then competes with SKEENS for supremacy and eventual merging. This rich physics was not observed when strong seeding of the backscattered wave prevented any detection of these intermediate processes.

The accurate representation and evolution of the particles distribution function provided by the Eulerian Vlasov code offers a powerful tool to study highly nonlinear nonstationary processes in high energy density plasmas. We have uncovered some distinctive features of KEEN waves participating in the Raman process, using a 1D Eulerian Vlasov-Maxwell code that relativistically evolves both ions and electrons. To avoid any interference from artificially distorted distribution functions or imposed linear wave seeding, we start the code from an initial Maxwellian distribution, and a very weak scattered light field standing wave pattern which is enough to trigger both SRFS and then SKEENS. The system evolves under the influence of a pump light wave which provides fluctuations from which SRBS eventually develops. We identify in the early phase of the Raman interaction a reflected light that matches the backscattering of the pump laser off a KEEN wave whose fundamental harmonic has the same wavelength as the forward scattered light, and its appearance precedes the growth and saturation of SRBS. The evolution of the system is however modified with the results presented in section 4.2, close to the center of the simulation domain. In this region, the round-off errors have reached a level where they act as a perturbation, leading to the simultaneous appearance and growth of the SRBS process, in addition to the KEEN wave (see Figure (14)). So we have two distinct evolution scenarios of Raman scattering in the domain we study. To the right of the region

In future studies, we propose to investigate the physics of the interaction between SKEENS and SRBS, but eliminating the need for SRFS initiation. This can be achieved by driving the KEEN wave directly by the ponderomotive force generated by the beating of the pump and the appropriate seed electromagnetic wave. Driving KEEN waves directly and electromagnetically generalizes the work of Afeyan et al. [18-20] which has been based on the Vlasov-Poisson system of equations. We expect to find interesting resonances, even with KEEN waves that have significantly lower phase velocities than the electron plasma waves. The work here showed the coevolution of SKEENS and SRBS for electrostatic waves whose phase velocities were so close that their vortical structures in phase-space directly overlapped and were eventually mixed.