**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 wavewave interaction picture much more intricate via wave-particle and particle-particle interac‐ tions (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

© 2014 Shoucri and Afeyan; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

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

Stimulated Raman Scattering with a Relativistic Vlasov-Maxwell Code: Cascades of Nonstationary Nonlinear…

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253

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 selforganization 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

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

code studies.

accompanying chaotic, bursty behaviour.

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.

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

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

phase-space (see also [17]), without being inundated by numerical artifacts.

the 2003 IFSA meeting [18].

252 Computational and Numerical Simulations

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 selforganization 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.

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

Stimulated Raman Scattering with a Relativistic Vlasov-Maxwell Code: Cascades of Nonstationary Nonlinear…

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

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

, length is normalized to *l*

are normalized respectively to the velocity of light *c* and to *Mec*, where *Me* is the electron mass and *c* is the velocity of light. We have the following Vlasov equations for the electrons and the

<sup>0</sup> =*cω<sup>p</sup>* −1

<sup>r</sup> <sup>r</sup> (2)

, velocity and momentum

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−1

(*x*, *pxe*,*<sup>i</sup>*

*t x* m g

*μi* =*Mi* / *Me* is the ratio of ion to electron masses.

)<sup>2</sup> + (*a*<sup>⊥</sup> / *μe*,*<sup>i</sup>*

and *ϕ* is given by Poisson's equation, which is given here by:

2

*x* ¶ f

equations. Defining *<sup>E</sup>* <sup>±</sup> <sup>=</sup>*Ey* <sup>±</sup> *Bz*, we have:

, *t*):

<sup>2</sup> , ,, ,

*e i xe i e i e i x ei ei e i e i xe i*

*fpf <sup>a</sup> <sup>f</sup> <sup>E</sup>*

^ ¶ ¶ ¶ ¶

)2)1/2 .

and *<sup>x</sup> <sup>a</sup> E E*

f

, , ,, ,

m g

(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 *μe* =1 and

*x t*

<sup>2</sup> (, ) (, ) *e xe xe i xi xi f x p dp f x p dp*

The transverse electromagnetic fields *Ey*, *Bz* for the linearly polarized wave obey Maxwell's

 ^ ^ ¶ ¶ =- =- ¶ ¶

<sup>1</sup> ( )0 <sup>2</sup>

*x p*

+ +- <sup>=</sup> ¶ ¶ ¶ ¶ <sup>m</sup> (1)

<sup>=</sup> - ¶ ò ò (3)

will be discussed.

the inverse plasma frequency *ω<sup>p</sup>*

ions distribution functions *f <sup>e</sup>*,*<sup>i</sup>*

where *γe*,*<sup>i</sup>* =(1 + (*pxe*,*<sup>i</sup>* / *μe*,*<sup>i</sup>*

**scheme**

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.
