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
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
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
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
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
where
(the upper sign in Eq.(1) is for the electron equation and the lower sign for the ion equation, and subscripts
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
We assume that at the time
where
Put
Equations (7) and (8) can be rewritten as:
Which are implicit equations for
Again the shifted values in Eq.(13) are calculated with a two-dimensional interpolation using a tensor product of cubic
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
A characteristic parameter of laser beams is the normalized vector potential or quiver momentum
The frequency and wavenumber (
with
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
Then
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
Figure (5) presents the spatial Fourier spectrum at time
At this stage, we look in Figure (5b) to the wavenumber spectrum of the forward electromagnetic wave
In free space, the forward propagating wave
We present in Figure (6a) the frequency spectrum of the forward propagating wave
Figure (7,left) shows the frequency spectrum for the longitudinal electric field recorded at the position
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
In Figure (9a) we present in the wavenumber spectrum of the longitudinal electric field in the domain
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
In Figure (11c) we present the wavenumber spectrum for the backward wave
We present in Figure (12) the frequency spectra between
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
In Figure (15) we present in the domain
So the pattern of excitation of the modes to the right of the domain, for instance in
Figure (17) shows the longitudinal electric field profile at
In Figure (18b,c) we present the wavenumber spectrum of the longitudinal electric field, the forward wave
To identify the frequency spectra at the end of the simulation, we present in Figure (19) the frequency spectra between
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.
Acknowledgments
The authors are grateful to the Centre de calcul scientifique de l’IREQ (CASIR) for computer time for the simulations presented in this work. BA would like to acknowledge the financial assistance of DOE OFES HEDP program through a subcontract via UCSD.
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