Kinetics of the Raman Instability in a Laser Plasma

Transcription

1 WDS'05 Proceedings of Contributed Papers, Part II, , ISBN MATFYZPRESS Kinetics of the Raman Instability in a Laser Plasma M. Mašek and K. Rohlena Institute of Physics, Academy of Sciences of the Czech Republic, Prague, Czech Republic. Abstract. The phase space evolution in a non-relativistic and homogeneous laser plasma in the presence of the stimulated Raman scattering is investigated. The mechanism of the electron acceleration in the potential wells of the plasma wave accompanying the Raman backscattering is analyzed by Vlasov-Maxwell simulations. In addition to the interaction with the primary wave a cascading is observed consisting in a secondary scattering of the primary Raman backscattered wave. The corresponding backward propagating plasma wave can accelerate electrons away from the target against the direction of the heating laser beam. Moreover, there is an indication of a non-linear quasi-mode combined from both the electrostatic waves with a strong tendency for trapping. The transform method is used for a solution of the set of partial differential equations which consists of the Vlasov equation and of the full set of Maxwell equations. To overcome numerical instabilities during the simulation, a simplified Fokker-Planck collision term was introduced. Introduction The stimulated Raman scattering (SRS) is one of the parametric instabilities, which develop in the underdense region (below quarter-critical density) of laser-produced plasmas. SRS can be simply characterized as a decay of incident pumping electromagnetic wave in another electromagnetic wave and in a fast forward going electrostatic Langmuir wave. Very well known feature of large amplitude electrostatic waves propagating through a plasma is the trapping and the accelerating electrons. Discovery of the wave acceleration has led to an intense study of this phenomenon mainly in the context of laser fusion experiments in the recent years. In direct drive fusion experiments the generation of hot electrons is an important issue, because it causes the target preheating and prevents the efficient compression of the fuel capsule [Rousseaux et al., 1992]. In the fast ignition scheme an ultra short ( 10 ps) and ultra intense heating laser pulse follows another longer ( 100 ps) hole boring pulse or it is applied through a guiding cone. The idea is to separate the processes of implosion and heating of the fuel [Tabak et al., 1994]. The both pulses may interact with the corona plasma surrounding the core earlier than they reach it. One of the possible interactions is SRS, which causes an energy loss of the primary electromagnetic wave and decreases the energy conversion from the laser pulse to the fuel capsule. Then the Raman reflectivity becomes a very important parameter. Last but not least, laser plasmas can be used as an effective accelerator of the plasma expansion. In this configuration the fast electrons guide the plasma expansion and decrease the time available for the destruction of high-charge ions by the recombination [Rohlena et al., 1996]. The Raman scattered electromagnetic wave can propagate in the backward and the forward direction with respect to the direction of motion of the laser pulse. In both the Raman backscattering (SRS-B) and the forward Raman scattering (SRS-F) the electron plasma waves travel in the forward direction, but their velocities may differ considerably. Slower SRS-B electrostatic wave in the laser-produced plasmas usually dominates over SRS-F, whose phase velocity is often comparable with the speed of light. With a high phase velocity of SRS-F electrostatic wave a small Landau damping rate is connected and vice versa, in the well underdense region of the plasma corona or in the high temperature plasma, where the SRS-B electrostatic wave is strongly Landau damped, SRS-F can become comparable with SRS-B in spite of its small growth rate [Karttunen et al., 1992]. As it was mentioned, the phase velocity of SRS-B electrostatic wave is closer to the thermal velocity then in the case of SRS-F. Hence, the large amplitude SRS-B plasma wave can interact with a significantly larger amount of electrons, which can get trapped in the wave. SRS can also couple to other plasma instabilities, which leads to a cascading, whereas the deformation of electron distribution function and the occurrence of population of the trapped accelerated electrons gives rise to new nonlinear modes of which the impinging laser may also be scattered [Salceda et al., 2003]. The simultaneous existence of SRS-B and SRS-F plasma wave in the plasma may lead 383

2 to a non-resonant difference quasi-mode formation generated by a nonlinear interaction of participating electrostatic waves [Malá, 2002] with the phase velocity ω B ω F k B k F, (1) where ω B, resp. ω F are the frequency of electrostatic wave corresponding to the SRS-B and SRS-F, respectively and k B, resp. k F are the wave numbers of these wave modes. Phase velocity of quasimode lies well within the bulk of the electron distribution and thus a strong interaction with the plasma electrons and a modulation of the electron distribution function is to be expected. The other possibility (+ sign) in (1) falls outside the range of interest. This paper is devoted to the study of the Raman instability using a numerical model based on the Fourier-Hermite expansion of the electron distribution function [Armstrong et al., 1970]. This approach is well applicable for a simulation using a model of a periodic slab and it is a good fit to a long scale length plasmas. The most crucial feature of the code is its numerical instability. On the other hand, using this method it is possible to model the initial condition quite naturally as a low amplitude white noise and no artificial initial condition is necessary. Moreover, the time evolution of the wave mode interacting with other modes is treated directly in the Fourier space. The resulting k spectra could be directly compared with the experiment. The numerical stabilization of the method is ensured by employing a simplified Fokker-Planck collision term [Grant et al., 1967]. Then it is possible to push the solution to region of nonlinear wave-wave interaction. Nowadays, a number of other Vlasov solvers have been developed. A good survey and also results of testing of these approaches is given e.g. in [Filbet et al., 2003]. These solution methods of the kinetics of the nonlinear wave-plasma interactions have also been applied in a number of other papers, for instance [Gagné et al., 1977], [Shoucri et al., 1978], [Ghizzo et al., 1990], [Bertrand et al., 1994], [Ghizzo et al., 1995] and [Califano et al., 2003]. Compared to the transform methods (in our case represented by the Fourier-Hermite expansion method) they have numerous advantages like a need for less computer time and a better numerical stability. The stability is achieved by a numerical smoothing of the electron distribution function after each time step, which leads to the spurious entropy production even in the absence of collisions. From this point of view, it seems better to use the transform method stabilized by the collision term with realistic value of the collision frequency [Mašek et al., 2004]. Moreover, this method also allows to use a more realistic initial condition in the form of low level white noise. The main advantage of the direct Vlasov equation solution in comparison to the very popular Particle-in-Cell (PIC) method is the absence of the numerical noise. For instance, the acceleration process is a relatively weak effect in the phase space, which may become comparable with the noise generated by the PIC methods. Similarly, if we are looking for non-resonant quasi-mode interaction or a loss of periodicity of the wave the PIC methods need not always be adequate [Bergmann et al., 1993]. This fact then could make problematic a correct interpretation of some kinetic effects obtained with the PIC models. Vlasov-Maxwell Model of Laser Corona The model presented in this contribution is centered on the PALS experiment, which represents a laser system with a subnanosecond pulse (400 ps) in a near infrared region (1.315 µm) generating a power density on the target typically W/m 2. Electron temperature of the corona achievable by the laser is estimated to be T e = 10 7 K ( 0.9keV ). Table 1 summarizes all the parameters of the laser and of the plasma in the corona. The laser pulse duration is relatively long and the incident electromagnetic field is not strong enough to cause the oscillatory motion of electrons to become relativistic, so we are able to simulate these Table 1. Parameters of the incident laser light and of the plasma in the corona, which are typical for the PALS experiment and which are used in the simulation. (I L - laser power density on the target, λ vac - laser vacuum wavelength, ω L - laser frequency, τ - pulse duration, T e - electron temperature in corona, θ = KT e /mc 2, n e - electron density, ω pe - electron plasma frequency.) Parameter Value Parameter Value I L W/m 2 T e K λ vac µm θ ω L s 1 n e m 3 τ 0.5 ns ω pe s 1 384

3 experiments as non-relativistic. The facility is working in the near infrared region, thus the plasma is in the corona almost collisionless with respect to the frequency of the laser light. In the case of SRS, the ions behave as a homogeneous neutralizing background. Because of computational capacity we restricted the calculation to one dimension only. Since we are interested mainly in the time evolution of electron distribution function a set of equations consisting of Vlasov equation for the distribution function and of full set of Maxwell equations was solved. The Maxwell equations are transformed using the Coulomb gauge ( A = 0) with a non-zero electrostatic potential appearing in the Poisson equation: f t + v f x x + e ( ϕ m e x e m e A A x ) f v = ν c ( (vf) v ) + 2 f v 2, (2) [ 2 x ] c 2 t 2 ω2 pe n e c 2 A = 0, (3) n 0 2 ϕ x 2 = e m e (n e n 0 ), (4) where A is the only non vanishing transverse component of vector potential, ϕ is the electrostatic potential, c is the speed of light, x the spatial coordinate (propagation direction), t is the time, v x is the velocity in the parallel direction, m e is the electron mass, e the electron charge and n 0 is the initial density of electrons and ions. In the Vlasov equation, the velocity in the perpendicular direction is replaced by the mean oscillatory velocity in the field of incident laser light v y = ea/m e. It means that a monokinetic fluid description is considered in the perpendicular direction [Bertrand et al., 1995]. The simplified Fokker-Planck collision term on the right hand side of the Vlasov equation (2), where ν c is collision frequency, is added for a temporal prolongation of the solution. For the normalization of the electron distribution function f we assume n e = f dv. (5) n 0 Usual transformation to dimensionless quantities is used. Time is compared with the electron plasma frequency ω pe, spatial coordinate is measured in the Debye lengths λ D. Then unit of velocity will be electron thermal velocity v T = KT e /m e, where K is the Boltzmann constant. The transformation method based on the Fourier-Hermite expansion of the distribution function is used for the solution. Numerical stabilization is ensured by employing the simplified Fokker-Planck collision term. An adequate number of terms of Hermite serie (700) and Fourier serie (100) is used to reach a satisfactory accuracy of computations. Detailed description of the solution method and discussion of its numerical stability can be found in an earlier paper [Mašek et al., 2004]. Results of Vlasov-Maxwell Model The solution of the above system renders time dependence of the electromagnetic as well as of the electrostatic spectra and the evolution of the distribution function in the phase space. The results of Maxwell-Vlasov simulations under condition of PALS experiment (see Table 1) are summarized in Table 2. In Table 2, one can find the dependence of the wave numbers and of the phase velocities of the SRS-B and SRS-F electron plasma waves on the ratio of electron density over the critical density (n crit ) of plasma obtained from the Vlasov-Maxwell simulations. As it follows from the values of SRS-B plasma wave phase velocities, the plasma wave is strongly Landau damped in the region of plasma corona Table 2. Wave numbers and phase velocities of electrostatic wave modes and SRS-B plasma wave growth rates (γ B ) obtained from Vlasov-Maxwell simulations in different regions of plasma corona. ω pe [s 1 ] n/n crit k B λ D k F λ D v phb /v T v phf /v T γ B [s 1 ] v phquasi /v T

4 0,16 0,14 0,5 Electrostatic spectrum 0,12 0,10 0,08 0,06 0,04 0,02 0,4 0,3 0,2 0,1 Electromagnetic spectrum 0,00 0,0-0,02 0,0 0,1 0,2 0,3 0,4 0,5 k λ D 0,0 0,1 0,2 0,3 0,4 0,5 k λ D Figure 1. Perturbed part of electrostatic (left) and electromagnetic wave number spectra at ω pe t = 200. with electron density approximately below n e /n crit = It is then clear that the most complex structure of the electrostatic and electromagnetic wave number spectra will be found in the case of this well underdense plasma, where SRS-B and SRS-F plasma waves exist simultaneously. They can nonlinearly interact with each other and combine in a non-resonant electrostatic quasimode. Moreover, another electrostatic mode appears in the wave number spectrum at a still lower electron density. We interpret it as an electrostatic mode connected with the secondary decay of the backward going scattered electromagnetic wave. The accompanying electrostatic mode then travels backward with respect to the impinging laser pulse with the phase velocity a little greater than the phase velocity of the SRS-B plasma wave. It has also a strong tendency to trap electrons but it accelerates them in the direction away from the target. The distribution function is thus affected most significantly because of existence of the mentioned plasma wave modes with the phase velocities close to the thermal velocity of electrons. Note that the region of simultaneous existence of SRS-B and SRS-F is broader in higher temperature plasmas [Karttunen et al., 1992], which are achievable e.g. in Tokamaks. As it follows from the above discussion the most interesting for our simulations is the well underdense region (below the value of electron density n e /n crit = 0.05) of plasma corona. To demonstrate the results of the code we have chosen the value of electron plasma frequency ω pe = s 1 and for the numerical stabilization of the computation the value of collision frequency ν c = 0.05 ω pe, which is realistic for the conditions of the simulation mainly for high-z plasmas. The electrostatic and electromagnetic wave number spectra obtained from such simulation at ω pe t = 200 are depicted in Fig. 1. We observed a strong growth of the SRS-B plasma wave and a weaker growth of SRS-F for the above parameters. The most visible peaks in both spectra thus corresponds to the SRS-B and their positions match the condition for the wave numbers exactly. The wave number of incident electromagnetic wave is k L λ D = and the wave numbers of the plasma wave and of the backscattered electromagnetic wave are k B λ D = and k λ D = 0.144, respectively. Smaller peaks in the electrostatic spectrum corresponds to SRS-F and secondary scattering of backscattered electromagnetic wave. A more detailed view can be found in Fig. 2, where the electrostatic spectrum is depicted in a logarithmic scale. It is easily seen there the peak corresponding to the quasi-mode, which is caused by non-linear interaction between SRS-B and SRS-F electrostatic wave, with the wave number k quasi λ D = A peak corresponding to the SRS cascade is also denoted there. A considerably broadening of the electrostatic spectrum is caused by the nonlinear effects, which become important at this time. In Fig. 3, the fast electron generation is illustrated by a phase-space plot of the electron distribution function at (a) ω pe t = 0, (b) ω pe t = 100, (c) ω pe t = 120, (d) ω pe t = 140, (e) ω pe t = 160, (f) ω pe t = 200. A linear scale of the contours is used and the values of distribution function are in the interval between f = 0.0 and f = 0.1. The small initial perturbations are growing quickly as the amplitude of forward going SRS-B plasma wave grows (Fig. 3(a) and (b)). The motion of electrons with the velocity in vicinity 386

5 -1 SRS-F quasimode SRS-B -2 secondary SRS-B log E ,0 0,1 0,2 0,3 0,4 0,5 k λ D Figure 2. Perturbed part of electrostatic wave number spectra in a logarithmic scale at ω pe t = 200. of the phase velocity v phb /v T = 3.45 starts to be strongly affected by the SRS-B plasma wave potential (Fig. 3(c)) and they are gradually accelerated. Due to a strictly periodic structure of electrostatic field, the motion of resonant electrons becomes organized and they travel with higher velocity than the phase velocity of plasma wave to the right, where its motion can be decelerated by another potential maximum and their energy is transferred back to the wave (Fig. 3(d)). When the amplitude of the electrostatic wave reaches a higher value, these electrons become trapped. Closed loops in the phase space plot of distribution function (Fig. 3(e) and (f)) indicate well trapped electrons. From a simple analysis of particle dynamics it is possible to find a relation for relative velocity which the particle gains from the electrostatic wave: 2eEB v rel =, m e k B where E B is the amplitude of SRS-B plasma wave. At ω pe t = 160, when the electrostatic wave reaches its maximum, the value of plasma wave amplitude is E B = V/m. It means that resulting relative velocity of accelerated electrons is v rel /v T = 1.5. Fig. 3(g) and (h) shows two distribution functions at ω pe t = 100 and ω pe t = 200, respectively, corresponding to the spatial averages over the simulation box. The dominant wave-particle interaction is due to the plasma wave created by SRS-B. A plateau is formed in the distribution function in the vicinity of the phase velocity of SRS-B plasma wave v phb /v T = In later stages of phase-space evolution the amplitude of plasma wave is gradually decreased as its energy is transferred to the electrons and the motion of the accelerated electrons is no longer affected by the wave potential, thus they stream freely with a velocity larger than the phase velocity of the SRS-B plasma wave. The second possible mechanism of accelerated electrons detrapping is a loss of electrostatic field spatial periodicity due to the coexistence of several other electrostatic wave mode in the plasma, e.g. by the cascading. This chaotization leads to a removal of potential barriers for the trapped electrons and they become free. Contrary to the acceleration scheme proposed in the paper [Bertrand et al., 1995], the plasma in the corona in the PALS experiment has to a low temperature ( 0.9keV ) with too little electrons in the relativistic range, thus the two-stage electron acceleration is not possible because of very high phase velocity (compared to the speed of light) of SRS-F plasma wave. Under the condition of the PALS experiment, the SRS-B growth rate remains too small, so as for the amplitude of SRS-B plasma wave to reach values allowing a phase space overlap of the trapping regions of both SRS-B and SRS-F plasma wave modes [Malá et al., to be published]. Moreover, considering the negligible amount of electrons 387

6 (a) (e) (b) (f) (c) (g) (d) (h) Figure 3. Phase-space contour plot of electron distribution function evolution at times (a) ω pe t = 0 (b) ω pe t = 100 (c) ω pe t = 120 (d) ω pe t = 140 (e) ω pe t = 160 (f) ω pe t = 200. Linear scale is used and the depicted values of distribution are in the interval between f = 0.0 and f = 0.1. Formation of plateau in vicinity of SRS-B plasma wave phase velocity is demonstrated by (g) at time ω pe t = 100 and (h) at time ω pe t = 200. A spatially averaged distribution function over the simulation box is depicted there. Note that logarithmic scale is used in these graphs. 388

7 in the vicinity of SRS-F plasma wave phase velocity and the small SRS-F growth rate in the plasma generated by the PALS facility, the acceleration in SRS-F plasma wave is not effective without some pre-acceleration process, e.g. by SRS-B plasma wave. As it was mentioned above, we observed the growth of the electrostatic non-resonant quasi-mode. The phase velocity of this mode lies on the body of the electron distribution v ph /v T = 0.54, where it can interact with the relatively large number of electrons. Because of its relatively small amplitude we recorded no significant modification of electron distribution function caused by this mode. The main effect of the quasi-mode thus remains the pre-acceleration of electrons from the body of distribution. A larger gain of SRS-B acceleration process then should be achieved with the pre-acceleration in the electrostatic quasi-mode. The amplitude of the primary backscattered electromagnetic wave reaches so high value that it can undergo, in turn, a subsequent scattering process, which leads to origin of backward going (with respect to the impinging laser pulse) electrostatic wave. The amplitude and the phase velocity of this wave is comparable to the values of SRS-B plasma wave. The phase velocity of this secondary scattered wave is v ph /v T = This value of phase velocity allows the wave in the considered plasma to feed particles into the tail of distribution function. These electrons move in the direction away from the target and accelerate the plasma expansion. In experiments, where the laser-produced plasma is used as a source of charged and accelerated particles, mechanisms speeding up the plasma expansion are helpful. SRS-B coupling to the SRS cascade is one of such mechanisms, which generates fast particles, which, in turn, accelerate the plasma expansion and decrease the time available for the ion recombination. The charged particle yield thus increases. Conclusion In our simulations, we assumed a homogeneous and a collisionless plasma. It is a good approximation of a real laser plasma with a long scale length. The restriction to a non-relativistic case permits to study the fusion relevant plasmas up to the temperature of a few kev. The transform method was used to solve the set of equations consisting of the Vlasov and Maxwell equations in a 1D geometry. The difficulties of the method were overcome employing the simplified Fokker-Planck collision term and it was possible to demonstrate its ability to give a meaningful description of phase space evolution in the presence of the stimulated Raman scattering. For the chosen plasma parameters a growth of both SRS-B and SRS-F electron plasma waves was observed. Within the discrete Fourier expansion the linearly resonant modes are showing the fastest growth rates for both the scattered modes even in a strongly non-linear regime, however, with a finite line width. The model allowed the detail study of the particle dynamics trapped in the potential wells of the electrostatic wave. The plateau was formed in the distribution function as a consequence of the electron acceleration in SRS-B plasma wave. Moreover, the SRS-B coupling to the SRS cascade was observed with a significant acceleration also in the direction away from the target against the heating beam. Non-linear interaction between SRS-B and SRS-F plasma waves led to a formation of non-resonant quasi-mode with the phase velocity close to the electron thermal velocity. Vice versa, the two-stage electron acceleration is not possible under the condition of the PALS experiment. The present work was centered on the PALS experiment keeping all the parameter values within realistic estimates corresponding to this kind of laser plasma, including the collision frequency. We thus think that, in spite of the intrinsic limitations of the transform method, our results reproduce reasonably well the true conditions in the electron phase space. Acknowledgments. Support by the grant No. 202/05/2745 of the Grant Agency of the Czech Republic is gratefully acknowledged. References Armstrong, T. P., Harding, R. C., Knorr, G., Montgomery, D., Solution of Vlasov s Equation by Transform Methods, Methods in Computational Physics vol. 9, Edited by B. Alder, S. Fernbach, M. Rotenberg, Academic Press, New York and London, 1970 Bergmann, A., Mulser, P., Breaking of Resonantly Excited Electron Plasma Wave, Phys. Rev E 47, 3538, 1993 Bertrand, P., Ghizzo, A., Karttunen, S. J., Pättikangas, T. J. H., Salomaa, R. R. E., Shoucri, M., Twostage Electron Acceleration by Simultaneous Stimulated Raman Backward and Forward Scattering, Phys. Plasmas 2 (8), ,

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