Suppression of stimulated Brillouin scattering by two perpendicular linear polarization lasers

Stimulated Brillouin scattering (SBS) is a basic problem for laser–plasma interactions. In this work, two perpendicular linear polarization lasers with different frequencies are combined to form a new beam. The polarization of the new beam varies between linear and ellipse, while the intensity remains constant. By adopting this method, a significant suppression of SBS is predicted due to the reduction in the effective wave–wave interaction lengths. Additionally, two linearly polarized beams would be easier to use in an experiment than an alternate approach using two circularly polarized beams. The suppression of SBS is modeled with a nonlinear wave–wave coupling model, and the model is verified with 1D particle-in-cell simulations.


I. INTRODUCTION
Stimulated Raman scattering (SRS) and stimulated Brillouin scattering (SBS) are basic problems for laser-plasma interactions. 1 They scatter incident laser energy and cause other harmful effects in both direct and indirect drivers. [2][3][4][5] In indirect inertial confinement fusion (ICF), SRS is dominant for the inner beams, and SBS is dominant for the outer beams. 6,7 Many methods have been proposed to reduce the effect of laser-plasma interactions by controlling the laser parameters or plasma parameters. Spatial smoothing, spectral dispersion, 8 and polarization smoothing 9,10 are multiple incident laser manipulation techniques to reduce SBS and SRS. Spike trains of uneven duration and delay (STUD), 11 alternating-polarization light, 12 and polarization rotation 13 are adopted to suppress the growth rate and the reflectivity of SRS or SBS by decreasing the effective interaction length.
In this article, a new form of incident light is proposed, which is a combination of two perpendicular linear polarization lasers with different frequencies. The polarization varies between linear and ellipse due to the relative phase accumulations of the two incident lights, while the intensity of the new incident light is constant. Moreover, it is easier to be set or controlled by two linear polarization lasers than two circularly polarized waves.
This article is organized as follows. The theoretical approach of the suppression of SBS by two perpendicular linear polarization lasers based on the wave-wave model is presented in Sec. II. Onedimension particle-in-cell (PIC) simulation to study the effect of the suppression of SBS caused by different incident lasers is given in Sec. III. Summaries are presented in Sec. IV.

A. Wave-wave equations
The incident light is a combination of two linear polarization lasers. Consider a vector potential, A 0 , of the form

ARTICLE scitation.org/journal/adv
where A 0 and a 0 are the vector potential and dimensionless amplitude of the incident light, respectively, with A 0 in the unit of mec 2 /e, me and e are the mass and charge of the electron, respectively, c is the speed of light in vacuum, and Ay, Az, ω 1 , ω 2 , k 1 , and k 2 are the vector potentials, frequencies, and wave vectors of incident lights with the polarization in the y and z direction, respectively. We define Ω = ω 1 − ω 2 , which is the frequency difference of these two incident lights. A simple schematic of the perpendicular linear polarization laser propagation is shown in Fig. 1. The polarization of this incident light varies between linear and ellipse due to the relative phase accumulations. The evolution of SBS in one-dimension can be described by the following wave equations: 1 A ′ is the vector potential of the backscattering light, which is combined by the backscattering lights of the incident lights, νLi is the Landau damping rate, n 0 is the electron (ion) density, ωpe = √ 4πn 0 e 2 m e is the plasma frequency, mi and Z are the mass and proton number of the ion, respectively, cs = √ ZT e mi , by neglecting the ion thermal velocity, n ′ e = δn 1 e i(k i1 x−ω i1 t) + δn 2 e i(k i2 x−ω i2 t) is the axially rapidly varying part which is responsible for backward Brillouin, and ωi 1 , ωi 2 , ki 1 , and ki 2 are the frequencies and wave vectors of ion acoustic waves stimulated in the y and z direction, respectively. A ′ can be simplified as where by, bz, ω ′ 1 , and ω ′ 2 are the vector potentials and frequencies of the backscattering lights corresponding to the incident lights in the y and z direction, respectively.
Substituting Eq. (1) into Eqs. (2) and (3), we obtain where s ω i2 are the group velocities of the waves, and they can be neglected in the homogeneous plasma. The matching relations are B. The growth rate of SBS In the simulation, a hydrogen plasma is used, and the electron (ion) temperature is Te = 1000 eV (Ti = 50 eV). The Landau damp can be calculated by the following equation in a harmonic plasma: The Landau damping is νLi = 0.0146ω 1,2 .
The threshold of the SBS absolute growth rate can be calcu- In the simulation, the growth rate of SBS is γ 0 = 1.9 × 10 12 s −1 , which is much bigger than γa.
Thus, δn 1 and δn 2 are mainly determined by t only. δn 1 and δn 2 can be solved by Eqs. (7) and (8), γy and γz are the growth rates of SBS from different directions, respectively. When SBS is saturated and for a hydrogen plasma, γy can be solved by Eq. (5) through δn 1 and δn 2 , Similarly, γz can be solved by Eq. 6, In one direction, the growth of SBS is synchronous in the whole simulation space. Before saturation, SBS would increase in a growth interval, and the growth interval tsat is determined by the temporal growth rate γ 0 . When Ω changes, γy varies with different Ω and t, but the growth time of SBS is still limited in the growth interval. Thus, the growth rate of SBS can be calculated by the time average of γy or γz in the growth interval, where I Ω=0 and I seed are the intensities of the backscattering light when Ω = 0 and the seed light, respectively,

. (21)
Thus, the intensity of the backscattering light can be accumulated by Ir = I seed eγ y t , where Ir is the intensity of the backscattering light.
We define R = I r I 0 , where I 0 is the intensity of the incident light. The reflectivity can be calculated through R Ω=0 , From these expressions, it is predicted that SBS is suppressed when Ω increases because of the monotonicity of the function e (γ y −γ 0 )t sat . Whenγy =γy min , Rs = R Ω=0 e (γ y min −γ 0 )t sat , which is the first smallest value of reflectivity for the periodicity of the cosine function, and the minimum is determined by tsat = ln I R Ω=0 When Ω is big enough, many cycles would be involved in the growth interval, and SBS can only grow in every cycle separately. In one cycle, the reflectivity of SBS can be derived fromγy and R Δ = e ( 2 π −1)γ 0 t sat R 0 . Thus, R will be saturated when Ω is too big and it will equal R Δ .

III. ONE-DIMENSION SIMULATIONS
The main mechanism in this interaction is stimulated Brillouin backscattering, and when the reflectivity is in the opposite direction of propagation, one-dimensional simulation is sufficient. A series of one-dimensional PIC simulations are performed with the code EPOCH, 14 and it runs with 120 cells per μm and 1000 particles per cell. The same resolution is used in all simulations. In the simulations, the laser wavelength and intensity are λ = 0.351 μm and I 0 = 5 × 10 14 W/cm 2 , respectively, the plasma length and density are L = 0.351 mm and n 0 = 5 × 10 20 cm 3 , respectively, and the electron (ion) temperature is Te = 1000 eV (Ti = 50 eV), which satisfies the weakly damped case.
The reflectivity variation in SBS versus time when the frequency difference Ω = 0 GHz is shown in Fig. 2(a), and the spectrum map of the electrostatic field is presented in Fig. 2(b), which shows the SBS is dominant. The growth rate of R Ω=0 is γ 0 = 1.9 × 10 12 s −1 , which corresponds to 302 GHz. The saturation value of SBS reflectivity is chosen to represent the reflectivity of different frequencies. As shown in Fig. 2(a), in the growth interval, the reflectivity grows monotonically until the saturation is about 0.16.
The reflectivity of SBS when Ω = 400 GHz is shown in Fig. 3(a), and Fig. 3(b) shows the spectrum map of the electrostatic field. The reflectivity of SBS when Ω = 400 GHz is suppressed obviously to about 0.04. Figure 4 gives the reflectivity of SBS and spectral analysis when Ω = 3000 GHz, which shows the SBS is not suppressed any better when the frequency difference is too big. On the one hand, these figures show the suppression of SBS with different frequency differences. On the other hand, it proves the hypothesis that the growth of SBS is limited in the same growth interval. The relations of the reflectivity of SBS versus different frequency differences are obtained and plotted in Fig. 5. The reflectivity reduced monotonically before the lowest reflectivity when Ω = 400 GHz which is near the value predicted by theoretical equations, π 1 t sat = 360 GHz. The smallest reflectivity R = 0.04 is also consistent with the theoretical value, R = 0.0448. In theory, the  saturation will appear after π 1 t sat /2π = 360 GHz, and the simulation is consistent with it. The reflectivity increases a little after 360 GHz because more than one cycle is involved in the growth interval and the SBS returns to be resonant.
Moreover, a series of simulations with different plasma lengths are performed. The plasma length is set to L = 0.175 mm, about 500λ, and the lowest reflectivity appears at about Ω = 400 GHz, as shown in Fig. 6, which is near the value of Ω = 469 GHz, which proves the correctness of the theory.
The variation in reflectivity versus Ω can be explained physically as well. SBS will increase when the backscattering light and polarization direction are in the same direction. The variation in polarization between linear and ellipse will reduce the effective interaction length of SBS. When Ω = 0, the backscattering light and polarization direction are in the same direction all the time, and the reflectivity of SBS reaches the highest value. When Ω is small, polarization of some incident lights is not in the same direction as the backscattering light, and the growth of SBS is suppressed. When Ω is big enough, the polarization changes fast enough where many cycles are involved in the growth interval. The growth of SBS is the same as in a single cycle, and the reflectivity will saturate at a low level all the time.

IV. CONCLUSIONS
In order to suppress SBS, a new form of incident light, which is a combination of two perpendicular linear polarization lasers with different frequencies, is proposed. A new theory of it ARTICLE scitation.org/journal/adv is proposed and proved. The reflectivity of SBS is suppressed from 0.16 to 0.04, which is about four times smaller than that of the single linear incident light with a suitable choice of frequency difference. The reflectivity of SBS reduces monotonically before saturation, and it will saturate when the frequency difference is too large. The lowest reflectivity can be predicted by theory when Ω = π 1 t sat , and the variation in reflectivity can be described precisely as well. Additionally, two linearly polarized beams would be easier to use in an experiment than an alternate approach using two circularly polarized beams.