Laser plasma instabilities (LPIs) are important but not well-understood processes in inertial confinement fusion (ICF) and include among others, stimulated Raman scattering (SRS), stimulated Brillouin scattering (SBS), and two-plasmon decay (TPD).1–6 The SRS process has attracted much attention because of the associated loss of laser energy and of generation of hot electrons to preheat the fuel.7–13 On entering an inhomogeneous plasma, the incident laser light undergoes scattering at a wide range of angles, and large-angle scattering was observed in early experiments.14,15 The scattering geometries can be categorized into two types: stimulated Raman backscattering (SRBS), where the scattered light is in the opposite direction to the incident light,16,17 and stimulated Raman side scattering (SRSS), where the scattered light is not collinear with the incident light.14,18,19

In recent years, SRSS, especially when propagating nearly perpendicular to the density gradient, has been widely observed at the National Ignition Facility (NIF),20 the OMEGA EP Laser System,21 and the SG-II laser facility,22 in multibeam LPI in hohlraums,23,24 in laser interactions with long-scale-length foam targets,25 and at the Prague Asterix Laser System (PALS) at intensities relevant to shock ignition.26 For certain parameter conditions, SRSS even predominates over over SRBS.20,22,25 It has been shown that SRSS has a great influence on laser energy loss, hot-electron generation, and transport, as well as other effects in inhomogeneous plasmas, and so it has attracted great interest among the ICF community. Theoretical studies aimed at tackling linear problems of SRSS have been carried out based on the k-space method27–32 and a path integral formula.20 Numerical studies of SRSS have been conducted with various simulation techniques, such as ray-based simulations21,22 and particle-in-cell (PIC) simulations.33,34 However, very few of these have focused on the nonlinear phenomena associated with SRSS or on the properties of the hot electrons generated by side-propagated Langmuir waves.

One of those nonlinear phenomena in particular may have a great impact on the side-scattering process, namely, rescattering. Rescattering of SRBS has been reported both theoretically and experimentally for many parameter conditions,35–39 but it has not yet been found in SRSS. Anomalous hot-electron generation due to rescattering of SRBS has also attracted much attention.35 Previous studies have shown that electron plasma waves (EPWs) will produce anomalous hot electrons, and hot electrons produced by SRBS have also been observed experimentally.40,41 The EPWs generated by the rescattering of SRBS may further heat the hot electrons generated during the SRBS process through a cascade acceleration mechanism,42 leading to the emergence of more energetic particles.35 Whether rescattering can also be found in SRSS with the possibility of accelerating electrons to superhot temperatures is still an open question, the answer to which requires an examination of the nonlinear properties of SRSS via kinetic simulations.

In this paper, we construct a theoretical model for the rescattering of SRSS. On the basis of this model, we derive the region of occurrence and threshold for higher-order SRSS, which are presented in Sec. II. Two-dimensional (2D) PIC simulations are used to verify the existence of rescattering of SRSS and to study its properties. It is shown that rescattering of SRSS occurs in the typical parameter space of direct-drive-based scenarios and is consistent with our theoretical model. A verification via 2D PIC simulations is presented in Sec. III. In Sec. IV, we explore the effect of second-order SRSS on the hot electrons. We find that heating by second-order SRSS has nearly the same consequence as that of first-order SRSS, but no cascade acceleration phenomenon is observed. In Sec. V, rescattering of SRSS is simulated under different intensities, density scale lengths, and electron temperatures, and the results are discussed in detail. The observation of higher-order SRSS in the high-intensity regime and the predominance of side scattering in almost all cases show that SRSS and the associated rescattering are robust and important processes in ICF. Finally, conclusions and a discussion are presented in Sec. VI.

An s-polarized pump laser (with frequency ω₀ and wave vector ${\mathbf{k}}_0=k_0\widehat{\mathbf{x}}$ in plasma) enters an inhomogeneous plasma at normal incidence. The density of the plasma varies linearly as n_e(x) = n_er(1 + x/L), where n_er is the resonant density and L = n_er/[dn_e(x)/dx] is its scale length. This setting is susceptible to near-90° side scattering, propagating near the y direction and polarized along z, owing to the small detuning effect induced by the density inhomogeneity.27,43 We assume such scattered light to have a frequency ω_s and wave vector ${\mathbf{k}}_s=k_{sx}\widehat{\mathbf{x}}+k_{sy}\widehat{\mathbf{y}}$. The sideward Raman scattering satisfies the three-wave matching condition$$\begin{array}{c}{\omega }_0={\omega }_p+{\omega }_s,\\ {\mathbf{k}}_0={\mathbf{k}}_p+{\mathbf{k}}_s,\end{array}$$(1)where ω_p and k_p are the frequency and wave vector, respectively, of the EPW.

After resonant growth near its turning point, the side-scattered light then bends toward the lower-density region, since the transverse density is uniform and the longitudinal density is nonuniform, as shown in Fig. 1(a). Under certain circumstances, the side-scattered light may interact with the plasma as a new pump laser, and SRS will be excited again when the renewed three-wave matching condition (1), is again satisfied. We call this rescattering of SRSS.

A schematic of SRSS rescattering in an inhomogeneous plasma is shown in Fig. 1. The incident light wave vector k₀, scattered light wave vector k_s, and EPW vector k_p can excite first-order SRSS when the matching condition is satisfied, as shown in Fig. 1(b). When the near-90° side-scattered light k_s is refracted backward to a certain density n₂, its wavenumber acquires a small component δk_x in the x direction. This provides the possibility of side scattering being excited again, since the it can be regarded as side scattering from an obliquely incident light wave. We call the SRSS excited by scattered light of SRSS second-order SRSS. There are two types of excited side scattering, namely, backward side scattering and forward side scattering, represented in Fig. 1(c) by k_s1 and k_s2, respectively. The corresponding plasma wave vectors are k_p1 and k_p2. As the scattered light propagates downward, the second-order SRSS can also excite side scattering if conditions permit, leading to third-order SRSS. This cascade phenomenon continues to higher orders, as shown in Fig. 1(a).

The conditions for the excitation of rescattering of SRSS are derived as follows. SRS occurs below the quarter-critical density of its pump laser, n_c/4, where $n_c=1.1\times 10^{21}/{\lambda }_{0\mu \mathrm{m}}^2$ cm⁻³ is the critical density of incident light and λ_0  μm is the vacuum wavelength of incident light in units of micrometers. The downward-propagating side-scattered light, having acquired a small increment δk_x, can be treated as a new incident light, and it has a new quarter-critical density $n_c^{\prime }/4$, which can be expressed as$$\frac{n_c^{\prime }}{4}=\frac{{\omega }_s^2}{{\omega }_0^2}\frac{n_c}{4}\approx \frac{n_c}{4}\left[{\left(1-\sqrt{\frac{n_1}{n_c}}\right)}^2\right],$$(2)at the point x = 0, where n₁ is the resonant density of the first-order SRSS, and the matching condition (1) is used. The resonant density of the second-order SRSS must be below this quarter-critical density: $n_2\le n_c^{\prime }/4$.

At the quarter-critical density of the first-order scattered light, $n_c^{\prime }/4$, the excited second-order waves satisfy$${\omega }_s={\omega }_{s2}+{\omega }_{L2}\approx 2{\omega }_{p2},$$(3)where both the second-order scattered light frequency ω_s2 and the second-order EPW frequency ω_L2 are near the local plasma frequency ω_p2. This gives us the density condition $1-\sqrt{n_1/n_c}=2\sqrt{n_2/n_c}$. Since the n₂ ≤ n₁, the highest density at which second-order SRSS can occur is n₂ ≤ n_c/9 = 0.11n_c. For third-order SRSS, the critical condition occurs at quarter-critical density of the second-order light, at which density first-, second-, and third-order SRSS are triggered simultaneously, and so we will have n₃ ≤ n_c/16 = 0.0625n_c. Therefore, it is easy to derive the region of occurrence for mth-order SRSS: n_m ≤ n_c/(m + 1)² (m ≥ 1). Note that the side-scattered light can only propagate to lower-density regions, in contrast to the rescattering of backward-scattered light.38

In addition, for the occurrence of SRSS, the threshold condition must also be exceeded. As already indicated, SRSS can be convectively unstable or absolutely unstable.29,30,33 The absolute threshold for SRSS is a good criterion to estimate the thresholds for rescattering. The absolute threshold is given by29$$\frac{k_p{a}_{0}c{\omega }_{pr}}{2{\omega }_0}\left[1-\frac{(2n+1){\omega }_{pr}^{1/2}}{k_p^{3/2}{a}_0^{3/2}{c}^{1/2}L}\right]-\frac{{\omega }_s{\nu }_s+{\omega }_{pr}{\nu }_p}{{\omega }_0}>0,$$(4)where k_p is the wavenumber of the EPW, $a_0=8.5\times 10^{-10}{(I{\lambda }_0^2)}^{1/2}$ is the normalized laser intensity, I is the laser intensity, λ₀ is the laser wavelength, c is the speed of light, and ω_pr is the plasma frequency at the resonant point. ${\nu }_s={\nu }_{ei}{\omega }_{pr}^2/2{\omega }_0^2$ and ν_p = ν_LW + ν_ei, where ν_ei and ν_LW are the collisional damping and Landau damping amplitudes, respectively, of the EPW. Finally, n = 0, 1, 2, …, is an integer that denotes the order of the unstable eigenstate.

In this work, we take the side-scattered light as a new “incident light,” and thus we obtain the threshold for mth-order SRSS. The quantities in Eq. (4) are changed to those at the resonant point of mth-order SRSS:$$\begin{array}{ll}\hfill & \frac{k_{p,m}{a}_{s,m-1}c{\omega }_{p,m}}{2{\omega }_{s,m-1}}\left[1-\frac{(2n+1){\omega }_{p,m}^{1/2}}{k_{p,m}^{3/2}{a}_{s,m-1}^{3/2}{c}^{1/2}{L}_m}\right]\hfill \\ \hfill & -\frac{{\omega }_{s,m-1}{\nu }_{s,m}+{\omega }_{p,m}{\nu }_{p,m}}{{\omega }_{s,m-1}}>0,\hfill \end{array}$$(5)where subscript m denotes quantities of the mth-order SRSS, and a_s,m−1 and ω_s,m−1 are the normalized laser intensity and frequency, respectively, of scattered light of the (m − 1)th-order SRSS. k_p,m is the wavenumber of the EPW and ω_p,m is the plasma frequency at the resonant point of the mth-order SRSS. If m = 1, then a_s,m−1 = a₀, ω_s,m−1 = ω₀, and Eq. (5) reduces to Eq. (4). For simplicity, damping is ignored, and, taking n = 0, we get the amplitude threshold for the mth-order side-scattered light:$$a_{s,m-1}^{3/2}>\frac{{\omega }_{p,m}^{1/2}}{k_{p,m}^{3/2}{c}^{1/2}{L}_m}.$$(6)This means that the excited scattered light needs to exceed a certain value to excite the next-order SRSS. This theoretical model is verified in Sec. III using 2D PIC simulations.

We have carried out a series of 2D PIC simulations to explore SRSS using the EPOCH code.44 A z-polarized laser propagating along the x axis in the x–y plane is assumed. The laser wavelength is λ₀ = 0.351 μm, and its intensity is I = 2 × 10¹⁵ W/cm². The simulation box has length 300 μm along the x axis and width 200 μm along the y axis; the total number of cells is 9000 × 6000, with 60 electrons per cell, and the plasma density is chosen in the range n₀ ∈ (0.05n_c, 0.2n_c), with a linear density profile (scale length L = 100 μm, Debye length λ_D = 15 nm at 0.05n_c and L = 400 μm, λ_D = 8 nm at 0.2n_c). Open boundary conditions are adopted for the fields, and a thermal boundary condition is adopted for the particles on all four sides. The electron temperature T_e = 2 keV, and the ions are fixed. The incident laser light is as wide as the transverse size of the box, and so finite-width effects on side scattering are negligible.29,33

The laser light enters the plasma from x = 0, and first-order SRSS occurs. The dispersion relation of SRSS in an increasing-density plasma is obtained as$$k_{sy}=\sqrt{\frac{{\omega }_s^2-{\omega }_p^2}{c^2}}\approx \frac{{\omega }_0}{c}\sqrt{1-2\sqrt{\frac{n_e(x)}{n_c}}},$$(7)where n_e(x) is a linear function of x. The dispersion relation for k_y is clearly seen in the time-averaged spectrum of E_z(x, k_y) in Fig. 2(a). The first-order side-scattered light spectrum is spread across the whole density range from 0.05n_c to 0.2n_c and satisfies the dispersion relation (7). In addition, there is an additional spectral feature that has a wave vector smaller than that of the first-order SRSS and starts just below 0.11n_c (red dashed vertical line). This special light wave mode can be inferred from the averaged k_x–k_y spectrum of E_z presented in Fig. 2(b). This shows that there are two kinds of wave vector, both of which are at nearly 90° to the x direction: the one with the larger wave vector k_s corresponds to first-order SRSS, and the other smaller one to second-order SRSS with wave vectors k_s1 and k_s2. The rescattering has a weaker intensity and a narrower spectral width than that of the first-order scattered light. At t = 2 ps, the scattered light intensity is about I_s = 1.6 × 10¹⁵ W/cm², and, by Eq. (6), the second-order SRSS meets the threshold condition. Thus, rescattering of SRSS has been observed in our simulation, and is consistent with our theoretically determined region of occurrence and threshold.

Further analysis of the EPW spectrum E_x indicates that the three-wave matching conditions are satisfied. In Fig. 2(c), we plot the time-averaged spectrum E_x(k_x, k_y), showing the spectrum of EPWs excited by SRSS. To explicitly demonstrate the matching condition, we have added the pump wave vector k₀ (yellow arrow) and scattered light wave vectors extracted from Fig. 2(b). There are two distinct EPWs dispersed in the spectrum. One is k_p(k_x ≈ 1, k_y ≈ 0.6), which is associated with k₀ and k_s and satisfies the wave vector matching condition for first-order SRSS as illustrated in Fig. 1(b). The other EPW lies in the middle of Fig. 2(c), and so we zoom in on the region outlined by the red dashed rectangle and plot it in Fig. 2(d). Two kinds of EPWs with k_p1(k_x ≈ − 0.1, k_y ≈ 0.9) and k_p2(k_x ≈ − 0.1, k_y ≈ 0.4) are excited by downward-propagating side-scattered light, generating a δk_x ≈ 0.1 (red arrow), and the corresponding wave vectors of the second-order scattered light (green arrows) are exactly those in Fig. 2(b). Thus, the following new wave vector matching conditions for the second-order SRSS are satisfied:$$\begin{array}{c}{\mathbf{k}}_s^{\prime }={\mathbf{k}}_{p1}+{\mathbf{k}}_{s1},\\ {\mathbf{k}}_s^{\prime }={\mathbf{k}}_{p2}+{\mathbf{k}}_{s2}.\end{array}$$(8)

The ω–k_y dispersion relation of the light waves is shown in Fig. 3, where the spatiotemporal behavior of E_z at n_e = 0.09n_c is Fourier-analyzed in the time interval between 1.4 and 1.5 ps. Two bright spots can be observed in Fig. 3. The first, with higher frequency ω_s ≈ 0.64ω₀ and larger wave vector k_sy ≈ 0.64ω₀/c, is the scattered light of the first-order SRSS, whose wave vector satisfies Eq. (7). The second, with lower frequency ${\omega }_s^{\prime }\approx 0.4{\omega }_0$ and smaller wave vector $k_{sy}^{\prime }\approx 0.3{\omega }_0/c$, is the scattered light of the second-order SRSS. The pure spectrum shows that other nonlinear effects, such as sidebands45 and spectral broadening,46 are not important in this regime, although the wave evolution does show that the system has already saturated in this time period, as shown in Fig. 4.

Finally, the time evolutions of the first- and second-order SRSS are depicted at various locations in Fig. 4. The first-order SRSS increases exponentially after the laser enters the plasma, initially exciting the SRSS in the low-density region, and saturates at about 1.2 ps. The growth rate of first-order SRSS in Fig. 4 is γ = 7.62 ps⁻¹, while the theoretical growth rate of SRSS can be calculated as γ = 8 ps⁻¹. The second-order SRSS increases slowly before 1.2 ps, and then reaches saturation quickly at about 1.7 ps. This is consistent with the fact that the second-order SRSS is excited by the first-order SRSS only when a certain intensity threshold has been exceeded according to Eq. (6). From this, together with the pure spectrum in Fig. 3, we conclude that rescattering of SRSS contributes to saturation of the Raman instability. The saturation strength of second-order SRSS is about an order of magnitude lower than that of first-order SRSS.

In previous work, it has been found that rescattering of backscattered light has a great influence on hot-electron production, since the EPWs generated by rescattering can further heat the electrons to a higher energy.35 The production of hot electrons is harmful to ICF, and so we are curious about the effect of higher-order rescatterings of SRSS on the electrons.

Figure 5(a) shows the electron distribution function f_e(p_x, p_y) in momentum space at t = 1 ps, where only first-order SRSS is observed. The arrows show the propagation directions of EPWs of the first-order SRSS (k_x ≈ 1, k_y ≈ 0.6, red arrow) and the second-order SRSS (k_x ≈ − 0.1, k_y ≈ 0.4, yellow arrow). At this early stage, electrons are clearly captured and accelerated by the first-order EPW along its propagation direction (red arrow), and no obvious hot-electron behavior is observed along the yellow arrow. Figure 5(b) shows the distribution function at t = 2 ps, where rescattering occurs. In addition to the bulk of forward-propagating hot electrons generated by the wide spectral range of the first-order SRSS, there are plenty of hot electrons propagating nearly perpendicular to the density gradient (along the yellow arrow), which have been produced by the second-order SRSS.

Figures 5(c) and 5(d) show the electron distribution function in x–p_x phase space at 1 and 2 ps, respectively. This part of p_x is generated mainly by the first-order SRSS, which is first excited in the low-density region as shown in Fig. 5(c), and gradually spreads and propagates to the higher-density region until the hot electrons fill in the entire space as shown in Fig. 5(d). This is consistent with the conclusion from Fig. 4 that the first-order SRSS is initially excited in the low-density region. Meanwhile, the hot electrons generated in the low-density region will also move forward and propagate into the high-density region. We have also plotted the electron distribution function in x–p_y phase space at 1 and 2 ps in Figs. 5(e) and 5(f), respectively. It can be seen that the perpendicular momentum p_y is generated mainly by the second-order SRSS, concentrated in a narrow rescattering region below 0.11n_c. This may lead to accumulation of hot electrons in the y direction.

The evolution of the hot-electron energy spectrum along the two propagation directions is demonstrated in Fig. 6. It can be seen that the EPW of the first-order SRSS slowly heats the electrons, and the energy spectrum becomes saturated at 2 ps. By contrast, the hot electrons heated by the EPW of the second-order SRSS barely change before 1 ps, but increase rapidly between 1 and 2 ps. The maximum particle energy is slightly lower than that of the first-order SRSS, which means that the second-order SRSS could also generate higher electron energy, although not as effectively as the first-order SRSS. It appears that the fraction is less, though, which makes sense. The effective hot-electron temperatures in the two cases are nearly the same, about T_h ≈ 15 keV. The phase velocity of the first-order EPW is between 0.2c and 0.4c, but that of the second-order EPW is about (0.6–0.7)c, and the propagation directions are different, and so there is little room for cascade acceleration via continuous phase velocities, which is why the hot-electron fraction of the second-order SRSS is smaller than that of the first-order SRSS. Hence, rescattering of SRSS will not generate hot-electron bursts as expected in the rescattering of backscattered light.

In addition, some other simulations have shown that greater transverse beam size facilitates the growth of side scattering and its rescattering, thus the wider the beam, the more hot electrons in the transverse direction will be observed. There also exists a competition between hot electrons generated by the second-order SRSS and those generated by the first-order SRSS, since the we find that the hot electrons behave differently in the presence of only first-order SRSS in other simulations.

In summary, the rescattering of SRSS could quickly trap and accelerate electrons in the perpendicular direction, but in a relatively small amount, since the phase velocity and propagation directions of the EPWs of the two side scatterings are different, which, in contrast to the rescattering of backscattered light, does not allow any effect of cascade acceleration. The generated hot electrons propagate perpendicular to the density gradient, which seems to be less hazardous to ICF experiments, but they are more concentrated in the lower-density regions, and the influence of this concentration requires further investigation.

To confirm that the occurrence of rescattering of SRSS is not an occasional phenomenon, we have simulated side scattering under other conditions, each time changing only one parameter while keeping the others the same. The time-averaged transverse electric field spectra E_z(x, k_y) are presented below to compare the excitation of rescattering under different conditions.

Figure 7 shows side-scattering excitations under different laser intensities ranging from I = 1 × 10¹⁴ W/cm² to 1 × 10¹⁶ W/cm². Other parameters are the same as in the previous case. As can be seen in Figs. 7(a) and 7(b), at the order of 10¹⁴ W/cm², there is an increasingly growing amplitude of the first-order SRSS, but no obvious rescattering is found. However, in Fig. 7(c), when the laser intensity is increased to I = 1 × 10¹⁵ W/cm², both first- and second-order SRSS are clearly observed, and the rescattering is located immediately below 0.11n_c. As the pump intensity increases to I = 1 × 10¹⁶ W/cm², not only the first- and second-order SRSS, but even third-order rescattering of SRSS are observed, as shown in Fig. 7(d). The third-order rescattering is excited below 0.0625n_c (the red-dashed line to the left), the theoretical maximum density for third-order SRSS. This is because, as shown by the threshold formula (6), the scattered light generated by the (m − 1)th-order SRSS can excite the mth-order SRSS if the threshold conditions are met. Moreover, the rescattering of SRSS is concentrated in the lower-density regions owing to pump depletion. This shows that at higher pump intensity, further higher-order SRSS and more nonlinear effects begin to appear.

We now discuss the dependence of the rescattering of SRSS on the density scale length L, to which SRSS is sensitive as indicated in Eq. (6). Figure 8 shows a series of simulations with the density scale length at 0.2n_c ranging from L = 100 μm to L = 800 μm, while the intensity is chosen as I = 2 × 10¹⁵ W/cm². Figure 8(c) is the same case as discussed in Fig. 2. It is clear that rescattering will occur at these four density scale lengths. As L increases, the amplitude of rescattering increases slowly, indicating that has only a weak dependence on L. When the density scale length increases to 800 μm, SRSS and its rescatterings become weak, probably because ofthe competition between side scattering and backscattering.

We also consider the effects of electron temperature on the rescattering of SRSS as illustrated in Fig. 9. The original case is shown in Fig. 9(b). As the electron temperature is not very high, T_e ≤ 2 keV, we easily find that rescattering occurs, since the Landau damping is small enough for the threshold (5) to be exceeded. However, as the electron temperature increases to T_e = 5 keV [Fig. 9(c)] and 10 keV [Fig. 9(d)], only first-order SRSS can be observed, owing to the dramatically increased Landau damping. Even the region of occurrence of first-order SRSS shrinks to the higher-density region, where damping is not that large.

Thus, it has been shown that rescattering of SRSS is more likely to be observed in regions of parameter space with high laser intensity, moderate density scale length, and low electron temperature. The intensity has the paramount effect, since the excitation of rescattering relies on the amplitude of the previous order. At relatively high intensity, a cascade of rescatterings will be observed, along with more nonlinear effects. It should also be noted that in almost all cases, SRSS and its rescattering are the dominant processes. Since the we have excluded the quarter-critical-density region where Raman backscattering is absolutely unstable and two-plasmon decay occurs, it is much easier for the threshold for SRSS to be exceeded than that for convective Raman backscattering. The 200 μm width of the laser beam also facilitates the growth of SRSS, making it the dominant process in our simulations. The conditions discussed in this paper are likely to be found in direct-drive-based ICF scenarios, and therefore we believe that SRSS will inevitably occur in ICF experiments.

In summary, the above theoretical and simulation results show that the emergence of rescattering of SRSS is a fundamental process associated with SRSS. Theoretically, we have presented a matching condition and threshold for rescattering of SRSS. The condition for occurrence of mth-order SRSS is that n_m ≤ n_c/(m + 1)², where n_c is the critical density of the pump laser. The thresholds for rescattering depend on the amplitude of the previous order and are estimated from the absolute threshold for SRSS. We have used 2D PIC simulations to verify the existence of rescattering of SRSS over a wide range of parameters relevant to ICF. The simulation results for the region of occurrence, the matching condition, and the threshold are consistent with the theoretical analysis. Hot electrons generated by the EPWs of the second-order SRSS propagate nearly perpendicular to the density gradient and are as hot as those generated by first-order SRSS. However, unlike the rescattering of SRBS, a cascade acceleration process cannot be created in the case of SRSS.

Finally, we have described the simulation results for a range of parameters and have found that rescattering of SRSS is not an occasional phenomenon. Higher-order SRSS occur preferentially in regions of parameter space with high pump intensity, moderate density scale length, and low electron temperature. As well as involving interesting physics, the rescattering of SRSS may be a common process in the context of direct-drive ICF.

Acknowledgment. This work was supported by the Strategic Priority Research Program of the Chinese Academy of Sciences (Grant No. XDA25050700), the Fund of the National Key Laboratory of Plasma Physics (Grant No. 6142A04230103), the National Natural Science Foundation of China (Grant No. 11805062), the China Postdoctoral Science Foundation (Grant No. 2022M720513), and the Anhui Provincial Natural Science Foundation (Grant No. 2308085QA25). The numerical calculations in this paper have been done at the Hefei Advanced Computing Center.