2 Particle-in-cell simulation setups

Two simulations, named cases 1 and 2, are performed in this paper. Except for the laser intensity, all the simulation parameters are the same in both cases. The laser intensities in cases 1 and 2 are 4.45 $\times {10}^{15}$ and 1 $\times {10}^{16}$ W/cm², respectively. Correspondingly, the normalized vector potentials (which are defined as ${a}_0={eE}_{\mathrm{L}}/{m}_{\mathrm{e}}{\omega}_0c$ , where ${a}_0$ is the normalized vector potential and ${E}_{\mathrm{L}}$ is the electric field of the laser) are 0.02 and 0.03, respectively. Other simulation parameters are listed as follows. The electron temperature is ${T}_{\mathrm{e}}=2.5$ keV and the ion temperature is ${T}_{\mathrm{i}}=200$ eV. Tritium plasma is used in the simulation and the electron density is ${n}_{\mathrm{e}}=0.095{n}_{\mathrm{c}}$ . The laser wavelength is ${\lambda}_0=$ 351 nm, the laser is propagating in the $x$ direction and its electric field lies in the $y$ direction (p-polarized). The simulation box is $120{\lambda}_0\times 40{\lambda}_0$ in the $x\times y$ directions, and the grid steps are $0.02{\lambda}_0\times 0.02{\lambda}_0$ in the $x\times y$ directions. In each simulation cell, 200 electrons and 200 ions are used. The plasma is located between $x=10{\lambda}_0$ and $110{\lambda}_0$ along the $x$ direction and between $y=-20{\lambda}_0$ and $y=20{\lambda}_0$ along the $y$ direction. The total simulation time is $3500{T}_0$ , where ${T}_0=c/{\lambda}_0\approx 1.17$ fs is the laser cycle.

Since ${k}_{\mathrm{epw}}{\lambda}_{\mathrm{d}}$ is a very important parameter, before discussing the simulations, an estimation of the parameter ${k}_{\mathrm{epw}}{\lambda}_{\mathrm{d}}$ is given according to the following equations:

(1) $$\begin{align}{\omega}_0^2={\omega}_{\mathrm{pe}}^2+{c}^2{k}_{\mathrm{L}}^2,\end{align}$$

(2) $$\begin{align}{\omega}_{\mathrm{s}}^2={\omega}_{\mathrm{pe}}^2+{c}^2{k}_{\mathrm{s}}^2,\end{align}$$

(3) $$\begin{align}{\omega}_{\mathrm{epw}}^2={\omega}_{\mathrm{pe}}^2+3{v}_{\mathrm{te}}^2{k}_{\mathrm{epw}}^2,\end{align}$$

(4) $$\begin{align}{\mathbf{k}}_{\mathrm{L}}={\mathbf{k}}_{\mathrm{epw}}+{\mathbf{k}}_{\mathrm{s}},\end{align}$$

(5) $$\begin{align}{\omega}_0={\omega}_{\mathrm{epw}}+{\omega}_{\mathrm{s}},\end{align}$$

where ${\omega}_0=2\pi c/{\lambda}_0$ is the laser frequency, ${\mathbf{k}}_{\mathrm{L}}$ is the laser wave vector in plasma, ${\omega}_{\mathrm{s}}$ is the frequency of the scattered light, ${\mathbf{k}}_{\mathrm{s}}$ is the wave vector of the scattered light and ${\omega}_{\mathrm{epw}}$ is the frequency of the EPW. Supposing the angle between ${\mathbf{k}}_{\mathrm{L}}$ and ${\mathbf{k}}_{\mathrm{s}}$ is $\theta$ ( $0\le \theta \le \pi$ ), then $\theta =0$ represents forward SRS (FSRS) and $\theta =\pi$ represents BSRS. The theoretical wave numbers for simulation were calculated, as shown in Figure 1. From the above equations, ${k}_{\mathrm{epw}}{\lambda}_{\mathrm{d}}$ is estimated to be 0.35 for BSRS and $0.08$ for FSRS. The simulation results are shown in Figures 2–5.

Figure 1 Calculated wave numbers of the scattered lights and EPWs of SRS for ${n}_{\mathrm{e}}=0.095{n}_{\mathrm{c}}$ and ${T}_{\mathrm{e}}=2.5$ keV according to Equations (1)–(5). In the figure, ${k}_0={\omega}_0/c$ is the laser wave number in vacuum and the laser is propagating in the ${k}_x$ direction. Both the laser and the scattered light have polarization, which means the direction of their electric fields is parallel to ${k}_x-{k}_y$ . When discussing SRS, ${k}_{\mathrm{L}}={k}_{\mathrm{epw}x}+{k}_{\mathrm{s}x}$ and ${k}_{\mathrm{epw}y}+{k}_{\mathrm{s}y}=0$ should be firstly satisfied; two examples are given in this figure, where the magenta is forward SSRS matching and the cyan is backward SSRS matching.

3 Analysis of simulation results

Since the p-polarized laser is propagating in the $x$ direction, it is reasonable to use the electric field component ${E}_x$ to reflect the information of the electrostatic modes (for side scattering, it also contains information of electromagnetic (EM) modes), such as the EPWs and the magnetic field ${B}_z$ to reflect the information of the EM modes. However, before discussing the simulation results, differences among the TPD of the backscattered light, SSRS of the laser and filamentation of the EPWs need to be reviewed firstly. It is known that TPD is a process in which a laser decays into two EPWs. In TPD, the matching conditions are ${\omega}_0={\omega}_{\mathrm{epw1}}+{\omega}_{\mathrm{epw2}}$ and ${\mathbf{k}}_{\mathrm{L}}={\mathbf{k}}_{\mathrm{epw1}}+{\mathbf{k}}_{\mathrm{epw2}}$ , where ( ${\omega}_{\mathrm{epw1}}$ , ${\mathbf{k}}_{\mathrm{epw1}}$ ), ( ${\omega}_{\mathrm{epw2}}$ , ${\mathbf{k}}_{\mathrm{epw2}}$ ) are the frequencies and wave vectors, respectively, of the two TPD-generated EPWs. For the TPD generated by the backscattered light, ( ${\omega}_0$ , ${\mathbf{k}}_{\mathrm{L}}$ ) should be exchanged with ( ${\omega}_{\mathrm{bs}}$ , ${\mathbf{k}}_{\mathrm{bs}}$ ), where ( ${\omega}_{\mathrm{bs}}$ , ${\mathbf{k}}_{\mathrm{bs}}$ ) are the frequency and wave vector, respectively, of the backscattered light. In SSRS, where a laser decays into a scattered light and an EPW, the matching conditions of the two daughter waves are ${\omega}_0={\omega}_{\mathrm{epw}}+{\omega}_{\mathrm{s}}$ and ${\mathbf{k}}_{\mathrm{L}}={\mathbf{k}}_{\mathrm{epw}}+{\mathbf{k}}_{\mathrm{s}}$ . Since the matching conditions are different, we can distinguish these two instabilities according to the wave number spectra of the electric or magnetic fields. Before comparing the wave number spectra, we will give some theoretical estimations. Suppose the angle between ${\mathbf{k}}_{\mathrm{L}}$ and ${\mathbf{k}}_{\mathrm{s}}$ is $\theta$ , both ${\mathbf{k}}_{\mathrm{s}}$ and ${\mathbf{k}}_{\mathrm{epw}}$ for SRS can be numerically calculated according to Equations (1)–(5). With the angle $\theta$ , Equation (4) can be rewritten as ${k}_{\mathrm{L}}={k}_{\mathrm{epw}x}+{k}_{\mathrm{s}x}$ and ${k}_{\mathrm{epw}y}+{k}_{\mathrm{s}y}=0$ , where ${k}_{\mathrm{s}x}={k}_{\mathrm{s}}$ cos $\theta$ and ${k}_{\mathrm{s}y}={k}_{\mathrm{s}}$ sin $\theta$ . For the plasma density and temperature used in our simulations, the calculated results for ${\mathbf{k}}_{\mathrm{epw}}=\left({k}_{\mathrm{epw}x},{k}_{\mathrm{epw}y}\right)$ and ${\mathbf{k}}_{\mathrm{s}}=\left({k}_{\mathrm{s}x},{k}_{\mathrm{s}y}\right)$ are shown in Figure 1. From this figure we can easily find out different SRS matchings, for example, forward SSRS matching and backward SSRS matching are shown in the figure. Filamentation of the EPW is not a three-wave process, so the wave number matching conditions are not necessarily satisfied. Although the EPW also has transverse components in wave number space (k-space), it is easily distinguished from TPD or SSRS according to the matching conditions.

Figure 2 Snapshots of ${E}_x$ in k-space for case 1 (a) and case 2 (b) at the early stage ( $t=500{T}_0$ ). Snapshots of ${E}_x$ in k-space for case 1 (c) and case 2 (d) at the latter stage ( $t=2500{T}_0$ ). The intensities of the spectra are in arbitrary units. The arrows denote the excited instabilities and the dashed lines denote the theoretical wave numbers of the EPW (red) and the scattered light (black) shown in Figure 1. It should be mentioned that ${E}_x$ has both an EM component and an electrostatic component for side scattering.

Figure 3 Snapshots of ${B}_z$ in k-space: (a), (b) cases 1 and 2, respectively, at $t=500{T}_0$ ; (c), (d) cases 1 and 2, respectively, at $t=2500{T}_0$ . The magenta dashed curve represents the theoretical wave numbers of back scattered or back side scattered light and the red dashed curve represents the theoretical wave numbers of forward scattered or forward side scattered light. In the figure, the intensities of the spectra are in arbitrary units.

Based on the above analysis, Figure 2 gives four snapshots of the ${E}_x$ fields in k-space for both cases 1 and 2 to decide what instabilities are excited. Figures 2(a) and 2(b) are ${E}_x\left({k}_x,{k}_y\right)$ at $t=500{T}_0$ (early stage) and Figures 2(c) and 2(d) are at $t=2500{T}_0$ (latter stage). In case 1, the theoretical wave number of the EPW of BSRS is ${k}_{\mathrm{bsrs}}\approx 1.514{k}_0$ , where ${k}_0={\omega}_0/c$ is the wave number of the laser in vacuum, and the theoretical wave number of the backscattered light is ${k}_{\mathrm{bs}}\approx -0.562{k}_0$ . In Figure 2(a), both the signals of TPD and BSRS are observed, but both of them are very weak. The wave numbers of two TPD-generated EPWs are ${\mathbf{k}}_{\mathrm{epw1}}/{k}_0=\left(\mathrm{0.221,0.575}\right)$ and ${\mathbf{k}}_{\mathrm{epw2}}/{k}_0=\left(-0.783,-0.575\right)$ which satisfy the matching condition ${\mathbf{k}}_{\mathrm{bs}}={\mathbf{k}}_{\mathrm{epw1}}+{\mathbf{k}}_{\mathrm{epw2}}$ . Furthermore, from Equations (2) and (3), the frequencies are calculated as ${\omega}_{\mathrm{bs}}/{\omega}_0\approx 0.641$ , ${\omega}_{\mathrm{epw1}}/{\omega}_0\approx 0.317$ and ${\omega}_{\mathrm{epw2}}/{\omega}_0\approx 0.330$ , which also satisfy the frequency matching condition ${\omega}_{\mathrm{bs}}={\omega}_{\mathrm{epw1}}+{\omega}_{\mathrm{epw2}}$ with a very small error. However, ${\mathbf{k}}_{\mathrm{epw1}}$ and ${\mathbf{k}}_{\mathrm{epw2}}$ are not definitely EPW signals because an EM wave also has ${E}_x$ components when it is obliquely propagating. Complementary evidence is that ${B}_z$ in k-space has no such signals, as shown in Figure 3(a), so it is reasonable to conclude that TPD of the backscattered light is excited. In case 2, TPDs of the backscattered light and BSRS also coexist in the early stage, as is shown in Figure 2(b). However, there are several differences between cases 1 and 2. Firstly, the EPW of BSRS is much more intense in case 2, and the frequency (or wave number) of the EPW is broadened in both the ${k}_x$ direction and the ${k}_y$ direction. Secondly, obliquely propagating EM components with a maximum ${k}_x/{k}_0\approx -0.562$ are observed and ${B}_x$ in k-space shown in Figure 3(b) further proves they are EM components. Thirdly, the EPWs also possess transverse components and part of them satisfy the wave number matching conditions of SSRS. The theoretical curve shows that backward SSRS ( $\theta >90{}^{\circ}$ ) is excited in this stage, which is also proved by Figure 3(b). The unmatched transverse parts are the result of filamentation of the EPWs. A short conclusion for this stage can be made as follows: if the driver of BSRS is relatively weak, the scattered light will excite the TPD and thus mitigate BSRS. However, if the driver is intense enough, the TPD cannot dissipate all energy of the backscattered light, then BSRS and backward SSRS will get an opportunity to grow, and the EPWs generated by them will grow intense enough to excite filamentation instability. In the latter stage ( $t=2500{T}_0$ ), the EPWs generated by TPD in case 1 grow larger and the EPW of the BSRS is still strongly mitigated, as is shown in Figure 2(c). In addition, Figure 3(c) shows that there are still no obliquely propagating EM components generated. In case 2, Figure 2(d) shows that the TPD of the backscattered light is suppressed, as well as frequency broadening of the EPW generated by BSRS. SSRS is also observed in Figure 1(d); however, in this stage the theoretical curves shown in Figures 2(d) and 3(d) imply that it is forward SSRS ( $\theta <90{}^{\circ}$ ) now. The unmatched part is also due to filamentation instability. It can be concluded that in the case of a stronger driver, both BSRS and TPD will be suppressed in the latter stage. Instead, FSRS and forward SSRS will grow larger and filamentation instability will be excited.

Figure 4 Snapshots of ${E}_x$ in real space, k-space and electron density perturbation $\delta {n}_{\mathrm{e}}/{n}_{\mathrm{c}}={n}_{\mathrm{e}}/{n}_{\mathrm{c}}-0.1028$ in real space. The left column represents the early stage at $t=500{T}_0$ and the right column represents the latter stage at $t=2500{T}_0$ . (a), (b) ${E}_x$ in real space, where the value is normalized by ${E}_0=9.16\times {10}^{12}$ W/cm². (c), (d) ${E}_x$ in k-space for region I. (e), (f) ${E}_x$ in k-space for region II. The intensities of the spectra are in arbitrary units. (g), (h) $\delta {n}_{\mathrm{e}}$ in real space.

Several reasons, such as Langmuir wave collapse and TPMI, could be responsible for filamentation of the EPWs. Langmuir wave collapse is excluded for BSRS because it occurs when ${k}_{\mathrm{epw}}{\lambda}_{\mathrm{d}}<0.3$ , but in our simulations, the parameter ${k}_{\mathrm{epw}}{\lambda}_{\mathrm{d}}$ for BSRS is 0.35. However, TPMI should be excluded for FSRS because ${k}_{\mathrm{epw}}{\lambda}_{\mathrm{d}}$ for FSRS is 0.08 in the simulations. From the above results, the reason why filamentation and side scattering are suppressed by TPD is easily analyzed. Take filamentation, for example. The growth rate of filamentation caused by TPMI is obtained from the formula ${\gamma}_{\mathrm{TP MI}}=\mid \Delta {\omega}_{\mathrm{TP}}/4\mid$ ^[ Reference Rose ^{25 ]}, where ${\omega}_{\mathrm{TP}}=h\left({k}_{\mathrm{epw}}{\lambda}_{\mathrm{d}}\right){\omega}_{\mathrm{b}}$ , ${\omega}_{\mathrm{b}}={k}_{\mathrm{epw}}{v}_{\mathrm{te}}\sqrt{e\phi /{T}_{\mathrm{e}}}$ is the electron bounce frequency and $\phi$ is the potential of the EPW, with $h<0$ . The TPMI has a larger growth rate with a larger ${k}_{\mathrm{epw}}{\lambda}_{\mathrm{d}}$ or a larger $\mid \phi \mid$ . For case 1, the TPD will dissipate the energy of the backscattered light all the time, which results in a decrease of the EPW intensity. As long as the TPD is not saturated, ${\gamma}_{\mathrm{TPMI}}$ in case 1 is always too small to excite filamentation instability. However, in case 2, since the driver is intense enough, BSRS, FSRS and SSRS have larger growth rates. The backscattered light can grow intense enough to saturate the TPD. When the TPD is saturated, other instabilities will get opportunities to grow and cause nonlinearities.

Figure 5 Snapshots of the electron distribution functions in x-p_x space: (a), (b) distribution functions for case 1 in the early and latter stages, respectively; (c), (d) those for case 2 in the early and latter stages, respectively.

To analyze why TPD is suppressed, we give a more detailed analysis for case 2 in Figure 4. In homogeneous plasmas, SRS is subject to both absolute and convective instability. For BSRS, the scattered light is counter-propagating with the laser, so the EPW near the laser incident side is more amplified and has a higher intensity. For FSRS, the intensity distribution is opposite to that of BSRS because the scattered light has the opposite propagation direction to BSRS. As a result, nonlinearities should be initially excited near the left-hand boundary (defined as region I in this paper, i.e., $x/{\lambda}_0<70$ ) for BSRS but initially excited near the right-hand boundary (region II, i.e., $x/{\lambda}_0>70$ ) for FSRS. So, Figures 4(a) and 4(b) show that filaments of the EPW of BSRS are firstly observed in region I in the early stage, but filamentation of the EPW of FSRS is more serious in region II in the latter stage. EPWs of BSRS and FSRS are distinguished by their wavelengths in real space; according to Figure 1, the EPW of BSRS has a shorter wavelength (or larger wave number). In Figures 4(c) and 4(e), ${E}_x$ values in k-space are given for regions I and II at $t=500{T}_0$ , respectively. The results show that TPD is excited only in region II in this stage. The reason is that the EPW of BSRS is intense enough to excite nonlinearities in region I in this stage. Theoretically, the quarter critical density of the backscattered light is about $0.1028{n}_{\mathrm{c}}$ in this paper, which means TPD will not be excited when the plasma density exceeds $0.1028{n}_{\mathrm{c}}$ . In region I, Figure 4(g) shows that stronger BSRS and filamentation of the EPW cause larger electron density perturbation, which make the electron density locally exceed $0.1028{n}_{\mathrm{c}}$ ( ${n}_{\mathrm{e}}-0.1028{n}_{\mathrm{c}}>0$ ), so TPD is suppressed in region I. In the latter stage, FSRS and forward SSRS grow strong enough to generate nonlinearities in both regions I and II. The wave number spectra shown in Figures 4(d) and 4(f) imply that forward SSRS is mainly excited in region I and filamentation of the EPW of FSRS mainly occurs in region II. As mentioned above, EPW collapse^[ Reference Xiao, Liu, Wu, Zheng and He ^{29 ]} should be the reason for filamentation because ${k}_{\mathrm{epw}}{\lambda}_{\mathrm{d}}<0.3$ for FSRS. Filamentation of the EPW of FSRS also changes the electron density distribution in region II, which also makes the electron density locally exceed $0.1028{n}_{\mathrm{c}}$ , as shown in Figure 4(h). As a result, TPD is suppressed in region II in the latter stage, as shown in Figure 4(f).

Distributions of electrons in phase space are employed to reflect the trapping process. In phase space, the trapping structure is a series of ‘phase islands’ with their center near the phase velocity of the EPWs. For the simulation parameters used in this paper, the phase velocity of the EPW is ${v}_{\mathrm{bs}}/c=0.2369$ for BSRS and ${v}_{\mathrm{fs}}/c=0.9281$ for FSRS. The phase velocities of the two EPWs generated by the TPD of the backscattered light are calculated as ${v}_{\mathrm{ph1}}/c=0.5146$ and ${v}_{\mathrm{ph2}}/c=0.3397$ according to Figure 2(c), and their $x$ components are ${v}_{1x}\approx 0.1846c$ and ${v}_{2x}\approx 0.2738c$ . As shown in Figures 5(a) and 5(b), in the early stage, the trapping structure is not obvious for case 1 since both TPD and BSRS are not quite intense in this stage. Near the plasma boundary, electrons accelerated by the sheath fields are observed. However, in the latter stage when the EPWs of TPD grow larger, trapping structures for both EPWs are observed in region II (the ‘island’ centers are near their $x$ component of the phase velocity). In case 2, Figure 5(c) shows that trapping structures are observed in region I in the early stage. This explains the filamentation of the EPW of BSRS in region I in the early stage. In region II, since BSRS is suppressed, the trapping structure is also suppressed. In the latter stage, Figure 5(d) shows trapping structures with both positive and negative ${p}_x$ . Theoretically, electrons trapped by the EPWs of SRS have only positive ${p}_x$ , as ${k}_x$ of the EPWs of SRS are usually positive (see Figure 1). The trapping structure with negative ${p}_x$ could be generated by reflection of the EPW on the plasma boundary. On the plasma–vacuum boundaries, strong electrostatic fields (which are named sheath fields) will be generated after a long-term simulation because of charge separation. The sheath field on the right-hand boundary can reflect EPWs, so the abnormal trapping structures in Figure 5(d) are mainly in region II. So, the trapping structures in Figures 5(c) and 5(d) are strong evidences for TPMI.