Laser plasma instabilities (LPIs) such as stimulated Raman scattering (SRS), stimulated Brillouin scattering (SBS), and two-plasmon decay (TPD), are among the most critical factors hindering inertial confinement fusion (ICF) ignition. Precise modeling and control of LPIs is of crucial importance for the ICF programs carried out at the National Ignition Facility (NIF),¹ at Laser Mégajoule (LMJ),² and at the ShenGuang (SG) Facilities.^3,4 SRS, through which the incident light decays into scattered light paired with a plasma wave, can cause both driver energy loss and fuel-preheating risk due to the energetic (hot) electrons accelerated by its daughter plasma wave.

In the ignition-relevant experimental conditions on the NIF with longer electron density scale length and hotter electron temperature than the short-scale-length OMEGA parameters, SRS has been identified as being dominant over TPD.⁵ The coexistence and competition of absolute SRS and TPD modes near the quarter-critical-density region of the incident laser was discussed in Ref. 6 via both theory and simulation. While contemporary facilities may not have sufficient laser energy to achieve ignition, it is believed that under the long-density-scale-length experimental conditions produced by the significantly larger energy drivers and longer pulse durations that will be available in the future, SRS will be dominant over TPD. It is therefore important to investigate SRS under long-density-scale-length conditions.

With developments in short-pulse laser systems, it has become possible to explore the interaction between laser pulses lasting just a few picoseconds with laser-created plasmas. In contrast with the fast-ignition regime,⁷ where ultra-intense short pulses are used, a regime using a moderately high-intensity (I ∼ 10¹⁵–10¹⁶ W/cm²) short pulse is more relevant to ICF implosions and has particular advantages for the investigation of LPI compared with the nanosecond implosion-relevant LPI regime.

All types of LPIs will reach a highly nonlinear, saturated, and correlated stage when driven by a nanosecond implosion-scale laser pulse, which complicates analysis and understanding of the resulting phenomena. In addition, performing kinetic simulations for LPI over a nanosecond period can be numerically challenging and in any case is not very physically realistic, since the evolution of hydrodynamic conditions needs to be taken into account during a nanosecond period. However, in the short-pulse LPI (SP-LPI) regime, not only do the large-scale background hydrodynamic profiles not change significantly during the short interaction time span, but also different types of LPIs can be largely decoupled by the use of appropriate experimental designs, since SRS and SBS have distinctly different growth times. Therefore, this regime can facilitate kinetic modeling, help in the understanding of LPI signals, and provide a bridge between experiments and hydrodynamic and LPI simulations.

Pioneering SP-LPI experimental work has been done since the 1990s,^8,9 with the primary motivation being the avoidance of nonlinear saturation of SBS by using short pulses lasting a few picoseconds. SRS in SP-LPI was first observed experimentally in 1995 at CEA,¹⁰ where the linear SRS theory was experimentally validated using a sub-picosecond pulse with peak intensity <2 × 10¹⁶ W/cm². Systematic studies have been made of the forward SRS (FSRS)-like reflection of a relativistic laser pulse,¹¹ time-resolved signals on electron plasma waves (EPWs) and ion-acoustic waves (IAWs) driven by SRS and SBS,¹² and the interaction of two spatially^13,14 or temporally¹⁵ separated short pulses via EPWs and/or IAWs.

SP-LPI with the moderately high laser intensities relevant to ICF experiments is of interest with regard to ignition. It has been proposed that a high-intensity laser pulse could be used to launch an ultrastrong shock to further compress the target at the end of a long lower-intensity main compression pulse in novel ignition schemes such as shock ignition (SI)^16,17 and hybrid drive.¹⁸ Hot electrons generated via LPI may benefit the formation of the shock rather than representing a fuel-preheating risk if they can be controlled at an appropriate energy level. However, such high-intensity ignition schemes are expected to involve rather complicated LPI processes, whose details can be comprehensively studied and understood with the help of SP-LPI simulations and experiments.

When the incident laser is intense enough, the backward SRS (BSRS) daughter light wave can be intense enough to further excite a secondary SRS in its own resonant density region. These so-called SRS rescattering modes alter the interpretation of experimental diagnostics such as scattering losses and generation of energetic electrons. Numerical investigations of rescattering^19–23 have been carried out using particle-in-cell (PIC) simulations. Backward SBS (BSBS) of the FSRS light in hot plasmas (T_e > 10 keV) was first reported and considered as the main saturation mechanism of FSRS in Ref. 19. It was found that owing to the high phase velocities of EPWs via SRS rescattering (Re-SRS) and the Langmuir decay of the Re-SRS EPWs, Re-SRS and its secondary processes can anomalously generate ∼100 keV hot electrons in a homogeneous plasma.²⁰ Absolute Re-SRS modes were reported to develop near one-sixteenth of the critical density of the incident laser^21,22 in the short-density-scale-length regime relevant to shock ignition.^16,17 The density range where absolute Re-SRS modes can be stimulated was derived in Ref. 23, and PIC simulations were performed for an inhomogeneous plasma near one-ninth of the critical density of the incident laser, the upper density limit at which absolute Re-SRS modes can develop. Hot electrons, typically of 170 keV, were generated via Re-SRS in this region.²³

Previous work on SRS rescattering has mostly been done in plasmas with a narrow density range^19,20,23 or a short density scale length.^21,22 Nevertheless, it is to be expected that in the large-scale plasmas relevant to ignition, the effects on rescattering of both a broad density range and a long density scale length will have to be taken into account. First, the pump sources of rescattering modes come from not only the locally generated BSRS light but also the BSRS light generated in higher-density regions. Second, when rescattering light propagates away from its generation location, it will possibly be amplified by coupling with primary BSRS light. The coupling distance could be sufficiently great in a large-density-scale-length plasma to give a high convective gain.

In this article, we present a novel scenario in the SP-LPI regime in a long-density-scale-length inhomogeneous plasma, with simulation parameters relevant to the SG-II planar-target experimental conditions. It is found that the secondary BSRS (Re-BSRS) stimulated by primary BSRS light can be convectively amplified during its propagation toward the higher-density side of the plasma in the bath of primary BSRS light. This process can be energetically important, depleting a significant amount of energy carried by the primary BSRS light and generating a considerable number of hot electrons.

The rest of this article is organized as follows. In Sec. II, the resonant region, dispersion relation, and matching conditions for SRS and Re-SRS are briefly discussed. The simulation setup is described in Sec. III and the simulation results and analysis in Sec. IV. Finally, a summary is given in Sec. V.

A schematic representation of BSRS and Re-SRS is presented in Fig. 1. BSRS light generated primarily at n_cr/4 encounters its quarter critical density at n_cr/16, where it can excite absolute SRS rescattering modes if it is intense enough. BSRS light generated primarily at a slightly lower density than n_cr/4 encounters its corresponding quarter critical density point slightly above n_cr/16, and so on. The respective resonance points for primary and secondary SRS become closer as the primary SRS occurs at lower densities, until they converge to a density at which the BSRS light encounters its corresponding quarter critical density locally. This density is readily found to be n ≈ n_cr/9 using the matching conditions and dispersion relations for primary and secondary SRS: ω₀ ≈ ω_p + ω_s1 and ω_p ≈ ω_s1/2 yield ω_p ≈ ω₀/3, i.e., n ≈ n_cr/9.

Therefore, in the special density range [n_cr/16, n_cr/9], absolute modes of Re-SRS can be stimulated, given sufficient intensity of the BSRS light coming from the higher-density side.

Below n_cr/16, absolute Re-SRS modes cannot be stimulated, since primary BSRS light is unable to find its corresponding quarter-critical-density point in this region. Convective Re-SRS modes are still allowed to be stimulated, and convective amplification along their propagation path can still be energetically important.

We have performed a series of 1D and 2D PIC simulations of SP-LPI with the parameters appropriate to SG-II planar-target experimental conditions, using the electromagnetic PIC code OSIRIS.²⁵ The detailed parameters are listed in Table I. The findings from the SG-II experiments will be published in another paper.

For the 1D simulations, the electron density ranges from 0.01n_cr to 0.22n_cr. An exponential density profile is used: n(x) = n₀ exp(x/L), where n₀ = 0.01n_cr and the density scale length L = 1180 µm. Two species of ions, C and H, are used, with a number density ratio 1:1. The electron and ion temperatures are T_e = 1 keV and T_C = T_H = 0.5 keV, respectively. A typical 1D simulation [case (i)] region is 3.35 mm long with grid size Δx = 0.2c/ω₀. Open (i.e., nonreflection) boundary conditions are applied for fields and absorbing boundary conditions for particles in the 1D simulations. The 2D simulation [case (vi)] box is 2012 × 60 µm², with grid 60 000 × 1800 and Δx = Δy = 0.2c/ω₀, covering an electron density range from 0.04n_cr to 0.26n_cr. Open boundary conditions are applied for fields and thermal reflection boundary conditions for particles along the longitudinal (x) direction. The transverse (y) direction is set to be periodic, which is appropriate for a large laser spot regime as in the SG-II experiments. The same L and temperatures are used in the 2D as in the 1D simulations. In both 1D and 2D simulations, 100 particles per species per cell are used, with electron–ion collisions taken into account.

Laser propagation in a large-scale-length plasma takes time. Therefore, although short laser pulses last for only ∼4 ps, the simulations are performed for ∼35 ps to allow all of the light signals to be eventually collected by the diagnostic at the left boundary so that the integrated fraction of the scattered light can be obtained; see Fig. 2. This configuration is adopted to allow a fair comparison with future experiments where time-integrated SRS signals will be measured.

The 1D simulations restrict the SRS modes to be either backward or forward. Thus, clean BSRS and FSRS signals can be obtained in 1D simulations. Also, in our 1D simulations, very high spatial and temporary resolutions can be achieved for sophisticated spectral analysis. The 2D simulations are performed not only for validation of the physical picture found in the 1D simulations, but also to evaluate 2D effects such as TPD modes generated by rescattered light.

The scope of the LPI evolution in the simulation of case (i) is shown in Fig. 2. In case (i), the incident laser intensity is sufficiently high that many types of LPI are stimulated. The incident laser pulse shines from the left boundary and propagates to the right. The various types of scattered light and their generation regions can be identified clearly from a combination of the light propagation information in Fig. 2 and the frequency spectra in Figs. 3 and 4. On the incident laser’s path, BSRS and BSBS are first excited in the low-density plasma that first encounters the incident light. These two types of light propagate directly to the left and are caught early by the collector located at the left boundary. The BSRS light is intense enough to stimulate secondary FSRS (Re-FSRS) in the low-density region below n_cr/16.

The dominance of Re-FSRS over Re-BSRS in the low-density region is due to Landau damping of the EPWs. The Re-BSRS EPWs in that density region are heavily Landau-damped because their phase velocities v_p = ω_EPW2/k_EPW2 are close to the local ambient electron thermal velocity. However, as the incident light propagates toward higher-density regions, k_EPW2 decreases gradually and ω_EPW2 increases, which leads to a decrease in Landau damping. The dominant rescattering mode then switches from Re-FSRS to Re-BSRS as n_e increases.

Then, Re-BSRS starts to appear in the density region near n_cr/16 and above. The Re-BSRS light is generated propagating to the right. It is then reflected at its corresponding critical-density point, propagating to the left, low-density, direction and contributing to the spectrum collected at the left boundary at later times. Only part of the early rising phase of the incident laser pulse can be transmitted through this large-scale plasma and reach the right boundary, while most of the incident light is depleted, as shown in Fig. 2.

Re-SRS can be stimulated once the primary BSRS is intense enough, with its level depending on I₀. The same kind of spectra for cases (ii) and (iii) are shown in Figs. 3(b) and 3(c), where I₀ is 5 × 10¹⁵ W/cm² and 3 × 10¹⁵ W/cm², respectively. In case (ii), all the LPI signals are weaker. Re-FSRS is barely visible, while Re-BSRS is observable but weaker. In case (iii), only the BSRS signals can be clearly observed in the spectrum, with a dim signal at about ω_s2 = 0.33ω₀ due to the absolute modes of Re-BSRS, which indicates that I₀ = 3 × 10¹⁵ W/cm² is close to the threshold for Re-BSRS under this specified set of plasma parameters. Despite the lowest threshold of the absolute Re-SRS mode at ω_s2 = 0.33ω₀, this mode is not energetically dominant in the high-I₀ cases, as shown in Figs. 3(a) and 3(b).

By application of a windowed Fourier transform in both space and time, the detailed in-flight spectra showing the evolution of all LPI signals at different stages in case (i) are obtained and are shown in Fig. 4. Forward- and backward-propagating light can be distinguished via negative and positive ω in the spectra. The incident laser pulse is located around ω = −ω₀. Figure 4(a) shows the early stage of the simulation: the incident laser pulse is propagating in the low-density region of the plasma (n < 0.1n_cr); on its way, backward-propagating light is generated via primary BSRS and BSBS and by Re-FSRS in different frequency ranges. The BSBS modes are at ω_s1 ≈ ω₀, since the ion-acoustic frequency is orders of magnitude lower than ω₀. The BSRS modes generated in the low-density region (n < 0.1n_cr) have a high frequency (ω_s1 ≈ ω₀ − ω_p > 0.7ω₀). The BSRS light is intense enough to generate a secondary FSRS locally, whose frequencies then lie in the range 0.4ω₀ < ω_s2 < 0.7ω₀, consistent with the spectrum shown in Fig. 4(a).

As the incident laser pulse reaches the higher-density region, the frequency of the BSRS light shifts to the red. The BSRS light starts to generate Re-BSRS, whose daughter light waves propagate forward; see Fig. 4(b). The forward-propagating Re-BSRS light follows the incident pulse and propagates continuously in a bath of the primary broadband BSRS light. The broadband BSRS spectrum contains resonant components that are able to satisfy the three-wave matching conditions with the Re-BSRS light, such that the Re-BSRS light can be convectively amplified during its propagation until the local electron density is too high and the primary BSRS is red-shifted too far to be resonant. The amplification of Re-BSRS is energetically important and depletes the BSRS light, as demonstrated by the darker region (x from 7000c/ω₀ to 10 000c/ω₀) in the BSRS spectrum in Fig. 4(b). While BSRS light stimulates significant Re-SRS in the low-density regions, we do not observe secondary scattering simulated by BSBS. The growth of BSBS is likely limited by the short pulse duration of the incident laser, and its secondary processes are not expected to be significant.

The Re-BSRS light continues to propagate forward until it encounters its corresponding critical density. It is then reflected and eventually collected by the diagnostic at the left boundary after t = 20 ps; see the reflected light in Fig. 2 and the spectra in Figs. 3(a) and 4(c). The critical-density surfaces for the Re-BSRS lights are shown by the dashed lines in Fig. 4(c). The time at which Re-BSRS light reaches the left boundary depends on the distance along its optical path, which is determined by its group velocity and the position of its critical density. That is the reason for the higher-frequency signal being caught at a later time in Figs. 3(a) and 3(b).

The energy carried by rescattered light is sensitive to the peak intensity of the incident laser pulse. Figure 5(a) shows the time-integrated energy fraction carried by the backward-scattered light (i.e., collected by the left boundary) at two different frequency ranges in a series of simulations with a number of incident laser intensities as listed in Table I. The energy fraction of the high-frequency ([0.5, 0.95]ω₀) components, usually considered experimentally as the frequency range of the BSRS light, is plotted as circles in the figure, while the energy fraction of the low-frequency ([0.32, 0.5]ω₀) components is plotted as stars. The high-frequency fraction rises sharply at I₀ ∼ (1–3) × 10¹⁵ W/cm², then saturates and even slightly drops for higher intensities. When I₀ < 3 × 10¹⁵ W/cm², SRS is likely to be in the linear regime, and so the SRS reflectivity is sensitive to I₀. The I₀ dependence of the high-frequency light fraction is qualitatively consistent with the experimental observations for SRS reflectivity vs laser intensity shown in Fig. 5 of Ref. 27. The low-frequency fraction (mostly attributed to Re-BSRS but partly to Re-FSRS) continues to increase with I₀ and becomes more energetically important for higher I₀ (5.9% at I₀ = 1.2 × 10¹⁶ W/cm²). Both Re-BSRS and Re-FSRS deplete the primary BSRS light, consequently reducing the energy fraction of the high-frequency light signals. They may also help explain the slight drop in SRS reflectivity vs I₀ shown in Fig. 5 of Ref. 27 at high laser intensities. The depletion of the primary BSRS light due to Re-BSRS could lead to an underestimate of the primary BSRS process.

The reason for classifying the scattered light by frequency is to provide a bridge between the simulation results and the time-integrated experimental diagnostics. However, in this way, the Re-FSRS modes are also mostly counted in the high-frequency spectra, which will affect the accuracy of measuring the pure primary BSRS levels. Since all types of LPI modes are distinguishable in our spectra (see Fig. 3), we are able to plot the energy fraction for each type in Fig. 5(b) by integrating the spectra of Fig. 3. This is also an advantage of SP-LPI studies, since it is usually very challenging to distinguish the LPI signals in the spectrum of a long-pulse simulation. The fraction of BSRS continues to decrease when I₀ is above 3 × 10¹⁵ W/cm², since the BSRS light transfers more and more energy to Re-BSRS and Re-FSRS. The fractions of both Re-BSRS and Re-FSRS increase with I₀. Re-BSRS appears to have a lower threshold, whereas Re-FSRS becomes dominant over Re-BSRS for larger pump intensities.

Both stronger drivers and longer amplification distance would facilitate convective amplification. However, there is not yet any indication of saturation for Re-BSRS modes up to I₀ = 1.2 × 10¹⁶ W/cm², while the BSRS light that serves as the driver of Re-BSRS has already reached a saturated level when I₀ > 3 × 10¹⁵ W/cm², as shown in Fig. 5(b). The spatial-frequency distributions of the primary BSRS modes are shown at different times in Figs. 6(a) and 6(c) for case (i) and in Figs. 6(b) and 6(d) for case (ii). It can be seen that the amplitude of the primary BSRS light does not increase much when I₀ increases from 5 × 10¹⁵ W/cm² to 1.2 × 10¹⁶ W/cm², while Fig. 5(b) tells us that the total fraction of Re-BSRS increases from 2.5% to 5.0%.

As an example, the convective amplification of an Re-BSRS mode at ω_s2 = 0.47ω₀ is shown for cases (i) and (ii) in Fig. 6(e). In both cases, the mode has distinguished amplification distances and gains. The spectral broadening of the BSRS EPW for higher I₀ also contributes to the amplification boosting of the Re-BSRS modes. As Re-BSRS light travels in a bath of primary BSRS light, the BSRS light satisfying ω_s1 = ω_s2 + ω_EPW2 (indicated by stars in Fig. 6) can contribute to the convective amplification. In Figs. 6(a) and 6(b), the matching-condition stars are both located inside the BSRS light spectrum, which indicates that the ω_s2 = 0.47ω₀ mode can be amplified at this moment in both cases. The amplification stops when the Re-BSRS light reaches a high-density point at which it is not able to pick up any pump component from the BSRS spectrum. As can be seen from Figs. 6(c) and 6(d), although the matching-condition star is still located inside the BSRS light spectrum in case (i), it is outside the spectrum in case (ii). Therefore, the ω_s2 = 0.47ω₀ mode is still being amplified in case (i), but has already saturated in case (ii) when t > 10 ps, as shown in Fig. 6(e).

The temporal–spatial distributions of the Re-BSRS modes and the backward-moving hot electrons >50 keV are shown in Figs. 7(a) and 7(b), respectively. There is a strong correlation between the two. We denote by α₅₀ the time-integrated kinetic energy carried by the backward-moving hot electrons >50 keV normalized to the total incident laser energy. The dependence of α₅₀ on I₀ is shown in Table I.

We also perform a 2D PIC simulation under the same plasma conditions to verify the convective amplification process of Re-SRS in two dimensions [case (vi) of Table I]. It is numerically challenging to apply the analysis in 2D simulations with the same temporal and spatial resolutions as in our 1D analysis, and so a spectrum in k_x space is used to identify different types of LPI modes, as shown in Fig. 8(a), which depicts the temporal evolution of the spatial Fourier transform of the transversely averaged E_y field applied in a spatial domain near n_cr/9. The incident and BSBS light both contribute to the signals at k ≈ 0.91ω₀/c. Wave numbers k_s1 in the range from 0.4ω₀/c to 0.6ω₀/c represent the BSRS light, while the Re-BSRS light starts to show up at k_s2 ≈ 0.2ω₀/c after 4 ps, by which time the BSRS signals are well established.

Significant convective amplification of the Re-BSRS light has also been observed in the 2D simulation. The Re-BSRS light at different frequencies propagates forward, passes the signal collecting domain near n_cr/9 from 4 ps to 8 ps, and then revisits this domain at t > 8 ps after it has been reflected at its corresponding critical density, as indicated in the spectrum in Fig. 8(a). Figure 8(b) shows the convective amplification of Re-BSRS light with ω_s2 = 0.4ω₀ along its propagation path, consistent with the 1D simulation results.

The spectra of light and plasma waves in Figs. 9(a) and 9(b) show rich 2D features in addition to the BSRS and Re-BSRS processes. The LPI processes are identified and indicated on Fig. 9. Without the limitations of 1D geometry, both BSRS and Re-BSRS have larger spread angles. Secondary TPD driven by the primary BSRS near n_cr/9 can be observed in the plasma wave spectrum in Fig. 9(b). TPD involves pairs of EPWs and generates no light waves. The EPWs with smaller k_x generated by the secondary TPD can be seen near k_x ≈ 0.2ω₀/c, and the EPWs with larger k_x are mixed with the EPWs associated with Re-BSRS near k_x ≈ 0.6ω₀/c. The hot-electron fraction is a few times larger than in the 1D simulation for case (i), which could be attributed to the secondary TPD. The transversely periodic configuration in our 2D simulation is relevant to a large-focal-spot scenario where the laser spot size is much larger than the transverse size of the simulation box. Important higher-dimensional physics such as spot-size effects should be studied with appropriate boundary conditions in future work.

To summarize, we have studied a novel scenario in the SP-LPI regime with parameters relevant to SG-II experiments. The plasma has a large density scale length together with a broad density range in this regime, which facilitates the generation and convective amplification of Re-BSRS modes. It is found that when the incident laser intensity is well above its BSRS threshold, the backscattered daughter light wave of BSRS can also excite secondary BSRS and FSRS in the region below n_cr/9 of the incident laser. The daughter light wave can be amplified via Re-BSRS as it propagates toward the higher-density region in the bath of broadband light generated by the primary BSRS process.

Sensitively correlated with the incident laser intensity I₀, convective amplification makes the Re-BSRS modes energetically important for high I₀. Re-BSRS can deplete the primary BSRS light significantly, which can be considered an energy sink that could lead to an underestimate of the primary BSRS levels in experiments. Higher I₀ not only drives more intense BSRS light, but also broadens its spectrum and thus enhances its amplification distance. Both effects increase the convective gain of Re-BSRS. A significant number of backward-moving hot electrons are generated. This hot-electron generation is found to be strongly correlated with the EPWs associated with Re-BSRS.

The findings of this work indicate that Re-SRS could be considered a concern for higher-intensity ignition-scale regimes. First, the depletion of the primary BSRS due to Re-SRS may lead to an underestimate of the primary BSRS level in an SP-LPI experiment and possibly in a long-pulse ignition-scale experiment. Second, the backward-moving hot electrons also represent a significant energy loss, but one likely to be missed by the conventional hard-x-ray diagnostics that are usually deployed on the higher-density side of the plasmas. Last, but not least, the EPWs generated via Re-SRS will eventually pass their energy to the ambient plasma via collisional damping and change the local plasma temperature on a longer time scale.