In inertial confinement fusion (ICF) experiments, multiple high-intensity laser beams generate a large-scale high-temperature plasma around the pellet.1–3 As these lasers propagate through the plasma, some unstable wave modes can be excited. These processes are called laser–plasma instabilities (LPIs).4 Back-stimulated Raman scattering (BSRS) is one of the main LPIs that occur in ICF experiments. BSRS transfers the energy of the laser to a back-propagating scattered light wave and a forward-propagating electron plasma wave (EPW). Both waves obstruct fusion ignition. The scattered light wave decreases the incident laser energy and introduces an asymmetry in the radiation field. The EPW produces energetic electrons that preheat the pellet. Therefore, mitigating BSRS is a critical factor in achieving high-gain ignition.5–7 Studies over the past four decades have led to a number of approaches being proposed for suppressing BSRS, including spike trains of uneven duration and delay (STUD),8–10 strong DC magnetic fields,11 polarization rotation,12 and broadband lasers.13–17

The approach of suppressing BSRS using broadband lasers was proposed in the 1970s.18 With the development of broadband laser facilities,19,20 this approach has recently attracted great interest. However, there is no general agreement on whether existing and upcoming broadband laser facilities, such as KunWu19 and FLUX20 (with bandwidth Δω/ω₀ ≈ 0.6%–1.5%) can mitigate BSRS in the kinetic regime. Some studies have shown that large-bandwidth broadband lasers (Δω/ω₀ > 3%) can significantly reduce the growth rate and increase the threshold of BSRS.21–25 By contrast, other studies have observed that moderate-bandwidth broadband lasers (Δω/ω₀ < 3%) do not suppress BSRS26 and actually lead to violent BSRS bursts.27,28 These bursts originate from the interaction of the high-intensity pulses in the broadband laser with the EPWs. Variations in temporal coherence of broadband lasers generate these pulses. Some high-intensity pulses can reach a peak intensity that is six to eight times higher than the average intensity of broadband lasers,27 which is over 1 × 10¹⁶ W/cm². In the kinetic regime, EPWs can trap electrons and reduce the Landau damping in their propagation path. When the propagation paths of the high-intensity pulses overlap with the EPWs, violent BSRS bursts occur. However, a systematic understanding of how electron kinetic affects the burst process is still lacking. The major challenge is the random fluctuation of the intensity envelope of broadband lasers. This randomness makes it hard to separate the physical details of the burst from the complex interaction of multiple pulses.

This paper presents a detailed study of electron kinetic effects in broadband laser-driven BSRS bursts. In long-duration simulations, we have observed that the evolutionary pattern of broadband laser-driven BSRS differs from that of a monochromatic laser. In the nonlinear stage, BSRS bursts triggered by broadband lasers generate intense scattered light and many hot electrons. On the basis of the physical mechanism of the burst events, a simplified model has been established. In this model, the incident laser consists of a flat laser and a strong pulse. The flat laser is used to generate EPW packets, and the strong pulse is used to excite violent bursts. This simplified design eliminates interference from the randomness of broadband lasers and provides a suitable environment for analyzing kinetic effects in bursts. Through a series of numerical simulations, some typical kinetic phenomena have been observed. The EPW intensity increases abruptly during the BSRS burst. Accordingly, the EPW spectrum changes dramatically from a sideband structure to a turbulence-like structure. The burst also causes violent vortex-merging in the electron phase space. In addition, we have found a significant asymmetry in the EPW spectrum. During the burst, the asymmetry is amplified and transferred to the electron phase space, which causes an increase in the center trap velocity. The turbulence-like EPWs modify the electron distribution function and produce a large number of hot electrons after bursting. Simulations show that a burst triggered by a single high-intensity pulse can increase the proportion of hot electrons from 6.76% to 14.78%. These findings enhance the comprehension of broadband LPI and provide guidance for the design of upcoming broadband laser facilities and experiments.

Several numerical models of broadband lasers have been proposed in the last decade. The most widely used models are the frequency modulation (FM) model25,26 and the multi-frequency beamlets (MFB) model.24,28,29 In our simulations, the MFB model was used. This model uses many discrete beamlets with different frequencies to approximate a continuous frequency spectrum. The approximation is highly accurate when the number of beamlets is large (typically over a hundred). Figure 1(a) shows a schematic of the MFB model. The electric field of the MFB broadband laser can be written as$$E_{\text{MFB}}=\sum _{i=1}^N{E}_i\cos ({\omega }_{i}t+{\varphi }_i),$$(1)where N is the number of beamlets, and ω_i and ϕ_i are the frequency and phase of the ith beamlet. ω_i is randomly selected within [ω₀ − Δω/2, ω₀ + Δω/2], where Δω and ω₀ are the bandwidth and center frequency of the broadband laser. ϕ_i is randomly selected within [−π, π]. We use a RECT spectrum laser with bandwidth Δω/ω₀ = 1.5%. As shown in Fig. 1(b), each beamlet has the same E_i in the RECT spectrum laser.

Traditional beam smoothing techniques, such as continuous phase plates (CPPs) and random phase plates (RPPs),30 degrade the laser’s spatial coherence, resulting in spatial inhomogeneity (speckles). Similarly, the bandwidth degrades the broadband laser’s temporal incoherence, resulting in temporal fluctuations in laser intensity, as shown in Fig. 1(c). These fluctuations can be treated as a train of local pulses with random peak intensity and lifetime. The peak intensity of some strong pulses can reach four to seven times the average intensity ⟨I⟩, as marked by the red arrows in Fig. 1(c). On the basis of statistical optics theory,31 the average lifetime of these pulses ⟨T_lifetime⟩ is inversely proportional to their bandwidth. For broadband lasers with a RECT spectrum, ⟨T_lifetime⟩ ≈ τ_c = 2π/Δω, where τ_c is the laser’s coherence time.23,27

Current broadband laser facilities have bandwidths Δω/ω₀ = 0.6% (in KunWu19) and 1.5% (in FLUX32). Their average intensity ⟨I⟩ can reach up to ∼10¹⁵ W/cm². In such facilities, strong local pulses have peak intensities up to ∼10¹⁶ W/cm² and average lifetimes ⟨T_lifetime⟩ ≈ 100–300 fs. These pulses may affect the evolution of LPIs. For instabilities coupled with the IAWs, such as stimulated Brillouin scattering (SBS) and cross-beam energy transfer (CBET), the growth rates are relatively low. The lifetime of these high-intensity pulses (∼0.1 ps) is too short compared with the growth time of these instabilities (∼10 ps). Broadband lasers can suppress these instabilities, as reported in recent KunWu experiments.33,34 For instabilities coupled with EPWs, such as BSRS and two-plasmon decay (TPD), the growth rate is high, and these strong local pulses are anticipated to impact the evolution of BSRS.

To investigate the differences in the evolution of BSRS driven by monochromatic and broadband lasers, two one-dimensional simulations were performed. These simulations used the electromagnetic particle-in-cell (PIC) code ASCENT.35 The temporal and spatial units were normalized to the period τ₀ = 1.17 fs and wavelength λ₀ = 351 nm of the incident laser. One simulation used a monochromatic laser, while the other used a broadband laser with bandwidth Δω/ω₀ = 1.5% (similar to FLUX). The two lasers’ average intensity was ⟨I⟩ = 2.8 × 10¹⁵ W/cm². Both simulations lasted 9000τ₀ (∼10.5 ps) in a 200λ₀ (∼70 μm) box. A uniform hydrogen plasma with density n_e = 0.13n_c and length L_plasma = 180λ₀ was placed in the simulation box. Vacuums with length L_vac = 10λ₀ were placed on each side of the plasma. The electrons had temperature T_e = 3 keV, and the ions were immobile as a fixed positive charged background. Such plasma parameters enabled the rapid growth and saturation of BSRS. We obtained kλ_d ≈ 0.30. This implies that kinetic effects have a significant impact on the BSRS evolution.36

Figure 2 presents the simulation results. The first difference between the two simulation is in the lasers’ intensity envelopes, as shown in Figs. 2(a1) and 2(a2). The intensity envelope of the monochromatic laser remains constant, while that of the broadband laser varies significantly. In Fig. 2(a2), the strongest local pulse, marked with a red arrow, reaches six times the average intensity, i.e., I_peak ≈ 1.8 × 10¹⁶ W/cm². We also observe two different reflectivity modes. In the monochromatic laser case, the reflectivity achieves a stable mode after the BSRS saturation, as shown in Fig. 2(b1). In the broadband laser case, the reflectivity displays a series of spike structures after saturation, as shown in Fig. 2(b2). These reflectivity spikes correspond to local pulses. The most violent burst occurs at t = 4500τ₀, with a peak reflectivity R ≈ 6, which corresponds to the strongest local pulse shown in Fig. 2(a2). Although the broadband laser slightly delays the saturation of the BSRS from 600τ₀ to 900τ₀, the average reflectivity ⟨R⟩ is approximately the same at 9000τ₀ in the two simulations, namely, around 30%. Previous studies have also observed similar bursts.27,28 These studies have demonstrated that in broadband laser-driven BSRS, the nonlinear bursts are the primary source of back-scattered light.

The spatiotemporal evolution of EPWs also exhibits different patterns in the two simulations, as shown in Figs. 2(c1) and 2(c2). In the monochromatic laser case, EPW signals generated by BSRS are concentrated in the plasma’s surface layer, as marked by the dashed lines in Fig. 2(c1). These EPWs are rapidly damped during propagation. In the broadband laser case, the high-intensity pulses continuously transfer energy to the EPWs and back-scattered light via the BSRS in the propagation path, as marked by the dashed lines in Fig. 2(c2). The violent burst at 4500τ₀ produces significant EPW signals, as marked by the white arrow. These results confirm that the evolution of BSRS is affected by the variations in the broadband laser’s intensity envelope. If the average intensity of the broadband laser exceeds 10¹⁵ W/cm², then high-intensity local pulses can trigger violent bursts of BSRS. These bursts are considered to be the primary source of scattered light and hot electrons. Given the parameters of current broadband laser facilities, nonlinear BSRS bursts represent a problem worthy of attention.

The long-duration simulations reveal violent burst events during the nonlinear stage of the broadband laser-driven BSRS. However, quantitative assessment of these bursts still poses challenges. The random nature of the broadband laser’s intensity envelope is an obstacle to our obtaining a deeper understanding of the burst. We cannot predict when the bursts will occur or how intense they will be. Complex interactions between multiple local pulses also mask some of the kinetic processes involved in the bursts. Owing to the massive computational cost that would be incurred, performing thousands of long-time simulations to find quantitative modes is not feasible. Thus, it is necessary to construct a simplified model that can reproduce burst events while excluding the influence of randomness.

To design a simplified model of the burst, we need to comprehend its mechanisms. According to our previous work,27 the burst is not solely driven by one high-intensity local pulse, but originates from the interaction between the pulse and the EPW packets. As shown in Fig. 3(a), in the kinetic regime, the EPW packets (marked with orange arrows) trap electrons as they propagate. Trapping flattens the electron distribution function and leads to a reduction in the Landau damping γ_L. When a high-intensity pulse (marked with a blue arrow) is incident on the plasma and propagates to the region where former EPWs exist, a violent BSRS burst occurs at the crossing region (marked with light orange shading). The burst generates back-scattered light (marked with a green arrow) and enhances the intensity of the EPW packets.

We use a two-step approach to design a simplified model based on the burst process analysis, as shown in Fig. 3(b). In the first step, we inject a flat laser into the plasma slab. The laser’s intensity I_flat = 2.8 × 10¹⁵ W/cm² and duration t_flat = 800τ₀ are carefully selected to create a cluster of EPW packets in the plasma surface layer.37,38 In the second step, after 50τ₀, we inject a Gaussian pulse into the plasma with an average intensity ⟨I_pulse⟩ = 6 × 10¹⁵ W/cm²$(\approx 2I_{\text{flat}})$. The time interval between the flat laser and the pulse distinguishes the back-scattered light signals produced by the two. The pulse has a peak intensity I_peak = 1.8 × 10¹⁶ W/cm²$(\approx 6I_{\text{flat}})$ and a duration t_pulse = 200τ₀. Such pulses are frequently observed in broadband lasers with moderate bandwidths, as shown in Figs. 1(c) and 2(a2). In the future broadband laser facilities aimed at ignition (MJ level), the peak intensity of the pulses may be even higher. Since the group velocity of the laser pulse v_laser = 0.93c is much larger than that of EPW packets v_packet = 0.07c, the pulse and EPW packets encountered each other in the plasma and drive an intense BSRS burst. As shown in Fig. 3(c), the flat laser generates a weak back-scattered signal during 600τ₀–800τ₀, indicating initial saturation of the BSRS and the formation of EPW packets. After the high-intensity pulse has been injected into the plasma, a spike of back-scattered light is observed, providing evidence of the BSRS burst. This simplified model allows us to isolate and characterize the kinetic effects during bursts.

An overview of the BSRS burst process is given in Fig. 4. By analyzing the spatial and temporal evolution of the laser, back-scattered light, and EPWs, we can locate two BSRS events in the simulation, marked with the numbers “1” and “2.” The first BSRS event is driven by the flat laser. As can be seen in Fig. 4(a), a weak BSRS occurs at t = 650τ₀. The energy of the laser is transferred to the back-scattered light and EPWs, causing a “notch” structure (marked with “1”). At the same location in Fig. 4(b) and 4(c), back-scattered light and EPW packets are observed. The generation of EPW packets suggests that BSRS is entering the nonlinear stage. These EPW packets propagate deeper into the plasma at a group velocity of v_packet = 0.07c and encounter the high-intensity pulse at t = 1000τ₀. Violent EPW bursts occur at the location of the encounter.

We observe from Fig. 4(c) that the evolution of the EPWs in the simulation can be clearly divided into three stages. (1) During 0–650τ₀, the EPWs are in the linear stage. Their signal is so weak that it is barely visible. (2) During 650τ₀–1000τ₀, the intensity of the EPWs increases, and a wave packet structure emerges. (3) During 1000τ₀–1600τ₀, the high-intensity pulse, in synergy with these EPW packets, drives the BSRS burst. The intensity of the EPW packets sharply increases during the burst. After the burst, the EPW packets have a complex spatial structure.

The emergence of the BSRS burst demonstrates the success of our simplified modeling. In previous studies,27,28 multiple pulses in broadband lasers excited chaotic EPWs. Although similar bursts were observed in these studies, the chaotic EPWs hindered further analysis of kinetic effects. Our simplified model overcomes this limitation. The flat laser creates stable and homogeneous EPW packets, which enables us to use spectral analysis tools to detect kinetic effects during the burst.

The alteration in morphology of the EPW wave packets following the BSRS burst [shown in Fig. 4(c)] prompted us to analyze the spectrum of the EPWs. Applying a fast Fourier transform (FFT) to E_x gives the k-spectrum of EPWs over time, as shown in Fig. 5(a). The transform spatial window is [20λ₀, 180λ₀], and the wavenumber k_e is normalized to ω₀/c. Figures 5(b)–5(d) present the k-spectra at t = 600τ₀, 750τ₀, and 1100τ₀, which correspond to three different stages of the EPWs. On the basis of the respective features of these k-spectra, the three stages are named (1) the linear stage, (2) the sideband stage, and (3) the turbulence-like stage. A noteworthy finding is that the k-spectrum morphology of the EPWs changes from the sideband stage to the turbulence-like stage very rapidly, owing to the high-intensity pulses in broadband lasers.

As shown in Figs. 5(a) and 5(b), during 0–650τ₀, the wave spectrum has only one peak signal k_lw ≈ 1.41ω₀/c. The position of k_lw can be obtained by directly solving the dispersion relation of BSRS:$${\omega }^2-{\omega }_{\text{e}}^2=\frac{{\omega }_{\text{pe}}^2{k}^2{v}_{\text{os}}^2}{4}\left[\frac{1}{D(\omega -{\omega }_0,k-k_0)}\right],$$(2)where $D(\omega ,k)={\omega }^2-{\omega }_{\text{pe}}^2-k^2{c}^2$, ω_e is the frequency of the EPWs in the linear stage, ω₀ and k₀ are the circular frequency and wavenumber of the laser, and ω_pe is the electron plasma frequency. On substituting ${\omega }_{\text{e}}^2={\omega }_{\text{pe}}^2+3k^2{v}_{\text{th}}^2$ into Eq. (2), we get the wavenumber of the EPWs in the linear stage as$$k_{\text{lw}}=k_0+\frac{{\omega }_0}{c}{\left(1-\frac{2{\omega }_{\text{pe}}}{{\omega }_0}\right)}^{1/2}.$$(3)

The solution of Eq. (3) is k_lw = 1.41ω₀/c. The calculated value is in good agreement with k_lw observed in the simulation, which is marked as a white dashed line in Fig. 5(a). The good agreement between the simulation and the theoretical calculation indicates that the EPW intensity is weak at this stage, and the electron distribution function has not yet deviated significantly from the initial Maxwellian distribution.

During 650τ₀–1000τ₀, EPW packets appear. Their intensity in this stage is nearly constant, indicating that the BSRS is saturated. However, the k-spectrum shows new features compared with the linear stage. As shown in Fig. 5(a), some sideband structures near the linear mode k_lw are observed in this stage. The locations of these sidebands on the k-spectrum exhibit symmetry around the linear mode k_lw, but there is a difference in the intensities of the sidebands. The intensity of the lower-wavenumber sidebands is higher than that of the higher-wavenumber sidebands. In Fig. 5(c), it can be observed clearly that there are two main sideband peaks around k_lw, named k_down (sideband with lower wavenumber) and k_up (sideband with higher wavenumber). It has been shown that sidebands are a signature of a secondary instability called trapped particle instability (TPI).39

TPI is the result of electrons trapped by high-intensity EPWs. The number of trapped electrons N_trap increases with the rise in EPW intensity |E_x|. We approximate all the trapped electrons as a macroparticle bouncing in the EPW potential well ϕ. The bounce frequency is ${\omega }_{\text{be}}={(k_{\text{lw}}^{2}e\varphi /m_{\text{e}})}^{1/2}$.39,40 The oscillations of the macroparticle beat with the EPWs, and thus sidebands emerge. The dispersion relation for TPI is39,41,42$$1=\frac{{\omega }_{\text{t}}^2}{{\mathrm{\Omega }}^2-{\omega }_{\text{be}}^2}\left[\frac{1}{\epsilon (k,\omega )}+\frac{1}{\epsilon (k-2k_0,\omega -2{\omega }_0)}\right],$$(4)where Ω = ω − kv_p, and ω_t is the electron plasma frequency of the trapped electrons. We define the ratio of trapped electrons f_t = N_trap/N_e, where N_e is the total number of electrons. We denote the density of trapped electrons by n_t = f_tn_e. We then obtain the trapped electrons’ plasma frequency39${\omega }_{\text{t}}=\sqrt{n_{\text{t}}{e}^2/{\epsilon }_0{m}_{\text{e}}}=\sqrt{f_{\text{t}}{n}_{\text{e}}{e}^2/{\epsilon }_0{m}_{\text{e}}}=\sqrt{f_{\text{t}}}{\omega }_{\text{pe}}$. ɛ(k, ω) is the dielectric function of the background electrons, given by$$\epsilon (k,\omega )=1-\frac{{\omega }_{\text{pe}}^2}{{\omega }^2-3k^2{v}_{\text{th}}^2},$$(5)and v_th is the electron thermal velocity.

By integrating the electron distribution function during 600τ₀–1000τ₀, we get the average ratio of trapped electrons as ⟨f_t⟩ ≈ 0.001 in the sideband stage, giving ⟨ω_t⟩ ≈ 0.0316ω_pe, and the electron bounce frequency is ⟨ω_be⟩ ≈ 0.23ω_pe. We then obtain the distribution of the sideband modes ω_TPI(k_e) by numerically solving Eq. (4). There are two sideband peaks in ω_TPI(k_e), which are symmetrically distributed on both sides of the linear mode k_lw. The two maximum-growth sideband modes are k_down = 1.05ω₀/c and k_up = 1.77ω₀/c. As shown by the white dotted line in Fig. 5(a), the theoretical result is generally consistent with our simulation. The slight difference arises because we actually take the averages ⟨ϕ⟩ and ⟨f_t⟩ instead of the true values ϕ(t) and f_t(t).

The BSRS burst leads to a dramatic transition in the EPW k-spectrum, namely, from the sideband stage to the turbulence-like stage. During the burst, the pulse energy is transferred to EPW packets, leading to rapid growth of |E_x|. After the burst, the EPW k-spectrum shows a very complex, turbulence-like structure. In Fig. 5(a), it can be seen that the sidebands merge with each other in this stage, and their wavenumbers change with time. There is no longer a clear boundary between the sideband modes and the linear mode k_lw. We can only estimate upper and lower bounds for the k-spectrum of the EPWs, giving k_lb ≈ 0.81ω₀/c and k_ub ≈ 1.81ω₀/c, as shown in Fig. 5(d).

The rapid transition of the k-spectrum may come from wave-breaking of EPWs.43 Using the water bag model, the wave-breaking threshold E_thr is given by$$E_{\text{thr}}=\frac{m_{\text{e}}{v}_{\text{p}}{\omega }_{\text{e}}}{e}\sqrt{1+2{\beta }^{1/2}-\frac{8}{3}{\beta }^{1/4}-\frac{1}{3}\beta },$$(6)where $\beta =3v_{\text{th}}^2/v_{\text{p}}^2$. The threshold calculated using Eq. (6) is eE_thr/m_eω₀c = 0.0046, which is similar to the maximum value for EPWs that we observed in our simulations. Thus, the conversion of the EPW k-spectrum from the sideband stage to the turbulence-like stage is most likely due to the wave-breaking driven by high-intensity broadband laser pulses.

In the turbulence-like stage, the TPI model cannot accurate describe the EPW k-spectrum. This is because in a chaotic electric field, it is not possible to approximate all trapped electrons as a single macroparticle in simple harmonic motion.44,45 However, we still try to estimate k_lb and k_ub using the TPI model. The average ratio of trapped electrons ⟨f_t⟩ during 1000τ₀–1600τ₀ increases from 0.001 to 0.01, and its bounce frequency ⟨ω_be⟩ grows from 0.23ω_pe to 0.34ω_pe. Substituting ⟨f_t⟩ = 0.01 and ⟨ω_be⟩ = 0.34ω_pe into Eq. (4), we get the growth rate function of the sideband modes ω_TPI(k_e), as shown in Fig. 6(a). The theoretical results for the two maximum sideband modes are k_down = 0.82ω₀/c and k_up = 1.95ω₀/c. It can be seen that k_down is basically consistent with the simulation lower boundary k_lb = 0.81ω₀/c. However, k_up and the upper boundary k_ub = 1.81ω₀/c have larger deviations. Moreover, during the turbulence-like stage, the k-spectrum shows a strong asymmetric feature. The intensity of the low-wavenumber modes is significantly higher than that of the high-wavenumber modes. This asymmetry appears in the sideband stage and is further amplified during BSRS bursts. The discrepancy between the simulation upper bound k_ub and the theoretical result k_up is due to the amplified k-spectrum asymmetry.

The sideband asymmetry in the TPI was discovered and explained by Brunner and Valeo.46 Similar phenomena were observed in the simulations by Mašek and Rohlena47 and Yang et al.42 However, it is noteworthy that the asymmetry also exists in broadband laser-driven BSRS and is significantly enhanced after BSRS bursts. This asymmetry arises from the variation of the Landau damping γ_L with the wavenumber of the EPWs, k_e. If the electrons in the plasma satisfy a Maxwellian distribution, then the Landau damping of the EPWs is$${\gamma }_{\text{L0}}\left(k_{\text{e}}\right)=-\sqrt{\frac{\pi }{8}}\frac{{\omega }_{\text{pe}}}{{\left(k_{\text{e}}{\lambda }_{\text{de}}\right)}^3}\exp \left(-\frac{1}{2k_{\text{e}}^2{\lambda }_{\text{de}}^2}-\frac{3}{2}\right).$$(7)

γ_L0(k_e) is plotted in Fig. 6. The origin of the k-spectrum asymmetry is easy to understand when we compare the damping rate γ_L0(k_e) with the growth rate ω_TPI(k_e). In the green shaded region in Fig. 6, the Landau damping γ_L0(k_e) < ω_TPI(k_e), and these low-wavenumber EPW modes can grow easily. By contrast, when γ_L0(k_e) > ω_TPI(k_e), it is almost impossible for EPW modes with high wavenumbers to grow. Although ω_TPI(k_e) exhibits a symmetrical distribution around k_lw, the nonlinear growth of γ_L0(k_e) favors the lower wavenumbers in mode competition. The difference in growth rates between EPW modes gives rise to the k-spectrum asymmetry in the kinetic regime.

It is worth noting that the restriction of the high-wavenumber modes is loosened when the BSRS enters the nonlinear stage. In the kinetic regime, with the increased number of trapped electrons, the flattening of the electron distribution causes the Landau damping to decrease dramatically.48–50 Yin et al.50 found that the Landau damping in the high-intensity speckle decreased to one-tenth of its initial value γ_L0 (marked as the dotted line in Fig. 6). Liu et al.27 also observed that the Landau damping decreased to one-half of its initial value with a broadband laser (marked as the dashed line in Fig. 6). The decrease in γ_L gives these high-wavenumber EPW modes a chance to grow, as shown in the yellow shaded region in Fig. 6. Nevertheless, the energy of these modes is still relatively low, and the modes in the pink shaded region in Fig. 6 hardly grow at all.

High-intensity pulses in broadband lasers act to amplify the asymmetry of the EPW k-spectrum even more significantly. The energy of the high-intensity pulse is transferred to the low-wavenumber sideband modes through bursts (see Appendix A). In Appendix B, we compare the BSRS bursts driven by pulses of different intensities and find that the k-spectrum asymmetry is proportional to the pulse intensity. The asymmetric k-spectrum affects electron trapping, which in turn alters the electron phase space structure.

High-intensity pulses in broadband lasers significantly change both the k-spectrum structure of the EPWs, and also trigger violent vortex-merging in electron phase space. Electrons can be trapped when their velocity v_e is close to the EPW phase velocity v_p. These trapped electrons create vortices in electron phase space. The center velocity of these vortices v_center is equal to v_p. The height of these vortices is given by51$$\mathrm{\Delta }{v}_{\text{tr}}=2v_{\text{th}}\sqrt{\frac{e\varphi }{T_{\text{e}}}}.$$(8)

To illustrate the evolution of these vortices, we present in Fig. 7 snapshots of the electron phase space, for various times (t = 600τ₀, 800τ₀, 1050τ₀, and 1100τ₀). The x space of these snapshots moves with the EPW packets. As shown in Fig. 7(a), in the linear stage, there is only one EPW mode, with k_lw = 1.41ω₀/c. The linear mode forms the periodic structure of the potential. As a result, each electron vortex is independent of the others, and the trapped electron orbits are closed in phase space. In the sideband stage, while the intensity of the sideband mode E(k_sideband) grows close to that of the linear mode E(k_lw), the potential between the two neighboring vortices is decreased. Correspondingly, we find in Fig. 7(b) that the periodic potential structure is disrupted and some vortices start to merge.

During the BSRS burst, two factors lead to a more violent vortex-merging process. The first is the rapid increase in EPW intensity, which further increases the height of the vortex Δv_tr, as well as the number of trapped electrons N_trap. The second factor is the transition of the EPW k-spectrum from the sideband to the turbulence-like structure. Thus, the periodic potential is further disrupted. As shown in Fig. 7(c), compared with the linear phase, the height of the vortices increases, the potential barriers between some vortices have disappeared, and the electron orbits in phase space are no longer periodic closed orbits. At t = 1100τ₀, the BSRS burst has ended. As shown in Fig. 7(d), the initial periodic vortex no longer exists, and the vortex-merging process stabilizes. The electron phase space becomes chaotic, and the electron orbits almost fill the interior of the vortices. Vortex-merging induced by the BSRS burst is more violent than in the sideband stage. In practical broadband laser experiments, this process may occur many times and cause the yield of hot electrons to exceed linear theoretical predictions.

In addition to the increase in trap width Δv_tr, another interesting phenomenon appears in Fig. 7. We find that along with the vortex-merging process, the central velocity of the vortex v_center deviates from the initial phase velocity v_epw, leading to an increasing trend. In Figs. 8(a)–8(c), we show partial enlargements of the electron phase-space snapshots for the linear stage (t = 600τ₀), the sideband stage (t = 800τ₀), and the turbulence-like stage (t = 1050τ₀), respectively. The details of the electron vortices are carefully measured. In the linear stage, v_center is equal to v_epw, and Δv_tr ≈ 0.23c. In the sideband stage, v_center ≈ 0.33c is shifted away from v_epw, and Δv_tr is increased to 0.52c. After the burst, the gap between v_center ≈ 0.38c and v_epw becomes larger, and Δv_tr increases to 0.60c.

These changes are also manifest in the electron distribution function. Figure 8(d) shows that the BSRS burst has enlarged the width of the trap platform (by increasing Δv_tr) and extended its high-energy tail (by increasing v_center). There is no doubt that the increase in v_center is an important mechanism for hot-electron generation. Thus, a key question arises, namely, how can the increase in v_center be explained?

Using the Bohm–Gross dispersion relation, the phase velocity v_p of different k_e modes can be expressed as$$v_{\text{p}}(k_{\text{e}})=\frac{\sqrt{{\omega }_{\text{pe}}^2+3v_{\text{th}}^2{k}_{\text{e}}^2}}{k_{\text{e}}}.$$(9)

Equation (9) links the EPW k-spectrum and the electron phase space. As can be seen in Fig. 9(a), these EPW modes with lower wavenumbers have higher phase velocities (marked with the black line). In the linear stage, the electrons are trapped only by the linear mode k_lw = 1.41ω₀/c, and the center trapping velocity v_center is equal to v_epw (marked as the blue dashed line). After the BSRS burst, the intensity of the lower-wavenumber modes increases. In the turbulence-like k-spectrum, electrons are no longer trapped by the linear mode. The center trapping velocity v_center rises from v_p(k_lw) = 0.29c to v_p(k_center). Here, k_center represents the actual center position of the EPW k-spectrum during the turbulence-like stage, which is given by$$k_{\text{center}}(t)=\frac{{\int }_{k_{\text{lb}}}^{k_{\text{ub}}}{kE}_x(k,t)dk}{{\int }_{k_{\text{lb}}}^{k_{\text{ub}}}{E}_x(k,t)dk},$$(10)where k_lb and k_ub are the lower and upper boundaries of the k-spectrum. Using Eq. (10), we get k_center(1050τ₀) = 1.15ω₀/c. Substitution of k_center(1050τ₀) into Eq. (8) shows that the phase velocity v_p(k_center = 1.15ω₀/c) increases to 0.35c, which is marked with the red dashed line in Fig. 9(a). Equations (9) and (10) demonstrate that the asymmetry in the EPW k-spectrum is transferred to electron phase space through wave–particle interaction.

The increases in center trapping velocity v_center and trap width Δv_tr raise the high-energy tail of the electron distribution function from 75 to 90 keV, as can be seen in Fig. 9(b), and the ratio of hot electrons (u_e > 20 keV) increases from 6.76% to 14.78%. Note that one burst takes only 0.2 ps, but generates a large number of hot electrons. This may help to explain the hot-electron anomaly observed in recent experiments at the KunWu facility.33 In these experiments, broadband lasers lasted several nanoseconds. Multiple similar burst events can cause a significant increase in the hot-electron yield.

Furthermore, the BSRS burst and saturation mechanisms are related to the plasma density.52 In Appendix C, we perform simulations using the simplified model for various plasma densities and observe a transition from trapping-domination (kinetic regime) to Langmuir-wave decay-domination (fluid regime) as the plasma density increases. In addition, the burst behavior may change in high-dimensional simulations. In high-dimensional schemes, lasers and EPWs are localized in the speckle structure, reducing the probability of bursts. However, the diffusion of hot electrons and side-scattered light also introduces the risk of self-organized bursts.49 These considerations suggest that the long-term evolution of the instabilities coupled to EPWs driven by broadband lasers is complex, and further investigation requires systematic experiments on facilities such as KunWu and FLUX.

Violent burst events have been observed in numerical simulations of BSRS driven by a broadband laser with bandwidth Δω/ω₀ = 1.5%. The bandwidth degrades the broadband laser’s temporal incoherence, resulting in high-intensity local pulses. The interaction of EPWs and these high-intensity pulses causes BSRS bursts. By developing a simplified model of bursts, we have excluded the interference of broadband laser randomness and revealed the presence of rich kinetic effects during BSRS evolution. The BSRS bursts both increase the intensity of the EPWs and change their spectral structure. After a burst, the EPW k-spectrum has a turbulence-like structure, and chaotic structures caused by vortex-merging are observed in the electron phase space. We have also found that high-intensity pulses amplify the asymmetry of the EPW k-spectrum. The asymmetry is transferred to electron phase space, causing an increase in the electron trapping center velocity, which in turn generates a large number of hot electrons. The ratio of hot electrons increases from 6.76% to 14.78% during one violent burst event. These results should help to improve understanding of broadband laser-driven instabilities.

Further investigation is required to determine the relationship between the kinetic bursts and laser intensity on the nanosecond time scale. Experimental results on KunWu34 have shown that a 527 nm broadband laser with bandwidth Δω/ω₀ = 0.6% can effectively suppress BSRS when the average intensity is lower than 5 × 10¹⁴ W/cm². However, the suppression fades as the laser intensity increases. Future broadband laser facilities for ignition have higher energies (MJ level). Whether these high-energy laser facilities can effectively suppress LPIs at moderate bandwidths is a topic of interest and needs to be verified by further experiments. In addition, some ingenious broadband laser schemes have shown a reliable ability to suppress LPIs.13,15,17 From another perspective, broadband lasers have the potential to be used as drivers for shock ignition. More work remains to be done to study how the results may change in 2D and 3D geometries and to develop statistical methods for quantitatively assessing hot electron yields that can be used in broadband laser experiments.

Acknowledgment. This project is supported by the National Key R&D Program of China (Grant No. 2022YFA1603204) and the National Natural Science Foundation of China (Grant Nos. 12325510, 12235014, and 11975055). Valuable discussions with Professor Chunyang Zheng, Professor Zhanjun Liu, Dr. Enhao Zhang, and Dr. Hongyu Zhou have been beneficial for this research.