Mitigation of parametric instabilities in laser–plasma interactions is crucial for the success of laser-driven inertial confinement fusion (ICF).1–3 The most important instabilities include stimulated Raman scattering (SRS), stimulated Brillouin scattering (SBS), and two-plasmon decay (TPD),4,5 which not only inhibit energy deposition by scattering light away from plasmas, but also reduce fuel compressibility owing to the generation of suprathermal electrons.2,6 Since the 1960s, various laser techniques and schemes have been proposed to mitigate parametric instabilities by increasing the incoherence of the driving lasers, such as beam smoothing,7–12 temporal shaping,13 frequency chirping,14 and using broadband lasers.15–17

With the use of broadband lasers,15–24 the instability growth rate is expected to be reduced by a factor of Γ/Δω when the laser bandwidth Δω is larger than the linear growth rate Γ of the instabilities.18 Experiments have demonstrated that SBS can be efficiently suppressed by using spatially and temporally incoherent laser beams.20 Frequency detuning of the drive laser beams is expected to suppress the TPD instability and hot-electron generation in direct-drive ICF, even with a relatively narrow bandwidth ∼0.7%.22 Numerical simulations have also demonstrated the efficacy of multi-terahertz laser bandwidth in suppressing cross-beam energy transfer in direct-drive ICF.23 Furthermore, the thresholds of absolute SRS and TPD can be significantly increased when 1% laser bandwidth is introduced for ICF-relevant conditions.24,25

SRS, in which a laser light wave decays into an electron plasma wave and a scattered light wave, has received most attention, because it usually has a larger growth rate than other instabilities. Previous studies have shown that broadband lasers can effectively reduce SRS,16,21,26–28 although the effects of laser bandwidth on the scattering angle of SRS have rarely been studied. In particular, some simulations have revealed that forward SRS could develop when backward SRS is suppressed.29 Further, sideward SRS with a monochromatic laser beam has been studied extensively,30–33 and it has been shown that the scattering angle varies with the laser spot size. Importantly, sideward SRS can not only generate hot electrons,34,35 but also seed instabilities by interacting with other laser beams.36 Therefore, it becomes particularly important to investigate the angular distributions of SRS when backward SRS may already be suppressed by using a broadband laser.

In this work, the competition between sideward and backward SRS in the interactions between broadband lasers and homogeneous plasmas is investigated theoretically and numerically. Our study suggests that a much larger bandwidth is required to suppress side-scattering as compared to backscattering. The remainder of the paper is organized as follows. In Sec. II, a theoretical model of SRS side-scattering for the coupling of two monochromatic laser beams with a certain frequency difference is presented. Extension to broadband lasers is done by numerical simulations in Sec. III. The paper concludes with a summary in Sec. IV.

We start from the general dispersion relation for SRS with a monochromatic laser A_L = A₀ cos(k₀ · x − ω₀t):37,38$$\frac{4({\omega }^2-{\omega }_{pe}^2)}{{\omega }_{pe}^2{k}^2{c}^2{a}_0^2}=\frac{1}{D(\mathbf{k}-{\mathbf{k}}_0,\omega -{\omega }_0)}+\frac{1}{D(\mathbf{k}+{\mathbf{k}}_0,\omega +{\omega }_0)},$$(1)where $D(\mathbf{k}\pm {\mathbf{k}}_0,\omega \pm {\omega }_0)={(\omega \pm {\omega }_0)}^2-{(k_x\pm k_0)}^2{c}^2-k_y^2{c}^2-{\omega }_{pe}^2$, k₀ = k₀i is the laser wave vector in the plasma, a₀ = eA₀/m_ec² is the normalized laser intensity, and k = k_xi + k_yj and ω_pe are the wave vector and frequency of the electron plasma wave, respectively. For the sake of simplicity here, the effect of the plasma temperature is ignored in the theoretical model. For backward SRS, k_y = 0.37 For sideward SRS, however, k_y ≠ 0.

The linear growth rate Γ = Im(ω) can be obtained from Eq. (1). As an example, this is displayed as a function of the scattering angle θ, defined by θ = 180° − arctan[k_y/(k_x − k₀)] = arctan(k_sy/k_sx), in Fig. 1(a), where a₀ = 0.02 and n₀ = 0.128n_c, and k_sx and k_sy are the two components of k_s. Combining Eq. (1) with the frequency and wave vector matching conditions between the pump laser (ω₀, k₀), the scattered light (ω_s, k_s), and the electron plasma wave (ω_pe, k), one can obtain k_sr = 0.533 ω₀/c and ω_s = 0.642 ω₀ at the resonance point for a cold plasma.

The probability distribution of the electron plasma wave vector k at resonance is a circle of radius k_sr = 0.533 ω₀/c and center at k₀ = 0.934 ω₀/c, and the maximum growth rate Γ_max is obtained for backward SRS with θ = 180°. For a monochromatic laser, the growth rate Γ decreases monotonically with decreasing scattering angle θ.

SRS can develop within certain region around the resonance points, the boundary of which is sketched by the dashed lines in Fig. 1(a). Here, polar coordinates are used, where the center is the resonance point k_sr = 0.533ω₀/c, and $\mathrm{\Delta }k=\sqrt{{(k_x-k_0)}^2+k_y^2}-k_{sr}$ is the deviation of the wavenumber of the electron plasma wave from the resonance point. The boundary can be obtained from Eq. (1) using Γ = 0 as$$\mathrm{\Delta }{k}_b=a_0{k}_r\sqrt{\frac{{\omega }_{pe}({\omega }_0-{\omega }_{pe})}{{\omega }_0^2-2{\omega }_0{\omega }_{pe}}},$$(2)where $k_r=\sqrt{k_0^2+k_{sr}^2-2k_0{k}_{sr}\cos \theta }$ is the wavenumber of the electron plasma wave at the resonance points.

We now consider a broadband laser, which can be considered as the superposition of many monochromatic lasers with different frequencies. To simplify the analysis, we first consider the case with two monochromatic laser beams at frequencies ω₀ ± 0.5δω. These two beams have their own resonance points with the electron plasma waves of different wavenumbers k_r ± 0.5δk_r, where the wavenumber difference δk_r can be evaluated as$$\frac{\delta k_r}{\delta {\omega }_0}\approx \frac{d{k}_{rx}}{d{\omega }_0}=\frac{{\omega }_0}{c\sqrt{{\omega }_0^2-{\omega }_{pe}^2}}-\frac{({\omega }_0-{\omega }_{pe})\cos \theta }{c\sqrt{{\omega }_0^2-2{\omega }_{pe}{\omega }_0}},$$(3)where the electron plasma wave component along the x direction, k_rx, is given by$$k_{rx}=c^{-1}\left(\sqrt{{\omega }_0^2-{\omega }_{pe}^2}-\cos \theta \sqrt{{\omega }_0^2-2{\omega }_{pe}{\omega }_0}\right).$$(4)

Analogous to Eq. (1), the dispersion relation for this two-color laser light with A_L = A₁ cos(k₁ · x − ω₁t) + A₂ cos(k₂ · x − ω₂t) can be obtained as$${\omega }^2-{\omega }_{pe}^2=\frac{k^2{c}^2{\omega }_{pe}^2}{4}[{\mathrm{\Pi }}_0+f(\theta ){\mathrm{\Pi }}_{\times }],$$(5a)where$${\mathrm{\Pi }}_0=\sum _{i=1}^2\left[\frac{a_i^2}{D(\mathbf{k}+{\mathbf{k}}_i,\omega +{\omega }_i)}+\frac{a_i^2}{D(\mathbf{k}-{\mathbf{k}}_i,\omega -{\omega }_i)}\right],$$(5b)$${\mathrm{\Pi }}_{\times }=\sum _{i=1}^2\left[\frac{a_1{a}_2}{D(\mathbf{k}+{\mathbf{k}}_i,\omega +{\omega }_i)}+\frac{a_1{a}_2}{D(\mathbf{k}-{\mathbf{k}}_i,\omega -{\omega }_i)}\right].$$(5c)

Here, Π₀ is the direct contribution of two individual monochromatic laser beams, while Π_× is the contribution caused by the coupling of these two beams. The coupling factor f(θ) is a function of the scattering angle, which depends on the frequency difference δω and other laser–plasma parameters. The derivation of the above dispersion relation is detailed in Appendix A.

To examine the coupling condition, let us consider the two-color light case with δω/ω₀ = 1.2%, a_1,2 = 0.0141, and n₀ = 0.128n_c. In Fig. 1(b), the resonance points obtained from Eq. (3) for two individual monochromatic beams are shown by the red and blue lines, respectively, while the instability regions with Γ > 0 obtained from Eq. (2) are shown by the red and blue shading, respectively. If these two instability regions overlap, then these two beams couple with each other and f(θ) = 1. Otherwise, the two beams are decoupled and f(θ) = 0. Figure 1(b) indicates that the two instability regions overlap when the scattering angle θ is less than a certain angle θ_th. This threshold θ_th can be evaluated using Eqs. (2) and (3) as$$\mathrm{\Delta }{k}_b({\theta }_{\text{th}})=\delta k_r({\theta }_{\text{th}}).$$(6)Correspondingly, the coupling factor is$$f(\theta )=\left\{\begin{array}{cc}1,\hfill & \theta <{\theta }_{\text{th}},\hfill \\ 0,\hfill & \theta \ge {\theta }_{\text{th}}.\hfill \end{array}\right.$$(7)

For convenience, the notation used in the derivation of the theoretical model has been summarized in the Nomenclature list at the start of the paper. For given electron density and laser intensity, Fig. 1(d) shows that θ_th decreases with increasing frequency difference δω. As shown in Fig. 1(b), one has θ_th ≃ 90° for δω/ω₀ = 1.2%, a_1,2 = 0.0141, and n₀ = 0.128 n_c. By combining Eqs. (5) and (7), the growth rate Γ for two-color laser light can be calculated numerically. As shown in Fig. 1(c), it is now found that the maximum value of Γ is achieved at θ ≃ θ_th ≃ 90°, and so sideward SRS will dominate over backward SRS. In other words, stimulated side-scattering can develop even if backscattering is suppressed by using two-color laser light.

For the sake of simplicity, the coupling factor in Eq. (5) is set to be f(θ) = 1 if these two laser beams are coupled. Otherwise, f(θ) = 0. Such a step function form of the coupling factor f(θ) induces a discontinuity in the growth rate as shown in Fig. 1(c). In reality, however, the growth rate may not drop so suddenly if the transition from coupling to decoupling takes place in a finite region.

The above analysis was carried out for two different monochromatic laser beams, but it can in principle be extended to many monochromatic laser beams at different frequencies. On the other hand, broadband laser light can be considered as the superposition of many monochromatic laser beams. Therefore, the phenomenon discussed above, namely, the development of sideward SRS and its dominance over backward SRS, is expected to arise with a broadband laser as well. In the case of a broadband laser with a continuous frequency spectrum, however, the transition from backward to sideward SRS will take place at a fractional bandwidth that is different from the required frequency difference in the theoretical analysis using two-color laser beam.

To verify the above analysis, a series of 2D simulations were carried out by using the particle-in-cell (PIC) code EPOCH.39 The laser was incident along the x direction, with a Gaussian profile in the transverse direction. The laser wavelength λ = 0.35 μm, the spot size w₀ = 10λ, and the normalized amplitude a₀ = 0.02. A semi-infinite pulse was adopted, with a linear ramp at the initial 60 fs. The simulation box was 110λ × 40λ and was resolved with a cell dimension of dx × dy = 0.02λ × 0.05λ. There were 100 macroparticles per species per cell, and absorbing boundary conditions were used along each direction. The initial electron temperature was T_e = 3 keV, and the ions were fixed. Since side-scattering preferentially occurs out of the polarization plane,37 S-polarized laser fields were assumed in the simulations.

For comparison, we first show the spatial distribution of E_x at t = 800T₀ with a monochromatic laser beam in Fig. 2(a). It can be seen that the scattering due to SRS is largely concentrated around θ = 180°. The wave-vector distribution of E_x integrated over 0 ≤ t ≤ 1500T₀ is plotted in Fig. 2(b), which demonstrates that the scattering has its maximum amplitude roughly at θ_peak = 180°. This confirms that backward SRS is the dominant scattering for a monochromatic laser beam.

For comparison, the integrated wave-vector distributions of E_x for two-color laser light with δω/ω₀ = 0.6% and 1.2% are shown in Figs. 2(c) and 2(d), respectively. Here, the amplitudes of two individual monochromatic laser beams are set to be $a_1=a_2=a_0/\sqrt{2}=0.0141$, so that their total laser energy is equal to that in the case of a single monochromatic laser beam. As observed in previous studies,16,21,40Fig. 2(c) indicates that the backward SRS is significantly reduced when two-color laser light is used. More importantly, Fig. 2(d) demonstrates that significant sideward SRS occurs at roughly θ ≈ 90°, whereas backward SRS is effectively suppressed by the use of two-color laser light with δω/ω₀ = 1.2%. This phenomenon is consistent with the theoretical prediction shown in Fig. 1.

We also performed PIC simulations to study the angular distribution of SRS with three-color S-polarized laser light. In this case, the transition from backward to sideward SRS is also observed with increasing frequency difference between individual monochromatic laser beams. In addition, we studied the angular distribution of SRS for the P-polarized case. Figure 3(a) shows that backward SRS is also dominant for a P-polarized monochromatic laser beam. For P-polarized two-color laser light with δω/ω₀ = 1.2%, Fig. 3(b) shows that backward SRS is effectively suppressed. However, no obvious sideward SRS is observed in any of the 2D simulations with P-polarized two-color laser light that we performed, suggesting that sideward SRS is azimuthally anisotropic and preferentially occurs out of the plane of polarization.

Figure 4 demonstrates the transition from backward to sideward SRS when a generalized broadband laser source is used. In these simulations, the broadband laser light was assumed to have a flat-top frequency spectrum, and it was modeled as a combination of an ensemble of monochromatic laser beams25 (see Appendix B). The energy of the broadband laser was the same as that of the monochromatic laser, and the other simulation parameters were the same as those in Fig. 2.

As expected, the backward SRS is weakened with increasing laser bandwidth Δω. Figure 4(a) indicates that backward SRS is already suppressed effectively by a broadband laser with Δω/ω₀ = 2.6%. However, sideward SRS obviously develops around θ_peak ≈ 90°. The angle θ_peak at which the scattering amplitude peaks is shown as a function of Δω in Fig. 4(b), which clearly illustrates the transition from backward to sideward SRS with increasing laser bandwidth. The averaged reflectivity and transmittivity shown in Fig. 4(c) also reveal the transition from backward to sideward SRS with increasing laser bandwidth. At Δω/ω₀ = 2.6%, the reflectivity is nearly zero for both S- and P-polarized laser light, i.e., backward SRS has been effectively suppressed, regardless of polarization. However, there is a noticeable gap between the transmittivities of the P-polarized (∼100%) and S-polarized (∼87%) broadband sources. The lost laser energy in the case of S-polarization is due to side-scatter, whereas the latter cannot occur with P-polarization in 2D simulations. Interestingly, we find that the reflectivity at Δω/ω₀ ∼ 0.5% is higher than that in the monochromatic laser case. This may be caused by the coupling effect in the linear stage, as well as the burst behavior in the nonlinear stage.41

In addition, Fig. 4(d) shows that sideward SRS is very efficient at generating harmful hot electrons. Generally, the use of broadband lasers tends to reduce hot electron generation via suppression of backward SRS, as shown for the two cases with Δω/ω₀ = 2.6% for P-polarized and S-polarized lasers. However, the number of hot electrons with kinetic energy E_k > 80 keV is much higher in the case of S-polarized broadband lasers, owing to transition of the predominant scattering angle θ from 180°to 90°. This can also be seen in momentum phase space, as presented in Fig. 5. Further, owing to the occurrence of forward SRS as shown in Fig. 4(a), some electrons can attain kinetic energies E_k > 150 keV. The efficient hot electron generation due to sideward SRS in the S-polarization case can be partly alleviated with a larger bandwidth Δω/ω₀ = 4.0%.

To better understand this, the hot electron distributions in the p_x–p_y phase space are displayed in Figs. 5(a) and 5(c) for monochromatic laser light and broadband laser light with Δω/ω₀ = 2.6%, respectively. It is found that the hot electrons generated by monochromatic laser light are roughly collimated along the laser axis, whereas those generated by broadband laser light diverge significantly from the laser axis. The corresponding energy–angle distributions of the hot electrons with kinetic energies E_k > 30 keV are compared in Figs. 5(b) and 5(d). It is clear that broadband laser light is more efficient in generating hot electrons with kinetic energies E_k > 80 keV and that these more energetic hot electrons are mainly distributed along directions roughly at ±30° with respect to the laser axis. Such an angular distribution of hot electrons with higher energies is closely related to the direction of the plasma wave vector k_epw stimulated by the broadband laser light, as shown in Fig. 4(a). More importantly, the plasma wave of the sideward SRS in the broadband laser case has a smaller wavenumber k_epw than that of the backward SRS in the monochromatic laser case, as can be seen in Figs. 2(b) and 4(a). Correspondingly, the plasma wave phase velocity ω_epw/k_epw of the sideward SRS stimulated by the broadband laser is higher than that of the backward SRS stimulated by the monochromatic laser. Consequently, the hot electrons generated by the broadband laser with Δω/ω₀ = 2.6% are more energetic.

For the sake of simplicity, Landau damping has been ignored in our theoretical model. However, it is worth pointing out that Landau damping depends strongly on the ratio of plasma wave phase velocity to electron thermal velocity. Consequently, it will be affected significantly by the scattering direction. That is to say, the Landau damping rates of obliquely propagating electron plasma waves will be smaller than those for backward SRS. Considering Landau damping, therefore, the threshold of backward SRS will be enhanced more obviously than that of sideward SRS. The influence of Landau damping on the competition between backward and sideward scattering will be studied in more detail in future work. In addition, the amplification of the side-scattered light may be suppressed with the use of a finite laser spot, because they propagate out from the laser beam axis.42 With an infinite laser spot, sideward SRS would become stronger and surpass backward SRS with a smaller laser bandwidth.

Moreover, in this study, the SRS is mainly discussed in the context of interaction of a broadband laser and a homogeneous plasma. In an inhomogeneous plasma, the sideward SRS at angles near 90° has the lowest threshold, owing to the largest resonant region being along the transverse direction. This threshold at densities below n_c/4 can increase with increasing laser bandwidth.43 Experiments with normal lasers at intensities relevant to shock ignition schemes have shown that sideward SRS cannot be ignored, since it not only scatters laser energy at large angles, but also damps TPD and backward SRS driven in the higher-density region.44,45 Therefore, there is a need for further study of the competition between sideward SRS, backward SRS, and TPD in the region close to n_c/4.

In conclusion, we have found that the angular distribution of SRS depends strongly on laser bandwidth. First, the growth rate of SRS has been derived as a function of the scattering angle for two-color laser light at frequencies ω₀ ± 0.5δω. It is found that the two frequency components will be decoupled when the scattering angle is larger than a threshold angle θ_th, which decreases with increasing δω. Consequently, the strongest growing SRS will not always be located at θ = 180°. It turns out that sideward SRS may dominate over backward SRS for two-color laser light, as demonstrated by numerical simulation. Second, the transition from backward to sideward SRS has been confirmed generally for broadband lasers, which can be considered as the superposition of many monochromatic laser beams. Because sideward SRS can generate hot electrons with higher energy than backward SRS, special attention should be paid to sideward SRS when broadband lasers are considered as future ICF drivers.46,47

Acknowledgment. This work was supported by the Strategic Priority Research Program of the Chinese Academy of Sciences (Grant No. XDA25050100), the National Natural Science Foundation of China (Grant Nos. 11991074, 11975154, 12005287, and 12135009), and the Science Challenge Project (Grant No. TZ2018005). X. F. Li was supported by the China and Germany Postdoctoral Exchange Program from the Office of the China Postdoctoral Council and the Helmholtz Centre (Grant No. 20191016) and the China Postdoctoral Science Foundation (Grant No. 2018M641993). Y. Zhao was also supported by Guangdong Basic and Applied Basic Research Foundation (Grant No. 2023A1515011695). Simulations were carried out on the JURECA and JUWELS supercomputers at the Jülich Supercomputing Centre, which are granted from the Projects JZAM04 and LAPIPE.