In inertial confinement fusion (ICF), owing to the limited energy of a single laser beam, a large number of beams are needed to deliver the megajoule laser energy to the target required for both indirect-drive and direct-drive schemes.1–3 The ubiquitous overlapping of laser beams leads to complex multibeam laser–plasma interaction (LPI) instabilities, including crossed-beam energy transfer (CBET) between different beams,4–8 seeded multibeam instability due to seeds generated elsewhere in the plasma1 and by glint,9 and collective instability with shared daughter waves.3,10,11 Among the various LPI instabilities in ICF, stimulated Brillouin scattering (SBS) and stimulated Raman scattering (SRS) instabilities are of primary concern, since they can scatter significant amounts of light, leading to a great energy loss from the incident lasers as well as degradation of the irradiation symmetry.12–17 The collective modes with common daughter waves deserve particular attention owing to their great temporal growth rate and convective gain, which scale up with the number of pump beams.18–20 Experimentally, collective SRS and SBS result in significant scattered light losses in novel directions,19–21 which can be located far from the apertures of the beams where diagnostics are usually set up12,22 and are hence quite hard to detect. Understanding these processes is essential for better identifying, modeling, and diagnosing multibeam SRS or SBS processes and is helpful to optimize ICF implosions.

The collective SRS or SBS processes include shared plasma (SP) wave modes and shared light (SL) wave modes, depending on whether the shared daughter wave is a common Langmuir/ion acoustic wave or a common scattered wave. Previous theoretical studies of the homogeneous temporal growth rate for collective SP and SL modes of multiple beams have been conducted using a fluid description.23–25 In addition, some two-dimensional (2D) particle-in-cell simulations have verified the importance of in-plane collective SRS modes of two overlapped beams,25,26 where the wavevectors of daughter waves lie in the plane of incidence of two overlapped beams. The SP modes of collective SBS in the spatial convective regime, which is typical of practical ICF conditions,27–29 have recently been studied, and it has been found that the out-of-plane modes can be quite important for some polarization states of the laser beams.30 In the present work, the SL modes of collective SBS in the convective regime are studied, and the impacts of the crossing angle, polarization states, and finite beam overlapping volume of the two laser beams on SL modes of SBS are investigated systematically for both zero and nonzero wavelength differences between the two pump beams. Compared with the SP modes, the SL modes are found to be much more sensitive to the polarization states and wavelength difference. Nevertheless, the out-of-plane modes can still be quite important for some polarization states and beam crossing angles. The results of this work should be helpful in comprehending and estimating the importance of collective SBS with shared scattered wave in ICF experiments.

The remainder of the paper is organized as follows. In Sec. II, the theoretical model for SL modes is presented, where an analytic solution for the convective gain coefficient is given. In Sec. III, the impacts of the crossing angle, polarization states, and finite beam overlapping volume of the two laser beams on the scattered wavelength and spatial amplification of the collective SBS modes with shared scattered wave are investigated for both zero and nonzero wavelength differences between the pump lasers, and the importance of out-of-plane modes relative to in-plane modes is discussed. In Sec. IV, the conclusions are given, together with some discussions.

The SL modes of two crossing beams incorporate five coupled waves: the two pump light waves and one common scattered light wave, as well as two plasma waves corresponding to the coupling between each pump light and the common scattered light. The phase matching conditions can be written as$${\omega }_{0\alpha }={\omega }_{\mathrm{s}}+{\omega }_{e{s}_{\mathrm{\alpha }}},$$(1)$${\mathbf{k}}_{0\alpha }={\mathbf{k}}_{\mathrm{s}}+{\mathbf{k}}_{e{s}_{\mathrm{\alpha }}},$$(2)where the ω_i with subscripts i = 0α, es_α, and s (α = 1, 2) are the wave frequencies of laser beam α, plasma wave α, and the common scattered wave, respectively, and k_i with i = 0α, es_α, and s are the corresponding wavevectors. The geometry of the collective SL modes for two overlapped beams with crossing angle 2θ_h is shown in Fig. 1(a), where the xy plane is defined as the (k₀₁, k₀₂) plane with the x direction along the bisector of k₀₁ and k₀₂. The direction of the wavevector k_s for the scattered wave can be specified by (θ_s, φ_s), where the out-of-plane angle −90° ≤ φ_s ≤ 90° is the altitude of k_s measured from the xy plane, and the azimuthal angle −180° ≤ θ_s ≤ 180° is the angle from the x axis to the orthogonal projection of k_s onto the xy plane. The wavevector k_esα for the plasma wave driven by beam α is determined by the matching condition k_esα = k_0α − k_s, yielding$$\begin{array}{c}\hfill k_{{\mathrm{e}\mathrm{s}}_1}=\sqrt{k_{01}^2+k_{\mathrm{s}}^2-2{\mathbf{k}}_{01}\cdot {\mathbf{k}}_{\mathrm{s}}}\approx 2k_{01}\sin \frac{1}{2}{\vartheta }_1,\\ \hfill k_{{\mathrm{e}\mathrm{s}}_2}=\sqrt{k_{02}^2+k_{\mathrm{s}}^2-2{\mathbf{k}}_{02}\cdot {\mathbf{k}}_{\mathrm{s}}}\approx 2k_{02}\sin \frac{1}{2}{\vartheta }_2,\end{array}$$(3)where the approximate equality is applicable for collective SBS modes with a common scattered wave since k_s ≈ k₀₁ ≈ k₀₂ is taken for this approximation. ϑ₁ and ϑ₂ are the scattering angles of beams I and II and are defined by$$\begin{array}{cc}\hfill \cos {\vartheta }_1& =\frac{{\mathbf{k}}_{01}\cdot {\mathbf{k}}_{\mathrm{s}}}{k_{01}{k}_{\mathrm{s}}}=\cos {\phi }_{\mathrm{s}}\cos ({\theta }_{\mathrm{s}}+{\theta }_{\mathrm{h}}),\hfill \\ \hfill \cos {\vartheta }_2& =\frac{{\mathbf{k}}_{02}\cdot {\mathbf{k}}_{\mathrm{s}}}{k_{02}{k}_{\mathrm{s}}}=\cos {\phi }_{\mathrm{s}}\cos ({\theta }_{\mathrm{s}}-{\theta }_{\mathrm{h}}).\hfill \end{array}$$(4)

For most practical cases in ICF, both SRS and SBS are spatial problems,27,31,32 for which the convective amplification properties are of great importance. To study the convective amplification of the SL modes, the envelope approximation for the five coupled waves can be adopted. In the strong-damping regime, the equations for the complex vector amplitudes of the laser beams (a_0α) and the common scattered wave (a_s), and for the complex amplitude of the density perturbation of plasma waves (δn_esα) can be written as30,31$$\frac{\delta n_{e{s}_{\mathrm{\alpha }}}}{n_0}=-{\gamma }_{{pm}_{\mathrm{\alpha }}}\frac{k_{{es}_{\mathrm{\alpha }}^2}{c}^2}{2{\omega }_{\mathrm{p}\mathrm{e}}^2}{\mathit{a}}_{0\alpha }\cdot {\mathit{a}}_{\mathrm{s}}^{*}$$(5)and$${\mathbf{k}}_{\mathrm{s}}\cdot \nabla {\mathit{a}}_{\mathrm{s}}=-\frac{j{\omega }_{\mathrm{p}\mathrm{e}}^2}{4c^2}\sum _{\alpha =1,2}\frac{\delta n_{{es}_{\mathrm{\alpha }}^{*}}}{n_0}{a}_{0\alpha }{\mathbf{e}}_{0\alpha \perp {\mathbf{n}}_{\mathrm{s}}},$$(6)where a_i ≡ eA_i/m_ec is the normalization of the magnetic vector potential A, e is the electron charge, m_e is the electron mass, n₀ is the unperturbed electron density, and c is the speed of light in vacuum. ${\mathbf{e}}_{0\alpha \perp {\mathbf{n}}_{\mathrm{s}}}\equiv {\mathbf{n}}_{\mathrm{s}}\times ({\mathbf{e}}_{0\alpha }\times {\mathbf{n}}_{\mathrm{s}})={\mathbf{e}}_{0\alpha }-({\mathbf{e}}_{0\alpha }\cdot {\mathbf{n}}_{\mathrm{s}}){\mathbf{n}}_{\mathrm{s}}$ is the projection of e_0α ≡ a_0α/a_0α onto the plane perpendicular to n_s ≡ k_s/k_s, which arises because only this component of a_0α can excite the electromagnetic component (perpendicular to n_s) of the scattered wave. γ_pmα is the ponderomotive response function33 for plasma wave α, defined as$${\gamma }_{\mathrm{p}\mathrm{m}}({\omega }_{\mathrm{e}\mathrm{s}},k_{\mathrm{e}\mathrm{s}})=\frac{(1+{\chi }_{\mathrm{I}}){\chi }_{\mathrm{e}}}{1+{\chi }_{\mathrm{I}}+{\chi }_{\mathrm{e}}},$$(7)where χ_I(ω_es, k_es) = Σ_βχ_iβ(ω_es, k_es) and χ_e(ω_es, k_es) are the ion susceptibility (summed over ion species β) and electron susceptibility,34 respectively. In this paper, for simplicity, the flow velocity is assumed to be zero for all species. Nevertheless, if species β were to flow with velocity u_β, then this nonzero flow velocity could easily be considered by replacing ω_es in χ_iβ(ω_es, k_es) with ω_es − k_es · u_β.35

As illustrated in Fig. 1(b), the projections of the polarization directions e₀₁ and e₀₂ onto the plane perpendicular to k_s can be different in direction, and then, according to Eq. (6), the direction of a_s depends on the competition between the drives by beams I and II. Defining e_∥ as the unit vector parallel to ${\mathbf{e}}_{01\perp {\mathbf{n}}_{\mathrm{s}}}$, and e_⊥ ≡ n_s × e_∥ as the unit vector perpendicular to e_∥ and n_s, as shown in Fig. 1(b), and writing ${\mathit{a}}_{\mathrm{s}}=a_{{\mathrm{s}}_{\parallel }}{\mathbf{e}}_{\parallel }+a_{{\mathrm{s}}_{\perp }}{\mathbf{e}}_{\perp }$, Eqs. (5) and (6) can be written as$$\frac{\delta n_{{\mathrm{e}\mathrm{s}}_1}}{n_0}=-{\gamma }_{{\mathrm{p}\mathrm{m}}_1}\frac{k_{{\mathrm{e}\mathrm{s}}_1}^2{c}^2}{2{\omega }_{\mathrm{p}\mathrm{e}}^2}{a}_{01}{a}_{{\mathrm{s}}_{\parallel }}^{\ast }\cos {\phi }_1,$$(8)$$\frac{\delta n_{{\mathrm{e}\mathrm{s}}_2}}{n_0}=-{\gamma }_{{\mathrm{p}\mathrm{m}}_2}\frac{k_{{\mathrm{e}\mathrm{s}}_2}^2{c}^2}{2{\omega }_{\mathrm{p}\mathrm{e}}^2}{a}_{02}\cos {\phi }_2(a_{{\mathrm{s}}_{\parallel }}^{\ast }\cos {\delta }_{\perp }+a_{{\mathrm{s}}_{\perp }}^{\ast }\sin {\delta }_{\perp })$$(9)and$${\mathbf{k}}_{\mathrm{s}}\cdot \nabla a_{{\mathrm{s}}_{\parallel }}=-\frac{j{\omega }_{\mathrm{p}\mathrm{e}}^2}{4c^2{n}_0}(\delta n_{{\mathrm{e}\mathrm{s}}_1}^{\ast }{a}_{01}\cos {\phi }_1+\delta n_{{\mathrm{e}\mathrm{s}}_2}^{\ast }{a}_{02}\cos {\phi }_2\cos {\delta }_{\perp }),$$(10)$${\mathbf{k}}_{\mathrm{s}}\cdot \nabla a_{{\mathrm{s}}_{\perp }}=-\frac{j{\omega }_{\mathrm{p}\mathrm{e}}^2}{4c^2}\frac{\delta n_{{\mathrm{e}\mathrm{s}}_2}^{\ast }}{n_0}{a}_{02}\cos {\phi }_2\sin {\delta }_{\perp },$$(11)where φ_α is the angle between ${\mathbf{e}}_{0\alpha \perp {\mathbf{n}}_{\mathrm{s}}}$ and e_0α, which satisfies cos φ_α = |e_0α × n_s|, and δ_⊥ is the angle between ${\mathbf{e}}_{01\perp {\mathbf{n}}_{\mathrm{s}}}$ and ${\mathbf{e}}_{02\perp {\mathbf{n}}_{\mathrm{s}}}$, which satisfies$$\sin {\delta }_{\perp }=\frac{{\mathbf{n}}_{\mathrm{s}}\cdot ({\mathbf{e}}_{01}\times {\mathbf{e}}_{02})}{\cos {\phi }_1\cos {\phi }_2}.$$(12)Inserting Eqs. (8) and (9) into Eqs. (10) and (11), equations for $a_{{\mathrm{s}}_{\parallel }}$ and $a_{{\mathrm{s}}_{\perp }}$ can be obtained as$${\partial }_{\eta }{a}_{{\mathrm{s}}_{\parallel }}={\kappa }_1{a}_{{\mathrm{s}}_{\parallel }}+{\kappa }_2\cos {\delta }_{\perp }(a_{{\mathrm{s}}_{\parallel }}\cos {\delta }_{\perp }+a_{{\mathrm{s}}_{\perp }}\sin {\delta }_{\perp }),$$(13)$${\partial }_{\eta }{a}_{{\mathrm{s}}_{\perp }}={\kappa }_2\sin {\delta }_{\perp }(a_{{\mathrm{s}}_{\parallel }}\cos {\delta }_{\perp }+a_{{\mathrm{s}}_{\perp }}\sin {\delta }_{\perp }),$$(14)where the coordinate η is along the direction of k_s, and the single-beam gain coefficients are ${\kappa }_1\equiv \mathrm{I}\mathrm{m}[{\gamma }_{{\mathrm{p}\mathrm{m}}_1}]k_{{\mathrm{e}\mathrm{s}}_1}^2{|a_{01}|}^2{\cos }^2{\phi }_1/8k_{\mathrm{s}}$ and ${\kappa }_2\equiv \mathrm{I}\mathrm{m}[{\gamma }_{{\mathrm{p}\mathrm{m}}_2}]k_{{\mathrm{e}\mathrm{s}}_2}^2{|a_{02}|}^2{\cos }^2{\phi }_2/8k_{\mathrm{s}}$. The gain coefficient κ_c of the common scattered wave can be obtained from Eqs. (13) and (14) by taking the solution form $a_{{\mathrm{s}}_{\parallel }},a_{{\mathrm{s}}_{\perp }}\propto e^{{\kappa }_{\mathrm{c}}\eta }$, yielding$${\kappa }_{\mathrm{c}}^2-{\kappa }_{\mathrm{c}}({\kappa }_1+{\kappa }_2)+{\kappa }_1{\kappa }_2{\sin }^2{\delta }_{\perp }=0.$$(15)Usually, there are two solutions for κ_c, with the larger one satisfying max[κ₁, κ₂] ≤ κ_c ≤ κ₁ + κ₂ and the smaller one satisfying 0 ≤ κ_c ≤ min[κ₁, κ₂]. The polarization directions of a_s corresponding to these two modes are orthogonal to each other and are determined by$$\frac{a_{{\mathrm{s}}_{\perp }}}{a_{{\mathrm{s}}_{\parallel }}}=\frac{{\kappa }_2\cos {\delta }_{\perp }\sin {\delta }_{\perp }}{{\kappa }_{\mathrm{c}}-{\kappa }_2{\sin }^2{\delta }_{\perp }}.$$(16)The mode with larger κ_c will dominate the convective amplification of the scattered wave, except when the polarization direction of the seed for a_s is exactly along the polarization direction of the mode with smaller κ_c. Therefore, it is the SL mode with larger κ_c that is mainly discussed in this work.

In this section, we investigate the impacts of crossing angle, polarization states, and the finite overlapping volume of the two laser beams on collective SBS modes with shared scattered wave for both zero and nonzero wavelength differences between the two pump beams.

Assuming zero flow velocity, the ion acoustic waves satisfy the dispersion relation ${\omega }_{a_{\alpha }}=k_{a_{\alpha }}{c}_{\mathrm{s}}$, where c_s is the ion acoustic velocity. (Note that in the context of SBS, the subscript “a” is used to denote quantities related to ion acoustic waves, which corresponds to the symbol “es” for plasma waves in Sec. II.) Then, from the matching condition$${\omega }_{\mathrm{s}}={\omega }_{01}-{\omega }_{a_1}={\omega }_{02}-{\omega }_{a_2},$$(17)we obtain the requirement$$\Delta {\omega }_0\equiv {\omega }_{01}-{\omega }_{02}={\omega }_{a_1}-{\omega }_{a_2}=c_{\mathrm{s}}(k_{a_1}-k_{a_2}).$$(18)Thus, for two laser beams of the same wavelength, i.e., Δω₀ = 0, it is required that $k_{a_1}=k_{a_2}$, leading to θ_s = 0° or θ_s = 180°, while for Δω₀ ≠ 0, the possible directions of k_s are determined by Eq. (18) in combination with Eqs. (3) and (4), which is much more complicated. In the following, we discuss these two cases separately.

For the SL modes of collective SBS, since $k_{a_1}=k_{a_2}$ when the two pump beams have the same wavelength, k_s is located on the bisecting plane between k₀₁ and k₀₂ (the xz plane in Fig. 1), and the ponderomotive response ${\gamma }_{{\mathrm{p}\mathrm{m}}_1}={\gamma }_{{\mathrm{p}\mathrm{m}}_2}$ owing to the symmetric matching condition. Thus, the single-beam gain coefficient ${\kappa }_{\alpha }=(\mathrm{I}\mathrm{m}[{\gamma }_{\mathrm{p}\mathrm{m}}]k_a^2/8k_{\mathrm{s}}){|a_{0\alpha }|}^2{\cos }^2{\phi }_{\alpha }$. Considering two beams with the same intensity, the gain coefficient for the SL mode has the upper limit ${\kappa }_{\mathrm{c}}\le {\kappa }_1+{\kappa }_2\le {\kappa }_{\mathrm{c}}^{\mathrm{U}}\equiv (\mathrm{I}\mathrm{m}[{\gamma }_{\mathrm{p}\mathrm{m}}]k_a^2/4k_{\mathrm{s}})|a_0{|}^2$. According to Eq. (15), ${\kappa }_{\mathrm{c}}/{\kappa }_{\mathrm{c}}^{\mathrm{U}}$ is the larger root of the following equation:$$\begin{array}{l}\hfill {\left(\frac{2{\kappa }_{\mathrm{c}}}{{\kappa }_{\mathrm{c}}^{\mathrm{U}}}\right)}^2-2\left({\cos }^2{\phi }_1+{\cos }^2{\phi }_2\right)\frac{{\kappa }_{\mathrm{c}}}{{\kappa }_{\mathrm{c}}^{\mathrm{U}}}+{(\cos {\phi }_1\cos {\phi }_2\sin {\delta }_{\perp })}^2=0,\end{array}$$(19)where the factors cos φ₁, cos φ₂ and sin δ_⊥ depend solely on the geometry (θ_h, φ_s, θ_s = 0° or 180°) of the SL mode and the polarization states of the laser beams, which are denoted by the polarization angle β_α (−90° < β_α ≤ 90°), as shown in Fig. 1. Therefore, ${\kappa }_{\mathrm{c}}/{\kappa }_{\mathrm{c}}^{\mathrm{U}}$ is determined completely by the beam crossing angle θ_h, the out-of-plane angle φ_s, and the polarization angles β₁ and β₂, while the dependence of ${\kappa }_{\mathrm{c}}^{\mathrm{U}}$ on the scattered wavelength is reflected in the gain spectrum of ${\kappa }_{\mathrm{c}}^{\mathrm{U}}$.

For a typical plasma condition at the laser entrance hole of a He plasma,36${\kappa }_{\mathrm{c}}^{\mathrm{U}}$ is shown in Fig. 2, in which the gain coefficient is normalized by I₁₅ = I₀₁[W/cm²]/10¹⁵. For θ_s = 0° and −90° < φ_s < 90°, where k_s is in the quadrants x > 0, the scattered wavelength increases with increasing θ_h, while the peak value of ${\kappa }_{\mathrm{c}}^{\mathrm{U}}$ decreases with increasing θ_h. This is because the term $k_a^2\mathrm{I}\mathrm{m}[{\gamma }_{\mathrm{p}\mathrm{m}}]$ in ${\kappa }_{\mathrm{c}}^{\mathrm{U}}$ peaks at ω_a ≈ k_ac_s (λ_B − λ₀ ∝ ω_a ∝ k_a), with its peak value decreasing with increasing k_a,30 and k_a = 2k₀ sin[arccos( cos φ_s cos θ_h)/2] for θ_s = 0° [obtained from Eqs. (3) and (4)] increases with increasing θ_h. For θ_s = 180° and −90° < φ_s < 90°, where k_s is in the quadrants x < 0, the scattered wavelength decreases with increasing θ_h, while the peak value of ${\kappa }_{\mathrm{c}}^{\mathrm{U}}$ increases with increasing θ_h. This is because k_a = 2k₀ cos[arccos(cos φ_s cos θ_h)/2] for θ_s = 180° decreases with increasing θ_h. Besides, k_a increases as the angle between k_s and $\widehat{x}$ increases from zero to 180°, which corresponds to φ_s varying from 0 to 90° for θ_s = 0° and then from 90° to 0 for θ_s = 180°, as shown in Fig. 1. Consequently, the scattered wavelength increases while the peak gain value decreases as the angle between k_s and $\widehat{x}$ increases from 0° through 90° to 180°, as shown in Fig. 2.

The influence of polarization on κ_c of the SL modes can be evaluated through ${\kappa }_{\mathrm{c}}/{\kappa }_{\mathrm{c}}^{\mathrm{U}}$ determined by Eq. (19). Owing to the symmetric relation ${\kappa }_{\mathrm{c}}/{\kappa }_{\mathrm{c}}^{\mathrm{U}}{|}_{{\theta }_{\mathrm{s}}=180°,{\phi }_{\mathrm{s}}}={\kappa }_{\mathrm{c}}/{\kappa }_{\mathrm{c}}^{\mathrm{U}}{|}_{{\theta }_{\mathrm{s}}=0,-{\phi }_{\mathrm{s}}}$, in the following discussion of the modification of κ_c by polarization states, we assume θ_s = 0. First, it is observed that the range of ${\kappa }_{\mathrm{c}}/{\kappa }_{\mathrm{c}}^{\mathrm{U}}$ attainable by adjusting the polarization angles β₁ and β₂ depends on the out-of-plane angle φ_s. In particular, ${\kappa }_{\mathrm{c}}/{\kappa }_{\mathrm{c}}^{\mathrm{U}}=1$ can be attained only for in-plane scattering (φ_s = 0) when both beams are s-polarized (β₁ = β₂ = 0), while ${\kappa }_{\mathrm{c}}/{\kappa }_{\mathrm{c}}^{\mathrm{U}}<1$ for all combinations of β₁ and β₂ when φ_s ≠ 0. The upper and lower bounds of the range of ${\kappa }_{\mathrm{c}}/{\kappa }_{\mathrm{c}}^{\mathrm{U}}$ at each φ_s can be obtained analytically, as given by Eqs. (A1)–(A4) in Appendix A. Figure 3 shows the variation of the range of ${\kappa }_{\mathrm{c}}/{\kappa }_{\mathrm{c}}^{\mathrm{U}}$ with φ_s for different crossing angles. As can be seen, the range of variation of ${\kappa }_{\mathrm{c}}/{\kappa }_{\mathrm{c}}^{\mathrm{U}}$ is quite broad, especially for large obtuse crossing angles, indicating the significant role played by the polarization in the SL modes. As a result, the SL mode at some out-of-plane angle φ_s can be enhanced or reduced effectively by adjusting β₁ and β₂. For a specified φ_s, the upper bound of ${\kappa }_{\mathrm{c}}/{\kappa }_{\mathrm{c}}^{\mathrm{U}}$ depends on the best achievable alignment between the polarization directions of the two laser beams and the common scattered wave. Complete alignment among these three waves is only possible for in-plane scattering (φ_s = 0), where the polarization direction perpendicular to k₀₁ and k₀₂ is also orthogonal to k_s. For out-of-plane scattering, the best achievable polarization alignment decreases with increasing out-of-plane angle, making ${\kappa }_{\mathrm{c}}/{\kappa }_{\mathrm{c}}^{\mathrm{U}}$ drop with φ_s. Notice that for an acute crossing angle where θ_h < 45°, at small φ_s, for the best achievable polarization alignment, the SL modes with a_s along the direction of $\widehat{y}\times {\mathbf{n}}_{\mathrm{s}}$ have larger gain coefficient, while at large φ_s, the SL modes with a_s along the direction of $\widehat{y}$ have larger gain coefficient, and the best alignment occurs when both laser beams are p-polarized, and their alignment with a_s along $\widehat{y}$ is independent of φ_s, leading to a constant ${\kappa }_{\mathrm{c}}/{\kappa }_{\mathrm{c}}^{\mathrm{U}}$. This results in a curvature inflection of the upper bound of ${\kappa }_{\mathrm{c}}/{\kappa }_{\mathrm{c}}^{\mathrm{U}}$, as shown in Fig. 3.

To see the relative importance of SL modes with different out-of-plane angles for specified polarization states, we consider the relation between ${\kappa }_{\mathrm{c}}/{\kappa }_{\mathrm{c}}^{\mathrm{U}}$ and φ_s when β₁ and β₂ are specified. Several special cases can be identified:(a)When both beams I and II are s-polarized (β₁ = β₂ = 0), it is found that ${\kappa }_{\mathrm{c}}/{\kappa }_{\mathrm{c}}^{\mathrm{U}}={\cos }^2{\phi }_{\mathrm{s}}$. Consequently, for this polarization state, the in-plane SL mode is favored.(b)When both beams are p-polarized (β₁ = β₂ = 90°), it is found for |sin φ_s| ≤ 1/tan θ_h that ${\kappa }_{\mathrm{c}}/{\kappa }_{\mathrm{c}}^{\mathrm{U}}\equiv {\cos }^2{\theta }_{\mathrm{h}}$ over −90° < φ_s ≤ 90°, where a_s is along the $\widehat{y}$ direction, and hence its alignment with the p-polarized laser beams is independent of the out-of-plane angle; otherwise, the orthogonal mode with a_s along the $\widehat{y}\times {\mathbf{n}}_{\mathrm{s}}$ direction has a larger gain coefficient, rendering ${\kappa }_{\mathrm{c}}/{\kappa }_{\mathrm{c}}^{\mathrm{U}}={\sin }^2{\theta }_{\mathrm{h}}{\sin }^2{\phi }_{\mathrm{s}}$, more favorable for out-of-plane modes.(c)When the polarization directions of the two pump beams are orthogonal (e₀₁ · e₀₂ = 0), it is found that a_s lies in the (e₀₁, e₀₂) plane, and the orthogonal drives of the two beams complement each other, making ${\kappa }_{\mathrm{c}}/{\kappa }_{\mathrm{c}}^{\mathrm{U}}\equiv 1/2$ over −90° < φ_s ≤ 90°. Thus, κ_c is just the same as the gain coefficient of the single-beam side-scatter at the same scattering angle, for which the mode with k_s perpendicular to e_0α, and hence complete polarization alignment, is always allowed. The gain enhancement by sharing of the scattered wave vanishes for this polarization state, indicating that the SL mode can be effectively suppressed by tuning the polarization directions of the two pump beams to be orthogonal.

For other combinations of β₁ and β₂, the typical variation of ${\kappa }_{\mathrm{c}}/{\kappa }_{\mathrm{c}}^{\mathrm{U}}$ with φ_s is shown in Fig. 4. Generally, there exists one most favored mode corresponding to the maximum value of ${\kappa }_{\mathrm{c}}/{\kappa }_{\mathrm{c}}^{\mathrm{U}}$ at some out-of-plane angle ${\phi }_{\mathrm{s}}^{\mathrm{M}}$. Further analysis shows that this maximum value is completely determined by the polarization alignment between the two pump beams,$${\max }_{{\phi }_{\mathrm{s}}}\left[\frac{{\kappa }_{\mathrm{c}}}{{\kappa }_{\mathrm{c}}^{\mathrm{U}}}\right]=\frac{1}{2}(1+|\cos {\delta }_{\mathrm{p}\mathrm{o}\mathrm{l}}|),$$(20)where$$\begin{array}{ll}\hfill \cos {\delta }_{\mathrm{p}\mathrm{o}\mathrm{l}}& \equiv {\mathbf{e}}_{01}\cdot {\mathbf{e}}_{02}\hfill \\ \hfill & ={\cos }^2{\theta }_{\mathrm{h}}\cos ({\beta }_1-{\beta }_2)+{\sin }^2{\theta }_{\mathrm{h}}\cos ({\beta }_1+{\beta }_2).\hfill \end{array}$$(21)The detailed derivation is given in Appendix B. This maximum value is attained when the polarization direction of a_s is along the bisector of the acute angles between e₀₁ and e₀₂ (i.e., along e₀₁ + e₀₂ for  cos δ_pol > 0 and e₀₁ − e₀₂ for cos δ_pol < 0). This condition, together with the requirement a_s ⊥ k_s, then gives$$\tan {\phi }_{\mathrm{s}}^{\mathrm{M}}=\left\{\begin{array}{cc}-\tan [({\beta }_1-{\beta }_2)/2]\sin {\theta }_{\mathrm{h}},\hfill & \cos {\delta }_{\mathrm{p}\mathrm{o}\mathrm{l}}>0,\hfill \\ \frac{\sin {\theta }_{\mathrm{h}}}{\tan [({\beta }_1-{\beta }_2)/2]},\hfill & \cos {\delta }_{\mathrm{p}\mathrm{o}\mathrm{l}}<0.\hfill \end{array}\right.$$(22)Therefore, the out-of-plane angle $|{\phi }_{\mathrm{s}}^{\mathrm{M}}|$ of the most favored SL mode is largely determined by β₁ − β₂, which characterizes the overall deviation from s-polarization of beams I and II. Typically, there is a jump in ${\phi }_{\mathrm{s}}^{\mathrm{M}}$ at cos δ_pol = 0, where the sign of ${\phi }_{\mathrm{s}}^{\mathrm{M}}$ becomes opposite. Before this jump, $|{\phi }_{\mathrm{s}}^{\mathrm{M}}|$ increases with increasing |β₁ − β₂|, and after it, $|{\phi }_{\mathrm{s}}^{\mathrm{M}}|$ decreases with increasing |β₁ − β₂|. For an acute crossing angle with θ_h < 45°, the upper limit of $|{\phi }_{\mathrm{s}}^{\mathrm{M}}|$ is $\arctan (\sin {\theta }_{\mathrm{h}}/\sqrt{\cos 2{\theta }_{\mathrm{h}}})$, corresponding to 15.5° and 35.3° for θ_h = 15° and 30°, respectively, while for an obtuse crossing angle with θ_h > 45°, $|{\phi }_{\mathrm{s}}^{\mathrm{M}}|$ can reach 90° when β₁ = β₂ ≥ 90° − arccos(1/ tan²θ_h)/2. Thus, for small acute crossing angles, the out-of-plane modes with relatively small out-of-plane angles can be favored for some polarization states, while for large obtuse crossing angles, the large-angle out-of-plane SL modes can also be favored.

In practice, the overlapping volume of the laser beam is finite, limiting the amplification length l_amp (along the n_s direction) of the SL modes. If it is assumed that the laser width is w_b, then, to enclose the amplification length inside the overlapping volume, it is required that $l_{\mathrm{a}\mathrm{m}\mathrm{p}}|{\mathbf{n}}_{s\perp {\mathbf{k}}_{01}}|\le w_{\mathrm{b}}$ and $l_{\mathrm{a}\mathrm{m}\mathrm{p}}|{\mathbf{n}}_{s\perp {\mathbf{k}}_{02}}|\le w_{\mathrm{b}}$, where ${\mathbf{n}}_{s\perp {\mathbf{k}}_{0\alpha }}$ is the projection of n_s onto the plane perpendicular to the laser propagation direction k_0α. This gives the greatest amplification length $l_{\mathrm{a}\mathrm{m}\mathrm{p}}=w_{\mathrm{b}}/\sqrt{1-{\cos }^2{\phi }_{\mathrm{s}}{\cos }^2{\theta }_{\mathrm{h}}}$ for two beams at the same wavelength. It can be seen that l_amp decreases with increasing out-of-plane angle; however, the rate of decrease drops with increasing crossing angle, corresponding to a decrease of about 74%, 50%, 29%, 13%, and 3% when φ_s increases from 0° to 90°, for θ_h at 15°, 30°, 45°, 60° and 75°, respectively. Considering this effect, the achievable gain κ_cl_amp/w_b of the SL modes with different out-of-plane angles is shown in Fig. 5 for three polarization combinations β₁ = β₂ = 0, β₁ = −β₂ = 45°, and β₁ = β₂ = 90°, where the value of κ_c at the peak wavelength is used. For small crossing angle θ_h = 15°, the gains of the in-plane modes are always larger than those of the out-of-plane modes, irrespective of the beam polarization, owing to the rapidly falling amplification length with increasing φ_s. For larger crossing angles, however, the relative importance of the out-of-plane modes with respect to the in-plane modes depends on the polarization states of the laser beams. Especially for large obtuse crossing angles, the gains of the out-of-plane SL modes can significantly exceed those of the in-plane modes for certain polarization states, even when the effects of finite beam overlapping volume have been taken into account.

For two beams with nonzero wavelength difference, on substituting the expression (3) for $k_{{\mathrm{e}\mathrm{s}}_1}=k_{a_1}$ and $k_{{\mathrm{e}\mathrm{s}}_2}=k_{a_2}$ into the matching requirement (18) for the SL mode, it is found that$$\frac{\Delta {\omega }_0}{4k_{01}{c}_{\mathrm{s}}}=\cos \frac{{\vartheta }_1+{\vartheta }_2}{4}\sin \frac{{\vartheta }_1-{\vartheta }_2}{4},$$(23)where the scattering angles ϑ₁ for beam I and ϑ₂ for beam II are functions of θ_h, θ_s, and φ_s. Without loss of generality, we designate the beam with shorter wavelength as beam I, and then Δω₀ = ω₀₁ − ω₀₂ ≥ 0 and $\Delta {\lambda }_0={\lambda }_{02}-{\lambda }_{01}\approx {\lambda }_{01}^2\Delta {\omega }_0/2\pi c\ge 0$, where λ₀₁ and λ₀₂ are the vacuum wavelengths of beams I and II, respectively. Since the maximum value of the right-hand side of Eq. (23) as a function of θ_s and φ_s is $\frac{1}{2}\sin {\theta }_{\mathrm{h}}$ located at θ_s = θ_h and φ_s = 0, it is required that Δω₀ ≤ 2k₀₁c_s sin θ_h and hence $\Delta {\lambda }_0\le 2{\lambda }_{\mathrm{01}}{c}_{\mathrm{s}}\sin {\theta }_{\mathrm{h}}\sqrt{1-n_{\mathrm{e}}/n_{\mathrm{c}}}/c$ for the SL modes to exist. For given Δλ₀, the possible directions of k_s for the SL modes as obtained from Eq. (23) constitute a loop on the (θ_s, φ_s) sphere, as shown in Fig. 6. When Δλ₀ = 0, the k_s loop constitutes a great circle in the xz plane perpendicular to k₀₁ − k₀₂. With increasing Δλ₀, the k_s loop contracts to a smaller and smaller loop encircling the wavevector direction of the laser beam with longer wavelength (here the direction of k₀₂ at θ_s = θ_h and φ_s = 0), until at the greatest allowed wavelength difference $2{\lambda }_{01}{c}_{\mathrm{s}}\sin {\theta }_{\mathrm{h}}\sqrt{1-n_{\mathrm{e}}/n_{\mathrm{c}}}/c$, the k_s loop retracts to one point corresponding to the direction of k₀₂. The contraction of the k_s loop is more severe for a smaller beam crossing angle, for which the greatest allowed wavelength difference is smaller. By tuning the wavelength difference between the two pump beams greater than $2{\lambda }_{01}{c}_{\mathrm{s}}\sqrt{1-n_{\mathrm{e}}/n_{\mathrm{c}}}/c$, SL modes are diminished for any beam crossing angle. This provides an efficient way to suppress the SL modes of SBS.

The k_s loop can be parameterized by −180° < α_⊥ ≤ 180°, the angle from (k₀₁, k₀₂) plane to (k_s, k₀₂) plane, where the in-plane SL modes correspond to α_⊥ = 0 or 180°.44 One upper bound of κ_c for all possible polarization states is ${\kappa }_{\mathrm{c}}^{\mathrm{U}}={\sum }_{\alpha =1,2}\mathrm{I}\mathrm{m}[{\gamma }_{p{m}_{\mathrm{\alpha }}}]k_{a_{\alpha }}^2|a_{0\alpha }{|}^2/8k_{\mathrm{s}}$. We can again can use ${\kappa }_{\mathrm{c}}^{\mathrm{U}}$ to characterize the dependence of κ_c on the scattered wavelength, as shown in Fig. 7 for SL modes of two laser beams with different wavelength differences. Since the system is symmetric with respect to reflection at the (k₀₁, k₀₂) plane, 0 ≤ α_⊥ ≤ 180° is shown. In Fig. 7(b), the contributions of beams I and II to ${\kappa }_{\mathrm{c}}^{\mathrm{U}}$ are also displayed for α_⊥ = 180°. Since $k_{a_2}<k_{a_1}$, the contribution of beam II has a greater peak value yet a narrower width compared with beam I. Hence, beam II with the longer wavelength contributes more to the peak of ${\kappa }_{\mathrm{c}}^{\mathrm{U}}$, whereas beam I with the shorter wavelength contributes more to the wing of ${\kappa }_{\mathrm{c}}^{\mathrm{U}}$. For each Δλ₀, because the scattering angles ϑ₁ and ϑ₂ and hence $k_{a_1}$ and $k_{a_2}$ increase with increasing α_⊥, the peak wavelength increases with increasing α_⊥, while the peak value of ${\kappa }_{\mathrm{c}}^{\mathrm{U}}$ decreases. Furthermore, with increasing Δλ₀, the shortest peak wavelength of the SL mode with α_⊥ = 0 increases, while the longest peak wavelength of the SL mode with α_⊥ = 180° decreases, because of the contraction of the k_s loop. This leads to a narrower wavelength range for the possible SL modes when the laser wavelength difference is enlarged. Also, it can be shown that the shortest peak wavelength of the SL mode with α_⊥ = 0 increases with increasing θ_h, while the longest peak wavelength of the SL mode with α_⊥ = 180° increases with increasing θ_h when ${\theta }_{\mathrm{h}}<2\arcsin (\sqrt{|\Delta {\omega }_0|/4k_0{c}_{\mathrm{s}}})$, and decreases with increasing θ_h for a larger beam crossing angle.

Taking into account the effects of the polarization states and the finite beam overlapping volume of the two laser beams, the achievable κ_cl_amp/w_b can be calculated for an arbitrary allowed Δλ₀, similar to the case for Δλ₀ = 0. Combining the results for different Δλ₀ and taking the value of κ_c at the peak wavelength, a map of κ_cl_amp/w_b vs the direction of k_s can be obtained, as shown in Fig. 8, where a view along the $-\widehat{y}$ direction (cf. Fig. 1) is taken. It is clear that the polarization states can significantly modify the gain of the SL modes. Especially for large beam crossing angles and relatively small Δλ₀, for which the k_s loop is relatively large, the out-of-plane SL modes with φ_s deviating from zero can be quite important, similar to the case with zero laser wavelength difference discussed above.

In summary, based on a linear kinetic model, an analytic convective solution has been derived for the SL modes of two overlapped laser beams. The effects of crossing angle, polarization states, and the finite overlapping volume of the two beams on the collective SBS modes with shared scattered waves have been discussed in detail for both zero and nonzero wavelength differences between the two laser beams. When the two beams are of the same wavelength, the wavevectors of the shared scattered waves lie on a circle in the bisecting plane between the wavevectors of the two laser beams. The wavelength of the scattered waves varies with the beam crossing angle and the out-of-plane angle of the SL modes. The gain coefficients of the SL modes, on the other hand, are also subject to the polarization states of the laser beams. When the two laser beams are both s-polarized, the gain coefficient of the SL mode is twice the gain coefficient of a single beam, while when the polarization directions of the two beams are orthogonal to each other, the gain coefficient of the collective SBS modes becomes the same as the single-beam side-scatter with the same scattering angle. Furthermore, for some polarization states and especially for obtuse crossing angles, the out-of-plane SL modes can become more important than the in-plane modes. With increasing wavelength difference between the two laser beams, the possible directions of the wavevectors of the common scattered wave contract toward the wavevector direction of the pump beam with longer wavelength. This changes the scattered wavelengths and the gain coefficients of the SL modes. Nevertheless, depending on the polarization state and the beam crossing angle, the out-of-plane modes can still be quite important. Finally, for sufficiently large vacuum wavelength difference $\Delta {\lambda }_0>2{\lambda }_{01}{c}_{\mathrm{s}}\sqrt{1-n_{\mathrm{e}}/n_{\mathrm{c}}}/c$, the SL modes of SBS no longer exist, which provides an efficient way to suppress the SL modes of SBS.

In this work, uniform plasma conditions with zero flow velocity have been assumed for an illustrative analysis. A nonzero flow velocity effectively leads to an additional wavelength difference between the two laser beams, which can also be accounted for by our model. Furthermore, in ICF, various laser smoothing techniques, such as kinoform/random phase plate (KPP/RPP),37 smoothing by spectral dispersion (SSD),38 polarization smoothing (PS),39 and some new methods,40–42 are often used to suppress LPI. Consequently, the laser beam intensity distribution can be highly nonuniform with many high-intensity speckles, and the induced temporal/spatial incoherence of the laser beam introduces additional mismatching into SBS, leading to a modified ponderomotive response γ_pm.43 To obtain a precise gain by integrating the local gain coefficient, these two factors, along with the realistic overlapping pattern of the laser beams,20,32 should be properly taken into account in further simulations. The collective SL modes can have much higher gain coefficient than single-beam SBS, and consequently they can be amplified to a great magnitude over a short distance. Especially for practical inhomogeneous plasmas, when the resonance length is limited by the inhomogeneity of the flow velocity or temperature,5,32 the collective SL mode could dominate over the single-beam SBS mode. Finally, from a comparison with the collective SP modes investigated in Ref. 30, it is found that depending on the crossing angle, polarization states, and the wavelength difference between the two laser beams, either the SP or the SL mode can be more important. Simulations under realistic plasma and laser conditions are required to assess the importance of the SL modes, for which this work provides valuable theoretical references.

Acknowledgment. This work was supported by the National Key R&D Program of China (Grant No. 2017YFA0403204), the Science Challenge Project (Grant No. TZ2016005), the National Natural Science Foundation of China (Grant Nos. 11 875 093 and 11 875 091), and the Project supported by CAEP Foundation (Grant No. CX20210040).