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Collective stimulated Brillouin scattering modes of two crossing laser beams with shared ion acoustic wave

The overlapping of multiple beams is common in inertial confinement fusion (ICF), making the collective stimulated Brillouin scattering (SBS) with shared ion acoustic wave (IAW) potentially important because of the effectively larger laser intensities to drive the instability. In this work, based on a linear kinetic model, an exact analytic solution for the convective amplification of SBS with the shared IAW modes stimulated by two overlapped beams is presented. From this solution, effects of the wavelength difference, crossing angle, polarization states, and finite beam overlapping volume of the two laser beams on the shared IAW modes are studied. It is found that a wavelength difference of several nanometers between the laser beams has negligible effects, except for a very small crossing angle about one degree. However, the crossing angle, beam polarization states, and finite beam overlapping volume can have significant influences on the shared IAW modes. Furthermore, the out-of-plane modes, in which the wavevectors of daughter waves lie in the different planes from the two overlapped beams, are found to be important for certain polarization states and crossing angles of the laser beams with the finite beam overlapping volume. This work is helpful to comprehend and estimate the collective SBS with shared IAW in ICF experiments.


I. INTRODUCTION
In inertial confinement fusion (ICF), the overlapping of multiple beams is common for both indirect-drive and direct-drive schemes [1,2]. This leads to complex multibeam laserplasma instabilities. One important example is crossed-beam energy transfer (CBET) between two beams [3][4][5][6][7], which can redistribute laser energy, alter drive symmetry, and modify hydrodynamic conditions. Apart from CBET, the collective stimulated Brillouin scattering (SBS) and the collective stimulated Raman scattering (SRS) of multiple beams with shared daughter waves can also be important, since the temporal growth rate and convective gain are expected to depend on the combined laser intensities. Experimentally, it is observed that the energy amplification factor increases with the number of pump beams and significant scattered light losses are produced in novel backward directions due to the collective coupling [8][9][10][11][12]. Thus the laser-target coupling is significantly impaired. Understanding, and in some cases mitigating these endemic processes, is essential for optimizing ICF implosions.
The shared daughter wave can be a common Langmuir/ion acoustic wave or scattered light wave, termed as the shared plasma wave (SP) mode and shared scattered light wave (SL) mode, respectively. Theoretically, the homogeneous temporal growth rate for collective SP and SL modes of multiple beams has been investigated in a general framework under fluid description [13][14][15]. However, for most practical cases in ICF, SRS and SBS instabilities are spatial problems [16][17][18][19], for which theoretical work for the collective modes is lacking.
Although two-dimensional (2D) particle-in-cell simulations have been performed recently to study SRS of two overlapped laser beams [15,20], the out-of-plane modes, where the wavevectors of daughter waves lie in the different planes from the plane of two overlapped beams, have not been considered, since two-dimensional simulations restrict all wavevectors in the same plane. In this work, theoretical study for the SP modes of collective SBS in convective regime is performed. By a linear kinetic model, an exact analytic solution for convective amplification of SP modes of two overlapped beams is obtained in the limit of strong damping of plasma waves and negligible pump depletion. Based on this solution, impacts of the wavelength difference, crossing angle, polarization states, and finite beam overlapping volume of the two laser beams on the collective SBS modes with shared ion acoustic wave (IAW) are studied systematically. Especially, the out-of-plane modes are discussed, which might be important in ICF experiments. This work is helpful to comprehend and estimate the collective SBS with shared IAW in ICF experiments. This paper is organized as follows: In Section II, the theoretical model for SP modes is presented, where an analytic solution for its convective amplification is given. In Section III, impacts of the wavelength difference, crossing angle, polarization states, and finite beam overlapping volume of two laser beams on the scattered wavelength and spatial amplification of the collective SBS modes with shared IAW are investigated, and the importance of outof-plane modes relative to in-plane modes are also discussed. In Section IV, the conclusions as well as some discussions are given.

II. THEORETICAL MODEL FOR SP MODES OF TWO BEAMS
A. Matching geometry of the SP modes The SRS or SBS stimulated by a single laser beam is a resonant three-wave parametric instability process where the incident wave decays into a plasma wave and a scattered wave.
It occurs when the plasma wave is resonant with the ponderomotive force created by beating of the laser wave and scattered light wave, and the scattered wave is resonant with the transverse current created by beating between the laser wave and the density perturbation of the plasma wave. Thus, the phase matching condition is required, where K i ≡ (ω i /c, k i ) with subscripts i = 0, s, es are the four-wavevectors (comprising the wave frequency ω i and wavenumber vector k) of the laser beam (i = 0), scattered wave (i = s) and the plasma wave (i = es) respectively.
When there are two overlapped beams at four-wavevectors K 01 and K 02 , the beating of these two waves with a common plasma wave at four-wavevector K es would generate scattered waves at K s 1 = K 01 − K es and K s 2 = K 02 − K es . (The anti-Strokes components are ignored here.) The coupling of these generated scattered waves with the laser beams (K s 1 with K 02 or K s 2 with K 01 ) generates two new plasma waves at K es ± ∆K 0 , where ∆K 0 ≡ K 01 − K 02 . Again, the coupling of these two plasma waves with the pump beams generates two new scattered wave at K s 1 + ∆K 0 and K s 2 − ∆K 0 , etc. Finally, a series of scattered waves at K s 1 + n∆K 0 and K s 2 − n∆K 0 , and plasma waves at K es ± n∆K 0 (n = 0, 1 · · · ) can be generated. However, only those plasma waves and scattered waves with four-wavevectors nearly resonant with the natural modes are important. Assuming that K es , K s 1 and K s 2 are resonant with the natural modes, then the waves with n ≥ 1 can be ignored when the two laser beams are incoherent in the sense that either ∆ω 0 or |∆k 0 | is large enough to exceed the resonant peak widths of the natural modes of plasma and scattered waves. In such cases, each laser beam develops its scattered wave independently, and there are five coupled waves remained, including two laser waves and their corresponding scattered waves, and one common plasma wave. The five-wave matching condition can be written as where the subscripts 0α , sα (α = 1, 2) and es represent the incident wave, scattered wave and plasma wave, respectively.
The matching condition (2), together with the dispersion relation of the electromagnetic waves (EMWs), can determine k es as a function of k 0α , ω 0α , ω es , and the out-of-plane angle α ⊥ defined as the angle between the (k 01 , k 02 )-plane and the (k s 1 , k s 2 )-plane. Intuitively, the five-wave coupling geometry of SP modes is shown in Fig. 1, where the two spheres formed by k s 1 and k s 2 have radii k sα = (ω 0α − ω es ) 2 − ω 2 pe /c (α = 1, 2) according to the matching condition (2) and the dispersion relation (3), and k es is determined by the intersection of these two spheres, which generally defines a circle in a plane perpendicular to ∆k 0 . Different SP modes (corresponding to different points on the k es -circle) can be parameterized by the out-of-plane angle −π < α ⊥ ≤ π, where the in-plane modes, for which k es , k s 1 and k s 2 are coplanar with k 01 and k 02 , have α ⊥ = 0 or α ⊥ = π.
Along this line, the wavevector of the shared plasma wave of three overlapped beams is determined by intersection between the three spheres formed by k sα (α = 1, 2, 3), which defines two points when existing. For more than three beams, a common plasma wave exists only when the intersection points between the three spheres formed by k sα (α = 1, 2, 3) happen to be located on spheres formed by other k sα (α = 4, · · · ). This imposes a stringent requirement on the beam symmetry as noticed by [14]. For an indirect drive, the required beam symmetry can be satisfied only in a quite limited volume near the laser entrance hole (LEH), when the incident laser beams are angularly distributed in a highly symmetric configuration [1]. Beyond this region, not only the beam symmetry condition is hard to be satisfied, but also the overlapping volume of multiple beams drops rapidly. Consequently, compared to multiple beam overlapping, effects of two beam overlapping would become more important. angle −π < α ⊥ ≤ π is defined as the angle between the (k 01 , k 02 )-plane and the (k s 1 , k s 2 )-plane.

B. Convective amplification of the SP modes
The equation for the EMW is where A is the potential vector and J is the transverse current. Using the cold-fluid approximation for the transverse electron motion m e v e = eA, the transverse current J = −en e v e = −e 2 n e A/m e , where m e , e, v e , n e are the electron mass, electron charge, electron quiver velocity and electron number density, respectively. Substituting J into Eq. (4) and taking the normalization a = eA/m e c yield where n e = n 0 + δn es is decomposed into the unperturbed electron density n 0 and the perturbed electron density δn es , and ω pe = e 2 n 0 /ǫ 0 m e is the plasma frequency.
In the envelope approximation for five-wave coupling of SP modes, and δn es (t, r) = 1 2 (δñ es e jΨes + cc.), whereã i is the complex vector amplitude of EMW, δñ es is the complex amplitude of density perturbation, and the phase Ψ i ≡ −jω i t + jk i · r with subscript i = 0α,s α ,es for the laser wave, scattered wave and plasma wave, respectively. The envelope approximation holds when |∇a i |/|a i | ≪ k i , |∂ t a i |/|a i | ≪ ω i , |∇n es |/n es ≪ k es and |∂ t n es |/n es ≪ ω es . Using the phase matching condition (2), equations for the electromagnetic components ofã i , which satisfy k i ·ã i = 0, can be derived from Eqs. (5-7) as For simplicity, the tilde in the complex amplitude is dropped in Eq. (8) and the following paper. n i = k i /k i is defined as an unit vector along k i , thus the term n i × (a j × n i ) = a j − (n i · a j )n i is the projection of a j onto the plane perpendicular to k i . The operator L em i is defined by where the ∇ · k term arises from the plasma inhomogeneity. Then, for a steady-state convective solution in the strong damping regime of homogeneous plasma, L em i = c 2 k i · ∇/ω i .
While the plasma response to the ponderomotive drive has the following expression [21], where the ponderomotive response function γ pm is where ǫ = 1 + χ I + χ e is the dielectric function, and χ I (ω es , k es ) = β χ iβ (ω es , k es ) and χ e are the ion susceptibility (summed over ion species β) and electron susceptibility, respectively. For simplicity, the flow velocity is assumed to be zero for all species in the following.
Nevertheless, the above formula can be applied to the non-zero flow case by replacing ω es appearing in χ iβ (ω es , k es ) with ω es − k es · u β if species β were to flow with velocity u β .
Considering the polarization states of the EMWs, the EMW complex vectors can be written as a sα = a sα e sα and a 0α = a 0α e 0α , where e i is a unit vector along the polarization direction. Using the fact n sα · e sα = n 0α · e 0α = 0, it can be proven According to Eq. (8), the component of a sα that can be convectively amplified is along the direction of n sα × (e 0α × n sα ), so e sα is parallel to n sα × (e 0α × n sα ), yielding the polarization alignment factor where ϕ α is the angle between e 0α and e sα . Then, using Eqs. (12)(13), for the steady-state convective solution, Eqs. (8-10) can be simplified as Substituting δn es into equations for a sα , we get where r a ≡ a * 02 cos ϕ 2 /a * 01 cos ϕ 1 , and the gain coefficient for single beam α is Eqs. (15)(16) are two-dimensional in nature since n s 1 and n s 2 are along different directions.
They can be solved for a beam overlapping volume over which the pump depletion of a 0α is negligible. It is convenient to choose a (in most cases) non-orthogonal coordinate system x = x 1 n s 1 + x 2 n s 2 , in which n s 1 · ∇ = ∂/∂x 1 and n s 2 · ∇ = ∂/∂x 2 . As derived in Appendix B, the solution can be written as  where is the SP mode response to seed of a s 1 at x = 0, while is the SP mode response to seed of a s 2 at x = 0, where I 0 and I 1 are the zero-order and first-order modified Bessel functions of the first kind, respectively.
G 1 and G 2 comprise two terms. The smaller second term of G 1 gives us a s 1 (x 1 , x 2 ) = e κ 1 x 1 a s 1 (x 1 = 0, x 2 ), and thus describes the one-dimensional (1D) amplification along x 1direction of the sidescatter due to beam I alone. This term can ignite the two-dimensional amplification of the SP modes. For example, from seed of a s 1 at x = 0, firstly the sidescatter of beam I generates perturbations δn es along the 1D straight line x 2 = 0; this again serves as seeds to a s 2 and amplified along the n s 2 direction, generating perturbations δn es over the 2D x-space; finally the seeding of 2D perturbations of δn es results in the amplification of both a s 1 and a s 2 over the 2D x-space. The steady-state two-dimensional amplification due to sharing of plasma waves is described by the first terms of G 1 and G 2 , where the 2D volume V amp (x) ≡ η 1 x 1 n 1 + η 2 x 2 n 2 (0 ≤ η 0,1 ≤ 1) participates in amplifying scattered waves from the seed at x = 0 to the point x = x 1 n 1 + x 2 n 2 . At sufficient gain length κ 1 κ 2 x 1 x 2 ≫ 1, for the first term of G 1 , we have Similar results can be obtained for G 2 . So all the response components G 11 , G 12 , G 21 and which is adopted in the following illustrative analysis instead of the accurate expression for G 1,2 .
From the asymptotic form G A of the SP mode response, we can define a representative gain coefficient to quantify the overall amplification ability of the SP modes. There is a direction n A parallel to ∇G A , along which the required amplification distance is minimum (in the asymptotic sense) for the seed at a given point (e.g., x = 0) to achieve a specific n s1 and (b) θ s 12 = 120 • , when κ 2 /κ 1 = 1/2 (in blue) and κ 2 = κ 1 (in red). The direction n A that is parallel to ∇G is indicated. The parallelogram marked by dashed lines is the 2D gain volume that takes parts in amplifying seeds from the origin to the corresponding point.
gain, as shown in Fig. 2. The representative gain coefficient the typical size of the corresponding 2D gain volume for points along n A , can be defined as in Appendix C. In our definition, κ A is approximately of the same magnitude as that the amplification by one beam dominates. The ratio R A [κ 2 /κ 1 , θ s 12 ] ≡ κ A /(κ 1 + κ 2 ) and the direction of n A , can be determined by κ 2 /κ 1 and the angle θ s 12 between n s 1 and n s 2 , as detailed in Appendix C.

III. ANALYSIS OF SHARED IAW MODES OF TWO BEAMS
In this section, impacts of the wavelength different, crossing angle and polarization states of the two laser beams on the collective SBS modes with shared IAW are investigated.
Because k s 1 ≈ k 01 and k s 2 ≈ k 02 for SBS, the wave coupling geometry can be simplified significantly. As shown in Figure 3, choosing point 'O' as the initial point, the terminal point of k a is approximately located on the circle C a (k a -circle), which lies in the plane perpendicular to k 02 − k 01 , having a center located at the point 'O 0 ' and a radius of the length of |OO 0 |. According to definitions of the angles shown in Fig. 3, the relation can be obtained. Defining the crossing angle between k 01 and k 02 as θ 12 = θ h 1 + θ h 2 , k a = 2k 01 k 02 sin θ 12 cos(α ⊥ /2) k 2 01 + k 2 02 − 2k 01 k 02 cos θ 12 (23) can be derived from Eq. (22).
The geometry of shared IAW modes of two overlapping beams. In the presented coordinate system, xy-plane is chosen to be the (k 01 , k 02 )-plane with x-axis perpendicular to k 01 − k 02 and y-axis along k 02 −k 01 , and z-axis is along k 01 ×k 02 . The circle C a (lying in the xz-plane), which can be parameterized by the out-of-plane angle α ⊥ , comprises all possible locations of the endpoints of k a . Beam I or beam II is said to be s-polarized when a 0α is along the ±s (z-axis) direction, and p-polarized when a 0α is along the direction of ±p α = ±s × k 0α . Other linear polarization states of beam I or II are described by the polarization angle 90 • ≥ β α ≥ −90 • , which is the angle from s to a 0α .
The representative gain coefficient κ A can be written as Here θ s 12 ≈ θ 12 is taken because k sα ≈ k 0α for SBS, and the polarization alignment factor cos ϕ α as can be calculated from Eq. (13) depends on the polarization states (β 1 , β 2 ) of the laser beams and geometry (α ⊥ , θ 12 ) of the SP mode. For two laser beams with the same intensity and small wavelength difference, |a 0 | = |a 01 | = |a 02 | and k s = k s 1 ≈ k s 2 , the upper bound of κ A for all possible polarization combinations of the two laser beams is κ U A can be achieved when the polarization direction e 0α k a × k 0α (α = 1, 2), which results in e 0α ⊥ k sα and hence cos ϕ α = 1, a complete alignment between e 0α and e sα .
For two overlapped laser beams with the same vacuum wavelength λ 01 = λ 02 , the k acircle is located on the bisecting plane between k 01 and k 02 with θ h 1 = θ h 2 = θ 12 /2. For a non-zero wavelength difference ∆λ 0 ≡ λ 02 − λ 01 ≪ λ 01 , λ 02 , the plane in which the k a -circle lies deviates from the bisecting plane between k 01 and k 02 , with an angle From the expression for ∆θ, it can be expected that the effects of the laser wavelength difference on the SP modes are significant only when ∆λ 0 /λ 0 is comparable to 2 tan(θ 12 /2). For ∆λ 0 /λ 0 ≪ 2 tan(θ 12 /2), ∆θ is quite small, therefore, θ h1 ≈ θ h2 ≈ θ 12 /2 and the dependence of k a on ∆λ 0 is rather weak. Consequently, the effects of the laser wavelength difference on the SP modes are negligible. As an example, κ U A versus λ Bα − λ 0α are shown in Fig. 4(a) and Fig. 4(b) for ∆λ 0 = 0 nm and ∆λ 0 = 6 nm, respectively, where a typical plasma condition at LEH [22] n e = 0.06 n c , T e = 2.5 KeV and T e /T i = 3.5 is chosen. For simplicity, the flow velocity which only leads a Doppler wavelength shift is assumed to be zero, and the gain coefficient is normalized by I 15 = I 01 [ Wcm −2 ]/10 15 . Here, n c is the critical density for Beam I. In Fig. 4(a), θ 12 = 1 • , so for ∆λ 0 = 6 nm, ∆λ 0 /λ 0 ≈ 2 tan(θ 12 /2). As expected, κ U A changes significantly when ∆λ 0 varies from zero to 6 nm. In Fig. 4(b), θ 12 = 10 • , so for ∆λ 0 = 6 nm, ∆λ 0 /λ 0 < 0.2 tan(θ 12 /2). As expected, the change of κ U A is negligible when ∆λ 0 varies from zero to 6 nm. The small crossing angle cases with 2 tan(θ 12 /2) ∆λ 0 /λ 0 should be of interest in the attempt to suppress laser-plasma parametric instabilities with broadband lasers [23,24]. In most cases of ICF, the crossing angle θ 12 is typically not too small, but the laser wavelength difference is much shorter than the vacuum wavelength [25], making its effects on the SP modes negligible. Therefore, in the following analysis, mainly the case of ∆λ 0 = 0 is discussed for the SP modes. When ∆λ 0 = 0, Eq. (23) can be further simplified as k a = 2k 01 cos(θ 12 /2) cos(α ⊥ /2), (27) and the scattering angle between k sα and k 0α is In the limiting case α ⊥ ∼ 180 • , θ scat ∼ 0 and k sα is nearly in the forward scattering direction of k 0α . Such near-forward SBS needs very long time to develop, and the final saturation stage usually contains significant contribution from the anti-Strokes waves [26,27]. Hence, we do not consider these cases here, and restrict θ 12 ≤ 150 • and α ⊥ ≤ 150 • , which guarantees θ scat > 7.6 • . Fig. 5 shows κ U A /I 15 versus λ B − λ 0 in He plasma for (a) SP modes with different laser beam crossing angles θ 12 and (b) SP modes with different out-of-plane angles α ⊥ , when two laser beams with the same intensity and vacuum wavelength are assumed. The scattered wavelength at which κ U A peaks decreases with increasing θ 12 or α ⊥ , while the corresponding peak value increases with θ 12 or α ⊥ . Since κ U A ∝ k 2 a Im[γ pm ] and k a decreases with increasing θ 12 or α ⊥ from Eq. (27), these characteristics are mainly due to properties of k 2 a Im[γ pm ], which peaks at ω a ∝ k a and its peak value increases with decreasing k a , as discussed in Appendix A.
κ A /κ U A depends only on the geometry (α ⊥ , θ 12 ) of the SP mode, and the polarization alignment between the laser beams and scattered waves. The most favored SP mode by beam α, for which the polarization of laser beam and the scattered light is in full alignment (cos ϕ α = 1), has e 0α ⊥ k a , as discussed above. Denoting an arbitrary polarization state of the laser beam by the polarization angle β α (−90 • < β α ≤ 90 • ) as shown in Fig. 3, the most favored out-of-plane angle by beam I is α ⊥ = −2 arctan[sin(θ 12 /2) tan β 1 ], while α ⊥ = 2 arctan[sin(θ 12 /2) tan β 2 ] is most favored by beam II. Except for s-polarization (β α = 0), the SP modes with non-zero out-of-plane angles are accentuated. When β 1 = −β 2 , the polarization states of beam I and beam II are symmetric with respect to the bisecting plane between k 01 and k 02 , therefore, at one and the same α ⊥ , both beam I and beam II are fully aligned with their respective scattered waves in polarization, making the value of κ A /κ U A reach one for this α ⊥ . For a nonzero β 1 +β 2 , the symmetry between beam I and beam II is broken after taking into account their polarization states, the most favored out-of-plane angle α ⊥ that results in a full polarization alignment between beam I and its scattered wave (cos ϕ 1 = 1) deviates from the most favored α ⊥ by beam II. Consequently, the maximum should be less than one. Fig. 6 shows the typical variation of κ A /κ U A with α ⊥ for the cases θ 12 = 60 • and θ 12 = 120 • , where effects of the degree of polarization symmetry breaking (characterized by β 1 + β 2 ) between beam I and II are illustrated in panels (a) and (c), and the overall effects of deviation from s-polarization of beam I and II (characterized by β 1 − β 2 ) are illustrated in panels (b) and (d). One remarkable feature is that different polarization states can modify the gain coefficient κ A significantly for |α ⊥ | 90 • . As a consequence, the SP mode with some outof-plane angle, for example, the angle for which the overlapping geometry permits a long gain length, can be effectively suppressed by choosing a proper combination of β 1 and β 2 .
However, for large |α ⊥ |, the modification by the polarization states becomes insignificant, because the scattering angles between k 0α and k sα are small, and thus the polarization alignment factors cos ϕ α are less sensitive to β α . Furthermore, it can be noticed that the polarization modification is more significant for an acute crossing angle θ 12 ≤ 90 • than an obtuse crossing angle θ 12 > 90 • . In fact, for the former, cos ϕ 1,2 can varies from zero to one, where cos ϕ 1,2 = 0, which corresponds to a complete polarization misalignment between the laser beams and the scattered waves, occurs when β 1 = −β 2 = ± arcsin[tan θ h ] and α ⊥ = arctan[(sin θ h tan β 1 ) −1 ]; while for the latter, cos ϕ 1,2 can only vary from | cos θ 12 | to one, where the minimum of cos ϕ 1,2 occurs when both beams are p-polarized (β 1 = β 2 = ±90 • ) and α ⊥ = 0. Finally, there is a local maximum of κ A /κ U A in the interior of our considered interval −150 • < α ⊥ < 150 • , and at this maximum the value of κ A /κ U A is mainly determined  IG. 6: κ A /κ U A versus α ⊥ for shared IAW modes of two laser beams with crossing angle (a-b) θ 12 = 60 • and (c-d) θ 12 = 120 • . by β 1 + β 2 (the degree of polarization symmetry breaking); while the corresponding out-ofplane angle α M ⊥ is mainly determined by β 1 − β 2 (the overall polarization deviation from s-polarization). The mode corresponding to the interior local maximum of κ A /κ U A is of great interest, since among the SP modes with relatively small out-of-plane angles and hence large temporal growth rate γ 0 ∝ √ k a ∝ cos(α ⊥ /2) [28], it is most vulnerable to be excited.
Notice that curves for θ 12 < 90 • and for 180 • − θ 12 coincide except that for the former, the local maximum exists over a larger range of β 1 + β 2 . For comparison, the larger one of κ A /κ U A at the endpoints α ⊥ = ±150 • is also displayed as the dashed curves. (f) α M ⊥ versus β 1 −β 2 , when β 1 = −β 2 is taken.
for case θ 12 = 120 • respectively. Fig. 7(e) shows the local max A has a local maximum value in the interior of interval −150 • < α ⊥ < 150 • as shown in Fig. 6, and this local maximum value can be larger than κ A /κ U A at the endpoints α ⊥ = ±150 • . With the increase of β 1 + β 2 , the interior local maximum value decreases until it disappears at some β 1 + β 2 as shown in Fig. 6. As a result, increasing β 1 + β 2 to decrease max α ⊥ [κ A /κ U A ] for relatively small |α ⊥ | can be an effective method to suppress the collective SBS modes with shared IAW, especially for θ 12 ∼ 90 • .
On the other hand, |α M ⊥ | increases from zero towards 180 • when |β 1 − β 2 | increases from zero to 180 • . Therefore, due to modification of polarization states on κ A , depending on β 1 − β 2 the most favored out-of-plane angle of the SP modes can deviate significantly from zero, especially for a large crossing angle. This implies the potentially important roles of out-of-plane SP modes.
The two laser beams are at the same wavelength (351 nm) and intensity (10 15 W/cm 2 ). The plasma condition n e = 0.06 n c , T e = 2.5 KeV, T e /T i = 3.5, and zero flow velocity for He plasma is taken.
In practice, the laser beam width is finite, and usually much smaller than the beam length along the laser propagation direction. Thus, a finite beam overlapping volume dependent on the width and crossing angle of the two overlapped beams is formed. For such case, the parallelogram gain volume V amp (x) = η 1 x 1 n s 1 +η 2 x 2 n s 2 (0 ≤ η 1,2 ≤ 1), that takes part in the amplification of the collective SP modes, must be enclosed in the beam overlapping volume.
Then, the largest achievable asymptotic gain G A,able for the SP modes is the maximum value For larger crossing angles, however, the relative importance of the out-of-plane modes with respect to the in-plane modes are determined by the beam polarization states to a greater extent. As shown in Fig. 8, with consideration of the finite beam overlapping volume, the gain of out-of-plane SP modes for large crossing angle can still be larger than the in-plane modes when either beam I or beam II deviates from s-polarization significantly. Another noticeable thing is that for a given beam width w b , the beam overlapping region in the (k 01 , k 02 )-plane is a rhombus with side length w b / sin θ 12 , which increases with θ 12 when θ 12 ≥ 90 • . Together with the fact that κ U A increases with increasing θ 12 as discussed above, G A,able at θ 12 = 150 • is much larger than other crossing angles, as shown in Fig. 8. As a consequence, the larger obtuse crossing angle is beneficial to SP mode amplification. Finally, it is worth to point that the plasma inhomogeneity can result in different phase matching length along different spatial directions, imposing more limitation on the available volume that can take part in the SP mode amplification. Depending on the specific plasma condition, this can significantly modify the relative importance of different SP modes.

IV. DISCUSSION AND SUMMARY
In summary, based on a linear kinetic model for the shared plasma modes of two overlapped laser beams, an analytic convective solution is derived. From this solution, effects of wavelength difference, crossing angle and polarization states of the two beams on the collective SBS modes with shared IAW are discussed in details. A small wavelength difference (∼nm) is found to have negligible effects on the SP modes except for very small crossing angle (∼ 1 • ). For two beams with nearly equal wavelength, wavevectors of the shared IAW of all possible collective SBS modes lie on a circle in the bisecting plane between wavevectors of the two laser beams. For a specified plasma condition, the wavelength of the scattered waves decreases with increasing beam crossing angle θ 12 or increasing out-of-plane angle α ⊥ of the SP mode. However, the strength of the SP modes are subject to more factors. Depending on the polarization states of the laser beams, and the geometry and relative orientation of the gain volume relative to the beam overlapping volume, the out-of-plane SP modes can be important. Our results suggest that in a realistic simulation for the collective SBS modes, all these factors must be properly accounted for to obtain a reliable result.
In this work, uniform plasma conditions with zero flow velocity are assumed for the illustrative analysis. However, it can be easily extended to the non-zero flow velocity case, which would lead to a wavelength shift of the scattered waves. For practical ICF conditions, the plasma inhomogeneity, together with the practical laser intensity distribution and overlapping pattern of the laser beams, would complicate the situation significantly. Accurate account of all these factors require simulations, for which this work provides valuable theoretical references. Furthermore, the analytical convective solution presented here can help construct the numerical solution by applying it over spatial regions of the grid size, as done previously for single beam LPIs [21,29].  peak value of k 2 a Im[γ pm ]. This can be illustrated in the fluid limit v the ≫ c s ≫ v thI , where v the and v thI are the electron and ion thermal velocity respectively, and there are relatively simple analytical formulae for χ I and χ e . In the fluid limit, where ω pi is the ion plasma frequency. For He plasma with Maxwellian EEDF, the ion acoustic velocity [30] is given by Using the approximation ν a = Im[ǫ]/(∂ǫ r /∂ω), it can be obtained So with the decrease of k a , c s (weakly) increases, and hence the exponential part (corresponding to the tail of the ion energy distribution function) of ν a /ω a decreases. The change in the exponential part is generally stronger, leading to decreasing ν a /ω a and c 2 s ν a /ω a with decreasing k a . As a consequence, max[k 2 a Im[γ pm ]] as given by Eq. (A3) increases (slowly) with decreasing k a . The two-dimensional Laplace transform of a function f (x, y) [31] is defined by where q = (q x , q y ), and q x and q y are the complex frequencies corresponding to x 1 and x 2 , respectively. The associated one-dimensional Laplace transforms along x-axis and y-axis are defined by The Laplace transform changes differentiation to multiplication, When L q [f (x, y)] is known, f (x, y) can be restored by the inverse Laplace transform, The operation of L −1 q on the one-dimensional Laplace transforms yields The inverse Laplace transform changes convolution into multiplication, where the symbol '*' denotes the convolution operation defined as (f * g)(x, y) = where is determined by the boundary conditions specified by a s 1 (x 1 = 0, x 2 ) and a s 2 (x 1 , x 2 = 0).
Then, a s 1 and a s 2 can be restored from L q [a s 1 ] and L q [a s 2 ] by the inverse Laplace transform.
Using the identities (B6-B8), it can be obtained and two diagonals of the parallelogram have lengths x 1 1 + x 2 r ± 2x r cos θ s 12 . Using the root mean square of the two diagonals as the typical size (L m ) of the required gain volume, we have L m = x 1 1 + x 2 r , and the gain coefficient κ A ≡ ln G A /L m defined for this typical size satisfies (1 + κ r ) 1 + x 2 r . (C4) The ratio R A is a function of κ r and θ s 12 , since x r in the RHS is a function of κ r and θ s 12 as specified by Eq. (C2). R A varies between 1 and √ 2. The upper bound of √ 2 appears with n A along the bisector of n s 1 and n s 2 when κ 1 = κ 2 and hence x 1 = x 2 , while the lower bound of one can be approached when κ 1 ≪ κ 2 or κ 2 ≪ κ 1 .
Appendix D: The projection of the gain volume onto one direction n r and perpendicular to n r The parallelogram gain volume V amp (x) = η 1 x 1 n s 1 + η 2 x 2 n s 2 (0 ≤ η 1,2 ≤ 1) is spanned by n s 1 and n s 2 . Denoting the projection of n sα onto one direction n r as n nr sα , and the projection of n sα onto the plane perpendicular to n r as n ⊥nr sα , then n sα = n nr sα + n ⊥nr sα , yielding V amp = V nr amp + V ⊥nr amp , V nr amp ≡ x 1 η 1 n nr s 1 + x 2 η 2 n nr s 2 V ⊥nr amp ≡ x 1 η 1 n ⊥nr s 1 + x 2 η 2 n ⊥nr s 2 (D2) It can be seen that the longitude scale of the gain volume defined as the length of V nr amp is Proj nr [V amp ] = x 1 |n nr s 1 | + x 2 |n nr s 2 | for η 1,2 ranging from zero to one. While V ⊥nr amp is itself a parallelogram in the plane perpendicular to n r , so the transverse scale of the gain volume defined as the longest spatial scale of V ⊥nr amp is the longer diagonal, Proj ⊥nr [V amp ] = x 2 1 |n ⊥nr s 1 | 2 + x 2 2 |n ⊥nr s 2 | 2 + 2x 1 x 2 |n ⊥nr s 1 · n ⊥nr s 2 |.