Mitigation of multibeam stimulated Raman scattering with polychromatic light

Without loss of generality, we firstly consider that two beams drive SRS sharing the same Langmuir wave in a homogeneous plasma. When the laser bandwidth Δω₀ is much smaller than the growth rate Γ [39], the coupling equations for plasma density perturbation $\tilde{n}_{e}$ and scattering lights are:

$$\begin{align} \left(\partial_{tt}-3v_{th}^2\nabla^2+\omega_{pe}^2\right)\tilde{n}_{e} = \frac{\omega_{pe}^2}{4\pi ec}\nabla^2 &(\vec{\tilde{A}}_1\cdot\vec{a}_{1}+\vec{\tilde{A}}_2\cdot\vec{a}_{2}+\vec{\tilde{A}}_1\cdot\vec{a}_{2}\nonumber\\ & + \vec{\tilde{A}}_2\cdot\vec{a}_{1}), \end{align} \tag{ 1 }$$

$$\begin{equation} \left(\partial_{tt}-c^2\nabla^2+\omega_{pe}^2\right)\vec{\tilde{A}}_1 = -4\pi ec\tilde{n}_{e}\vec{a}_{1}, \end{equation} \tag{ 2 }$$

$$\begin{equation} \left(\partial_{tt}-c^2\nabla^2+\omega_{pe}^2\right)\vec{\tilde{A}}_2 = -4\pi ec\tilde{n}_{e}\vec{a}_{2}, \end{equation} \tag{ 3 }$$

where $\vec{a}_1$ and $\vec{a}_2$ are the amplitude of beam 1 and beam 2, respectively, v_th is the electron thermal velocity, and ω_pe is the frequency of electron plasma wave. $\vec{\tilde{A}}_1$ and $\vec{\tilde{A}}_2$ are the scattering perturbations developed by beam 1 and beam 2, respectively. The relation between laser intensity I_i and a_i is given by $I_{i}(\mathrm{W}\,\mathrm{cm}^{-2}) = 1.37\times 10^{18}a_{i}^2/[\lambda(\mu \mathrm{m})]^2$, where i denotes the ith beam.

The coupling terms $\vec{\tilde{A}}_1\cdot\vec{a}_{2} = \tilde{A}_1{a}_{2}\cos(\psi)$ and $\vec{\tilde{A}}_2\cdot\vec{a}_{1} = \tilde{A}_2{a}_{1}\cos(\psi)$ indicate that one of the pump beams is coupled by the scattering light developed by the other laser to excite the Langmuir wave, where ψ is the angle between the polarizations of the two beams. Therefore, two Langmuir waves $\vec{k}_{L1} = \vec{k}_1-\vec{k}_{s2}$ and $\vec{k}_{L2} = \vec{k}_2-\vec{k}_{s1}$ are developed from the partially shared scattering lights. One notes that equations (1)–(3) can be reduced to the single laser case under $\vec{k}_1 = \vec{k}_2$. The interaction of two pump beams can be projected into the plane of their wavevectors, where we always have cos(ψ) = 1 for s-polarized (electric field of light is perpendicular to the plane) lasers. However, one has cos(ψ) ≤ 1 for p-polarized (electric field of light is parallel to the plane) lasers, where the equality only exists for the special case that $\vec{k}_{1}/|k_{1}| = -\vec{k}_{s2}/|k_{s2}|$. In the following discussions, two pump lasers both with s-polarization are considered due to the relatively larger growth rate.

Multibeam SRS developed by pump beams with a same frequency have been discussed in the previews works [11, 47]. Two resonant modes $\vec{k}+\vec{k}_1-\vec{k}_2$ and $\vec{k}+\vec{k}_2-\vec{k}_1$ can be found from the Fourier analysis of the coupling terms $\vec{\tilde{A}}_1\cdot\vec{a}_{2}$ and $\vec{\tilde{A}}_2\cdot\vec{a}_{1}$. Here we consider that the two beams are along the same propagation direction $\vec{k}_1/|k_1| = \vec{k}_2/|k_2|$ and with different frequencies $\delta\omega_0 = \omega_1-\omega_2$. By doing the Fourier analysis of equations (1)–(3) in the plane of $\vec{k}_1$ and $\vec{k}_2$, one obtains the dispersion relation:

$$\begin{align} \omega^2-\omega_{pe}^2-3k^2v_{th}^2 = \frac{\omega_{pe}^2k^2c^2}{4}& \left[a_1^2\left(\frac{1}{D_{e1-}}+\frac{1}{D_{e1+}}\right) + a_2^2\left(\frac{1}{D_{e2-}}+\frac{1}{D_{e2+}}\right)\right]\nonumber\\ & \quad +\frac{\omega_{pe}^2k^2c^2}{4} \left[a_1a_2\left(\frac{1}{D_{e1-}}+\frac{1}{D_{e1+}}+\frac{1}{D_{e2-}}+\frac{1}{D_{e2+}}\right)\right], \end{align} \tag{ 4 }$$

where $D_{e1\pm} = (\omega\pm\omega_1)^2-(\vec{k}\pm\vec{k}_1)^2c^2-\omega_{pe}^2 = \omega_{1\pm}^2-\vec{k}_{1\pm}^2c^2-\omega_{pe}^2$, $D_{e2\pm} = (\omega\pm\omega_2)^2-(\vec{k}\pm\vec{k}_2)^2c^2-\omega_{pe}^2 = \omega_{2\pm}^2-\vec{k}_{2\pm}^2c^2-\omega_{pe}^2$, ω_i and $\vec{k}_i$ are the frequency and wavevector of ith beam, respectively. We have $D_{e2-}\approx D_{e1-}+2(\delta\omega_0\omega_{1-}-\delta\vec{k}_0\cdot\vec{k}_{1-})$ with $\delta \vec{k}_0 = \omega_1\vec{k}_1\delta\omega_0/(\omega_1^2-\omega_{pe}^2)$. The modulus of $D_{e1-}$ is in the order of $|D_{e1-}|\sim\Gamma$, and $|\delta\omega_0\omega_{1-}+\delta \vec{k}_0\cdot\vec{k}_{1-}|\sim\delta\omega_0$, where Γ is the imaginary part of ω. When $\Gamma\gg\delta\omega_0$, the backscattering mode $1/D_{e2-}\sim(1-\delta_1)/D_{e1-}$ with $\delta_1 = 2(\delta\omega_0\omega_{1-}-\delta\vec{k}_0\cdot\vec{k}_{1-})/D_{e1-}$. Therefore, the two beams are strongly coupled under a small bandwidth $\delta\omega_0\ll\Gamma$. On the contrary, $2(\delta\omega_0\omega_{1-}-\delta\vec{k}_0\cdot\vec{k}_{1-})$ is the dominant term of $D_{e2-}$ under a broad bandwidth $\delta\omega_0\gg\Gamma$, and correspondingly the two beams are decoupled [29, 35].

Figure 1(a) shows the wavenumber distribution of the SRS growth rate in the rectangular coordinates by solving equation (4). The plasma density is $n_e = 0.188n_{c1}$, and electron temperature is T_e  = 2keV, where n_c1 is the critical density for beam 1. The incident angles of the two beams in plasma are equal $\theta_{p1} = \theta_{p2} = 10^{\circ}$, and with the same amplitude $a_1 = a_2 = 0.012$ and different frequencies $\delta\omega_0 = \omega_{1}-\omega_{2} = 0.2\%\omega_1$. The two scattering lights have a small frequency difference equal to 0.2%ω₁, and both of them are shared by the pumps. One finds that the instability regions of the two beams are overlapped under this condition. Therefore, the instability region width Δk is the major factor for the coupling of different instability modes.

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The instability regions of the scattering lights are decoupled $|k_{s1}-k_{s2}|\gt\Delta k$, when the frequency difference satisfies $\delta\omega_0\gtrsim{3}\Gamma$ [39]. Under this condition, the pump lasers will develop their respective scattering lights independently, and equations (1)–(3) are reduced to:

$$\begin{equation} \left(\partial_{tt}-3v_{th}^2\nabla^2+\omega_{pe}^2\right)\tilde{n}_{e} = \frac{\omega_{pe}^2}{4\pi ec}\nabla^2(\vec{\tilde{A}}_1\cdot\vec{a}_{1}+\vec{\tilde{A}}_2\cdot\vec{a}_{2}), \end{equation} \tag{ 5 }$$

$$\begin{equation} \left(\partial_{tt}-c^2\nabla^2+\omega_{pe}^2\right)\vec{\tilde{A}}_1 = -4\pi ec\tilde{n}_{e}\vec{a}_{1}, \end{equation} \tag{ 6 }$$

$$\begin{equation} \left(\partial_{tt}-c^2\nabla^2+\omega_{pe}^2\right)\vec{\tilde{A}}_2 = -4\pi ec\tilde{n}_{e}\vec{a}_{2}. \end{equation} \tag{ 7 }$$

By doing the Fourier analysis of equations (5)–(7) in the plane of $\vec{k}_1$ and $\vec{k}_2$, one obtains the dispersion relation:

$$\begin{align} \omega^2-\omega_{pe}^2-3k^2v_{th}^2 = \frac{\omega_{pe}^2k^2c^2}{4}& \left[a_1^2\left(\frac{1}{D_{e1-}}+\frac{1}{D_{e1+}}\right)\right.\nonumber\\ & \quad \left.+ a_2^2\left(\frac{1}{D_{e2-}}+\frac{1}{D_{e2+}}\right)\right]. \end{align} \tag{ 8 }$$

Figure 1(b) shows the phase space of two beams with different incident angles $\theta_{p1} = 10^{\circ}$, $\theta_{p2} = -10^{\circ}$ and the frequency difference $\delta\omega_0 = 1.4\%\omega_1$. The wavenumber difference of scattering lights $\delta k_s = k_{s1}-k_{s2}$ is larger than the instability region width Δk_i of ith beam. Therefore, their scattering lights are no longer shared by each other. Two common Langmuir waves can be found in figure 1(b), where the backscattered common wave $\vec{k}_{LB}$ with the largest growth rate is the major instability mode when the scattering lights are decoupled. Figure 1(c) presents the growth rate as a function of angle between the vectors of the two pumps for different δω₀. One finds that the growth rate of the collective SRS slightly decreases with the enhancement of δω₀, when the scattering lights have been decoupled. However, the growth rate Γ can be significantly reduced by the incident angle, and therefore the frequency difference can separate the two instability regions at a certain large θ.

Here we introduce the polychromatic light, which is composed of many beamlets with different frequencies $\vec{a} = \sum_i^N \vec{a}_{i}\exp(\vec{k}_ic-\omega_{i}t+\phi_{i})$ [39, 40], where a_i is the normalized amplitude of ith beamlet with a carrier frequency ω_i and a random phase φ_i , and N is the beamlets number. The bandwidth of the polychromatic light is $\Delta\omega_0 = \omega_N-\omega_1 = (N-1)\delta\omega_0$. The beamlets of polychromatic light can propagate in different directions, such as the case with N = 2 shown in figure 1(b). And also, they can propagate in the same direction, as the example with N = 2 shown in figure 1(a). To keep the pump energy conserved between normal laser beam a₀ and polychromatic light a, the beamlet amplitude is $a_{i} = a_0/\sqrt{N}$.

In analogy to the above two-beam analysis, the dispersion relation is easily extended to N beams case which can describe the collective SRS driven by polychromatic light

$$\begin{equation} \omega^2-\omega_{pe}^2-3k^2v_{th}^2 = \frac{\omega_{pe}^2k^2c^2}{4}\sum_i^Na_i^2\left(\frac{1}{D_{ei-}}+\frac{1}{D_{ei+}}\right). \end{equation} \tag{ 9 }$$

Equation (9) also can be obtained from the generalized Wigner–Moyal statistical theory with the photon distribution function $f(\vec{k}) = \Sigma a_i^2\delta(\vec{k}-\vec{k}_i)$, where δ function has been used here [36]. An example is displayed in figure 1(d), where the polychromatic beam 2 is composed of two beamlets with different frequencies, and the beam 1 is a normal laser with uniform amplitude a₁ = 0.012. The beamlet amplitude of the polychromatic light is a_2i ≈ 0.00855, which has an equal energy to beam 1. There are four common electrostatic modes can be found in figure 1(d), and the highest growth rate is reduced with comparing to figure 1(b). One also finds that the mitigation effects on multibeam SRS are different from the single beam model. The instability regions can be completely separated by the decoupled beamlets in the same propagation direction. However, they may still have intersections in the multibeam configurations with different incident angles. One notes that the wavenumber difference δk_L of backward SRS is larger than that of forward SRS, as shown in figure 1(d). This is mainly because the frequency difference not only reduces the radius of the wavenumber circle $\delta r_i = (\omega_i-\omega_{pe})\delta\omega_0/c\sqrt{\omega_i^2-2\omega_i\omega_{pe}}$, but also shifts the center to the left $\vec{k}_i(1-\omega_i\delta\omega_0/k_i^2c^2)$. The frequency difference for the mitigation of forward SRS is $\delta r_i-\omega_i\delta\omega_0/k_ic^2\gt\Delta k$ which is larger than the threshold for backward SRS [36]. Therefore, forward SRS and modulation instability are less sensitive to the small bandwidth. Generally, backward SRS dominates in the kinetic regime $k_L\lambda_D\sim{0.3}$ due to the high growth rate, where λ_D is the Debye length. Forward SRS is insufficient to heat a large number of electrons independently, and the control of backward SRS is important for the suppression of hot electrons in the convective regime. Note that forward SRS may be excited by the beat of beamlets when the bandwidth is larger than the frequency of the plasma wave [48]. In this work, we mainly consider the bandwidth effect around $\Delta\omega_0\sim{3}\%\omega_1$, where the corresponding plasma density for the beat wave excitation is $n_e\sim{0.001}n_{c1}$. Note that this density is much smaller than the usually concerned plasma density in ICF.

In a brief summary, the scattering lights are decoupled from each other when the frequency difference of pump beams satisfies $\delta\omega_0\gtrsim{3}\Gamma\sim{1}\%\omega_1$, and therefore the linear growth rate is reduced. For further mitigation, polychromatic light can be used as the pump beams, which can significantly mitigate the multibeam SRS by dividing the common mode to multiple decoupled common modes, and the weak modes are heavily damping in hot plasmas.

According to the above analysis about multibeam SRS in homogeneous plasmas, the wavevector matching condition $\vec{k}_i = \vec{k}_{Li}+\vec{k}_{si}$ and frequency matching condition $\omega_i = \omega_{Li}+\omega_{si}$ of ith beam are the necessary conditions for the excitation of collective SRS, where $\vec{k}_{si}$ and ω_si are the wavevector and frequency of the scattering light, respectively. In inhomogeneous plasmas, multibeam SRS is developed in the resonant region where the matching conditions are satisfied, and then convective amplified in the propagation. For the multibeam SRS with a common electron plasma wave, the frequency matching condition $\omega_{1}-\omega_{s1} = \omega_{2}-\omega_{s2} = \omega_L = \sqrt{\omega_{pe}^2+3k_L^2v_{th}^2}$ will be satisfied when the wavevector matching condition $\vec{k}_{1}-\vec{k}_{s1} = \vec{k}_{2}-\vec{k}_{s2} = \vec{k}_{L}$ is satisfied at plasma density n_e . Therefore, we study the multibeam parametric instability in inhomogeneous plasmas based on the wavevector matching condition in the rectangular coordinates $k_{ix}-k_{Lx} = k_{six}$ and $k_{iy}-k_{Ly} = k_{siy}$. According to the dispersion relation of scattering light $k_{six}^2c^2+k_{siy}^2c^2 = (\omega_i-\omega_L)^2-\omega_{pe}^2$, one obtains the relation for ($k_{Lx},k_{Ly}$) as follows:

$$\begin{align} & \left[k_{Lx}c-k_{ix}c\left[1+\frac{3v_{th}^2}{c^2}\left(1-\frac{\omega_i}{\omega_{pe}}\right)\right]\right]^2 \nonumber\\ &\quad + \left[k_{Ly}c-k_{iy}c\left[1+\frac{3v_{th}^2}{c^2} \left(1-\frac{\omega_i}{\omega_{pe}}\right)\right]\right]^2 \nonumber\\ & = \omega_i^2-2\omega_i\omega_{pe}+\frac{3v_{th}^2}{c^2}\left(1-\frac{\omega_i}{\omega_{pe}}\right)(k_i^2c^2+\omega_i^2-2\omega_i\omega_{pe}). \end{align} \tag{ 10 }$$

Equation (10) indicates that the wavenumber distribution of k_L is a circle with its center $O_i(k_{ix}[1+3v_{th}^2/c^2(1-\omega_i/\omega_{pe})],k_{iy}[1+3v_{th}^2/c^2(1-\omega_i/\omega_{pe})])$. One notes that the center O_i is the wavevector of incident light $\vec{k}_{i} = (k_{ix},k_{iy})$, and the radius is $r_i = \sqrt{\omega_i^2-2\omega_i\omega_{pe}}/c$ for cold plasma v_th  = 0. The thermal modification for both O_i and r_i are in the order of $10^{-2}\omega_i$. Considering a small change of the plasma density δn_e , the change of wavevector radius is

$$\begin{equation} \delta r_i/\omega_i = -\frac{\delta n_e/n_{ci}}{2r_ic^2\omega_{pe}/\omega_i^2}, \end{equation} \tag{ 11 }$$

where n_ci is the critical density for ith beam. Based on equation (11), we know that the radius r_i decreases with the plasma density n_e .

The above discussion is valid for both homogeneous and inhomogeneous plasmas. Considering for a plasma with inhomogeneous density in the longitudinal direction $\vec{x}$, one has a relation for the incident angle in vacuum θ_v and the angle of the pump beam in plasma $\sin\theta_{p} = \omega_i\sin\theta_v/\sqrt{\omega_i^2-\omega_{pe}^2}$. As an example, we firstly consider two pump beams with the same frequency $\omega_{1} = \omega_{2}$, and the incident angles $\theta_{v1} = 18^{\circ}$ and $\theta_{v2} = -18^{\circ}$, respectively. The wavevector distribution of multibeam SRS at $n_e = 0.188n_{c1}$ and T_e  = 0 is presented in figure 2(a). Two intersections $\vec{k}_{LF}$ and $\vec{k}_{LB}$ can be found, which are similar to figure 1(b). The longitudinal projection of the wavevector $\vec{k}_{sF}$ is in the forward direction, and the longitudinal projection of the wavevector $\vec{k}_{sB}$ is in the backward direction. Note that the scattering light of beam 1 $\vec{k}_{sB1}$ is a seed mode for beam 2 to drive a concomitant electrostatic mode $\vec{k}_{L2}$. Therefore, parts of the scattering lights are shared in the process of multibeam SRS coupling by a common electron plasma wave. One obtains $|\vec{k}_{LF}c| = \sqrt{\omega_1^2\cos^2\theta_v-\omega_{pe}^2}-\sqrt{\omega_1^2\cos^2\theta_v-2\omega_1\omega_{pe}}$ and $|\vec{k}_{LB}c| = \sqrt{\omega_1^2\cos^2\theta_v-\omega_{pe}^2}+\sqrt{\omega_1^2\cos^2\theta_v-2\omega_1\omega_{pe}}$ from equation (10) at k_Ly  = 0 and v_th  = 0. A critical point is found at $|\vec{k}_{sF}| = |\vec{k}_{sB}|$, i.e. $\omega_{pe} = \omega_1\cos^2\theta_v/2$, where the propagation of the scattering light is perpendicular to the density gradient, and therefore the SRS instability is absolute [23]. The multibeam SRS sharing a common Langmuir wave will be developed only when $\omega_{pe}\leq\omega_1\cos^2\theta_v/2$ is satisfied.

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Now we consider the case when there is a frequency difference between the two beams. A similar example to the above case with $\delta\omega_0 = 7.7\%\omega_1$ and T_e  = 2 keV is displayed in figure 2(b). The plasma density n_e is $n_{e2} = n_e/n_{c2} = 0.221$ for beam 2, which is larger than $n_{e1} = n_e/n_{c1} = 0.188$ due to $n_{e2}/n_{e1} = \omega_1^2/\omega_2^2\gt1$. As discussed above, the radius r_i decreases with the increasing of plasma density. One finds a smaller circle O₂ as compared to O₁ in figure 2(b), and the two circles no longer intersect. Therefore, the frequency difference of pump beams can change the position of absolute SRS region where $|r_{O_1O_2}| = r_{O_1}+r_{O_2}$, i.e.

$$\begin{equation} n_{ec}/n_{c1} = \frac{\cos^4\theta_v}{4}\left(1-\frac{\delta\omega_0}{\omega_{1}}\right). \end{equation} \tag{ 12 }$$

Equation (12) shows that the density for the multibeam absolute SRS is slightly reduced by the frequency difference. As an example, the density for absolute SRS is $n_{ec} = 0.205n_{c1}$ for beams with δω₀ = 0, $\theta_{v1} = 18^{\circ}$ and $\theta_{v2} = -18^{\circ}$. When the frequency difference is increased to $\delta\omega_0 = 2\%\omega_1$, the absolute SRS region is changed to $n_{ec} = 0.2n_{c1}$. Considering the absolute SRS developed by N beams, the absolute SRS regions are different for every two beams, when different frequency differences are introduced to the multibeam. The instability is mitigated due to the intense one common electrostatic field is divided to multiple submodes. Moreover, the frequency difference can also suppress the concomitant electrostatic modes as $|k_{sB1}|\ne|k_{sB2}|$.

The wavenumber distribution of SRS driven by two polychromatic lights with bandwidth $\Delta\omega_0 = 4.4\%\omega_1$ and beamlets number N = 2 is shown in figure 2(c). The plasma density is $n_e = 0.188n_{c1}$ and the incident angles are $\theta_{v1} = 18^{\circ}$ and $\theta_{v2} = -18^{\circ}$. One finds that the common mode is divided to multiple submodes with different frequency $|\delta\omega_{LB}| = |\sqrt{(\omega_{pe}^2+3k_{LB1}^2v_{th}^2)/\omega_1^2}-\sqrt{(\omega_{pe}^2+3k_{LB2}^2v_{th}^2)/\omega_1^2}|$. When $\omega_{pe}/\omega_{1}\ll\cos^2\theta_v/2$, we have:

$$\begin{align} |\delta\omega_{LB}/\omega_1|& \approx\frac{3k_{LB1}v_{th}^2\delta\omega_0}{\omega_{L1}\omega_1c}\nonumber\\ & \times\left(\frac{\cos^2\theta_v}{\sqrt{\cos^2\theta_v-\omega_{pe}^2/\omega_{1}^2}}+\frac{\cos^2\theta_v-\omega_{pe}/\omega_{1}}{\sqrt{\cos^2\theta_v-2\omega_{pe}/\omega_{1}}}\right). \end{align} \tag{ 13 }$$

Based upon our previous work [40], the coupling of Langmuir waves will be reduced if there is a frequency difference between the pump beams, which leads to the reduction of Rosenbluth gain saturation coefficient [49, 50]. Considering a linear density profile in the longitudinal direction $n_e(x) = n_0(1+x/L)$, where L is the density scale length. The saturation coefficient of the common mode k_LB driven by two normal lasers $\omega_1 = \omega_2$ with an equal amplitude $a_1 = a_2$ is $G_{nor} = \pi La_1^2k_{LB}^2/2k_{sB}$ [23]. The saturation coefficient is reduced by the frequency difference between different common modes $G_{brb} = G_{nor}[1+\cos(\delta\omega_{LB}\beta)]$, where $\beta = 2~\textrm{ L}\omega_1\nu_p^2\omega_{LB}^3/3k_{LB}v_{th}^2n_0\cos\theta_v(\omega_1^2-\omega_{pe}^2)^{3/2}$ for low density plasmas, and ν_p is the damping rate of plasma wave [40]. Considering polychromatic pumps with N beamlets, Langmuir waves are weakly coupled due to their frequency shifts equation (13), and multibeam SRS is heavily damped if $G_{brb-i}\lt1$ for every two common modes. Here we consider that G_{brb − i}  = 1, and every beamlet shares different common modes with all the other beamlets, i.e. there are N² common Langmuir waves. Note that the parity of N has little effect on the final result under $N^2\gg1$. The amplification coefficient for polychromatic light can be estimated as $\alpha_{p} = N^2\exp(1)/2$, and the amplification coefficient for normal lasers with the same energy is α_n  = exp(N/2) based upon $a_i = a_1/\sqrt{N}$. The ratio $\alpha_{p}/\alpha_{n}$ is found to be less than one $\alpha_{p}/\alpha_{n}\lt1$ for $N\gtrsim{10}$. Generally, the amplification coefficients of polychromatic light α_p is a function of N^m , and the amplification coefficients of normal lasers is $\alpha_{n}\propto\exp(\mu N)$, where m is a finite exponent, and $\mu\gt0$ is a constant. The relation $\alpha_{p}/\alpha_{n}\lt1$ always can be satisfied for a large enough N. Therefore, the saturation level of Langmuir waves can be reduced by broadband light with large numbers of beamlets.

The wavenumber distribution of the scattering light driven by ith beam is $k_{six}^2c^2+k_{siy}^2c^2 = (\omega_i-\omega_L)^2-\omega_{pe}^2$, which can be reduced to $k_{six}^2c^2+k_{siy}^2c^2 = \omega_{i}^2-2\omega_i\omega_{pe}$ for cold plasma. Different from the Langmuir wave, the circle center of the scattering light wavemumber is the origin of the coordinates (0, 0), and the radius is $r_s = \sqrt{\omega_{i}^2-2\omega_i\omega_{pe}}/c$. The phase space of scattering lights can be separated when the two beams have a frequency difference $\delta r_s/\omega_i\sim(1-\omega_{pe}/\omega_i)\delta\omega_0/r_sc^2$. An example is shown in figure 2(d), where the scattering lights are driven by two-color pumps with frequency difference $\delta\omega_0 = 2\%\omega_1$. Therefore, polychromatic light can detune the scattered seed lights in the development of concomitant electrostatic field.

When the incident angles are large enough $|r_{O_1O_2}|\gt r_{O_1}+r_{O_2}$, multibeam SRS will no longer share with the same Langmuir wave. Sidescattering in the perpendicular direction of the density gradient is an absolute instability in inhomogeneous plasmas [51]. According to equation (1), most of the scattering lights are detuned from the polychromatic pump beam. The scattering light $\vec{k}_{s1} = (0,-k_{s1y})$ cannot be shared by beam 2 if $\omega_2\ne\omega_1$, as shown in figure 2(d). Therefore, SRS sidescattering can be mitigated by the polychromatic light when the intense sidescattering wave is broken up into many weak beamlets. In one-dimensional geometry, there exists a secondary amplification of the scattering lights with different frequency in inhomogeneous plasmas. For an example, the scattering light $\omega_{s1} = 0.6\omega_1$ developed by beam 1 ω₁ at $n_{e1} = 0.16n_{c1}$ will be amplified again by beam 2 $\omega_2 = 0.98\omega_1$ at $n_{e2} = 0.144n_{c1}$ where the frequencies of the scattering lights are equal $\omega_{s2} = \omega_{s1} = 0.6\omega_1$. However, the 90^∘ sidescattering light $(\vec{k}_{s1},\omega_{s1})$ developed at $n_{e1}$ is no longer perpendicular to the density gradient at $n_{e2}$. The angle between the two scattering lights is

$$\begin{equation} \theta_{12} = \arccos\left(\sqrt{\frac{\omega_{s1}^2/\omega_1^2-n_{e1}/n_{c1}}{\omega_{s1}^2/\omega_1^2-n_{e2}/n_{c1}}}\right). \end{equation} \tag{ 14 }$$

The wavevector mismatch $\vec{k}_{s1}\ne\vec{k}_{s2}$ mitigates the secondary amplification even though their frequencies are equal $\omega_{s2} = \omega_{s1}$.

Multibeam can be coupled by a same scattering light under k_sy  = 0 in inhomogeneous plasmas. A particular region is the density around $n_e\sim{0.25}n_{c1}$, where the wavevector of SRS scattering light is $\vec{k}_s\approx0$, and SRS is an absolute instability due to the group velocity of the scattering light is $\vec{v}_{gs}/c = \vec{k}_sc/\omega_s = 0$. Considering an example that the plasma density is $n_e = 0.243n_{c1}$ and electron temperature is T_e  = 3 keV. The incident angles of the two normal lasers are $\theta_{v1} = 28^{\circ}$ and $\theta_{v2} = -28^{\circ}$, respectively. The wavevector of the scattering light is $\vec{k}_s\approx0$, and $\vec{k}_{Li} = \vec{k}_i$ is satisfied for the wavevector matching condition. Therefore, both beam 1 and beam 2 can share with the same scattering light $(\vec{k}_sc,\omega_s) = (0,0.494)\omega_1$. One notes that the frequency of the scattering light satisfies $\omega_{pe} = 0.493\omega_1\lt\omega_{s1}\lt0.5\omega_1$. The absolute instability region $[\omega_1/2-a_1|k_1|c/4,\omega_1/2+a_1|k_1|c/4]$ for beam 1 can be obtained from equation (8) under v_th  = 0, a₂ = 0 and $\vec{k} = \vec{k}_1$ [52]. The pump beams will no longer share with a common scattering light when $\omega_1/2-a_1|k_1|c/4\gt\omega_2/2+a_2|k_2|c/4$, i.e.

$$\begin{equation} \delta\omega_0\gt\frac{k_1^2c^2(a_1+a_2)}{2|k_1|c+a_2\omega_1}\approx0.433(a_1+a_2). \end{equation} \tag{ 15 }$$

The above threshold indicates that multiple pump beams will be decoupled by the frequency difference in the sharing with a common scattering light. For example, comparing to the normal lasers with intensity $I\sim 10^{15}\,\mathrm{W}\ \mathrm{cm}^{-2}$, absolute multibeam SRS with a common scattering light will be mitigated by the polychromatic lights with N = 20 and bandwidth $\Delta\omega_0 = (N-1)\delta\omega_0 = 3.8\%\omega_1$. According to the threshold for developing absolute SRS in inhomogeneous plasmas, one knows that absolute SRS can be considerably suppressed when the decoupled beamlets satisfy $a_i\lesssim(k_iL)^{-2/3}$ [50]. Note that two-plasmon decay instability also can be effectively mitigated when equation (15) is satisfied [40, 53].

To validate the mitigation of multibeam SRS by polychromatic light, we have performed several two-dimensional (2D) simulations mainly for direct drive by using the OSIRIS code [54, 55]. The space and time given in the following are normalized by the laser wavelength in vacuum λ and the laser period τ. The plasma occupies a region from x = 5λ to x = 115λ with the density profile $n_e(x) = 0.18\exp[(x-5)/1000]n_{c1}$. The mass of the ion is $m_i = 1836m_e$ with a charge Z = 1. The initial electron and ion temperatures are T_e  = 2 keV and T_i  = 700 eV, respectively. Two s-polarized semi-infinite pump lasers with a uniform amplitude are incident from the left boundary of the simulation box.

Firstly, we consider the case of two normal beams with an equal amplitude $a_1 = a_2 = 0.012$, and different frequencies $\delta\omega_0 = \omega_1-\omega_2 = 4.5\%\omega_1$. The incident angles are $\theta_{v1} = 18^{\circ}$ and $\theta_{v2} = -18^{\circ}$. The spatial distribution of the pump beams at t = 1200τ is displayed in figure 3(a). One finds that an intense electrostatic field has been excited in the crossed region of the two beams based on figure 3(b). According to equation (10), the beams are still coupled by the common Langmuir wave under $\delta\omega_0 = 4.5\%\omega_1$. Therefore, an intense Langmuir wave is developed around $(k_{Lx}c,k_{Ly}c)\sim(0.8,0)\omega_1$ as shown in figure 3(c). Note that the wavenumber circle of beam 1 is larger than that of beam 2 as discussed above. The pump beams no longer share with the common Langmuir wave when the frequency difference is increased to $\delta\omega_0 = 7.7\%\omega_1$, as exhibited in figure 3(d). The larger growth rate of beam 2 leads to the more intense electrostatic field around $k_{Ly}\lt0$ by comparing to beam 1. The simulation results presented in figures 3(c) and (d) validate that the common electrostatic modes are different for every two beams, when different frequency differences are introduced to the pump beams. Based upon figure 3(e) we know that the growth of electrostatic energy excited by normal lasers is reduced by the frequency difference $\delta\omega_0 = 4.5\%\omega_1$, due to the mitigation of the concomitant electrostatic fields. When the two instability regions are totally separated at $\delta\omega_0 = 7.7\%\omega_1$, the growth rate is further reduced.

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Next, we study the mitigation of SRS with polychromatic light. As an comparison, we have performed a simulation for two normal lasers $a_1 = a_2 = 0.01$ with an equal frequency $\omega_1 = \omega_2$. The incident angles are $\theta_{v1} = 10^{\circ}$ and $\theta_{v2} = -10^{\circ}$. The plasma parameters are the same to the above simulation example. According to equations (8) and (10), we know that the wavevector of beam 1 is $O_1(0.87,0.17)\omega_1/c$, and the backscattering common mode is the major instability mode with wavenumber $k_{LB}c\sim{1.17}\omega_1$ at plasma density $n_e\sim{0.188}n_{c1}$, which are very close to the simulation results presented in figure 4(a), where $O_1(0.88,0.18)\omega_1/c$ and $k_{LB}c\sim{1.15}\omega_1$. The wavevector of the scattering light developed by beam 1 is $\vec{k}_{s1}c = \vec{k}_{1}c-\vec{k}_{LB}c\approx(-0.27,0.18)\omega_1$, and the wavevector of the other scattering light is $\vec{k}_{s2}c = \vec{k}_{2}c-\vec{k}_{LB}c\approx(-0.27,-0.18)\omega_1$ as shown in figure 4(b). One notes that two concomitant electrostatic modes $\vec{k}_{L1}c = \vec{k}_{1}c-\vec{k}_{s2}c\approx(1.17,0.36)\omega_1$ and $\vec{k}_{L2}c = \vec{k}_{2}c-\vec{k}_{s1}c\approx(1.17,-0.36)\omega_1$ are induced by the scattering light from the other beam.

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To keep the energy conservation of normal laser a₁ and polychromatic light, the amplitude of each beamlet is $a_{1i} = a_{2i} = a_1/\sqrt{N}\sim0.0023$ under N = 19. We find that the intensities of the common Langmuir wave and the concomitant electrostatic field are significantly reduced by the polychromatic lights with bandwidth $\Delta\omega_0 = 2\%\omega_1$ at t = 1500τ, with comparing figure 4(c) to figure 4(a). As discussed above, the different submodes are weakly coupled, and therefore the whole instability saturation level is reduced by the polychromatic lights. However, the multibeam SRS has not been totally suppressed, due to the weak coupling of the beamlets under a relatively small bandwidth $\Delta\omega_0 = 2\%\omega_1$. Therefore, one finds multiple electrostatic submodes around $k_{LB}c\sim{1.15}\omega_1$ at t = 3000τ as shown in figure 4(d). When the bandwidth is increased into $\Delta\omega_0 = 3\%\omega_1$, only a much weak common Langmuir wave is developed at t = 3000τ, known from figure 4(e). The temporal growth of the electrostatic energies developed by pump beams with different bandwidth are shown in figure 4(f). One finds that the growth rate in the linear regime $t\lt2500\tau$ and the saturation level at t = 3000τ are reduced by the bandwidth, which are consistent with our theoretical conclusions. When the interactions evolve into the non-linear regime for Δω₀ = 0 and $\Delta\omega_0 = 2\%\omega_1$, the electrostatic energies slowly grow with time. Note that the linear growth of the electrostatic energies of Δω₀ = 0 is larger than that of $\Delta\omega_0 = 2\%\omega_1$, i.e. $g_1\gt g_2$. Therefore, the mitigation of multibeam SRS by polychromatic light can also be found in the non-linear regime. When the bandwidth is enhanced to $\Delta\omega_0 = 3\%\omega_1$, the collective SRS is found to be still in the linear regime at t = 6000τ. The intensity of Langmuir waves directly leads to the strength of hot electrons. The single common electrostatic field is divided to many weak modes with different wavevectors, and the hot electron productions are mitigated when the weak coupling modes are heavily damped. The electron energy distribution at the end of linear regime t = 3000τ and the fully non-linear regime t = 6000τ are presented in figures 4(g) and (h). For normal laser beams, the common electron plasma wave heats a large number of electrons together with the two concomitant electrostatic modes. Polychromatic lights with bandwidth $\Delta\omega_0 = 2\%\omega_1$ can effectively reduce the electron hot tail even in the fully non-linear regime. The electron hot tail is sufficiently suppressed by polychromatic lights with bandwidth $\Delta\omega_0 = 3\%\omega_1$. Different from homogeneous plasmas, the simulation results conform that an effective mitigation can be found at a relatively small bandwidth $\Delta\omega_0 = 3\%\omega_1$ in inhomogeneous plasmas in both linear and non-linear regimes, because the electrostatic strength is reduced by the wavenumber detuning outside the resonant region [40, 48, 56].

To validate the mitigation effects in a relatively broad plasma density range, we have performed the simulations for the beam overlapping region [0.14, 0.19]n_c1 , with L = 900λ. The mass of the ion is $m_i = 3672m_e$ with an effective charge Z = 1, and the other parameters are the same to the above simulations. The wavenumber distributions of Langmuir wave at the same time for different bandwidth have been displayed in figure 5. One still finds a well mitigation of multibeam SRS by polychromatic light with bandwidth $\Delta\omega_0 = 3\%\omega_1$. Generally, the whole level of SRS can be controlled if the intensities of Langmuir wave and scattering light are mitigated in the high density region where the growth rate and saturation level are relatively higher.

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The above simulations indicate that polychromatic lights with bandwidth $\Delta\omega_0 = 3\%\omega_1$ can effectively mitigate the intensity of multibeam SRS coupling by a common Langmuir wave. Here we study the excitation of multibeam SRS sharing with the same scattering light near 0.25n_c1 . The plasma occupies a region from x = 5λ to x = 75λ with the density profile $n_e(x) = 0.236\exp[(x-5)/1000]n_{c1}$. The electron and ion temperatures are T_e  = 3 keV and T_i  = 1 keV, respectively.

We consider first that SRS excited by normal laser beams. Figure 6(a) shows the wavenumber distribution of scattering light developed by a single normal laser with a uniform amplitude a₁ = 0.01 and incident angles $\theta_{v1} = 28^{\circ}$. One finds that the wavevector of the scattering lights is around $\vec{k}_s\sim0$. Next, we perform the simulation with two normal laser beams $\omega_1 = \omega_2$, their uniform amplitudes are equal $a_1 = a_2 = 0.01$, with the incident angles $\theta_{v1} = 28^{\circ}$ and $\theta_{v2} = -28^{\circ}$. The wavevectors of the scattering lights driven by two normal lasers are also around $\vec{k}_s\sim0$, and the collective instability has been fully developed at t = 800τ as presented in figure 6(b). The intensity of the scattering lights for two normal beams is higher than that of single beam, known from the comparison between figures 6(a) and (b). Therefore, the two beams share with the same scattering light in the development of SRS. Figure 6(c) shows that the two beams develop their respective Langmuir waves independently, where $\vec{k}_{L1}c = \vec{k}_1c\approx(0.73,0.47)\omega_1$ and $\vec{k}_{L2}c = \vec{k}_2c\approx(0.73,-0.47)\omega_1$.

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Finally, we validate the mitigation effect of polychromatic light. To keep the energy conservation between normal laser a₁ and polychromatic light, the amplitude of each beamlet is $a_{1i} = a_{2i} = a_1/\sqrt{N}\sim0.0023$ for N = 19. According to figure 6(d), we know that the intensity of Langmuir waves has been reduced by the polychromatic lights with bandwidth $\Delta\omega_0 = 3\%\omega_1$. The wavevector projection k_Lx has a broad range, due to the coupling of beamlets with different frequencies. When the bandwidth is increased to $\Delta\omega_0 = 5\%\omega_1$, no instability can be found at t = 1500τ in figure 6(e). Therefore, weak coupling modes are developed under $\Delta\omega_0 = 3\%\omega_1$, where the bandwidth is smaller than the threshold equation (15) $\Delta\omega_0\lt3.8\%\omega_1$. However, no obvious instability mode can be found when the threshold is satisfied $\Delta\omega_0 = 5\%\omega_1\gt3.8\%\omega_1$, where the submodes are much weak and heavily damped by the high electron temperature. The temporal growth of the electrostatic energies shown in figure 4(f) further validates that the bandwidth can reduce the growth rate and the saturation level of multibeam SRS with sharing a common scattering light. Polychromatic light can effectively mitigate the collective SRS in both linear and non-linear regimes. The electron energies are diagnosed at the end of linear regime t = 1500τ and the fully non-linear regime t = 3000τ, and the distributions are shown in figures 6(g) and (h). The electron energy distribution of $\Delta\omega_0 = 5\%\omega_1$ is almost overlapped by that of $\Delta\omega_0 = 3\%\omega_1$ at t = 1500τ, when their electrostatic energies have little difference. A comparison has been made for the mitigation effects of single beam and two beams at t = 3000τ. Generally, the bandwidth threshold for mitigation of multibeam SRS is larger than that of one-beam case, due to the relatively higher growth rate of the former. The intensity of SRS driven by two polychromatic beams with bandwidth $\Delta\omega_0 = 3\%\omega_1$ is comparable to that of one normal laser. Note that SRS is considerably suppressed by one beam with $\Delta\omega_0 = 3\%\omega_1$, and the same mitigation effect is found at a larger bandwidth $\Delta\omega_0 = 5\%\omega_1$ for two beams, where their distributions are almost overlapped and no hot electrons can be found at this time. Therefore, polychromatic light with bandwidth $\Delta\omega_0 = 3\%\omega_1$ is enough to reduce the hot electron production even in the non-linear regime.