Stimulated Raman Scattering of X-Mode Laser in a Plasma Channel

Abstract

Stimulated Raman forward scattering (SRFS) of an intense X-mode laser pump in a preformed parabolic plasma density profile is investigated. The laser pump excites a plasma wave and one/two electromagnetic sideband waves. In Raman forward scattering, the growth rate of the parametric instability scales as two-third powers of the pump amplitude and increases linearly with cyclotron frequency.

1. Introduction

The propagation of intense laser pulses in plasma [1, 2] is relevant to various applications including laser-driven acceleration [3–5], optical harmonic generation [6, 7], X-ray laser [8], laser fusion [9, 10], and magnetic field generation [11]. The laser-plasma interaction leads to a number of relativistic and nonlinear effects such as self-focusing [12–14], parametric Raman and Brillouin instabilities [15], filamentation and modulational instabilities [16–24], and soliton formation.

Stimulated Raman scattering (SRS) is an important parametric instability in plasmas. In stimulated Raman forward scattering (SRFS), a high phase velocity Langmuir wave is produced that can accelerate electrons to high energy [25–35]. In stimulated Raman backward scattering (SRBS), electron plasma wave (EPW) has smaller phase velocity and can cause bulk heating of electrons. The high-amplitude laser wave enables the development of this instability in plasmas with a density significantly higher than the quarter critical density (n_c/4), which is usually considered as the SRS density limit for lower laser intensities. Shvets et al. [36] have demonstrated that SRS is strongly modified in a plasma channel, where the plasma frequency varies radially through the radial dependence of the plasma density n ₀(r). Liu et al. [37] have studied forward and backward Raman instabilities of a strong but nonrelativistic laser pump in a preformed plasma channel in the limit when plasma thermal effects may be neglected. Mori et al. [38] have developed an elegant formalism of SRFS in one dimension (1D). Panwar et al. [39] have studied SRFS of an intense pulse in a preformed plasma channel with a sequence of two pulses. They have studied the guiding of the main laser pulse through the plasma channel created by two lasers. Sajal et al. [40] have studied SRFS of a relativistic laser pump in a self-created plasma channel. Hassoon et al. [41] have studied the effect of a transverse static magnetic field on SRFS of a laser in a plasma. The X-mode excites an upper hybrid wave and two sidebands. They found that the growth rate of SRFS increases with the magnetic field. Gupta et al. [42] have investigated SRS of laser in a plasma with energetic drifting electrons generated during laser-plasma interaction. They showed numerically that the relativistic effects increase the growth rate of the Raman instability and enhance the amplitude of the decay waves significantly.

Liu et al. [43] developed 1D Vlasov–Maxwell numerical simulation to examine RBS instability in unmagnetized collisional plasma. Their results showed that RBS is enhanced by electron-ion collisions. Kalmykov et al. [44] have studied RBS in a plasma channel with a radial variation of plasma frequency. Paknezhad et al. [45] have investigated RBS of ultrashort laser pulse in a homogeneous cold underdense magnetized plasma by taking into account the relativistic effect of nonlinearity up to third order. The plasma is embedded in a uniform magnetic field. Kaur and Sharma [46] developed nonlocal theory of the SRBS in the propagation of a circularly polarized laser pulse through a hot plasma channel in the presence of a strong axial magnetic field. They established that the growth rate of SRBS of a finite spot size significantly decreases by increasing the magnetic field. Paknezhad et al. [47] have studied the Raman shifted third harmonic backscattering of an intense extraordinary laser wave through a homogeneous transversely magnetized cold plasma. Due to relativistic nonlinearity, the plasma dynamics is modified in the presence of transverse magnetic field, and this can generate the third harmonic scattered wave and electrostatic upper hybrid wave via the Raman scattering process.

In this paper, we examine the SRFS of an X-mode laser pump in a magnetized plasma channel including nonlocal effects. In many experiments in high-power laser-plasma interaction, transverse magnetic fields are self-generated [11]. Laser launched from outside travels in X-mode in such magnetic fields and often creates parabolic density profiles. Thus, the current problem is relevant to experimental situations.

The ponderomotive force due to the front of laser pulse pushes the electrons radially outward on the time scale of a plasma period ${\omega }_p^{-1}$ , creating a radial space charge field, and modifies the electron density, where ω_p is the electron plasma frequency. Laser and the sidebands exert a ponderomotive force on electrons driving the plasma wave. The density perturbation due to plasma wave beats with the oscillatory velocity due to laser pump to produce nonlinear currents, driving the sidebands.

In Section 2, we analyse the SRFS of a laser pump in a preformed channel with nonlocal effects. In Section 3, we discuss our results.

2. Raman Forward Scattering

Consider a two-dimensional plasma channel with a parabolic density profile immersed in a static magnetic field $B_s\hat{z}$ . The electron plasma frequency in the channel varies as

(1) $\begin{array}{}{\omega }_p^2={\omega }_{p0}^2\left(1+\frac{y^2}{L_n^2}\right),\end{array}$

where ω_p0 is the plasma frequency at y = 0 and L_n is the density scale length. An X-mode laser propagates through the channel (cf. Figure 1),

(2) $\begin{array}{}{E}_0=A_0\left(y\right)\left(\hat{y}-\frac{{\epsilon }_{0xy}}{{\epsilon }_{0xx}}\hat{x}\right)e^{-i\left({\omega }_{0}t-k_{0x}x\right)}.\end{array}$

Figure 1: Schematic of excitation of a Langmuir wave and two sidebands by an X-mode laser in a plasma channel.

The plasma permittivity at ω ₀ is a tensor. Its components are

(3) $\begin{array}{c}{\epsilon }_{0xx}={\epsilon }_{0yy}=1-\frac{{\omega }_p^2}{\left({\omega }_0^2-{\omega }_c^2\right)},{\epsilon }_{0xy}=-{\epsilon }_{0yx}=i\frac{{\omega }_c}{{\omega }_0}\frac{{\omega }_p^2}{\left({\omega }_0^2-{\omega }_c^2\right)},\\ {\epsilon }_{0zz}=1-\frac{{\omega }_p^2}{{\omega }_0^2},\\ {\epsilon }_{0xz}={\epsilon }_{0zx}={\epsilon }_{0yz}={\epsilon }_{0zy}=0,\end{array}$

where ${\omega }_c=\left(e{B}_s/m\right)$ is electron cyclotron frequency and e and m are the electronic charge and mass, respectively. Maxwell’s equations $\nabla \times E_0=i{\omega }_0{\mu }_0{H}_0$ and $\nabla \times H_0=-i{\omega }_0{\epsilon }_s\underset{=0}{\epsilon }\cdot E_0$ combine to give the wave equation

(4) $\begin{array}{}{\nabla }^2{E}_0-\nabla \left(\nabla \cdot E_0\right)+\frac{{\omega }_0^2}{C^2}\underset{=}{{\epsilon }_0}\cdot E_0=0,\end{array}$

where ε_s is the free space permittivity. Here, we chosen the plasma to be uniform and expressed the spatial-temporal variation of E₀ as $e^{-i\left({\omega }_{0}t-k_{0x}x-k_{0y}y\right)}$ we would obtain from equation (4), on replacing ∇ by $\left(i{k}_{0y}\hat{y}+i{k}_{0x}\hat{x}\right),$

(5) $\begin{array}{}\underset{=}{D_0}\cdot E_0=0,\end{array}$

where $\underset{=}{D_0}=\left({\omega }_0^2/c^2\right)\underset{=}{{\epsilon }_0}-k_0^2\underset{=}{I}+k_0{k}_0$ . Its matrix form is expressed as follows:

(6) $\begin{array}{}\underset{=0}{D}=\left(\begin{array}{ccc}\frac{{\omega }_0^2}{c^2}{\epsilon }_{0xx}-k_{0y}^2& \frac{{\omega }_0^2}{c^2}{\epsilon }_{0xy}+k_{0x}{k}_{0y}& 0\\ -\frac{{\omega }_0^2}{c^2}{\epsilon }_{0xy}+k_{0x}{k}_{0y}& \frac{{\omega }_0^2}{c^2}{\epsilon }_{0xx}-k_0^2& 0\\ 0& 0& \frac{{\omega }_0^2}{c^2}{\epsilon }_{0zz}-k_0^2\end{array}\right).\end{array}$

The local dispersion relation of the mode is given by $\left(\underset{=}{D_0}\right)=0$ , which can be written as

(7) $\begin{array}{}{k}_{0y}^2=\frac{{\omega }_0^2}{c^2}\frac{{\epsilon }_{0xx}^2+{\epsilon }_{0xy}^2}{{\epsilon }_{0xx}}-k_{0x}^2=\frac{{\omega }_0^2}{c^2}\left(1-\frac{{\omega }_p^2}{{\omega }_0^2}\frac{{\omega }_0^2-{\omega }_p^2}{{\omega }_0^2-{\omega }_p^2-{\omega }_c^2}\right)-k_{0x}^2.\end{array}$

Equation (2) gives $E_{0x}=-\left({\epsilon }_{0xy}/{\epsilon }_{0xx}\right)E_{0y}$ . To incorporate nonlocal effects, one may deduce the mode structure equation from the above equation by replacing k _0y by $\left(-i\partial /\partial y\right)$ and operating over E _0y,

(8) $\begin{array}{}\frac{{\partial }^2{E}_{0y}}{\partial y^2}+\left(\frac{{\omega }_0^2}{c^2}\left(1-\frac{{\omega }_p^2}{{\omega }_0^2}\frac{{\omega }_0^2-{\omega }_p^2}{{\omega }_0^2-{\omega }_p^2-{\omega }_c^2}\right)-k_{0x}^2\right)E_{0y}=0.\end{array}$

Define

(9) $\begin{array}{c}{\beta }_0^{\left(1/4\right)}y={\xi }_0,{\beta }_0=\frac{{\omega }_{p0}^2}{{\omega }_0^2}\left(1+\frac{{\omega }_c^2\left({\omega }_0^2-{\omega }_c^2\right)}{{\left({\omega }_0^2-{\omega }_{p0}^2-{\omega }_c^2\right)}^2}\right)\frac{{\omega }_0^2}{L_n^2{c}^2},\\ {\lambda }_0=\left(\frac{{\omega }_0^2}{c^2}\left(1-\frac{{\omega }_{p0}^2}{{\omega }_0^2}\frac{{\omega }_0^2-{\omega }_{p0}^2}{{\omega }_0^2-{\omega }_{p0}^2-{\omega }_c^2}\right)-k_{0x}^2\right)\frac{1}{{\beta }_0^{\left(1/2\right)}}.\end{array}$

Equation (8) can be written as

(10) $\begin{array}{}\frac{{\partial }^2{E}_{0y}}{\partial {\xi }_0^2}+\left({\lambda }_0-{\xi }_0^2\right)E_{0y}=0.\end{array}$

The eigenvalue for the fundamental mode is λ₀ = 1, and the eigenfunction is given by $E_{0y}=A_0{e}^{-{\xi }_0^2/2}$ . From the eigenvalue condition, we obtain

(11) $\begin{array}{}{k}_{0x}^2=\frac{{\omega }_0^2}{c^2}\left(1-\frac{{\omega }_{p0}^2}{{\omega }_0^2}\frac{{\omega }_0^2-{\omega }_{p0}^2}{{\omega }_0^2-{\omega }_{p0}^2-{\omega }_c^2}\right)-{\beta }_0^{\left(1/2\right)}.\end{array}$

The pump wave produces oscillatory electron velocity,

(12) $\begin{array}{c}{v}_{0x}=-\frac{e}{m}\frac{{\omega }_c{E}_{0y}}{\left({\omega }_0^2-{\omega }_c^2-{\omega }_p^2\right)},v_{0y}=-i\frac{e}{m}\frac{\left({\omega }_0^2-{\omega }_p^2\right)E_{0y}}{{\omega }_0\left({\omega }_0^2-{\omega }_c^2-{\omega }_p^2\right)},\\ v_{0z}=0,\end{array}$

and parametrically excites a Langmuir wave with electrostatic potential,

(13) $\begin{array}{}\varphi =\varphi \left(y\right)e^{-i\left(\omega t-kx\right)},\end{array}$

and two electromagnetic sidebands with electric fields,

(14) $\begin{array}{}{E}_j=A_j\left(\hat{y}-\frac{{\epsilon }_{1xy}}{{\epsilon }_{1xx}}\hat{x}\right)e^{-i\left({\omega }_{j}t-k_{j}x\right)},\end{array}$

where j = 1,2 and ω_1,2 = ω ∓ ω₀, and k _1,2 = k ∓ k₀. The sideband waves produce oscillatory electron velocities,

(15) $\begin{array}{c}{v}_{jx}=-\frac{e}{m}\frac{{\omega }_c{E}_{jy}}{\left({\omega }_j^2-{\omega }_c^2-{\omega }_p^2\right)},v_{jy}=-i\frac{e}{m}\frac{\left({\omega }_j^2-{\omega }_p^2\right)E_{jy}}{{\omega }_j\left({\omega }_j^2-{\omega }_c^2-{\omega }_p^2\right)},\\ v_{jz}=0,\end{array}$

where j = 1,2.

The sideband waves couple with the pump to exert a ponderomotive force on electrons at ω, k, $F_{p\omega }=-\left(m/2\right)\left(v_0\cdot \nabla v_1+v_0^{\ast }\cdot \nabla v_2+v_1\cdot \nabla v_0+v_2\cdot \nabla v_0^{\ast }\right)-\left(e/2\right)\left(v_0\times B_1+v_1\times B_0+v_0^{\ast }\times B_2+v_2\times B_0^{\ast }\right).$ In the limit, $k_0\gg k$ , k₁ and k₂ are largely along $\hat{x}$ , and the x and y components of the ponderomotive force turn out to be

(16) $\begin{array}{c}{F}_{p\omega x}=\frac{e^{2}i\left(P{E}_{0y}{E}_{1y}+Q{E}_{0y}\ast E_{2y}\right)}{2m{\omega }_0{\omega }_1{\omega }_2\left({\omega }_0^2-{\omega }_c^2-{\omega }_p^2\right)\left({\omega }_1^2-{\omega }_c^2-{\omega }_p^2\right)},F_{p\omega y}=\frac{e^2{\omega }_c\left(R{E}_{0y}{E}_{1y}+S{E}_{0y}\ast E_{2y}\right)}{2m{\omega }_0{\omega }_1{\omega }_2\left({\omega }_0^2-{\omega }_c^2-{\omega }_p^2\right)\left({\omega }_1^2-{\omega }_c^2-{\omega }_p^2\right)},\end{array}$

where

(17) $\begin{array}{c}P={\omega }_2\left(\left(k_1-k_0\right){\omega }_1{\omega }_0{\omega }_c^2+k_1\left({\omega }_0^2-{\omega }_p^2\right)\left({\omega }_1^2-{\omega }_c^2-{\omega }_p^2\right)-k_0\left({\omega }_1^2-{\omega }_p^2\right)\left({\omega }_0^2-{\omega }_c^2-{\omega }_p^2\right)\right),Q={\omega }_1\left(\left(k_0-k_2\right){\omega }_2{\omega }_0{\omega }_c^2-k_2\left({\omega }_0^2-{\omega }_p^2\right)\left({\omega }_2^2-{\omega }_c^2-{\omega }_p^2\right)+k_0\left({\omega }_2^2-{\omega }_p^2\right)\left({\omega }_0^2-{\omega }_c^2-{\omega }_p^2\right)\right),\\ R={\omega }_2\left(k_0{\omega }_1+k_1{\omega }_0\right){\omega }_c^2,\\ S={\omega }_1\left(k_0{\omega }_2+k_2{\omega }_0\right){\omega }_c^2,\end{array}$

For $\omega \ll {\omega }_0$ , $F_{p\omega }=e\nabla {\phi }_p$ ; one may write

(18) $\begin{array}{c}{\varphi }_p=V_{01}\left(E_{1y}-\frac{{\eta }_1}{{\eta }_2}{E}_{2y}\right),V_{01}=\frac{e{E}_{0y}\left({\omega }_1\left({\omega }_0^2-{\omega }_p^2\right)\left({\omega }_1^2-{\omega }_c^2-{\omega }_p^2\right)-{\omega }_c^2{\omega }_p^2{\omega }_0\right)}{2m{\omega }_0{\omega }_1^2\left({\omega }_0^2-{\omega }_c^2-{\omega }_p^2\right)\left({\omega }_1^2-{\omega }_c^2-{\omega }_p^2\right)},\end{array}$

where

(19) $\begin{array}{c}{\eta }_1={\omega }_1^2\left({\omega }_1^2-{\omega }_c^2-{\omega }_p^2\right)\left({\omega }_2\left({\omega }_0^2-{\omega }_p^2\right)\left({\omega }_2^2-{\omega }_c^2-{\omega }_p^2\right)+{\omega }_0{\omega }_c^2{\omega }_p^2\right),{\eta }_2={\omega }_2^2\left({\omega }_2^2-{\omega }_c^2-{\omega }_p^2\right)\left({\omega }_1\left({\omega }_0^2-{\omega }_p^2\right)\left({\omega }_1^2-{\omega }_c^2-{\omega }_p^2\right)-{\omega }_0{\omega }_c^2{\omega }_p^2\right).\end{array}$

The drift velocity of plasma electrons due to electrostatic wave of potential φ in the component form is given by

(20) $\begin{array}{c}{v}_{\omega x}=-\frac{e}{m}\frac{\omega k}{\left({\omega }^2-{\omega }_c^2\right)}\varphi ,v_{\omega y}=-i\frac{e}{m}\frac{{\omega }_{c}k}{\left({\omega }^2-{\omega }_c^2\right)}\varphi ,\\ v_{\omega z}=0.\end{array}$

The nonlinear velocity $v_{\omega }^{NL}$ due to ponderomotive force can be obtained from equation (20) by replacing φ by φ_p.

Using the equation of continuity, the linear and nonlinear density perturbations at ω, k can be obtained as

(21) $\begin{array}{c}{n}_{\omega }^L=\frac{\nabla \cdot \left(n_0^0\left(y\right)v_{\omega }\right)}{i\omega },n_{\omega }^{NL}=\frac{\nabla \cdot \left(n_0^0\left(y\right)v_{\omega }^{NL}\right)}{i\omega }.\end{array}$

Using these in Poisson’s equation, ${\nabla }^2\phi =\left(e/{\epsilon }_0\right)\left(n_{\omega }^L+n_{\omega }^{NL}\right),$ we obtain

(22) $\begin{array}{}\varphi =-\frac{{\omega }_p^2\left(y\right)}{\left({\omega }^2-{\omega }_c^2-{\omega }_p^2\left(y\right)\right)}{\varphi }_p.\end{array}$

The nonlinear current densities at the sidebands can be written as

(23) $\begin{array}{c}{J}_1^{NL}=-\frac{1}{2}ne{v}_0^{\ast },J_2^{NL}=-\frac{1}{2}ne{v}_0.\end{array}$

Using equation (23) in the wave equation, we obtain

(24) $\begin{array}{}\underset{=1}{D}\cdot E_1=\left(-k_1^2\underset{=}{I}+k_1{k}_1+\frac{{\omega }_1^2}{c^2}\underset{=}{{\epsilon }_1}\right)\cdot E_1=\frac{k^{2}e\phi {\omega }_p^2{\omega }_1{E}_{0y}}{2m{c}^2{\omega }_0\left({\omega }^2-{\omega }_c^2\right)\left({\omega }_0^2-{\omega }_c^2-{\omega }_p^2\right)}\left(i{\omega }_0{\omega }_c\hat{x}+\left({\omega }_0^2-{\omega }_p^2\right)\hat{y}\right),\end{array}$

where ${\underset{=}{D}}_1=-k_1^2\underset{=}{I}+k_1{k}_1+\left({\omega }_1^2/c^2\right)\underset{=}{{\epsilon }_1}$ , and

(25) $\begin{array}{}\underset{=2}{D}\cdot E_2=\left(-k_2^2\underset{=}{I}+k_2{k}_2+\frac{{\omega }_2^2}{c^2}\underset{=}{{\epsilon }_2}\right)\cdot E_2=\frac{k^{2}e\varphi {\omega }_p^2{\omega }_2{E}_{0y}}{2m{c}^2{\omega }_0\left({\omega }^2-{\omega }_c^2\right)\left({\omega }_0^2-{\omega }_c^2-{\omega }_p^2\right)}\left(-i{\omega }_0{\omega }_c\hat{x}+\left({\omega }_0^2-{\omega }_p^2\right)\hat{y}\right),\end{array}$

where $\underset{=2}{D}=-k_2^2\underset{=}{I}+k_2{k}_2+\left({\omega }_2^2/c^2\right)\underset{=}{{\epsilon }_2}.$

Equations (24) and (25) on replacing k_1y by $-i\partial /\partial y$ for the sidebands can be written as

(26) $\begin{array}{}\frac{d^2{E}_{1y}}{d{\xi }_1^2}+\left({\lambda }_1-{\xi }_1^2\right)E_{1y}=\frac{W_{01}^2{e}^{-{\xi }_1^2}}{{\xi }_{10}^2-{\xi }_1^2}\left(E_{1y}-\frac{{\eta }_1}{{\eta }_2}{E}_{2y}\right),\end{array}$

(27) $\begin{array}{}\frac{d^2{E}_{2y}}{d{\xi }_2^2}+\left({\lambda }_2-{\xi }_2^2\right)E_{2y}=\frac{W_{02}^2{e}^{-{\xi }_2^2}}{{\xi }_{20}^2-{\xi }_2^2}\left(E_{2y}-\frac{{\eta }_2}{{\eta }_1}{E}_{1y}\right),\end{array}$

(28) $\begin{array}{c}{\xi }_1^2={\beta }_1^{\left(1/2\right)}{y}^2,{\xi }_2^2={\beta }_2^{\left(1/2\right)}{y}^2,\\ {\xi }^2={\beta }_1^{\left(1/2\right)}{y}^2\approx {\beta }_2^{\left(1/2\right)}{y}^2=\frac{{\beta }_1^{\left(1/2\right)}}{{\beta }_0^{\left(1/2\right)}}{\xi }_0^2\approx {\xi }_0^2,\\ {\xi }_{10}^2=\frac{\left({\omega }^2-{\omega }_c^2-{\omega }_{p0}^2\right)L_n^2{\beta }_1^{\left(1/2\right)}}{{\omega }_{p0}^2},\\ {\xi }_{20}^2=\frac{\left({\omega }^2-{\omega }_c^2-{\omega }_{p0}^2\right)L_n^2{\beta }_2^{\left(1/2\right)}}{{\omega }_{p0}^2},\\ {\alpha }_1=\frac{{\omega }_1^2{\left({\omega }_1^2-{\omega }_c^2-{\omega }_{p0}^2\right)}^2-k_{1x}^2{c}^2\left({\omega }_1^2-{\omega }_c^2\right)\left({\omega }_1^2-{\omega }_c^2-{\omega }_{p0}^2\right)-{\omega }_c^2{\omega }_{p0}^4}{c^2\left({\omega }_1^2-{\omega }_c^2\right)\left({\omega }_1^2-{\omega }_c^2-{\omega }_{p0}^2\right)},\\ {\alpha }_2=\frac{{\omega }_2^2{\left({\omega }_2^2-{\omega }_c^2-{\omega }_{p0}^2\right)}^2-k_{2x}^2{c}^2\left({\omega }_2^2-{\omega }_c^2\right)\left({\omega }_2^2-{\omega }_c^2-{\omega }_{p0}^2\right)-{\omega }_c^2{\omega }_{p0}^4}{c^2\left({\omega }_2^2-{\omega }_c^2\right)\left({\omega }_2^2-{\omega }_c^2-{\omega }_{p0}^2\right)},\\ {\beta }_1=\frac{{\omega }_c^2{\omega }_{p0}^4\left(2{\omega }_1^2-2{\omega }_c^2-{\omega }_{p0}^2\right)+{\omega }_1^2{\omega }_{p0}^2{\left({\omega }_1^2-{\omega }_c^2-{\omega }_{p0}^2\right)}^2}{c^2\left({\omega }_1^2-{\omega }_c^2\right){\left({\omega }_1^2-{\omega }_c^2-{\omega }_{p0}^2\right)}^2{L}_n^2}\\ =\frac{{\omega }_{p0}^2}{{\omega }_1^2}\left(1+\frac{{\omega }_c^2\left({\omega }_1^2-{\omega }_c^2\right)}{{\left({\omega }_1^2-{\omega }_{p0}^2-{\omega }_c^2\right)}^2}\right)\frac{{\omega }_1^2}{L_n^2{c}^2},\\ {\beta }_2=\frac{{\omega }_c^2{\omega }_{p0}^4\left(2{\omega }_2^2-2{\omega }_c^2-{\omega }_{p0}^2\right)+{\omega }_2^2{\omega }_{p0}^2{\left({\omega }_2^2-{\omega }_c^2-{\omega }_{p0}^2\right)}^2}{c^2\left({\omega }_2^2-{\omega }_c^2\right){\left({\omega }_2^2-{\omega }_c^2-{\omega }_{p0}^2\right)}^2{L}_n^2}\\ =\frac{{\omega }_{p0}^2}{{\omega }_2^2}\left(1+\frac{{\omega }_c^2\left({\omega }_2^2-{\omega }_c^2\right)}{{\left({\omega }_2^2-{\omega }_{p0}^2-{\omega }_c^2\right)}^2}\right)\frac{{\omega }_2^2}{L_n^2{c}^2},\\ {\lambda }_1=\frac{{\alpha }_1}{{\beta }_1^{\left(1/2\right)}}=\left(\frac{{\omega }_1^2}{c^2}\left(1-\frac{{\omega }_{p0}^2}{{\omega }_1^2}\frac{{\omega }_1^2-{\omega }_{p0}^2}{{\omega }_1^2-{\omega }_{p0}^2-{\omega }_c^2}\right)-k_{0x}^2\right)\frac{1}{{\beta }_1^{\left(1/2\right)}},\\ {\lambda }_2=\frac{{\alpha }_2}{{\beta }_2^{\left(1/2\right)}}=\left(\frac{{\omega }_2^2}{c^2}\left(1-\frac{{\omega }_{p0}^2}{{\omega }_2^2}\frac{{\omega }_2^2-{\omega }_{p0}^2}{{\omega }_2^2-{\omega }_{p0}^2-{\omega }_c^2}\right)-k_{0x}^2\right)\frac{1}{{\beta }_2^{\left(1/2\right)}},\\ V_{01}=\frac{e{E}_{0y}\left({\omega }_1\left({\omega }_0^2-{\omega }_p^2\right)\left({\omega }_1^2-{\omega }_c^2-{\omega }_p^2\right)-{\omega }_c^2{\omega }_p^2{\omega }_0\right)}{2m{\omega }_0{\omega }_1^2\left({\omega }_0^2-{\omega }_c^2-{\omega }_p^2\right)\left({\omega }_1^2-{\omega }_c^2-{\omega }_p^2\right)},\\ V_{02}=-\frac{e{E}_{0y}\left({\omega }_1\left({\omega }_0^2-{\omega }_p^2\right)\left({\omega }_1^2-{\omega }_c^2-{\omega }_p^2\right)-{\omega }_c^2{\omega }_p^2{\omega }_0\right)}{2m{\omega }_0{\omega }_1^2\left({\omega }_0^2-{\omega }_c^2-{\omega }_p^2\right)\left({\omega }_1^2-{\omega }_c^2-{\omega }_p^2\right)}\frac{{\eta }_1}{{\eta }_2},\\ W_{01}^2=\frac{e{k}^2{E}_{0y}{L}_n^2{\omega }_p^4{V}_{01}\left({\omega }_p^2{\omega }_0{\omega }_c^2-{\omega }_1\left({\omega }_1^2-{\omega }_c^2-{\omega }_p^2\right)\left({\omega }_0^2-{\omega }_p^2\right)\right)}{2m{c}^2{\omega }_0{\omega }_{p0}^2\left({\omega }^2-{\omega }_c^2\right)\left({\omega }_1^2-{\omega }_c^2-{\omega }_p^2\right)\left({\omega }_0^2-{\omega }_c^2-{\omega }_p^2\right)},\\ W_{02}^2=\frac{e{k}^2{E}_{0y}{L}_n^2{\omega }_p^4{V}_{02}\left({\omega }_p^2{\omega }_0{\omega }_c^2+{\omega }_2\left({\omega }_2^2-{\omega }_c^2-{\omega }_p^2\right)\left({\omega }_0^2-{\omega }_p^2\right)\right)}{2m{c}^2{\omega }_0{\omega }_{p0}^2\left({\omega }^2-{\omega }_c^2\right)\left({\omega }_2^2-{\omega }_c^2-{\omega }_p^2\right)\left({\omega }_0^2-{\omega }_c^2-{\omega }_p^2\right)}.\end{array}$

We solve equations (26) and (27) by the first-order perturbation theory. In the absence of nonlinear terms, the eigenfunctions for the fundamental mode satisfying equations (26) and (27) are

(29) $\begin{array}{}{E}_{\mathrm{1,2}}=A_{\mathrm{1,2}}{e}^{-{\xi }_1^2/2}.\end{array}$

Corresponding eigenvalues are λ_1,2 = 1. When the RHS of equations (26) and (27) are finite, we assume the eigenfunctions to remain unmodified; only the eigenvalues are changed a little. We substitute for E _1,2 from equation (29) in equations (26) and (27), multiply the resulting equations by $e^{-{\xi }^2/2}$ , and integrate over ξ from −∞ to ∞. Then, we obtain

(30) $\begin{array}{}\left({\lambda }_1-1\right)A_1=-W_{01}^2\frac{Z\left({\xi }_{10}\sqrt{2}\right)}{{\xi }_{10}}\left(A_1-\frac{{\eta }_1}{{\eta }_2}{A}_2\right),\end{array}$

(31) $\begin{array}{}\left({\lambda }_2-1\right)A_2=-W_{02}^2\frac{Z\left({\xi }_{10}\sqrt{2}\right)}{{\xi }_{10}}\left(A_2-\frac{{\eta }_2}{{\eta }_1}{A}_1\right),\end{array}$

where $Z\left({\xi }_{10}\sqrt{2}\right)$ is the plasma dispersion function. Equations (30) and (31) give the nonlinear dispersion relation

(32) $\begin{array}{}1=-\frac{Z\left({\xi }_{10}\sqrt{2}\right)}{{\xi }_{10}}\left(\frac{W_{01}^2}{{\lambda }_1-1}+\frac{W_{02}^2}{{\lambda }_2-1}\right).\end{array}$

This nonlinear dispersion relation is a function of the DC magnetic field. The dispersion relation is modified if one changes the value of the DC magnetic field. In the absence of the DC magnetic field,

(33) $\begin{array}{}1=-\frac{Z\left({\xi }_{10}\left({\omega }_c=0\right)\sqrt{2}\right)}{{\xi }_{10}\left({\omega }_c=0\right)}\left(\frac{W_{01}^2\left({\omega }_c=0\right)}{{\lambda }_1\left({\omega }_c=0\right)-1}+\frac{W_{02}^2\left({\omega }_c=0\right)}{{\lambda }_2\left({\omega }_c=0\right)-1}\right),\end{array}$

where ${\beta }_1^{1/4}\left({\omega }_c=0\right)y={\xi }_1\left({\omega }_c=0\right),$ ${\beta }_2^{1/4}\left({\omega }_c=0\right)y={\xi }_2\left({\omega }_c=0\right),$ ${\beta }_0^{1/4}\left({\omega }_c=0\right)y={\xi }_0\left({\omega }_c=0\right),$

(34) $\begin{array}{c}{\xi }_{10}\left({\omega }_c=0\right)=\frac{\left({\omega }^2-{\omega }_{p0}^2\right)L_n^2{\beta }_1^{1/2}\left({\omega }_c=0\right)}{{\omega }_{p0}^2},{\lambda }_1\left({\omega }_c=0\right)=\left(\frac{{\omega }_1^2}{c^2}\left(1-\frac{{\omega }_{p0}^2}{{\omega }_1^2}\frac{{\omega }_1^2-{\omega }_{p0}^2}{{\omega }_1^2-{\omega }_{p0}^2}\right)-k_{0x}^2\right)\frac{1}{{\beta }_1^{1/2}\left({\omega }_c=0\right)},\\ {\lambda }_2\left({\omega }_c=0\right)=\left(\frac{{\omega }_2^2}{c^2}\left(1-\frac{{\omega }_{p0}^2}{{\omega }_2^2}\frac{{\omega }_2^2-{\omega }_{p0}^2}{{\omega }_2^2-{\omega }_{p0}^2}\right)-k_{0x}^2\right)\frac{1}{{\beta }_2^{1/2}\left({\omega }_c=0\right)},\\ {\beta }_1\left({\omega }_c=0\right)=\frac{{\omega }_{p0}^2}{L_n^2{c}^2}={\beta }_2\left({\omega }_c=0\right)={\beta }_0\left({\omega }_c=0\right),\\ V_{01}\left({\omega }_c=0\right)=\frac{e{E}_{0y}\left({\omega }_1\left({\omega }_0^2-{\omega }_p^2\right)\left({\omega }_1^2-{\omega }_p^2\right)\right)}{2m{\omega }_0{\omega }_1^2\left({\omega }_0^2-{\omega }_p^2\right)\left({\omega }_1^2-{\omega }_p^2\right)},\\ V_{02}\left({\omega }_c=0\right)=-\frac{e{E}_{0y}\left({\omega }_1\left({\omega }_0^2-{\omega }_p^2\right)\left({\omega }_1^2-{\omega }_p^2\right)\right)}{2m{\omega }_0{\omega }_1^2\left({\omega }_0^2-{\omega }_p^2\right)\left({\omega }_1^2-{\omega }_p^2\right)}\frac{{\eta }_1\left({\omega }_c=0\right)}{{\eta }_2\left({\omega }_c=0\right)},\\ {\eta }_1\left({\omega }_c=0\right)={\omega }_1^2\left({\omega }_1^2-{\omega }_p^2\right)\left({\omega }_2\left({\omega }_0^2-{\omega }_p^2\right)\left({\omega }_2^2-{\omega }_p^2\right)\right),\\ {\eta }_2\left({\omega }_c=0\right)={\omega }_2^2\left({\omega }_2^2-{\omega }_p^2\right)\left({\omega }_1\left({\omega }_0^2-{\omega }_p^2\right)\left({\omega }_1^2-{\omega }_p^2\right)\right),\\ W_{01}^2\left({\omega }_c=0\right)=\frac{e{k}^2{E}_{0y}{L}_n^2{\omega }_p^4{V}_{01}\left({\omega }_c=0\right)\left(-{\omega }_1\left({\omega }_1^2-{\omega }_p^2\right)\left({\omega }_0^2-{\omega }_p^2\right)\right)}{2m{c}^2{\omega }_0{\omega }_{p0}^2{\omega }^2\left({\omega }_1^2-{\omega }_p^2\right)\left({\omega }_0^2-{\omega }_p^2\right)},\\ W_{02}^2\left({\omega }_c=0\right)=\frac{e{k}^2{E}_{0y}{L}_n^2{\omega }_p^4{V}_{02}\left({\omega }_c=0\right)\left({\omega }_2\left({\omega }_2^2-{\omega }_p^2\right)\left({\omega }_0^2-{\omega }_p^2\right)\right)}{2m{c}^2{\omega }_0{\omega }_{p0}^2{\omega }^2\left({\omega }_2^2-{\omega }_p^2\right)\left({\omega }_0^2-{\omega }_p^2\right)}.\end{array}$

Taylor expanding λ₁ − 1 with respect to variables ω₁, k ₁ and λ₂ − 1 with respect to variables ω₂,k ₂ and substituting in equation (32), one obtains

(35) $\begin{array}{}1=-\frac{Z\left({\xi }_{10}\sqrt{2}\right)}{{\xi }_{10}}\frac{W_{01}^2}{\left(\partial {\lambda }_0/\partial {\omega }_0\right)}\left(\frac{1}{\left(\omega -k{v}_{g0}+\Delta \right)}-\frac{1}{\left(\omega -k{v}_{g0}-\Delta \right)}\right),\end{array}$

where

(36) $\begin{array}{c}\frac{\partial {\omega }_0}{\partial k_0}=-\frac{\partial {\lambda }_0/\partial k_0}{\partial {\lambda }_0/\partial {\omega }_0}=v_{g0},\Delta =\frac{{\Delta }^{\prime }}{2\left(\partial {\lambda }_0/\partial {\omega }_0\right)},\\ {\Delta }^{\prime }=\frac{{\partial }^2{\lambda }_0}{\partial {\omega }_0^2}{\omega }^2+\frac{{\partial }^2{\lambda }_0}{\partial k_0^2}{k}^2,\\ \frac{\partial {\lambda }_0}{\partial {\omega }_0}=\frac{\partial \left({\lambda }_0-1\right)}{\partial {\omega }_0}\cong \frac{1}{\sqrt{{\beta }_0}}\left(2\frac{{\omega }_0}{c^2}+\frac{2{\omega }_{p0}^2}{c^2}\frac{{\Delta }_1}{{\Delta }_2}-\frac{{\omega }_{p0}}{L_{n}c}\frac{{\Delta }_3}{{\Delta }_4}\right),\\ \frac{{\partial }^2{\lambda }_0}{\partial {\omega }_0^2}=\frac{1}{\sqrt{{\beta }_0}}\left(\frac{2}{c^2}+\frac{2{\omega }_{p0}^2}{c^2}\frac{{\Delta }_2{d}_1-{\Delta }_1{d}_2}{{\Delta }_2^2}-\frac{{\omega }_{p0}}{L_{n}c}\frac{{\Delta }_4{d}_3-{\Delta }_3{d}_4}{{\Delta }_4^2}\right),\\ \frac{\partial {\lambda }_0}{\partial k_0}=-\frac{2k_0}{\sqrt{{\beta }_0}},\\ \frac{{\partial }^2{\lambda }_0}{\partial k_0^2}=-\frac{2}{\sqrt{{\beta }_0}},\\ {\Delta }_1={\omega }_0\left(\left({\omega }_0^2-{\omega }_{p0}^2-{\omega }_c^2\right)\left({\omega }_{p0}^2-2{\omega }_0^2\right)-\left({\omega }_{p0}^2-{\omega }_0^2\right)\left(2{\omega }_0^2-{\omega }_{p0}^2-{\omega }_c^2\right)\right),\\ {\Delta }_2={\omega }_0^2{\left({\omega }_0^2-{\omega }_{p0}^2-{\omega }_c^2\right)}^2,\\ {\Delta }_3=-{\omega }_0{\omega }_c^2\left({\omega }_0^2-{\omega }_{p0}^2-{\omega }_c^2\right),\\ {\Delta }_4={\left({\left({\omega }_0^2-{\omega }_{p0}^2-{\omega }_c^2\right)}^2+{\omega }_c^2\left({\omega }_0^2-{\omega }_c^2\right)\right)}^{1/2}{\left({\omega }_0^2-{\omega }_{p0}^2-{\omega }_c^2\right)}^2,\\ \frac{\partial {\Delta }_1}{\partial {\omega }_0}=d_1=\left({\omega }_0^2-{\omega }_{p0}^2-{\omega }_c^2\right)\left({\omega }_{p0}^2-6{\omega }_0^2\right)-2{\omega }_0^2{\omega }_{p0}^2-\left(2{\omega }_0^2-{\omega }_{p0}^2-{\omega }_c^2\right)\left({\omega }_{p0}^2-3{\omega }_0^2\right),\\ \frac{\partial {\Delta }_2}{\partial {\omega }_0}=d_2=2{\omega }_0{\left({\omega }_0^2-{\omega }_{p0}^2-{\omega }_c^2\right)}^2,\\ \frac{\partial {\Delta }_3}{\partial {\omega }_0}=d_3=-{\omega }_c^2\left(3{\omega }_0^2+{\omega }_{p0}^2-{\omega }_c^2\right),\\ \frac{\partial {\Delta }_4}{\partial {\omega }_0}=d_4=4{\omega }_0\left({\omega }_0^2-{\omega }_{p0}^2-{\omega }_c^2\right){\left({\left({\omega }_0^2-{\omega }_{p0}^2-{\omega }_c^2\right)}^2+{\omega }_c^2\left({\omega }_0^2-{\omega }_c^2\right)\right)}^{1/2}\\ +{\omega }_0{\left({\omega }_0^2-{\omega }_{p0}^2-{\omega }_c^2\right)}^2\left({\omega }_0^2-{\omega }_{p0}^2\right){\left({\left({\omega }_0^2-{\omega }_{p0}^2-{\omega }_c^2\right)}^2+{\omega }_c^2\left({\omega }_0^2-{\omega }_c^2\right)\right)}^{-1/2}.\end{array}$

Equation (35) gives

(37) $\begin{array}{}{\left(\omega -k{v}_{g0}\right)}^2-{\Delta }^2=\frac{2\Delta W_{01}^{2}Z\left({\xi }_{10}\sqrt{2}\right)}{\left(\partial {\lambda }_0/\partial {\omega }_0\right){\xi }_{10}}.\end{array}$

We substitute the value of ξ₁₀ and $Z\left({\xi }_{10}\sqrt{2}\right)\approx -\sqrt{2}$ , also write $\omega ={\omega }_r+i\gamma$ , where ${\omega }_r=k{v}_{g0}=\sqrt{\left({\omega }_c^2+{\omega }_{p0}^2\right)}$ , and obtain

(38) $\begin{array}{}{\left(-1\right)}^{1/2}{\gamma }^3=\frac{2\sqrt{2}{W}_{01}^2{\omega }_{p0}\Delta }{\left(\partial {\lambda }_0/\partial {\omega }_0\right)L_n{\beta }_1^{1/4}}{e}^{2il\pi }.\end{array}$

Equation (38) gives the growth rate

(39) $\begin{array}{}\gamma ={\left(\frac{2\sqrt{2}{W}_{01}^2{\omega }_{p0}\Delta }{\left(\partial {\lambda }_0/\partial {\omega }_0\right)L_n{\beta }_1^{1/4}}\right)}^{1/3}\frac{\sin 2l\pi }{3}.\end{array}$

For l = 1, the growth rate turns out to be

(40) $\begin{array}{}\gamma =\frac{\sqrt{3}}{2}{\left(\frac{2\sqrt{2}{W}_{01}^2{\omega }_{p0}\Delta }{\left(\partial {\lambda }_0/\partial {\omega }_0\right)L_n{\beta }_1^{1/4}}\right)}^{1/3}.\end{array}$

In Figure 2, we have displayed the normalized growth rate $\left(\gamma /{\omega }_{p0}\right)$ of the Raman forward scattering instability as a function of normalized pump wave amplitude $\left(e{E}_0/m\omega c\right)$ for parameters: $\left({\omega }_{p0}/{\omega }_0\right)=0.2$ , $\left(L_n{\omega }_{p0}/c\right)=8$ , and $\left({\omega }_c/{\omega }_{p0}\right)=1$ . The growth rate increases with the amplitude of the laser pump. It does not go linearly but nearly as two-third powers of normalized laser amplitude. This is due to the fact that the width of the driven Langmuir mode is dependent on ponderomotive potential hence on pump amplitude. In Figure 3, we have displayed the normalized growth rate $\left(\gamma /{\omega }_{p0}\right)$ of the Raman forward scattering as a function of normalized static magnetic field $\left({\omega }_c/{\omega }_{p0}\right)$ for $\left({\omega }_{p0}/{\omega }_0\right)=0.2$ , $\left(L_n{\omega }_{p0}/c\right)=8$ , and $\left(e{E}_0/m\omega c\right)=0.1$ . In the absence of DC magnetic field, the growth rate $\left(\gamma /{\omega }_{p0}\right)$ is minimum. It rises with magnetic field. The magnetic field raises the frequency of the driven plasma wave and brings in cyclotron effects in nonlinear coupling leading to enhancement of growth rate.

Figure 2: Normalized growth rate as a function of normalized amplitude of the pump wave in Raman forward scattering for (ω_p0/ω₀) = 0.2, $\left(L_n{\omega }_{p0}/c\right)=8$ , and $\left({\omega }_c/{\omega }_{p0}\right)=1$ .

Figure 3: Normalized growth rate as a function of normalized cyclotron frequency for $\left({\omega }_{p0}/{\omega }_0\right)=0.2$ , $\left(L_n{\omega }_{p0}/c\right)=8$ , and $\left(e{E}_0/m\omega c\right)=0.1$ .

3. Discussion

The plasma channel with a parabolic density profile localizes the electromagnetic eigenmodes involved in the SRFS process within a width of the order ${\left(c{L}_n/{\omega }_{p0}\right)}^{1/2}$ . The Langmuir wave is more strongly localized, thus limiting the region of parametric interaction and reducing the growth rate. The static magnetic field modifies the electron response to these eigenmodes and significantly influences the nonlinear coupling. In the limit when the normalized growth rate $\left(\gamma /{\omega }_{p0}\right)>k{v}_{th}$ , one may neglect thermal effects. The growth rate roughly scales as $E_{0y}^{2/3}$ with pump amplitude and goes linearly with ambient magnetic field. For typical parameters, $\left({\omega }_{p0}/{\omega }_0\right)=0.2$ , $\left(L_n{\omega }_{p0}/c\right)=8$ , $\left({\omega }_c/{\omega }_{p0}\right)=0.5,$ and $\left(e{E}_0/m\omega c\right)=0.1,$ the normalized growth rate is $0.26$ whereas for $\left({\omega }_{p0}/{\omega }_0\right)=0.2$ , $\left(L_n{\omega }_{p0}/c\right)=8$ , $\left({\omega }_c/{\omega }_{p0}\right)=1,$ and $\left(e{E}_0/m\omega c\right)=0.8,$ the normalized growth rate is $0.028.$

In the earlier work by Liu et al. [37] for the parameters $\left({\omega }_{p0}/{\omega }_0\right)=0.2$ and $\left(L_n{\omega }_{p0}/c\right)=8$ $\left(v_0/c\right)=0.6,$ the normalized growth rate is $0.06$ whereas in our work for the parameters $\left({\omega }_{p0}/{\omega }_0\right)=0.2$ , $\left(L_n{\omega }_{p0}/c\right)=8$ , $\left({\omega }_c/{\omega }_{p0}\right)=1,$ and $\left(e{E}_0/m\omega c\right)=0.6,$ the normalized growth rate is $0.025$ Clearly, our results are in good agreement with the earlier work by Liu et al.