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

Authors: Sanjay Babu; Asheel Kumar; Ram Jeet; Arvind Kumar; Ashish Varma
Journal: Laser and Particle Beams
 (2021)

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## 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.


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## Body

## 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 nc/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 n0r. 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 periodωp−1, creating a radial space charge field, and modifies the electron density, where ωp 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 Section2, 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 fieldBsz^. The electron plasma frequency in the channel varies as(1)ωp2=ωp021+y2Ln2,where ωp0 is the plasma frequency at y=0 and Ln is the density scale length. An X-mode laser propagates through the channel (cf. Figure 1),(2)E0=A0yy^−ε0xyε0xxx^e−iω0t−k0xx.Figure 1
Schematic of excitation of a Langmuir wave and two sidebands by an X-mode laser in a plasma channel.The plasma permittivity atω0 is a tensor. Its components are(3)ε0xx=ε0yy=1−ωp2ω02−ωc2,ε0xy=−ε0yx=iωcω0ωp2ω02−ωc2,ε0zz=1−ωp2ω02,ε0xz=ε0zx=ε0yz=ε0zy=0,where ωc=eBs/m is electron cyclotron frequency and e and m are the electronic charge and mass, respectively. Maxwell’s equations ∇×E0=iω0μ0H0 and ∇×H0=−iω0εsε=0⋅E0 combine to give the wave equation(4)∇2E0−∇∇⋅E0+ω02C2ε0=⋅E0=0,where εs is the free space permittivity. Here, we chosen the plasma to be uniform and expressed the spatial-temporal variation of E0 as e−iω0t−k0xx−k0yy we would obtain from equation (4), on replacing ∇ by ik0yy^+ik0xx^,(5)D0=⋅E0=0,where D0==ω02/c2ε0=−k02I=+k0k0. Its matrix form is expressed as follows:(6)D=0=ω02c2ε0xx−k0y2ω02c2ε0xy+k0xk0y0−ω02c2ε0xy+k0xk0yω02c2ε0xx−k02000ω02c2ε0zz−k02.The local dispersion relation of the mode is given byD0==0, which can be written as(7)k0y2=ω02c2ε0xx2+ε0xy2ε0xx−k0x2=ω02c21−ωp2ω02ω02−ωp2ω02−ωp2−ωc2−k0x2.Equation (2) gives E0x=−ε0xy/ε0xxE0y. To incorporate nonlocal effects, one may deduce the mode structure equation from the above equation by replacing k0y by −i∂/∂y and operating over E0y,(8)∂2E0y∂y2+ω02c21−ωp2ω02ω02−ωp2ω02−ωp2−ωc2−k0x2E0y=0.Define(9)β01/4y=ξ0,β0=ωp02ω021+ωc2ω02−ωc2ω02−ωp02−ωc22ω02Ln2c2,λ0=ω02c21−ωp02ω02ω02−ωp02ω02−ωp02−ωc2−k0x21β01/2.Equation (8) can be written as(10)∂2E0y∂ξ02+λ0−ξ02E0y=0.The eigenvalue for the fundamental mode isλ0=1, and the eigenfunction is given by E0y=A0e−ξ02/2. From the eigenvalue condition, we obtain(11)k0x2=ω02c21−ωp02ω02ω02−ωp02ω02−ωp02−ωc2−β01/2.The pump wave produces oscillatory electron velocity,(12)v0x=−emωcE0yω02−ωc2−ωp2,v0y=−iemω02−ωp2E0yω0ω02−ωc2−ωp2,v0z=0,and parametrically excites a Langmuir wave with electrostatic potential,(13)ϕ=ϕye−iωt−kx,and two electromagnetic sidebands with electric fields,(14)Ej=Ajy^−ε1xyε1xxx^e−iωjt−kjx,where j=1,2 and ω1,2=ω∓ω0, and k1,2=k∓k0. The sideband waves produce oscillatory electron velocities,(15)vjx=−emωcEjyωj2−ωc2−ωp2,vjy=−iemωj2−ωp2Ejyωjωj2−ωc2−ωp2,vjz=0,where j=1,2.The sideband waves couple with the pump to exert a ponderomotive force on electrons atω,k, Fpω=−m/2v0⋅∇v1+v0∗⋅∇v2+v1⋅∇v0+v2⋅∇v0∗−e/2v0×B1+v1×B0+v0∗×B2+v2×B0∗. In the limit, k0≫k, k1andk2 are largely along x^, and the x and y components of the ponderomotive force turn out to be(16)Fpωx=e2iPE0yE1y+QE0y∗E2y2mω0ω1ω2ω02−ωc2−ωp2ω12−ωc2−ωp2,Fpωy=e2ωcRE0yE1y+SE0y∗E2y2mω0ω1ω2ω02−ωc2−ωp2ω12−ωc2−ωp2,where(17)P=ω2k1−k0ω1ω0ωc2+k1ω02−ωp2ω12−ωc2−ωp2−k0ω12−ωp2ω02−ωc2−ωp2,Q=ω1k0−k2ω2ω0ωc2−k2ω02−ωp2ω22−ωc2−ωp2+k0ω22−ωp2ω02−ωc2−ωp2,R=ω2k0ω1+k1ω0ωc2,S=ω1k0ω2+k2ω0ωc2,Forω≪ω0, Fpω=e∇φp; one may write(18)ϕp=V01E1y−η1η2E2y,V01=eE0yω1ω02−ωp2ω12−ωc2−ωp2−ωc2ωp2ω02mω0ω12ω02−ωc2−ωp2ω12−ωc2−ωp2,where(19)η1=ω12ω12−ωc2−ωp2ω2ω02−ωp2ω22−ωc2−ωp2+ω0ωc2ωp2,η2=ω22ω22−ωc2−ωp2ω1ω02−ωp2ω12−ωc2−ωp2−ω0ωc2ωp2.The drift velocity of plasma electrons due to electrostatic wave of potentialφ in the component form is given by(20)vωx=−emωkω2−ωc2ϕ,vωy=−iemωckω2−ωc2ϕ,vωz=0.The nonlinear velocityvωNL due to ponderomotive force can be obtained from equation (20) by replacing φ by φp.Using the equation of continuity, the linear and nonlinear density perturbations atω, k can be obtained as(21)nωL=∇⋅n00yvωiω,nωNL=∇⋅n00yvωNLiω.Using these in Poisson’s equation,∇2φ=e/ε0nωL+nωNL, we obtain(22)ϕ=−ωp2yω2−ωc2−ωp2yϕp.The nonlinear current densities at the sidebands can be written as(23)J1NL=−12nev0∗,J2NL=−12nev0.Using equation (23) in the wave equation, we obtain(24)D=1⋅E1=−k12I=+k1k1+ω12c2ε1=⋅E1=k2eφωp2ω1E0y2mc2ω0ω2−ωc2ω02−ωc2−ωp2iω0ωcx^+ω02−ωp2y^,where D=1=−k12I=+k1k1+ω12/c2ε1=, and(25)D=2⋅E2=−k22I=+k2k2+ω22c2ε2=⋅E2=k2eϕωp2ω2E0y2mc2ω0ω2−ωc2ω02−ωc2−ωp2−iω0ωcx^+ω02−ωp2y^,where D=2=−k22I=+k2k2+ω22/c2ε2=.Equations (24) and (25) on replacing k1y by −i∂/∂y for the sidebands can be written as (26)d2E1ydξ12+λ1−ξ12E1y=W012e−ξ12ξ102−ξ12E1y−η1η2E2y,(27)d2E2ydξ22+λ2−ξ22E2y=W022e−ξ22ξ202−ξ22E2y−η2η1E1y,(28)ξ12=β11/2y2,ξ22=β21/2y2,ξ2=β11/2y2≈β21/2y2=β11/2β01/2ξ02≈ξ02,ξ102=ω2−ωc2−ωp02Ln2β11/2ωp02,ξ202=ω2−ωc2−ωp02Ln2β21/2ωp02,α1=ω12ω12−ωc2−ωp022−k1x2c2ω12−ωc2ω12−ωc2−ωp02−ωc2ωp04c2ω12−ωc2ω12−ωc2−ωp02,α2=ω22ω22−ωc2−ωp022−k2x2c2ω22−ωc2ω22−ωc2−ωp02−ωc2ωp04c2ω22−ωc2ω22−ωc2−ωp02,β1=ωc2ωp042ω12−2ωc2−ωp02+ω12ωp02ω12−ωc2−ωp022c2ω12−ωc2ω12−ωc2−ωp022Ln2=ωp02ω121+ωc2ω12−ωc2ω12−ωp02−ωc22ω12Ln2c2,β2=ωc2ωp042ω22−2ωc2−ωp02+ω22ωp02ω22−ωc2−ωp022c2ω22−ωc2ω22−ωc2−ωp022Ln2=ωp02ω221+ωc2ω22−ωc2ω22−ωp02−ωc22ω22Ln2c2,λ1=α1β11/2=ω12c21−ωp02ω12ω12−ωp02ω12−ωp02−ωc2−k0x21β11/2,λ2=α2β21/2=ω22c21−ωp02ω22ω22−ωp02ω22−ωp02−ωc2−k0x21β21/2,V01=eE0yω1ω02−ωp2ω12−ωc2−ωp2−ωc2ωp2ω02mω0ω12ω02−ωc2−ωp2ω12−ωc2−ωp2,V02=−eE0yω1ω02−ωp2ω12−ωc2−ωp2−ωc2ωp2ω02mω0ω12ω02−ωc2−ωp2ω12−ωc2−ωp2η1η2,W012=ek2E0yLn2ωp4V01ωp2ω0ωc2−ω1ω12−ωc2−ωp2ω02−ωp22mc2ω0ωp02ω2−ωc2ω12−ωc2−ωp2ω02−ωc2−ωp2,W022=ek2E0yLn2ωp4V02ωp2ω0ωc2+ω2ω22−ωc2−ωp2ω02−ωp22mc2ω0ωp02ω2−ωc2ω22−ωc2−ωp2ω02−ωc2−ωp2.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)E1,2=A1,2e−ξ12/2.Corresponding eigenvalues areλ1,2=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 E1,2 from equation (29) in equations (26) and (27), multiply the resulting equations by e−ξ2/2, and integrate over ξ from −∞ to ∞. Then, we obtain(30)λ1−1A1=−W012Zξ102ξ10A1−η1η2A2,(31)λ2−1A2=−W022Zξ102ξ10A2−η2η1A1,where Zξ102 is the plasma dispersion function. Equations (30) and (31) give the nonlinear dispersion relation(32)1=−Zξ102ξ10W012λ1−1+W022λ2−1.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)1=−Zξ10ωc=02ξ10ωc=0W012ωc=0λ1ωc=0−1+W022ωc=0λ2ωc=0−1,where β11/4ωc=0y=ξ1ωc=0,β21/4ωc=0y=ξ2ωc=0,β01/4ωc=0y=ξ0ωc=0,(34)ξ10ωc=0=ω2−ωp02Ln2β11/2ωc=0ωp02,λ1ωc=0=ω12c21−ωp02ω12ω12−ωp02ω12−ωp02−k0x21β11/2ωc=0,λ2ωc=0=ω22c21−ωp02ω22ω22−ωp02ω22−ωp02−k0x21β21/2ωc=0,β1ωc=0=ωp02Ln2c2=β2ωc=0=β0ωc=0,V01ωc=0=eE0yω1ω02−ωp2ω12−ωp22mω0ω12ω02−ωp2ω12−ωp2,V02ωc=0=−eE0yω1ω02−ωp2ω12−ωp22mω0ω12ω02−ωp2ω12−ωp2η1ωc=0η2ωc=0,η1ωc=0=ω12ω12−ωp2ω2ω02−ωp2ω22−ωp2,η2ωc=0=ω22ω22−ωp2ω1ω02−ωp2ω12−ωp2,W012ωc=0=ek2E0yLn2ωp4V01ωc=0−ω1ω12−ωp2ω02−ωp22mc2ω0ωp02ω2ω12−ωp2ω02−ωp2,W022ωc=0=ek2E0yLn2ωp4V02ωc=0ω2ω22−ωp2ω02−ωp22mc2ω0ωp02ω2ω22−ωp2ω02−ωp2.Taylor expandingλ1−1 with respect to variables ω1,k1 and λ2−1 with respect to variables ω2,k2 and substituting in equation (32), one obtains(35)1=−Zξ102ξ10W012∂λ0/∂ω01ω−kvg0+Δ−1ω−kvg0−Δ,where(36)∂ω0∂k0=−∂λ0/∂k0∂λ0/∂ω0=vg0,Δ=Δ′2∂λ0/∂ω0,Δ′=∂2λ0∂ω02ω2+∂2λ0∂k02k2,∂λ0∂ω0=∂λ0−1∂ω0≅1β02ω0c2+2ωp02c2Δ1Δ2−ωp0LncΔ3Δ4,∂2λ0∂ω02=1β02c2+2ωp02c2Δ2d1−Δ1d2Δ22−ωp0LncΔ4d3−Δ3d4Δ42,∂λ0∂k0=−2k0β0,∂2λ0∂k02=−2β0,Δ1=ω0ω02−ωp02−ωc2ωp02−2ω02−ωp02−ω022ω02−ωp02−ωc2,Δ2=ω02ω02−ωp02−ωc22,Δ3=−ω0ωc2ω02−ωp02−ωc2,Δ4=ω02−ωp02−ωc22+ωc2ω02−ωc21/2ω02−ωp02−ωc22,∂Δ1∂ω0=d1=ω02−ωp02−ωc2ωp02−6ω02−2ω02ωp02−2ω02−ωp02−ωc2ωp02−3ω02,∂Δ2∂ω0=d2=2ω0ω02−ωp02−ωc22,∂Δ3∂ω0=d3=−ωc23ω02+ωp02−ωc2,∂Δ4∂ω0=d4=4ω0ω02−ωp02−ωc2ω02−ωp02−ωc22+ωc2ω02−ωc21/2+ω0ω02−ωp02−ωc22ω02−ωp02ω02−ωp02−ωc22+ωc2ω02−ωc2−1/2.Equation (35) gives(37)ω−kvg02−Δ2=2ΔW012Zξ102∂λ0/∂ω0ξ10.We substitute the value ofξ10 and Zξ102≈−2, also write ω=ωr+iγ, where ωr=kvg0=ωc2+ωp02, and obtain(38)−11/2γ3=22W012ωp0Δ∂λ0/∂ω0Lnβ11/4e2ilπ.Equation (38) gives the growth rate(39)γ=22W012ωp0Δ∂λ0/∂ω0Lnβ11/41/3sin2lπ3.Forl=1, the growth rate turns out to be(40)γ=3222W012ωp0Δ∂λ0/∂ω0Lnβ11/41/3.In Figure2, we have displayed the normalized growth rate γ/ωp0 of the Raman forward scattering instability as a function of normalized pump wave amplitude eE0/mωc for parameters: ωp0/ω0=0.2, Lnωp0/c=8, and ωc/ωp0=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 γ/ωp0 of the Raman forward scattering as a function of normalized static magnetic field ωc/ωp0 for ωp0/ω0=0.2, Lnωp0/c=8, and eE0/mωc=0.1. In the absence of DC magnetic field, the growth rate γ/ωp0 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 forωp0/ω0=0.2, Lnωp0/c=8, and ωc/ωp0=1.Figure 3
Normalized growth rate as a function of normalized cyclotron frequency forωp0/ω0=0.2, Lnωp0/c=8, and eE0/mωc=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 ordercLn/ωp01/2. 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 γ/ωp0>kvth, one may neglect thermal effects. The growth rate roughly scales as E0y2/3 with pump amplitude and goes linearly with ambient magnetic field. For typical parameters, ωp0/ω0=0.2, Lnωp0/c=8, ωc/ωp0=0.5, and eE0/mωc=0.1, the normalized growth rate is 0.26 whereas for ωp0/ω0=0.2, Lnωp0/c=8, ωc/ωp0=1, and eE0/mωc=0.8, the normalized growth rate is 0.028.In the earlier work by Liu et al. [37] for the parameters ωp0/ω0=0.2 and Lnωp0/c=8v0/c=0.6, the normalized growth rate is 0.06 whereas in our work for the parameters ωp0/ω0=0.2, Lnωp0/c=8, ωc/ωp0=1, and eE0/mωc=0.6, the normalized growth rate is 0.025 Clearly, our results are in good agreement with the earlier work by Liu et al.



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