#### 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 , 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 . 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 , creating a radial space charge field, and modifies the electron density, where 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 . The electron plasma frequency in the channel varies aswhere is the plasma frequency at and is the density scale length. An X-mode laser propagates through the channel (cf. Figure 1),

The plasma permittivity at is a tensor. Its components arewhere is electron cyclotron frequency and and are the electronic charge and mass, respectively. Maxwell’s equations and combine to give the wave equationwhere is the free space permittivity. Here, we chosen the plasma to be uniform and expressed the spatial-temporal variation of as we would obtain from equation (4), on replacing by where . Its matrix form is expressed as follows:

The local dispersion relation of the mode is given by , which can be written as

Equation (2) gives . To incorporate nonlocal effects, one may deduce the mode structure equation from the above equation by replacing by and operating over ,

Define

Equation (8) can be written as

The eigenvalue for the fundamental mode is and the eigenfunction is given by . From the eigenvalue condition, we obtain

The pump wave produces oscillatory electron velocity,and parametrically excites a Langmuir wave with electrostatic potential,and two electromagnetic sidebands with electric fields,where and , and . The sideband waves produce oscillatory electron velocities,where .

The sideband waves couple with the pump to exert a ponderomotive force on electrons at , In the limit, , are largely along , and the *x* and *y* components of the ponderomotive force turn out to bewhere

For , ; one may writewhere

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

The nonlinear velocity due to ponderomotive force can be obtained from equation (20) by replacing by .

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

Using these in Poisson’s equation, we obtain

The nonlinear current densities at the sidebands can be written as

Using equation (23) in the wave equation, we obtainwhere , andwhere

Equations (24) and (25) on replacing by for the sidebands can be written as

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

Corresponding eigenvalues are 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 from equation (29) in equations (26) and (27), multiply the resulting equations by , and integrate over from to . Then, we obtainwhere is the plasma dispersion function. Equations (30) and (31) give the nonlinear dispersion relation

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,where

Taylor expanding with respect to variables and with respect to variables and substituting in equation (32), one obtainswhere

Equation (35) gives

We substitute the value of and , also write , where , and obtain

Equation (38) gives the growth rate

For , the growth rate turns out to be

In Figure 2, we have displayed the normalized growth rate of the Raman forward scattering instability as a function of normalized pump wave amplitude for parameters: , , and . 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 of the Raman forward scattering as a function of normalized static magnetic field for , , and . In the absence of DC magnetic field, the growth rate 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.

#### 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 . 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 , one may neglect thermal effects. The growth rate roughly scales as with pump amplitude and goes linearly with ambient magnetic field. For typical parameters, , , and the normalized growth rate is whereas for , , and the normalized growth rate is

In the earlier work by Liu et al. [37] for the parameters and the normalized growth rate is whereas in our work for the parameters , , and the normalized growth rate is Clearly, our results are in good agreement with the earlier work by Liu et al.

#### Data Availability

The data used to support the findings of this study are available upon request from the authors.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.