Consider the propagation of a cosh Gaussian laser beam (CGB) of angular frequency (\({\omega }_{0+}\)) and wave vector (\({k}_{0+}\)) along the direction of the static magnetic field in a collisionless magneto plasma. The external applied static magnetic field (B0) is perpendicular to the propagation direction (towards z-axis) of laser beam. The electric field of cosh-Gaussian laser beam can be written as
$${E}_{0+}={E}_{x}+i{E}_{y}={A}_{0+}\left(r,z\right)\text{e}\text{x}\text{p}\left[i\left({\omega }_{0+}t-{k}_{0+}z\right)\right]$$
1
where+ sign denotes the right circular mode of propagation, \(A\)0+ is the amplitude of the electric field, and \(k\)0+ is the propagation wave vector of the laser beam.
The initial amplitude of the cosh-Gaussian laser beam at z = 0 is given by (Lü et al. 1999; Konar et al. 2007)
$$\left({A}_{0+}\right)=\frac{{E}_{00+}}{2}{e}^{\frac{{b}^{2}}{4}}\left\{\text{e}\text{x}\text{p}\left[-{\left(\frac{r}{{r}_{0}}+\frac{b}{2}\right)}^{2}\right]+\text{e}\text{x}\text{p}\left[-{\left(\frac{r}{{r}_{0}}-\frac{b}{2}\right)}^{2}\right]\right\}$$
2
where E00+ is the amplitude of cosh-Gaussian laser beam for the central position at r = z = 0, b is the decentred parameter of the beam, r is the radial coordinate of the cylindrical coordinate system, and r0 is the initial beam width.
The dielectric constant (\({\epsilon }_{+}\)) corresponding to wave propagation vector (\(k\)0+) is given by
$${\epsilon }_{+}(r,z)=1-\frac{{\omega }_{pe}^{2}}{{\omega }_{0+}^{2}}{\left(1-\frac{{\omega }_{ce}}{{\omega }_{0+}}\right)}^{-1}$$
3
where \({\omega }_{pe}={\left(\frac{4\pi {n}_{0}{e}^{2}}{{m}_{e}}\right)}^{1/2}\) is the electron plasma frequency, and \({\omega }_{ce}=\frac{e{B}_{0}}{{m}_{e}c}\) is the electron cyclotron frequency respectively, e is the electronic charge, \({m}_{e}\)is the rest mass of electron, c is the light velocity and n0 is the electron density of the plasma in the absence of laser beam.
The relativistic motion equation of an electron in the presence of a static magnetic field B0 and intense laser field E0+ is written as (Hassoon et al. 2010)
$${m}_{e}\frac{\partial }{\partial t}{\gamma }_{ }{\upsilon }_{0+}=-e\left({E}_{0+}+\frac{1}{c} ({\upsilon }_{0+}\times {B}_{0})\right)$$
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where γ is the relativistic factor and \({\upsilon }_{0+}\) is the oscillation velocity imparted by laser beam. The Lorentz force factor \(\left(-\frac{1}{c} ({\upsilon }_{0+}\times {B}_{0})\right)\) is not considered here because for ultra-short intense laser pulses relativistic nonlinearity becomes set up almost instantaneously.
The oscillatory drift velocity of electrons \(\left({\upsilon }_{0+}\right)\) for right circularly polarized mode of relativistic laser beam is given by
$${\upsilon }_{0+}={\upsilon }_{x}+i{\upsilon }_{y}=\frac{ie{E}_{0+}}{{m}_{e}\gamma {\omega }_{0+}\left(1 - \frac{{\omega }_{ce}}{\gamma {\omega }_{0+}}\right)}$$
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The relativistic factor (γ) is given by
$$\gamma \approx 1+{\alpha }_{+}{A}_{0+}{A}_{0+}^{*}$$
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where\({\alpha }_{+}=\frac{{e}^{2}}{{2m}_{e}^{2}{\omega }_{0+ }^{2}{c}^{2}}{\left(1 - \frac{{\omega }_{ce}}{{\omega }_{0+}}\right)}^{-2}\)
The propagation of cosh-Gaussian laser beam in magnetized plasma is governed by the general wave equation (Sodha et al. 1974)
$$\frac{{\partial }^{2}{E}_{0+}}{{\partial z}^{2}}-{\nabla (\nabla .E}_{0+})+\frac{{\omega }_{0+}^{2}}{{c}^{2}}{\epsilon }_{+}{(r,z)E}_{0+}=0$$
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where ε+ is the effective dielectric constant of the plasma.
The effective dielectric constant (ε+) of the plasma in the presence of relativistic nonlinearity for the right circularly polarized laser beam is given by
$${\epsilon }_{+}={\epsilon }_{xx}-i{\epsilon }_{xy}=1-\frac{{\omega }_{pe}^{2}}{{\omega }_{0+}^{2}\gamma }{\left(1-\frac{{\omega }_{ce}}{{\omega }_{0+}\gamma }\right)}^{-1}$$
8
where εxx and εxy are the components of plasma dielectric constant tensor. Putting the value of relativistic factor (γ) in above equation, one may get
$${\epsilon }_{+}=1-\frac{{\omega }_{pe}^{2}}{{\omega }_{0+}^{2}}{\left(1-\frac{{\omega }_{ce}}{{\omega }_{0+}}\right)}^{-1}+\frac{{\omega }_{pe}^{2}}{{\omega }_{0+}^{2}}{\left(1-\frac{{\omega }_{ce}}{{\omega }_{0+}}\right)}^{-2}{\alpha }_{+}{A}_{0+}{A}_{0+}^{*}$$
9
By using Eq. (1), Eq. (7) can be written in terms of A0+ is as
$$\frac{{\partial }^{2}{A}_{0+}}{{\partial z}^{2}}-2i{k}_{0+}\frac{\partial {A}_{0+}}{\partial z}-i{A}_{0+}\frac{\partial {k}_{0+}}{\partial z}\frac{1}{2}\left(1+\frac{{\epsilon }_{0+}}{{\epsilon }_{0zz}}\right)+\frac{{\omega }_{0+}^{2}}{{c}^{2}}\left[{\epsilon }_{+}\right(r,z)-{\epsilon }_{0+}]{A}_{0+}=0$$
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where
$${\epsilon }_{0+}=1-\frac{{\omega }_{pe}^{2}}{{\omega }_{0+}^{2}}$$
The solution of Eq. (10) can be written as
$${A}_{0+}={A}_{00+}\left(r, z\right)exp\left[-i{k}_{0+}{S}_{+}(r, z)\right]$$
11
where A00+ is a real function, \(k\)0+ \(= \frac{{\omega }_{0+}}{c}{\left({\epsilon }_{0+}\right)}^{1/2},\)and S+ is the eikonal which shows slight converging/diverging behavior of the beam in the plasma.
The expansion of \({\epsilon }_{+}\left(r,z\right)\) under the extended-paraxial region can be written as
$${\epsilon }_{+}(r,z)={\epsilon }_{0+}\left(z\right)-\frac{{r}^{2}}{{r}_{0}^{2}}{\epsilon }_{2+}\left(z\right)-\frac{{r}^{4}}{{r}_{0}^{4}}{\epsilon }_{4+}\left(z\right)$$
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where \({\epsilon }_{0+},\) \({\epsilon }_{2+},\)and \({\epsilon }_{4+}\) are the expansion coefficients. Substituting Eqs. (11) and (12) in Eq. (10) and neglecting the term \(\frac{{\partial }^{2}{A}_{0+}}{{\partial z}^{2}}\), real and imaginary parts can be obtained as
$$2\left(\frac{\partial {S}_{+}}{\partial z}\right)+\frac{2{S}_{+}}{{k}_{0+}}+{\left(\frac{\partial {S}_{+}}{\partial r}\right)}^{2}=\frac{1}{{k}_{0+}^{2}{A}_{00+}}\left(1+\frac{{\epsilon }_{0+}}{{\epsilon }_{0zz}}\right)\left(\frac{{\partial }^{2}{A}_{00+}}{\partial {r}^{2}}+\frac{1}{r}\frac{\partial {A}_{00+}}{\partial r}\right)-\frac{{r}^{2}{\epsilon }_{2+}\left(z\right)}{{r}_{0}^{2}{\epsilon }_{0+}\left(z\right)}-\frac{{r}^{4}{\epsilon }_{4+}\left(z\right)}{{r}_{0}^{4}{\epsilon }_{0+}\left(z\right)}$$
13
and
$$\frac{\partial {A}_{00+}^{2}}{\partial z}+{A}_{00+}^{2}\left(1+\frac{{\epsilon }_{0+}}{{\epsilon }_{0zz}}\right)\left(\frac{{\partial }^{2}{S}_{+}}{\partial {r}^{2}}+\frac{1}{r}\frac{\partial {S}_{+}}{\partial r}\right)+\frac{\partial {A}_{00+}^{2}}{\partial r}\frac{\partial {S}_{+}}{\partial r}+\frac{{A}_{00+}^{2}}{{k}_{0+}}\frac{d{k}_{0+}}{dz}=0$$
14
The solution of Eqs. (13) and (14) for cosh-Gaussian beam under extended-paraxial theory can be written as (Sodha and Faisal 2008: Purohit et al. 2021)
$${A}_{00+}^{2}\left(r,z\right)=\frac{{E}_{00+}^{2}}{4{f}_{0+}^{2}}exp\left(\frac{{b}^{2}}{2}\right)\times {\left(\text{e}\text{x}\text{p}\left[-{\left(\frac{r}{{r}_{o}{f}_{0+}}+\frac{{b}^{2}}{2}\right)}^{2}\right]+\text{e}\text{x}\text{p}\left[-{\left(\frac{r}{{r}_{o}{f}_{0+}}-\frac{{b}^{2}}{2}\right)}^{2}\right]\right)}^{2}$$
$$\times \left(1+\frac{{r}^{2}}{{r}_{0}^{2}{f}_{0+}^{2}}{a}_{2}\left(z\right)+\frac{{r}^{4}}{{r}_{0}^{4}{f}_{0+}^{4}}{a}_{4}\left(z\right)\right)$$
15
and
$${S}_{+}\left(r,z\right)={S}_{0+}\left(z\right)+\frac{{r}^{2}}{{r}_{0}^{2}}{S}_{2+}\left(z\right)+\frac{{r}^{4}}{{r}_{0}^{4}}{S}_{4+}\left(z\right)$$
16
where \({a}_{2}\left(z\right)\)and \({a}_{4}\left(z\right)\)are the coefficients of \({r}^{2}\)and \({r}^{4}\), which characterizing the extended-paraxial region contribution to the beam intensity and indicating the departure of the beam from the Gaussian nature, \({S}_{0+}\left(z\right)\)is the axial phase shift, \({S}_{2+}\left(z\right)\) and \({S}_{4+}\left(z\right)\) indicate the spherical curvature of the wavefront and the wavefront departure from the spherical nature respectively.
Substituting the value of \({A}_{00+}^{2}\)and \({S}_{+}\)from Eqs. (15) and (16) in Eq. (14), and equating the coefficients of \({r}^{0}\), \({r}^{2}\), and \({r}^{4}\) on both sides of the resulting equation, one can obtain
$${S}_{2}\left(z\right)=\frac{{r}^{2}}{{f}_{0+}^{2}}{\left(1+\frac{{\epsilon }_{0+}}{{\epsilon }_{0zz}}\right)}^{-1}\frac{{df}_{0+}}{dz}$$
17
$$\frac{{da}_{2}}{d\xi }=-16{S}_{04+}{f}_{0+}^{2}\left(1+\frac{{\epsilon }_{0+}}{{\epsilon }_{0zz}}\right)$$
18
and
$$\frac{{da}_{4}}{d\xi }=8{S}_{04+}{f}_{0+}^{2}\left({b}^{2}+3{a}_{2}-2\right)\left(1+\frac{{\epsilon }_{0+}}{{\epsilon }_{0zz}}\right)$$
19
where
$${S}_{04+}={S}_{4+}\left(\frac{{\omega }_{0+}}{c}\right)$$
20
The relation between a4 and a2 can be obtained by eliminating \({S}_{04+}\) in Eqs. (18) and (19) and integrating the result with the initial conditions a4 = 0 and a2 = 0 at \(\xi\) = 0 i.e.
$${a}_{4}=\frac{\left({4b}^{2}{a}_{2}+{3a}_{2}^{2}-{4a}_{2}\right)}{4}$$
21
After replacing Eq. (21) in Eq. (15), one may obtain the amplitude of cosh-Gaussian laser beam as:
$${A}_{00+}^{2}\left(r,z\right)=\frac{{E}_{00+}^{2}}{4{f}_{0+}^{2}}exp\left(\frac{{b}^{2}}{2}\right)\times {\left(\text{e}\text{x}\text{p}\left[-{\left(\frac{r}{{r}_{o}{f}_{0+}}+\frac{{b}^{2}}{2}\right)}^{2}\right]+\text{e}\text{x}\text{p}\left[-{\left(\frac{r}{{r}_{o}{f}_{0+}}-\frac{{b}^{2}}{2}\right)}^{2}\right]\right)}^{2}$$
$$\times \left(1+\frac{{r}^{2}}{{r}_{0}^{2}{f}_{0+}^{2}}{a}_{2}\left(z\right)+\frac{{r}^{4}}{{r}_{0}^{4}{f}_{0+}^{4}}\frac{\left({4b}^{2}{a}_{2}+{3a}_{2}^{2}-{4a}_{2}\right)}{4}\right)$$
22
Similarly, by replacing Eqs. (15) and (16) in Eq. (13) and equating the coefficients of r2 and r4 in the resulting equation, following equations have been obtained for the beam width parameter (\({f}_{0+}\)) of cosh-Gaussuan laser beam and for \({S}_{04+}\):
$$\frac{{d}^{2}{f}_{0+}}{d{\xi }^{2}}={\left(1+\frac{{\epsilon }_{0+}}{{\epsilon }_{0zz}}\right)}^{2}\frac{{\chi }_{2}}{{\epsilon }_{0+}{f}_{0+}^{3}}-\left(1+\frac{{\epsilon }_{0+}}{{\epsilon }_{0zz}}\right)\frac{2{\epsilon }_{2+}{f}_{0+}{\rho }^{2}}{{\epsilon }_{0+}} \left(23\right)$$
and
$$\frac{{dS}_{04+}}{d\xi }=\frac{1}{4}\left(1+\frac{{\epsilon }_{0+}}{{\epsilon }_{0zz}}\right)\frac{{\chi }_{4}}{{\epsilon }_{0+}{f}_{0+}^{6}}-\frac{{\epsilon }_{4+}{\rho }^{2}}{2{\epsilon }_{0+}}-\frac{2{S}_{04+}}{{f}_{0+}}\frac{d{f}_{0+}}{d\xi } \left(24\right)$$
where
$${\chi }_{2}=8{a}_{4}+2{a}_{2}{b}^{2}-3{a}_{2}^{2}-4{a}_{2}-\frac{{b}^{4}}{3}-4{b}^{2}+4$$
and
$${\chi }_{4}={4a}_{2}^{3}+{4a}_{2}^{2}+{4a}_{4}{b}^{2}-{2a}_{2}^{2}{b}^{2}-8{a}_{4}-14{a}_{2}{a}_{4}-2{a}_{2}\frac{{b}^{4}}{3}+17\frac{{b}^{4}}{6}+\frac{{b}^{6}}{12}$$
\(\xi =\frac{cz}{{r}_{0 }^{2}{\omega }_{0+}}\) is the dimensionless distance of propagation, and \(\rho =\frac{{r}_{0}{\omega }_{0+}}{c}\) is dimensionless initial beam width of the beam. The first term on the right-hand side in Eq. (23) represents the diffraction effect, and the second term is a nonlinear term on the account of the relativistic nonlinearity, including the magnetic field.
2.1 Evaluation of the Effective Dielectric Constant
In order to solve the value of the effective dielectric constant in extended-paraxial region, one can expand the solution for \({A}_{00+}^{2} \text{a}\text{s} \text{a} \text{p}\text{o}\text{l}\text{y}\text{n}\text{o}\text{m}\text{i}\text{a}\text{l} \text{i}\text{n}\) r2 and r4 as
$${A}_{00+}^{2}={g}_{0}+{g}_{2}{r}^{2}+{g}_{4}{r}^{4}$$
25
where
$${g}_{0}=\frac{{E}_{00+}^{2}}{{f}_{0+}^{2}}$$
$${g}_{2}={g}_{0}\frac{\left({b}^{2}+{a}_{2}-2\right)}{{r}_{0}^{2}{f}_{0+}^{2}}$$
and
$${g}_{4}={g}_{0}\frac{\left({a}_{4}+{a}_{2}{b}^{2}-2{a}_{2}+\frac{{b}^{4}}{3}-2{b}^{2}+2\right)}{{r}_{0}^{4}{f}_{0+}^{4}}$$
To write ε+ explicitly in the extended-paraxial approximation, γ can be expanded as
$$\gamma ={\gamma }_{0}+{\gamma }_{2 }\frac{{r}^{2}}{{r}_{0}^{2}}+{\gamma }_{4 }\frac{{r}^{4}}{{r}_{0}^{4}}$$
26
where the values of \({\gamma }_{0}\), \({\gamma }_{2 }\)and \({\gamma }_{4}\) can be obtained from Eq. (6) are as:
$${\gamma }_{0}={\left(1+\frac{{\gamma }_{0}^{2}\alpha {g}_{0}}{{\left({\gamma }_{0}- \frac{{\omega }_{ce}}{{\omega }_{0+}}\right)}^{2}}\right)}^{1/2}$$
$${\gamma }_{2}=\frac{\alpha {g}_{2}}{2\left({\gamma }_{0}- \frac{{2\omega }_{ce}}{{\omega }_{0+}} - \frac{2{\omega }_{ce}}{{\omega }_{0+}{\gamma }_{0}^{2}} + \frac{2{\omega }_{ce}^{2}}{{\omega }_{0+}^{2}{\gamma }_{0}^{3}}\right)}$$
and
$${\gamma }_{4}=\frac{\alpha {g}_{4}+\left(3 - \frac{{\omega }_{ce}}{{\omega }_{0+{\gamma }_{0}}} -\frac{{2\omega }_{ce}^{2}}{{\omega }_{0+}^{2}{\gamma }_{0}^{2}} + \frac{{\omega }_{ce}^{2}}{{\omega }_{0+}^{2}{\gamma }_{0}^{4}}\right)}{2\left({\gamma }_{0}- \frac{{\omega }_{ce}}{{\omega }_{0+}} - \frac{{\omega }_{ce}}{{\omega }_{0+}{\gamma }_{0}^{2}} + \frac{{\omega }_{ce}^{2}}{{\omega }_{0+}^{2}{\gamma }_{0}^{3}}\right)}$$
By substituting the value of γ from Eq. (26) in Eq. (8), expanding ε+ as a series of power \(\frac{r}{{r}_{0}}\), and comparing the result with Eq. (13), one obtains
$${\epsilon }_{0+}\left(z\right)=1-\frac{{\omega }_{pe}^{2}}{{\omega }_{0+}^{2}}\frac{1}{\left({\gamma }_{0 }-\frac{{\omega }_{ce}}{ {\omega }_{0+}}\right)}$$
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$${\epsilon }_{2+}\left(z\right)=\frac{{\omega }_{pe}^{2}}{{\omega }_{0+}^{2}}\frac{{\gamma }_{2 }}{{\left({\gamma }_{0 }-\frac{{\omega }_{ce}}{ {\omega }_{0+}}\right)}^{2}}$$
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and
$${\epsilon }_{4+}\left(z\right)=\frac{2{\omega }_{pe}^{2}}{{\omega }_{0+}^{2}}\left[\frac{\left({\gamma }_{0 }-\frac{{\omega }_{ce}}{ {\omega }_{0+}}\right){\gamma }_{4}-{\gamma }_{2}^{2}}{{\left({\gamma }_{0 }-\frac{{\omega }_{ce}}{ {\omega }_{0+}}\right)}^{3}}\right]$$
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