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 (*B**0*) 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 *E*00+ 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 *r*0 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 *n**0* 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 *B*0 and intense laser field *E*0+ is written as (Hassoon et al. 2010)

$${m}_{e}\frac{\partial }{\partial t}{\gamma }_{ }{\upsilon }_{0+}=-e\left({E}_{0+}+\frac{1}{c} ({\upsilon }_{0+}\times {B}_{0})\right)$$

4

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

$${\upsilon }_{0+}={\upsilon }_{x}+i{\upsilon }_{y}=\frac{ie{E}_{0+}}{{m}_{e}\gamma {\omega }_{0+}\left(1 - \frac{{\omega }_{ce}}{\gamma {\omega }_{0+}}\right)}$$

5

The relativistic factor (*γ*) is given by

$$\gamma \approx 1+{\alpha }_{+}{A}_{0+}{A}_{0+}^{*}$$

6

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)

$$\frac{{\partial }^{2}{E}_{0+}}{{\partial z}^{2}}-{\nabla (\nabla .E}_{0+})+\frac{{\omega }_{0+}^{2}}{{c}^{2}}{\epsilon }_{+}{(r,z)E}_{0+}=0$$

7

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 *ε**xx* and *ε**xy* 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

$$\frac{{\partial }^{2}{A}_{0+}}{{\partial z}^{2}}-2i{k}_{0+}\frac{\partial {A}_{0+}}{\partial z}-i{A}_{0+}\frac{\partial {k}_{0+}}{\partial z}\frac{1}{2}\left(1+\frac{{\epsilon }_{0+}}{{\epsilon }_{0zz}}\right)+\frac{{\omega }_{0+}^{2}}{{c}^{2}}\left[{\epsilon }_{+}\right(r,z)-{\epsilon }_{0+}]{A}_{0+}=0$$

10

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 *A*00+ 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

$${\epsilon }_{+}(r,z)={\epsilon }_{0+}\left(z\right)-\frac{{r}^{2}}{{r}_{0}^{2}}{\epsilon }_{2+}\left(z\right)-\frac{{r}^{4}}{{r}_{0}^{4}}{\epsilon }_{4+}\left(z\right)$$

12

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 *a*4 and *a*2 can be obtained by eliminating \({S}_{04+}\) in Eqs. (18) and (19) and integrating the result with the initial conditions *a*4 = 0 and *a*2 = 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 *r*2 and *r*4 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 \({A}_{00+}^{2} \text{a}\text{s} \text{a} \text{p}\text{o}\text{l}\text{y}\text{n}\text{o}\text{m}\text{i}\text{a}\text{l} \text{i}\text{n}\) *r**2* and *r**4* 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

$${\epsilon }_{0+}\left(z\right)=1-\frac{{\omega }_{pe}^{2}}{{\omega }_{0+}^{2}}\frac{1}{\left({\gamma }_{0 }-\frac{{\omega }_{ce}}{ {\omega }_{0+}}\right)}$$

27

$${\epsilon }_{2+}\left(z\right)=\frac{{\omega }_{pe}^{2}}{{\omega }_{0+}^{2}}\frac{{\gamma }_{2 }}{{\left({\gamma }_{0 }-\frac{{\omega }_{ce}}{ {\omega }_{0+}}\right)}^{2}}$$

28

and

$${\epsilon }_{4+}\left(z\right)=\frac{2{\omega }_{pe}^{2}}{{\omega }_{0+}^{2}}\left[\frac{\left({\gamma }_{0 }-\frac{{\omega }_{ce}}{ {\omega }_{0+}}\right){\gamma }_{4}-{\gamma }_{2}^{2}}{{\left({\gamma }_{0 }-\frac{{\omega }_{ce}}{ {\omega }_{0+}}\right)}^{3}}\right]$$

29