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Second harmonic generation of high power Cosh-Gaussian beam in cold collisionless plasma

  • Keshav Walia , 1, ,
  • Kulkaran Singh 1 ,
  • Deepak Tripathi , 2
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  • 1Department of Physics, DAV University, Jalandhar, India
  • 2Department of Physics, AIAS, Amity University, Noida, India

Author to whom any correspondence should be addressed.

Received date: 2022-05-25

  Revised date: 2022-07-25

  Accepted date: 2022-07-27

  Online published: 2022-09-26

Copyright

© 2022 Institute of Theoretical Physics CAS, Chinese Physical Society and IOP Publishing

Abstract

The purpose of this study is to explore the second harmonic generation (SHG) of a high power Cosh-Gaussian beam in cold collisionless plasma. The ponderomotive force causes carrier redistribution from high field to low field region in presence of a Cosh-Gaussian beam thereby producing density gradients in the transverse direction. The density gradients so produced the results in electron plasma wave (EPW) generation at the frequency of the input beam. The EPW interacts with the input beam resulting in the production of 2nd harmonics. WKB and paraxial approximations are employed for obtaining the 2nd order differential equation describing the behavior of the beam's spot size against normalized distance. The impact of well-established laser-plasma parameters on the behavior of the beam's spot size and SHG yield are also analyzed. The focusing behavior of the beam and SHG yield is enhanced with an increase in the density of plasma, the radius of the beam and the decentred parameter, and with a decrease in the intensity of the beam. The results of the current problem are really helpful for complete information of laser-plasma interaction physics.

Cite this article

Keshav Walia , Kulkaran Singh , Deepak Tripathi . Second harmonic generation of high power Cosh-Gaussian beam in cold collisionless plasma[J]. Communications in Theoretical Physics, 2022 , 74(10) : 105502 . DOI: 10.1088/1572-9494/ac846e

1. Introduction

Several theoretical and experimental research groups are interested in exploring laser-plasma interaction physics as a result of its connection with a variety of applications including laser-driven fusion, plasma-based accelerators and higher harmonic generation [18]. One can achieve success in the above-mentioned applications through much deeper transition of laser beam inside plasma and acquiring minimum spot size so that maximum energy from the laser beam to the system could be transferred. Several nonlinear phenomena such as harmonic generation, scattering instabilities, self-focusing etc are produced on intense laser interaction with plasma [921]. Researchers are exploring these instabilities theoretically as well as experimentally for detailed information of intense laser interaction with plasma [2227]. Amongst these nonlinear phenomena, the phenomenon of self-focusing occupies a distinctive place. This phenomenon was first time discovered by Askaryan in 1962 [28]. The self-focusing phenomenon is receiving major attention of many researchers on account of its direct relevance to other nonlinear phenomena. This phenomenon arises on account of a change in the plasma's overall dielectric function. The overall plasma's dielectric function can change as a result of three main mechanisms namely relativistic effects, collisions and ponderomotive force.
The most important research area in the laser-plasma interaction process is the production of harmonics. In fact, plasma is the most promising medium for the production of harmonics. It results in the conversion of the laser beam fundamental frequency into several harmonics. Harmonic production strongly influences laser beam transition through plasma medium. Generation of harmonics helps in finding several plasma parameters including local electron density, opacity and electrical conductivity [29, 30]. The transit of beam through plasma medium can be easily tracked through second harmonic generation (SHG). Earlier work on SHG was carried out by Sodha and Kaw [31]. In the field of spectroscopy, harmonic radiations play a commanding role [3235]. The production of harmonics can be done through several mechanisms including plasma wave excitation electron plasma wave (EPW), plasma instabilities and resonant absorption [8, 3639]. However, the commonly used method for harmonics production is EPW excitation. In this method, EPW generated at input wave frequency interacts nonlinearly with input wave resulting in the production of SHG. Major work on this mechanism has been done by various theoretical and experimental research groups in the past [4051]. The commanding role is played by plasma wave production in laser-induced fusion due to the generation of high velocity electrons. The plasma wave interacts with plasma particles and transfers energy to particles further causing particle acceleration [52]. The past research work confirms that the majority of research on SHG is carried out either through uniform beam or through Gaussian beam. SHG of intense Cosh-Gaussian beam in cold collisionless plasma is explored in the present problem. The 2nd order differential equation reporting the behavior of laser beam spot size is set up in section 2. The 2nd harmonic source term and expression for second harmonic yield is derived in sections 3 and 4 respectively. The discussion of findings obtained is given in section 5.

2. Evolution of laser beam spot size

One can write the initial intensity profile for the Cosh-Gaussian beam at $z=0$ as
$\begin{eqnarray}E\left(r,0\right)={E}_{0}\exp \left(-\displaystyle \frac{{r}^{2}}{{w}_{0}^{2}}\right)\cosh ({\rho }_{0}r).\end{eqnarray}$
The starting beam waist and starting field amplitude at $r=z=0$ in equation (1) is given by ${w}_{0}$ and ${E}_{0}$ respectively. The Cosh-factor is expressed as ${\rho }_{0}.$ Equation (1) can be re-written as
$\begin{eqnarray}\begin{array}{l}E\left(r,0\right)={E}_{0}\exp \left(-\displaystyle \frac{{b}^{2}}{4}\right)\left\{\exp \left(-{\left[\displaystyle \frac{r}{{w}_{0}}+\displaystyle \frac{b}{2}\right]}^{2}\right)\right.\\ \,+\left.\exp \left(-{\left[\displaystyle \frac{r}{{w}_{0}}-\displaystyle \frac{b}{2}\right]}^{2}\right)\right\},\end{array}\end{eqnarray}$
$b={w}_{0}{\rho }_{0}$ is the decentred parameter. The intensity profile for the given beam profile at $z\gt 0$ is written as
$\begin{eqnarray}\begin{array}{l}E\left(r,z\right)=\displaystyle \frac{{E}_{0}}{2f}\exp \left(\displaystyle \frac{{b}^{2}}{4}\right)\left\{\exp \left(-{\left[\displaystyle \frac{r}{{w}_{0}f}+\displaystyle \frac{b}{2}\right]}^{2}\right)\right.\\ \,+\left.\exp \left(-{\left[\displaystyle \frac{r}{{w}_{0}f}-\displaystyle \frac{b}{2}\right]}^{2}\right)\right\}.\end{array}\end{eqnarray}$
The transition of the laser beam through plasma causes variation in plasma's effective dielectric function as
$\begin{eqnarray}\varepsilon ={\varepsilon }_{0}+{\rm{\Phi }}(E\cdot {E}^{* }),\end{eqnarray}$
where the linear and nonlinear terms of dielectric function are expressed by ${\varepsilon }_{0}=1-\tfrac{{\omega }_{p}^{2}}{{\omega }^{2}}$ and ${\rm{\Phi }}\left(E\cdot {E}^{* }\right)=\tfrac{{\omega }_{p}^{2}}{{\omega }^{2}}\left[1-\tfrac{{N}_{0e}}{{N}_{0}}\right]$ respectively. There is generation of ponderomotive force on account of the transition of the laser beam passing through cold collisionless plasma. Electrons get redistributed as a result of this ponderomotive force. In a steady state, the plasma pressure gradient balances the ponderomotive force. One can obtain modified plasma density [53, 54] as
$\begin{eqnarray}{N}_{0e}={N}_{0}\exp \left(-\displaystyle \frac{{e}^{2}E{E}^{* }}{m{\omega }^{2}{T}_{e}}\right).\end{eqnarray}$
In equation (5), $-e,$ $m$ and ${T}_{e}$ denote charge, rest mass and temperature of the electron. For cold collisionless plasma
$\begin{eqnarray}{\rm{\Phi }}\left(E\cdot {E}^{* }\right)=\displaystyle \frac{{\omega }_{p}^{2}}{{\omega }^{2}}\left[1-\exp \left(-\displaystyle \frac{{e}^{2}E{E}^{* }}{m{\omega }^{2}{T}_{e}}\right)\right].\end{eqnarray}$
The wave equation satisfied by the electric field vector of the laser beam is written as
$\begin{eqnarray}{{\rm{\nabla }}}^{2}E-{\rm{\nabla }}\left({\rm{\nabla }}.E\right)+\displaystyle \frac{{\omega }^{2}}{{c}^{2}}\varepsilon E=0.\end{eqnarray}$
By making use of assumption $\tfrac{{c}^{2}}{{\omega }^{2}}\left|\tfrac{1}{\varepsilon }{{\rm{\nabla }}}^{2}\,\mathrm{ln}\,\varepsilon \right|\ll 1,\,$one can ignore the ${\rm{\nabla }}\left({\rm{\nabla }}.E\right)$ term in equation (7). So, equation (7) reduces to
$\begin{eqnarray}{{\rm{\nabla }}}^{2}E+\displaystyle \frac{{\omega }^{2}}{{c}^{2}}\varepsilon E=0.\end{eqnarray}$
The solution for equation (8) can be written as [55, 56],
$\begin{eqnarray}E={E}_{0}(r,z)\exp \left[{\rm{i}}(\omega t-k\left(S+z\right))\right],\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{E}_{0}\left(r,z\right)=\displaystyle \frac{{E}_{00}}{2f}\exp \left(\displaystyle \frac{{b}^{2}}{4}\right)\left\{\exp \left(-{\left[\displaystyle \frac{r}{{w}_{0}f}+\displaystyle \frac{b}{2}\right]}^{2}\right)\right.\\ \,+\left.\exp \left(-{\left[\displaystyle \frac{r}{{w}_{0}f}-\displaystyle \frac{b}{2}\right]}^{2}\right)\right\},\end{array}\end{eqnarray}$
$\begin{eqnarray}S=\displaystyle \frac{1}{2}{r}^{2}\displaystyle \frac{1}{f}\displaystyle \frac{{\rm{d}}f}{{\rm{d}}z}+{{\rm{\Phi }}}_{0}(z),\end{eqnarray}$
$\begin{eqnarray}k=\displaystyle \frac{\omega }{c}\sqrt{{\varepsilon }_{0}}.\end{eqnarray}$
In equation (10), $f$ denotes the beam waist and satisfies the 2nd order differential equation
$\begin{eqnarray}\begin{array}{l}\displaystyle \frac{{{\rm{d}}}^{2}f}{{\rm{d}}{\eta }^{2}}=\displaystyle \frac{4}{{f}^{3}}-{\left(\displaystyle \frac{{\omega }_{p}{w}_{0}}{c}\right)}^{2}\left(2-{b}^{2}\right)\displaystyle \frac{{e}^{2}{E}_{00}^{2}}{m{\omega }^{2}{T}_{e}}\\ \,\times \exp \left(-\displaystyle \frac{{e}^{2}{E}_{00}^{2}}{m{\omega }^{2}{T}_{e}{f}^{2}}\right)\displaystyle \frac{1}{{f}^{3}},\end{array}\end{eqnarray}$
where $f=1$ and $\tfrac{{\rm{d}}f}{{\rm{d}}\eta }=0$ at $\eta =0.$

3. EPW excitation mechanism

We know that the Cosh-Gaussian beam is self-focused on account of collisionless plasma if the power of the beam gets larger than the critical beam power. Moreover, the production of density gradients in the perpendicular direction takes place. There is excitation of EPW at input beam frequency due to these density gradients. EPW excitation mechanism can be understood through below mentioned set of equations:
$\begin{eqnarray}\displaystyle \frac{\partial {N}_{e}}{\partial t}+{\rm{\nabla }}\cdot \left({N}_{e}V\right)=0,\end{eqnarray}$
$\begin{eqnarray}{\rm{\nabla }}\cdot E=4\pi \left(Z{N}_{oi}-{N}_{e}\right)e,\end{eqnarray}$
$\begin{eqnarray}\displaystyle \frac{P}{\,{N}_{e}^{\gamma }}={\rm{Constant}},\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}m\left[\displaystyle \frac{\partial V}{\partial t}+\left(V\cdot {\rm{\nabla }}\right)V\right]=-e\left[E+\displaystyle \frac{1}{c}V\times B\right]\\ \,-2{\rm{\Gamma }}\,{mV}-\displaystyle \frac{\gamma }{{N}_{e}}{\rm{\nabla }}{P}_{e}.\end{array}\end{eqnarray}$
By solving the above set of equations and further using linear perturbation theory, we have
$\begin{eqnarray}\begin{array}{l}-{\omega }^{2}{{n}}_{1}+{v}_{{\rm{th}}}^{2}{{\rm{\nabla }}}^{2}{{n}}_{1}+2{\rm{\iota }}{\rm{\Gamma }}\omega {{n}}_{1}+{\omega }_{p}^{2}{\left[\displaystyle \frac{{N}_{0e}}{{N}_{0}}\right]}^{2}{{n}}_{1}\\ \,\cong \displaystyle \frac{e}{m}({{n}}_{0}\mathop{{\rm{\nabla }}.}\limits^{\longrightarrow}\vec{E}).\end{array}\end{eqnarray}$
One can solve equation (18) in order to have the source equation of SHG as
$\begin{eqnarray}\begin{array}{l}{n}_{1}=\displaystyle \frac{e{n}_{0}}{m}\displaystyle \frac{{E}_{00}}{2f}{\rm{\exp }}\left(\displaystyle \frac{{b}^{2}}{4}\right)\left\{{\rm{\exp }}\left(-{\left[\displaystyle \frac{r}{{w}_{0}f}+\displaystyle \frac{b}{2}\right]}^{2}\right)\right.\\ \,+\left.{\rm{\exp }}\left(-{\left[\displaystyle \frac{r}{{w}_{0}f}-\displaystyle \frac{b}{2}\right]}^{2}\right)\right\}\left\{\displaystyle \frac{r}{{w}_{0}^{2}{f}^{2}}\right\}\\ \,\times \displaystyle \frac{1}{\left\{{\omega }^{2}-{k}^{2}{v}_{{\rm{t}}{\rm{h}}}^{2}-{\omega }_{p}^{2}{\left({\rm{\exp }}\left(-\displaystyle \frac{{e}^{2}{E}_{00}^{2}}{m{\omega }^{2}{T}_{e}{f}^{2}}\right)\right)}^{2}\right\}}.\end{array}\end{eqnarray}$

4. Second harmonic generation

Excited EPW interacts nonlinearly with the input beam thereby producing the 2nd harmonics. One can begin from the well known Maxwell's equations for obtaining the 2nd harmonic field equation ${E}_{2}$ as
$\begin{eqnarray}{{\rm{\nabla }}}^{2}{{E}}_{2}+\displaystyle \frac{{\omega }_{2}^{2}}{{c}^{2}}{\varepsilon }_{2}\left({\omega }_{2}\right){{E}}_{2}=\displaystyle \frac{{\omega }_{p}^{2}}{{c}^{2}}\displaystyle \frac{{n}_{1}}{{n}_{0}}{{E}}_{0}.\end{eqnarray}$
In equation (20), the frequency and dielectric function of the 2nd harmonic is denoted by ${\omega }_{2}=2\omega $ and ${\varepsilon }_{2}$ respectively. From equation (20), it is possible to obtain amplitude for the 2nd harmonic as
$\begin{eqnarray}\begin{array}{l}{E}_{2}=\displaystyle \frac{{\omega }_{p}^{2}}{{c}^{2}}\displaystyle \frac{{n}_{1}}{{n}_{0}}\displaystyle \frac{{E}_{00}}{2f}\exp \left(\displaystyle \frac{{b}^{2}}{4}\right)\left\{\exp \left(-{\left[\displaystyle \frac{r}{{w}_{0}f}+\displaystyle \frac{b}{2}\right]}^{2}\right)\right.\\ \,+\left.\exp \left(-{\left[\displaystyle \frac{r}{{w}_{0}f}-\displaystyle \frac{b}{2}\right]}^{2}\right)\right\}\displaystyle \frac{1}{({k}_{2}^{2}-4{k}_{0}^{2})}.\end{array}\end{eqnarray}$
The power associated with the 2nd harmonic is written as
$\begin{eqnarray}{P}_{2}=\displaystyle \iint | {E}_{2}{| }^{2}{\rm{d}}x{\rm{d}}y.\end{eqnarray}$
The power associated with the main beam is written as
$\begin{eqnarray}{P}_{0}=\displaystyle \iint | {E}_{0}{| }^{2}{\rm{d}}x{\rm{d}}y.\end{eqnarray}$
The yield of SHG is given as
$\begin{eqnarray}\begin{array}{l}{Y}_{2}=\displaystyle \frac{{P}_{2}}{{P}_{0}} =\displaystyle \frac{{\displaystyle \iint \left(\displaystyle \frac{{\omega }_{p}^{2}}{{c}^{2}}\displaystyle \frac{e}{m}{\left(\displaystyle \frac{{E}_{00}}{2f}\exp \left(\displaystyle \frac{{b}^{2}}{4}\right)\left\{\exp \left(-{\left[\displaystyle \frac{r}{{w}_{0}f}+\displaystyle \frac{b}{2}\right]}^{2}\right)+\exp \left(-{\left[\displaystyle \frac{r}{{w}_{0}f}-\displaystyle \frac{b}{2}\right]}^{2}\right)\right\}\right)}^{2}\left\{\displaystyle \frac{r}{{w}_{0}^{2}{f}^{2}}\right\}\displaystyle \frac{1}{\left\{{\omega }_{0}^{2}-{k}^{2}{v}_{{\rm{t}}{\rm{h}}}^{2}-{\omega }_{p}^{2}{\left(\exp \left(-\displaystyle \frac{{e}^{2}{E}_{00}^{2}}{m{\omega }^{2}{T}_{e}{f}^{2}}\right)\right)}^{2}\right\}}\displaystyle \frac{1}{({k}_{2}^{2}-4{k}_{0}^{2})}\right)}^{2}r{\rm{d}}r{\rm{d}}\theta }{\displaystyle \iint {\left(\displaystyle \frac{{E}_{00}}{2f}\exp \left(\displaystyle \frac{{b}^{2}}{4}\right)\left\{\exp \left(-{\left[\displaystyle \frac{r}{{w}_{0}f}+\displaystyle \frac{b}{2}\right]}^{2}\right)+\exp \left(-{\left[\displaystyle \frac{r}{{w}_{0}f}-\displaystyle \frac{b}{2}\right]}^{2}\right)\right\}\right)}^{2}r{\rm{d}}r{\rm{d}}\theta }\end{array}\end{eqnarray}$

5. Discussion

The dependence of beam width $f$ and yield of SHG ${Y}_{2}$ against normalized distance in cold collisionless plasma is represented by equations (13) and (24). Since the analytical solution of equations (13) and (24) cannot be obtained. Hence the numerical solution of these equations is carried out through the RK4 method for well-established laser and plasma parameters;
$\begin{eqnarray*}{E}_{00}=3.0\times {10}^{9}\,{\rm{V}}\,{{\rm{m}}}^{-1},\,4.0\times {10}^{9}\,{\rm{V}}\,{{\rm{m}}}^{-1},\,5.0\times {10}^{9}\,{\rm{V}}\,{{\rm{m}}}^{-1},\end{eqnarray*}$
$\begin{eqnarray*}\displaystyle \frac{{\omega }_{p}^{2}}{{\omega }^{2}}=0.04,\,0.05,\,0.06,\end{eqnarray*}$
$\begin{eqnarray*}b=0,1,\,2,\end{eqnarray*}$
$\begin{eqnarray*}{w}_{0}=30\,\mu {\rm{m}},\,40\,\mu {\rm{m}},\,50\,\mu {\rm{m}},\end{eqnarray*}$
$\begin{eqnarray*}{T}_{e}=2,3,4\,{\rm{eV}}.\end{eqnarray*}$
There is some physical interpretation linked with two terms on the RHS of equation (13). The first and second terms in equation (13) correspond to divergence and convergence of the beam. The overall diffraction and convergence is decided on the basis of relative values of these terms.
The dependence of beam waist $f$ on $\eta $ at distinct values of ${E}_{00}\left({\rm{i}}.{\rm{e}}.\,{E}_{00}=3.0\times {10}^{9}\,{\rm{V}}\,{{\rm{m}}}^{-1},4.0\times {10}^{9}\,{\rm{V}}\,{{\rm{m}}}^{-1},5.0\,\times {10}^{9}\,{\rm{V}}\,{{\rm{m}}}^{-1}\right)$ at fixed values of other parameters is depicted in figure 1. Green, red and black correspond to ${E}_{00}\,=3.0\times {10}^{9}\,{\rm{V}}\,{{\rm{m}}}^{-1},4.0\times {10}^{9}\,{\rm{V}}\,{{\rm{m}}}^{-1}\,\mathrm{and}\,5.0\times {10}^{9}\,{\rm{V}}\,{{\rm{m}}}^{-1}$ respectively. An increase in value of ${E}_{00}$ causes a decrease in focusing behavior of beam due to supremacy of diffraction term over convergence term at increasing values of ${E}_{00}.$
Figure 1. Dependence of beam waist $f$ on $\eta $ at distinct values of ${E}_{00}\left({\rm{i}}.{\rm{e}}.{E}_{00}=3.0\times {10}^{9}\,{\rm{V}}{{\rm{m}}}^{-1},\,4.0\times {10}^{9}\,{\rm{V}}\,{{\rm{m}}}^{-1},\,5.0\times {10}^{9}\,{\rm{V}}\,{{\rm{m}}}^{-1}\right)$ at fixed values of other parameters. Green, red and black curves correspond to ${E}_{00}=3.0\times {10}^{9}\,{\rm{V}}\,{{\rm{m}}}^{-1},\,4.0\times {10}^{9}\,{\rm{V}}\,{{\rm{m}}}^{-1}\,\mathrm{and}\,5.0\times {10}^{9}\,{\rm{V}}\,{{\rm{m}}}^{-1}$ respectively.
The dependence of beam waist $f$ on $\eta $ at distinct values of $\tfrac{{\omega }_{p}^{2}}{{\omega }^{2}}\left({\rm{i}}.{\rm{e}}.\tfrac{{\omega }_{p}^{2}}{{\omega }^{2}}=0.04,\,0.05,\,0.06\right)$ at fixed values of other parameters is depicted in figure 2. Green, red and black curves correspond to $\tfrac{{\omega }_{p}^{2}}{{\omega }^{2}}=0.04,\,0.05\,{\rm{and}}\,0.06$ respectively. Increase in value of $\tfrac{{\omega }_{p}^{2}}{{\omega }^{2}}$ causes enhancement in beam's focusing ability due to the supremacy of convergence term over diffraction term at increasing values of $\tfrac{{\omega }_{p}^{2}}{{\omega }^{2}}.$
Figure 2. Dependence of beam waist $f$ on $\eta $ at distinct values of $\tfrac{{\omega }_{p}^{2}}{{\omega }^{2}}\left({\rm{i}}.{\rm{e}}.\,\tfrac{{\omega }_{p}^{2}}{{\omega }^{2}}=0.04,0.05,0.06\right)$ at fixed values of other parameters. Green, red and black curves correspond to $\tfrac{{\omega }_{p}^{2}}{{\omega }^{2}}=0.04,0.05\,{\rm{and}}\,0.06$ respectively.
The dependence of beam waist $f$ on $\eta $ at distinct values of beam radius ${w}_{0}\left({\rm{i}}.{\rm{e}}.\,{w}_{0}=30\,\mu {\rm{m}},40\,\mu {\rm{m}},50\,\mu {\rm{m}}\,\right)$ at fixed values of other parameters is depicted in figure 3. Green, red and black curves correspond to ${w}_{0}=30\,\mu {\rm{m}},40\,\mu {\rm{m}}\,{\rm{a}}{\rm{n}}{\rm{d}}\,50\,\mu {\rm{m}}$ respectively. An increase in value of beam radius ${{w}}_{0}$ causes enhancement in beam's focusing ability due to supremacy of convergence term over diffraction term at increasing values of ${w}_{0}.$
Figure 3. Dependence of beam waist $f$ on $\eta $ at distinct values of beam radius ${w}_{0}\left({\rm{i}}.{\rm{e}}.\,{w}_{0}=30\,\mu {\rm{m}},\,40\,\mu {\rm{m}},\,50\,\mu {\rm{m}}\,\right)$ at fixed values of other parameters. Green, red and black curves correspond to ${w}_{0}=30\,\mu {\rm{m}},\,40\,\mu {\rm{m}}\,{\rm{and}}\,50\,\mu {\rm{m}}$ respectively.
The dependence of beam waist $f\,$on $\eta $ at distinct values of the decentred parameter $^{\prime} b^{\prime} \,\left({\rm{i}}.{\rm{e}}.\,b=0,\,1,\,2\right)$ at fixed values of other parameters is depicted in figure 4. Green, red and black curves correspond to $b=0,1\,{\rm{and}}\,2$ respectively. An increase in value of decentred parameter $b$ causes enhancement in beam's focusing ability due to supremacy of convergence term over diffraction term at increasing values of $b.$
Figure 4. Dependence of beam waist $f\,$on $\eta $ at distinct values of decentred parameter $^{\prime} b^{\prime} \,\left({\rm{i}}.{\rm{e}}.\,b=0,\,1,\,2\right)$ at fixed values of other parameters. Green, red and black curves correspond to $b=0,\,1\,{\rm{and}}\,2$ respectively.
The dependence of SHG yield ${Y}_{2}$ against $\eta $ at distinct values of ${E}_{00}\left({\rm{i}}.{\rm{e}}.\,{E}_{00}=3.0\times {10}^{9}\,{\rm{V}}\,{{\rm{m}}}^{-1},\,5.0\times {10}^{9}\,{\rm{V}}\,{{\rm{m}}}^{-1}\right)$ at fixed values of other parameters is depicted in figure 5. Green and red curves correspond to ${E}_{00}=3.0\times {10}^{9}\,{\rm{V}}\,{{\rm{m}}}^{-1}\,\mathrm{and}\,5.0\,\times {10}^{9}\,{\rm{V}}\,{{\rm{m}}}^{-1}$ respectively. An increase in value of ${E}_{00}$ causes a decrease in SHG yield ${Y}_{2}$ due to a decrease in focusing behavior at higher values of ${E}_{00}.$
Figure 5. The dependence of SHG yield ${Y}_{2}$ against $\eta $ at distinct values of ${E}_{00}\left({\rm{i}}.{\rm{e}}.\,{E}_{00}=3.0\times {10}^{9}\,{\rm{V}}\,{{\rm{m}}}^{-1},\,5.0\times {10}^{9}\,{\rm{V}}\,{{\rm{m}}}^{-1}\right)$ at fixed values of other parameters. Green and red curves correspond to ${E}_{00}=3.0\times {10}^{9}\,{\rm{V}}\,{{\rm{m}}}^{-1}\,\mathrm{and}\,5.0\times {10}^{9}\,{\rm{V}}\,{{\rm{m}}}^{-1}$ respectively.
The dependence of SHG yield ${Y}_{2}$ against $\eta $ at distinct values of $\tfrac{{\omega }_{p}^{2}}{{\omega }^{2}}\left({\rm{i}}.{\rm{e}}.\,\tfrac{{\omega }_{p}^{2}}{{\omega }^{2}}=0.04,\,0.06\right)$ at fixed values of other parameters is depicted in figure 6. Green and red curves correspond to $\tfrac{{\omega }_{p}^{2}}{{\omega }^{2}}=0.04\,{\rm{and}}\,0.06$ respectively. An increase in value of $\tfrac{{\omega }_{p}^{2}}{{\omega }^{2}}$ causes an increase in SHG yield ${Y}_{2}$ due to enhancement in focusing ability at higher values of $\tfrac{{\omega }_{p}^{2}}{{\omega }^{2}}.$
Figure 6. The dependence of SHG yield ${Y}_{2}$ against $\eta $ at distinct values of $\tfrac{{\omega }_{p}^{2}}{{\omega }^{2}}\left({\rm{i}}.{\rm{e}}.\,\tfrac{{\omega }_{p}^{2}}{{\omega }^{2}}=0.04,\,0.06\right)$ at fixed values of other parameters. Green and red curves correspond to $\tfrac{{\omega }_{p}^{2}}{{\omega }^{2}}=0.04\,{\rm{and}}\,0.06$ respectively.
The dependence of SHG yield ${Y}_{2}$ against $\eta $ at distinct values of beam radius ${w}_{0}\left({\rm{i}}.{\rm{e}}.\,{w}_{0}=30\,\mu {\rm{m}},50\,\mu {\rm{m}}\,\right)$ at fixed values of other parameters is depicted in figure 7. Green and red curves correspond to ${w}_{0}=30\,\mu {\rm{m}}\,{\rm{and}}\,50\,\mu {\rm{m}}$ respectively. An increase in the value of beam radius ${{\rm{w}}}_{0}$ causes an increase in SHG yield ${Y}_{2}$ due to enhancement in focusing ability at higher values of ${{w}}_{0}.$
Figure 7. The dependence of SHG yield ${Y}_{2}$ against $\eta $ at distinct values of beam radius ${w}_{0}\left({\rm{i}}.{\rm{e}}.\,{w}_{0}=30\,\mu {\rm{m}},50\,\mu {\rm{m}}\,\right)$ at fixed values of other parameters. Green and red curves correspond to ${w}_{0}=30\,\mu {\rm{m}}\,{\rm{and}}\,50\,\mu {\rm{m}}$ respectively.
The dependence of SHG yield ${Y}_{2}\,$against $\eta $ at distinct values of the decentred parameter $b\,\left({\rm{i}}.{\rm{e}}.\,b=0\,\mathrm{and}\,2\,\right)$ at fixed values of other parameters is depicted in figure 8. Green and red curves correspond to $b=0\,{\rm{and}}\,2\,$respectively. An increase in the value of decentred parameter $b$ causes an increase in SHG yield ${Y}_{2}$ due to enhancement in focusing ability at higher values of $b.$
Figure 8. The dependence of SHG yield ${Y}_{2}$ against $\eta $ at distinct values of the decentred parameter $b\,\left({\rm{i}}.{\rm{e}}.\,b=0\,\mathrm{and}\,2\,\right)$ at fixed values of other parameters. Green and red curves correspond to $b=0\,{\rm{and}}\,2$ respectively.

6. Conclusions

SHG of high power Cosh-Gaussian beam in cold collisionless plasma is explored in the present work with the help of WKB and paraxial theory approach by considering ponderomotive nonlinearity. The outcome of the current research work is mentioned below:

(1)The focusing behavior of the beam is enhanced with an increase in density of plasma, the radius of the beam and decentred parameter, and with a decrease in the intensity of the beam.

(2)SHG yield is enhanced with an increase in density of plasma, the radius of the beam and decentred parameter, and with a decrease in intensity of the beam.

The results of the current problem are really helpful for complete information of laser-plasma interaction physics.
1
Singh A Walia K 2010 Relativistic self-focusing and self-channeling of Gaussian laser beam in plasma Appl. Phys. B 101 617

DOI

2
Walia K Singh A 2021 Non-linear interaction of Cosh-Gaussian beam in thermal quantum plasma under combined influence of relativistic-ponderomotive force Optik 247 167867

DOI

3
Corkum P B Rolland C Rao T 1986 Supercontinuum generation in gases Phys. Rev. Lett. 57 2268

DOI

4
Utlaut W F Cohen R 1971 Modifying the ionosphere with intense radio waves Science 174 245

DOI

5
Brueckner K A Jorna S 1974 Laser-driven fusion Rev. Mod. Phys. 46 325

DOI

6
Sprangle P Esarey E Krall J 1996 Laser driven electron acceleration in vacuum, gases, and plasmas Phys. Plasmas 3 2183

DOI

7
Faure J Glinec Y Pukhov A Kiselev S Gordienko S Lefebvre E Rousseau J P Burgy F Malka V 2004 A laser-plasma accelerator producing monoenergetic electron beams Nature 431 541

DOI

8
Wilks S C Dawson J M Mori W B Katsouleas T Jones M E 1989 Photon accelerator Phys. Rev. Lett. 62 2600

DOI

9
Walia K 2018 Effect of self-focused high power elliptical laser beam on second harmonic generation in unmagnetized plasma Nonlinear Opt. Quantum Opt. 49 209

10
Walia K 2018 Second harmonic generation of high power elliptical laser beam in underdense plasma: relativistic effects Nonlinear Opt. Quantum Opt 49 91

11
Walia K 2018 Nonlinear interaction of elliptical laser beam with thermal quantum plasma: relativistic effects Nonlinear Opt. Quantum Opt 49 143

12
Lemoff B E Yin G Y GordonIII C L Barty C P J Harris S E 1995 Demonstration of a 10-Hz femtosecond-pulse-driven XUV Laser at 41.8 nm in Xe IX Phys. Rev. Lett. 74 1574

DOI

13
Bers A Shkarofsky I P Shoucri M 2009 Relativistic Landau damping of electron plasma waves in stimulated Raman scattering Phys. Plasmas 16 022104

DOI

14
Walia K Tyagi Y Tripathi D Alshehri A M Ahmad N 2019 Stimulated Raman scattering of high power beam in thermal quantum plasma Optik 195 163166

DOI

15
Walia K Tripathi D Tyagi Y 2017 Investigation of weakly relativistic ponderomotive effects on self-focusing during interaction of high power elliptical laser beam with plasma Commun. Theor. Phys. 68 245

DOI

16
Walia K Kaur S 2016 Nonlinear interaction of elliptical laser beam with collisional plasma: effect of linear absorption Commun. Theor. Phys. 65 78

DOI

17
Walia K 2021 Self-focusing of laser beam in weakly relativistic-ponderomotive thermal quantum plasma Optik 225 165889

DOI

18
Walia K 2020 Stimulated Brillouin scattering of high power beam in unmagnetized plasma: effect of relativistic and ponderomotive nonlinearities Optik 221 165365

DOI

19
Walia K 2020 Self-focusing of high power beam in unmagnetized plasma and its effect on Stimulated Raman scattering process Optik 225 165592

DOI

20
Walia K 2020 Nonlinear interaction of high power beam in weakly relativistic and ponderomotive cold quantum plasma Optik 219 165040

DOI

21
Walia K Tripathi D 2019 Self-focusing of elliptical laser beam in cold quantum plasma Optik 186 46

DOI

22
Liu X Umstadter D Esarey E Ting A 1993 Harmonic generation by an intense laser pulse in neutral and ionized gases IEEE Trans. Plasma Sci. 21 90

DOI

23
Kaw P Schmidt G Wilcox T 1973 Filamentation and trapping of electromagnetic radiation in plasmas Phys. Fluids 16 1522

DOI

24
Young P E Baldis H A Drake R P Campbell E M Estabrook K G 1988 Direct evidence of ponderomotive filamentation in a laser-produced plasma Phys. Rev. Lett. 61 2336

DOI

25
Deutsch C Bret A Firpo M C Gremillet L Lefebvre E Lifschitz A 2008 Onset of coherent electromagnetic structures in the relativistic electron beam deuterium-tritium fuel interaction of fast ignition concern Laser Part. Beams 26 157

DOI

26
Walia K 2016 Nonlinear interaction of high power elliptical laser beam with cold collisionless plasma J. Fusion Energy 35 446

DOI

27
Walia K 2014 Enhanced Brillouin scattering of Gaussian laser beam in collisional plasma: moment theory approach J. Nonlinear Opt. Phys. Mater. 23 1450011

DOI

28
Askaryan G A 1962 Effects of the gradient of strong electromagnetic beam on electrons and atoms JETP 15 1088

29
Teubner U Gibbon P 2009 High-order harmonics from laser-irradiated plasma surfaces Rev. Mod. Phys. 81 445

DOI

30
Stamper J A Lehmberg R H Schmitt A Herbst M J Young F C Gardner J H Obenschain S P 1985 Phys. Rev. E 28 2563

31
Sodha M S Kaw P K 1969 Harmonics in Plasmas in Advances in Electronics and Electron Physics vol 27 Marton L New York Academic 187293

32
Bauer M Lei C Read K Tobey R Gland J Murnane M M Kapteyn H C 2001 Phys. Rev. Lett. 87 025501

DOI

33
Winterfeldt C Spielmann C Gerber G 2008 Colloquium: optimal control of high-harmonic generation Rev. Mod. Phys. 80 117

DOI

34
Tobey R I Siemens M E Cohen O Murnane M M Kapteyn H C Nelson K A 2007 Ultrafast extreme ultraviolet holography: dynamic monitoring of surface deformation Opt. Lett. 32 286

DOI

35
Gisselbrecht M Descamps D Lyng C L'Huillier A Wahlstrm C G Meyer M 1999 Absolute photoionization cross sections of excited He states in the near-threshold region Phys. Rev. Lett. 82 4607

DOI

36
Brunel F 1990 Harmonic generation due to plasma effects in a gas undergoing multiphoton ionization in the high-intensity limit J. Opt. Soc. Am. B 7 521 526

DOI

37
Parashar J Pandey H D 1992 Second-harmonic generation of laser radiation in a plasma with a density ripple IEEE Trans. Plasma Sci. 20 996

DOI

38
Sodha M S Sharma J K Tewari D P Sharma R P Kaushik S C 1978 Plasma wave and second harmonic generation Plasma Phys. 20 825

DOI

39
Erokhin N Zakharov V E Moiseev S S 1969 Second harmonic generation by an electromagnetic wave incident on inhomogeneous plasma Sov. Phys. JETP 29 101

40
Purohit G Rawat P Gauniyal R 2016 Second harmonic generation by self-focusing of intense hollow Gaussian laser beam in collisionless plasma Phys. Plasmas 23 013103

DOI

41
Salih H A Sharma R P 2004 Plasma wave and second harmonic generation of intense laser beams due to relativistic effects Phys. Plasmas 11 3186

DOI

42
Sharma P Sharma R P 2012 Study of second harmonic generation by high power laser beam in magnetoplasma Phys. Plasmas 19 122106

DOI

43
Bhatia A Walia K Singh A 2021 Influence of self-focused Laguerre–Gaussian laser beam on second harmonic generation in collisionless plasma having density transition Optik 245 167747

DOI

44
Walia K Sharma P Singh A 2021 Second harmonic generation of Cosh-Gaussian beam in unmagnetized plasmas: effect of relativistic-ponderomotive force Optik 245 167627

DOI

45
Walia K Verma R K Singh A 2021 Second harmonic generation of laser beam in quantum plasma under collective influence of relativistic-ponderomotive nonlinearities Optik 255 165745

DOI

46
Wadhwa J Singh A 2019 Generation of second harmonics of intense Hermite–Gaussian laser beam in relativistic plasma Laser Part. Beams 37 79

DOI

47
Wadhwa J Singh A 2019 Generation of second harmonics by a self-focused Hermite-Gaussian laser beam in collisionless plasma Phys. Plasmas 26 062118

DOI

48
Burnett N H Baldis H A Richardson M C Enright G D 1977 Harmonic generation in CO2 laser target interaction Appl. Phys. Lett. 31 172

DOI

49
Walia K 2016 Effect of self-focusing of elliptical laser beam on second harmonic generation in collisionless plasma Optik 127 6618

DOI

50
Walia K Kakkar V Tripathi D 2020 Second harmonic generation of high power laser beam in cold quantum plasma Optik 204 164150

DOI

51
Bhatia A Walia K Singh A 2021 Second harmonic generation of intense Laguerre–Gaussian beam in relativistic plasma having an exponential density transition Optik 244 167608

DOI

52
Fibich G 1996 Small beam nonparaxiality arrests self-focusing of optical beams Phys. Rev. Lett. 76 4356

DOI

53
Wang Y Zhou Z 2011 Propagation characters of Gaussian laser beams in collisionless plasma: effect of plasma temperature Phys. Plasmas 18 043101

DOI

54
Gupta D N Suk H 2011 Enhanced thermal self-focusing of a Gaussian laser beam in a collisionless plasma Phys. Plasmas 18 124501

DOI

55
Sodha M S Ghatak A K Tripathi V K 1976 Progress in Optics Amsterdam North Holland

56
Akhmanov S A Sukhorukov A Khokhlov R 1968 Self-focusing and diffraction of light in a nonlinear medium Sov. Phys. Usp. 10 609

DOI

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