1. Introduction
In recent years, technological advancement has led to the production of intense lasers with intensities exceeding ${10}^{18}\,{\rm{W}}\,{{\rm{cm}}}^{-2}$ [1, 2]. Several researchers have been investigating the interaction of intense lasers with plasmas due to its applicability to distinct applications, such as inertial confinement fusion, particle acceleration and relativistic nonlinear optics [3-15]. Success can be achieved in these applications on the basis of deeper laser beam transition through plasmas. The interaction of lasers with plasma causes the generation of many nonlinear phenomena, including self-focusing, filamentation and scattering instabilities. Laser-plasma coupling efficiency is greatly reduced as a result of these nonlinear phenomena [16-24]. Therefore, experimental/theoretical research groups are working on these instabilities so that laser-plasma coupling efficiency can be improved. Out of the various instabilities mentioned above, a crucial role is played by scattering instabilities, such as stimulated Raman scattering (SRS) and stimulated Brillouin scattering (SBS), as they cause a reduction in the coupling efficiency of laser-plasma interaction. The input beam is split into an electron plasma wave (EPW) and scattered wave in the SRS process. The relativistic electrons are produced due to this EPW; there is also the possibility of the target core being preheated due to these relativistic electrons. The information regarding the wasted energy is obtained through the scattered beam. Therefore, Raman reflectivity is basically useful for getting the information of useful and dissipated energies in laser-plasma coupling.
A literature survey has already confirmed that the past research work on SRS was explored through the concept of plane waves. If a pump beam with non-uniform irradiance is taken, then a phenomenon like self-focusing becomes dominant. Many other nonlinear processes, including SRS, SBS and electrostatic waves, are strongly affected due to the self-focusing phenomenon. Therefore, the consideration of self-focusing is an essential requirement while exploring the SRS process in laser-driven fusion. Many researchers have already explored the interplay between scattering instabilities and self-focusing [25-34]. Barr et al [35] explored the growth rate of SRS while incorporating the self-focusing phenomenon. Rozmus et al [36] explored the generation of thermal electrons through EPWs in the SRS process. Bulanov et al [37] explored the interplay between self-focusing in the SRS process in underdense plasma. Tzeng and Mori explored the inter-connection between SRS and self-focusing [38]. They also found that the SRS process causes suppression of both cavitation and self-focusing. The effect of self-focusing on SRS has also been explored by Russell et al [39]. Fuchs et al [40] explored the growth of SRS under distinct conditions via variation in suitable laser-plasma parameters. Mahmoud and Sharma [41] explored the impact of self-focusing and pump depletion on the SRS process. In their work, they followed the approach of modified Raman gain. Rose et al [42] found that Landau damping associated with electrostatic waves gets increased due to the small density of superthermal electrons. Matsuoka et al [43] explored the formation of filaments via particle in cell (PIC) simulations and further investigated the correlation between these filaments and the SRS process. Many researchers have also explored the impact of self-focusing on the SRS process in different plasma environments. The incitement of the current study is to explore the self-focusing effect on SRS in collisional plasma.
The present work explores the effects of a self-focused intense beam on SRS in collisional plasma. The pump wave (${\omega }_{0},{k}_{0})$ interacts nonlinearly with the pre-excited EPW $(\omega ,k$), resulting in production of a back-scattered wave $({\omega }_{0}-\omega ,$ ${k}_{0}-k$). Here, we have considered $(k\approx 2{k}_{0})$ as a special back-scattering case. The nonlinearity in the dielectric function of plasma is created as a result of non-uniform heating of carriers. This further leads to focusing of the input wave. There is also a change in the dispersion relation for the EPW, and focusing of the EPW is observed under special conditions. As the intensity distribution of the input wave and EPW is linked with the back-scattered wave, enhanced back-reflectivity is found with the focusing of waves. The well-established approximations, namely, paraxial approximation and Wentzel-Kramers-Brillouin (WKB) approximation, are utilized to derive nonlinear ordinary differential equations (ODEs) for the beam waists of distinct waves involved in the process and SRS back-reflectivity. The present paper is structured in three sections: in section 2 , second-order ODEs for the beam waist of the EPW are set up. In section 3 , second-order ODEs for beam waists of the input wave, scattered wave and the expression for SRS back-reflectivity are set up. A discussion and conclusions are presented in sections 4 and 5, respectively.
2. Wave equation solution for the EPW
The high-power beam with angular frequency ${\omega }_{0}$ and a propagation vector ${k}_{0}$ is assumed to be propagating in unmagnetized plasma. Suppose the direction of the propagation of the beam is along the z-axis. Initial irradiance distribution associated with such a beam is represented as3 ), the various types of collisions existing in collisional plasma are governed through parameter $\chi .$ If $\chi =-3,$ then collisions between ions and electrons are dominant. If $\chi =2,$ then collisions between electrons and diatomic molecules are dominant. If $\chi =0,$ then collisions are dependent on speed. In equation (3 ), the nonlinearity coefficient $ \mbox{`} \alpha \mbox{'}$ is denoted as $\alpha =\frac{{e}^{2}M}{6{K}_{{\rm{B}}}{T}_{0}\gamma {m}^{2}{\omega }_{0}^{2}}.$ Here, ${N}_{0},$ $e,$ $m,{K}_{{\rm{B}}}$ and ${T}_{0}$ correspond to the number density in the beam's absence, electronic charge, electronic mass, Boltzmann's constant and the temperature of plasma in equilibrium, respectively. In the hydrodynamic approach, the motion of plasma particles can be described through the following fluid equations [47]
7 ) may be expressed as8 ), $\omega $ denotes angular frequency, $k$ denotes the wave vector for the EPW and $S$ denotes the eikonal for the EPW. Further, one can write the dispersion relation for the EPW as8 ) in equation (7 ) and equate the real and imaginary parts separately,10 ) and (11 ) is represented as12 ), the damping factor is denoted by ${k}_{i}.$ The beam waist for the EPW is denoted by $ \mbox{`} f\mbox{'}$ and it satisfies the following differential equation21 )).
$\begin{eqnarray}E\bullet {E}^{* }{\rm{|}}={E}_{0}^{2}\exp \left(-\frac{{r}^{2}}{{r}_{0}^{2}}\right),\end{eqnarray}$
$\begin{eqnarray}{k}_{0}=\frac{{\omega }_{0}}{c}\sqrt{{\varepsilon }_{0}}=\frac{{\omega }_{0}}{c}\sqrt{1-\frac{{\omega }_{{\rm{p}}}^{2}}{{\omega }_{0}^{2}}}.\end{eqnarray}$
In the above equations, ${r}_{0},$ $c,{\varepsilon }_{0}$ and $r$ denote the initial beam radius, the speed of light, the linear part of the dielectric function and the radial coordinate of the cylindrical coordinate system, respectively. One can express the modified electron concentration for collisional plasma as [44-46] $\begin{eqnarray}{N}_{0e}={N}_{0}{\left[1+\frac{1}{2}\alpha E{E}^{* }\right]}^{\frac{\chi }{2}-1}.\end{eqnarray}$
In equation (| (1) Equation of continuity $\begin{eqnarray}\frac{\partial N}{\partial t}+{\rm{\nabla }}\bullet \left({NV}\right)=0,\end{eqnarray}$ | |
| (2) Equation of motion $\begin{eqnarray}\begin{array}{c}m\left[\frac{\partial V}{\partial {\rm{t}}}+\left(V\bullet {\rm{\nabla }}\right)V\right]=-e\left[E+\frac{1}{c}\left(V\times B\right)\right]\\ -2{\rm{\Gamma }}{mV}-\frac{\gamma }{N}{\rm{\nabla }}P,\end{array}\end{eqnarray}$ | |
| (3) Poisson's equation |
$\begin{eqnarray}{\rm{\nabla }}\bullet E=-4\pi {eN}.\end{eqnarray}$
In the above equations., $N$ corresponds to instantaneous electron density, while $V$ and $P$ denote the speed of electron fluid and pressure, respectively. Here, we have taken $\gamma =3,$ which is the ratio of specific heats, and ${\rm{\Gamma }}$ is the Landau damping factor. By using a perturbation approach and further following standardized techniques, one can write the equation for electron density variation as $\begin{eqnarray}\begin{array}{c}\frac{{{\rm{\partial }}}^{2}n}{{\rm{\partial }}{t}^{2}}+2{\rm{\Gamma }}\frac{{\rm{\partial }}n}{{\rm{\partial }}t}-3{v}_{{\rm{th}}}^{2}{{\rm{\nabla }}}^{2}n\\ +\frac{{\omega }_{{\rm{p}}}^{2}}{\gamma }\frac{{N}_{0e}}{{N}_{0}}n=0.\end{array}\end{eqnarray}$
Following [44-46], the solution of equation ( $\begin{eqnarray}\begin{array}{c}n={n}_{0}\left(r,z\right)\\ \times \exp \left[\,,[,{\rm{i}}\left(\omega t-k(z+S\left(r,z\right))\right)\right].\end{array}\end{eqnarray}$
In equation ( $\begin{eqnarray}{\omega }^{2}={\omega }_{{\rm{p}}}^{2}\left(\frac{{N}_{0e}}{{N}_{0}}\right)+3{k}^{2}{v}_{{\rm{th}}}^{2}.\end{eqnarray}$
Now, put equation ( $\begin{eqnarray}\begin{array}{c}2\frac{\partial S}{\partial z}+{\left(\frac{\partial S}{\partial r}\right)}^{2}=\frac{1}{{k}^{2}{n}_{0}}{{\rm{\nabla }}}_{\perp }^{2}{n}_{0}\\ +\frac{{\omega }_{{\rm{p}}}^{2}}{3{k}^{2}{v}_{{\rm{th}}}^{2}}\left[1-\frac{{N}_{0e}}{{N}_{0}}\right],\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{c}\frac{\partial {n}_{0}^{2}}{\partial z}+\frac{\partial S}{\partial r}\frac{\partial {n}_{0}^{2}}{\partial r}+{n}_{0}^{2}{{\rm{\nabla }}}_{\perp }^{2}S\\ +\frac{2{\rm{\Gamma }}}{3{v}_{{\rm{th}}}^{2}}\frac{\omega {n}_{0}^{2}}{k}=0.\end{array}\end{eqnarray}$
Following [44-46], the solution of equations ( $\begin{eqnarray}\begin{array}{c}{n}_{0}^{2}=\frac{{n}_{00}^{2}}{{f}^{2}}\\ \times \exp \left(-\frac{{r}^{2}}{{a}^{2}{f}^{2}}\right)\exp \left(-2{k}_{{\rm{i}}}z\right),\end{array}\end{eqnarray}$
$\begin{eqnarray}S=\frac{1}{2}{r}^{2}\frac{1}{f}\frac{{\rm{d}}f}{{\rm{dz}}}+{\rm{\Phi }}\left(z\right).\end{eqnarray}$
In equation ( $\begin{eqnarray}\begin{array}{c}\frac{{{\rm{d}}}^{2}f}{{\rm{d}}{z}^{2}}=\frac{1}{{k}^{2}{a}^{4}{f}^{3}}-\frac{{\omega }_{{\rm{p}}}^{2}f}{3{k}^{2}{v}_{{\rm{th}}}^{2}}\\ \times \left(1-\frac{\chi }{2}\right)\frac{\alpha {E}_{00}^{2}}{2{r}_{0}^{2}{f}_{0}^{4}}{\left(1+\frac{\alpha {E}_{00}^{2}}{2{f}_{0}^{2}}\right)}^{\frac{\chi }{2}-2}.\end{array}\end{eqnarray}$
The boundary condition is $f=1$ and $\frac{{\rm{d}}f}{{\rm{d}}z}=0$ at $z=0.$ Further, ${f}_{0}$ and ${r}_{0}$ denote the beam waist and initial radius of the beam for the input beam. (cf equation (3. Wave equation solution for the pump wave and scattered wave
We know that the total electric field (${E}_{{\rm{T}}})$ is the addition of fields due to pump wave ($E)$ and scattered wave $({E}_{s})$ 16 ), the total current density is expressed as ${J}_{T}={N}_{e}V.$ By equating 0th-order and first-order terms, one can get18 ), the complex conjugate of $n$ is ${n}^{* }.$ We have ignored the pump depletion in equation (17 ). This is because there is a depletion in energy due to pump depletion, which produces a major impact on the self-focusing of various waves involved and SRS reflectivity. Further, to solve equation (18 ), we ignored the term ${\rm{\nabla }}\left({\rm{\nabla }}\bullet E\right)$ in comparison to ${{\rm{\nabla }}}^{2}{E}_{s},$ assuming that the scale length of change in the dielectric function in the radial direction is greater than the wavelength of the main beam. Also, we can say that ${r}_{0}\gg 2\pi /{k}_{0}.$ These approximations are actually helpful for mathematical simplicity. Following [44-46], one can write the solution of equation (17 ) as18 ) is mentioned below22 ), ${k}_{S0}^{2}=\frac{{\omega }_{S}^{2}}{{c}^{2}}\left[1-\frac{{\omega }_{{\rm{p}}}^{2}}{{\omega }_{S}^{2}}\right]=\frac{{\omega }_{s}^{2}}{{c}^{2}}{\epsilon }_{S0},$ with ${\omega }_{S}={\omega }_{0}-\omega $ and ${k}_{S1}={k}_{0}-k.$
$\begin{eqnarray}\begin{array}{c}{E}_{{\rm{T}}}=E\exp \left({\rm{i}}{\omega }_{0}t\right)\\ +{E}_{s}\exp \left({\rm{i}}{\omega }_{s}t\right).\end{array}\end{eqnarray}$
The wave equation for the field vector ${E}_{{\rm{T}}}$ may be expressed as $\begin{eqnarray}\begin{array}{c}{{\rm{\nabla }}}^{2}{E}_{{\rm{T}}}-{\rm{\nabla }}\left({\rm{\nabla }}\bullet {E}_{{\rm{T}}}\right)\\ =\frac{1}{{c}^{2}}\frac{{\partial }^{2}{E}_{{\rm{T}}}}{\partial {t}^{2}}+\frac{4\pi }{{c}^{2}}\frac{\partial {J}_{{\rm{T}}}}{\partial t}.\end{array}\end{eqnarray}$
In equation ( $\begin{eqnarray}{{\rm{\nabla }}}^{2}E+\frac{{\omega }_{0}^{2}}{{c}^{2}}\left[1-\frac{{\omega }_{{\rm{p}}}^{2}}{{\omega }_{0}^{2}}\frac{{N}_{0e}}{{N}_{0}}\right]E=0,\end{eqnarray}$
$\begin{eqnarray}\begin{array}{c}{{\rm{\nabla }}}^{2}{E}_{s}+\frac{{\omega }_{s}^{2}}{{c}^{2}}\left[1-\frac{{\omega }_{{\rm{p}}}^{2}}{{\omega }_{s}^{2}}\frac{{N}_{0e}}{{N}_{0}}\right]{E}_{s}\\ =\left[\frac{{\omega }_{p}^{2}{\omega }_{s}{n}^{* }}{2{c}^{2}{\omega }_{0}{N}_{0}}\right]E-{\rm{\nabla }}\left({\rm{\nabla }}\bullet E\right).\end{array}\end{eqnarray}$
In equation ( $\begin{eqnarray}{E}_{0}^{2}=\frac{{E}_{00}^{2}}{{f}_{0}^{2}}\exp \left[-\frac{{r}^{2}}{{r}_{0}^{2}{f}_{0}^{2}}\right],\end{eqnarray}$
$\begin{eqnarray}{S}_{0}=\frac{1}{2}{r}^{2}\frac{1}{{f}_{0}}\frac{{\rm{d}}{f}_{0}}{{\rm{d}}z}+{{\rm{\Phi }}}_{0}\left(z\right).\end{eqnarray}$
In the above equations., ${f}_{0}$ denotes the beam waist for the input beam, and the differential equation satisfied by it is mentioned below; $\begin{eqnarray}\begin{array}{c}\frac{{{\rm{d}}}^{2}{f}_{0}}{{\rm{d}}{z}^{2}}=\frac{1}{{k}_{0}^{2}{r}_{0}^{4}{f}_{0}^{3}}-\frac{{\omega }_{{\rm{p}}}^{2}}{{\omega }_{0}^{2}{\varepsilon }_{0}}\frac{\alpha {E}_{00}^{2}}{2{r}_{0}^{2}{f}_{0}^{3}}\\ \times {\left(1-\frac{\chi }{2}\right)\left(1+\frac{\alpha {E}_{00}^{2}}{{f}_{0}^{2}}\right)}^{\frac{\chi }{2}-2}.\end{array}\end{eqnarray}$
Now, the solution of equation ( $\begin{eqnarray}\begin{array}{c}{E}_{s}={E}_{s0}\left(r,z\right){{\rm{e}}}^{+{\rm{i}}{k}_{s0}z}\\ +{E}_{s1}\left(r,z\right){{\rm{e}}}^{-{\rm{i}}{k}_{s1}z}.\end{array}\end{eqnarray}$
In equation (Upon putting equation (22 ) into equation (18 ) and further equating terms with the same phases24 ) has a solution of the form25 ) into equation (24 ), further ignoring space differentials, one can get23 ) has a solution of the form27 ) in equation (23 ) and further equating real terms and imaginary terms separately,28 ) and (29 ) may be written as
$\begin{eqnarray}\begin{array}{c}-{k}_{{\rm{S}}0}^{2}{E}_{{\rm{S}}0}^{2}+2{\rm{i}}{k}_{{\rm{S}}0}\frac{\partial {E}_{{\rm{S}}0}}{\partial z}+{{\rm{\nabla }}}_{{\rm{\perp }}}^{2}{E}_{{\rm{S}}0}\\ +\frac{{\omega }_{S}^{2}}{{c}^{2}}\left[{\epsilon }_{{\rm{S}}0}+\frac{{\omega }_{{\rm{p}}}^{2}}{{\omega }_{S}^{2}}\left(1-\frac{{N}_{0e}}{{N}_{0}}\right)\right]{E}_{{\rm{S}}0}=0,\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{c}-{k}_{{\rm{S}}1}^{2}{E}_{{\rm{S}}1}^{2}+2{\rm{i}}{k}_{{\rm{S}}1}\frac{\partial {E}_{{\rm{S}}1}}{\partial z}\\ +{{\rm{\nabla }}}_{{\rm{\perp }}}^{2}\,{E}_{{\rm{S}}1}+\frac{{\omega }_{S}^{2}}{{c}^{2}}\left[{\epsilon }_{S0}+\frac{{\omega }_{{\rm{p}}}^{2}}{{\omega }_{S}^{2}}\left(1-\frac{{N}_{0e}}{{N}_{0}}\right)\right]{E}_{{\rm{S}}1}\\ =\frac{1}{2}\frac{{\omega }_{{\rm{p}}}^{2}}{{c}^{2}}\frac{{n}^{* }}{{N}_{0}}\frac{{\omega }_{S}}{{\omega }_{0}}{E}_{0}\exp \left(-{\rm{i}}{k}_{0}{S}_{0}\right).\end{array}\end{eqnarray}$
Equation ( $\begin{eqnarray}{E}_{{\rm{S}}1}={E}_{{\rm{S}}1}^{{\prime} }\left(r,z\right){{\rm{e}}}^{-{\rm{i}}{k}_{0}{S}_{0}}.\end{eqnarray}$
By substituting equation ( $\begin{eqnarray}\begin{array}{c}{E}_{{\rm{S}}1}^{{\prime} }=-\frac{1}{2}\frac{{\omega }_{{\rm{p}}}^{2}}{{c}^{2}}\frac{{n}^{* }}{{N}_{0}}\frac{{\omega }_{s}}{{\omega }_{0}}\\ \times \frac{\hat{E}{E}_{0}}{\left[{k}_{s1}^{2}-{k}_{s0}^{2}-\frac{{\omega }_{{\rm{p}}}^{2}}{{c}^{2}}\left(1-\frac{{N}_{0e}}{{N}_{0}}\right)\right]}.\end{array}\end{eqnarray}$
Further, equation ( $\begin{eqnarray}{E}_{{\rm{S}}0}={E}_{{\rm{S}}00}{{\rm{e}}}^{{\rm{i}}{k}_{{\rm{S}}0}{S}_{c}}.\end{eqnarray}$
Using equation ( $\begin{eqnarray}\begin{array}{c}2\frac{\partial {S}_{c}}{\partial z}+{\left(\frac{\partial {S}_{c}}{\partial r}\right)}^{2}\\ =\,\frac{1}{{k}_{s0}^{2}{E}_{s00}}{{\rm{\nabla }}}_{{\rm{\unicode{x027D8}}}}^{2}{E}_{{\rm{S}}00}+\frac{{\omega }_{{\rm{p}}}^{2}}{{\epsilon }_{s0}{\omega }_{s}^{2}}\left[1-\frac{{N}_{0e}}{{N}_{0}}\right],\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{c}\frac{\partial {E}_{s00}^{2}}{\partial z}+\frac{\partial {S}_{c}}{\partial r}\frac{\partial {E}_{s00}^{2}}{\partial r}\\ +\,{E}_{s00}^{2}{{\rm{\nabla }}}_{{\rm{\unicode{x027D8}}}}^{2}{S}_{c}=0.\end{array}\end{eqnarray}$
Following [44-46], the solution of equation ( $\begin{eqnarray}{E}_{{\rm{S}}00}^{2}=\frac{{B}_{1}^{2}}{{f}_{S}^{2}}\exp \left[-\frac{{r}^{2}}{{b}^{2}{f}_{s}^{2}}\right],\end{eqnarray}$
$\begin{eqnarray}{S}_{c}=\frac{1}{2}{r}^{2}\frac{1}{{f}_{S}}\frac{{\rm{d}}{f}_{S}}{{\rm{d}}z}+{{\rm{\Phi }}}_{S}\left(z\right).\end{eqnarray}$
In the above equations, $b$ and ${f}_{s}$ represent the initial radius and beam waist of the back-scattered beam. Further, the differential equation satisfied by ${f}_{s}$ is given below; $\begin{eqnarray}\begin{array}{c}\frac{{{\rm{d}}}^{2}{f}_{s}}{{\rm{d}}{{\rm{z}}}^{2}}=\frac{1}{{k}_{s0}^{2}{b}^{4}{f}_{s}^{3}}-\frac{{\omega }_{{\rm{p}}}^{2}}{{\omega }_{S}^{2}{\varepsilon }_{s0}}\frac{\alpha {E}_{00}^{2}{f}_{s}}{2{r}_{0}^{2}{f}_{0}^{4}}\\ {\times \left(1-\frac{\chi }{2}\right)\left(1+\frac{\alpha {E}_{00}^{2}}{{f}_{0}^{2}}\right)}^{\frac{\chi }{2}-2}.\end{array}\end{eqnarray}$
The initial conditions are ${\rm{d}}{f}_{s}/{dz}=0$ and ${f}_{s}=1$ at $z=0.$ To obtain the value of ${B}_{1},$ let us assume that ${E}_{s}=0$ at $z={z}_{c},$ i.e. $\begin{eqnarray}\begin{array}{c}{E}_{s}={E}_{s0}\left(r,z\right){{\rm{e}}}^{+{\rm{i}}{k}_{s0}z}\\ +\,{E}_{s1}\left(r,z\right){{\rm{e}}}^{-{\rm{i}}{k}_{s1}z}=0.\end{array}\end{eqnarray}$
At $z={z}_{c},$ the amplitude of the scattered wave is zero, i.e. $\begin{eqnarray}\begin{array}{c}{B}_{1}=\frac{{\omega }_{{\rm{p}}}^{2}{\omega }_{s}{N}_{00}}{2{c}^{2}{\omega }_{0}{N}_{0}}\frac{{E}_{00}{{\rm{e}}}^{-{\rm{i}}{k}_{{\rm{i}}}{z}_{c}}}{\left[{k}_{s1}^{2}-{k}_{s0}^{2}-\frac{{\omega }_{{\rm{p}}}^{2}}{{c}^{2}}\left(1-\frac{{N}_{0e}}{{N}_{0}}\right)\right]}\\ \times \frac{{f}_{s}({z}_{c})}{{f}_{s}({z}_{c})f({z}_{c})}\frac{\exp {\rm{}}(-{\rm{i}}\left({k}_{0}{S}_{0}+{k}_{s1}{z}_{c}\right))}{\exp {\rm{}}(+{\rm{i}}\left({k}_{s0}{S}_{c}+{k}_{s0}{z}_{c}\right))},\end{array}\end{eqnarray}$
with the condition that $\frac{1}{{b}^{2}{f}_{s}^{2}}=\frac{1}{{a}^{2}{f}^{2}}+\frac{1}{{r}_{0}^{2}{f}_{0}^{2}}.$Now, one can express SRS back-reflectivity as,
$\begin{eqnarray}\begin{array}{c}R=\frac{1}{4}{\left(\frac{{\omega }_{{\rm{p}}}^{2}}{{c}^{2}}\right)}^{2}{\left(\frac{{\omega }_{s}}{{\omega }_{0}}\right)}^{2}{\left(\frac{{N}_{00}}{{N}_{0}}\right)}^{2}\\ \times \frac{\left({L}_{1}-{L}_{2}-{L}_{3}\right)}{{\left[{k}_{s1}^{2}-{k}_{s2}^{2}-\frac{{\omega }_{{\rm{p}}}^{2}}{{c}^{2}}\left(1-{\left(1+\frac{\alpha {E}_{00}^{2}}{{f}_{0}^{2}}\right)}^{-5/2}\right)\right]}^{2}},\end{array}\end{eqnarray}$
where
$\begin{eqnarray*}\begin{array}{c}{L}_{1}={\left(\frac{{f}_{s}}{{f}_{0}f}\right)}_{z={z}_{c}}^{2}\frac{1}{{f}_{s}^{2}}\\ \times \exp \left(-2{k}_{{\rm{i}}}{z}_{c}-\frac{{r}^{2}}{{b}^{2}{f}_{s}^{2}}\right),\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{c}{L}_{2}=-2{\left(\frac{{f}_{s}}{{f}_{0}f}\right)}_{{z}_{c}}\frac{1}{f{f}_{0}{f}_{s}}\\ \times \exp \left(-\frac{{r}^{2}}{2{b}^{2}{f}_{s}^{2}}-\frac{{r}^{2}}{2{a}^{2}{f}^{2}}-\frac{{r}^{2}}{2{r}_{0}^{2}{f}_{s}^{2}}\right)\\ \left(\times \exp (-{k}_{{\rm{i}}}\left(z+{z}_{c}\right)\right){\rm{\cos }}\left({k}_{s0}+{k}_{s1}\right)\left[z-{z}_{c}\right],\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{c}{L}_{3}=\frac{1}{{f}^{2}{f}_{0}^{2}}\\ \times \exp \left(-\frac{{r}^{2}}{{a}^{2}{f}^{2}}-\frac{{r}^{2}}{{r}_{0}^{2}{f}_{s}^{2}}-2{k}_{{\rm{i}}}{z}_{c}\right).\end{array}\end{eqnarray*}$
4. Discussion
The second-order ODE for the beam waists $f,$ ${f}_{0}$ and ${f}_{s}$ of an EPW, main beam and scattered beam are expressed by equations (14 ), (21 ) and (32 ), respectively. It is not possible to obtain an analytical solution of these equations. Therefore, these equations are numerically solved for established laser-plasma parameters;
$\begin{eqnarray*}\begin{array}{l}\alpha {E}_{00}^{2}=2.0,2.5,3.0;\\ \frac{{\omega }_{{\rm{p}}}^{2}}{{\omega }_{0}^{2}}=0.15,0.20\,{\rm{and}}\,0.25;\\ {r}_{0}=a=20\,\mu {\rm{m}},25\,\mu {\rm{m}},30\,\mu {\rm{m}};\\ b=40\,\mu {\rm{m}},50\,\mu {\rm{m}},60\,\mu {\rm{m}}.\end{array}\end{eqnarray*}$
In all these three equations, the first term causes a diffraction phenomenon, while the second term causes convergence of the beam. The dominance of the first term in each equation causes diffraction of the beam, while dominance of the second term causes convergence of the beam. When both terms of each equation balance each other, then neither convergence nor diffraction takes place and the beam goes into self-trapped mode.
The impact of the intensity of the laser beam on the beam widths ${f}_{0},$ $f$ and ${f}_{s}$ of the pump wave, EPW and scattered wave is shown in figures 1, 2 and 3, respectively. The black, red, and green curves represent $\alpha {E}_{00}^{2}=2.0,2.5\,{\rm{and}}\,3.0$, respectively. The beam widths of each beam get shifted towards higher $\eta $ with a rise in the $\alpha {E}_{00}^{2}$ parameter. The focusing behavior of these waves is reduced with a rise in $\alpha {E}_{00}^{2}$ values. This is due to the supremacy of the divergence term in comparison to the convergence term at larger $\alpha {E}_{00}^{2}$ values.
Figure 1. The impact of the intensity of the laser beam on the beam width ${f}_{0}$ of the pump wave. The black, red, and green curves represent $\alpha {E}_{00}^{2}=2.0,2.5\,{\rm{and}}\,3.0$, respectively. |
Figure 2. The impact of the intensity of the laser beam on the beam width $f$ of the EPW. The black, red, and green curves represent $\alpha {E}_{00}^{2}=2.0,2.5\,{\rm{and}}\,3.0$, respectively. |
Figure 3. The impact of the intensity of the laser beam on the beam width ${f}_{s}$ of the scattered wave. The black, red, and green curves represent $\alpha {E}_{00}^{2}=2.0,2.5\,{\rm{and}}\,3.0$, respectively. |
The impact of the density of plasma electrons $\frac{{\omega }_{{\rm{p}}}^{2}}{{\omega }_{0}^{2}}$ on the beam widths ${f}_{0},$ $f$ and ${f}_{s}$ of the pump wave, EPW and scattered wave is shown in figures 4, 5 and 6, respectively. The green, red and black curves represent $\frac{{\omega }_{{\rm{p}}}^{2}}{{\omega }_{0}^{2}}=0.15,0.20\,{\rm{and}}\,0.25$, respectively. The beam widths of each beam get shifted towards smaller $\eta $ with a rise in $\frac{{\omega }_{{\rm{p}}}^{2}}{{\omega }_{0}^{2}}$ values. The focusing behavior of these waves is increased with a rise in $\frac{{\omega }_{{\rm{p}}}^{2}}{{\omega }_{0}^{2}}$ values. This is due to the supremacy of the convergence term in comparison to the divergence term at larger $\frac{{\omega }_{{\rm{p}}}^{2}}{{\omega }_{0}^{2}}$ values.
Figure 4. The impact of the density of plasma electrons $\frac{{\omega }_{{\rm{p}}}^{2}}{{\omega }_{0}^{2}}$on the beam width ${f}_{0}$ of the pump wave. The green, red and black curves represent $\frac{{\omega }_{{\rm{p}}}^{2}}{{\omega }_{0}^{2}}=0.15,0.20\,{\rm{and}}\,0.25$, respectively. |
Figure 5. The impact of the density of plasma electrons $\frac{{\omega }_{{\rm{p}}}^{2}}{{\omega }_{0}^{2}}$ on the beam width $f$ of the EPW. The green, red and black curves represent $\frac{{\omega }_{{\rm{p}}}^{2}}{{\omega }_{0}^{2}}=0.15,0.20\,{\rm{and}}\,0.25$, respectively. |
Figure 6. The impact of the density of plasma electrons $\frac{{\omega }_{{\rm{p}}}^{2}}{{\omega }_{0}^{2}}$ on the beam width ${f}_{s}$ of the scattered beam. The green, red and black curves represent $\frac{{\omega }_{{\rm{p}}}^{2}}{{\omega }_{0}^{2}}=0.15,0.20\,{\rm{and}}\,0.25$, respectively. |
The impact of the pump radius of the beam ${r}_{0}$ on the beam widths ${f}_{0},$ $f$ and ${f}_{s}$ of the pump wave, EPW and scattered wave is shown in figures 7, 8 and 9, respectively. The black, red and green curves represent ${r}_{0}=20\,\mu {\rm{m}},25\,\mu {\rm{m}},30\,\mu {\rm{m}}$, respectively. The beam widths of each beam get shifted towards smaller $\eta $ values with a rise in ${r}_{0}$ values. The focusing behavior of these waves is increased with a rise in ${r}_{0}$ values. This is due to the supremacy of the convergence term in comparison to the divergence term at larger ${r}_{0}$ values.
Figure 7. The impact of the pump radius of the beam ${r}_{0}$ on the beam width ${f}_{0}$ of the pump wave. The black, red and green curves represent ${r}_{0}=20\,\mu {\rm{m}},25\,\mu {\rm{m}},30\,\mu {\rm{m}}$, respectively. |
Figure 8. The impact of the pump radius of the beam ${r}_{0}$ on the beam width $f$ of the EPW. The black, red and green curves represent ${r}_{0}=20\,\mu {\rm{m}},25\,\mu {\rm{m}},30\,\mu {\rm{m}}$, respectively. |
Figure 9. The impact of the pump radius of the beam ${r}_{0}$ on the beam width ${f}_{s}$ of the scattered wave. The black, red and green curves represent ${r}_{0}=20\,\mu {\rm{m}},25\,\mu {\rm{m}},30\,\mu {\rm{m}}$, respectively. |
The impact of intensity of the laser beam $\alpha {E}_{00}^{2}$ on SRS back-reflectivity ‘R' is shown in figure 10. The intensity parameters $\alpha {E}_{00}^{2}=2.0\,{\rm{and}}\,3.0$ are for the red curve and black curve, respectively. An increment in the $\alpha {E}_{00}^{2}$ parameter causes a reduction in SRS back-reflectivity ‘R'. This is because the extent of the focusing ability of distinct waves involved in the SRS process is reduced at larger $\alpha {E}_{00}^{2}$ values. The results are in agreement with experimental results [48].
Figure 10. The impact of the intensity of the laser beam $\alpha {E}_{00}^{2}$ on SRS back-reflectivity ‘R'. The intensity parameters $\alpha {E}_{00}^{2}=2.0\,{\rm{and}}\,3.0$ are for the red curve and black curve, respectively. |
The impact of the density of plasma electrons $\frac{{\omega }_{{\rm{p}}}^{2}}{{\omega }_{0}^{2}}$ on SRS back-reflectivity ‘R' is shown in figure 11. The plasma density parameters $\frac{{\omega }_{{\rm{p}}}^{2}}{{\omega }_{0}^{2}}=0.15\,{\rm{and}}\,0.25$ are for the black curve and red curve, respectively. An increment in the $\frac{{\omega }_{{\rm{p}}}^{2}}{{\omega }_{0}^{2}}$ parameter produces an increase in SRS back-reflectivity ‘R'. This is because the extent of the focusing ability of distinct waves involved in the SRS process is increased at larger $\frac{{\omega }_{{\rm{p}}}^{2}}{{\omega }_{0}^{2}}$ values. The results are in agreement with experimental results [49].
Figure 11. The impact of the density of plasma electrons $\frac{{\omega }_{{\rm{p}}}^{2}}{{\omega }_{0}^{2}}$ on SRS back-reflectivity ‘R'. The plasma density parameters $\frac{{\omega }_{{\rm{p}}}^{2}}{{\omega }_{0}^{2}}=0.15\,{\rm{and}}\,0.25$ are for the black curve and red curve, respectively. |
The impact of the pump radius of the beam ${r}_{0}$ on SRS back-reflectivity ‘R' is shown in figure 12. The beam radius parameters ${r}_{0}=20\,\mu m\ \mathrm{and}\ 25\,\mu {\rm{m}}$ are for the red curve and black curve, respectively. An increment in the ${r}_{0}$ parameter produces an increase in SRS back-reflectivity ‘R'. This is because the extent of the focusing ability of distinct waves involved in the SRS process is increased at larger ${r}_{0}$ values.
Figure 12. The impact of the pump radius of the beam ${r}_{0}$ on SRS back-reflectivity ‘R'. The beam radius parameters ${r}_{0}\,=20\,\mu {\rm{m}}\ \mathrm{and}\ 25\,\mu {\rm{m}}$ are for the red curve and black curve, respectively. |
5. Conclusions
The influence of a self-focused beam on the SRS process in collisional plasma is explored in the present study. The focusing ability of various waves involved in the process is found to get enhanced with a decrease in the $\alpha {E}_{00}^{2}$ parameter and with a rise in $\frac{{\omega }_{{\rm{p}}}^{2}}{{\omega }_{0}^{2}}$ and ${r}_{0}$ values. SRS back-reflectivity is found to get enhanced with a decrease in the $\alpha {E}_{00}^{2}$ parameter and with a rise in $\frac{{\omega }_{{\rm{p}}}^{2}}{{\omega }_{0}^{2}}$ and ${r}_{0}$ values. The results of the present study are really useful in inertial confinement fusion. This is because a decrease in SRS back-reflectivity causes improvement in laser-plasma coupling efficiency.