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.
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 [1–8]. 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 [9–21]. Researchers are exploring these instabilities theoretically as well as experimentally for detailed information of intense laser interaction with plasma [22–27]. 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 [32–35]. The production of harmonics can be done through several mechanisms including plasma wave excitation electron plasma wave (EPW), plasma instabilities and resonant absorption [8, 36–39]. 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 [40–51]. 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
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
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
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
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:
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
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
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;
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.
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