Niti Kant¹, Shivani Vij², Sudarshan Kumar Chakravarti³, Juleshwar Prasad Kushwaha³, Vishal Thakur^,11 Department of Physics, Lovely Professional University, G.T. Road, Phagwara 144411, Punjab, India
2 Department of Applied Sciences, DAV Institute of Engineering & Technology, Jalandhar-144008, Punjab, India
3 Department of Physics, T.M.B. University, Bhagalpur, India

Corresponding authors: *E-mail:vishal20india@yahoo.co.in

Received:2019-05-21Online:2019-12-1


Abstract
In the present manuscript, we analyse the effect of exponential plasma density ramp for relativistic self-focusing of Hermite-cosh-Gaussian laser pulse in magnetoplasma. The exponential plasma density ramp is found to be more prominent in achieving the stronger self-focusing of Hermite-cosh-Gaussian laser beam in comparison to the tangential plasma density ramp. We propose a theoretical model for propagation of Hermite-cosh-Gaussian laser pulse in magnetoplasma with exponential density ramp. The nonlinearity in the medium arises because of the relativistic motion of electrons, being responsible for relativistic self-focusing. Equation of the beam width parameter is set up by taking the expression for the dielectric function and following Wentzel-Kramers-Brillouin (WKB) with paraxial ray approximations for mode indices $m$ = 0, 1 and 2. Effect of decentered parameter is also analysed, which results in stronger self-focusing of the Hermite-cosh-Gaussian laser beam. Stronger self-focusing of laser beam is more pronounced in high density plasma with higher magnetic field.
Keywords： self-focusing;decentered parameter;exponential density ramp;Hermite-cosh-Gaussian beam


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Niti Kant, Shivani Vij, Sudarshan Kumar Chakravarti, Juleshwar Prasad Kushwaha, Vishal Thakur. Relativistic Self-Focusing of Hermite-cosh-Gaussian Laser Beam in Magnetoplasma with Exponential Plasma Density Ramp. [J], 2019, 71(12): 1469-1474 doi:10.1088/0253-6102/71/12/1469

1 Introduction

In the last few years, interaction of the laser beam with plasma has been world widely studied because of its unique applications, which include inertial confinement fusion,^[1-3] the laser electron acceleration,^[4-6] self-focusing, harmonic generation^[7] etc. Among these nonlinear effects one of the most important effects is self-focusing, which has been very fascinating area of research for past few years. Whenever high power laser beam penetrates through plasma, dielectric function of the plasma is modified due to the oscillatory velocity of electrons and hence results in relativistic self-focusing. Focusing and defocusing of the first six TEM$_{0p}$ Hermite-cosh-Gaussian laser beam in collisionless plasma was studied by Takale $et al.$^[8] and it was observed that, the modes with the odd p-values defocuses whereas, with even p-values show oscillatory along with defocusing behavior of the beam.

"Gaussian beam'' normally implies radiation confined to the fundamental (TEM$_{00}$) mode, is a of monochromatic light having transverse magnetic and amplitude given by the Gaussian function. However, in case of Hermite cosh Gaussian (HchG) beam, paraxial solutions of the Helmoltz equation as Hermite-sinusoidal-Gaussian beams, which was given by Casperson and Tovar.^[9] Special case of the Hermite-sinusoidal Gaussian beams is the distribution at $z$ = 0 is known as HchG beam. Hermite-cosh-Gaussian beam can be obtained in the laboratory by the superposition of two decentered Hermite-Gaussian beams as cosh-Gaussian ones. Propagation of such HchG beam in plasmas was studied theoretically by Belafhal and Ibnchaikh^[10] and Patil $et al.$^[11]

A theoretical model was presented to produce THz radiation by two cross-focused copropagating Gaussian laser beams in a density rippled magnetized plasma.^[12] They concluded that with the increase in magnetic field, amplitude of the THz radiation generation increases significantly. Optimization of the laser-plasma parameters gives the radiated normalized THz power of the order of 10 kW. Also, it was observed that circularly or elliptically polarized THz radiation can be generated when a static magnetic field is imposed on a gas target along the propagation direction of a two-color laser driver.^[13] Fujioka $et al.$^[14] observed the kilo Tesla (kT) magnetic fields using a capacitor-coil target, in which two nickel disks are connected by a U-turn coil. A magnetic flux density of 1.5 kT was measured during the Faraday effect 650 $\mathrm{\mu}$m away from the coil, when the capacitor was driven by two beams from the GEKKO-XII laser. Wang $et al.$^[15] investigated the fast ignition via integrated particle-in-cell simulation including both generation and transport of fast electrons. With this scheme it is demonstrated that two counter propagating, 6 ps, 6 kJ lasers along the magnetic field transfer 12% of their energy to the core, which is then heated to 3 keV.

The importance of the density transition and the decentered parameter for stronger self-focusing of the laser beam was successfully reported by Nanda and Kant.^[16] Also, it has been observed that, under the influence of magnetic field and plasma density ramp, the self-focusing becomes stronger up to great extent.^[17] Kant and Wani,^[18] examined the behavior of the beam width parameter for normalized distance of propagation at optimized values of decentered parameters, plasma density and various absorption levels. Further, in quantum plasma self-focusing of cosh-Gaussian laser beam was presented by Habibi and Ghamari^[19] by following higher order paraxial theory. They got better results for self-focusing of cosh-Gaussian laser beams in comparison to Gaussian beams by selecting suitable values of decentered parameter. By considering the ponderomotive nonlinearity, self-focusing of HchG laser beams in magnetoplasma was theoretically observed by Patil $et al.$^[11] It was noticed that, self-focusing is significantly increased by the presence of mode index and decentered parameters. Singh and Walia^[20] followed the moment theory to explain nonlinear differential equation of beam width parameter to analyze the self-channeling of Gaussian laser beam and relativistic self-focusing.

In the past few years the paraxial wave family of the laser beams has become a considerable area of interest. HchG laser beam is found to be major solutions of the paraxial wave equation. Effective and early relativistic self-focusing of cosh-Gaussian laser beam with cold quantum plasma was observed by Nanda $et al.$^[21] They reported a comparative study of self-focusing of cosh-Gaussian laser in classical relativistic case and relativistic cold quantum plasma. It was noticed that, when the laser beam penetrates deeper inside the relativistic cold quantum plasma, self-focusing of laser beam increases significantly with the early effects because of the quantum contribution. Aggarwal $et al.$^[22] observed the importance of quantum effects for enhanced self-focusing with combined effect of ponderomotive and relativistic nonlinearity.

Gill $et al.$^[23] exploited an analysis of self-phase modulation and self-focusing of the cosh-Gaussian laser beam in plasma by considering the combined effect of ponderomotive and relativistic nonlinearity. The occurrence of self-focusing with strong effects was seen and concluded that for optimized value of decentered parameters stronger effects occur in weakly relativistic ponderomotive case in comparison to relativistic case alone.^[24]

For last few years, in order to have stronger self-focusing during the laser plasma interaction slowly increasing plasma density ramp is introduced. But it is noticed that self-focusing by using such type of density profile i.e. tangential density ramp $(n(\xi)=n_0\tan(\xi/d))$ is limited to the value of $\xi<\xi_d$, where $\xi_d$ is that value of the distance of propagation at which density will approach to infinity. Hence, in order to overcome these limitations, exponential density ramp profile was proposed by Sen $et al.$^[25] In this investigation, self-focusing of HchG laser is examined by developing the equation of beam width parameter in magnetoplasma with exponential density transition under relativistic effects. Effect of the exponential density transition is observed and it is observed that, the laser beam penetrates up to larger distances without getting highly diverged hence, resulting in stronger self-focusing of laser beam. Efficient self-focusing is noticed with the exponential density ramp in comparison with tangential plasma density ramp. A rise in self-focusing of laser has been noticed with the enhancement in decentered parameter, plasma frequency and magnetic field with the early effects. The result of the present work may be helpful in knowing the physics behind high power laser driven fusion and can add to the explanation of the devices involved in the beam propagation in various applications.

The manuscript is defined as follows: Section 2 is devoted to the equations governing nonlinear dielectric function and the characteristics of the beam width parameter with normalized propagation distance. Section 3 presents the results and discussions. Final conclusion of the investigation is presented in Sec. 4.

2 Theoretical Considerations

Field distribution of Hermite-cosh-Gaussian laser beam travelling along z-axis in plasma can be written as

(1) $$ E(r,z) = \frac{{{E_0}}}{{f(z)}}\Big[ {{H_m}\Big( {\frac{{\sqrt 2r}}{{{r_0}f(z)}}} \Big)} \Big] {e^{{{{b^2}}}/{4}}}\nonumber\\ \times\Big\{ {{\exp\Big[{\! -\! {{\Big({\frac{r}{{{r_0}f(z)}}\! +\! \frac{b}{2}} \Big)}^2}}\Big]}\! +\! {\exp\Big[{\! -\! {{\Big({\frac{r}{{{r_0}f(z)}}\! -\! \frac{b}{2}} \Big)}^2}}\Big]}}\! \Big\}, $$
where $E_{0}$ represents amplitude of HchG laser, $H_m$ is the Hermite polynomial of the $m^{\rm th}$ order, $f(z)$ represents dimensionless beam width parameter, $b$ shows the decentered parameter of beam and $r_0$ is the spot size of laser beam. Oscillatory velocity related to the electrons is $v=eE/m_0\omega\gamma$. Here $\omega, e$, and $m_0$ are the angular frequency of the incident laser beam, electronic charge, and rest mass respectively. Also $\gamma=\sqrt{1+α EE^*}$ represents the relativistic factor where $α =e^{2}/m^2_0\omega^2c^2$, stands for the speed of light in vacuum. The dielectric function for the nonlinear medium can be written as

(2) $$ \varepsilon=\varepsilon_0+\phi(EE^*), $$
where $\varepsilon_0=1-\omega_P^2/\omega^2$ represents linear part of dielectric function with $\omega_P$ as the frequency of plasma. Taking the relativistic motion of electron, one can write $m_e=m_0\gamma$. Therefore, the dielectric function of plasma is taken in the form, $\varepsilon_{\rm rel}=1-\omega_P^2/\gamma\omega^2$ with equilibrium plasma frequency $\omega_P=(4\pi e^2n(\xi)/m_0)^{1/2}$.

The intensity dependent nonlinear part of the dielectric function can be given as

$ \phi(EE^*)=\Big(\frac{\omega_{P0}^2}{\omega^2}\Big)\Big[1-\frac{1}{(1+α EE^*)^{1/2}}\Big].$ ({)3a}$(3a)

Plasma density ramp can be assumed as, $n(\xi)=n_0\exp(\xi/d)$ and hence, $\varepsilon_{\rm rel}=1-(\omega_{P0}^2\exp(\xi/d)/\gamma\omega^2)$,where $\xi=z/R_d$ stands for normalized propagation distance, $d$ is the adjustable constant, and $R_d$ is diffraction length.

Thus, nonlinear part of the dielectric function can be given as

$ \phi(EE^*)=\frac{\omega_{P0}^2\exp(\xi/d)}{\omega^2\gamma(1-\omega_c/\omega)},$ (3b)

where $\omega_c(=eB_0/mc)$, represents the cyclotron frequency. By using Maxwell's equations, one can derive the wave equation for laser propagation as

(4) $$ \frac{\partial^2\vec{E}}{\partial{}z^2}+\frac{\partial^2\vec{E}}{\partial{}r^2}+\frac{1}{r}\frac{partial\vec{E}}{\partial{}r}+\frac{\varepsilon\omega^2}{c^2}\vec{E}=0. $$
Solution of Eq. (4) can be written as

(5) $$ \vec{E}=A(r, z)\exp[i(\omega{}t-kz)], $$
where $A(r, z)$ represents complex amplitude of electric field, $k=\sqrt{\omega^2-\omega_{P0}^2\exp(\xi/d)/\gamma(1-\omega_c/\omega)}$ and $A(r, z)=A_0(r, z)\exp[-i{}kS(r, z)],$ where $A_0$ and $S$ represent real functions of $r$ and $z$ correspondingly. Putting the expressions for $\vec{E}$ and $A(r, z)$ in Eq. (4), and hence separate the real and imaginary parts of resulting equation as

(6) $$ - 2\frac{{\partial S}}{{\partial z}} + \frac{{S{\omega ^2}\exp (z/d{R_d})}}{{{c^2}{k^2}d{R_d}}}\frac{{\omega _{P0}^2}}{{\gamma {\omega ^2}\left({1 - {\omega _c}/\omega } \right)}} + \frac{{z{\omega ^2}\exp(z/d{R_d})}}{{{c^2}{k^2}d{R_d}}}\frac{{\omega _{P0}^2}}{{\gamma {\omega ^2}\left({1 - {\omega _c}/\omega } \right)}}\frac{{\partial S}}{{\partial z}}\\- \frac{{zS{\omega ^4}\exp (z/d{R_d})}}{{2{c^4}{k^4}{d^2}R_d^2}}{\left({\frac{{\omega _{P0}^2}}{{\gamma {\omega ^2}\left( {1 - {\omega _c}/\omega }\right)}}} \right)^2} + \frac{{z{\omega ^2}\exp(z/d{R_d})}}{{{c^2}{k^2}d{R_d}}}\frac{{\omega _{P0}^2}}{{\left( {1 - {\omega_c}/\omega } \right)\gamma {\omega ^2}}}\\- \frac{{{z^2}{\omega ^4}\exp (z/d{R_d})}}{{4{c^4}{k^4}{d^2}R_d^2}}{\left({\frac{{\omega _{P0}^2}}{{\left( {1 - {\omega _c}/\omega } \right)\gamma {\omega^2}}}} \right)^2} - {\left( {\frac{{\partial S}}{{\partial r}}} \right)^2} +\frac{1}{{2A_0^2{k^2}}}\frac{{{\partial ^2}A_0^2}}{{\partial {r^2}}} -\frac{1}{{4A_0^4{k^2}}}{\left( {\frac{{\partial A_0^2}}{{\partial r}}}\right)^2}\\ + \frac{1}{{2rA_0^2{k^2}}}\left( {\frac{{\partial A_0^2}}{{\partial r}}}\right) + \frac{{\phi(A_0^2)}}{{{\varepsilon _0}}} =0, $$
(7) $$ - \frac{1}{{A_0^2}}\left( {\frac{{\partial A_0^2}}{{\partial z}}} \right) + \frac{{{\omega ^2}\exp (z/d{R_d})}}{{{c^2}{k^2}d{R_d}}}\frac{{\omega_{P0}^2}}{{\gamma {\omega ^2}\left( {1 - {\omega _c}/\omega } \right)}} +\frac{{z{\omega ^2}\exp (2z/d{R_d})}}{{{c^2}{k^2}{d^2}R_d^2}}\frac{{\omega_{P0}^2}}{{\gamma {\omega ^2}\left( {1 - {\omega _c}/\omega } \right)}}\\+ \frac{{z{\omega ^4}\exp (z/d{R_d})}}{{4{c^4}{k^4}{d^2}R_d^2}}\frac{{\omega _{P0}^2}}{{\gamma {\omega ^2}\left( {1 - {\omega _c}/\omega } \right)}} -\frac{{z{\omega ^2}\exp (z/d{R_d})}}{{2{c^2}{k^2}d{R_d}A_0^2}}\frac{{\omega_{P0}^2}}{{\gamma {\omega ^2}\left( {1 - {\omega _c}/\omega }\right)}}\frac{{\partial A_0^2}}{{\partial z}} - \frac{{{\partial^2}S}}{{\partial {r^2}}}\\- \frac{1}{{A_0^2}}\frac{{\partial S}}{{\partial r}}\frac{{\partial A_0^2}}{{\partial r}} - \frac{1}{r}\frac{{\partial S}}{{\partial r}} =0. $$
The solutions of Eqs. (6) and (7) are of the form

$ A_0^2 = \frac{{E_0^2}}{{{f^2}(z)}}{\Big[ {{H_m}\Big( {\frac{{\sqrt 2r}}{{{r_0}f}}} \Big)} \Big]^2}\exp{ \Big( \frac{b^2}{2} \Big) }\Big\{ {{\exp{\! \Big[ -\! 2\Big({\frac{r}{{{r_0}f(z)}}\! +\! \frac{b}{2}} \Big)}}^2 \Big] \! +\! {\exp{\! \Big[ -\! 2\Big({\frac{r}{{{r_0}f(z)}}\! -\! \frac{b}{2}} \Big)}}^2\Big]\! +\! 2{\exp{\! \Big[ -\! \Big({\frac{{2{r^2}}}{{r_0^2{f^2}(z)}}\! +\! \frac{{{b^2}}}{2}} \Big)}}}\Big]\! \Big\},$ (8)} S=\frac{r^2}{2}\beta(z)+\varphi(z),$ (9)$

where $\beta(z)=(1/f(z))d{}f/d{}z$, $\phi(z)$ represents arbitrary function of $z$. Substituting these values in Eq. (6), we get the equations governing the evolution of beam width parameter.

For $m$ = 0

(10) $$ \left[ {\frac{{\xi \exp (\xi d)}}{{2d\left\{ {1 - \frac{{\omega _{P0}^2\exp (\xi d)}}{{\gamma {\omega ^2}\left( {1 - {\omega _c}/\omega } \right)}}} \right\}}}\left( {\frac{{\omega _{P0}^2}}{{\gamma {\omega ^2}}}} \right) - 1} \right]\frac{{{d^2}f}}{{d{\xi ^2}}} - \frac{{\xi \exp (\xi d)}}{{2d\left\{ {1 -\frac{{\omega _{P0}^2\exp (\xi d)}}{{\gamma {\omega ^2}\left( {1 - {\omega_c}/\omega } \right)}}} \right\}}}\left( {\frac{{\omega _{P0}^2}}{{\gamma {\omega^2}}}} \right)\frac{1}{f}{\left( {\frac{{d{}f}}{{d\xi }}} \right)^2} +\frac{{\left( {4 - 4{b^2}} \right)}}{{{f^3}}}\\- \frac{{4α E_0^2}}{{{f^3}}}\left( {\frac{{\omega _{P0}^2\exp (\xi/d)}}{{{\omega ^2}}}} \right){\left( {\frac{{\omega {r_0}}}{c}} \right)^2}{\left({1 + \frac{{4α E_0^2}}{{{f^2}}}} \right)^{-({3}/{2})}}{e^{({{{b^2}}}/{2})}} =0. $$
For $m$ = 1

(11) $$ \left[ {\frac{{\xi \exp (\xi d)}}{{2d\left\{ {1 - \frac{{\omega _{P0}^2\exp (\xi d)}}{{\gamma {\omega ^2}\left( {1 - {\omega _c}/\omega } \right)}}} \right\}}}\left( {\frac{{\omega _{P0}^2}}{{\gamma {\omega ^2}}}} \right) - 1} \right]\frac{{{d^2}f}}{{d{\xi ^2}}} - \frac{{\xi \exp (\xi d)}}{{2d\left\{ {1 -\frac{{\omega _{P0}^2\exp (\xi d)}}{{\gamma {\omega ^2}\left( {1 - {\omega _c}/\omega } \right)}}} \right\}}}\left( {\frac{{\omega _{P0}^2}}{{\gamma {\omega^2}}}} \right)\frac{1}{f}{\left( {\frac{{d{}f}}{{d\xi }}} \right)^2} + \frac{{\left( {4 - 4{b^2}} \right)}}{{{f^3}}}\\ - \frac{{8α E_0^2}}{{{f^3}}}\left( {\frac{{\omega _{P0}^2\exp (\xi/d)}}{{{\omega ^2}}}} \right){\left( {\frac{{\omega {r_0}}}{c}}\right)^2}{e^{({{{b^2}}}/{2})}}\left( {{b^2} - 2} \right) =0. $$
For $m$ = 2

(12) $ \left[ {\frac{{\xi \exp (\xi d)}}{{2d\left\{ {1 - \frac{{\omega _{P0}^2\exp(\xi d)}}{{\gamma {\omega ^2}\left( {1 - {\omega _c}/\omega } \right)}}}\right\}}}\left( {\frac{{\omega _{P0}^2}}{{\gamma {\omega ^2}}}} \right) - 1}\right]\frac{{{d^2}f}}{{d{\xi ^2}}} - \frac{{\xi \exp (\xi d)}}{{2d\left\{ {1 -\frac{{\omega _{P0}^2\exp (\xi d)}}{{\gamma {\omega ^2}\left[ {\left( {1 -{\omega _c}/\omega } \right)} \right]}}} \right\}}}\left( {\frac{{\omega_{P0}^2}}{{\gamma {\omega ^2}}}} \right)\frac{1}{f}{\left( {\frac{{d{}f}}{{d\xi }}}\right)^2}\\+ \frac{{\left( {4 - 4{b^2}} \right)}}{{{f^3}}} - \frac{{16α E_0^2}}{{{f^3}}}\left( {\frac{{\omega _{P0}^2\exp (\xi d)}}{{{\omega ^2}}}}\right){\left( {\frac{{\omega {r_0}}}{c}} \right)^2}{\left( {1 + \frac{{16α E_0^2}}{{{f^2}}}} \right)^{3/2}}{e^{{b^2}/2}}\left( {5 - 2{b^2}} \right) =0.$
In the similar way, Eq. (7) gives the boundary conditions, at $\xi=0, f=1$, and $(d{}f/d\xi)=0$.

3 Results and Discussion

Consider incident Nd:YAG laser beam with frequency and the spot size as $\omega=1.778\times{}10^{15}$ rad/s and $r_0=80.82 \mu$m to do the numerical analysis. Equations (10), (11), and (12) have been solved to study the effects of exponential density ramp and decentered parameter on self-focusing of the HchG laser beam for mode indices $m$=0, 1, and 2. Presently we have taken the plasmas density ramp as $n(\xi)=n_0\exp(\xi/d)$ having initial electron density as $n_0=0.503\times{}10^{21}{\rm cm}^{-3}, d=0.05$. Here Fig. 1 shows the comparative study of dependence of the beam width parameter ($f$) on normalized distance of propagation ($\xi$) for mode indices $m=0, 1$, and 2 with exponential and tangential plasma density ramps. In Fig. 1(a), one may notice the effect of exponential density transition in comparison to tangential density ramp on the relativistic self-focusing of the HchG laser beam through magnetoplasma. Similar results with localized upward plasma density transition for the relativistic self-focusing of highly intense laser were shown by Gupta $et al.$^[26] where the stronger self-focusing occurs nearly at $\xi=0.5$. In this manuscript, strong self-focusing of the HchG laser beam occurs at $\xi=0.06$. In the similar way for $m$=1, in Fig. 1(b), it is noticed that, the beam passing through plasma with exponential as well as tangential density transition gets diffracted. In Fig. 1(c), strong self-focusing with early effects is noticed for $m=2$. Previously, Kant $et al.$^[27] have observed self-focusing of a laser beam with ponderomotive nonlinearity under plasma density transition and found strong self-focusing nearly at $\xi=0.5$. Here in this work, stronger self-focusing is noticed even at lower values of the normalized distance of propagation.

Figure 2 reveals the dependence of the beam width parameter ($f$) on normalized distance of propagation ($\xi$) for various values of the decentered parameters $b$ at mode indices $m$=0, 1, and 2. One may clearly observe that, self-focusing of the HchG laser increases with increase in the value of decentered parameter and shifted towards the lower values of distance of propagation. This is because, for higher values of the decentered parameter diffraction term in Eq. (12) becomes dominant and hence, leading to strong self-focusing effects. Strong self-focusing of the HchG laser beam under collisionless magnetoplasma was studied by Patil $et al.$^[11] which supports our results.

Figure 3 depicts dependence of beam width parameter ($f$) on normalized distance of propagation ($\xi$) for various values of the external magnetic field at mode indices $m$=0, 1, 2. One may clearly see that with the presence of static magnetic field, self-focusing of the laser beam enhances. Dynamics of the oscillating electrons are changed because of Lorentz force which further alters the plasma wave and hence, stronger self-focusing occurs. As the magnetic field increases, the nonlinear term begins to control over the diffractional divergence term. It further, changes the propagation characteristics of the plasma. Due, to strong interaction between the laser and magnetic fields, the laser beam is strongly focused. Thus, magnetic field plays a key role in the enhancement of the self-focusing of laser beam in plasma. The transverse magnetic field also reduces the critical power, which is a primary requirement for self-focusing. Therefore, magnetic field makes the self-focusing effect stronger to a larger extent. Magnetic field also confines the electron trajectory.^[28] Nanda $et al.$^[17] have observed effective self-focusing of the Gaussian beam at $\xi=0.1$. However, in our case stronger self-focusing for HchG laser beam occurs at $\xi=0.03$ for $\omega_c/\omega$=0.8 with early effects.

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Fig. 1Comparison between effects of exponential and tangential plasma density ramps on the dependence of beam width parameter ($f$) on normalized propagation distance ($\xi$) for various values of (a) $m=0$, (b) $m=1$, and (c) $m=2$. Rest of the parameters are considered as $α {}E_0^2=0.1, \omega_{P0}/\omega=0.75, \omega_c/\omega=0.8, b=0.9, \omega{}r_0/c=470$, and $d$=0.05.

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Fig. 2Dependence of beam width parameter ($f$) on normalized propagation distance ($\xi$) at various values of decentered parameter for (a) $m=0$, (b) $m=1$, and (c) $m=2$.



Rests of the parameters are same as considered in Fig. 1.

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Fig. 3Dependence of beam width parameter ($f$) on normalized propagation distance ($\xi$) at various values of $\omega_c/\omega$ for (a) $m=0$, (b) $m=1$, and (c) $m=2$. Rests of the parameters are same as considered in Fig. 1.

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Fig. 4Dependence of the beam width parameter ($f$) on normalized propagation distance ($\xi$) at different values of $\omega_{p0}/\omega$ for (a) $m=0$, (b) $m=1$, and (c) $m=2$. The restof parameters are same as considered in Fig. 1.



Figure 4 depicts the dependence of beam width parameter ($f$) on normalized distance of propagation ($\xi$) for various values of $\omega_{P0}/\omega$ at mode indices $m=0, 1, 2$. One may clearly notice that, self-focusing of HchG laser beam enhances and occurs earlier with the rise in the relative density parameter $\omega_{P0}/\omega$ for $m$=2. Patil $et al.$^[29] have observed effective self-focusing of the Gaussian beam at $\xi=0.5$. Here, in the current analysis, strong self-focusing of the HchG laser beam occurs at $\xi=0.03$.

4 Conclusion

Present manuscript describes the relativistic self-focusing of the HchG laser beam in the plasma under exponential density transition. Employing WKB approximation along with paraxial ray approach, equation of the beam width parameter is deduced and enhancement in self-focusing of the HchG beam is seen. A comparative study of exponential and tangential plasma density ramps for relativistic self-focusing of the HchG laser beam in magnetoplasma has been done. The former is found to be more prominent in achieving stronger self-focusing of Hermite-cosh-Gaussian laser beam as compared to the later one. Self-focusing of HchG laser beam for mode indices $m=0, 1$, and 2 is analyzed and found that self-focusing capability of the beam increases and happens with early effects. Also effective self-focusing is observed with rise in values of the decentered parameter at a specific intensity. It is noticed that with density transition the spot size of HchG beam shrinks considerably as it passes deeper inside the plasma. Present work may be employed in the analysis of quantum dots, laser induced fusion etc.

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