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.

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In present paper the determination of the spot size of an ultra-short laser beam in uniform magnetized plasma with a dominant cold plasma has been studied. The liner dispersion relation of the laser beam propagating in Magnetized plasma have been found. Magnetic field is set up and source dependent expansion method is applied to determining the spot size of the intense laser beam with gaussian profile. The transverse magnetization of plasma and its impact on the self-focusing property of the dense laser governing to reduction in critical power necessary to self-focusing beam is shown.

In this article we present a discussion and overview of mathematical result of the self-focusing of a Langmuir wave which governs Zakharov system and has studied the self- focusing of a Langmuir wave following by Gaussian distribution. Langmuir wave propagates through uncharged plasma which governed by Zakharov systems. The phenomenon plays a vital role in the Dynamics. We present the article mathematical model with effect of Landou damping. Relativistic mass oscillation and ponderomotive force on electrons of the ionized plasma encouraged the Langmuir wave which resists the self-focusing effect when damping is ignored. The Beam radius gets narrow. when it further propagates considering the paraxial ray’s approximation, the self-focusing length Rn. It shows that characteristics of varying bandwidth distance of propagation in relativistic plasma.

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