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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