This paper presents theoretical investigation of effect of relativistic self-focusing of cosh-Gaussian (ChG) laser beam on second-harmonic generation in an underdense plasma. Steep transverse density gradients are produced in the plasma by the electron plasma wave excited by relativistic self-focusing of ChG laser beam. The generated plasma wave interacts with the pump beam to produce its second harmonics. Following Jeffrey Wentzel Kramers Brillouin (J.W.K.B) approximation and moment theory the differential equation governing the evolution of spot size of laser beam with distance of propagation has been derived. The differential equation so obtained has been solved numerically by the Runge–Kutta method to investigate the effect of decentered parameter, intensity of laser beam as well as density of plasma on self-focusing of the ChG laser beam, and generation of its second harmonics. It has been observed that the peak intensity of the laser beam shifts in the transverse direction by changing the decentered parameter and a noticeable change is observed on focusing of the laser beam as well as on conversion efficiency of second harmonics.

Self-shaping and propagation of intense laser beams of different radial profiles in plasmas is investigated. It is shown that when a relativistic elliptically Gaussian beam propagates through an underdense plasma, its radial profile will self-organize into a circularly symmetric self-similar smooth configuration. Such self-similar propagation can be attributed to a soliton-like structure of the laser pulse. The anisotropic electron distribution results in a circular electric field that redistributes the electrons and modulates the laser pulse to a circular radial shape.