This article presents the cross-focusing of two high power dark hollow Gaussian beams in plasma when relativistic nonlinearity is operative. A paraxial like approach has been used in the present analysis. In this study, the non-linear dielectric function has been expanded in terms of radial distance from the maximum of the irradiance, rather than from the axis, as is the case of Gaussian beams. The nature of propagation of a hollow Gaussian beam propagating in plasmas has been studied under the influence of relativistic non-linearity. The effect on the order (n) of hollow Gaussian beam on the cross-focusing of two beams has been explored in relativistic non-linearity in this publication.

In the present paper, we have investigated the growth of a Gaussian perturbation superimposed on a Gaussian laser beam. The nonlinearity we have considered is of relativistic type. We have setup the nonlinear differential equations for beam width parameter of the main beam, growth and width of the laser spike by using the WKB and paraxial ray approximation. These are coupled ordinary differential equations and therefore these are simultaneously solved numerically using the Runge Kutta method. It has been observed from the analysis that self-focusing/defocusing of the main beam and the spike determine the growth dynamic of the spike.

This paper presents an effect of relativistic mutual interaction of two laser beams of different frequencies on the growth of a laser ripple in laser produced plasmas. The nonlinearity due to relativistic mass variation depends not only on the intensity of one laser but also on the second laser. Therefore, one laser beam affects the dynamics of the second beam and hence a mutual nonlinear interaction (cross-focusing) takes place. The dynamical equations governing the laser intensity of two laser beams and the perturbation present on one laser beam (ripple) have been set up and a numerical solution has been presented for typical laser plasma parameters. It is found that a change in the intensity of the second laser beam can affect the growth of the laser ripple significantly. This study is important in plasma beat wave excitation and collective laser particle accelerators.

This article presents an effect of cross focusing of two laser beams on the growth of a laser ripple in laser-produced plasmas. The mechanism of nonlinearity is assumed to be ponderomotive force, arising because of the Gaussian intensity distribution of the laser beams. The dynamical equation governing the laser ripple intensity has been set up and a numerical solution has been presented for typical laser plasma parameters. It is found that the change in the intensity of the second laser beam can affect the growth of the laser ripple significantly.