Relativistic and ponderomotive self-focusing of a laser beam in a radially inhomogeneous plasma. I. Paraxial approximation

Abstract

The propagation of a high-irradiance laser beam in a plasma whose optical index depends nonlinearly on the light intensity is investigated through both theoretical and numerical analyses. The nonlinear effects examined herein are the relativistic decrease of the plasma frequency and the ponderomotive expelling of the electrons. This paper is devoted to focusing and defocusing effects of a beam assumed to remain cylindrical and for a plasma supposed homogeneous along the propagation direction but radially inhomogeneous with a parabolic density profile. A two-parameter perturbation expansion is used; these two parameters, assumed small with respect to unity, are the ratio of the laser wavelength to the radial electric-field gradient length and the ratio of the plasma frequency to the laser frequency. The laser field is described in the context of a time envelope and spatial paraxial approximations. An analytical expression is provided for the critical beam power beyond which self-focusing appears; it depends strongly on the plasma inhomogeneity and suggests the plasma density tailoring in order to lower this critical power. The beam energy radius evolution is obtained as a function of the propagation distance by numerically solving the paraxial equation given by the two-parameter expansion. The relativistic mass variation is shown to dominate the ponderomotive effect. For strong laser fields, self-focusing saturates due to corrections of fourth order in the electric field involved by both contributions.