Effects of self-steepening on femtosecond laser filamentation in nonlinear transparent materials

Self-steepening refers to a shape distortion experienced by an optical pulse propagating in a nonlinear transparent medium, due to the competition between a nonlinear dispersion and the intensity-dependent refractive index of the propagation medium. Although its contribution to the phenomenon of laser filamentation in nonlinear optical media has been highlighted in the literature, a theoretical evidence of the impact of self-steepening on the optical field profile has not been provided so far. In this study we report results of theoretical investigations predicting the emergence of envelope solitons and optical kinks in a Kerr optical medium, characterized by a nonlinear dispersion and high-order optical nonlinearities. The mathematical model consists of a K-order nonlinear complex Ginzburg-Landau equation with self-steepening coupled to a rate equation. Such model features a femtosecond laser filamentation occuring in a nonlinear transparent material, in the presence of K-order photon absorption processes which favor the formation of a free-electron plasma with time-varying density. With the help of an appropriate ansatz and introducing a specific coordinate that merges time and space variables into one single spatio-temporal coordinate, the complex Ginzburg-Landau equation together with the rate equation are mapped onto a system of coupled first-order ordinary differential equations. The system of coupled first-order ordinary differential equations are solved numerically, by means of a sixth-order Runge-Kutta algorithm with fixed step. Numerical results suggest that self-steepening promotes an optical-shock structure with a kink shape, that causes a rapid saturation of the density of induced electron plasma as it varies with time. The kink distortion of the propagating optical field steepens as the strength of self-steepening is increased, becoming insensitive to a variation of the parameter K.