Modulational instability and nonlinear dynamics of femtosecond lasers in transparent materials with non-Kerr nonlinearities

Femtosecond laser inscription in transparent materials is a physical process that finds widespread applications in material engineering, particularly in laser micromachining technology. In this process, the nonlinear optical response of the transparent material can be either intrinsic or induced by multiphoton ionization processes. In this work, a generic model is considered to describe the dynamics of femtosecond laser filamentation in transparent materials characterized by non-Kerr nonlinearities, focusing on the influence of multiphoton ionization processes in the generation of an electron plasma of inhomogeneous density. The mathematical model consists of a complex Ginzburg-Landau equation with a generalized saturable nonlinearity, besides the residual nonlinearity related to multiphoton ionization processes. This generalized complex Ginzburg-Landau equation is coupled to a rate equation for time evolution of the electron plasma density, where multiphoton ionizations are assumed to be the sole processes controlling the generation of the electron plasma. Dynamical properties of the model are discussed starting from the continuous-wave regime, where a modulational-instability analysis enables us to determine the stability conditions of continuous-wave modes in the system. The analysis reveals a dominant tendency of continuous-wave stability for relatively large values of the multiphoton ionization order K, provided the femtosecond laser operates in the anomalous dispersion regime. Numerical simulations of the mathematical model feature a family of wavetrains composed of self-focused, well-separated, pulse-shaped optical filaments whose repetition rates are shortened but amplitudes are increased, with an increase in K. Simulations suggest that such nonlinear wavetrain structures do not need the transparent material to be intrinsically nonlinear and that they may also be favored solely by the nonlinearity induced by multiphoton ionization processes in a linear transparent material.