A semi-analytic model for tightly focused ultrashort pulses in the nonlinear regime

When intense ultrashort pulses are focused into a transparent solid, the large intensities achieved in the focal region can lead to permanent modifications. Whereas such a technique has been widely employed to realize photonic devices [1], there has been a relative lack of theoretical models capable of describing the light-matter interaction in this highly nonlinear regime. The simulation of the nonlinear optical propagation is indeed quite demanding: a spatio-temporal computation of the field is required to describe the interplay between spatial diffraction, temporal dispersion, Kerr self-focusing, plasma defocusing, nonlinear absorption, to cite the most important phenomena at work [2]. Although the propagation of single pulses is relatively simple, the full problem requires the inclusion of additional effects occurring in a slower scale, such as thermal diffusion and emission of acoustic waves. From a theoretical point of view, a simplified model for the optical propagation in the nonlinear regime would greatly simplify the overall modeling of the physics at work. On more empirical grounds, even a rough knowledge of the deposited energy can help in improving the laser-writing. Here we present a model for the optical propagation solely based on ordinary different equations (ODEs), thus much easier and faster to solve than models based upon partial different equations, such as the nonlinear Schrödinger equation (NLSE) or nonlinear FDTD [3]. We test our results with respect to the NLSE, showing the reliability of our model for a wide interval of excitations.