Simplified approach to the modeling of nonlinear optical propagation of tightly focused beams

We propose a simplified model based on ordinary differential equations (ODEs) to describe the nonlinear optical propagation of tightly focused beams in the case of negligible temporal dispersion. We first find an analytical form for the power decay due to multi-photon ionization (MPI). In a second step, we couple such a result with an ODE for the beam second moment accounting for the self-phase modulation. The accuracy of our method is tested versus numerical simulations based on a beam propagation method (BPM). Our results find direct application in the vast field of femtosecond writing and laser material processing.