A perturbative approach to self-phase modulation and self-steepening of short laser pulses propagating in nonlinear media

The solution of the wave equation in the envelope approximation with temporal corrections for a laser pulse propagating in a medium where the Kerr effect, field ionization, and associated absorption take place, is obtained through a first order perturbative approach. Dispersion is, however, neglected to decouple the equations for the amplitude and the phase. The closed-form expressions so obtained clarify the influence of the various terms of the equation on the laser amplitude and on the frequency generation as a function of the retarded time. Furthermore, they allow extracting scaling parameters which size the nonlinear effects. The results are illustrated quantitatively in the case of a femtosecond pulse focused in the air with typical parameters. (paper)