Analytical insights into self-phase modulation: beyond the basic theory

We present a closed-form analytical description of the early stages of spectral broadening of ultrashort laser pulses beyond the basic theory of self-phase modulation (SPM). In the limit of short propagation paths, approximate analytical expressions derived as a part of our treatment recover the canonical SPM-theory results for the nonlinear shift and spectral broadening. For longer propagation paths, these expressions shed light on how dispersion effects enter the scene, decelerating the spectral broadening in the regime of normal dispersion and giving rise to an explosion-like bandwidth growth in anomalous-dispersion high-soliton-number pulse evolution scenarios. Based on this formalism, we will provide an analytical derivation for the relation between the maximum soliton self-compression length and the soliton number, which has been previously treated as purely empirical.