(2+1)-D propagation of spatio-temporal solitary waves including higher-order corrections

We study the propagation of bright two-dimensional spatio-temporal solitary waves using a higher-order multi-dimensional non-linear Schrödinger equation. Starting directly from Maxwell's equations, a multiple-scales derivation is presented which results in a generalized first-order vectorial evolution equation that is valid for the non-linear spatio-temporal propagation of a predominantly linearly polarized electric field with large spatial and temporal bandwidths. A reduced version of this full equation including the higher-order linear and non-linear effects of third- and fourth-order dispersion, space–time focusing, shock, stimulated Raman scattering, and ultrafast quintic index saturation, is solved numerically via a modified split-step algorithm. Material parameters corresponding to those of fused silica at λf=1.55 m are used, with the addition of a negative quintic saturation term. Without quintic saturation, the non-linear spatio-temporal wave broadens under the action of the higher-order space–time effects. In addition, in the absence of Raman scattering, the wave undergoes collapse until arrested by the remaining higher-order terms. Frequency down-shifting and spatio-temporal broadening due to Raman scattering are found to have the greatest effect on non-linear spatio-temporal wave propagation. Nevertheless, we demonstrate that quintic saturation effectively stabilizes the wave such that broadening is reduced considerably, permitting nearly stationary propagation over many confocal distances, albeit with substantial down-shift. The resulting spatio-temporal solitary waves should be useful for applications in ultrafast all-optical switching and logic, and the generalized evolution equations will provide a refined starting point for the study of spatio-temporal phenomena in other areas as well.