(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 Schrodinger equati on. Starting directly from Maxwell's equations, a multiple-scales derivatio n is presented which results in a generalized first-order vectorial evoluti on equation that is valid for the non-linear spatio-temporal propagation of a predominantly linearly polarized electric field with large spatial and t emporal bandwidths. A reduced version of this full equation including the h igher-order linear and non-linear effects of third- and fourth-order disper sion, spacetime 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 la mbda(f) = 1.55 mu m are used, with the addition of a negative quintic satur ation term. Without quintic saturation, the non-linear spatio-temporal wave broadens under the action of the higher-order space-time effects. In addit ion, in the absence of Raman scattering, the wave undergoes collapse until arrested by the remaining higher-order terms. Frequency down-shifting and s patio-temporal broadening due to Raman scattering are found to have the gre atest 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 prop agation over many confocal distances, albeit with substantial down-shift. T he resulting spatio-temporal solitary waves should be useful for applicatio ns in ultrafast all-optical switching and logic, and the generalized evolut ion equations will provide a refined starting point for the study of spatio -temporal phenomena in other areas as well.