Simulation of Two-Dimensional Nonlinear Envelope Pulse Dynamics by a Two-Step Spatiotemporal Angular Spectrum Method

We present an extension of our previous nonlinear beam-simulation method to the propagation and interaction of pulse envelopes. The extra time dimension is applied in the context of a dispersive nonlinear medium that is described by a Klein–Gordon equation with an added cubically nonlinear, self-focusing term. Pulse propagation in this medium is modeled as the evolution of a spatiotemporal spectrum—i.e., the frequency-dependent angular spectrum of the pulse envelope—traversing a sequence of self-induced, thin, weak phase filters. Preliminary simulation experiments show agreement with known behavior in the absence of nonlinearity, confirm the existence of an (apparently unstable) stationary solution, and demonstrate mutual pulse attraction with subsequent destruction, initial survival following oblique collision, and mutual repulsion of out-of-phase pulses

Split-Step-Type Angular Plane-Wave Spectrum Method for the Study of Self-Refractive Effects in Nonlinear Wave Propagation

We introduce an efficient algorithm to simulate nonlinear self-induced wave refraction effects. The algorithm is applied to the case of self-focusing, defocusing, and predicted steady-state behavior. The results show reasonably good agreement with available approximate theory and excellent agreement when exact analytic solutions are available for comparison