Four-wave mixing simulation in weakly nonlinear Bragg gratings using the grating dispersion operator in the nonlinear Schrödinger equation

Abstract

While the nonlinear Shrödinger equation (NLSE) and its solving via the split-step Fourier method are well established when studying the Kerr interactions in waveguides, it is typically not applied when modeling a nonlinear interaction in a Bragg grating (BG). In that specific case, the solving of a set of coupled equations is preferred as they form the natural framework to deal with co- and contra-propagating waves. This, however, has limitations for input spectra much larger than this bandgap, e.g., for frequency combs or multispectral pump schemes. In order to deal with those in a Bragg grating, we adapt the usual NLSE solving via split-step Fourier by embedding the Bragg resonance into the dispersion operator. Although it requires that the total nonlinearity along the propagation remains moderate, i.e., the nonlinear phase shift γPL < 2π, and the pump(s) frequency(ies) to be outside of the bandgap, this modeling allows us to retrieve established results and points towards the BG ability to tune and quench four-wave mixing processes.