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

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 <i>γPL</i> < 2<i>π</i>, 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.