A Reliable Split-Step Fourier Method for the Propagation Equation of Ultra-Fast Pulses in Single-Mode Optical Fibers

The extension to the split-step Fourier method (SSFM) for Schrödinger-type pulse propagation equations that we propose in this article is designed with the accurate simulation of pulses in the femtosecond regime in single-mode communication fibers in mind. We show that via an appropriate operator splitting scheme, Kerr nonlinearity and the self-steepening and stimulated Raman scattering terms can be combined into a single sub-step consisting of an inhomogeneous quasilinear first-order hyperbolic system for the real-valued quantities intensity and phase. First- and second-order accurate shock-capturing upwind schemes have been developed specifically for this nonlinear sub-step, which enables the accurate and oscillation-free simulation of signals under the influence of Raman scattering and extreme self-steepening with the SSFM. Benchmark computations of ultra-fast Gaussian pulses in fibers with strong nonlinearity demonstrate the superior approximation properties of the proposed approach.