Robust split-step Fourier methods for simulating the propagation of ultra-short pulses in single- and two-mode optical communication fibers

Abstract: Extensions of the split-step Fourier method (SSFM) for Schr\"odinger-type pulse propagation equations for simulating femto-second pulses in single- and two-mode optical communication fibers are developed and tested for Gaussian pulses. The core idea of the proposed numerical methods is to adopt an operator splitting approach, in which the nonlinear sub-operator, consisting of Kerr nonlinearity, the self-steepening and stimulated Raman scattering terms, is reformulated using Madelung transformation into a quasilinear first-order system of signal intensity and phase. A second-order accurate upwind numerical method is derived rigorously for the resulting system in the single-mode case; a straightforward extension of this method is used to approximate the four-dimensional system resulting from the nonlinearities of the chosen two-mode model. Benchmark SSFM computations of prototypical ultra-fast communication pulses in idealized single- and two-mode fibers with homogeneous and alternating dispersion parameters and also high nonlinearity demonstrate the reliable convergence behavior and robustness of the proposed approach.