Numerical modeling considerations for an applied nonlinear Schrödinger equation

• Published in: Applied optics (2004) 2015-02, Vol.54 (6), p.1426-1435
• Main Authors: Pitts, Todd A, Laine, Mark R, Schwarz, Jens, Rambo, Patrick K, Hautzenroeder, Brenna M, Karelitz, David B
• Format: Article
• Language: English
• Subjects: Computation; Finite difference method; Ionization; Mathematical analysis; Mathematical models; Nonlinearity; Operators; Schroedinger equation
• ISSN: 1559-128X; 2155-3165
• DOI: 10.1364/AO.54.001426

A model for nonlinear optical propagation is cast into a split-step numerical framework via a variable stencil-size Crank-Nicolson finite-difference method for the linear step and a choice of two different nonlinear integration schemes for the nonlinear step. The model includes Kerr, Raman scattering, and ionization effects (as well as linear and nonlinear shock, diffraction, and dispersion). We demonstrate the practical importance of numerical effects when interpreting computational studies of high-intensity optical pulse propagation in physical materials. Examples demonstrate the significant error that can arise in discrete, limited precision implementations as one attempts to improve practical operator accuracy through increased operator support size and sampling frequency. We also demonstrate the effect of the method used to obtain the finite-difference operator coefficients defining the equations ultimately used in the discrete model. Smooth, plausible, but incorrect solutions may result from these numerical effects. This implies the necessity of a complete, precise description of all numerical methods when reporting results of computational physics investigations in order to ensure proper interpretation and reproducibility.