Efficient shooting algorithms for solving the nonlinear one-dimensional scalar Helmholtz equation

摘要

We study numerically the effect of nonlinearity on the transmission of monochromatic scalar optical fields through one-dimensional stratified dielectric media. We implement two different efficient shooting algorithms to solve the boundary value problem associated to the governing Helmholtz equation. One is based upon a recursive two-dimensional bisection procedure acting on the vector space of all allowed starting field phase space quadratures to solve the two-dimensional root-finding problem. This method is valid for any kind of nonlinearity. The other one makes use of a particular phase invariance of the specific equations and boundary conditions considered, reducing thereby the number of independent variables for the shooting method. We compare the accuracy and the computational expenses of both algorithms.