High-order numerical method for the nonlinear Helmholtz equation with material discontinuities in one space dimension

TL;DR

This paper introduces a robust Newton-based numerical method with a novel fourth order discretization for solving the nonlinear Helmholtz equation with material discontinuities, enabling high nonlinearity simulations in one dimension.

Contribution

It develops a Newton-type solver combined with a new fourth order finite-volume discretization for the nonlinear Helmholtz equation with discontinuities.

Findings

Newton's method converges rapidly for high nonlinearity levels.

The new discretization handles material discontinuities effectively.

The approach sets a foundation for multi-dimensional problem analysis.

Abstract

The nonlinear Helmholtz equation (NLH) models the propagation of electromagnetic waves in Kerr media, and describes a range of important phenomena in nonlinear optics and in other areas. In our previous work, we developed a fourth order method for its numerical solution that involved an iterative solver based on freezing the nonlinearity. The method enabled a direct simulation of nonlinear self-focusing in the nonparaxial regime, and a quantitative prediction of backscattering. However, our simulations showed that there is a threshold value for the magnitude of the nonlinearity, above which the iterations diverge. In this study, we numerically solve the one-dimensional NLH using a Newton-type nonlinear solver. Because the Kerr nonlinearity contains absolute values of the field, the NLH has to be recast as a system of two real equations in order to apply Newton's method. Our numerical simulations show that Newton's method converges rapidly and, in contradistinction with the iterations based on freezing the nonlinearity, enables computations for very high levels of nonlinearity. In addition, we introduce a novel compact finite-volume fourth order discretization for the NLH with material discontinuities.The one-dimensional results of the current paper create a foundation for the analysis of multi-dimensional problems in the future.