A High-Order Numerical Method for the Nonlinear Helmholtz Equation in Multidimensional Layered Media

TL;DR

This paper introduces a high-order finite-difference method with nonlocal boundary conditions for solving the nonlinear Helmholtz equation in layered media, enabling accurate simulation of nonparaxial laser propagation and solitons.

Contribution

It develops a novel computational approach that extends previous techniques to multidimensional, nonparaxial regimes, addressing solution singularities and discontinuities in nonlinear optics.

Findings

Disappearance of solution singularities in the scalar NLH model with nonparaxial effects.

First numerical evidence of nonparaxial effects removing singularities.

Simulation of wavelength-scale spatial solitons and counter-propagating solitons.

Abstract

We present a novel computational methodology for solving the scalar nonlinear Helmholtz equation (NLH) that governs the propagation of laser light in Kerr dielectrics. The methodology addresses two well-known challenges in nonlinear optics: Singular behavior of solutions when the scattering in the medium is assumed predominantly forward (paraxial regime), and the presence of discontinuities in the % linear and nonlinear optical properties of the medium. Specifically, we consider a slab of nonlinear material which may be grated in the direction of propagation and which is immersed in a linear medium as a whole. The key components of the methodology are a semi-compact high-order finite-difference scheme that maintains accuracy across the discontinuities and enables sub-wavelength resolution on large domains at a tolerable cost, a nonlocal two-way artificial boundary condition (ABC) that simultaneously facilitates the reflectionless propagation of the outgoing waves and forward propagation of the given incoming waves, and a nonlinear solver based on Newton's method. The proposed methodology combines and substantially extends the capabilities of our previous techniques built for 1Dand for multi-D. It facilitates a direct numerical study of nonparaxial propagation and goes well beyond the approaches in the literature based on the "augmented" paraxial models. In particular, it provides the first ever evidence that the singularity of the solution indeed disappears in the scalar NLH model that includes the nonparaxial effects. It also enables simulation of the wavelength-width spatial solitons, as well as of the counter-propagating solitons.