Simulations of the Nonlinear Helmholtz Equation: Arrest of Beam Collapse, Nonparaxial Solitons, and Counter-Propagating Beams

TL;DR

This paper numerically investigates the (2+1)D and (1+1)D nonlinear Helmholtz equations, demonstrating collapse arrest due to nonparaxiality, stable sub-wavelength solitons, and detailed dynamics of counter-propagating beams, surpassing previous models.

Contribution

It provides the first numerical evidence that nonparaxiality and backscattering can prevent collapse and explores sub-wavelength solitons and counter-propagating beam dynamics using the comprehensive NLH model.

Findings

Nonparaxiality and backscattering arrest beam collapse.

Sub-wavelength solitons propagate undistorted over many diffraction lengths.

Backscattered fields are computed for the first time.

Abstract

We solve the (2+1)D nonlinear Helmholtz equation (NLH) for input beams that collapse in the simpler NLS model. Thereby, we provide the first ever numerical evidence that nonparaxiality and backscattering can arrest the collapse. We also solve the (1+1)D NLH and show that solitons with radius of only half the wavelength can propagate over forty diffraction lengths with no distortions. In both cases we calculate the backscattered field, which has not been done previously. Finally, we compute the dynamics of counter-propagating solitons using the NLH model, which is more comprehensive than the previously used coupled NLS model.