Coupled Mode Equation Modeling for Out-of-Plane Gap Solitons in 2D Photonic Crystals

TL;DR

This paper derives coupled mode equations to model out-of-plane gap solitons in 2D photonic crystals, providing a theoretical framework and numerical examples for localized optical beams near spectral gaps.

Contribution

It presents a systematic derivation of coupled mode equations for out-of-plane gap solitons in 2D photonic crystals using Bloch variables, including numerical validation.

Findings

Coupled mode equations are derived as nonlinear Schrödinger systems with cross derivatives.

Numerical examples demonstrate the applicability to hexagonal photonic crystal structures.

The model accurately approximates gap solitons near spectral edges.

Abstract

Out-of-plane gap solitons in 2D photonic crystals are optical beams localized in the plane of periodicity of the medium and delocalized in the orthogonal direction, in which they propagate with a nonzero velocity. We study such gap solitons as described by the Kerr nonlinear Maxwell system. Using a model of the nonlinear polarization, which does not generate higher harmonics, we obtain a closed curl-curl problem for the fundamental harmonic of the gap soliton. For gap solitons with frequencies inside spectral gaps and in an asymptotic vicinity of a gap edge we use a slowly varying envelope approximation based on the linear Bloch waves at the edge and slowly varying envelopes. We carry out a systematic derivation of the coupled mode equations (CMEs) which govern the envelopes. This derivation needs to be carried out in Bloch variables. The CMEs are a system of coupled nonlinear stationary Schr\"odinger equations with an additional cross derivative term. Examples of gap soliton approximations are numerically computed for a photonic crystal with a hexagonal periodicity cell and an annulus material structure in the cell.