Photonic systems with two-dimensional landscapes of complex refractive index via time-dependent supersymmetry

TL;DR

This paper introduces a novel framework using time-dependent supersymmetry to construct exactly solvable two-dimensional optical models with complex refractive index landscapes, enabling analysis of localized light states and guided modes.

Contribution

It develops a method for creating solvable 2D optical models with complex refractive index landscapes using time-dependent supersymmetry, including non-$ ext{PT}$-symmetric systems.

Findings

Construction of localized light dots and guided modes.

Analysis of $ ext{PT}$-symmetry and localization properties.

Application to various wave guide configurations.

Abstract

We present a framework for the construction of solvable models of optical settings with genuinely two-dimensional landscapes of refractive index. The solutions of the associated non-separable Maxwell equations in paraxial approximation are found with the use of the time-dependent supersymmetry. We discuss peculiar theoretical aspects of the construction. Sufficient conditions for existence of localized states are discussed. Localized solutions vanishing for large $|\vec{x}|$, that we call light dots, as well as the guided modes that vanish exponentially outside the wave guides are constructed. We consider different definitions of the parity operator and analyze general properties of the $\mathcal{PT}$-symmetric systems, e.g. presence of localized states or existence of symmetry operators. Despite the models with parity-time symmetry are of the main concern, the proposed framework can serve for construction of non-$\mathcal{PT}$-symmetric systems as well. We explicitly illustrate the general results on a number of physically interesting examples, e.g. wave guides with periodic fluctuation of refractive index or with a localized defect, curved wave guides, two coupled wave guides or a uniform refractive index system with a localized defect.