We present a framework for the construction of solvable models of optical settings with genuinely two-dimensional landscapes of refractive index. Solutions of the associated nonseparable Maxwell equations in paraxial approximation are found using the time-dependent supersymmetry. We discuss peculiar theoretical aspects of the construction. In particular, we focus on the existence of localized solutions specific for the new systems. Sufficient conditions for their existence are discussed. Localized solutions vanishing for large $|\vec{x}|$, which 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.

• Received 7 September 2018
• Revised 27 March 2019

DOI:https://doi.org/10.1103/PhysRevA.99.053812