Modeling of surface waves in photonic crystal structures with a refractive index profile decreasing with distance from the surface

Objectives. Identification of the propagation patterns of surface waves in inhomogeneous and nonlinear crystal structures using mathematical models is an important fundamental problem in condensed matter physics, specifically waveguide optics. Models of waveguide structures used to establish an exact analytical solution are of particular significance. The aim of this work is to carry out a theoretical study of transversely polarized surface electric waves propagating along a photonic crystal with a certain refractive index profile.

Methods. The methods of mathematical physics, analysis, differential equations, and theory of special functions, as well as physical models of waveguide optics, were used in this study.

Results. A generalized hyperbolic permittivity profile was proposed to describe the spatially inhomogeneous distribution of the optical properties of a photonic crystal. This profile has a wide range of possibilities for varying its shape, allowing it to be used for a wide range of problems not limited to waveguide optics. An exact analytical solution of the wave equation with the selected permittivity profile was found in terms of the Whittaker function. Frequent cases of the generalized profile for which exact analytical solutions were indicated were also considered. These are expressed through the Whittaker and Macdonald functions. The study also describes surface transverse electric waves, where the field is localized near the surface of the photonic crystal and decreases with distance from it. The solution obtained also describes waveguide modes in which the field decreases with distance from the surface of the photonic crystal with oscillations. New features of surface wave localization were established. These were caused by a change in the parameters of the generalized hyperbolic profile modeling the dependence of the permittivity. It was also established that the maximum intensity of the surface wave is located in the photonic crystal.

Conclusions. The results of the description of the characteristics of surface waves obtained expand the theoretical concepts of waveguide optics. They can be useful in predicting the optical properties of various photonic crystal structures, as well as in designing various waveguide structures with the required dispersion-optical characteristics.

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