Mathematical modeling of smoothly-irregular integrated-optical waveguide and mathematical synthesis of waveguide luneburg lens

Abstract

In this paper we consider a class of multilayer integrated-optical waveguides consisting of homogeneous dielectric layers of constant or variable thickness, which are being systematically numerically studied using the cross-section method. The method is based on the adiabatic approximation of the asymptotic expansion on the one hand and the expansion in the complete system of modes of regular comparison waveguide. The paper discusses the problems of numerical implementation of the cross-section method to the transformation of particular mode in a smooth transition from one planar regular open waveguide to another. Luneburg proposed a model of the ideal optical instrument (in the framework of geometrical optics), afterwards called Luneburg lens. Later classical Luneburg lens was included in the family of the ideal optical instruments - generalized Luneburg lenses. Zernike in his work showed that a local increase in thickness of the waveguiding layer leads to a local deceleration of phase velocity of the propagating waveguide mode. This effect has led to the idea of manufacturing the waveguide (two-dimensional) Luneburg lenses instead of volume (three-dimensional) lenses. In this work we synthesized mathematically the thickness profiles of the additional (irregular in thickness) waveguide layer forming the thin-film generalized waveguide Luneburg lens.