Waveguide Modes of a Planar Gradient Optical Waveguide

The mathematical model of light propagation in a planar gradient optical waveguide consists of the Maxwell's equations supplemented by the matter equations and boundary conditions. In the coordinates adapted to the waveguide geometry, the Maxwell's equations are separated into two independent sets for the TE and TM polarizations. For each polarization there are three types of waveguide modes in a regular planar optical waveguide: guided modes, substrate radiation modes, and cover radiation modes. In this work we implement the numerical-analytical calculation of all types of waveguide modes. For the eigenvalue problem with a piecewise linear-constant potential we used the Airy functions to calculate the cover radiation modes and substrate radiation modes. We took advantage of reducing the initial potential scattering problem (in the case of the continuous spectrum) to the equivalent ones for the Jost functions: the Jost solution from the left for the substrate radiation modes and the Jost solution from the right for the cover radiation modes.