The propagation of TE- and TM-polarized electromagnetic (EM) waves through a dielectric layer with spatial variation of the refractive index is reported in this paper. The relative permittivity of the layer is assumed to be graded along the thickness direction, and its spatial variation is described by a quartic polynomial. The corresponding mode equations are Helmholtz equations with variable coefficients that can be transformed to a triconfluent Heun equation, a local exact solution of which is expressed in terms of triconfluent Heun functions. The solutions cover many particular cases owing to a variability of four optional parameters (coefficients) of the quartic polynomial. The general local solution for a TE-polarized EM wave is employed for the calculation of transmission properties of a periodic one-dimensional photonic crystal using the Floquet theory.
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