Trapped waves in varying dielectric media

In this paper the propagation characteristics of a variable dielectric medium are investigated. Specifically, the medium is an infinite dielectric slab imbedded in free space but whose dielectric constant merges continuously into the free space value at the boundaries while attaining a maximum value at the center of the slab. In solving Maxwell´s Equations in such a medium, it is shown that certain solutions of the fields in the medium reveal the existence of trapped modes within the slab. The solutions are expressible in terms of hypergeometric functions which simplify to hyperbolic functions for the lowest ordered modes. The governing conditions which establish the existence of these modes are also used to determine the appropriate cutoff frequency of each mode.