S-polarized guided electromagnetic waves at a planar interface between vacuum and a graded-index dielectric

Analytic expressions for the dispersion relation and electric field profiles of guided waves supported by an asymmetric graded-index dielectric waveguide are derived. The system studied consists of vacuum in the region , and a dielectric medium in the region whose dielectric permittivity decreases continuously with increasing distance into it from the interface according to . Here is the (real) index of refraction of the medium at , and it is assumed that . It is found that for frequencies below a certain critical value given by , where is the speed of light in vacuum, the dispersion curve consists of a single branch that exists in a narrow spectral range. Its electric field decays exponentially with increasing distance into each medium. For frequencies above this critical value, the dispersion curve possesses several branches. The corresponding electric field decays exponentially with increasing distance into the vacuum, and decays in an oscillatory fashion with increasing distance into the grad...