Study of eigen value equation of hyperbolic waveguide under slow wave consideration

The modal characteristics of a step-index waveguide with a hyperbolic core-cladding boundary have been studied. For the study of this new waveguide we select an elliptical coordinate system ,so that cross-sectional boundary is represented by a hyperbola .Using suitable boundary condition, and the field components,determined for a fourth order determinantal characteristic equation is obtained. The characteristic equation involve modified Mathieu functions, Mathieu functions and their derivative.