Exact eigenvalue equations of modes in elliptical fibers of step-index profile

The exact eigenvalue equations of modes in elliptical fibers of step-index profile are not available in the literature. By using a rigorous method, an elliptical fiber mode of step-index profile is expressed in terms of the even and odd Mathieu wave functions of the first kind in its core and in terms of their decaying even and odd modes in its cladding. Then, by matching the tangential components of the total electromagnetic fields across the core-cladding boundary and using the orthogonalities of the circumferential Mathieu functions, the eigenvalue equations are derived in closed forms. Results are shown for the propagation characteristics of some low orders of the elliptic modes.