Mode determination for planar waveguide using the four-sheeteddispersion relation

A numerical method for the determination of the modes of planar, multilayer waveguides is presented. It is possible, with this algorithm, to guarantee that all the solutions within any finite region of the complex plane representing the propagation constant are found; neither prior knowledge of the number of solutions nor their approximate locations within the region is required. This method is formulated using the four-sheeted Riemann surface of the analytic function that defines the waveguide dispersion relation and whose zeros correspond to the proper and improper modes of the structure. Both TE and TM modes can be determined for guides that incorporate materials with complex indexes