Hybrid composed method associating conformal transformation with matrix formulation for computing eigenvalues and eigenvectors in bended optical waveguides

A fast hybrid composed and numerical method for the calculation of eigenvalues (effective indices) and eigenvectors (optical modes) of a slab waveguide presenting a radius of curvature is presented in this letter. This compound method combines a conformal transformation of the complex plane plus a multilayer matrix formalism addressing directly the modified index profile obtained by the conformal transformation; the matrix formalism is then applied in second step so as to discretize and slice the profile in a virtual multilayer structure. This method being conveniently operable on a personal computer in a short amount of time and is easy to implement. Its results are then compared to values afforded with the commercial vectorial software COMSOL 3D so as to discuss their accuracy.