Spectral element method for optical planar waveguide modal analysis

A spectral element method is developed for determining the propagation constants of guided and leaky modes in planar multilayer waveguides. The originality of the technique lies in what we believe to be a novel combination of a hierarchical basis built with modified Legendre polynomials and a rational mapping that efficiently handles semi-infinite computational subdomains. To handle leaky waves, we extend the algebraic mapping to complex coordinates that act as perfectly matched layers (PMLs). The method thus naturally enforces both radiation conditions at infinity and boundary conditions at material interfaces, leading to improved convergence properties without additional constraints or numerical treatments. The efficiency and numerical precision of our method are demonstrated through comparisons with results from the literature for high-index-contrast dielectric and plasmonic waveguides, ARROW anisotropic structures, and quantum well waveguides.