Numerically Stable and Reliable Computation of Electromagnetic Modes in Multilayered Waveguides Using the Cauchy Integration Method With Automatic Differentiation

A robust and efficient method is presented for the computation of the electromagnetic modes supported by planar multilayer waveguides that may comprise lossy, active, plasmonic, and uniaxial media, including graphene sheets. Pole-free and numerically stable dispersion functions (DFs) are developed for various shielding configurations using the S-matrix formulation. The modal propagation constants are computed by the Cauchy integration method on the four-sheeted Riemann surface, using the derivative of the DF for greater reliability. Since analytical derivatives of the S-parameters are difficult to obtain, automatic differentiation is employed, implemented by operator overloading in modern Fortran. The method is validated using various benchmark problems found in the literature.