Degeneracy-Discriminating Modal FEM Computation in Higher Order Rotationally Symmetric Waveguides

An accurate and efficient method is proposed to characterize waveguides with arbitrary cross sections exhibiting discrete rotational symmetries, known in the literature as
$C_{N}$
symmetries. This new method is based on a novel tailored 2-D finite element method (FEM) formulation exploiting rotationally periodic boundary conditions involved in the minimum sector that defines the waveguide cross section. While retaining the proven accuracy and convergence rate of FEM, it allows for a reduction in dimensionality of the problem and classifies the modes with respect to their rotational symmetry. Most importantly, it discriminates reliably and unambiguously degenerate modes (no matter how high their order is) with the exact same computed cutoff frequency value. This is especially interesting in modal analysis of devices involving circular polarization, since the formulation inherently provides different mode sets for clockwise and counterclockwise circularly polarized modes. The method has been tested on a wide variety of waveguides and computed higher order modes have been compared against analytical solutions and general-purpose FEM.