Traceable 2-D Finite-Element Simulation of the Whispering-Gallery Modes of Axisymmetric Electromagnetic Resonators

This paper explains how a popular commercially available software package for solving partial-differential-equations (PDEs), as based on the finite-element method, can be configured to efficiently calculate the frequencies and fields of the whispering-gallery (WG) modes of axisymmetric dielectric resonators. The approach is traceable; it exploits the PDE-solver´s ability to accept the definition of solutions to Maxwell´s equations in so-called weak form. Associated expressions and methods for estimating a WG mode´s volume, filling factor(s), and in the case of closed (open) resonators, its wall (radiation) loss, are provided. As no transverse approximation is imposed, the approach remains accurate even for quasi-transverse-magnetic/electric modes of low finite azimuthal mode order. The approach´s generality and utility are demonstrated by modeling several nontrivial structures, i.e., 1) two different optical microcavities (one toroidal made of silica, the other an AlGaAs microdisk), 2) a third-order sapphire:air Bragg cavity, and 3) two different cryogenic sapphire WG-mode resonators; both 2) and 3) operate in the microwave X-band. By fitting one of 3) to a set of measured resonance frequencies, the dielectric constants of sapphire at liquid-helium temperature have been estimated.