Ex-house 2D finite-element simulation of the whispering-gallery modes of arbitrarily shaped axisymmetric electromagnetic resonators

TL;DR

This paper demonstrates how to configure commercial PDE-solving software to accurately simulate whispering-gallery modes in arbitrarily shaped axisymmetric dielectric resonators without transverse approximation, enabling detailed analysis of complex resonator structures.

Contribution

It introduces a traceable finite-element method approach using PDE software for modeling WG modes in arbitrary axisymmetric resonators, including loss estimation and application to various structures.

Findings

Successfully modeled optical and microwave resonators with complex geometries.

Estimated dielectric constants of sapphire at cryogenic temperatures.

Validated approach with experimental resonance frequency fitting.

Abstract

It is described, explicitly, how a popular, commercially-available software package for solving partial-differential-equations (PDEs), as based on the finite-element method (FEM), can be configured to 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 approxi-mation is imposed, the approach remains accurate even for so-called quasi-TM and -TE modes of low, finite azimuthal mode order. The approach's generality and utility are demonstrated by modeling several non-trivial structures: (i)two different optical microcavities [one toroidal made of silica, the other an AlGaAs microdisk]; (ii) a 3rd-order microwave Bragg cavity containing alumina layers (iii) two different cryogenic sapphire X-band microwave resonators. By fitting one of the latter to a set of measured resonance frequencies, the dielectric constants of sapphire at liquid-helium temperature have been estimated.