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