In this talk, we are interested in the resonance frequencies of a two
dimensional optical cavity, and more specially to those corresponding to
whispering gallery modes (WGM). WGM are optical waves with high angular
frequencies circling around the cavity, almost perfectly guided round by
total internal reflection that meet a resonance condition: after one
round-trip they return to the same point with the same phase.
Our main result consists in an asymptotic expansion of resonances as the
angular frequency becomes large for a cavity of a quite general geometry
obtained using a WKB method. This approach includes the case of a
varying optical index.
The interest of our formulas is twofold. They provide a very accurate
approximation of resonances and modes with high angular frequencies
where standard numerical methods such as the Finite Element Method are
inefficient. For lower angular frequencies, where a numerical approach
is required, they provide information on the localization in the
spectrum of the sought-out resonance that is used to improve the
efficiency of the computations.
This work is done in collaboration with Stéphane Balac and Monique Dauge.