Mathematical analysis of whispering gallery modes in graded index optical micro-disk resonators

We study from a theoretical point of view whispering gallery modes (WGM) in graded index micro-disk resonators where the refractive optical index varies with the radial position. Using a quantum mechanical analogy, we highlight three different behaviors for the WGM depending on the sign of a key parameter expressed as the ratio of the refractive index value to its derivative at the cavity boundary. This results in three asymptotic expansions of the resonances for large polar mode index providing first-approximations of WGM in a simple and quick way. Besides, these expansions yield a theoretical fundation to considerations of V.S. Ilchenko, A.A. Savchenkov, A.B. Matsko, L. Maleki, J. Opt. Soc. Am. A 20, 157 (2003), about three sorts of effective potentials for TE modes in a dielectric sphere