https://doi.org/10.1140/epjd/e2020-10303-5
Regular Article
Mathematical analysis of whispering gallery modes in graded index optical micro-disk resonators
1
Department of Applied Mathematics, University of California, Merced, 5200 North Lake Road, Merced, CA 95343, USA
2
UNIV. RENNES, CNRS, FOTON-UMR 6082, Enssat, 22305 Lannion, France
a e-mail: stephane.balac@univ-rennes1.fr
Received:
28
May
2020
Received in final form:
4
September
2020
Accepted:
7
September
2020
Published online:
1
November
2020
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.
Key words: Optical Phenomena and Photonics
© EDP Sciences / Società Italiana di Fisica / Springer-Verlag GmbH Germany, part of Springer Nature, 2020
