Mathematical analysis of whispering gallery modes in graded index optical micro-disk resonators
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THE EUROPEAN PHYSICAL JOURNAL D
Regular Article
Mathematical analysis of whispering gallery modes in graded index optical micro-disk resonators St´ephane Balac1,a , Monique Dauge1 , Yannick Dumeige2 , Patrice F´eron2 , and Zo¨ıs Moitier1 1 2
Department of Applied Mathematics, University of California, Merced, 5200 North Lake Road, Merced, CA 95343, USA UNIV. RENNES, CNRS, FOTON-UMR 6082, Enssat, 22305 Lannion, France Received 28 May 2020 / Received in final form 4 September 2020 / Accepted 7 September 2020 Published online 1 November 2020 c EDP Sciences / Societ`
a Italiana di Fisica / Springer-Verlag GmbH Germany, part of Springer Nature, 2020 Abstract. 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 firstapproximations 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.
1 Introduction Optical micro-resonators are key devices for many applications in photonics and they have been widely studied in the past two decades [1,2]. Optical dielectric resonators supporting Whispering Gallery Modes (WGM) have gained much interest thanks to their capability to strongly confine the light in very compact volumes with exceptional properties like a very narrow spectral linewidth and high quality factor. WGM resonators are usually formed of dielectric materials with a constant refractive index. This results in a complex spectral pattern with unequal mode spacing and a high spectral density that may potentially limit their performance or the range of their applications. Optical micro-resonators with spatially varying refractive index offer new opportunities to improve or enlarge the field of applications of optical micro-resonators. This kind of optical micro-cavities falls under the category of GRaded INdex (GRIN) structures that are widely used in various fields in optics [3]. WGM in optical micro-resonators are specific resonances of electromagnetic waves inside a convex micrometric dielectric cavity with smooth surface. They correspond to light-waves circling around the cavity, almost perfectly guided round by optical total internal reflection, that meet a resonance condition (after one round-trip they return to the same point with the same phase and hence interfere constructively with themselves, forming standing waves). Use of graded index cavity can result in a shift of the spatial location of WGM away from the extea
e-mail: stephane.balac@univ-renn
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