Theoretical investigations on optical caustics of spherical microresonators: Analytical expressions of caustics and their asymptotic behaviors, computational simulations

This paper introduces analytical expressions of optical caustics r1 and r2 in micro-resonators (respectively called inner and outer caustics in micro-optics) in terms of radial and azimutal modal numbers n and l mutually. Such a theoretical study focuses on spherical micro-resonators and is carried out according to two complementary ways: Electromagnetic (E.M) approaches based respectively on an eigenvector problem resolution called modal field in integrated optics and an analogy with Quantum Mechanics (Q.M) using the notions of potential and geometric eikonal principle. Thus, using quantization and phase considerations, fundamental new formulation of caustics, that depend on the two previous quantum modal numbers simultaneously, have been shaped and confronted with numerical and graphical simulations. Both ways lead to results in good agreement (correlation coefficient superior to 0.99) and allow us to specify the evolution of such caustics according to each modal number of the quantification; such an analytical caustics formulation involves a specific function f whose study is expanded for various values of n ranging from 1 to 500. Moreover a whole study of asymptotic behaviors of f-function and its inverse f− 1 on specific n and l values, directly related to the previous caustics expressions, is conducted, allowing the expansion of new analytical expressions of caustics in various limit cases and asymptotic behaviors. Then, the evolution of the positions of caustics in terms of modal numbers is interpreted, and the physical meanings of optical properties such as quasi-resonance, position and confinement of field with resonance, and associated electromagnetic fields are discussed.