Optical resonances in graded index spheres: A resonant-state expansion study and analytic approximations

Recent improvements in the resonant-state expansion (RSE), focusing on the static mode contribution, have made it possible to treat transverse-magnetic (TM) modes of a spherically symmetric system with the same efficiency as their transverse-electric (TE) counterparts. We demonstrate here that the efficient inclusion of static modes in the RSE results in its quick convergence to the exact solution regardless of the static mode set used. We then apply the RSE to spherically symmetric systems with continuous radial variations of the permittivity. We show that in TM polarization, the spectral transition from whispering gallery to Fabry-Perot modes is characterized by a peak in the mode losses and an additional mode as compared to TE polarization. Both features are explained quantitatively by the Brewster angle of the surface reflection which occurs in this frequency range. Eliminating the discontinuity at the sphere surface by using linear or quadratic profiles of the permittivity modifies this peak and increases the Fabry-Perot mode losses, in qualitative agreement with a reduced surface reflectivity. These profiles also provide a nearly parabolic confinement for the whispering gallery modes, for which an analytical approximation using the Morse potential is presented. Both profiles result in a reduced TE-TM splitting, which is shown to be further suppressed by choosing a profile radially extending the mode fields. Based on the concepts of ray optics, phase analysis of the secular equation, and effective quantum-mechanical potential for a wave equation, we have further developed a number of useful approximations which shed light on the physical phenomena observed in the spectra of graded-index systems.