  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.
