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

In the related research it is demonstrated that the efficient inclusion of static modes in the resonant-state expansion (RSE) results in its quick convergence to the exact solution regardless of the static mode set used. Then, the RSE is applied to spherically symmetric systems with continuous radial variations of the permittivity. It is shown that in TM polarization, the spectral transition from whispering gallery to Fabry-Pérot 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-Pérot 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, a number of useful approximations are discussed which shed light on the physical phenomena observed in the spectra of graded-index systems.

Data is given separately for each figure in the paper. The software to calculate the resonances of graded index spherical resonators can be found at: http://langsrv.astro.cf.ac.uk/RSESpherical/RSESpherical.html.

**Figure 1**

Resonant-states wavenumbers of a homogeneous sphere with refractive index n_r = 2, radius R, angular momentum number l=20.

Approximation to the imaginary part of the wavenumbers based on the phase analysis.

Approximation to the imaginary part of the wavenumbers based on the reflection coefficient at the surface of the resonator.

**Figure 2**

Resonant-state associated with the Brewster peak, given as a function of resonator refractive index, for angular momentum number l=10, l=20 and l=80.

Analytic approximation for the position of the Brewster peak based on the Brewster angle at the surface of the resonator.

**Figure 3**

a) Wavenumber of the eigenmodes of a homogeneous sphere (as on figure 1.).

b) Real part of the effective potential, real part of the Y_1 component of the normalised field, for transverse-electric (TE) and transverse-magnetic (TM) polarisation, for the first whispering gallery mode.

c) Wavenumber of the eigenmodes of a sphere with linear permittivity profile (inset).

d) Real part of the effective potential, a Morse potential approximation for the TE potential, real part of the Y_1 component of the normalised field, for TE and TM polarisation, for the first whispering gallery mode.

e) Wavenumber of the eigenmodes of a sphere with quadratic permittivity profile (inset).

f) Real part of the effective potential, real part of the Y_1 component of the normlaised field, for TE and TM polarisation, for the first whispering gallery mode.

**Figure 4**

a) Absolute difference between neighboring TE and TM modes for constant, linear, and quadratic permittivity profiles, for l=20.

b) Real part of the difference between neighboring TE and TM modes for constant, linear, and quadratic permittivity profiles.

The dataset also contains the wavenumber of the modes.

**Figure 5**

a) Wavenumber of the eigenmodes for a permittivity profile (shown on inset) that creates a wide flat potential well for the fundamental WGM, for l=20.

b) Real part of the effective potential, real part of the Y_1 component of the normlaised field, for TE and TM polarisation, for the first whispering gallery mode.

c) Absolute difference between neighboring TE and TM modes for a permittivty profile with r_0=0.1 with approximate values based on perturbation of TE modes to TM modes.

Additionally, the absolute difference for , r_0=0.2, and r_0=0.1 with a sharp edge at the surface is shown.

d) Real part of the difference between neighboring TE and TM mode, for the same permittivity as on c. and with the same approximation.

**Figure 6**

Phase analysis of the wavenumber of the eigenmodes of a homogeneous sphere with n_r=2 and l=20.

The functions Psi, and Phi are given, along with the exact wavenumber values.

**Figure 7**

Same as figure 1. but for n_r=4 and l=10.

**Figure 8**

Wavenumbers of the eigenmodes of a nonmagnetic homogeneous slab, with permittivity = 4, at non-normal incidence, with in plane wavenumber of p=20. Both even and odd, and TE and TM modes are shown, along with an approximation for the imaginary part based on the reflection coefficient at the slab surface.

**Figure 9**

Top: TM resonant state of a sphere of n_r = 2, l=20 and radius R, and radius 0.8R.

Bottom: Relative error for the eigenmodes of a sphere with raduis 0.8R calculated with the RSE using Mittag-Leffler representation #3 (with static modes sets VC, VSC, MVSC) and #4.

**Figure 10**

Top: TM resonant state of a sphere of n_r = 2, l=20 and radius R, and n_r=sqrt(9).

Bottom: Relative error for the eigenmodes of a sphere with n_r=sqrt(9) calculated using the RSE with Mittag-Leffler representation #3 (with static modes sets VC, VSC, MVSC) and #4.

**Figure 11**

Real part of the modified effective potential and the Y_1 component of the normlaised field for a quadratic permittivity profile for the first 4 TM whispering gallery mode.

**Figure 12**

Same as figure 4, but for l=80, with the addition of the wavenumbers plotted.

**Figure 13**

The difference between consecutive modes, separately for TE and TM polarisation, for l=80, for homogeneous, linear, and quadratic permittivity profile.

Research results based upon these data are published at https://doi.org/10.1103/PhysRevA.105.033522