Continuous spectrum of modes for optical micro-sphere resonators

• Spherical dielectric resonators have a continuous spectrum of resonant frequencies.
• A pair of outward-inward traveling waves forms the mode field outside the sphere.
• Mode field of near-zero normalized frequencies is concentrated outside the sphere.
• The quality factor due to radiation degrades by increasing the dielectric loss.

This paper presents an improved modal analysis for the spherical dielectric resonator. This is commonly carried out by assuming an outgoing spherical Hankel function for the region surrounding the dielectric sphere. It will be shown that this assumption is incomplete and cannot lead to the entire spectrum of resonance frequencies. Following an analytical formulation, we prove that, like cylindrical resonators, the only choice for the outer region of the dielectric sphere can be a proper linear combination of an inward and an outward traveling wave. Starting from this formulation, we determine the complete spectrum of the resonance frequencies and the associated mode fields. In this analysis, the continuous spectrum of resonance frequencies is introduced and the properties of radiation modes are studied in detail. The proposed analytical formulation is thereafter employed to calculate the quality factor of the resonator due to radiation and dielectric loss.