Quality Factor of Dielectric Spherical Resonators

We develop a method to find the quality (Q) factor of the eigenmodes of a dielectric spherical resonator as a function of its complex refractive index. First, we analytically show that the Q factor of magnetic and electric multipolar modes in a lossless spherical resonator with high refractive index (n ≫ 1) scale as n^2j+1 and n^2j+3, respectively, where j denotes the multipolar order. We numerically validate these results and show that our high-n analytical relation is accurate for the dipolar modes when n > 5. For higher multipolar orders, the analytical relation becomes valid for increasingly lower n. We study the dependence of the Q factor on absorption losses and determine a general functional form that describes the Q factor of all multipolar modes as a function of any complex refractive index. Finally, we observe that this functional form predicts a multipolar-dependent singular value of optical gain, which gives rise to a lasing condition with an infinite Q factor.