Quality factor of dielectric optical resonators

We show analytically that the quality (Q) factor of magnetic and electric Mie modes in a lossless dielectric spherical resonator with high refractive index (n 1) scales as n²ʲ⁺¹ and n²ʲ⁺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 for all Mie modes for 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.