Purcell factor of spherical Mie resonators

Using a modal approach, we compute the Purcell factor for spherical resonators. The analytic expressions of the normal modes of the resonator are used to calculate the effective volumes. Then we show that the effective volume associated with each normal mode is related to the translation-addition coefficients of a displaced dipole. Finally, we consider a silicon resonator homogeneously doped with electric dipolar emitters, and we show that the average electric Purcell factor dominates over the magnetic one.