Exact mode volume and Purcell factor of open optical systems

The Purcell factor quantifies the change of the radiative decay of a dipole in an electromagnetic environment relative to free space. Designing this factor is at the heart of photonics technology, striving to develop ever smaller or less lossy optical resonators. The Purcell factor can be expressed using the electromagnetic eigenmodes of the resonators, introducing the notion of a mode volume for each mode. This approach allows to use an analytic treatment, consisting only of sums over eigenmode resonances, a so-called spectral representation. We show in the present work that the expressions for the mode volumes known and used in literature are only approximately valid for modes of high quality factor, while in general they are incorrect. We rectify this issue, introducing the exact normalization of modes. We present an analytic theory of the Purcell effect based on the exact mode normalization and resulting effective mode volume. We use a homogeneous dielectric sphere in vacuum, which is analytically solvable, to exemplify these findings.