A scalar product for the radiation of resonant modes

The study of the modes of optical resonators is beneficial for theoretical and practical endeavors. Here, we introduce a scalar product, originally devised for radiation fields, to such studies. This scalar product allows one to normalize and compare resonant modes using their corresponding radiation fields. Such fields are polychromatic fields free of divergences, which are determined for any given mode by its complex frequency and the modal field on the surface of the resonator. The scalar product is expressed as a surface integral involving the modal fields, multiplied by a closed-form factor incorporating the complex frequencies. In a practical application, we study the modes of disk-shaped whispering gallery resonators, and we show that the proposed scalar product accurately predicts the geometry-dependent coupling between modes.