A scalar product for the radiation of resonant modes

TL;DR

This paper introduces a conformally-invariant scalar product for resonant modes in optical resonators, enabling normalization and comparison of modes based on their radiation fields, with applications to whispering gallery resonators.

Contribution

It presents a new scalar product that uses radiation fields to analyze resonant modes, improving mode comparison and prediction of mode interactions.

Findings

Accurately predicts mode crossings and anti-crossings.

Provides a practical method for mode normalization.

Demonstrates application to disk-shaped whispering gallery resonators.

Abstract

We introduce the conformally-invariant scalar product, originally devised for radiation fields, to the study of the modes of optical resonators. 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 from the complex frequencies and the modal fields on the surface of the resonator. The scalar product is expressed as surface integrals involving the modal fields, multiplied by closed-form factors incorporating the complex frequencies. In a practical application, we study the modes of disk-shaped whispering gallery resonators, and show that the proposed scalar product accurately predicts the geometry-dependent crossings and anti-crossings between modes.