A duality relation between non-spherical mirror optical cavities and its application to gravitational-wave detectors

TL;DR

This paper proves a duality relation between non-spherical mirror optical cavities, linking their eigenspectra, and demonstrates its application to designing optical cavities for gravitational-wave detectors with flat beam profiles.

Contribution

It establishes a novel duality relation between cavities with mirror deviations h(r) and -h(r), applicable beyond small perturbations, aiding optical cavity design for gravitational-wave detection.

Findings

Proves a unique duality relation between eigenspectra of non-spherical mirror cavities.

Demonstrates the relation's application to gravitational-wave detector cavities.

Provides insights for designing cavities with flat intensity profiles.

Abstract

In this paper, we analytically prove a unique duality relation between the eigenspectra of paraxial optical cavities with non-spherical mirrors: a one-to-one mapping between eigenmodes and eigenvalues of cavities deviating from flat mirrors by $h(\vec{r})$ and cavities deviating from concentric mirrors by $-h(\vec{r})$, where $h$ need not be a small perturbation. We then illustrate its application to optical cavities, proposed for advanced interferometric gravitational-wave detectors, where the mirrors are designed to support beams with rather flat intensity profiles over the mirror surfaces. This unique mapping might be very useful in future studies of alternative optical designs for advanced gravitational waves interferometers or experiments employing optical cavities with non-standard mirrors.