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

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.