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Abstract

In his recent work on a tilt instability for advanced LIGO interferometers, P. Savov discovered numerically a unique duality relation between the eigenspectra of paraxial optical cavities with non-spherical mirrors: he found a one-to-one mapping between eigenstates and eigenvalues of cavities deviating from flat mirrors by h(r) and cavities deviating from concentric mirrors by -h(r), where h need not be a small perturbation. In this paper, we analytically prove and generalize this remarkable result. We then illustrate its application to interferometric gravitational-wave detectors; in particular, we employ it to confirm the numerical results of Savov and Vyatchanin for the impact of optical-pressure torques on LIGO's Fabry-Perot arm cavities (i.e. the tilt instability), when 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 LIGO interferometers, when an important feature is the intensity distribution on the cavity optics.