In situ correction of mirror surface to reduce round-trip losses in Fabry-Perot cavities

High-finesse resonant cavities play an important role in many metrology applications such as gravitational wave detectors. The performance of these cavities can be limited by round-trip losses (RTP) generated by light that is scattered by the mirror surface defects into higher-order modes that are close to resonance. In this paper we develop a detailed model of this effect and we study possible strategies to correct the mirror surface. We show that it is possible to restrict the correction to the combination of a reduced set of surface deformations that can be reproduced on the mirror using projected heating patterns. We show with an optical simulation that by acting on the cavity mirrors it is possible to reduce RTP to the large angle scattering limit. We also show that the optimal correction can be computed without any a priori knowledge of the mirror surface, but based only on measurements of the power stored inside the cavity, thus opening up the possibility of a simple implementation of the proposed algorithm.