Adaptive optics sensing and control technique to optimize the resonance of the Laguerre-Gauss 33 mode in Fabry-Perot cavities

Second and third generation gravitational wave interferometric detectors will be limited in their sensitivity by thermal noise of the core optics. One way to reduce this contribution is to use an input laser beam with a more uniform distribution of the power: for this reason the use of the Laguerre-Gauss LG(3,3) mode as interferometer input has been suggested. The main issue with this approach is the fact that in resonant cavities with spherical mirrors the input mode will be degenerate with nine other modes. This implies very stringent requirements on the mirror surface quality, beyond the present polishing technology capabilities: it is not possible to obtain mirrors with low enough surface roughness to meet the requirements for the operation of a gravitational wave detector. In a previous paper an approach to apply in situ thermal corrections to the main surface of the mirrors was proposed. In this paper we develop further the technique, showing that it is possible to compute the optimal correction using only the information that can be extracted from the intensity images in reflection of the resonant cavity, without any a priori knowledge of the mirror surface maps. We test our proposal using optical simulations and we are able to considerably improve the quality of the beam reflected from a cavity with realistic mirror surface maps: without any correction the purity of the reflected beam was degraded to below 90%; with the proposed adaptive optics system we could recover a purity of 99.96%. The implementation of the proposed system would allow the use of a LG(3,3) input mode with the mirror qualities available today. In addition we show that it is possible to correct the aberrations introduced by both mirrors acting only on one of the two. In this way it is possible to avoid introducing unwanted thermal lensing in the input mirrors.