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 ${\mathrm{LG}}_{3,3}$ 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 ${\mathrm{LG}}_{3,3}$ 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.

Figure 1

Mirror surface map, computed analytically, that gives the best coupling from a ${\mathrm{LG}}_{3,3}$ to a ${\mathrm{LG}}_{4,1}$ mode. The coated mirror diameter considered here is 51 cm, which is that needed to have a cavity capable of sustaining a ${\mathrm{LG}}_{33}$ mode with about 6 cm Gaussian radius with 1 ppm clipping losses (see also Sec. 2). The color scale has been clipped to best show the map features.

Figure 2

Maps to generate ninth order LG modes in addition to the input ${\mathrm{LG}}_{33}$. Only the maps that generate the mode with real coefficients are shown. The maps that generate those with imaginary coefficients are simply rotated by $\pi /4l$. These maps have been computed using parameters corresponding to an Advanced Virgo-like resonant cavity, see the main text for more details.

Figure 3

Simulated power image (a) in reflection of a Fabry-Perot cavity with a perfect end mirror and an input mirror with a map randomly generated (b) that introduces some coupling from the fundamental mode to the other degenerate modes. The two images have the same transverse scale and the diameter of the mirror map is 51 cm.

Figure 4

Results of the reconstruction of the full field from intensity image only. (a) amplitude image of the real field; (b) amplitude image of the reconstructed field; (c) phase image of the real field; (d) phase image of the reconstructed field; (e) difference between the amplitude images; (f) difference between the phase images.

Figure 5

Modal decomposition complex coefficient reconstructed from an intensity image, using the algorithm described in the text. They are compared with those obtained with a decomposition based on the full knowledge of the complex field.

Figure 6

(a) example of the dependence of all error signals on the amplitude of one selected map added to the input mirror of a Fabry-Perot cavity. (b) sensing matrix, composed of the slope of the linear dependence of all error signals on all map amplitudes.

Figure 7

(a) intensity image of the beam reflected by the cavity, without correction. (b) the same, with the correction computed by the adaptive optics control.

Figure 8

Performances of the simulated adaptive optics control system. (a) evolution with the iteration number of the absolute values of the error signals; (b) evolution of the map coefficients; (c) evolution of the percentage of the reflected beam power which is not in the fundamental mode; (d) final correction map.