LG33

Related literature since the ET design study, focusing on simulation with realistic mirrors (not about the table top experiments):

1. Higher order Laguerre-Gauss mode degeneracy in realistic, high finesse cavities by C. Bond et al, 2011 link
2. Effects of mirror aberrations on Laguerre-Gaussian beams in interferometric gravitational-wave detectors by T. Hong, 2011 link
3. Reduction of higher order mode generation in large scale gravitational wave interferometers by central heating residual aberration correction by R. Day et al., 2013 link
4. Adaptive optics sensing and control technique to optimize the resonance of the Laguerre-Gauss 33 mode in Fabry-Perot cavities by G. Vajente and R. A. Day 2013 link
5. In situ correction of mirror surface to reduce round-trip losses in Fabry–Perot cavities by G. Vajente, 2011 link

From 1,2 the main message was the surface of the mirrors should be around 10 times better than what we had for the second generation interferometers. However 2 demonstrated that we do not need to correct all the surface but some specific Zernike polynomials. 3,4,5 explained how specific spatial defect should be corrected to enable good purity of LG33.

• Higher-order Laguerre-Gauss interferometry for gravitational-wave detectors with in situ mirror defects compensation, Allocca et al. 2015 link

• Experimental test of higher-order Laguerre–Gauss modes in the 10 m Glasgow prototype interferometer, Sorazu et al. 2013 link

Create an updated Finesse file for the Einstein Telescope HF with proper RoC, including a realistic long recycling cavity. The same file will also be used for parametric instabilities.

1. Do the simulations with perfect spherical mirrors. Is the small astigmatism in the recycling cavity a show stopper ? The astigmatism is introduced in the PRC by the lightly converging beam passing through the beam splitter. Simulations have shown that astigmatism is slightly less than 0.1%. No power decreased or obvious distorted mode shape noticed.

1. Add mirror maps from second generation (with and without coating). How to get the right size (no real map with such size available) ? Several ways could be explored:
   • use current mirror map and fill the outside area with zero. Since most of the power is concentrated on the central area of the mirrors, it may be realistic way.
   • Stretching the map horizontal and vertically, that would shift the PSD of the surface toward low frequencies.
   • Expanding the map on the Zernike basis, recreate the maps with the same components but on a larger aperture.