Advanced LIGO's present baseline design uses arm cavities with Gaussian light beams supported by spherical mirrors. Because Gaussian beams have large intensity gradients in regions of high intensity, they average poorly over fluctuating bumps and valleys on the mirror surfaces, caused by random thermal fluctuations (thermoelastic noise). Flat-topped light beams (mesa beams) are being considered as an alternative because they average over the thermoelastic fluctuations much more effectively. However, the proposed mesa beams are supported by nearly flat mirrors, which experience a very serious tilt instability. In this paper we propose an alternative configuration in which mesa-shaped beams are supported by nearly concentric spheres, which experience only a weak tilt instability. The tilt instability is analyzed for these mirrors in a companion paper by Savov and Vyatchanin. We also propose a one-parameter family of light beams and mirrors in which, as the parameter alpha varies continuously from 0 to pi, the beams and supporting mirrors get deformed continuously from the nearly flat-mirrored mesa configuration ("FM") at alpha=0, to the nearly concentric-mirrored mesa configuration ("CM") at alpha=pi. The FM and CM configurations at the endpoints are close to optically unstable, and as alpha moves away from 0 or pi, the optical stability improves.
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