Finite mirror effects in advanced interferometric gravitational wave detectors

Thermal noise is expected to be the dominant source of noise in the most sensitive frequency band of second-generation, ground-based gravitational-wave detectors. Reshaping the beam to a flatter, wider profile which probes more of the mirror surface reduces this noise. The 'Mesa' beam shape has been proposed for this purpose and was subsequently generalized to a family of hyperboloidal beams with two parameters: twist angle α and beam width D. Varying α allows a continuous transition from the nearly flat (α=0) to the nearly concentric (α=π) Mesa beam configurations. We analytically prove that in the limit D→∞ hyperboloidal beams become Gaussians. The ideal beam choice for reducing thermal noise is the widest possible beam that satisfies the Advanced LIGO (Laser Interferometer Gravitational-wave Observatory) diffraction loss design constraint of 1 part per million (ppm) per bounce for a mirror radius of 17 cm. In the past the diffraction loss has often been calculated using the clipping approximation that, in general, underestimates the diffraction loss. We develop a code using pseudospectral methods to compute the diffraction loss directly from the propagator. We find that the diffraction loss is not a strictly monotonic function of beam width, but has local minima that occur due to finite mirror effects and leads to natural choices of D. For an α=π Mesa beam a local minimum occurs at D=10.67 cm and leads to a diffraction loss of 1.4 ppm. We then compute the thermal noise for the entire hyperboloidal family. We find that if one requires a diffraction loss of strictly 1 ppm, the α=0.91π hyperboloidal beam is optimal, leading to the coating thermal noise (the dominant source of noise for fused-silica mirrors) being lower by about 10% than for a Mesa beam while other types of thermal noise decrease as well. We then develop an iterative process that reconstructs the mirror to specifically account for finite mirror effects. This allows us to increase the D parameter to 11.35 cm for a nearly concentric Mesa beam and lower the coating noise by about 30% compared to the original Mesa configuration