The dependence of test-mass thermal noises on beam shape in gravitational-wave interferometers

TL;DR

This paper rigorously derives and analyzes how beam shape influences thermal noise in gravitational-wave detectors, improving understanding of noise scaling laws and effects of finite mirror size.

Contribution

It provides a unified, rigorous derivation of thermal noise scaling laws and investigates deviations caused by finite mirror size.

Findings

Unified derivation of four key thermal noise scaling laws

Identification of gaps in previous analyses

Analysis of finite mirror size effects on noise scaling

Abstract

In second-generation, ground-based interferometric gravitational-wave detectors such as Advanced LIGO, the dominant noise at frequencies $f \sim 40$ Hz to $\sim 200$ Hz is expected to be due to thermal fluctuations in the mirrors' substrates and coatings which induce random fluctuations in the shape of the mirror face. The laser-light beam averages over these fluctuations; the larger the beam and the flatter its light-power distribution, the better the averaging and the lower the resulting thermal noise. In semi-infinite mirrors, scaling laws for the influence of beam shape on the four dominant types of thermal noise (coating Brownian, coating thermoelastic, substrate Brownian, and substrate thermoelastic) have been suggested by various researchers and derived with varying degrees of rigour. Because these scaling laws are important tools for current research on optimizing the beam shape, it is important to firm up our understanding of them. This paper (1) gives a summary of the prior work and of gaps in the prior analyses, (2) gives a unified and rigorous derivation of all four scaling laws, and (3) explores, relying on work by J. Agresti, deviations from the scaling laws due to finite mirror size.