Internal thermal noise in the LIGO test masses : a direct approach

TL;DR

This paper introduces a new direct method based on the Fluctuation-Dissipation Theorem to analyze internal thermal noise in LIGO test masses, capable of handling inhomogeneous dissipation and arbitrary beam sizes, revealing surface dissipation effects.

Contribution

A novel direct approach to thermal noise analysis that overcomes limitations of previous normal-mode decomposition methods, applicable to inhomogeneous dissipation and various beam sizes.

Findings

The new method agrees with previous results in their domain of validity.

Surface dissipation effects scale as 1/r_0^2, potentially significant for small beam sizes.

Homogeneous dissipation scales as 1/r_0, less sensitive to beam size.

Abstract

The internal thermal noise in LIGO's test masses is analyzed by a new technique, a direct application of the Fluctuation-Dissipation Theorem to LIGO's readout observable, $x(t)=$(longitudinal position of test-mass face, weighted by laser beam's Gaussian profile). Previous analyses, which relied on a normal-mode decomposition of the test-mass motion, were valid only if the dissipation is uniformally distributed over the test-mass interior, and they converged reliably to a final answer only when the beam size was a non-negligible fraction of the test-mass cross section. This paper's direct analysis, by contrast, can handle inhomogeneous dissipation and arbitrary beam sizes. In the domain of validity of the previous analysis, the two methods give the same answer for $S_x(f)$, the spectral density of thermal noise, to within expected accuracy. The new analysis predicts that thermal noise due to dissipation concentrated in the test mass's front face (e.g. due to mirror coating) scales as $1/r_0^2$, by contrast with homogeneous dissipation, which scales as $1/r_0$ ($r_0$ is the beam radius); so surface dissipation could become significant for small beam sizes.