Thermal noise in half infinite mirrors with non-uniform loss: a slab of excess loss in a half infinite mirror

TL;DR

This paper calculates thermal noise in half-infinite mirrors with a lossy coating layer, providing formulas and thresholds for coating loss to limit noise increase in precision optical systems.

Contribution

It introduces a model for thermal noise in half-infinite mirrors with non-uniform loss layers, deriving scaling laws and practical loss thresholds for specific materials.

Findings

Thermal noise scales with coating loss and thickness relative to beam size.

For silica, coating loss must be below 3x10^-5 to keep noise increase under 10%.

Similar thresholds apply to sapphire test masses.

Abstract

We calculate the thermal noise in half-infinite mirrors containing a layer of arbitrary thickness and depth made of excessively lossy material but with the same elastic material properties as the substrate. For the special case of a thin lossy layer on the surface of the mirror, the excess noise scales as the ratio of the coating loss to the substrate loss and as the ratio of the coating thickness to the laser beam spot size. Assuming a silica substrate with a loss function of 3x10-8 the coating loss must be less than 3x10-5 for a 6 cm spot size and a 7 micrometers thick coating to avoid increasing the spectral density of displacement noise by more than 10%. A similar number is obtained for sapphire test masses.