# Brownian Thermal Noise in Functional Optical Surfaces

## TL;DR

This paper introduces a formalism to accurately compute Brownian thermal noise in complex optical surfaces by relating a specific readout variable to Maxwell's stress tensor, improving previous estimates.

## Contribution

It develops a method linking the readout variable to Maxwell's stress tensor for precise thermal noise calculations in optical surfaces.

## Key findings

- Exact analysis yields lower thermal noise estimates than previous simplified models.
- The formalism reveals a strong correlation between optical properties and thermal noise.
- Application to T-shape grating reflectors demonstrates improved accuracy.

## Abstract

We present a formalism to compute Brownian thermal noise in functional optical surfaces such as grating reflectors, photonic crystal slabs or complex metamaterials. Such computations are based on a specific readout variable, typically a surface integral of a dielectric interface displacement weighed by a form factor. This paper shows how to relate this form factor to Maxwell's stress tensor computed on all interfaces of the moving surface. As an example, we examine Brownian thermal noise in monolithic T-shape grating reflectors. The previous computations by Heinert et al. [Heinert et al., PRD 88 (2013)] utilizing a simplified readout form factor produced estimates of thermal noise that are tens of percent higher than those of the exact analysis in the present paper. The relation between the form factor and Maxwell's stress tensor implies a close correlation between the optical properties of functional optical surfaces and thermal noise.