# On the performance limits of coatings for gravitational wave detectors made of alternating layers of two materials

## TL;DR

This paper investigates the fundamental performance limits of multilayer dielectric coatings used in gravitational wave detectors, optimizing for minimal transmittance and thermal noise without assumptions on layer configuration.

## Contribution

It formulates the coating design as a multiobjective optimization problem and derives explicit bounds on transmittance and noise, revealing fundamental tradeoffs and limits.

## Key findings

- Derived a Pareto boundary for transmittance and noise.
- Identified a closed-form lower bound curve in the tradeoff plane.
- Showed the relation between transmittance and noise for quarter wavelength designs.

## Abstract

The coating design for mirrors used in interferometric detectors of gravitational waves currently consists of stacks of two alternating dielectric materials with different refractive indexes. In order to explore the performance limits of such coatings, we have formulated and solved the design problem as a multiobjective optimization problem consisting of the minimization of both coating transmittance and thermal noise. An algorithm of global optimization (Borg MOEA) has been used without any a priori assumption on the number and thicknesses of the layers in the coating. The algorithm yields to a Pareto tradeoff boundary exhibiting a continuous, decreasing and non convex (bump-like) profile, bounded from below by an exponential curve which can be written in explicit closed form in the transmittance-noise plane. The lower bound curve has the same expression of the relation between transmittance and noise for the quarter wavelength design where the noise coefficient of the high refractive index material assumes a smaller equivalent value. An application of this result allowing to reduce the computational burden of the search procedure is reported and discussed.