Brownian thermal noise in multilayer coated mirrors

We analyze the Brownian thermal noise of a multilayer dielectric coating used in high-precision optical measurements, including interferometric gravitational-wave detectors. We assume the coating material to be isotropic, and therefore study thermal noises arising from shear and bulk losses of the coating materials. We show that coating noise arises not only from layer thickness fluctuations, but also from fluctuations of the interface between the coating and substrate, driven by fluctuating shear stresses of the coating. Although thickness fluctuations of different layers are statistically independent, there exists a finite coherence between the layers and the substrate-coating interface. In addition, photoelastic coefficients of the thin layers (so far not accurately measured) further influence the thermal noise, although at a relatively low level. Taking into account uncertainties in material parameters, we show that significant uncertainties still exist in estimating coating Brownian noise.