Thermal noise and coating optimization in multilayer dielectric mirrors

Optical multilayer coatings of high-reflective mirrors significantly determine properties of Fabry-Perot resonators. Thermal (Brownian) noise in these coatings produce excess phase noise which can seriously degrade the sensitivity of high-precision measurements using these cavities. In particular, it is one of the main limiting factors at the current stage in laser gravitational-wave detectors (for example, project LIGO). We present a method to calculate this effect accurately and analyze different strategies to diminish it by optimizing the coating. Traditionally, the effect of the Brownian noise is calculated as if the beam is reflected from the very surface of the mirror's coating. However, the beam penetrates the coating, and Brownian expansion of the layers leads to dephasing of interference in the coating and consequently to an additional change in the reflected amplitude and phase. Fluctuations in the thickness of a layer change the strain in the medium and hence, due to a photoelastic effect, change the refractive index of this layer. This additional effect should also be considered. It is possible to reduce the noise by changing the total number and thicknesses of high and low refractive layers preserving the reflectivity. We show how an optimized coating may be constructed analytically rather than numerically as before. We also check the possibility of using internal resonant layers, an optimized cap layer, and double mirrors to decrease the thermal noise.