Optical multilayer coatings of high-reflective mirrors significantly determine the 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 with these cavities, in particular in laser gravitational-wave antennas (for example project LIGO), where at the current stage it is one of the main limiting factors. We present a method to calculate this effect accurately and analyze different strategies to diminish it by optimizing the coating. Traditionally this noise is calculated as if the beam is reflected from the surface of the mirror fluctuating due to the sums of the fluctuations of each layer. However the beam in fact penetrates a coating and Brownian expansion of the layers leads to dephasing of interference in the coating and consequently to additional change in reflected phase. Fluctuations in the thickness of a layer change the strain in the medium and hence due to photoelastic effect change the refractive index of this layer. This additional effect should be also considered. It is possible to make the noise smaller preserving the reflectivity by changing the total number of layers and thicknesses of high and low refractive ones. We show how this optimized coating may be constructed analytically rather then numerically as before. We also check the possibility to use internal resonant layers and optimized cap layer to decrease the thermal noise.
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