  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.
