  We report on thermal noise from the internal friction of dielectric coatings
made from alternating layers of Ta2O5 and SiO2 deposited on fused silica
substrates. We present calculations of the thermal noise in gravitational wave
interferometers due to optical coatings, when the material properties of the
coating are different from those of the substrate and the mechanical loss angle
in the coating is anisotropic. The loss angle in the coatings for strains
parallel to the substrate surface was determined from ringdown experiments. We
measured the mechanical quality factor of three fused silica samples with
coatings deposited on them. The loss angle of the coating material for strains
parallel to the coated surface was found to be (4.2 +- 0.3)*10^(-4) for
coatings deposited on commercially polished slides and (1.0 +- 0.3)*10^{-4} for
a coating deposited on a superpolished disk. Using these numbers, we estimate
the effect of coatings on thermal noise in the initial LIGO and advanced LIGO
interferometers. We also find that the corresponding prediction for thermal
noise in the 40 m LIGO prototype at Caltech is consistent with the noise data.
These results are complemented by results for a different type of coating,
presented in a companion paper.
