  The displacement noise in the test mass mirrors of interferometric
gravitational wave detectors is proportional to their elastic dissipation at
the observation frequencies. In this paper, we analyze one fundamental source
of dissipation in thin coatings, thermoelastic damping associated with the
dissimilar thermal and elastic properties of the film and the substrate. We
obtain expressions for the thermoelastic dissipation factor necessary to
interpret resonant loss measurements, and for the spectral density of
displacement noise imposed on a Gaussian beam reflected from the face of a
coated mass. The predicted size of these effects is large enough to affect the
interpretation of loss measurements, and to influence design choices in
advanced gravitational wave detectors.
