We calculate the thermal noise in half-infinite mirrors containing a layer of
arbitrary thickness and depth made of excessively lossy material but with the
same elastic material properties as the substrate. For the special case of a
thin lossy layer on the surface of the mirror, the excess noise scales as the
ratio of the coating loss to the substrate loss and as the ratio of the coating
thickness to the laser beam spot size. Assuming a silica substrate with a loss
function of 3x10-8 the coating loss must be less than 3x10-5 for a 6 cm spot
size and a 7 micrometers thick coating to avoid increasing the spectral density
of displacement noise by more than 10%. A similar number is obtained for
sapphire test masses.Comment: Passed LSC (internal) review. Submitted to Phys. Rev. D. (5/2001)
Replacement: Minor typo in Eq. 17 correcte
