Thermal noise in half-infinite mirrors with nonuniform loss: A slab of excess loss in a half-infinite mirror

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 $3\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}8},$ the coating loss must be less than $3\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}5}$ for a 6 cm spot size and a 7 \ensuremath{\mu}m thick coating to avoid increasing the spectral density of displacement noise by more than 10%. A similar number is obtained for sapphire test masses.