  The photo-thermal transfer function from absorbed power incident on a
dielectric mirror to the effective mirror position is calculated using the
coating design as input. The effect is found to change in amplitude and sign
for frequencies corresponding to diffusion length comparable to the coating
thickness. Transfer functions are calculated for the $Ti$-doped ${\rm
Ta_2O_5:SiO_2}$ coating used in Advanced LIGO and for a crystalline ${\rm
Al_xGa_{1-x}As}$ coating. The shape of the transfer function at high
frequencies is shown to be a sensitive indicator of the effective absorption
depth, providing a potentially powerful tool to distinguish coating-internal
absorption from surface contamination related absorption. The sign change of
the photo-thermal effect could also be useful to stabilize radiation
pressure-based opto-mechanical systems. High frequency corrections to the
previously published thermo-optic noise estimates are also provided. Finally,
estimating the quality of the thermo-optic noise cancellation occurring in
fine-tuned ${\rm Al_xGa_{1-x}As}$ coatings requires the detailed heat flow
analysis done in this paper.
