Photo-Thermal Transfer Function of Dielectric Mirrors for Precision Measurements

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.