Photothermal transfer function of dielectric mirrors for precision measurements

The photothermal 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 ${\mathrm{Ta}}_2{\mathrm{O}}_5:{\mathrm{SiO}}_2$ coating used in Advanced LIGO and for a crystalline ${\mathrm{Al}}_{\mathrm{x}}{\mathrm{Ga}}_{1-\mathrm{x}}\mathrm{As}$ 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 photothermal effect could also be useful to stabilize radiation pressure-based optomechanical 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 ${\mathrm{Al}}_{\mathrm{x}}{\mathrm{Ga}}_{1-\mathrm{x}}\mathrm{As}$ coatings requires the detailed heat flow analysis done in this paper.

Figure 1

A bode plot of the photothermal transfer function correction factor for a ${\mathrm{Ta}}_2{\mathrm{O}}_5:{\mathrm{SiO}}_2$ coating. The black (gray) traces are for a 19-doublet (eight-doublet) coating, corresponding to an Advanced LIGO end (input) test mass coating. For the solid traces the heat was deposited at the front surface of the coating. For the dashed traces it was deposited at the fourth interface layer, at a depth of $0.68\mu \mathrm{m}$. Finally, for the dash-dotted traces, the power was deposited in the coating according to the optical power present in each layer. At high frequencies the transfer function strongly depends on heat deposition depth, which in turn can be exploited to measure the absorption depth (see text). To get the full transfer function multiply with Eq. (2). The calculation is based on the parameters from Table 1.

Figure 2

A bode plot of the photothermal transfer function correction factor for a $\text{GaAs}:{\mathrm{Al}}_{0.92}{\mathrm{Ga}}_{0.08}\mathrm{As}$ coating with 40.5 and 18.5 $\lambda /4$ doublets (black and gray). For the solid traces the heat was deposited at the front surface of the coating. For the dash-dotted traces, the power was deposited in the coating according to the optical power present in each layer. To get the full transfer function multiply with Eq. (2). The calculation is based on the parameters from Table 2.

Figure 3

Thermo-optic noise of a ${\mathrm{Ta}}_2{\mathrm{O}}_5:{\mathrm{SiO}}_2$ coating with 19 and eight doublets (black and gray). The solid trace is based on the full heat flow calculation in the coating. The dashed and dash-dotted traces are the thin and thick coating approximations discussed in [9]. The calculation is based on the parameters from Table 1 and a beam spot size of $\mathrm{w}=6\mathrm{cm}$.

Figure 4

Thermo-optic noise of a $\text{GaAs}:{\mathrm{Al}}_{0.92}{\mathrm{Ga}}_{0.08}\mathrm{As}$ coating with 40.5 and 18.5 $\lambda /4$ doublets (black and gray), corresponding to power reflectivities of (1–2.5 ppm) and 0.9976. The solid trace is based on the full heat flow calculation in the coating. The dashed and dash-dotted traces are again the thin and thick coating approximations discussed in [9], applied to the crystalline coating. A cancellation of the noise coupling naturally occurs for a 18.5 layer doublet, but can be engineered for higher reflectivity coatings by deviating from the simple $\lambda /4$ structure [20]. The calculation is based on the parameters from Table 2 and a beam spot size of $\mathrm{w}=6\mathrm{cm}$.