Thermoelastic dissipation in inhomogeneous media: loss measurements and displacement noise in coated test masses for interferometric gravitational wave detectors

The displacement noise in the test-mass mirrors of interferometric gravitational wave detectors is proportional to their elastic dissipation at the observation frequencies. In this paper, we analyze one fundamental source of dissipation in thin coatings, thermoelastic damping associated with the dissimilar thermal and elastic properties of the film and the substrate. We obtain expressions for the thermoelastic dissipation factor necessary to interpret resonant loss measurements, and for the spectral density of displacement noise imposed on a Gaussian beam reflected from the face of a coated mass. The predicted size of these effects is large enough to affect the interpretation of loss measurements, and to influence design choices in advanced gravitational wave detectors.

Figure 1

Diagram showing substrate with multilayer coating where the properties of the coating have been approximated to those of a single layer coating. Shown for clarity are the linear thermal expansion coefficients of the film, ${\alpha }_{\mathrm{f}}$ and substrate ${\alpha }_{\mathrm{s}}$.

Figure 2

Calculated frequency dependence of the thermoelastic losses of fused-silica substrates with dielectric multilayer coatings formed from 30 alternating quarter-lambda layers of (a) ${\mathrm{S}\mathrm{i}\mathrm{O}}_2$ and ${\mathrm{T}\mathrm{a}}_2{\mathrm{O}}_5$, (b) ${\mathrm{A}\mathrm{l}}_2{\mathrm{O}}_3$ and ${\mathrm{T}\mathrm{a}}_2{\mathrm{O}}_5$, (c) ${\mathrm{S}\mathrm{i}\mathrm{O}}_2$ and ${\mathrm{T}\mathrm{a}}_2{\mathrm{O}}_{\mathrm{5}}$ or (d) $\mathrm{A}{\mathrm{l}}_{\mathrm{2}}{\mathrm{O}}_{\mathrm{3}}$ and $\mathrm{T}{\mathrm{a}}_{\mathrm{2}}{\mathrm{O}}_{\mathrm{5}}$. Shown are loss factors for frequencies up to 100 kHz.

Figure 3

Calculated thermoelastic thermal noise, $\sqrt{S_x(f)}$ for coatings of (a) an $\mathrm{S}\mathrm{i}{\mathrm{O}}_{\mathrm{2}}/\mathrm{T}{\mathrm{a}}_{\mathrm{2}}{\mathrm{O}}_{\mathrm{5}}$ coating applied to a silica substrate (b) an ${\mathrm{A}\mathrm{l}}_2{\mathrm{O}}_3/\mathrm{T}{\mathrm{a}}_{\mathrm{2}}{\mathrm{O}}_5$ applied to a silica substrate (c) an $\mathrm{S}\mathrm{i}{\mathrm{O}}_{\mathrm{2}}/\mathrm{T}{\mathrm{a}}_{\mathrm{2}}{\mathrm{O}}_{\mathrm{5}}$ coating applied to a sapphire substrate and (d) an ${\mathrm{A}\mathrm{l}}_2{\mathrm{O}}_3/\mathrm{T}{\mathrm{a}}_{\mathrm{2}}{\mathrm{O}}_5$ coating applied to a silica substrate. Shown in each case is the noise obtained using the explicit low-frequency limit for the noise (solid line), and the noise including the full frequency dependence (dashed line), as given in Eq. (10).

Figure 4

Frequency-dependent part of thermoelastic loss function, $g(\xi )$ as a function of frequency, where $\xi \equiv \sqrt{2\omega {\tau }_f}$ and ${\tau }_f=l^2{C}_f/k_f$. Curves are shown for three values of the parameter $R\equiv k_f{\gamma }_f/k_s{\gamma }_s=\sqrt{k_f{C}_f/k_s{C}_s}$. Note that peak height is only a weak function of $R$.

Figure 5

Peak value of the normalized dissipation, $g_{\mathrm{m}\mathrm{a}\mathrm{x}}\equiv g({\xi }_{\mathrm{m}\mathrm{a}\mathrm{x}})$ as a function of the parameter $R$. The peak value of the normalized dissipation is seen to be only a weak function of $R$, the only material parameter on which it depends.

Figure 6

Normalized frequency ${\xi }_{\mathrm{m}\mathrm{a}\mathrm{x}}$ at which the normalized dissipation takes its maximum value, as a function of the parameter $R$. ${\xi }_{\mathrm{m}\mathrm{a}\mathrm{x}}$ corresponds to the thermal diffusion time across the film for $R<1$, and decreases as $1/R$ (corresponding to ${\omega }_{\mathrm{m}\mathrm{a}\mathrm{x}}$ decreasing as $1/R^2$ for $R>1$).