Material loss angles from direct measurements of broadband thermal noise

We estimate the loss angles of the materials currently used in the highly reflective test-mass coatings of interferometric detectors of gravitational waves, namely Silica, Tantala, and Ti-doped Tantala, from direct measurement of coating thermal noise in an optical interferometer testbench, the Caltech TNI. We also present a simple predictive theory for the material properties of amorphous glassy oxide mixtures, which gives results in good agreement with our measurements on Ti-doped Tantala. Alternative measurement methods and results are reviewed, and some critical issues are discussed.

Figure 1

Ti-doped Tantala. Refraction index according to Bruggemann formula (top left), tensile (Young) modulus (top right), loss angle (bottom left), and Poisson modulus (bottom right) according to Barta EMT formula.

Figure 2

Comparison between Titania-doped Tantala loss angle distributions resulting from TNI measurements and EMT.

Figure 3

(Left) Joint distribution of ${\varphi }_{{\mathrm{SiO}}_2}$ and ${\varphi }_{{\mathrm{Ta}}_2{\mathrm{O}}_5}$ obtained from the measured loss angles of coatings 1 and 2 in Table 1. (Right) The 0.95, 0.9, 0.85 quantile ellipses of the same distribution.

Figure 4

Graphic construction for standard error propagation for coatings 1 and 2, showing the intersection between the uncertainty strips obtained from Eq. (2) upon letting ${\varphi }_C={\mu }_c^{(I,II)}\pm {\sigma }_c^{(I,II)}$. The resulting uncertainty intervals for ${\mathrm{SiO}}_2$, $I_L$, and ${\mathrm{Ta}}_2{\mathrm{O}}_5$, $I_H$ are indicated.

Figure 5

Titania doped Tantala loss angle distributions from coatings 3 (QWL) and 4 (OPT) and their pooled (maximum likelyhood) combination. The average and standard deviation of the pooled distribution are $3.66\times 10^{-5}$ and $0.26\times 10^{-5}$, respectively.

Figure 6

Graphic construction for standard error propagation for coatings 3 and 4. The red and blue strips are obtained from Eq. (2) using the measured loss angles of coatings 3 and 4 and their uncertainties. The green band is the Silica loss angle uncertainty strip obtained from measurements on coatings 1 and 2. The intersection of the green band with each coating measurement yields two uncertainty intervals for ${\mathrm{TiO}}_2\colon\colon {\mathrm{Ta}}_2{\mathrm{O}}_5$ loss angle, $I_{H,3}$ and $I_{H,4}$, respectively. The pooled uncertainty interval is $I_{H,3}\cap I_{H,4}$.