Since we measured thermal noise from the coating(QWL, SiO2/Ta2O5), we want to extract loss angles of each materials. The losses are about a factor of 2 higher than the numbers reported in the literature.
So far, there are 3 calculations we have been using for coating noise estimation.
 Nakagawa , this calculation assumes Young’s moduli , Poisson ratio of the coating and the substrate are the same. The coating is s a single thin layer. Sx ~ phiC . If we fit this calculation to our result, the coating loss is about 4.15e4. This number agrees with the result from Numata2003, and agree with both our measurement from short and long cavities.
 Harry2002: Sx is function of ( phi_perp, phi_para), both phi_perp and phi_para are functions of( Y,Yc,sigma,sumac, phiL, phiH)
 Hong2013, Sx is function of phi_bulk and phi_shear. If we assume that phi bulk and shear are the same for each material phi bulk L = phi shear L = phi L, Sx ~ const1 phi L + const2 * phi H,,
In essence, both Harry's and Hong's result can be written as a linear combination of phiL and phiH. I used Harry result to compare with Hong to see if there is any differences in the result or not, but both gave me the same answer.
==calculation==
The calculation is attached below. I made sure that the calculation from Hong and Harry are correct by choosing the elastic properties of the coatings to be the same as that of substrate and checking the the results agree with Nakagawa's. So the code should be correct.
Then, I varied phiL and phiH to match the measurement. The measurement is represented by the prediction by Nakagawa with the fitted loss (phiC = 4.15e4).
Both calculations gave the similar relation between phiH (phi tantala) and phiL (phi silica) to match the measurement:
phiH = 1.4 phiL + 9.7 (Hong)
phiH = 1.44 phi L + 9.77 (Harry) (assume sigma1 = sigma 2 = 0)
The problem is if we use the nominal numbers from various reports, phiL ~ 1e4, phiH ~ 4e4. The result will be off by almost a factor of 2. For example, for phiL = 1e4, this means phi H has to be 8.4e4. Or if phi H is chosen to be 4e4, phi L will be ~4e4 as well. It seems that our result is higher than the predictions (under some assumptions).
Table1 below shows some possible values of phiL and phiH extracted from our result and the calculation.
phiL x10^4 
phiH x10^4 
0.1 
9.6 
0.5 
9.05 
1 
8.33 
1.5 
7.61 
2 
6.89 
3 
5.35

But we have good evidences from Numata and short/long cavities (spot size dependent) to believe that the measurement is real coating thermal noise . The reason why the prediction is smaller than the measurement could be that the losses is actually higher in our coating. Most ring down measurements were done after 2002 while our coatings were fabricated around 1997. Coating vendors might become more careful about loss and improved their process. But the result from Numata was out in 2003, and it is about the same as ours, so I'm really not sure what can we say about this.
==numbers from literature==
Penn2003: (disc ring down) phiL = (0.5 +/ 0.3) x10^4 , phiH = (4.4 +/ 0.2) x10^4
Numata2003 (direct measurement) phiC = 4.4e4;
Crooks2004(disc ring down) phiL = 0.4+/0.3 x10^4, phiH = (4.2+/0.4)x10^4 (the frequency dependent part is ignored)
Crooks2006: (disc ring down) phiL = (1.0 +/ 0.2) x10^4 phiH = (3.8 +/ 0.2) x10^4 (small change in TE calculation from previous paper)
Martin2009 : (blade) phiH = 3+/ 0.5 x10^4 (at 300K)
Martin2010: (blade) phiH = (2.55) x10^4 ( heat treated at 600C, several frequencies)
LMA2014: (blade) phiL = (0.43+/0.02) x10^4 phiH = (2.28 +/ 0.2) x10^4
