I used results from ring down measurement in Penn 2003, without assuming the values of YL,YH. If the actual Young's moduli of both materials are about 60% of their nominal values, the calculation of BR noise will match our measurement within 3%. 

I used ring down drumhead mode from sample C2 and F2 since the phi_coating as reported in the paper is about the same as the phi_coating obtained from the analytical result (see previous entry). With these two eqs, I can write

Ysub * D/3 * phitot_1  = phiL*YL*dL_1 + phiH*YH*dH_1-------(1)  (see previous entry, last eq).

Ysub * D/3 * phitot_2  = phiL*YL*dL_2 + phiH*YH*dH_2-------(2)  .

phi tot_1 and _2 are 1/Qtot from the two samples. D is the thickness of the substrate (0.25 cm). dL and dH are the physical thickness of siO2 and Ta2O5 in each sample.

For any fixed values of YH and YL, the two eqs will solve for a pair of phiL and phiH.

First, I checked the validity of these two ring down measurements by using YL = 72 GPa, YH= 140GPa. The results are 

PhiL = 1.29e-4, phiH = 4.13e-4. These numbers agree with the reported values.

Then, I varied YH from 0.5*YH_0 to 2*YH_0 and YL from 0.5*YL_0 to 2*YL_0 ( YH_0 = 140GPa, YL_0 = 72GPa), and solved for the corresponding phiL and phiH. Then with all 4 parameters, BR noise can be calculated.

Below is a plot of ratio of BR calculation and our measurement, vs YH. Each trace represents different value of YL.

Each point on the plot will have information about phiL and phiH. If YL = 43 GPa (0.6*72GPa) and YH = 84 GPa (0.6*140GPa), the loss angles extracted from the ring down measurements are phiL = 2.15e-4 and phiH = 6.9 e-4. All these four parameters give the estimated BR noise comparable to our measurement to 2% (in PSD unit). 

==Conclusion==

 I'm trying to explain why our measurement is larger than the estimated calculation using numbers from literature. But we have good reasons to believe that the measurement is really BR coating since 

• The data has a correct slope, 1/f in PSD
• Scale with 1/w^2 in PSD, (BR noise from substrate/ spacer will have different scaling)
• Agree with Numata2003.

It is possible that loss angles in our coating is lossier than usual. But there are still other possible explanations. The results from ring down measurements rely on the values of Young's moduli of the coating materials. If the actual values divert from the nominal values, the losses will be changed as well. So I used the result from the ring down measurement, without assuming any values of YH and YL, then extracted values of phiH and phiL using different combinations of YH and YL and  calculated the coating noise according to each set of parameters.  If YL and YH have lower Young's moduli than their nominal values, coating BR noise will be higher and agree with our measurement. 

One might argue that 0.6 YL and 0.6 YH are too low.  Ta2O5 was measured with nano indentation to be ~ 140 GPa (Abernathy). Other references measured Ta2O5 ~ 100 GPa (see ref 16, 20 in Crooks2006 paper). So, uncertainty around 40% might be possible.  

In addition, this calculation also assume phi_bulk = phi_shear.  But the different value of phiB/phiS can also change the calculation between 0.5*S_0 to 1.6*S_0, for different values of phi bulk/phi shear ratio is varied by a factor of 5(see Hong2013). These values also change the noise level significantly.

So with the uncertainties in Young's moduli, the loss angles from ring down measurements can be changed significantly.  If the Young's moduli of the coatings are smaller than the nominal values, the loss angles calculated from a ring down result will be higher, and  it resuls in a higher level of coating BR noise calculation.

==Note== 

I'm surprised that for the value of 0.6*YL_0 and 0.6*YH_0 used above, with the loss angles of phiH = 6.89e-4 and phiL = 2.15 e-4, the calculated BR noise is almost the same as when I use the nominal value of YH,YL with the same loss (2.15 and 6.89e-4) see, PSL:1408.  I double checked the result, but I did not see anything wrong in the calculation. It turns out that the BR calculation is not very sensitive to YL, YH, but it is directly proportional to phiH, phiL. However, the values of phiH, phiL obtained from a ring down measurement are very sensitive to YL and YH  as we can see from the plot above.