I checked Brownian coating noise level with uncertainties in coating parameters. The measured result is barely at the edge of the confident interval.
Hong2013 look into coating noise level when materials' parameters are changed. One example is the Young's modulus of Ta2O5. With the assumption phi bulk = phi shear, if Y_Ta2O5 is varied between 70e9 to 280e9 (nominal value = 140e9), coating thermal noise can be changed by a factor of 0.9 to 1.5 from the nominal value (in PSD m^2/Hz unit). It seems that the range is quite large compared to the numbers measured by various groups, (see PSL895 for error in material parameters). I used a smaller range, but I varied other parameters as well.
==Note about uncertainties in calculation==
I used rand command in Matlab to generate random values. The reasons are 1) for reported loss angles, say 4+/- 2e-4, if I use Gaussian dist, with sigma =2, mean = 4, sometimes the generated value will be negative, and 2) since we are only trying to see the possible range of the estimated noise level, not the real statistic value, rand should be ok at this point.
==1:fixed loss angles==
First I checked how much the parameters effect the calculation if the loss angles are fixed (phi silica = 1e-4, phi tantala = 4e-4). Y tantala is chosen between (70-280 GPa), Y silica is varied between72e9 +/- 10%, Poisson's ratio are varied between 10percent for coating materials. All substrate parameters are fixed, since they should be relatively well measured compared to that of the coatings. The result is around 0.5-0.85 of the measurement (in PSD m^2/Hz).
For a more conservative value of Ta2O5 ( 140+/-40 GPa), the result is a factor of 0.5-0.64 of the measurement.
==2:varied loss angles==
In this study, I varied loss angles of phi_silica = [0.8-1.2] x10^-4, phi_ta2O5 = [3,5]x10^-4, these numbers are reported from several measurement. Then I change the uncertainties range of Y_Ta2O5 in my calculation
- Y_ta2o5 = [100,180] GPa, the result is around [0.4,0.88] of the measurement.
- Y_ta2o5 = [70,280] GPa, the result is around [0.4,1.09] of the measurement. A histogram of the ratio between noise level and the measurement is shown below (from 1e4 runs). The measurement value (Sx/Smeasured = 1) is barely at the edge of the confident interval (and not from Gaussian distribution either).
==note and comment==
Both Hong's and Harry's calculation provide quite the same value (within 3%). So I show only histogram obtained from Hong's calculation. I don't know why the study shown in Hong paper choose the value of Y tantala between 70-280 GPa, most of the measurements report smaller uncertainty. But with that higher value of Y_Ta2O5, it can explain the measured noise level from our measurement. However, I doubt that this argument is valid, since most of the ring down measurements to evaluate phi_Ta2O5 assume Y_tantala ~ 140GPa. Then the loss angle of Ta2O5 should carry some information about Y_Ta2O5 in it and cannot be treated as an independent parameter like this calculation. I'll look into the ring down papers to see how much Y_Ta2O5 affects the extraction of its loss angle.