I'm checking how loss angle of Ta2O5 is related to its Young's modulus (as used in ring down measurements), then I use that relation in error calculation for BR noise in coating. The uncertainties in Young's moduli of SiO2/Ta2O5 might lead to errors in loss angles and BR noise in coating.

==Background==

Many ring down measurements (see Penn2003, Crooks2004, Crooks2006), observed loss from a disc substrate with multilayer coatings of Ta2O5/SiO2. The loss in the coating (ring down mode) is written as  

phi_c = (phi1*Y1*d1 + phi2*Y2*d2 )/ (Y1*d1 + Y2*d2)    --------(1)

Where phi_c is determined from the measurement. Y is the young's modulus, phi is loss angle of material, d is physical thickness of the material.

Then phi1 and phi2 is determined with the assumption that Y1 and Y2 are known.  

So, the reported value of phi Ta2O5 is directly related to its Young's modulus. The uncertainty calculation of BR noise where Y, phi are varied independently might not reflect the real situation.

For example, I recalculated phi_c (of QWL structure) using phiH phiL of  4e-4, 1e-4. YH = 140 Gpa, YL = 72GPa. Then I rearranged eq(1) so that phiH can be written as a function of YH to see how the loss angle of Ta2O5 (H) will change with its Young's modulus assuming that YL and phiL are fixed.

fig1: How phi Ta2O5 changes with Young's modulus.

==BR calculation with loss parametrized by Young's modulus==

With the loss angles parametrized by the Young's modulus, I calculate the estimated thermal noise compared to our measurement (using Hong2013) 

fig2: ratio of BR calculated and our result. 

It is interesting that, even with the lower phiH as YH increases, the total BR noise increases. And the nominal value that we have been using (YTa2O5 = 140 GPa) yields almost the minimum value of BR noise calculation. 

==next step== 

  So far, the calculation is done assuming phiL = 1e-4, YL = 72e9.  The next step is to varied phiL, YL, phiH, YH all together ( with the constraint given by eq1) and see how BR noise changes.

  I'm also checking how large the errors are in the measurements for Young's modulus (both SiO2/Ta2O5). Crooks2006 reports the value of Young's modulus of Ta2O5 with the assumption that Y_SiO2 is 72e9. This might give another constraint.