In our meeting, Eric mentioned that there might be some uncertainty in how the average coating properties are calculated.
To see how much it matters, I set the average properties to either that of the highindex (H) or lowindex (L) material, and calculated the ratio of the new thermooptic noise to the original calculation (TO'/TO) and the ratio of the new thermooptic noise to the unchanged Brownian noise (TO'/Br) for Tara's optimized coating structure. The results are in the table below:
Change: 
TO'/TO 
TO'/Br 
No Change 
1 
0.015 
C_c = CH 
0.99 
0.014 
C_c = CL 
1.01 
0.015 
k_c = kH 
1.12 
0.016 
k_c = kL 
0.89 
0.013 
alphaBar_c = aH 
358 
5.17 
alphaBar_c = aL 
384 
5.55 
alphaBar_k = alphas 
372 
5.37 
alphaBar_k_TR = alphas 
3.12 
0.045 
alphaBar_c = alphaBar_kH 
2.88 
0.042 
alphaBar_c = alphaBar_kL 
2.047 
0.030 
alphaBar_k_TR = alphaBar_k 
5.775 
0.084 
alphaBar_k = alphaBar_k_TR 
145 
2.096 
C = Heat Capacity/Volume, k = thermal conductivity, alpha/a = thermal expansion
alphaBar_c and alphaBar_k are more complicated, since they take into account the Poisson ratio and Young's modulus of the coating materials, and may be wildly different from the thermal expansion coefficient. alphaBar_c is an average of alphaBar_k values, and when I use "alphaBar_k = alphas", I'm indicating that alphaBar_k is an array, and I have replaced that array with an array of the corresponding thermal expansion coefficients. As we can see in the final four rows of the table, alphaBar_c has a much smaller affect if we use an alphaBar_k value with all its added moduli and ratios instead of just regular thermal expansion. alphaBar_k_TR is the array of values used in the "Yamamoto Correction" to calculate the appropriate alphaBar for the thremorefractive noise.
This all indicates to me that while most of the averages won't have much effect on our cancellation, a mistake in the calculation of alphaBar_k will.
The difference between alphaB_k and alphaBar_k_TR (in the last two rows of the table) is also interesting. Kazuhiro Yamamoto tells us this equation is correct, and explains the correction here. It's apparently because there is no added strain in the substrate due to the change in the refractive index, while there is strain for the thermal expansion.
