I checked all the discrepancies in the calculations between GWINC and that of Cole. The issues are almost cleared, only the value of effective beta, (dn/dT) that still remains.
The PSD of TO noise in [m^2/Hz] is given by S_{x}(f)= S_{T }(f) x (dTE + dTR). See Evans etal Phys Rev D 78, 102003. Where:
- S
_{T }(f) = Temperature fluctuation as sensed by a Gaussian beam
- dTE = dx/dT, or rate of change of mirror position with respect to temperature change due to thermoelastic mechanism.
- dTR = rate of change of mirror position with respect to temp change due to thermo-refractive mechanism
S_{T}(f) can be calculated analytically, see BGV, Phys Lett A 271 , (2000) 303-307 eq9, this also assumes adiabatic approximation. In Mike Martin's thesis, the temp fluctuation is generalized to all frequency (by contour integral, I'll show the details later). The parameters for calculating S_{T}(f) are taken from that of substrate (in GWINC), but Cole's paper and Mike's thesis use that of the coatings. That makes Cole's result about a factor of 7 higher than that from GWINC. Matt and I discussed this with Mike, he thought that the calculation should use the substrate's properties since the thermal length in the frequency of interest is much larger than the coating thickness.
The issue with which parameters should be used might be a less serious problem if (dTE + dTR) can partially cancel out making the whole TO noise much smaller. Basically dTE is ~ alpha* coatings thickness, where alpha is the thermal expansion coefficient of the coatings. dTR is ~ beta_eff * lambda. The calculations for dTE from GWINC and Cole are about the same (1.1 x 10^-10) [m/K], where the effective beta are different by about and order of magnitude. Cole reports the value of beta effective to be -5.5 x10^-10 , meanwhile GWINC gives me 0.5x10^-10.
This means that the TE and TR,as calculated from GWINC are more comparable, and the TO result is reduced significantly. While the TO result from Cole is mostly TR. I calculated the TR following the 1/4 stack approximation in Evans paper and got the same result as in Cole. I'm checking what happen in GWINC code for TR calculation. |