I'm working on estimating aLIGO sensitivity when material uncertainties are taken into account. I have a result for a reference cavity, uncertainty due to Ta2O5's Young's modulus might have smaller effect than we previously expected. All plots and code are attached below.
==Intro==
GWINC does not take any uncertainties in material parameters into account, so its noise budget does not have any error bar. We want to know how the noise budget might change due to imprecise knowledge of the material parameters. One particular issue is coating thermal noise that is dominating around 30  200 Hz, so we want to know how its level will change with material parameters. Some import ant parameters are loss angles and Young's moduli of each material.
In Hong et al 2013 paper, there is a plot of the calculated coating Brownian noise vs Ta2O5's Young's modulus (YH). The calculated coating BR noise is calculated with the corresponding YH while other parameters are fixed. This would be ok if each parameters were independently measured. In reality, loss angles are measured from ring down measurements, and YH and YL are used to calculated the material loss angles (phiH/phiL), see Penn et al. 2003. So to make the calculation reflects the real situation, we should take the correlation between phiH/phiL and YH/YL into account when we calculate coating BR noise. So the goal is to estimate coating BR noise for aLIGO with some uncertainties from loss angles and Young's moduli of the coatings.
==calculation==
calculate BR noise vs YL and YH (see PSL:1408 for the original code) using numbers from our setup (these can be changed later when we want to apply for aLIGO calculation). The code calculates BR noise with phiL = 1e4, phiH = 8e4. the numbers are from our measurement and another ring down measurement. This does not take the correlation between loss any YL/YH into account. I do this to compare my code to Hong's result, and they agree. YL and YH are varied between 80% and 120% of their nominal values (YL = 72 GPa, YH = 140 GPa).
above: Fig1:Thermal noise level of 28 Layer QWL structure, spot size = 180 um as a function of YH and YL
above:Fig2: three slices from the 3d plot for different values of YL, Y_L min and Y_L max are 80% and 120% of the nominal value.
above:Fig3: three slices from the 3d plot for different values of YH, Y_H min and Y_H max are 80% and 120% of the nominal value.
From the plots, we see that the uncertainty in Y_L does not change the noise level that much compare to Y_H. This is because phiH is about a factor of 8 larger than phiL, the effect from the lower loss material does not show up much. If the losses of the two materials are comparable, then the uncertainties of their Young's moduli will equally change the BR noise.
We can see from the plot that, in the estimated range of YH, the higher YH leads to the higher BR noise level.
Next, let's assume that the values of SiO2 are well measured and the error is much smaller than those of Ta2O5, so we can fix phiL and YL. Then recalculate BR noise when phiH and YH are correlated. I use a calculation from ring down measurement (see PSL:1412 or Harry 2002 or Penn 2003). The equation is
constant = phi_parallel = (YL*dL*phiL + YH*dH*phiH) / (YL*dL + YH*dH)
from this equation, we can write phiH as a function of YH assuming that other parameters are constant. Currently, I'm using numbers from CTN setup.
above: Coating BR noise as phi_H is varied along Y_H (green) compared with the previous calculation (Blue) from fig 2. The two traces cross at YH = 140 GPa. Note that in this plot, YH is varied between 50% and 200% of the nominal value. We see that the uncertainty of coating noise due to YH becomes smaller compared to the previous calculation done in Hong paper.
==what to do next==
 Apply my calculation to GWINC: Right now it is a stand alone code, but I plan to write a code that uses parameters and function mostly from GWINC because it will be easy if we want to plot it later for aLIGO noise budget.
 Find out the uncertainties of YH and phiH for Ta2O5 doped with TiO2, check the amount of TiO2 for aLIGO coating.
 Plot aLIGO noise budget , and estimate the inspiral range due to the sensitivity.
==Note==
From fig3, uncertainty in YL does not change the BR noise level that much, but this calculation assumes no correlation between YL and phiL. I have not been able to include uncertainty in YL and see the effect on phiL yet, because that will need more constraint equation. But I should check if it will greatly change phiL and affect the total BR noise calculation or not.
