I updated the optimization and error analysis. The error in optimized structure is comparable to that of a standard quarter wave length structure.
After a discussion with Rana, Garrett, and Matt, I fixed the thermooptic calculation, and the error analysis done in PSL:PSL:1315. The modifications are
1) fix the TO calculation (Yamamoto TR correction): There is a modification for TR correction that is not in Evans etal 2008, paper. I contacted M. Evans to ask about the details of this correction which is done in GWINC.
2) Try another optimized coatings with the correct TO calculation: After the correction, I ran doAlGaAs.m code, cf PSL:1269 using fmincon function , to find another optimized structure. The result is shown below.
above) layer structure in optical thickness, the .fig and .mat file are attached below. Note .mat file contains 54 layers, you need to add 1/4 cap to the first entry to calculate the noise budget.
above) noise budget of the optimized coating.
3) Repeat the error analysis : This time I used the following assumptions (from G Cole)
 the error is not random among each layer
 the error is constant in each layer type, ie all the layers from the same material (nH or nL) have the same percentage of error,
 error from nH and nL have the same sign. If one is thicker, another one is thicker, but the magnitude are uncorrelated.
 nH (GaAs) has better thickness control with 2sigma = 1percent, while nL(AlGaAs), has 2sigma = 2 percent.
Fig1: Above, percentage of error distribution between the two materials used in the calculation. nH(red) has 2 sigma = 1% and nL(blue) has 2sigma=1%.The same error distributions are used for both optimized layers and QWL layers in comparion, see fig2.
The section below is the algorithm used to distribute the error, this one makes the error between the two materials to be the same sign. The whole code can be found on svn.
mu1 = 0;
sigma1 = 0.5; % 2sigma is 1percent;
mu2 = 0;
sigma2 = 1;
run_num = 5e4; % how many test we want
errH = normrnd(mu1,sigma1,[run_num,1]); %errH in percent unit
errL = normrnd(mu2,sigma2,[run_num,1]); %errL in percent unit
errL = abs(errL).*sign(errH); %make sure that errH and errL have the same sign
dOpt = xout; % xout from doAlGaAs (optimized layer)
dOpt = [ 1/4 ; dOpt]; % got 54 layer no cap from doALGaAs, need to add the cap back
dOpt_e = zeros(length(dOpt),1);
for ii = 1:run_num;
dOpt_e(1:2:end)= dOpt(1:2:end)*(1+ errH(ii)/100 );
dOpt_e(2:2:end)= dOpt(2:2:end)*(1+ errL(ii)/100 );

===Result==
This time I calculated the change in reflection phase (TOP left), the ratio between TO noise from the coatings with error and the coatings with no error(top right), transmission (bottom left), and ratio of BR noise ( bottom right). The result from the optimized coating(blue) is compared with the QWL coating (black).
Fig2: Error analysis, in 5e4 run. Blue: from optimized coatings Black:from 55 QWL coatings, from 5x10^4 runs.
Reflection phase: The reflection phase can be away up to ~6 degree. The power at the surface will be ~Finesse/pi * Power input * sin^2 (6degree) ~ 50mW. Seems high, but this is about a regular power used in the lab.
Ratio of PSD TO/TO_0 : At worse, it seems the PSD TO noise will be ~ a factor of 10 higher than the design. However, this will be still lower than BR noise. I plotted only the error distribution for optimized coatings because for QWL coatings, the ratio will be about the same, since TO is dominated by TE.
Transmission: Most of the results are within 197200 ppm. The optimized coating has transmission ~ 197ppm. The QWL with 55 layers has transmission ~100ppm.
Ratio of BR: not much change here.
