Since the optimized layer structure is designed, I'm checking how the coatings properties change with error in layer thickness.
G.Cole said that they can control each layer thickness within 0.3%. So I tested the optimized coatings properties by adding some random number within +/ 0.5% on each layer thickness. The results are shown below for 10 000 test.
The error check does the following:
 start from the optimized coating structure reported in PSL:1291.
 add random thickness to each layer, within 0.5% of each layer
 calculate the values of interest, then histogram them.
The figure below is an example of the varying layer thickness added by rand command. They are confined within 0.5%.
1) result from the error in thickness control
Above: histograms of the important values. top left, reflected phase. top right, ratio between PSD of Brownian noise and Thermo optic noise at 100 Hz. Bottom left, transmission. Bottom right, total coating thickness error.
comments: this test is chosen for 0.5% error which is almost a factor of 2 worse than what they claimed (0.3%), so the actual result should be better. I assumed 0.5% errof because of the irregular layer structure of the optimized coatings, there might be some more error in the manufacturing process.
 Reflected phase: we want the reflected phase to be close to 180, so that the Efield at the coating surface is close to 0. more than 50% of the results are within 179.5degree, this means that the power build up will be ~ Finesse/pi * Power input * sin^2 (0.5degree) ~ less than 0.4 mW, so there should be no problem about burning at the surface.
 ratio between PSD of Brownian/Thermo optic noise. This plot imply how well the cancellation works. Since Brownian noise will almost not change (both materials have the same loss, total thickness varies less than 1%), the ratio of Br/TO noise (at 100Hz) tells how much TO cancellation is. From the histogram we are quite sure that cancellation will work most of the time.
 Transmission is good around 200+/ 10ppm this is ok with the requirement.
 total physical error is ~5nm while the coatings thickness is ~ 4um. so the total error is <0.1% Brownian noise calculation will not change much.
2) result from different calculated Beta values:
Here I checked what happen if the beta calculation was wrong, and the error is still within 0.5% in each layer.
In Evans paper, the effect from "Thermorefractive" comes from the phase changes of the wave travels in each layer. So it includes the effect from dn/dT and dz. The effective beta for each layer is given as
[evan B8],
where alpha bar is
[evans A1]
Where s denotes substrate, k denotes the material in each layer (high or low indices).
So my, calculation & optimization have been using these equations.
However, in the original GWINC code for TO calculation, the calculation [B8], alphabark( used in dTR) is not the same as A1, but rather.
alphaH * (1 + sigH) / (1  sigH)
see getCoatLayerAGS.m. Line 1617.
This is used in the calculation for beta effective in getCoatTOphase. Line7374. Notice that for dTE, the alpha_bar_k is the same as used in Evans. (line 77).
the comment says "Yamamoto thermorefractive correction". I emailed kazuhiro yamamoto, but never got a response back. So I keep using the same formula as in Evans because I don't see the reason why the expansion contribution should be different between TE and TR.
So this is the nb plot for TO noise from the optimized coating, if using yamamoto TR correction.
Above: nb from the optimized coatings, using Yamamoto TR correction. The cancellation becomes worse, but TO is still lower than other noise.
Finally, I repeat the same error analysis for random noise in the thickness (+/ 0.5%).
Most of the parameters behave similarly, except the cancellation (upper right plot). Now BR is only ~ x12 larger than TO noise because of the worse cancellation. Good news is, it still below Brownian noise, the cancellation still somehow works.
=summary=
 From the optimized coating structure (T=200ppm), thickness control within 0.5% in each layer will make the coating work as expected.
 The yamamoto TR correction is still an unresolved issue, but the optimized coating will still work.
 we should be ready to order soon.
