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- The approximated Beta_eff, for 1/4 high reflective coatings, which is reported in BGV 2000, and Evans 2008 is given by B_eff ~ (nH^2 *BL + nL^2*BH) / (nH^2 - nL^2) (which was used in Cole's paper to calculated their TO noise). BGV gave a sketch of this calculation in their paper (which I have not yet thoroughly understood). One problem is that, the result for B_eff obtained from this formula is the same whether the coatings start with nH or nL. This should be wrong, since most of the TR effect comes from the very first layers. The order of nH/nL should matter.
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Beff ~ (nH^2 *BL + nL^2*BH) / (nH^2 - nL^2) is valid only if the top layer is 1/4 layer of nL, [Gorodetsky, Phys Lett A 372 (2008)]. The complete calculation for general case is given in the reference. If the layer starts with nH, beta eff is = (BetaH + BetaL) / (4x(nH^2 - nL^2) ). So, GWINC and analytical approximation agree, Yay!  .
The effective beta reported in Cole's paper is 5e-4, but it should be ~ 5e-5 for coatings start with nH. The real thermo optic noise for their setup will be lower ( because TE is about the same level as TR). Their real TO noise should be a factor of 5.5 below the reported one (in Hz^2/Hz unit).
Note: There are still issues about the thermal fluctuation and the cut off frequency. These will greatly change the shape of the TO noise and the total noise level. I'm still investigating it.
The 1/2 wavelength cap with nL does reduce the TO noise. But we need to know exactly how thick the nH film on top will be, so the real TO effect can be estimated accurately.
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