I found out why the calculated values of the coatings' effective beta from GWINC and Cole etal paper are different. The order of Low/High refractive index material have something to do with the beta effective calculation.

Here are some facts about the coatings and calculation:

• The AlGaAs coatings used in the paper have no 1/2 wave cap. The structure is consisted of only 1/4 wave layers. Start with nH on top, and end with nH at the substrate (SiO2).
•  The PSD from thermo-refractive is SdT x Beta_eff x lambda. Where SdT is temperature fluctuation, Beta_effective is the overall dn/dT of the coatings,see the entry below for more details.
• In GWINC, Beta_eff is calculated numerically, taking each layer and calculating the reflectivity, then sum all the effect together. The result for Beta_eff is different, if the first layer (the top one) is changed between nH or nL. ( 5e-5 and 5e-4, cf PSL1178).
• 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.
• Computed values of B_eff from Gwinc code and the simplified formula agree if both start with nL. This makes me think that there is some assumptions in the simplified B_eff formula that the first layer is nL (which is customary, in SiO2/Ta2O5 coatings ).

So, I believe that the calculation for TO noise I have right now is correct. And for 100 ppm transmission (56 layers) with 1/2 wave cap, the TO noise is significantly reduced (add plot).  We should be able to finalize what we want for the AlGaAs mirrors soon.