1) The coating term is account for thermal expansion only(TE), no thermo refractive has not been included yet.

Farsi treats the effect from TR by assuming that all the TR contribution comes from the first few layers with 180 degree phase different from TE. From their result, TR contribution is much smaller in our frequency band, so I haven't included it in the calculation yet.

2) I agree that our fixed spacer setup will have different result. I'll think about it. Since the coupling is high at low frequency, and the thermal diffusion length is large  ~sqrt(k/C/2pif), which is ~ 1.7 cm [sqrt(1Hz/f)],  (SiO2 material properties) This is certainly comparable to the radius of the mirrors, and any effect on the edge may not be negligible.

Mistake!!, the heat diffusion length is sqrt[ k/ (rho*C*2pif) ] ,  where k is heat conductivity (W/mK), rho is mass density, C is specific heat per kg. This gives 367 um x sqrt(1Hz/f) in fused silica, which is ~ 1.2 mm at 0.1 Hz, and it is still small compared to the mirror's size. Then the boundary condition of our optics might not be as bad as I thought.  I'm thinking about using COMSOL to estimate the effect due to heat escaping from the substrate to the spacer. However, I expect that the boundary effect will help us a bit. The contact area at the spacer will provide additional heat bath for the substrate and reduce the heat at the spot area, thus reducing the thermal expansion effect. Plus, the expansion of the spacer/the substrate at the contact area will be in the opposite direction of that on the spot area. The expansion at the spot area will make the cavity shorter, while the expansion of the spacer/contact area will make the cavity longer. So I think the calculation will still give us the upper bound level, the shape at low frequency ( around 0.5 Hz and below), may change due to the longer heat diffusion length and the boundary effect is not negligible.