Also, I found that the effective reflectivities from the top surface of layers calculated in Evans et a. PRD 78, 102003 (2008) were different from reflectivities calculated by matrix method in Hong et al. PRD 87, 082001 (2013). It turned out that the sign before phase in Eq. B3 and B4 in Evans et a. PRD 78, 102003 (2008) were opposite to what comes from the matrix method. After discussions with Gabriele, I came to the conclusion that this sign only creates a difference of giving complex conjugates of effective reflectivity. But to have consistency, I have corrected the sign of phase in the new code. Attached is a comparison of old effective reflectivity and the new one. One can see that the imaginary values are opposite in sign but everything else matches perfectly. Also, this has no effect on the effective coating coefficient of thermo-refraction.