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I know this post is three years old but I wanted to verify the formula used in glass material to calculate the transmittance and transmissivity. I post it here for future reference. Most of this is taken from "Medium and High Temperature" book p.53-54

For homogenous transparent media the light intensity decreases exponentially with layer thickness, as predicted by Beer’s law:

Therefore transmittance, the exited (or transmitted) light intensity over incidence intensity, is:

where s is the thickness of the material and k is extinction coefficient. This quantity applies to a single pass. Hence transmissivity (less used in literature) is a single pass transmittance. In practice a layer of glass will have multiple interreflections which must all be accounted for in calculating the total transmittance . The total wavelength-averaged reflectance leaving the top surface divided by the incident flux in below figure can be calculated by summing the infinite series of components making up the total flux.

Using algebra this simplifies to:

Which is what you get in the formula for glass in Behaviour of materials in RADIANCE. These equations were first developed by G. G. Stokes and are called Stokes’ equations.

As mentioned the in the above formula applies to a single pass. That’s why it is said that transmissivity excludes multiple interreflction within the medium. Now coming to derivation of transmissivity (which is needed as for glass material parameters calculated using the formula in trans.cal) from normal transmittance , we have:

This is in the form of quadratic equation . The answers are:

Which is same as what we find in trans.cal file:

sq(x) : xx; rn = sq((1-n)/(1+n)); tn = (sqrt(sq(sq(1-rn))+4sq(rnTn))-sq(1-rn)) / (2sq(rn)*Tn);

The only thing I didn't figured out is why, in Radiance behaviour of materials, the transmissivity later is taken as: