I have been working a project of creating a simulation of thin-film interference in a 3D modeling program, Blender, to be able to create materials that would use its effects. While uncommon, I would love for it to be able to correctly calculate reflectance and transmittance when the incident medium is absorbing. I was able to derive calculations for those situations such that R+T=1 (with what I thought was a generalized form of Snell's law), but sadly, I'm realizing that they are incorrect. I have been able to find an article, "Generalization of complex Snell-Descartes and Fresnel laws", that generalizes Snell's law, and I believe it holds the--partial--key to solving this problem (There's still the correction of the transmittance term by the Poynting vector). The problem is, I don't understand how the different angles used in the generalization are calculated. For example, in the article, β can be found from the wave vector, but the article describes a form of the wave vector that I'm not sure how to derive. I'd love to be able to continue with this project, but this problem of calculating for absorbing incident media has had me stuck for many months. If you would like to see my progress check out this Blender Artists post. For an idea of my understanding of this, I honestly don't understand much of the physics (creating materials to use in Blender is what got me into this topic), but I try to understand it through the math presented. So when there seems to be a gap in the math, there's likely a gap in my understanding. Thanks everyone!