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I understand that Fresnel tells us what percentage of light reflects vs refracts, but what law tells me the percentage of refracted light that is transmitted versus the percentage of refracted light that is scattered back out as diffuse?

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  • $\begingroup$ Do you mean subsurface scattering? $\endgroup$
    – Matthias
    Feb 26 '17 at 13:07
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Physically, the origin of diffuse light is subsurface scattering, which happens continuously as light travels through a material. So, the proportion of transmitted light depends on the thickness of the object.

There's no precise equivalent to the Fresnel law, but maybe the closest thing is the Beer–Lambert law. It states that the transmitted light falls off exponentially with distance, proportional to $e^{-\sigma x}$, where $x$ is the distance the light has traveled through the material and $\sigma$ is the extinction coefficient (with units of inverse length). The non-transmitted portion of the light may be partly absorbed, and partly scattered through the material.

You get the appearance of "diffuse" when typical length scale of scattering is smaller than you can see ($1/\sigma \ll 1\text{ pixel}$). Then the light gets scattered so many times in such a short distance that by the time it comes back out, its distribution looks isotropic (Lambertian). The apparent diffuse color is controlled by the wavelength-dependence of the scattering and attenuation coefficients.

When the scattering length scale is larger, you get a soft appearance to the material (as in wax, marble, human skin, etc), or partial translucency.

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