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2) Eventually the ray direction will become so close to
horizontal that the ray cannot refract. The angle of
incidence is so close to 90° that there is no angle of
refraction that satisfies Snell's equation. In this case
the ray undergoes total internal reflection. The angle
of reflection is equal to the angle of incidence. The
ray now goes back up through the layers. In the ideal
case, the upward path is exactly the mirror image of
downward path.
3) There will always be a Dedekind cut, some angle right
at the boundary between refraction and total internal
reflection. The ray exhibited a 90 degree angle of
refraction when it entered this layer.
Here's the key idea: We have been hornswoggled by paying
too much attention to this case. First of all, it only
happens on a subset of measure zero among the set of all
possible initial conditions. So we would be well justified
in ignoring it for this reason alone.
Secondly, the layers are an imaginary construction anyway.
I am free to shift they layer-boundary half a layer one
way or the other, whenever necessary, to make case (3)
go away.