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Well... I'm still bothered by it. If you go back to my original
message, I did consider what you're mentioning here. Mathematically,
if the index truly varies continuously, I don't see that total
internal reflection (TIR) can occur because there are no layers. The
angle smoothly and asymptotically bends toward 90 degrees.
So now I have two explanations in mind:
1. The index variation in the real world isn't mathematically smooth.
There will be "layers" at some sufficiently fine level and hence TIR
will occur.
2. The rays in the real world aren't mathematical lines of zero cross
section. The tops of the wavefronts will "refract" differently than
the bottoms, with the effect of bending the light waves upward.
Is one explanation better than the other? Are the two explanations
related to each other?
Which would you use to explain highway mirages in an intro course?