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A point light source S sits at the bottom of a stack of parallel transparent layers. The bottommost medium has refractive index mu0. Every layer above it has refractive index mu(n) = mu0 + mu0/(4n - 18), where n is the layer number counted upward from n=1. A ray leaves S at an angle of incidence i = 30 deg to the normal. At the upper surface of which layer does total internal reflection first occur?

1. n = 3
2. n = 2
3. n = 1
4. n = 4

Correct answer: n = 3

Solution

By Snell's law, mu0*sin(30 deg) = mu0/2 is the invariant. Evaluating mu(n) for n=1 through 5, mu(4) = mu0 + mu0/(16-18) = mu0 - mu0/2 = mu0/2, so the ray hitting the upper surface of layer n=3 encounters layer n=4 with mu(4)=mu0/2, satisfying the TIR condition exactly.

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