Chapter 43: Problem 50

Question: A light ray of given wavelength, initially in air, strikes a 90^o prism at P (see Fig 42, Chap 43) and is refracted there and at Q to such an extent that it just grazes the right hand prism surface at Q.

1.

Determine the index of refraction of this prism for this wavelength in terms of the angle of incidence $\theta_1$, that gives rise to this situation.

2.

Give a numerical upper bound for the index of refraction of the prism. Show, by ray diagrams, what happens if the angle of incidence at P is slightly less or slightly greater than $\theta$.

Solution: At Q we have that $n\sin\phi~=~1$, where $\phi$ is measured wrt the normal at point Q. At P we have that $\sin\theta~=~n\sin\theta_2~=~n\cos\phi~=~\sqrt{n^2-1}$, from which we see that $n~=~\sqrt{1+\sin^2\theta}$. Since $\sin\theta\le~1$, we have that $n\le~\sqrt{2}$. Now either side of the $\theta$, we have total internal reflection for less than, and transmission for greater than.