Reflection and Refraction — Question 4

Question 4

Parallel layers preserve one transverse ray invariant

A ray crosses three plane parallel, stationary, transparent layers. Their indices are n1, n2, and n3 and ray angles to the common normal are theta1, theta2, and theta3. The middle layer has thickness d. Assume all refractions propagate.

1. Eliminate the middle index to relate the entry and final angles.

2. Find the transit time through the middle layer and explain why changing n2 can change time even when the final angle is unchanged.

Question 4 - Solution

Parallel layers preserve one transverse ray invariant

(a) Method and calculation

$$n_1\sin\theta_1=n_2\sin\theta_2=n_3\sin\theta_3\equiv K.$$$$\boxed{\theta_3=\sin^{-1}(K/n_3)}.$$

(b) Interpretation and further calculation

$$\ell_2=d/\cos\theta_2,\quad t_2=\ell_2/(c/n_2).$$$$\boxed{t_2=\frac{n_2d}{c\sqrt{1-K^2/n_2^2}}}.$$

Physical check

The end direction depends on the conserved transverse phase gradient, while the internal speed and path length depend on the middle index.