Reflection and Refraction — Question 3

Question 3

A parallel slab shifts a ray without changing its final direction

A plate of thickness d and refractive index n is surrounded by air. A ray enters at angle i from the normal, refracts to r, and exits the parallel rear face. Assume n greater than one.

1. Derive the exit angle and the perpendicular separation between the outgoing ray and the undeviated continuation of the incoming ray.

2. Find the small-incidence approximation to the displacement, with i in radians.

Question 3 - Solution

A parallel slab shifts a ray without changing its final direction

(a) Method and calculation

$$\sin i=n\sin r,\quad n\sin r=\sin i_{\mathrm{out}}\Rightarrow\boxed{i_{\mathrm{out}}=i}.$$$$\Delta x=d(\tan i-\tan r),\quad D=\Delta x\cos i=\boxed{d\sin(i-r)/\cos r}.$$

(b) Interpretation and further calculation

$$r\simeq i/n,\quad\sin(i-r)\simeq i(1-1/n),\quad\cos r\simeq1.$$$$\boxed{D\simeq di(1-1/n)}.$$

Physical check

The lateral separation D is measured perpendicular to the parallel external rays; it is not the displacement measured along the rear surface.