Example 12.1: The corner-cube reflector

Question: Two mirrors are placed at right-angles to one another. Show that a light-ray incident from any direction in the plane perpendicular to both mirrors is reflected through $180^\circ$.
 
Answer: Consider the diagram. We are effectively being asked to prove that $\alpha=i_1$, for any value of $i_1$. Now, from trigonometry,

$$i_2 = 90^\circ - r_1.$$

But, from the law of reflection, $r_1=i_1$ and $i_2=r_2$, so

$$r_2=90^\circ - i_1.$$

Trigonometry also yields

$$\alpha = 90^\circ - r_2.$$

It follows from the previous two equations that

$$\alpha = 90^\circ-(90^\circ - i_1) = i_1.$$

Hence, $\alpha=i_1$, for all values of $i_1$.

It can easily be appreciated that a combination of three mutually perpendicular mirrors would reflect a light-ray incident from any direction through $180^\circ$. Such a combination of mirrors is called a corner-cube reflector. Astronauts on the Apollo 11 mission (1969) left a panel of corner-cube reflectors on the surface of the Moon. These reflectors have been used ever since to measure the Earth-Moon distance via laser range finding (basically, a laser beam is fired from the Earth, reflects off the corner-cube reflectors on the Moon, and then returns to the Earth. The time of travel of the beam can easily be converted into the Earth-Moon distance). The Earth-Moon distance can be measured to within an accuracy of $3\,{\rm cm}$ using this method.