Lenses work because of *Snell's law*, which expresses how much rays of light bend when entering or exiting a transparent material. Recall that the speed of light in a medium is less than the speed in an vacuum. For a given material, let its *refractive index* be defined as

in which is the speed of light in the medium. For example, means that light takes twice as long to traverse the medium than through a vacuum. For some common examples, for air, for water, and for crown glass.

Figure 4.9 shows what happens to incoming light waves and rays. Suppose in this example that the light is traveling from air into glass, so that . Let represent the incoming angle with respect to the surface normal, and let represent the resulting angle as it passes through the material. Snell's law relates the four quantities as

Typically, and are given, so that (4.3) is solved for to obtain

If , then is closer to perpendicular than . If , then is further from perpendicular. The case of is also interesting in that light may not penetrate the surface if the incoming angle is too large. The range of is 0 to , which implies that (4.4) provides a solution for only if

(4.5) |

If the condition above does not hold, then the light rays reflect from the surface. This situation occurs while under water and looking up at the surface. Rather than being able to see the world above, a swimmer might instead see a reflection, depending on the viewing angle.

Steven M LaValle 2016-12-31