Rainbow

Example

Model a raindrop as a sphere of water with refractive index $n > 1$ in air. A ray enters the drop with angle of incidence $i$ and is refracted to the angle $r$ given by Snell's law $\sin i = n \sin r$. It is reflected once inside and then leaves the drop. Its total deviation is $$D(i) = \pi + 2i - 4r.$$ Since $\cos i = n \cos r \, \frac{dr}{di}$, the condition $D'(i) = 0$ reads $n \cos r = 2 \cos i$. Squaring and using $n^2 \cos^2 r = n^2 - \sin^2 i$ gives $$\cos^2 i = \frac{n^2 - 1}{3}.$$ For $n = 4/3$ this gives $i \approx 59.4^\circ$, $r \approx 40.2^\circ$ and the minimum deviation $D \approx 138^\circ$. Near a minimum of $D$ many rays leave in almost the same direction. So the light is concentrated at the angle $180^\circ - 138^\circ = 42^\circ$ from the point opposite the Sun. This concentration of rays is a caustic, and it is the primary rainbow.