Rainbows!

The preferred direction of the scattered light depends on how many internal reflections the light ray undergoes in the raindrop. Let’s first consider the case of one internal reflection, as shown below. The light ray refracts into the raindrop, then reflects inside, and then refracts back out into the air. Let $β$ and $φ$ be defined as in the figure. Then the angles $2β$ and $2β - φ$ are shown.

Snell’s law at the refraction points gives $\sin (2β - φ) = \frac{4}{3} \sin β$ .... (1) we can set $\frac{dφ}{dβ} = 0$ . The result is $\Rightarrow β_{max} ≈ 40.2^◦$