Order of colors after reflection in a raindrop

Question

I studied two figures of sunlight reflecting in a raindrop:

In the first image, red is shown above violet, but in the second image, red is below violet. Are both cases possible?

Answer 1

Clearly both diagrams cannot be correct. Look closely at the refraction of the rays as they exit. In the left image, it appears as though the rays refract towards the normal while the light is transitioning from a region of high refractive index to a region of low refractive index:

As a general rule, refractive index decreases as wavelength increases (unless you are at a resonance, where it gets a 'bump'). So you expect red to be refracted less than violet -- but both of them have to be refracted away from the normal. A better diagram for showing what an observer sees (recalling that the sun is "very far away", the incoming rays are "parallel"):

Of course I have exaggerated the angles to make clear what is going on. In reality the difference in angle of deflection is quite small - the refractive index between 400 nm and 800 nm drops from about 1.57 to 1.55, and the angles are only different by about 1.7°. I am also ignoring the fact that since the sun is not a point source, you actually get a range of angles for both violet and red light: this smears out the colors of the rainbow a little bit.

Answer 2

The violet ray deviates through a greater angle in both diagrams.

The diagram on the left shows what the observer sees - different colours arriving at the eye from different angles. This is a misleading diagram because the red and violet reaching the eye do not come from the same incident ray of sunlight - they come from different rays.

The diagram on the right correctly shows how the colours in a single ray are dispersed.

Answer 3

Both diagrams imply that the raindrops work exactly like a prism does when it projects a spectrum on the wall, and that is wrong. The error is that sunlight hits the drop across an entire hemisphere, not just one point, and the angles are different at each point.

1. When light strikes a raindrop at an angle A relative to the surface normal, refraction makes it enter the drop at angle B=asin(sin(A)/n). Here, n is the index of refraction, about 1.331 for red light and 1.344 for violet. Note that B<=A. (They are equal at A=B=0.)
2. The difference in n makes the colors in any single ray separate, but they can (and do) recombine with the colors from other rays. Any comparison to a prism is inaccurate.
3. When it encounters the surface of the drop again, the internal ray makes the same angle B with the surface normal. Some exits, making angle A with the surface normal again, and some reflects. (The first diagram above makes it look like it exits at an angle C<B, which is wrong.) This process, in theory at least, continues endlessly.
4. The second diagram shows the total deviation of the original ray after one such reflection. Summing the deviations at each encounter with the surface, it is (A-B)+(180-2B)+(A-B)=180+2A-4B. The supplement of this, 4B-2A, is the angle between the original ray and the exiting ray.
5. If you plot this for 0<=A<90, using the indices I gave, you will see that the red light exits at all angles between 0 and 42.35, and the violet at all angles between 0 and 40.58. (Apparently, whoever made the second diagram used slightly different indices.)
6. The colors are not limited to a single band, they fill in an entire circle. But each circle is brightest in its outer 0.5 degrees. Since these bright bands are in different places for different colors, we see a rainbow.
7. Inside the colored bands, the rainbow still exists as a dim white disk. What is usually noticed, however, is that the sky outside the rainbow appears darker.