Dispersion on a Sphere and Rainbow: UNIZOR.COM - Physics4Teens - Waves - Phenomena of Light
Channel: Zor Shekhtman on UNIZOR Education
Date: 2022-06-27

UNIZOR.COM - Physics through a Looking Glass of Mathematician
Notes to a video lecture on http://www.unizor.com

Dispersion through a Sphere

Let's analyze how a rainbow is developed after a rain or above waterfalls.
We will model this process with a set of parallel rays of white light (from the Sun) going through air and falling onto a transparent sphere (a water droplet).

Generally speaking, when a ray of light falls on a border between two substances (like air and water in case of a rainbow), part of light goes through the border, refracting on its way, and part is reflected by a border surface.

Let's first examine the simplest case of a white light ray from the Sun going through a spherical water droplet without reflection.

Notice that the perpendicular to a surface of a sphere at its any point is a radius to this point.

Let α be the incident angle of the white light ray coming from the air onto a transparent sphere (a water droplet hanging in the air),
β will be a refraction angle (different for different colors because of the difference in the speed of light of different wavelengths),
β' will be an incident angle, as the light goes from inside of a sphere out into the air and
γ will be a refraction angle of the light that came out from the sphere into the air.

From the Law of Refraction:
sin(α)·nair = sin(β)·nwater
and
sin(β')·nwater = sin(γ)·nair

Incidentally, ∠β = ∠β'.
Therefore,
sin(α)·nair = sin(γ)·nair
from which follows that
∠α = ∠γ.

So, the white light ray is split by the refraction on two surfaces of the sphere, on entry and on exit, and we definitely have a dispersion of white light, as it goes from the air through a sphere of the water droplet and into the air again.

With a fixed position of the Sun in the sky and an observer on the ground, for each given droplet of water, there is only a narrow set of sun rays that go through a droplet, disperse into a conical spectrum of colors and reaches the eyes of an observer.

Moreover, if a particular component of a particular white sun ray, like red, is directed by refractions into an eye of an observer, the other component of the same white sun ray, like green, will miss this eye; it's a green component of another (neighboring) white sun ray, that is refracted by another water droplet, will reach the same eye.
That green light will fall into the eye at a different angle than the red described above, and that's why we see red light in one point of a sky and green in another, all colors in the same sequence, in order of their refractive index in water.

The refraction without internal reflection described above will produce colorful rainbow, but an observer at a position presented on the picture above might not be able to see it because the Sun will brightly shine straight into his eyes.
More practical rainbow observation can be obtained if the Sun is behind or on the side from an observer and, even better, when an observer is shielded from the direct sun rays by trees or buildings or mountains.

Consider what happens if the sun rays after penetrating the water droplet reflect from the inside of a droplet and then refract on the way out going into the eyes of an observer.

As mentioned before, not all the light goes along the path on the picture above. On each sphere surface, on entry into or exit from a droplet of water, the light is partially reflected. Moreover, light internally partially reflected inside a sphere can be partially reflected multiple times before it exits the sphere, each time scattering rays in different directions.

In this scenario the Sun does not directly flashes into the eyes of an observer, and the rays of different colors can be seen.
All other considerations about how we see different colors explained above are applicable to this case as well.

In theory, the light can be internally reflected more than once, and, correspondingly, the rainbow will be visible in more than one place on the sky.
However, we always see a circularly formed rainbow, which requires an additional explanation.