Mathematics 309
Part II - Introduction to light
3. Rainbows
A ray of light penetrating a drop of water has a number of choices to make in proceeding. It can decide to pass straight through, or it can reflect around inside a few times before exiting. In fact, light entering a drop chooses all of these options. Some part passes through, and some bounces around inside. One of the more interesting possibilities is that where it reflects exactly once before coming out again.
The way in which the rays arrange themselves is responsible for rainbows.
The geometry of a single ray
Suppose given in space a sphere filled with water, say of radius R. All rays of light passing within R of the centre will be deflected by refraction when they enter the sphere.
If y is the distance of the ray's path from the centre, the angle of incidence i is arcsin(y/R).
If r is the refraction angle, then the ray rotates i-r at the point of impact (see the discussion of refractions). At the back of the sphere is is reflected, and effectively rotates 180-2r. When it exits the sphere it again rotates i-r. The total angle it rotates through is 2(i-r)+(180-2r). The deflection angle between the entering and exiting rays is therefore 180-2(i-r)-(180-2r) = 4r-2i.
As i ranges from 0 to 90 the deflection angle increases, reaches a maximum, and then decreases.
Click on image to see PostScript source.
Here is the graph of deflection versus incidence:
Click on image to see PostScript source.
The maximum is about 43o. It can be computed exactly by calculus. Let h = y/R. Then
D = - 2i + 4r
= - 2 arcsin(h) + 4 arcsin(h/n)
dD / dh = -2 (1 - h2)-1/2 + (4/n) (1 - h2/n2)-1/2
and this will vanish when
h = (4-n2)1/2/31/2
h = 0.8624
i = 59.58o
r = 40.42o
D = 42.52o
The rainbow
Suppose that we are considering raindrops falling in the sky opposite to the sunset, which we assume to be pure red. Suppose that instead of asking how horizontal rays from the Sun scatter from a single raindrop, we ask which rays will be visible to an observer on the ground. He will see rays in the direction from which they come, which means that he will see a drop at an elevation of D only if it produces a ray with deflection D.
It turns out that most of the rays have a deflection near the maximum deflection, which means that he really sees only those drops at an elevation near that angle. In effect, he sees a (pure red) rainbow (because the raindrops are spherically symmetric).
Why are most deflections near the maximum? We can visualize this by imagining N rays from the sun, and plot which elevations they belong to, for various N.
N = 10
Click on image to see PostScript source.
N = 100 (zoom)
N = 250 (zoom)
As N gets larger and larger, the density of rays at angle about 42.5o goes to infinity. It's not so easy to see in the picture above, but here's a bar graph of the distribution of deflections for 1000 rays.
Real rainbows are coloured. That's because different colours have different indices of refraction, and correspond to different maximum deflections.