Physics of rainbows

A most charming example of chromatic dispersion is a rainbow. When white sunlight is intercepted by a drop of water in the atmosphere, some of the light refracts into the drop, reflects from the drop's inner surface, and then refracts out of the drop. As with the prism, the first refraction separates the sunlight into its component colors, and the second refraction increases the separation. The result is the rainbow. This applet shows the physics of the rainbow.

Instructions on how to use the animation below:
The black circle represents a drop of water in the atmosphere. Initially, red light is coming from the left; you can click inside the colored blocks to change the color of the incoming light.
The incoming ray is unpolarized, which can be represented as a mixture of two polarized waves whose planes of polarization are perpendicular to each other. So the notation " 50%| 50%+" means that half is polarized in the up-down direction and half is perpendicular to the screen.

Many things can happen to the light.

• Part of the incoming ray is reflected back to the atmosphere (indicated by ray number 1). The intensity of each polarized component is shown along the ray path.
• Part of the light refracts into the drop, then refracts back to the atmosphere (ray number 2).
• Some reflects from the drop's inner surface, and refracts back to the atmosphere (ray number 3). This gives rise to the ordinary rainbow.
• Some reflects twice inside the drop, then refracts back to the atmosphere.(ray number 4) This gives rise to the secondary rainbow that is sometimes seen.

You can drag the incoming ray, move it up and down, and watch how the relative intensities change. R is the radius of the water drop; b is the vertical distance of the incoming ray from the center of the circle.

The intensity of the light coming from rays 3 and 4 is plotted versus viewing angle. Click ^ or v to change the scale. For ray number 3 there is a maximum scattering angle, and for ray number 4 there is a minimum -- this is why there are strong peaks in the scattered intensity. The rainbow is actually a disk of scattered light, but it is brightest at the edge; the disk for different wavelengths is a different size, and that is why we see the color effects there.

You can click inside the white box, and see what will happen to white light.

When light refracts, it follows the law of refraction

where n is the index of refraction. The number in the left-bottom corner i is the angle of incidence, r is the angle of refraction.

Most of the light is refracted out by ray 2. When your eyes intercept the separated colors from raindrops, the red you see comes from drops angled slightly higher in the sky than does the blue. You see a circular arc of color, with red on the outside and blue on the inside. Click inside the colored box to show this effect. You can drop one of the rain drops, Try it!