In my garden, when I'm watering the plants I sometimes see a rainbow or two. How did two rainbows appear? Why can't I see three rainbows then, or how can I see three rainbows?
The two rainbows that are formed are the primary and secondary rainbows respectively, in order of their intensity or brightness, as you may call it. A primary rainbow is formed as a result of a three- step process: Refraction with dispersion, followed by total internal reflection and then refraction.
The secondary rainbow is formed due to a four- step process: Refraction with dispersion, followed by total internal reflection(twice in this case) and refraction again.
Check out the following:
It is found that in case of the primary rainbow, violet light emerges at an angle of 40 degrees relative to the incoming light and red light at an angle of 42 degrees; thus we see the primary rainbow with red at top and violet at bottom.
In case of the secondary rainbow, emergent angles are 50 degrees and 53 degrees with respect to the incoming light, for red and violet colors respectively. Thus, the violet color is at the top while red is at the bottom.
The intensity of the light is reduced at the second internal reflection, and hence the secondary rainbow is very faint in the sky.You may take a look at the following:
A third rainbow even if it is formed as a consequence of successive total internal reflections, will be too dim to be visible.
The third- and fourth-order rainbows actually appear in the sunward direction — when the light makes three reflections in a water droplet it exits going roughly the same way that it entered, rather than roughly reversing direction as it does for the single and double reflections in first- and second-order rainbows. There are some photos of third-order rainbows scattered around, and a faintly visible fourth-order bow in a highly-processed image. I'm not sure whether they'd be visible naked-eye or not if you didn't know just where to look.
I just made a complete answer for how rainbows are formed at What makes a rainbow happen?. I'm answering here because the current answers say that it involves Total Internal Reflection, which is not true. There are internal reflections, but they are not total. And each color of light does not exit the raindrop at just one angle, it covers a wide range of angles.
Every time the light in a raindrop encounters the surface, some reflects and some exits. The light that exits after just one reflection makes the primary rainbow, and the light that exits after two makes the secondary.
After one reflection, red light reflects back toward the sun at every angle from 0° (straight toward the sun) up to about 42.3°. But it is brightest, by far, in outer 0.3° or so. Violet light is similar, but the range is about 0° to 40.6°. It is the bright ranges that make the colored bands of the rainbow. There is still reflected light inside the colored bands, seen as a dimmer white region that extends to the horizon. This is pretty clear in the picture linked to in the question. Compare the sky color just inside of, to just outside of, the colored bands.
After two reflections, the light reflects at all angles from 180° (same direction as the original light) down to about 50° (red) to 53° (violet), which you see about 10° above the primary rainbow.
Rainbows are results of total internal reflection - that is, when light hits the surface of a refractive material (in this case water) at an angle that is less than the angle of refraction (which varies for different wavelengths of light), the light reflects off the surface like a mirror instead of entering the material (or, in this case, leaving the water droplet). The reason why we see the spectrum as a result of this is due to the relationship between the angle of refraction of light and its wavelength. When travelling through a refractive material, different wavelengths of light are refracted at different angles, which causes them to hit the surface of the droplet at slightly different locations, spreading them out upon reflection.
A double rainbow occurs when light enters a water droplet, is internally reflected off the surface, and is travelling at such an angle that, when it comes in contact with the surface of the droplet again, the angle of "impact" is shallow enough that it is internally reflected a second time. This results in a second rainbow alongside the first (you still see the first because some light escapes the droplet) that is more blurred and appears upside-down.
Strictly speaking, it should be possible to see a triple (or quadruple, quintuple, etc.) rainbow given water droplets with the correct geometry. We don't see this, though, because of the unlikelihood that raindrops will be the exact correct size to allow for more than two cases of total internal reflection, and because the water is not a perfect medium for light to travel through. Some light is scattered, which is why double rainbows usually appear faint and less defined than regular rainbows. A third or fourth rainbow would be even fainter, so I doubt it would be visible to the naked eye, even if triple internal reflection occurred.