Rainbows Are Optically Fascinating - Part 1
or, how to astound your friends with your amazing knowledge when you're all looking for the pot of gold
Next time a rainbow is visible, look at it very carefully. I took the above photo recently. It shows both the primary rainbow (the brighter one) and the secondary rainbow.
Our first question is: why are there two rainbows? But there are other questions, like: Why is the secondary one (furthest out) so much weaker than the primary one? Why are the colors reversed in the secondary rainbow from the primary one? Why is there a dark region between the two rainbows? Is the light in the rainbows polarized? If so, in what direction and how does it become polarized? We will try to answer the first set of questions in this Part 1. Part 2 has to do with the polarization properties...stay tuned.
Shown below are diagrams of how the primary and secondary rainbows are produced by refractions and reflections inside rain drops.
For the primary rainbow, the sunlight enters a water drop where it is refracted due to the varying indices of refraction with wavelength (see graph of the refractive index of water versus wavelength), with the blue refracted at a larger angle than the red. Inside the drop, the refracted light is reflected at the back of the drop. The reflected blue crosses the red as it heads out through the front of the drop, where it is refracted again. That crossing within the drop reverses the red and blue, so that the red emerges at a larger angle with respect to the sun than the blue.
For the secondary rainbow, the refracted light inside the drop is reflected twice before emerging from the front of the drop. Inside the drop, the red crosses the blue twice, so that the red emerges at a smaller angle with respect to the sun than the blue. That explains the reversed colors in the primary and secondary rainbows.
Note that the angle between incoming sunlight and the red in the primary rainbow is 42˚. For the secondary rainbow that angle is 51˚. Circles representing those angles are shown approximately on top of the rainbow photo below. Also shown at the "X" is the point directly opposite the position of the sun (therefore named the anti-solar point), which is also the center point of the circles.
So now to the question about why the secondary rainbow is so much weaker than the primary. Look again at the diagrams of each. At the reflections inside the rain drop, not all of the light is reflected. Some of it is transmitted outside the drop. That light is indicated by the lighter colored arrows. The secondary rainbow light is reflected twice, so that more light had the opportunity to be transmitted outside the drop, reducing the amount of light in the secondary rainbow compared to the primary.
What about that much darker region between the primary and secondary rainbows? How does that happen? Our eyes are only capable of detecting wavelengths between about 700 nm (red) and 400 nm (blue-violet). Note the red sides of the primary and secondary rainbows are facing each other with the dark region in between. If our eyes could detect the near-infrared (beyond 700 nm), we would see those near-infrared colors just beyond the red extending into the dark region. However, water strongly absorbs light at near-infrared and infrared wavelengths longer than about 900 nm, so those wavelengths don't make it out of the raindrops. That's the main reason the dark region between the two rainbows exists.