Now that we understand rainbows and how they are formed, we can go deeper into the science behind them, and explain the workings of something much more mysterious; the double rainbow. For ages it has been seen as a sign of good luck and something even more magical than a single rainbow. Mainly because it is even rarer than a single rainbow and much harder to spot.

To understand the process of the double rainbow we again have to look at the optics behind the problem, and see how a single ray of light passes through a water droplet. If you haven’t done so yet, please first read “How do rainbows work?” before continuing here.
So to come back to the problem, it is best to start with a few facts, namely that the second rainbow always lies a certain height above the first one and that its colors are inverted.

Having learned from the last article that higher droplets means that the light is reaching you from a larger 

Sunlight reflecting trough a waterdroplet.

angle, we quickly see that there must be a second process that makes light exit the droplets at an angle that’s larger than 42 degrees. However, we have learned that this is the maximal angle of the second reflection inside a water droplet, therefore this can not be the same process. This means that we need to find some other process which does allow larger exit angles.

One of these processes is the third reflection, where the light doesn’t bounce off once, but twice before leaving the droplet. Causing it to come out at a much steeper angle of 51 degrees, where just like with the first rainbow the angle between blue and violet light is 2 degrees. All of this is illustrated in the 2nd picture. This process is different from the second reflection, and much weaker because we lose a bit of light that leaves the drop at every reflection. It is important to note that these two processes happen in every raindrop instantaneously, however because the angle is so high, the drop needs to be very high for the light of the third reflection to reach your eye.

Once more extending this process to multiple drops, we can imagine how this all works as a whole. The higher drops get, the larger the angle of reflected light has to be to make it possible for it to reach your eye. For the first rainbow this goes from 40 to 42 degrees and for the second rainbow from 51 to 53 degrees. This means that there is a gap of 9 degrees between the two where no sunlight is reflected, we call this gap Alexander’s dark band. We can see all this back in the third image, which is an extension of the one in the last article.

Schematic diagram of a rainbow.

By this time you may think, if double rainbows exist, can higher order rainbows exist too? The answer is yes, although you won’t ever see anything above a triple rainbow, which by itself is a very rare sight. However, higher order rainbows are relatively easy to create in laboratory conditions. The current record has been set in 1998, by 

High order rainbows.

a group of scientists who managed to get a 200th order rainbow in a carefully calibrated lab setting. They achieved this by shining a green laser onto a single water droplet.
Some of the results can be seen in the last picture, obtained from the paper: Josa B. et al. (1998), “Observation of high-order rainbows formed by a pendant drop”.

So in conclusion, the second rainbow of a double rainbow pair work exactly the same way as the first, except that it comes to be by the third reflection of a beam of light. The two rainbows are separated by Alexanders dark band and the second rainbow is inverted, meaning that it starts with violet and ends with red. With higher orders the same process will repeat itself, all separated by a dark band and with colors inverting back and forth.

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