Mathematics 309 - Rainbow, Alexander's Dark Band & Supernumerary Bows

A rainbow is just a circular arc of several colors arranged in spectral order (red, orange, yellow, green, blue, and violet). It can be seen at the close of a shower and also in the spray of waterfalls.

Primary and Secondary Rainbows

Primary rainbows are formed when sunlight enters a raindrop, and it gets refracted. Then some of the rays are reflected at the back of the raindrop. At the end, the ray is refracted again and exits.

Similarly, secondary rainbows are formed if the sunlight refracted as it enters the raindrop, and then reflected twice at the back of the raindrop. Because of the two internal reflections, the rays exit the drop at an angle of about 50 degrees, rather than the 42 degrees for the primary rainbow. Blue light emerges at a larger angle of 53 degrees. As a result, a secondary rainbow has its colors reversed compared to the primary rainbow.

Light Sky under Primary Rainbow

There are many parallel rays entering a droplet; the primary rainbow ray is the highest angle that strikes our eyes and therefore many of the rays emerge at angles smaller than the rainbow ray {40-42 degrees}. For those that are less than 40-42 degrees, (approx. 40 for violet, and 42 for red), they will reach our eyes from lower drops. These rays reflected from lower drops will lighten the sky, and form the 'light sky'. This light is in white color because of a mix of all rainbow colors.

There are only a small portion of rays that contribute to form the primary rainbow; there are lots of other reflected rays which have a deflection angle not equal to 40-42 degrees.

The light sky is the region below the primary rainbow. It is formed by the lower drops which have an angle less than the maximum angle for primary rainbow {40-42 degrees}.

The ray internally reflects the upper raindrop, but we are unable to see the ray because the reflected ray (<40 degrees) does not reach the observer. The same ray internally reflects the lower raindrop, we can see the reflected ray this time, and it is below the primary rainbow.

Alexander's Dark Band

Alexander's Dark Band is the dark area between the primary and secondary rainbows. In addition to the fact of forming a light sky inside the primary rainbow, the secondary rainbow ray is the smallest angle that reaches our eyes and therefore many of the rays emerge at angles greater than this rainbow ray {51 degrees}. Combining these two facts together, it means that for droplets that are between the greatest angle of the primary rainbow and the smallest angle of the secondary angle, no internally reflected rays coming out of droplets can reach our eyes. Therefore, raindrops between the primary rainbow and the secondary rainbow cannot send us any light that can form either primary or secondary rainbow. This region in between looks comparatively dark. This dark region between the two rainbows is called Alexander's Dark Band, in honor of Alexander of Aphrodisias, who noticed and discussed it for over eighteen hundred years ago.

Supernumerary Bows

Supernumerary bows are extra bands (mainly pale pink or green in color) near the primary rainbow and very rarely near the secondary rainbow. They are formed when the reflected light exits the drop and folds over in itself, causing the colors to interfere with one another. This interference then causes the dark and light bands.

Supernumerary rainbow (interference rainbow) is often seen inside the primary rainbow. It is extremely rare to be seen outside the secondary rainbow.

Young's Interference - Ray Tracing with Interference

The cause of supernumerary rainbow was explained by Thomas Young in 1804. His wave theory of light concluded that two waves from different wave source will produce alternating brightness and darkness. Young also indicated that light in fact is a transverse wave and that when two rays emerge at the same scattering angle within the raindrop (the two reflected rays coming out of the raindrop are parallel); they may interfere with each other. A constructive interference produces brightness; a destructive interference produces darkness.

The red and green rays are parallel both before hitting the drop and after scattering inside the raindrop. The path traveled by the green ray is longer than the red one.

How to Determine if the Light Interferes Constructively or Destructively?

Acoording to Descartes and Newton, the intensities of the two rays would have been added together; however, Young pointed out that the interference of these two rays which have the same scattering angle within the drop will form the supernumerary rainbow.

To find out if they form a constructive or destructive interference, we first need to look at the distance they traveled within the raindrop. Clearly, the distances that are traveled by the two rays are different; the green ray is longer than the red ray. Therefore, there is more light left in the red ray than the green ray. By taking the difference in phase of the two rays, we can get the scattered intensity. It would be a constructive interference if the phase difference is n(360), (where n = 0, 1, 2, 3¡K), ie 0, 360, 720. On the other hand, it would be a destructive interference if the phase difference is an odd multiple of 180 degrees, ie 180, 540, 900.

Under What Circumstances can We See Supernumerary Bows?

Since the distance that the ray travels within the drop depends on the size of the raindrop, if the sizes of raindrops are not the same, the intensity is so low that we cannot see the supernumerary rainbow. The raindrops need to have the same size in order for us to view the interference rainbow. The color and the spacing of supernumerary bow are also related to the drop size. A large drop would have a larger space between bows. Therefore, if all the raindrops are relatively small, it would be easier for us to see the bows.

REFERENCES:

http://users.bart.nl/~boudewyn/Regenboog2.htm

http://hjem.get2net.dk/Hemmingsen/Rainbow/youngs.htm

http://hyperphysics.phy-astr.gsu.edu/hbase/atmos/supnum.html#c1

http://cse.ssl.berkeley.edu/rainbows/layouts/U2/U2.3.html