A rainbow is just a distorted image of the sun. It results from raindrops which rearrange the sunlight via reflection and refraction.





The Formation of a Rainbow



refractive bending of white light by a triangular glass prism

The formation of rainbows by raindrops was first clearly discussed by Rene Descartes.

Let's assume that the rays from the sun are parallel and that all raindrops are spherical. Of the many paths taken by the rays through a (spherical) water droplet, several rays become concentrated near a minimum deviation angle. These rays enhance the intensity at that particular angle to produce the primary rainbow which we actually see. The ray which is produced at the minimum deviation angle (the lowest ray) is called the Descartes ray.







Why Do We See Different Colors?

Droplet Size

Diameter of water drop Features of the Rainbow ~ 1-2mm The violet is very bright and the green is vivid. The rainbow contains pure red, but barely any blue. There are many spurious bows, violet-pink alternating with green without interruption into the primary bow. ~ 0.5mm The red is significantly weaker. There are fewer supernumerary bows, violet-pink and green are again alternating. ~ 0.2-0.3mm There is no more red. The bow is broad and well developed for the rest of the colors. The supernumerary bows become more yellow. If the diameter of the drops is around 0.2mm, a gap occurs between the supernumerary bows. If the diameter is less than 0.2mm, a gap is formed between the primary bow and the first supernumerary bow. ~ 0.08-0.1mm The bow is broader and paler, the only vivid color is violet. The first supernumerary bow is well seperated from the primary bow and visibly shows tints of white. ~ 0.06 A distinct white stripe is contained in the primary bow. < 0.05mm White Rainbows: Fogbows, Mistbows, Cloudbows











Supernumerary Rainbows



a primary rainbow, a fainter secondary rainbow, and several

supernumerary bows (pastel-shaded arcs) inside the primary rainbow

White Rainbows

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Direction of light in a fog droplet:

Light is refracted, reflected, and refracted again inside the droplet.

Here diffraction dominates, and there are no well defined ray paths.

Interference Theory



We can now use interference to explain the pale or pastel colors of white rainbows. First, recall that any bow is determined by refraction, with violet deviated the most and red deviated the least. Thus in the primary rainbow, different colors occupy different positions, with red on the outside. But the primary rainbow is just the first interference maximum, and for each color the width of that maximum depends on the size of the raindrops. For very large drops, the width of each color band will be narrow, therefore the various colors do not overlap too much, which results in fairly pure rainbow colors. On the other hand, for small drops each band of color can be so broad that all the colors overlap. This combining of overlapping colors yields a pale or white bow.











References

Craig Ryomoto

April 28 2003

Mathematics 309



all diagrams are drawn in Postscript

click on diagrams to view PS source



click on photographs for link to related website

