Mathematics 309 - Rainbows

The Formation of a Rainbow

Sir Isaac Newton found that white light is composed of all wavelengths of visible light. White light is a mixture of all the colors of the spectrum, which are: Red, Orange, Yellow, Green, Blue, Indigo, and Violet. If we break up white light we can see the various components. A glass prism can be used to split white light into various wavelengths. This split occurs because each color in the white light has a different index of refraction. Thus, the different colors will respond differently to the glass. For example, blue light will refract more than the wavelengths corresponding to yellow, and yellow light will refract more than the wavelengths corresponding to red. This effect is called dispersion.

refractive bending of white light by a triangular glass prism

You can see a rainbow whenever you look opposite the sun at sunlit raindrops (or water drops). The raindrop acts like a mirror in that it reflects some of the refracted light back towards you, while other rays leave directly from the opposite side. These refracted rays are the ones that you see as a rainbow.
Thus, when the white light from the sun hits a raindrop the light is dispersed as it enters (like in the prism). The different colors undergo refraction and reflection due to the change of index of refraction between the water and the air.

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?

As discussed earlier, the light from the sun is composed of white light which is a mixture of all the colors of the spectrum. And like prisms, the white light is also refracted and dispersed in a water droplet. Thus, when white light hits the water droplet (assume spherical) the angles at which each color emerges is different. Again, this is due to the fact that each color has different indices of refraction, for example blue will refract more than red. Red light emerges at about 42° and blue light emgerges at about 40.5° with respect to the incoming ray of light. If we recall the fact that we only see the Descartes ray from a single drop we can conclude that each water droplet contributes to only one color of the rainbow.

Since each raindrop produces only one color, a rainbow must consist of many raindrops. In fact, the raindrops lower in the sky contributes to the blue light, and the drops higher in the sky contributes red light. This is because we see red light at about 42° to our eye and blue light at about 40.5° to our eye. Simple geometry then shows that the blue droplet must be lower. So to see a rainbow of colors red, orange, yellow, green, blue, and violet, the red light must be seen from a droplet higher than the droplet for the other colors, and the violet light must be seen from a droplet lower than the rest. Thus when we see a rainbow we see red on the top of the arc and violet on the bottom respectively.

On occasions we see a secondary rainbow which is about twice as wide as the primary rainbow, and has its colors reversed. Higher order rainbows could also exist but I will not be covering the content of this material.

Droplet Size

The colors we see of the rainbow depends on the size of the raindrops. If the drops are large, 1 millimeter or more in diameter, the rainbow will be bright with well defined colors where red, green, and violet are bright but having little blue. As the droplets get smaller, red weakens. In fine mist, all colors except violet may disappear. Even smaller droplets, less than 0.05mm in diameter (ie. fog droplets) produce rainbows of overlapping colors that appears white. This is called the white rainbow or fog bow.

So far we have assumed that all raindrops are spherical, and thus we drew each drop's cross section as a circle. But raindrops are a mixture of many sizes and shapes. A raindrop will most likely not have a radius of more than a couple millimeters. This is because they will usually break up due to the collisions with other raindrops. But on occasions, in a rainstorm, when there are very few drops, drops of a few millimeters in radius have been observed.
As a raindrop falls through the air, the droplet undergoes aerodynamic distortion which causes the shape to distort, flattening the drop. Small drops of radius less than 0.14mm remain spherical, but as the size of the drop increases, the flattening becomes more obvious.
Spherical drops produce symmetrical rainbows. When the sun is near the horizon, the rainbow is observed to be brighter at their sides, than at the top. This is due to the fact that there is a complex mixture of size and shape of the raindrops. The reflection and refraction of light from a flattened water droplet is not symmetrical. For a flattened drop, some of the rainbow ray is lost at the top and bottom of the drop. This is why we see rays from these flattened drops only as we view them horizontally. Thus, the rainbow produced by these large drops is bright at its base. The fainter rainbow near the top of the arc are produced only from small spherical drops.

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

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

Some rainbows have an additional faint arc or series of arcs just inside the primary bow which are usually more obvious near the top of the bow than at its base. These are called Supernumerary Rainbows and were explained by Thomas Young in the early 1800's as a result from interference due to the wave nature of light.
See Interference Theory.
The spacing of a supernumerary bow depends on the size of the droplets. Small drops yield a larger spacing than larger drops. If there are a wide range of drop sizes, the supernumerary bow from each drop will have different spacings which tend to overlap and cancel out the clear variation of intensity that results from droplets all of the same size.

a primary rainbow, a fainter secondary rainbow, and several
supernumerary bows (pastel-shaded arcs) inside the primary rainbow

White Rainbows

As water droplets get smaller, the colors of the rainbow begin to disappear. In fine mist, all colors except violet may disappear. Even finer droplets which are smaller than 0.05mm in diameter, for example fog droplets, produce the white rainbow or fogbow. Fogbows are formed similarly to rainbows in that light is reflected inside a water droplet and emerges to form a bow opposite the sun. Other than that, there are major differences. Rainbows are formed by raindrops which are large enough so that the suns rays passing through the drop follow a well defined 'geometrical optics' path. Much smaller drops which form fogbows extensively diffract light to produce a broad pale bow.
When light enters a fog droplet, the light is partially refracted and reflected at each surface of the droplet. Most of the light entering a droplet leaves directly from the opposite side without being reflected. This process is similar to a rain droplet, but for a fog droplet the light is spread by diffraction and no precise ray paths can be traced within or near the droplet.
The color of a fogbow is pale because the fogbows formed in each color are overlapped significantly.
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Along the main light path, inside the main arc of the fogbow, widely spaced supernumerary bows are produced by constructive and destructive interference of overlapping wave crests.

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

In 1803, Thomas Young claimed that the wave theory of light offered a much superior explanation of the rainbow. Interference is the wave property that we use to explain supernumerary bows. If two light waves having either coinciding wave crests or troughs, the effect is constructive interference which produces a brighter color of light. If two light waves have a coinciding wave crest and trough, the light becomes darker than either wave's average brightness. This effect is called destructive interference.Thus, when one portion of a light wave passes through another light wave there may be either constructive or destructive intereference.

Supernumerary bows are caused by the interference of two different portions of the same light wave.
The light waves from the sun is refracted into the droplet, and some of it is reflected from the opposite side. Then, some of this reflected light is refracted out of the drop to form a rainbow. This is similar as before when we used light rays. But, as the wave front traverses through the drop, some waves folds over on itself. When the two portions of the wave are superimposed, interference occurs which produces a pattern of bright and dark bands. This interference pattern is drawn below.

The bright and dark bands which radiate from the droplet creates the bright and dark bands of light that form the rainbow. The region at the angle of minimum deviation form the primary rainbow. This region is seperated from the first supernumerary bow by a dark band and seperated from the second supernumerary bow by two bands of darkness. Thus, the supernumerary bows are a part of the primary bow.

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

Greenler, Robert, Rainbows, Halos, and Glories Peanut Butter Publishing 1999
Laven, Philip, Optics of a water drop
Lee, Raymond L., and Fraser, Alistair B., The Rainbow Bridge Penn State Press 2001
Lynch, David K., and Livingston, William, Color and Light in Nature Cambridge University Press 1995
Lynds, Beverly T., About Rainbows
Minnaert, Marcel G. J., Light and Color in the Outdoors Springer-Verlag 1974
Nave, R., Rainbow Concepts
Walker, Jearl, Light from the Sky Scientific American
Atmospheric Optics
Rainbow Reading

Craig Ryomoto
April 28 2003
Mathematics 309

all diagrams are drawn in Postscript
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