Water droplets and light form the basis of all rainbows, which are circular arcs of color with a common center. Because only water and light are required for rainbows, one will see them in rain, spray, or even fog.
A raindrop acts like a prism and separates sunlight into its individual color components through refraction, as light will do when it passes from one medium to another. When the white light of the sun strikes the surface of the raindrop, the light waves are bent to varying degrees depending on their wavelength. These wavelengths are reflected on the far surface of the water drop and will bend again as they exit. If the light reflects off the droplet only once, a single rainbow occurs. If the rays bounce inside and reflect twice, two rainbows will appear: a primary and a secondary. The second one will appear fainter because there is less light energy present. It will also occur at a higher angle.
Not all the light that enters the raindrop will form a rainbow. Some of the light, which hits the droplet directly at its center, will simply pass through the other side. The rays that strike the extreme lower portions of the drop will product the secondary bow, and those that enter at the top will produce the primary bow.
The formation of the arc was first discussed by Rene Descartes in 1637. He calculated the deviation for a ray of red light to be about 180 - 42, or 138°. Although light rays may exit the drop in more than one direction, a concentration of rays emerge near the minimum deviation from the direction of the incoming rays. Therefore the viewer sees the highest intensity looking at the rays that have minimum deviation, which form a cone with the vertex in the observer's eye and with the axis passing through the Sun.
The color sequence of the rainbow is also due to refraction. It was Sir Isaac Newton, however, 30 years after Descartes, who discovered that white light was made up of different wavelengths. Red light with the longest wavelength, bends the least, while violet, being the shortest wavelength, bends the most. The vertical angle above the horizon will be a little less than 41° for the violet (about 40°) and a little more for the red (about 42°). The secondary rainbow has an angular radius of about 50° and its color sequence is reversed from the primary. It is universally accepted that there are seven rainbow colors, which appear in the order: red, orange, yellow, green, blue, indigo, and violet. However, the rainbow is a whole continuum of colors from red to violet and even beyond the colors that the eye can see.
Supernumerary rainbows, faintly colored rings just inside of the primary bow, occur due to interference effects on the light rays emerging from the water droplet after one internal reflection.
No two people will see the same rainbow. If one imagines herself or himself standing at the center of a cone cut in half lengthwise and laid on the ground flatside down, the raindrops that bend and reflect the sunlight that reach the person's eye as a rainbow are located on the surface of the cone. A viewer standing next to the first sees a rainbow generated by a different set of raindrops along the surface of a different imaging cone.
Using the concept of an imaginary cone again, a viewer could predict where a rainbow will appear by standing with his back to the sun and holding the cone to his eye so that the extension of the axis of the cone intersects the sun. The rainbow will appear along the surface of the cone as the circular arc of the rainbow is always in the direction opposite to that of the sun.
A rainbow lasts only about a half-hour because the conditions that create it rarely stay steady much longer than this. In many locations, spring is the prime rainbow-viewing month. According to David Ludlum, a weather historian, rainfall becomes more localized in the spring and brief showers over limited areas are a regular feature of atmospheric behavior. This change is a result of the higher springtime sun warming the ground more effectively than it did throughout the previous winter months. This process produces local convection. These brief, irregular periods of precipitation followed by sunshine are ideal rainbow conditions. Also, the sun is low enough for much of the day to allow a rainbow to appear above the horizon—the lower the sun, the higher the top of a rainbow.
The "purity" or brightness of the colors of the rainbow depends on the size of the raindrops. Large drops or those with diameters of a few millimeters, create bright rainbows with well defined colors; small droplets with diameters of about 0.01 mm produce rainbows of overlapping colors that appear nearly white.
For refraction to occur, the light must intersect the raindrops at an angle. Therefore no rainbows are seen at noon when the sun is directly overhead. Rainbows are more frequently seen in the afternoon because most showers occur in mid day rather than morning. Because the horizon blocks the other half of a rainbow, a full 360° rainbow can only be viewed from an airplane.
The sky inside the arc will appear brighter than that surrounding it because of the number of rays emerging from a raindrop at angles smaller that those that are visible. But there is essentially no light from single internal reflections at angles greater than those of the rainbow rays. In addition to the fact that there is a great deal of light directed within the arc of the bow and very little beyond it, this light is white because it is a mixture of all the wavelengths that entered the raindrop. This is just the opposite in the case of a secondary rainbow, where the rainbow ray is the smallest angle and there are many rays that emerge at angles greater than this one. A dark band forms where the primary and secondary bows combine. This is known as the Alexander's dark band, in honor of Alexander of Aphrodisias who discovered this around 200 B.C.
If a viewer had a pair of polarizing sunglasses, he or she would see that light from the rainbow is polarized. Light vibrating horizontally at the top of the bow is much more intense than the light vibrating perpendicularly to it across the bow and it may be as much as 20 times as strong.
Although rare, a full moon can produce a lunar rainbow when it is bright enough to have its light refracted by raindrops just as is the case for the sun.
Danielson, Eric W., James Levin, and Elliot Abrams. Meteorology. 2nd ed. with CD-ROM. Columbus: McGraw-Hill Science/Engineering/Math, 2002.
Hancock P. L., and B. J. Skinner, eds. The Oxford Companion to the Earth. Oxford: Oxford University Press, 2000.
Lutgens, Frederick, Edward Tarbuck, and Dennis Tasa. The Atmosphere: An Introduction to Meteorology. Upper Saddle River, NJ: Prentice-Hall. 1997.
"Chasing Rainbows." Christian Science Monitor (July 27, 1999).
"The Near Sky: April Showers and Rainbows." Sky & Telescope 97, no. 4 (April 1999).