The rainbow actually consists of two rainbows, the primary and the secondary. Between the bows, the sky is darker than otherwise. This area is called Alexander's dark band after Alexander of Aphrodisias (A.D. 200), who was the first to describe the dark area.
The first to really try to describe the rainbow was Aristotle. He presumed that the rainbow was caused by reflection of sunlight in the clouds. The light was reflected at a certain angle. That means the rainbow consists of a cone of rays. Aristotle was thus the first to explain the rainbow's circular shape and that the rainbow is not located at a definite place on the sky, but is seen in a certain direction.
Roger Bacon measured the angle of the rainbow cone as 42° in 1266 (the secondary rainbow is 8° higher on the sky). In 1304 the German monk Theodoric of Freiberg proposed the hypothesis that each raindrop in the cloud makes its own rainbow. He verified his hypothesis by observing the diffraction of sunlight in a circular bottle. Notice that Aristotle believed that the rainbow originated from the entire cloud. Theodoric's results remained unknown for three centuries, until Descartes (in 1656) rediscovers diffraction in the drop. Both Theodoric and Descartes knew that the rainbow consisted of 2 bows. In the primary rainbow, the rays are reflected once within the drop, in the secondary rainbow, the rays are reflected twice within the drop. Furthermore, they noticed that only one color is seen when looking at the drop (the bottle) in a given direction. From this they concluded that the colors of the rainbow arrive (in the eye) from different drops in the cloud. Those were the principal ideas of the rainbow.
A more thorough study of the ray's traveling through the drop requires knowledge of the laws of reflection and refraction. The law of reflection is easy to »comprehend«, whereas the law of refraction takes knowledge of the velocity of light in different materials. The law of refraction is due to the Dutchman Willibrord Snell (1621).
The rainbow's most spectacular aspect is naturally its colors. They were explained by Newton. In 1666 Newton showed that white light being refracted in a prism is split up in colors. The color scattering results from the fact that the index of refraction is dependent on the wavelength (the color). Each color in the sunlight thus has its own rainbow. What we see is a collection of rainbows, each slightly displaced from to the rest. Newton worked out the angle of the red rainbow, 42° 2' and for the violet rainbow, 40° 17'. This gives a rainbow of 1° 45'. This would have been the width of the rainbow if the sunlight were parallel, but it isn't; the sun disk has a diameter of half a degree. Newton concluded that the width of the rainbow should be 2 degrees and 15 minutes (which agreed nicely with Newton's own measurements).
Notice that this does not mean that the colors of the rainbow are pure colors (spectral colors). They're actually not! Indeed, raindrops are not prism-shaped. The factual colors of the rainbow are a sum of colors (we will look at that in the program).
The rainbow even contains some supernumerary arcs (mainly red and green bands within the primary rainbow). Newton's theory could not explain these supernumerary arcs. In 1803, Thomas Young showed that the waves from two wave sources (e.g. two holes in a pier in a bowl of water) creates alternating light and darkness. In other words, in some directions the light interferes constructively, in other directions it interferes destructively. Young himself pointed out that the supernumerary arcs could be caused by constructive and destructive interference of sunrays which have traveled different ways through the raindrop, so the distances traveled can turn out to be an odd number of half wavelengths (darkness) or an even number of wavelengths (light).
Notice that the distance traveled by the light depends on the drop's size. Therefore, the drops must have the same size to make it possible to see the supernumerary arcs. Most commonly, this happens at the top of the rainbow. Raindrops commonly grow as they fall which results in drops of various sizes near the ground.
The light of the rainbow is almost 100% polarized which can be seen with a pair of polaroid glasses. The polarization is due to the fact that the angle of refraction in the drop is very close to Brewster's angle (David Brewster 1815). Therefore, most light of parallel polarization disappears out of the drop at the first reflection (and refraction) within the drop.