A rainbow is a curved band of colors in the sky, but it is not a thing in one place, and it is light that reaches your eyes only when you stand in the right spot. To see one, the sun must be behind you and the water drops in front of you. In other words, the arch is not painted on the sky at a fixed distance from anyone. It consists of sunlight arriving along particular directions, which depend on the position of the observer. Your rainbow is yours alone: each person has a different shadow point and horizon, and a different set of raindrops makes the rainbow each person sees. Because of this, the rainbow you see is determined by your own position. Two people standing side by side therefore look at slightly different rainbows.
The spot exactly opposite the sun, called the antisolar point, is the center of the rainbow's circle, and the main bow forms where you look about 42 degrees away from that point. A rainbow is round because every direction that is 42 degrees from the antisolar point makes a circle around it. If you can see your own shadow while it rains, and the sun is low enough, the rainbow circles around the shadow of your head. The higher the sun is in the sky, the lower the rainbow sits and the less of its arch you can see. At sunset the arch is a half circle with its top about 42 degrees above the horizon. If the sun is more than 42 degrees above the horizon, no rainbow can be seen from the ground.
Now look inside a drop, and a raindrop is not a prism. Raindrops are nearly round, and the light separates because it is bent twice at curved surfaces of the drop, and light bends when it goes from air into water. This bending is called refraction. Refraction is the bending of a wave as it passes from one material to another, such as light going from air to water. Light moves more slowly in water than in air, which is why it bends when it enters a raindrop, and sunlight enters a raindrop, bends, bounces off the back of the drop, and bends again as it leaves, heading back toward the sun's side of the sky. Different colors of light bend by different amounts when they enter water, so the colors spread apart, and violet bends the most and red the least.
The index of refraction of water changes with wavelength, which is why the drop spreads sunlight into colors, and dispersion means light is spread out by wavelength. Reflecting off the back of a drop does not disperse light, because reflection does not depend on wavelength, and human eyes can see wavelengths of light from about 380 to 700 nanometers, with violet at the short end and red at the long end. Red light leaves a raindrop turned about 42 degrees from the sunlight's path and blue-violet light about 40 degrees, so the colors line up in bands. The light that makes the main rainbow leaves a drop turned about 138 degrees from the direction it came from, which is the same as 42 degrees from the point opposite the sun. These angles are the numbers that place each color at its own height in the bow.
Every raindrop sends out all the colors, but each drop sends different colors in different directions, so your eye catches just one color from each drop. Red light comes out of the drops at a steeper angle than blue light, so you see red from drops higher up, at the top and outer edge of the bow, and blue from lower drops. In the main rainbow the colors always come in the same order, with red on the outside and blue and violet on the inside, and a red rainbow shows when the sun is very low, near sunrise or sunset. Scattering in the long path through the air removes the blue and green light and leaves mostly red, and air molecules scatter short wavelengths such as blue and violet more strongly than long wavelengths such as red, which is also why sunsets look red.
The ancient Greek thinker Aristotle explained the rainbow as sunlight reflecting off a cloud, which is not right. Around 1300, scholars in Persia and Germany worked out that a rainbow comes from light refracted and reflected inside single raindrops. Theodoric of Freiberg, in Germany, studied rainbows with a globe of water, treating it as a magnified raindrop. In Persia, Kamal al-Din al-Farisi studied the rainbow by filling a large glass vessel with water and putting it in a camera obscura, to stand in for one giant raindrop. In 1637 René Descartes explained the rainbow by tracing light through a raindrop, and with Newton he set up the basic ray theory of the rainbow. In Descartes' ray theory, a rainbow forms where the angle of the rays leaving the drop reaches an extreme value, a turning point. The simple ray picture of Descartes and Newton explains the main bow but not the faint extra bands. Those need the wave nature of light, which means that the ray picture, though it locates the main bow, is not the whole account of what is seen near the rainbow.
Before Newton, many people thought a prism changed sunlight and made the colors. In the mid-1660s Isaac Newton used glass prisms to show that white light is made of a spectrum of colors. Newton let a beam of sunlight pass through a prism onto a wall, and then showed that a second prism brings the colors back together into white light. In his crucial experiment, a board with a hole let only one color, such as red, pass into a second prism. That light bent but stayed the same color, so a prism does not make colors, it only bends each color by a different amount, and people had seen prisms make colors before Newton. His new idea was that white light is a mix of all the colors. Seen as a sequence, the experiments moved from one prism, to a board with a hole, to a lens that recombined the colors.
Newton also used a lens to bring a spread-out spectrum back into one beam, and the merged beam was white. Passed through another prism, it split into a spectrum again. Newton explained the rainbow as sunlight refracted and reflected by raindrops that act as tiny prisms, and a rainbow's colors and the colors that come out of a prism are the same, in the same order. He told the Royal Society about his prism findings in 1672, but he had made the discoveries years earlier, in the 1660s. Newton's telescope shows a related effect of color. Lenses bend each color by a different amount, so a lens telescope shows colored fringes around a star, and this blur is called chromatic aberration. Newton built a telescope with a curved mirror instead of lenses, because lenses split starlight into colors and mirrors reflect all colors equally.
The usual order of colors is red, orange, yellow, green, blue, indigo and violet, remembered as ROYGBIV. A rainbow is really a smooth, continuous range of colors. Many scientists think indigo is too close to blue to tell apart, so some modern lists merge indigo into blue and violet. In this list the word indigo names the color between blue and violet, which is the one that scientists find hardest to separate from its neighbor, and a continuous range means that there are no sharp edges between one color and the next. The list begins with red, which has the longest wavelength of visible light, and it ends with violet, which has the shortest
Theodoric also explained the double rainbow: the extra reflection reverses the order of the colors and weakens the light, and a double rainbow happens when light bounces twice inside the raindrops instead of once. In a double rainbow the second bow is outside the first, and its colors are in reverse order, and the secondary rainbow comes from light that reflects twice inside the drop and leaves at about 50 to 51 degrees, wider than the primary bow. A third-order, or tertiary, rainbow is a different case, and it is centered on the sun itself, so you have to face the sun to look for it, and it is very faint. Put another way, the first bow comes from one reflection inside each drop and the second bow from two reflections, which accounts for the flipped colors and the wider angle.
Sometimes two rainbows show at once: a bright one and a fainter one above it with its colors flipped, and the dark strip of sky between the two bows is called Alexander's band. It is named for Alexander of Aphrodisias, a Greek philosopher who studied it around 200 AD in connection with Aristotle's incorrect rainbow theory, and the band is dark because almost no sunlight is scattered toward your eyes from that part of the sky. The band lies between the primary bow, at about 42 degrees, and the secondary bow, at about 50 to 51 degrees, so it occupies a gap of roughly eight or nine degrees of sky. Alexander studied it around 200 AD, whereas the globe and vessel experiments came around 1300. The reverse order of colors and the wider angle are the two signs that tell a second bow from a first.
Fogbows are pale, nearly white rainbows that form in fog, mist or cloud, and they are broad and pale because fog droplets are far smaller than raindrops, so the light spreads out (diffracts) and smears the colors. A glory is a ring of colored light around the shadow of the viewer's head, often seen with fogbows, and it is physically more complex than a rainbow. A rainbow sends light back at about 42 degrees from the point opposite the sun, whereas a glory comes from light scattered almost straight back. The comparison is between two angles: about 42 degrees from the point opposite the sun for the rainbow, and almost straight back for the glory. In the fogbow, the small size of the droplets is the stated reason that the colors are smeared and the bow looks broad and pale.
Sometimes faint extra pastel arcs, called supernumerary bows, show just inside the main rainbow, and they come from interference: light waves that add together where they match and cancel where they do not. Interference means that two sets of waves meeting in the same place can strengthen one another where their crests match, and weaken one another where crest meets trough. In 1803 Thomas Young explained supernumerary bows as interference between light waves, which helped the wave theory of light win acceptance. Rainbows have served as a testing ground for theories of light: Descartes's rays, Newton's colors, and Young's waves. Young's explanation, dated 1803, therefore connects the faint arcs to the nature of light itself, and it belongs to a sequence in which rays, colors and waves were each tested against the rainbow.
Light leaves a raindrop in many directions, but the most light is concentrated near 42 degrees, so that is where the bright rainbow appears. One special ray, the rainbow ray, has the smallest angle of deviation of all rays entering the drop, and much of the sunlight is focused along it. Because the rays bunch up near that smallest angle, the light there is concentrated, and the concentration is what makes the bright arc, and physicists call this kind of bright line, where many light rays pile up at the same angle, a caustic. The rainbow is the most familiar caustic. Put plainly, the bright arc seen at about 42 degrees is a caustic: the rays leaving the drop pile up near one angle instead of spreading evenly across the directions in which light can leave.
Words to know
- caustic: bright ray pile-up
- dispersion: spreading by wavelength
- interference: waves adding together
- supernumerary: extra, additional
- antisolar: opposite the sun
