Image credit: Eric Rolph via Wikimedia Commons [CC BY-SA 2.5] (source)

There Is More to a Rainbow than Meets the Eye

In this episode of the Philipendium, we examine the hidden mysteries of rainbows — along with other natural phenomena that produce colors without pigments.

As a kid, I was fascinated by rainbows — as many kids are. My understanding at the time was that a rainbow appears only if it is raining and the sun is shining at the same time. So if I saw the sun come out before a rainstorm had fully stopped, I would dash outside to find the rainbow. Of course, this meant that I had to stand in the rain. Furthermore, it was often hard to find the rainbow, because I had no idea which direction to look. So I’d search the entire sky. And if I still did not see anything, I would try to adjust my position so that the trees and neighboring houses blocked a different part of the sky, opening up a new patch of sky for me to examine.

In my earliest years I still believed the myth that there is a pot of gold at the end of the rainbow. After all, this idea appeared over and over on TV and in movies and in conversations with other kids. But after my parents pooh-poohed the idea several times, I came to accept that there is no pot of gold. But even without the prospect of gold, the idea of finding the end of the rainbow was fascinating. I tried to imagine what it would be like to stand right next to where the rainbow reached the ground. How wide would the rainbow be? How intense would the colors be?

Later, as an adult in my early 20s, I came to associate rainbows with driving. Most of my rainbow sightings occurred while I was driving through the countryside, especially in agricultural areas where there were few trees to block the view. It seemed that the best chance to see a rainbow was in the late afternoon during the summer, whenever a rain shower had just passed by and the sun had re-emerged. By now I knew that rain in my current location was unnecessary — it simply needed to be raining someplace nearby. The ideal situation was when part of the sky was still black from the passing storm, because this would create the perfect background for the rainbow to be especially visible.

Consequently, my rainbow habit was only slightly changed from when I was a kid. Instead of running outside if the sun popped out while it was raining, my new habit was to stop the car the moment the sun came out after a rainstorm. I would scour the sky, primarily in the direction of the darkest clouds, to find a rainbow. I had two primary measures of success. First, I wanted to see a rainbow that formed a full arch across the sky. Second, I wanted to see a second, fainter rainbow above the first one — a double rainbow. On those rare occasions when I met both goals, I felt especially lucky.

Still, I could not shake the idea that the rainbow had a location. Perhaps the rainbow was a mile away, or perhaps it was four miles away. Eventually it became clear that the location was relative to me, and not relative to the earth. If I drove towards the rainbow, then the rainbow would not get bigger, and in fact the rainbow seemed to maintain a constant distance from me, receding at the same speed that I was driving. If I drove parallel to the rainbow instead of towards it, then it would follow me, just as the sun or moon does — which is quite different than a distant tree or hill, which changes angle as I drive parallel to it.

There was another phenomenon that caught my attention as a kid. If I stood out in the sun with a garden hose, and I set the nozzle to spray a fine mist into the air, then I could see what looked like a tiny rainbow. If I positioned myself and the nozzle just right, I could see an arc that formed nearly a full circle. Although I called it a rainbow, I knew perfectly well that it was not a real rainbow. In my model of the world, a real rainbow was really big and up in the sky and caused by rain, while my little artificial rainbow had none of those essential attributes.

As an adult, I eventually came to realize that my little garden hose rainbow was identical in all essential attributes to a “real” rainbow. But the bigger revelation came when I realized that a rainbow is always located in exactly the opposite direction from the sun. Or more precisely, the highest point in a full arch rainbow is always directly opposite the sun. Now, whenever I stopped the car to find a rainbow, I knew exactly where to look — in the same direction that my shadow pointed. (Note the shadow of the photographer in the opening photo.) Likewise, I knew that my odds were strongest if the darkest clouds were directly opposite from the sun.

However, this revelation also presented a mystery to me. I had heard that the drops of falling rain act like a prism, dividing visible light into its constituent colors. But in a prism, the light goes in one side and comes out the other side, more or less. Therefore, if the rainbow is in the opposite direction from the sun, then the raindrops are NOT acting just like a prism, because the sun is going in one side of the drop and then coming back out the same side, more or less.

Ultimately, I learned that raindrops act not only like a prism, but also like a mirror. A mirror is typically a pane of glass to which a silver backing has been applied. Light passes through the glass, reflects off the silver backing, and then passes through the glass again. A raindrop does not have a silver backing (even though a cloud is metaphorically said to have a silver lining) — but some of the light does indeed reflect off the inside back surface of the drop. So in that respect a drop of water is indeed like a mirror. But as the light enters the drop, the angle of travel is bent (refracted) by the change in material. As the light exits the drop, after reflecting off the back, the path of travel is bent again. Different wavelengths of light are bent slightly differently, which causes the colors to separate. So in that respect the drop is like a prism. The angle between the beam that enters the drop and the beam that exits the drop is approximately 41 degrees. The angle ranges from 40 degrees for violet light to 42 degrees for red light. The other rainbow colors fall between these two extremes.

Image credit: “What makes a rainbow? An explanation of atmospheric optical phenomena

When I learned this explanation, I was partially satisfied, but something still seemed to be missing. There are many raindrops in a rainbow, but I don’t see all the colors of a rainbow emanating from a single drop — I only see one color from any given drop. Furthermore, all the nearby drops produce exactly the same color. If a particular raindrop is in the green part of a rainbow, then where did the other colors go?

The answer is that the other colors emerging from that particular drop all missed me. Instead of striking my eye, those other colors just barely missed me, landing in front of me or behind me. In fact, someone standing 30 or 40 feet in front of me will see the same rainbow, but not the same colors coming from same drops. A drop that is part of the green band from my vantage point might be part of the orange band from my friend’s position. If my friend is standing 100 or 200 feet in front of me, then a completely different set of drops may produce the rainbow, even though we think we see exactly the same rainbow.

To put it another way — wherever I happen to stand, the rainbow I see contains only a tiny fraction of the raindrops that are involved in producing the entire rainbow. Other raindrops produce the rainbow for someone standing 100 feet or 1000 feet away. Even crazier, those raindrops are falling the whole time. Each raindrop passes through my rainbow for only a moment, and then other raindrops take its place. It requires a continuous parade of raindrops to make a rainbow.

A Bigger Brain-Teaser

If that isn’t mind-bending enough, now try to wrap your head around this next brain teaser. You may already know that raindrops in the air are spherical, not “raindrop-shaped”. Therefore raindrops are completely symmetrical — also called radially symmetric. Given that the rainbow colors exiting a raindrop come out at a 41 degree angle, compared to the entering ray of sunshine, then how does the light know which direction to turn? Does it turn 41 degrees to the left, or to the right, or up, or down? The diagram above shows the light bending downward by 41 degrees, but is that correct — and if so, then why?

Well in fact, the diagram is slightly misleading, because it only shows one beam of light entering the drop. When the light enters the drop, the direction of the reflection from the back is controlled by what part of the curved back of the drop the light strikes. Different beams of light will strike different parts of the back surface. The upshot is that for any given drop, some of the light is reflected 41 degrees to the left, some of it 41 degrees to the right, some of it 41 degrees upward, and some of it 41 degrees downward — and all points in between, forming a complete ring. From your particular vantage point looking at a rainbow, the only drops that reflect the light 41 degrees downward are those at the very top of the rainbow arc. If you draw a line from your eyes to the lower left edge of the rainbow and then back to the sun, you still have a 41 degree angle, but this angle is primarily to your left instead of up. But the same drops that are located at the left edge of the rainbow — from your vantage point — could be at the right edge of the rainbow for someone who is standing a long distance to your left. Those same drops send colored beams to both the left and the right at a 41 degree angle.

Because the drops also bend light 41 degrees upward, it is possible to see a full circle rainbow — under the right circumstances. If you are up in the air — perhaps in an airplane — and it is sunny where you are but raining nearby, then you might see a 360-degree rainbow, a complete circle. It helps if the sun is at a low angle in the sky. The colors you see in the very bottom of the rainbow circle have been bent upward by 41 degrees. (A great example of a full circle rainbow can be seen on the NASA website.)

Now let’s bring this argument full circle (ha ha). Each raindrop emits a rainbow of colors in a complete circle, and under the right circumstances, you can see a rainbow of colors in a complete circle. And yet these are two very different circles, although they both involve angles of 41 degrees. For every raindrop that forms a 41 degree angle between you and the sun, you only see a single color, and you only receive the tiniest sliver of the full circle of that color emanating from the drop. The rainbow arc that you actually see is due to the convergence of light on your eye, a tiny sample of the colored light emerging from each of countless little drops. In contrast, the rainbow of light coming from each drop is divergent, heading in many directions, unperceivable by any single observer — but tiny samples of the rainbow circle may be seen by observers standing in many different locations. Therefore, quite literally, there is much more to a rainbow than what meets the eye.

Rainbows That Are Not Rainbows

Now that your mind is thoroughly warped by trying to wrap around this concept, let’s move on to something slightly easier. Let’s talk about rainbows that aren’t really rainbows — because drops of water are not involved.

Sometimes you can see a faint rainbow in the sky when sunlight passes through high cirrus clouds — those wispy, feathery clouds that can decorate an otherwise blue sky. These clouds are so high that they are made of particles of ice, rather than droplets of water. Unlike a normal rainbow, these colorful arcs tend to be high in the sky, and in the same general direction as the sun. Because ice is involved, not rain, the preferred term for this phenomenon is halo. There are several variations on this phenomenon, with different results, but the key point is that some of the ice crystals act like prisms. Light is bent and split into its constituent colors as it passes through these tiny prisms. Unlike the case of water drops, light is not reflecting off the inner surface of the ice crystals — therefore the light passes through the crystal just once. The upshot is that you have to look in the general direction of the sun, instead of the opposite direction, in order to see the colors.

(Note: Clouds made of ice or tiny water droplets can also produce colors through diffraction. For a discussion of the many ways that clouds can produce optical effects, see the excellent article What makes a rainbow? An explanation of atmospheric optical phenomena by Meteorologist Claire Flynn.)

You also may have noticed rainbow colors in a patch of oil on a wet road. Or perhaps you saw the rainbow colors in an oil sheen floating on the surface of a lake or a puddle. In any of these cases, you were observing an extremely thin layer of oil that has spread out over water. It’s hard to imagine just how thin this layer of oil is, but if you can see rainbow colors, then it is probably in the range of 1 or 2 microns thick. (A micron is a millionth of a meter — and a meter is just over 3 feet.) This is in the same general range as the wavelength of visible light, which is around 0.4 microns (at the violet end of the spectrum) to around 0.7 microns (at the red end of the spectrum).

When light strike the surface of the oil film, some of it is reflected back up. As the remaining light passes through the oil and then the water, some of it is reflected when it meets the boundary between the oil and the water. This sets up the conditions for an interference pattern.

Before we go any farther, let’s look at an analogy — a different situation with a different kind of interference pattern. Imagine that you toss two small pebbles simultaneously into a still pool of water. Each pebble produces a set of circular ripples radiating outward. These tiny waves alternate between crests and troughs. So what happens when waves from the two pebbles meet? When two crests meet, they join to form a higher crest, and when two troughs meet, they join to form a deeper trough. We call this constructive interference. But when a crest meets a trough, the two cancel each other out, and the result is neither a crest nor a trough. We call this destructive interference.

Image credit: “Sea Turtle” via Flickr [CC BY-NC-ND 2.0] (source)

We have to be careful when translating this analogy to our oil film, because the intersecting circles are no longer relevant. In our new situation, we don’t have waves traveling in two different directions, crossing paths. Instead, we have light reflecting off two closely spaced surfaces, and then traveling in the same direction — towards our eyes. What IS relevant is that because light is made of waves (even though we cannot perceive the waves), and because the wavelength of visible light is only slightly smaller than the thickness of the oil, we have a situation where interference can occur. The light that reflects off the lower boundary of the oil merges with the light reflecting off the top surface. The light from both surfaces contains a mix of wavelengths. But some of those wavelengths will be in phase, comparable to two crests merging in our ripple analogy. Other wavelengths will be out of phase, comparable to a crest merging with a trough and canceling each other out.

So what determines whether a particular wavelength coming from the two surfaces is in phase or out of phase? It’s the thickness of the layer of oil — or more precisely, the distance that light has to travel through the layer of oil (considering both the trip down and the trip back up). If the distance traveled through the oil is an exact multiple of a particular wavelength, then you get constructive interference. But if the distance is halfway between two multiples (for example, 2.5 times the wavelength), then you get destructive interference. The upshot is that for any given point on the oil slick, some colors are reinforced due to interference, and other colors are attenuated. Of course, this is partly determined by the angle of your viewpoint. If you shift to a higher angle (more directly overhead) or a lower angle (more oblique to the surface of the oil), then you change the length of the path that the light travels through the oil to reach your eye. And therefore the colors change, even in those cases where the film has a uniform thickness.

This phenomenon is called thin film interference. Light interacts with soap bubbles in exactly the same way. Light reflects off both surfaces of the soapy film, setting up the potential for interference that produces bands of colors. A soap bubble tends to be thicker towards the bottom and thinner towards the top — and therefore both the thickness of the film and your angle of viewing will influence the colors that you see.

Colors Without Pigments

Some people, when considering the rainbow colors in a soap bubble or an oil film, will say that the colors “aren’t really there”. This line of thought confuses the concept of pigments with the concept of colors. It’s true that the colors we see in a soap film are not caused by pigments, but we certainly do see colors. Colors are due to the mix of wavelengths that enter the eye, as interpreted by the brain. Pigments work by selectively absorbing certain wavelengths of light, while allowing other wavelengths to be reflected or transmitted — and therefore available to be seen. Thin film interference is simply a different method for affecting which wavelengths of light reach our eyes. If the colors in a soap film are “not really there”, then we might as well say that the colors in a rainbow aren’t there either. After all, there are no pigments in a rainbow.

In those cases where colors are NOT caused by pigments, then do we always see rainbow stripes? No, there are interesting cases where we can see a solid color, despite the lack of any pigment. One case is the blue sky, which is not due to any pigment in the air, but instead due to how light is scattered as it strikes molecules of air in the atmosphere. Shorter wavelengths are more prone to scattering than longer wavelengths. Therefore, when we look into the sky during a sunny day, more blue is redirected towards our eyes than any of the other colors, causing the sky to appear a pastel blue.

Another case is the green color we see on the head of a male mallard. This color is not due to a green pigment, but to the structure of the feathers. Tiny parallel ridges and grooves, on a scale not much greater than the wavelength of visible light, trap certain wavelengths (due to destructive interference) while reflecting green light. A similar phenomenon, but even more spectacular, occurs in peacocks — where the brilliant blues, greens, and purples are caused by the structure of the feathers rather than pigments. Likewise, the brilliant metallic blue color in the wings of certain butterflies is caused by structure, not pigments.

In summary, the colors we see in nature are not only caused by pigments — which selectively absorb certain wavelengths of visible light — but also by other phenomena that have selective effects on different wavelengths of light. The sky is blue because more blue than red is scattered by air molecules, and therefore more blue is redirected towards our eyes. Peacocks and brilliant blue butterflies are colorful because microscopic structures in their feathers or scales have different interference effects on various wavelengths of light. Rainbows are colorful because different wavelengths of light are refracted (bent) to slightly different degrees. Each drop of water in a rainbow reflects and refracts all of the rainbow colors into a huge circle, but from any single viewpoint we only see a tiny sliver of a single color coming from that drop. Rainbows, because of their dependence upon the angle of 41 degrees, are located relative to the viewer, and will move with the viewer. Unfortunately, this means that none of us will ever reach the end of the rainbow, although we might have a good time trying.

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