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How to Create and Observe a Dozen Rainbows in a Single Drop of Water
by Jearl Walker
THE RAINBOW, at once grand and delicate, appeared in the sky as I sat swaying with my grandmother on her porch glider. Aledo, Tex., is a small town, and there is not much to do there but swing on my grandmother's front porch. As the arc of color took form she turned with a sly look to her grandson, the physicist, and asked me why the rainbow colors appear only in that one arc. Why is the entire sky not filled with colors?
I offered the conventional explanation, comparing the separation of colors in the rainbow to the dispersion of white light by a prism. A particular geometry of scattering, I explained, is needed in order for each of the colors to reach our eyes. Hence the colors are confined to a single arc, a fixed set of directions.
Feeling satisfied with my explanation, and feeling rather smart, I turned back to the sky just as a second rainbow appeared, somewhat higher than the first. One look at my grandmother and I knew I was in trouble. Almost in rhythm with our swinging, she asked question after question. Why is the higher bow wider? Why is the sequence of colors reverse in the higher bow? Why is the space between the bows darker than the surrounding sky? What are the faint, narrow arcs just below the first rainbow and just above the second one? And, again if there can be two rainbows, why not more? Why is the entire sky not covered with rainbows?
When my grandmother starts asking me how the world works, it often seems that my academic training comes out second best to her intuition. The question of why the sky is not covered with rainbows is a good example. In fact, the sky should be filled with rainbows, but reports of more than two are rare. The primary rainbow, the one most often seen, is formed by light rays that are reflected once inside raindrops and then scattered toward the observer. The occasional secondary rainbow requires two internal reflections of the rays. Additional internal reflections should give rise to higher-order rainbows, but apparently they are too faint compared with the background glare to be readily distinguished. They are there in the sky, hanging with the same delicate colors as the familiar rainbows, but they are not ordinarily seen.
The theory of the rainbow was discussed by H. Moysés Nussenzveig of the University of Sao Paulo in the April issue of Scientific American. He described attempts made since the time of Newton to explain the rainbow's appearance, culminating in recent mathematical theories. Here I shall be concerned mainly with the observation of rainbows, and in particular with an experiment in which many of the higher-order rainbows can be observed. With care, all the rainbows through the 13th should be observable. They appear not in the sky but in a single drop of water.
Only a handful of people have seen the higher-order rainbows. The French investigator Felix Billet was one of them; in 1868 he reported seeing 19 rainbows in a thin stream of falling water. Instead of a stream we shall employ a droplet suspended from a thin wire. The experiment is simple enough to be done in the kitchen, arid yet it enables one to join that small club of people who have seen the higher-order rainbows.
With an atomizer or a plant sprayer, gently spray water onto the end of a piece of wire that has been mounted vertically with its free end pointing down. Spray until a drop forms and hangs from the end of the wire. Obtaining such a drop may take practice and patience. It may help to coat the wire with black wax, which makes the water bead up and also reduces glare when the drop is later illuminated.
Once a drop has formed, cover the apparatus with a large box from which one side has been removed. Make a small hole in the box in a side adjacent to the open one and at the height of the water drop; the drop will be illuminated through the small hole and viewed from the open side of the box. The source of illumination can be a slide projector or a movie projector. It should be positioned so that the beam entering the box through the small hole strikes the drop horizontally and traverses a circular cross section. Ideally only the drop and not the wire will be lighted.
The main reason higher-order rainbows are not visible in the sky is the glare of sunlight directly transmitted or reflected from raindrops. In our experiment too glare tends to obscure the rainbows, but it is more easily controlled in the kitchen. Only the light falling on a part of the drop contributes to a given rainbow, and the bright glare spots often result from light striking another part. It is therefore possible to eliminate or at least reduce the glare by masking a part of the incident beam. That is most easily accomplished by adjusting an index card to block off part of the hole in the carton.
Center under the suspended droplet a full-circle protractor, with the zero-degree mark at the side of the drop opposite the incident beam. With the protractor aligned in this way, it measures the total angle through which the light rays are deflected in the drop. Rays passing straight through emerge at zero degrees: those reflected back into the light source emerge at 180 degrees. The first-order rainbow can be found between 137.6 degrees (red) and 139.4 degrees (blue), the colors appearing near the right-hand edge of the drop. The second-order bow comes into view at about 129 degrees, on the left side of the drop. Near the center in both views are glare spots.
The first-order rainbow is made up of rays reflected once inside the drop, but only some of those rays contribute; the rest emerge at the wrong angles. For the sake of simplicity suppose only half the drop is illuminated, so that incident rays are uniformly distributed from the center to one edge, where they graze it tangentially. All these rays are refracted as they enter the drop; they are then reflected by the interior surface, and finally they are refracted again as they emerge from the drop. The scattered light is spread over a large range of angles, the deviation of any particular ray being determined by where on the surface of the drop that ray first made contact. Rays incident at the very center pass through the drop, are reflected and are doubled back on their incident path, for a total deviation of 180 degrees. Rays incident farther from the center of the drop have smaller deviation angles, but only up to a point. As the incident ray nears the edge of the drop a minimum deviation is reached. The rainbow is simply the enhancement in the brightness of the scattered light at the angle where the largest number of rays emerge. That angle is the angle of minimum deviation.
The index of refraction of water, which determines how much the rays are bent on entering and leaving the drop, is different for each color, or wavelength, of light. (Blue light is refracted more than red light.) As a result each color has a different angle of least deviation, where the scattered light is most intense. The rainbow would still exist even if water did not disperse white light into its component colors, but it would be only a bright white band; dispersion creates a series of monochromatic arcs, each at a slightly different angle.
Even for the first-order rainbow from a single drop, the white glare spot near the rainbow colors is comparatively bright. The spot is made up of rays reflected from the outside surface of the drop. If you were more than about a meter from the drop, as you surely would be from raindrops forming a natural rainbow, the glare spot and the rainbow spot would be indistinguishable, and the effective brightness of the rainbow colors would be reduced. Natural higher-order rainbows are lost entirely in the glare (except, perhaps, under unusual circumstances).
In the kitchen you can move close enough to the drop to distinguish the separate bundles of glare and rainbow light; moreover, you can block some of the glare. Slowly move the vertical edge of an index card across the incident beam. If the card is inserted from the side of the beam opposite you, it soon blocks the incident rays that contribute to the rainbow spot. If it is started on the near side, then when its shadow is about a quarter of the way across the drop, it intercepts the rays that produce the glare spot. The selective blocking is easiest with a large drop, say a drop two millimeters in diameter. As the drop evaporates and shrinks, positioning the index card becomes difficult. The color separation is also clearest with a large drop and fades to white as the drop evaporates. Natural bows with washed-out colors are seen when sunlight is scattered by very small raindrops.
Rays reflected twice inside the drop, contributing to the second-order rainbow, are also brightest at the angle of least deviation; again, that angle is different for each color. If the second order rainbow is observed on the same side of the water drop as the first-order one, then blue falls at 126.5 degrees and red at 129.6. As my grandmother pointed out, the color sequence is opposite to that of the first-order rainbow, and the second-order rainbow is wider. The angular width is greater because the rays giving rise to the second-order rainbow enter the drop closer to the edge and undergo greater dispersion.
Between the first- and second-order rainbows relatively little light is scattered from the drop. Rays reflected once inside the drop all emerge at angles equal to or greater than the angle of the first-order rainbow. Rays internally reflected twice emerge at angles equal to or less than the angle of the second order bow. The angular region between the two rainbows is therefore dark. The only light emerging in that region is light that has been reflected more than twice and as a result is relatively dim. This dark area between the first two rainbows is called Alexander's dark band,after the Greek philosopher Alexander of Aphrodisias.
All the higher-order rainbows are formed by the same mechanism as the first two, but after additional internal reflections. Many of them are faint and are obscured by glare, but they can be found if you know where to look. By examining the drop at the angles calculated for the higher-order rainbows I was able to observe all of them through the 13th order. In several cases the incident rays contributing glare had to be blocked with an index card. The most difficult rainbows to see are the ones near the axis of the incident beam. If you must look toward the light source, as with the fourth, eighth and 12th rainbows, then you must contend not only with the glare reflected from the surface of the drop but also with light transmitted through the drop with no internal reflections. Both are white and relatively bright. When looking almost directly away from the light source, as in the case of the l0th-order rainbow, it is difficult to avoid blocking the incident beam with your head.
The 11th-order rainbow is surprisingly prominent, if part of the incident beam is blocked in order to eliminate the glare spot. The 12th and 13th rainbows require considerable patience in order to block just enough of the glare rays but not too many of the rainbow rays. If Billet saw the 19th-order rainbow in his falling-stream apparatus, either his patience or his eyesight was better than mine.
Colored light can appear on the suspended drop through one other phenomenon in addition to the rainbow. Some of the incident light is nearly tangent to the drop and enters almost parallel to the surface. For these rays, as for all others, blue light is refracted slightly more than red light. If the corresponding emerging rays are observed at the appropriate angle, only the blue light will be visible: the drop will have a blue edge. The correct angle is one slightly too large for the red rays to reach but within the range of the more strongly refracted blue light. The blue edges are too faint to be seen naturally because little of the incident light enters the raindrops tangentially. Each of the rainbows has its own associated blue-edge angle, but beyond the first two rainbows the edges are difficult to find. The first order blue edge appears on the right side of the drop at a scattering angle of 165 degrees.
At some angles near rainbow angles white spots appear that are not glare. They are internally reflected rays that emerge at angles other than the minimum-deviation angle. They are white because they are composed of approximately equal intensities of the colors. Two such spots can be seen on the drop from an angle of 145 degrees. Both result from light reflected once inside the drop, but with different points of incidence, one on each side of the point that gives rise to the primary rainbow. As you move your view from an angle of 145 degrees toward the rainbow angle of 138 degrees the two white spots move toward each other. At the rainbow angle they merge and take on color.
The natural rainbow is polarized parallel to its arc. You can determine the sense of polarization of the rainbow rays emerging from the suspended water drop by viewing them through a polarizing filter, such as one eyepiece of polarizing sunglasses (which are designed to exclude light with a horizontal plane of polarization). The incident light can be regarded as an equal mixture of two polarizations: one in the plane that includes the light source, the drop and the observer, the other perpendicular to that plane. More light with the second polarization is transmitted through the drop without reflection. The emerging rainbow is therefore polarized perpendicular to the defined plane and parallel to the rainbow arc.
The angular positions of the rainbows depend on the index of refraction of water, which is about 1.33 (at a wavelength of 5,890 angstroms, the wavelength of the yellow light from a sodium-vapor lamp). I searched my grandmother's kitchen for some other transparent liquid with a different index of refraction. The most convenient one I found was light corn syrup. By carefully applying small amounts of the syrup to the wire mount you can build up a suspended drop. You must work quickly, however, or the drop will harden in a distorted shape. Fresh corn syrup has an index of refraction of 1.47 or 1.48. As a result the angular positions of all the rainbows are shifted clockwise from their positions in water, although not by a uniform amount. As the drop dries over a period of several days and the syrup becomes more concentrated the refractive index increases. The rainbows shift slowly around the drop.
I have also seen rainbows formed by drops of diiodomethane, a transparent fluid with the relatively high index of and diiodomethane, the first-order rainbow shifts toward a scattering angle of 180 degrees. If the index of refraction were made exactly 2.0, the angle would reach 180 degrees and the rainbow would be unobservable. Any attempt to see it would block the incident beam. Even if the awkward problem of viewing angle could somehow be overcome, the first-order rainbow would not appear. With an index of 2.0 all the rays enter the drop, reflect from the opposite side and then return to the light source. There is no dispersion and therefore no rainbow. Each-of the higher-order rain bows reaches a similar limit at a higher index of refraction.
Not all my grandmother's questions have been answered by this experiment. For example, the faint bows below and above the natural rainbows, which are called supernumerary arcs, have not been explained. Neither has the fading of the rainbow colors as the drops evaporate and shrink. On the other hand, we have seen the delicate colors of more than a dozen rainbows within a single drop of water. What a fabulous sight it would be if all of them were visible in the sky!
THE RAINBOW: FROM MYTH TO MATHEMATICS. Carl B. Boyer. Thomas Yoseloff, 1959.
MULTIPLE RAINBOWS FROM SINGLE DROPS OF WATER AND OTHER LIQUIDS. Jearl D. Walker in American Journal of Physics, VOL. 44, No. 5. pages 421-433; May, 1976.
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