IN THIS DEPARTMENT FOR JULY, 1977, I described how the common first and second-order rainbows can be examined in a simple experiment. I did not discuss the extra bands of color, termed supernumerary arcs, that sometimes accompany a natural rainbow. This month I shall describe several experiments by which such arcs can be investigated.

The commonest natural rainbow is called the first-order rainbow because it requires one reflection of light rays inside a falling raindrop. The rarer second-order rainbow requires two internal reflections. Higher-order rainbows, involving even more internal reflections, are probably too dim to be seen in the sky but can be seen in the experiments I described in 1977. My observations were prompted by some questions my grandmother had asked me about a double rainbow she had seen. The questions were simple but the answers were not.

Two of the questions were left unanswered in my earlier discussion. My grandmother asked them after I had likened the colors of a rainbow to the colors thrown by a glass prism held in sunlight. My glib answer obviously did not sit well with her. One question was: If the separation of colors in a rainbow is the same as that in a prism, why do extra bands of color (usually purple) appear next to the expected colors? The additional bands lie just below the first-order rainbow and (more rarely) just above the second-order one. A prism does not yield such extra bands of color. The second question was: How can some rainbows be white? Again a prismatic separation of colors cannot be responsible.

The extra bands of color are the supernumerary arcs, a name that implies they should not be present. Indeed, if one believes water drops separate white light into the component colors as a prism does, the supernumerary arcs are unexpected. The experiments that elucidate the phenomenon are extensions of the ones I described earlier but now include the effects of the wave interference of the light being scattered from a droplet of water. The experiments are not difficult, and they reveal the beautiful optical pattern of which the common rainbow colors are a part.

In the earlier article I described how a water droplet can be suspended from the end of a wire held vertically and then can be illuminated with white light from a projector. Of all the rays of light on the droplet one ray (named the Cartesian ray after Rene Descartes, who first determined its nature) enters the droplet at the single point that enables it to emerge to form the first-order rainbow. The rays striking the droplet at other points do not contribute to that rainbow.

In the experimental setup shown in the illustration in Figure 4 the Cartesian ray contributing the first-order rainbow enters on the side of the droplet opposite the observer. After the ray is refracted into the droplet it reflects on the inner surface once and then leaves the droplet on the side of the observer. The path of the ray is sketched from an overhead view in the top illustration in Figure 5. If the incident beam of light is leveled properly (a task that may require a bit of patience), the ray crosses through a circular and horizontal cross section of the droplet. When the observer's head is in the right position to intercept the emerging ray, the colors of the rainbow are seen.

The Cartesian ray for the first-order rainbow is bent about 138 degrees from its initial direction of travel. All the other rays that enter the droplet and reflect once from the inner surface are scattered by larger angles, up to a full 180 degrees for the ray that enters the center of the droplet and is returned directly to the light source. No rays reflecting once inside the droplet can be scattered by an angle smaller than the angle of the Cartesian ray.

If white light illuminates the droplet, the component colors are refracted by slightly differing amounts and leave the droplet at differing angles. Blue is refracted a little more than red, and so the blue Cartesian ray is turned from its original direction of travel slightly more than the red one. The intermediate colors are bent by intermediate amounts.

When the droplet is viewed at an angle near the rainbow angle, the different colors emerge at different angles and therefore can be distinguished. The scattering angle, measured with respect to the initial direction of the light, is about 138 degrees for red and 139 for blue. If the drop were one of many drops falling through sunlight, an observer would see an arc of colors in the sky: the natural first-order rainbow. No such arc exists for a water droplet suspended from a wire because there is only a single droplet, but the colors can easily be seen on the edge of the droplet (the side opposite the light source) where the rays emerge.

Suppose the white light is replaced with monochromatic light from a source such as a helium-neon laser operating at a wavelength of 632.8 nanometers. One would expect only the red component of the rainbow to emerge from the droplet. What one actually sees, surprisingly, is a series of vertical red bands rather than a single red band. The brightest and widest band lies at the angular position of the red in the rainbow colors generated with white light. The other red bands are fainter and narrower and extend in a series that starts at the widest band and continues toward larger scattering angles. This series of extra bands holds the clue to the nature of the supernumerary arcs.

Sue Foroughi of Council Bluffs, Iowa, has investigated the pattern of red bands scattered by a suspended water droplet illuminated with a helium-neon laser. She improved on my experimental design by substituting a burette for the wire. With careful adjustment of the spigot on the burette she was able to form a water droplet at the lower end of the apparatus. She positioned the droplet over the center of a spectrometer so that she could use the telescope arm of the apparatus to aid her observations of the pattern. (The telescope effectively places the observer at a distance from the droplet. Without it the eye focuses the scattered light to form an image of the droplet, rather than the pattern, on the retina.)

Foroughi was particularly interested in how the size of the droplet affected the number of red bands per degree in the pattern. She formed a fresh droplet at the end of the burette and counted the angular density of the bands in the pattern by moving the telescope arm through an angle of one degree. She had to approximate the diameter of the droplet. After waiting for evaporation to reduce the droplet she repeated the process.

A droplet initially about four millimeters in diameter had a pattern density of 16 red bands per degree near the main rainbow band. After two hours the diameter had been reduced to about 2.5 millimeters and the density had dropped to eight bands. Foroughi found that as the droplet shrank, the bands widened and so the number of them per degree decreased. This result is a clue to the nature of the white rainbows my grandmother asked about.

In my experimental setup the ray that 9 will enable an observer to see the second-order rainbow enters the droplet on the same side as the observer. This Cartesian ray is refracted into the droplet, reflects twice off the inner surface and emerges toward the observer. As before, the different colors are refracted by slightly differing amounts and thus emerge at slightly differing angles. An observer moving through the range of angles of these colors will see the spectrum that in the natural environment gives the second-order rainbow.

With the experimental setup for the suspended droplet the colors appear on the edge of the droplet toward the light source. The red Cartesian ray emerges at an angle of about 130 degrees with respect to its initial direction of travel, the blue Cartesian ray at about 127 degrees. I shall call these angles the scattering angles for the second-order rainbow.

If the droplet is fully illuminated, the observer can move through the angles of the first-order rainbow and then those of the second-order rainbow by moving from a scattering angle of about 140 degrees to one of about 126 degrees. The color sequence will be blue through red and then, following a separation, red through blue, with the second-order colors occupying a larger angular range. Between the two rainbow sequences the droplet will be relatively dark.

This dark band, which also appears between the arcs of a natural rainbow, is named Alexander's band. The darkness results from a relative lack of light rays emerging from a water droplet into that 9 angular range. All the rays reflecting once inside the droplet must emerge at scattering angles of about 138 degrees or more. All the rays reflecting twice inside the droplet must emerge at scattering angles of about 130 degrees or less. For this reason the intermediate range of about eight degrees is left relatively dark.

When a source of white light is replaced with a helium-neon laser, the full second-order rainbow is replaced with another pattern of red bands. The brightest and widest band is again at the position of the red in the full rainbow. The vertical bands extend from the angular position of the normal red to smaller scattering angles, becoming progressively narrower as the angular distance from the brightest band is increased.

This type of pattern is distinctive for the first two orders of rainbow. Each additional order displays its own pattern when a droplet is illuminated with monochromatic laser light. For example, the fifth-order rainbow results from five internal reflections of light Although that rainbow is too dim to see in the sky, its colors are visible on a suspended droplet if the incident beam of light is carefully leveled. The fifth-order Cartesian ray enters on the side of the droplet opposite the observer, reflects five times inside the droplet and emerges from the edge of the droplet opposite the light source. The red emerges at an angle of about 127 degrees from the initial direction of the light, the blue at about 134 degrees. When the white-light source is replaced by the laser, the fifth order rainbow results in a pattern of red bands with the widest and brightest band at an angle of about 127 degrees. The pattern is dimmer than the patterns of the first two orders are. In general the higher the order of rainbow is, the dimmer and wider the main bands of its pattern are. The same is true for natural rainbows: the higher the order, the dimmer and wider the rainbow.

When a water droplet is illuminated with white light, each color produces its own pattern of bright and dark bands near the rainbow angles. The natural rainbow consists of the main (widest and brightest) bands for each color. Al though the narrower and dimmer bands in each color's pattern still exist, they are usually imperceptible in a natural rainbow. Occasionally, however, the dimmer bands overlap in just the right way to be visible. These bands are the supernumerary arcs. In the experimental set up and in nature they appear on the blue side of the rainbows.

When I repeated Foroughi's experiments and some of my earlier ones, I arranged for a droplet of water to hang from the needle of a syringe. I had ground the tip flat so that when the syringe was pointed down, a droplet could form and hang properly when I pushed the plunger. The optics of a rainbow be come much more complicated than I have indicated if the light rays do not pass through a circular cross section of a water droplet. Hence it is desirable to achieve a droplet that hangs straight down and has a circular and horizontal cross section.

The syringe was held in place by a test-tube clamp mounted on a ring stand. A piece of foam rubber in the clamp made the fit snug around the syringe. By gently depressing the plunger I could form either a droplet or a large drop on the blunted needle. The drops were illuminated with a 3 5-milliwatt helium-neon laser. If you repeat this work, I suggest that you either use a less powerful laser or put filters in the laser beam in order to dim it. The light reflecting from the needle and even from the droplet was uncomfortably and dangerously bright. If the light from your laser is polarized, the polarization should be vertical to create a proper scattering of light from the droplet.

By adjusting the illumination on the droplet I could make the pattern for either the first- or the second-order rain bow. Illuminating the side of the drop let closest to me created the pattern for the second-order rainbow. Illuminating the other side created the first-order pat tern. Because the beam was narrower than the droplet, at least for the larger droplets, I could not get both patterns simultaneously. When I inserted a lens in the beam, however, the beam was spread sufficiently to create both pat terns simultaneously.

Because my laser is quite bright the patterns were cast on the walls of my room with considerable brilliance. I could also display them easily by holding a sheet of white paper near the drop let, and I could examine them through the viewer on my camera. If I illuminated the far side of the droplet in just the right way, I could see the partly over] lapping patterns for the first- and fifth- order rainbows.

The light can best be aligned on the droplet by means of the shadow of the droplet on a wall. As I watched the shadow of the droplet I slowly maneuvered the laser until I had centered the shadow in the beam. If I wanted to illuminate one side of the droplet more than the other, I moved the laser accordingly. Sometimes I put a black card in the laser beam so that it blocked the light falling on one side of the droplet. I could also use the card to shade the needle, which otherwise produced a glare.

I photographed the patterns with my: 35 millimeter camera, using ASA 64 film at an exposure of 1/125 second, with no lens. To photograph the patterns of the first two orders of rainbow I positioned the camera and the tripod as close to the suspended water droplet as I could without blocking the laser beam with the camera. I could expand the pattern on the film by moving the camera farther from the droplet. This maneuver made the details of the pattern more apparent but enhanced the adverse effects of vibrations of the droplet. Air currents and vibrations of the building shake the droplet and momentarily ruin the ideal circular cross section traversed by the light rays. The pattern wavers. I minimized the vibrations by putting concrete blocks on the ring stand to steady it. The effects could be reduced further by moving the experiment to the basement or by mounting the rig on the kind of vibration-isolation stand employed in making holograms.

When I wanted to photograph the water droplet but not the red bands, I mounted my normal lens (50 millimeters and f/1.4) and several clasp lenses on my camera to focus the rays scattered from the droplet. The proper exposure had to be determined by experimentation. To be sure of getting a few good pictures I operated the camera at many exposures.

Instead of waiting for a droplet to shrink through evaporation I watched, the patterns of the first two orders of rainbow while slowly pushing the plunger on the syringe. As the droplet grew I could see the bands in the pattern become narrower. I also noted another effect. The angular positions of the main bands (the brightest and widest band in each pattern) shifted. As a droplet grew the main bands of the two orders moved toward each other, thereby narrowing the intermediate dark region. If the drop is large (four millimeters or so in diameter), the patterns are relatively close and consist of relatively narrow bands. As the drop shrinks, the patterns shift away from each other and the bands become wider.

Why does a monochromatic source of light produce a pattern of bright and dark bands near the normal angular position of the rainbows? Clearly the droplets are not separating colors as a prism would. In 1803 Thomas Young demonstrated that light waves can interfere with one another to produce patterns of bright and dark bands. The patterns Foroughi and I have seen are due to such an interference of light.

Consider two rays of light illuminating a suspended droplet close to each side of the Cartesian ray that eventually emerges in the first-order rainbow. Choose rays that reflect once inside the droplet (just like the Cartesian ray) and emerge in the same direction. (The direction will not be the same as it is for the Cartesian ray, of course, but if the observer is at the correct angle, the two rays will be seen.) When the rays leave the laser, they are in phase, that is, the waves they represent are exactly in step In scattering from the droplet, however, the rays travel over different path lengths. As a result they emerge from the droplet with a phase relation that may be changed from the original one.

Suppose one of the rays travels half a wavelength farther than the other ray. (A proper calculation would include the change in the wavelength of the light when it is in the water.) The first ray will lag behind the second by half a wavelength once the two rays emerge. Since an observer at the correct position will see both rays simultaneously, the ray interfere destructively, meaning that the observer sees darkness because the rays (differing by half a wavelength) are exactly out of step.

Next consider the situation when the observer moves slightly toward larger scattering angles. Again two rays of light (different ones this time) can be seen emerging from the droplet. Suppose the extra distance traveled by one of them is a full wavelength. When two identical rays are initially in step, a shift of one of them by a full wavelength puts them in step again. The rays interfere constructively and the observer sees a bright band. If the color of the light is red, the observer sees a bright red band.

At larger scattering angles the bright and dark bands alternate. The first dark band, the one next to the main bright one, arises because two rays are out of phase by half a wavelength. The second dark band is caused by another two rays m that are out of phase by 3/2 wavelengths, the third and fourth dark bands are caused by rays that are out of phase by 5/2 and 7/2 wavelengths respectively and so on.

The rays that emerge in the brightest and widest of the bright bands are in phase. The next bright band in the pattern results from two rays that emerge out of phase by a full wavelength. The other bright bands in the pattern result from still other rays that are out of phase by an additional wavelength each time. If an observer stands in the interference pattern cast by the water droplet, he will always see either a bright band or a dark one because of two rays emerging from the droplet. If he looks .. directly at the droplet, he sees two spots of light where the two interfering rays emerge. As he moves toward the rainbow angle the two spots move closer together and finally merge to form the red of the rainbow.

If the droplet is illuminated with white light, each color produces an interference pattern and only the bright main bands of each color are seen. The first calculations of the intensity of the light in the patterns were made by George B. Airy in 1838. His results for four colors are given in the illustration above. Each wavelength generates an interference pattern with its main band close to the angle that the theories of Young and Descartes had predicted for the corresponding color of the rainbow.

Airy's calculations include the size of the droplet because that determines whether the rays emerging from it at an angle different from the rainbow angle result in brightness or darkness. The size of the droplet also determines the exact angle at which a rainbow appears. The smaller the droplet, the greater the discrepancy between the angles of the colors and the angles predicted by the older theories. As a droplet shrinks through evaporation the patterns for the various colors change in two ways. The patterns for the first two orders of rainbow shift away from each other. All the bright bands become broader. Neither change is predicted by the earlier theories, which did not describe the rainbow as being an interference pattern.

When I depress the plunger on my syringe, I make the suspended droplet larger. The resulting shift of the patterns is easily seen. Since the droplet is growing larger, the patterns of the first two orders of rainbow shift toward each other. I can also easily see that the widths of the individual bright bands decrease. If I allow the droplet to slowly shrink by evaporation, the opposite changes take place, as Foroughi observed.

The dependence of the patterns on the drop size has another result. When the drop is relatively large (several millimeters in diameter), the main bright bands in the separate patterns for each color are distinguishable. Each band for each color is quite narrow. With white light entering the drop the observer sees a rainbow at the rainbow angle. With a smaller droplet the main bands are wider and so overlap. Therefore the observer is less able to distinguish the colors and the rainbow appears to be washed out.

If the droplet is even smaller, say .3 millimeter or less, the main bands overlap so much that the rainbow is white. This is the white rainbow my grandmother asked me about. It appears rarely and then only in mists with droplets small enough to produce overlapping of the main bands at the rainbow angle. When I let a droplet suspended from the syringe evaporate, I can see the colors of the rainbow slowly wash out.

Theories of the rainbow more modern and accurate than those of Descartes, Young and Airy have been proposed. In 1908 Gustav Mie devised a scheme for computing the amplitudes of the light waves scattered by a drop. The computation promised more accurate results than the older theories, but it requires so many calculations that even today's high-speed computers cannot complete them. For drops several millimeters in diameter the older theories gave results that were good enough. The trouble appeared when one considered smaller droplets. A more workable approach, called the complex-angular-momentum theory of the rainbow, was discussed in this magazine three years ago [see "The Theory of the Rainbow," by H. Moyses Nussenzveig; SCIENTIFIC AMERICAN, April, 1977]. Other references on theories of the rainbow are listed in the bibliography for this issue.

The natural rainbow is strongly polarized parallel to its arc. You can test the polarization by rotating a polarizing filter between your eye and a rainbow. At one orientation the rainbow is quite bright; with a 90-degree rotation of the filter the rainbow almost disappears.

A similar observation can be made with the light scattered from a suspended water droplet. If the droplet is illuminated with light from a projector, the incident light is unpolarized, but it can be thought of as composed of two orthogonal senses of polarization. One of them lies in the plane defined by the projector, the observer and the cross section of the droplet traversed by the light ray. The other is perpendicular to that plane. The incident light polarized in the first sense is poorly scattered by the droplet and contributes little to the rainbow. The incident light polarized perpendicularly to the plane is primarily responsible for the rainbow.

The light from my laser is polarized vertically because of the arrangement of the Brewster windows inside the laser cavity, a common arrangement in gas lasers. The light is therefore polarized properly to cross through a horizontal cross section of a suspended droplet and create a bright rainbow. When I rotate the laser tube, the rainbow pattern I see becomes dim. When I turn the laser on its side, the beam is polarized parallel to the plane of the laser, the droplet and the observer. The sense of polarization entering the droplet is hence wrong, and the rainbow pattern is so weak that it is almost invisible.

Bibliography

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.

ANGULAR SCATTERING AND RAINBOW FORMATION IN PENDANT DROPS. Kenneth Sassen in Journal of the Optical Society of America, Vol. 69, No. 8, pages 1083-1089; August, 1979.

COMPLEX ANGULAR MOMENTUM THEORY OF THE RAINBOW AND THE GLORY. H. M. Nussenzveig in Journal of the Optical Society of America, Vol. 69, No. 8, pages 1068-1079; August, 1979.

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