The Rainbow

Contents

Observing the Rainbow

The colors of the rainbow are a familiar sight to everyone, raising spirits and exciting feelings of well-being. Although often seen, the rainbow is seldom observed. A closer acquaintance increases the delights of the experience, as the full variety and extent of the display is recognized and appreciated.

The rainbow can be observed naturally in the rain falling from a thundershower, or artificially in the spray from a hose. In either case the sun must be shining (though the rainbow would be seen in artificial light as well) and not too high in the sky. Rainbows can be seen in the west in the morning, and in the east in the afternoon. Because thunderstorms are mostly an afternoon happening, rainbows are usually seen in the east under a storm that has drifted eastwards on the west wind. Thunderstorms are associated with rainbows because they are limited in area, and the sun shines between them. With continuous frontal rain, the sun is usually not visible, so there is no rainbow. Rainbows are made much easier to observe because they are opposite the sun, and the eye does not have to look into the glare.

The variables influencing the rainbow are the altitude of the sun, and the size of the raindrops in which the rainbow is seen. The rainbow is a cone making an angle of 42° with the antisolar direction, which is normally beneath the horizon for an observer on the surface. When the sun is at an elevation of 42°, only a little arc at the top is visible at the horizon. As the sun sinks, the arc of the rainbow rises in the sky, until at sunset it forms a semicircle. The eye interprets the light arriving on the cone as a circle somewhere in space, though the rainbow is really only a direction. If the observer is at a slight elevation, some of the rainbow can be seen beneath the horizon, in front of not too distant objects, which gives the impression of a physical existence not too far away. The story of the "pot of gold" at the end of the rainbow takes advantage of this illusion to send fools on a quest. The rainbow just keeps receding as they try to approach it. We should realize that the rainbow is a cooperative effect of many raindrops, with considerable depth, and that our visual impression is the best the visual sense can do with its data. When you see a rainbow, try to estimate its apparent distance from you, or its apparent size.

The figure at the right is a guide to the principal rainbow phenomena. The primary bow at 42° is only a part of the whole, but the brightest part. Impure spectral colors vary from red on the outside to blue (violet) on the inside, provided the incident light is white. The actual colors depend on the size of the raindrops and the constitution of the incident light. Outside the primary bow is the secondary bow at 51°, with the order of colors reversed. This bow is fainter than the primary bow, but is nearly always there with close observation. Between the two bows is a dark annular region, Alexander's Dark Space, where scattered light from the raindrops is a minimum. Inside the primary bow, and outside the secondary bow, are supernumerary bows of quite variable appearance and color. Usually several are visible inside the primary bow, colored alternately pink and green. The supernumerary bows outside the secondary bow are often too faint to be seen.

The colors in the rainbow are impure spectral colors, and depend on the size of the drop. A normal primary rainbow from large raindrops (greater than 1 mm diameter) contains easily distinguishable bands of green, yellow, orange and red, with very little blue, although violet does appear. There are almost always numerous supernumerary bows directly adjoining the violet. The red disappears for smaller drops, and is gone for 0.2 mm drops. Even smaller drops give a pale, broad bow with prominent violet, and separated supernumerary bows. Whiteness appears for drops less than 0.1 mm in diameter. The mist-bow occurs for drops less than 50 μm in diameter, typical cloud droplets.

If you examine the rainbow through a Nicol prism or a good Polaroid screen, such as a sunglass lens, the light of the primary bow will be found to be almost completely polarized. The direction of polarization is radial, from the antisolar direction. The secondary rainbow is also highly polarized, but not quite as much as the primary bow. The polarization shows conclusively that the rainbow is a result of refraction.

Descarte's Rainbow

Although it was known that the rainbow had something to do with clouds and rain, Theodoric of Saxony seems the first in modern times to suggest around 1300 that the rainbow was caused by the refraction of light in raindrops. At the time, nothing useful was known about refraction or the origin of color in light, so a detailed explanation was not possible. About 300 years later, Antonio de Dominis recalled Theodoric's suggestion, and so it came to the notice of René Descartes (1596-1650), who at the time was working in the Low Countries to escape the ignorant obscurantism of the Church. Willebrord Snell (1591-1626) had come across the true law of refraction in 1621. It was remarkable that this law had not been found in antiquity, and waited so long to be discovered. Descartes put Theodoric and Willebrord together, and in 1637 published his calculation of light ray paths in raindrops, in one of his Essais Philosophiques. Since Snell was dead, he saw no reason to give him credit for the refraction formula, since it might be thought Descartes' own work, as indeed it is in France.

Descartes gave a proof of the law of refraction, which, like practically all of his physical explanations, was completely erroneous. He asserted that light travelled faster in dense media, and was a kind of rotary pressure. It remained for Huygens and Fermat to give correct and satisfying explanations at a later date. The law of refraction is n' sin i = n" sin r, where i and r are the angles that the incident and refracted rays, lying in the same plane, make with the normal to the surface. The light travels from a medium in which the index of refraction is n' to a medium in which it is n". The index of refraction is c/v, where c is the velocity of light in a vacuum, and v its velocity in the medium. This formula can also be written sin i = n sin r, where n = n"/n' is the relative index of refraction. For water against air, n is about 4/3 for orange light, increasing slightly at shorter (bluer) wavelengths and decreasing slightly for longer (redder) wavelengths. More precisely, at 656.3 nm n = 1.3318, and at 404.7 nm, n = 1.3435. A Cauchy formula, n = 1.3246 + 3092/λ2, where λ is in nm, gives n to a reasonable accuracy in the visible range.

The deviation of a light ray on refraction at the surface of the drop is (i - r), while the deviation at an internal reflection is 180° - 2r. The rainbow is formed by rays that have been refracted on entry and exit, and reflected once at the back of the drop, so their total deflection D = 180° - 4r + 2i. Actually, most of the intensity passes through when the ray is refracted at the back of the drop, but we are interested in the part that is reflected, so these rays are not shown in the diagram. Easy calculations of D as a function of i give the results in the plot at the left. It is clear that there is a minimum value of D corresponding to the rainbow angle, with no rays deviated into Alexander's dark space. The rays with larger values of D appear within the primary bow, in the region of the supernumerary bows. For each value of D in this region, there are two possible angles of incidence, so two rays pass through every point. The raindrop creates two waves in this region, which we shall examine in detail below.

For the moment, we simply notice that if the incident light is uniformly spread over the cross-section of the drop, the scattered light will be concentrated at the angle of miniumum deviation, since light entering an annular area around the corresponding angle of incidence will all exit at this angle. This was Descarte's explanation of the rainbow, and for once it had some truth to it. The ray corresponding to minimum deviation is rightly called Descartes' Ray in his honor. The corresponding angle of incidence is 59.39°, close to the approximate value 60°, which is easier to remember.

The secondary bow has exactly the same explanation, except that the rays causing it have undergone two internal reflections, so that D = 2i - 6r, dropping the additive 360°. The two reflections account for its weakness relative to the primary bow. A larger number of internal reflections also correspond to rainbows of still smaller intensity, and the next couple are in the direction of the sun, not opposite to it, and are not visible in the glare. Therefore, we only have two rainbows, both opposite to the sun. For an index n, the angle of incidence for rainbow rays (minimum deviation rays) making k internal reflections is given by cos i = [(n2 - 1)/(2k + k2)]1/2.

The side view in the diagram at the left will help to clarify angular relations. The rays from the sun are parallel, and strike raindrops at various locations, as shown. Two raindrops are shown in the sky, one in a position to show part of the primary bow to the observer at O, the other showing part of the secondary bow. For scattered light to enter Alexander's dark space, the deviation would have to be less than the minimum, which it cannot be. A raindrop, or a dew drop, is shown below the observer. From an aircraft, a rainbow that is a complete circle can sometimes be seen, when there is rain both above and below the observer. This circular rainbow is easier to arrange with a garden hose. There is a color photo of a rainbow on the index page of the Physics section of this website. Supernumerary bows are just visible, and the contrast with Alexander's dark space is clear. The secondary bow shows on the original slide, but is difficult to make out on the image.

If you close your eyes alternately when viewing a rainbow that is nearby, as in the spray from a garden hose, you will note that there is one rainbow for each eye. When both eyes are open, you will get binocular fusion which should put the rainbow at an infinite distance. Rainbow light comes from a direction, not from a point in space.

One drop is shown on the ground. Dew drops often cover the ground in the morning. Where a dew drop is on the rainbow cone, rainbow colors will be sent to the eye. The result is called the dew bow, another delightful rainbow phenomenon. So far as visual stimulus is concerned, this is no different from raindrops in the sky, but the visual effect is quite different. With the dew bow, we perceive the bow to be in the plane of the ground, and its shape is the intersection of the cone and the horizontal plane of the ground, which is a hyperbola. Exactly the same stimulus coming from the sky is interpreted as a circle. This shows that the visual sense does the best it can to interpret the information it receives to reflect the actual surroundings, and does not simply repeat what it receives. Everything in our visual picture is a result of interpretation; it is not "reality" directly perceived.

Fresnel's formulas for the reflection and transmission coefficents can be expected to be accurate for the rainbow, since the drops are sufficiently large and are quite clean. These coefficients depend on the polarization, which is referred to a plane containing the eye, the antisolar point, and the drop. Parallel polarization has the E vector in this plane, while perpendicular polarization has the E vector normal to the plane. For reflection of parallel polarization, the amplitude ratio is sin(i - r)/sin(i + r). For perpendicular polarization the ratio of tan(i - r)/tan(i + r). The squares of these expressions give the intensity ratios. The intensity transmission coefficients are unity less the reflection ratios.

In the primary rainbow, the reflection coefficient is 0.115 for parallel polarization, and 0.0043 for perpendicular. When multiplied by the transmission coefficient for entry and exit, the intensity in the primary bow is 0.0897 for parallel, 0.0043 for perpendicular, polarization, a ratio of 21 to 1. The primary rainbow is strongly polarized parallel.

In the secondary rainbow, the reflection coefficients for one reflection are 0.23 and 0.055 for the two polarizations, so for two reflections the coefficients will be 0.053 and 0.003. Multiplying by the transmission coefficients, the intensities are 0.0313 and 0.0027. The secondary rainbow is also strongly polarized, but the ratio is now about 12 to 1.

For comparing the intensities of the primary and secondary rainbows, we must add the polarizations and multiply by the approximate pupil areas. The pupil areas will be taken proportional to the circumferences at the corresponding heights of incidence, which are 0.669 for the primary, and 0.777 for the secondary, bows. The final result is 0.0629 for the primary bow, and 0.0264 for the secondary. This makes the primary bow 2.4 times brighter than the secondary bow.

Airy's Rainbow

Descartes' explanation of the rainbow is attractive, but is very incomplete and does not give a good description of what is observed. The supernumerary bows are completely unexplained, as are the width of the rainbow and its colors. However, the explanation was attractive enough that its results were considered good enough that actual observations were not required, since the truth was already known. Newton's discovery of dispersion even made an explanation of the colors possible. As more and more became known about light, however, modification of the Descartes theory became mandatory.

The wave theory of light contributed polarization and Fresnel's equations for reflection and transmission. It is easy to combine these results with Descartes' rays, and explain the polarization of the rainbow. However, this was a somewhat later development. Explanation of the supernumerary bows was more pressing. Thomas Young, discoverer of interference, suggested that they were the result of the interference of the two waves indicated by ray tracing. Descartes' ray was a boundary between regions in which there was light, and in which there was no light. The wave surfaces, which are normal to the rays, had a cusp along this ray, where the two wavefronts on one side met in phase, and gave increased intensity over a conical surface generated by the Descartes' ray as the diagram is rotated about the axis of symmetry. A surface with this property is called a caustic (alluding to the increased intensity). Absolutely sharp terminations of waves cannot occur; there is always some spillover into shadows. Therefore, the intensity in the region of a caustic is worth further attention.

For an analogy, one might look at diffraction by a straight edge. Ideally, the incident radiation would be cut off sharply at the edge. In fact, beyond the edge there are two waves, the incident wave sharply terminated, and a cylindrical wave originating at the edge, and radiating both into the shadow and the light. These waves interfere to produce the typical fringes in the light region, and an exponentially decreasing wave as the shadow zone is entered. This description gives the same result as the Huygens picture of the wave front acting as sources of new waves as in the Fresnel diffraction theory. We expect that the intensity near a caustic will resemble that near a straight edge.

This problem was studied by George Biddell Airy (1801-1892), Astronomer Royal, and the results were published in 1849. The analysis is approximate, but it applies very well to the larger raindrops that make the common rainbow. For smaller raindrops, a more exact theory is necessary, and important advances have been made quite recently. However, Airy's theory gives a good account of actual rainbow phenomena, and is well worth knowing. The analysis is applicable to other matters than the rainbow, such as the WKB approximation in quantum mechanics.

Airy's procedure was to start from a wavefront that would give rise to the caustic upon further propagation. The later amplitude could be found by summing the infinitesimal amplitudes over the initial wavefront, as in Fraunhofer's treatment of diffraction, which was valid since the field would be observed at a great distance, while the initial wavefront was just outside the drop. To make the integrations possible, Airy considered a two-dimensional problem instead of the actual annular form of the initial wavefront. This approximation is only valid for large drops.

A cross-section of the initial wavefront is shown at the right. Its assumed form, a cubic parabola, is the lowest-order possible curve that has a vertical tangent at the origin O and crosses the vertical axis with a change in curvature reflecting the inclination of the rays on both sides of the condition of minimum deviation. We will not need the value of the dimensionless constant k in our analysis. We consider parallel rays at an angle θ to the axis, which can be considered to have a very slight convergence to some distant point, as in Fraunhofer diffraction, equivalent to focusing the rays at a finite point with the aid of a lens. The ray from O is the reference ray. A parallel ray from another point, say P, has a path difference of OS = OR - QT = x sin θ - y cos θ = x sin θ - (kx3/a2) cos θ.

This expression is simplified by introducing new variables u and z such that (2k/λa2)x3 cos θ = u3/2, and (2/λ)x sin θ = zu/2. Cubing the second expression and dividing it by the first, we find z3 = (2a2/kλ2) sin3θ/cos θ. Since θ is small in the region of interest, z is proportional to θ. Therefore, OS = (λ/4)(u3 - zu).

The path difference OS corresponds to a phase difference δ = (2π/λ)OS, which is then put into the integrand. Integration over the whole wavefront from x = -∞ to x = +∞ gives an amplitude proportional to ∫ cos δ dx, or to f(z) = ∫ cos[(π/2)(u3 - zu)du. The constants in the expression for the amplitude in terms of f(z) can be worked out, but we shall not need them. The intensity near the caustic cone will then be proportional to f2(z).

f(z) is proportional to the tabulated function Ai(-x) defined in Abramowitz and Stegun. The function Ai(x) is called the Airy function, which can be expressed in terms of Bessel functions of order 1/3. However, the integral is usually the most convenient definition. Ai(x) is defined so that it decreases exponentially for x > 0, and oscillates for x < 0. In our f(z), z is positive in the region of the supernumerary bows, within the caustic cone, and negative outside. This is just a matter of definition, but should be recognized. Using the integral representation 10.4.32 of A&S, I find that f(z) = 1.8739Ai(-0.93693z). Now table 10.11 can be used to evaluate f(z). For z = 0.6, I get f2(0.6) = 0.836, as given in Tricker, p. 179. I leave it to the reader to plot f2(z) as a function of z, as is done in Tricker. At z = 0, along Descartes's ray, f2(0) = 0.443. This is equivalent to the edge of the obstacle in the shadow of a straight edge. A sketch of the Airy function is shown in WKB Approximation.

In Descartes' theory, the increase in intensity at the caustic was only a result of the slow change of the deviation there. Now we have a much better expression for this intensity, and find that it is more complicated. The first maximum of f2 is responsible for the colors of the primary rainbow, since its position varies with the direction of Descartes' ray. The additional oscillations are responsible for the supernumerary bows. It is now possible to work out the perceived colors by CIE methods, and the results agree with observation. The colors change with the size of the drop.

Raindrops are generally larger than 0.5 mm in diameter, and range upwards to several millimetres. Very large raindrops are distorted by aerodynamic forces into an oblate shape that affects the rainbow, sometimes eliminating the secondary bow. Normal-sized drops are within the range of validity of Airy's theory, so the rainbow in rain is pretty well explained by it. For smaller drops, discrepancies are found that are certainly to be expected. Unfortunately, Airy's theory cannot be extended to them, and a new start must be made. The theoretical analysis is very difficult, and will not be treated here. For drops smaller than 0.2 mm, such as are found in fogs and clouds, there is still a bow, but it is nearly white since the prominent oscillations of the Airy theory are replaced by a steadier variation of the amplitude. The fog bow can be observed when the sun shines on a thin fog, and it is a beautiful display. A bow is not seen when the sun falls on a cloud, for the very good reason that its brightness makes very little change to the white brightness of the cloud behind. The whiteness of clouds is due to multiple scattering, making them quite bright in direct sunlight. Where a cloud is dark, the sun is not shining on it, and there will, of course, not be a bow. The eye responds to ratios of brightness, and background + bow is very little greater than background alone.

Moonlight can create a rainbow, if the moon is low and bright. However, the intensity is low, and colors cannot usually be distinguished. The red light from a low sun makes a rainbow that only contains red. Rainbows from the very small drops of drizzle may make a bow that is mostly bluish, on the way to the fogbow.

The Reflected Rainbow

The rainbow can be seen reflected in calm water. Since the rainbow is not a physical object, there are differences between its reflection and that of an object. Ray paths are shown in the diagram at the right. The observer is at O, a height h above the water surface. A horizontal line through O points to the observer's horizon. The light from the upper drop shown reaches O directly, but the light from the lower drop is reflected at the water surface. We imagine that the drops give light from the uppermost point of the rainbow, a, and the lowermost point, a', of its reflection. From the laws of reflection, the angles marked z are equal, so a' is seen as far below the horizon as a is above it. The diagram shows how the two arcs appear. They are exactly similar, and have a common chord at the horizon. This assumes that there are water droplets to scatter the necessary light. In nature, only parts of the arcs may be visible.

The small ellipse represents, say, a cloud at a height x above the water. Since it is a real object, its image in the water will be an equal distance below the water surface, as indicated. The cloud will be seen directly behind the actual rainbow, but will be seen below the reflected rainbow. This shows the difference between the reflection of a rainbow and an actual object.

In the preceding example, the light was reflected after scattering by the raindrop. Things are quite different if the light from the sun is reflected before scattering by raindrops. Clearly, the antisolar point for the rainbow produced by the reflected light will be as far above the horizon as the original antisolar point was below, which is the altitude of the sun as an angle. The new rainbow will surround the new antisolar point as the old rainbow surrounds the actual antisolar point. This is shown in the diagram at the left. R is the normal rainbow around the antisolar point A, R' is the rainbow in reflected sunlight around the reflected antisolar point A', and R" is the reflected rainbow. Note that R' and R" make a complete circle. As usual, only parts of the arcs may be visible depending on the location of raindrops. Drawing a ray diagram for this case is left as an exercise for the reader. If the observer is unaware of reflecting surfaces, this case can produce some unexpected surprises, with rainbow arcs appearing where they should not be.

Rainbow Experiments

Both Minnaert and Tricker recommend experiments with Florence flasks filled with water, which imitate water drops very well. A laser beam can be scanned across the flask, and the position of the scattered beams observed. A flask can be illuminated by beam of sunlight coming through a circular aperture in a large screen, and the backscattered light observed on the screen in the form of a circular rainbow (Minnaert, p. 175). The effects of various screens and masks on the rainbow can be seen. The reflections of a small lamp in the flask of water can be observed as the observer moves (Tricker, p. 45). These are easy and instructive experiments that will clarify the ray paths in a sphere of water.

Wood recommends viewing a small water sphere held at the end of a greased capillary tube. It is held close to the eye, but so that sunlight or arclight can reach it from behind you, and he says that all the principal phenomena are visible, and can be studied as a function of the size of the drop. Details are given in the Reference.

An entertaining experiment is shown at the right. All you need is a small Florence flask filled with water, a piece of cardboard, compasses, scale, calipers and an X-acto knife. My 125 ml flask was 32.5 mm in radius (measured with the calipers), but the drawing shows the distances for any radius. It is a good idea to trace the Descartes' ray for yourself, and verify the diagram. First, draw a circle on the cardboard where the Descartes' ray will fall. With the same centre, draw a circle 0.92a in radius (30 mm in my case) and cut it out carefully with the X-acto knife. The cardboard is easy to hold in place on the flask, but could be tacked to it with household cement or Blu-tack.

Hold the flask and cardboard in the sun, making the plane of the cardboard perpendicular to the sun's rays. This is easily done by watching the rainbow and making it circular. The colored ring of the rainbow should fall pretty much on the circle you first drew, red on the outside and blue on the inside. Scattered light will be noticed inside the ring of the rainbow, but no light outside in Alexander's dark space. The secondary bow will not be seen, because its ray would enter at a height of 0.95a, and so is masked off by the cardboard. If you take the trouble to make the radius of the aperture in the cardboard a little larger than a, and arrange for its support, then the secondary bow will appear.

A more ambitious experiment, in which both primary and secondary bows can be observed, is illustrated at the left. The flask is supported on a ringstand, and the screen, with an aperture of greater diameter than the flask, is fixed at one side. A mirror reflects the sunlight so that it is horizontal.

The method of finding the radii of the bows on the screen is shown at the right. The 42° rainbow leaves the axis at point P beyond the sphere, while the 51° rainbow departs from point P' in front of the sphere. Note that when the screen is close to the flask, the 51° rainbow falls within the 42° rainbow. When the screen is 58 cm from the centreline of a 125 ml flask, both rainbows have a radius of 41.6 cm. When the screen is farther away, the secondary bow finally falls outside the primary bow, and they are separated by Alexander's dark space. As the screen distance increases, the bows become fainter due to geometrical spreading, but both should be visible in sunlight.

These are excellent experiments, quite impressive in the classroom, if you have some way to bring a beam of sunlight into the room. A real heliostat is a wonderful advantage, for all kinds of experiments, not just rainbows. Sunspots and white-light diffraction are two other possibilities.

References

R. A. R. Tricker, Introduction to Meteorological Optics (London: Mills and Boon, 1970). Chapters III and VI.

M. Minnaert, The Nature of Light and Colour in the Open Air (New York: Dover, 1954). pp. 167-190.

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Washington, DC: U.S.G.P.O., 1964). pp. 446-452, Table 10.11, pp. 475-478.

H. M. Nussenzweig, The Theory of the Rainbow, Scientific American, April 1977.

R. W. Wood, Physical Optics, 3rd ed. (New York: Dover, 1967). Chapter XI.

Composed by J. B. Calvert

Created 31 July 2003

Last revised 14 August 2003