# Explaining the rainbow

A complete explanation of the rainbow, couched in the mathematics of the wave theory of light, is not for the fainthearted. Fortunately, it is possible to account for most of its principal features in terms of reflection and refraction of light, without invoking the wave nature of light.

People realised long ago that, since rainbows are always seen on the opposite side of the sky from the Sun, they are due to reflected sunlight. But if raindrops reflected light as efficiently as mirrors, rainbows would be blind-ingly bright. They would be amazing sights, though we wouldn't be able to look at them directly without endangering our eyesight. As it is, almost all the light that enters a raindrop passes straight through it because water is a transparent medium, so rainbows are rather faint. A rainbow is formed by the tiny amount of light reflected from the inside surface of each one of the multitude of drops that make up a rain shower.

Reflection alone can't explain why we see a circular arc of many colours when drops of water are illuminated. Refraction is also necessary. Refraction is the change in direction that occurs when light passes from one medium to another. Figure 5.15 shows the path of narrow bundles of light rays through a drop with a circular cross-section. Notice that two bundles of rays, in which rays are parallel to one another on entering the drop, diverge when they emerge from it. These diverging rays are responsible for the diffuse brightness that is sometimes noticeable within the arc of the primary bow. The rainbow itself is formed by a particular bundle of rays that emerge from the drop more or less parallel to one another. They make up what can be called the 'rainbow ray'.

The rainbow ray has a unique property: it is made up from rays that are deviated least from their original path. This angle of minimum deviation, as it is known, depends on the wavelength of light. For red light in water it is approximately 138°, and for violet light, 140°. In other words, the angle between light from the Sun and the rainbow ray responsible for the red band on the outside of the primary bow is about 42°, and for the inner violet band it is about 40°. The width of the bow is thus roughly 2°, about the width of your thumb held at arm's length, though natural bows are slightly broader than this for reasons given below.

The path of the rainbow ray represents a limit to the deviation, or change

No light emerges into this region from the drop

Figure 5.15 The rainbow ray. The diagram shows the path of some light rays through a drop with a circular cross-section. B is the rainbow ray. See text for a full explanation.

Light is reflected into this region from the drop --

No light emerges into this region from the drop

Figure 5.15 The rainbow ray. The diagram shows the path of some light rays through a drop with a circular cross-section. B is the rainbow ray. See text for a full explanation.

in direction, that light undergoes when reflected and refracted by a raindrop: rays can be deviated through larger angles, but they can't be deviated through a smaller one. Hence no light emerges from the drop into the region beyond the rainbow ray, which is why the sky between the primary and secondary bows sometimes appears distinctly darker than the rest of the sky.

More importantly, the absence of light in the region beyond the rainbow ray allows you to see the rainbow ray itself as a coloured fringe to the light reflected in your direction by each drop. Without the cut-off due to minimum deviation, a rainbow would not be a coloured arc. It would merely be a diffuse brightness around the antisolar point, rather like a heiligenschein, which most people would probably never notice.

You may find all this easier to understand if you carry out the following simple experiment. Fill a test tube with water - if you don't have a test tube a clear glass rod works just as well - stand with your back to the Sun, and hold the tube vertically at arm's length, within the shadow of your head. Keeping your arm outstretched, slowly swing it out from the shadow, ensuring that the Sun's reflection is always visible in the tube. At some point the Sun's image becomes strongly coloured: you are seeing the rainbow ray. Which colour do you see first: red or violet? Can you see more than one colour at a time? Through what angle have you moved your arm? See if you can identify the rainbow rays responsible for the primary and for the secondary arcs. Is the dark band between the primary and secondary arcs noticeable? If you have

Antisolar point

Primary bow

Supernumerary bows

Sunlight

### Sunlight

Figure 5.16 The primary arc. The primary arc of a rainbow is due to sunlight being reflected once within a raindrop. The outer edge of the arc is red, the inner edge blue or violet. Sometimes you will also notice a series of narrow coloured bands along the inner edge near the top of the arc. These are known as supernumerary bows.

tubes of different diameters you can also check if the brightness of the rainbow is due to size of the raindrops in which it is formed.

The most striking feature of a rainbow is its colours. These are due principally to refraction, which separates sunlight into its constituent colours as it passes though a raindrop. The resulting spectrum is not pure, and there are several reasons for this.

In the first place, sunlight is not perfectly parallel and so adjacent colours, because of refraction within a drop, overlap one another slightly. And rainbow rays of different colours diverge slightly on leaving a drop, which increases the amount of overlap. Colours in a pure spectrum, formed when a very narrow, parallel beam of sunlight is viewed through a glass prism, are well separated, and consequently are far more vivid than those in a bow.

Secondly, there are two mechanisms at work within a drop that cause colour: refraction and interference. Interference in drops with a diameter greater than about 1 mm is insignificant, and the dominant cause of colour in larger drops is refraction. But in very small drops the effect of interference between rays close to the rainbow ray becomes increasingly noticeable. This causes some colours, such as red and blue, to disappear and creates a succession of clearly separated bows, supernumerary bows (see section 5.4), just inside the primary arc. Depending on the range of drop sizes within a rain shower, the occurrence and relative intensity of various colours differ noticeably from one bow to another. Colours may also change during the lifetime of a particular bow if a large proportion of drops become smaller through evaporation.

Finally, drops are transparent, and so the colour and brightness of a bow is influenced by the colour and brightness of the background against which it is seen. Although the brightness of a rainbow increases with the depth of the shower in which it is formed, the shower itself also becomes brighter. And since the eye judges how bright a thing is by comparing it with its surroundings, an increase in the brightness of a shower makes a bow appear less bright than it really is.

Depending on the depth of the rain in which they are formed, the brightness of the primary and secondary bows, the space between them, and the space within the primary bow varies from one rainbow to another. In a shallow shower, the primary bow and the space it encloses will be faint, and the dark space hardly noticeable. The brightest bows are seen in light showers when the background is either a clear, blue sky or dull, uniformly grey clouds.

So much for what happens within a single drop. How does this explain the arc of a rainbow?

Have you ever have noticed, when looking at dew on a lawn early in the morning, that to see more than one colour in a particular dewdrop you have to move your head? The same is true of raindrops: your eye can only intercept a ray of a single colour from each drop. Other colours come from neighbouring drops. The bow itself is a mosaic of different colours that can be seen in its entirety only in a broad expanse of rain.

The bow's shape is due to the fact that the only rainbow rays that can enter your eye come from drops that all lie at the same angular distance from the antisolar point. This circle has an angular radius equal to the angle that rainbow rays emerging from each drop make with light from the Sun: 42° for red light, 40° for violet.

The resulting rainbow does not lie at a particular distance from you, though it appears to do so. It looks as if it's a flat arc, enclosing a region that is brighter than the surrounding sky, but it's actually three dimensional: a cone of light, fringed with colour, which stretches away from you into the rain. You can never see the cone itself because your eye is at its apex, and its base lies within the shower, so you are unaware of the rainbow's depth, of its three-dimensional nature. The arc is merely the most eye-catching aspect of this cone of light. We see it projected against the sky, which makes it appear flat.

Figure 5.17 Rainbow cone. Although a rainbow appears to be a semicircular arc it is really a cone of light that stretches away from the observer and into the shower of drops in which it is seen. The lower half of the cone is not seen because ground gets in the way: there are no drops below ground level, and no sunlight to illuminate them.

Figure 5.17 Rainbow cone. Although a rainbow appears to be a semicircular arc it is really a cone of light that stretches away from the observer and into the shower of drops in which it is seen. The lower half of the cone is not seen because ground gets in the way: there are no drops below ground level, and no sunlight to illuminate them.

The fact that the arc lies on the surface of this cone imposes a limit on the greatest distance at which you can see a complete rainbow in a shower. The apex of the bow is formed in drops that lie in the direction of the clouds from which they are falling. Hence it is the height of the clouds that limit how far away you can see a complete bow. For example, if the clouds are 300 m above the ground, and the Sun is 15° above the horizon, then the greatest distance at which you will see a complete bow is approximately 600 m. In this case, if you are more than 600 m from the rain, then you won't see the top of the rainbow, though you should see its ends.

The secondary bow is formed by two internal reflections of the rainbow ray. As with the primary bow, only a fraction of the beam is reflected each time. The amount of light remaining after two internal reflections is barely enough to be visible, which is why the secondary bow is so much fainter than the primary bow. Indeed, it is often not visible. At the same time, the colours in the secondary bow are in reverse order to those in the primary bow. Figure 5.18 shows that the secondary bow is due to drops that are further away from the ground than those in which the primary bow is formed. The angular radius of the secondary bow is approximately 50° and it is approximately 3.5° wide. The separation between the primary and secondary bows, i.e. Alexander's dark band, is about 9°.

Figure 5.18 The secondary arc. Sometimes a second arc is visible outside the primary one. The drops in which this secondary arc is seen are further from the antisolar point than those in which the primary arc is seen. The gap between the arcs is often noticeably darker than the sky outside the secondary arc and inside the primary arc. Notice that the order of colours seen in the secondary arc are the reverse of those seen in the primary arc.

Figure 5.18 The secondary arc. Sometimes a second arc is visible outside the primary one. The drops in which this secondary arc is seen are further from the antisolar point than those in which the primary arc is seen. The gap between the arcs is often noticeably darker than the sky outside the secondary arc and inside the primary arc. Notice that the order of colours seen in the secondary arc are the reverse of those seen in the primary arc.

Antisolar point

### Antisolar point

The principal weakness with the explanations that you have just read is that they are based on the passage of parallel beams of light through a idealised spherical raindrop. Although this explains the major features of the rainbow, there are many other features observed in natural rainbows that it cannot explain. The reason for this is that the theory does not take into account the shape of real raindrops, ignores the brightness and colour of background against which bows are actually seen, and assumes that the Sun is a point source of white light. Of these, the shape of the drop has the greatest effect on the appearance of the bow.

Most rain showers are composed of a range of drop sizes, from diameters as small as 0.2 mm to as large as 5 mm. Although the smallest drops are always spherical, larger ones become flattened as they fall, and so their vertical cross-section is not circular, though their horizontal one may be. Such a drop resembles a miniature bun, rounded on top and flattened underneath; raindrops are never pear-shaped. The rainbow ray that emerges from the vertical cross-section of a flattened drop undergoes greater deviation than the one that emerges from its horizontal cross-section. This should lead to a bow of varying curvature, with a vertical radius that is less than the horizontal one. However, smaller drops within the rain shower are not flattened to the same degree as the larger ones, and they give rise to a bow of constant curvature. The rainbow rays from the vertical sections of larger drops fall inside this bow and contribute to the general brightness inside the arc.

Distortion of raindrops also explains why the brightness of a rainbow is usually least along the apex of the arc. The proportion of drops that contribute rainbow rays is less there than it is closer to the ground, where drops of all sizes contribute rainbow rays to the vertical portions of the bow. Rainbow rays that form the vertical portion of a rainbow are more or less parallel to the ground. Since even the largest drops can have a circular horizontal cross-section, their rainbow rays can emerge in the same direction as those from smaller drops, giving rise to a brighter bow. Of course, this can be noticed only when a bow has a vertical portion, which occurs only when the Sun is close to the horizon.

The range of drop sizes within a rain shower makes it difficult to explain why we see supernumerary bows. These bows become more widely spaced, and are therefore more noticeable, as drop size decreases. In a shower composed of drops of many sizes, however, supernumerary bows due to one drop size should overlap those produced by other sizes. The net effect is that supernumerary bows should not be seen in most showers, they should be visible only when the drops are all of a similar size, for example in drizzle. Nevertheless, supernumerary bows are sometimes seen in heavy showers of rain, which are known to contain drops of different sizes.

(b) Secondary arc: refracted twice, reflected twice.

Figure 5.19 Diagrams (a), (b) and (c) show the path of rays through an idealised raindrop that has a circular cross-section. In practice, as raindrops grow larger, they become flattened as they fall to the ground. The rainbow ray undergoes greater deviation in the flattened raindrop (diagram e) than it does in the round raindrop (diagram d). Those parts of a rainbow formed in flattened drops therefore have a smaller diameter than the rainbow formed in spherical drops.

Figure 5.19 Diagrams (a), (b) and (c) show the path of rays through an idealised raindrop that has a circular cross-section. In practice, as raindrops grow larger, they become flattened as they fall to the ground. The rainbow ray undergoes greater deviation in the flattened raindrop (diagram e) than it does in the round raindrop (diagram d). Those parts of a rainbow formed in flattened drops therefore have a smaller diameter than the rainbow formed in spherical drops.

A possible explanation is that supernumerary bows due to smaller drops are noticeable because larger drops flatten slightly as they fall, and contribute their rainbow rays to the general brightness inside the primary bow. If this is the case, it also explains the fact that supernumerary bows are usually confined to the top of the bow. Lower down the arc supernumerary bows due to smaller drops in the vertical sections of a bow are overwhelmed by the light from larger drops.

The width of a natural rainbow is about half a degree greater than that predicted by theory because the Sun is not a point source and the rainbow ray is not precisely parallel light. Raindrops illuminated by a diverging beam not only produce a wider bow; the colours are less well separated, and therefore not spectrally pure, because they overlap slightly.