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JOHN CONSTABLE, a meticulous painter of the English sky, was a great
champion of meteorological accuracy in painting. His enthusiasm was, if
anything, more evident in print than on canvas. In 1830, he offered some
observations that ‘can hardly fail to be useful to the Landscape Painter’.
With an air of patient reproach, he notes that the ‘morning and evening
Bows are more frequent than those at noon, and are far more imposing and
attractive from their loftiness and span; the colours are also more brilliant,
‘Flashing brief splendour through the clouds awhile’ ‘. For us, as for Constable,
the rainbow evokes both curiosity and aesthetic delight: few other sights
in the sky are as memorable as the rainbow’s vivid colours. Our eyes tell
us that the rainbow is a paragon of colour, so much so that ‘all the colours
of the rainbow’ has become a byword for variety and vividness. But analysis
of the spectrum that we see in a rainbow does not bear this out.

To describe rainbow colours we must refer to some real, but seldom considered,
idiosyncrasies of colour vision. For example, most of us are sure we know
what ‘white’ means. Our explanations may range from ‘white means that there
is no colour’, to ‘white must be an equal mixture of all colours’. Surprisingly,
neither statement is true. We can call many different kinds of light white,
and our convictions will change with both time and place. Our descriptions
of other colours can be equally fluid, but we can begin to make sense of
the elusive colours of the rainbow, and improve on the state of affairs
described by the physicist W. J. Humphreys in 1940: ‘The ‘explanations’
generally given of the rainbow may well be said to explain beautifully that
which does not occur, and to leave unexplained that which does.’

Basically, a rainbow is an image of the Sun distorted by falling raindrops.
To explain how a rainbow is formed, we need to look in some detail at the
way light passes through a raindrop. Imagine a ray of sunlight hitting the
centre of a raindrop, at right angles to its surface. Some of this light
goes straight through the middle of the drop and some of it is reflected
back on itself – in other words, it is deviated by an angle of 180Degree
from its original path. Other rays of sunlight parallel to this one enter
the drop nearer to its edge, and therefore at more glancing angles to its
surface. The rays are refracted (bent) as they enter the raindrop, then
are reflected from its rear surface and refracted again as they emerge into
air. But these rays are not scattered evenly at all angles between 0Degree
and 180Degree : they all fall within 42Degree of the ray that passes through
the middle of the drop – that is, they deviate from their original path
by at least 138Degree (see Figure 1).

The ray that emerges at an angle of 138Degree from its original path,
the minimum deviation ray, is the key to understanding how a rainbow is
formed. Because of the optical properties of water, rays that are reflected
once inside a spherical raindrop cannot deviate by less than this angle.
As a result, the minimum deviation ray has many neighbours leaving the drop
at nearly the same angle. It is this concentration of rays, all deflected
by 138Degree , that forms the rainbow.

But why should the bow be a circular arc? Imagine that you are standing
with the Sun behind you, looking up at a rainbow. You are looking along
the surface of an imaginary 42Degree cone: as raindrops fall, they lie for
an instant on the cone, and the rainbow that you see is a mosaic of minimum
deviation rays from all these drops (see Figure 2). Only drops on the cone
can send you the rainbow light. For the bow to last more than an instant
the drops must fall continuously.

And because the edge of a shower of rain can pass quickly across the
position where the rainbow might form, the bow can appear or disappear rapidly.
As long as you see sunlit drops at the correct angle, the rainbow will be
with you. However, if any part of the circle where the rainbow can form
lacks either drops or direct sunlight, then that part of the bow will not
exist. This accounts for the fragments of rainbows we often see.

Because of the fixed relationship between the rainbow and the Sun, we
cannot see a rainbow in a distant shower if the Sun is higher than 42Degree
above the horizon (assuming that we are on level ground). Conversely, as
Constable noted, we can see more of the arc of a rainbow when the Sun is
lower in the sky. Because the larger segments of a rainbow arc are more
visually impressive, people are more likely to notice rainbows that occur
late or early in the day.

The rainbow’s colours arise because the minimum deviation ray is at
a slightly different angle for each colour. At the minimum deviation, a
ray of blue light is bent through about 139Degrees, and red through 138Degrees,
so red will be on the outside of the rainbow, closest to the Sun, and blue
will be towards the inside. In between the two, there will be a continuous
spread of colours, depending on the minimum deviation of different wavelengths
of light. The familiar sequence, red, orange, yellow, green, blue, indigo,
violet, suggests the hierarchy. But not all of these may be clear in a particular
rainbow, and colours may vary along the bow itself.

We can explain this variability in a quite straightforward way by treating
light as a wave. Light waves interfere in a fashion similar to waves in
water. If you drop two stones into a pool, the expanding rings of waves
will intersect. Where the wave crests coincide, they reinforce each other
to produce a larger wave than either of the originals. Alternatively, if
a wave crest of one ring is combined with a trough from the other, they
cancel out and the water stays level. Although light waves are electromagnetic,
not mechanical, and oscillate much faster than water waves, the interference
analogy holds on the scale of small raindrops. Cancellation of light produces
darkness and reinforcement yields more intense light than in the original
source. An interference pattern of dark and light bands results, and the
separated colours in the first bright band form the main, or primary, rainbow.
But the interference pattern also forms extra, or supernumerary, rainbows
with pastel colours that appear within the primary bow. Interference also
produces the dark bands that separate the primary and supernumerary bows.
The spacing and width of interference bows depends on the size of the raindrops;
smaller drops yield broader bows. As a result, the range of sizes of drops
in a shower partly determines the intensity and vividness of rainbow colours.

This picture of raindrops concentrating light to form a rainbow shows
that a primary rainbow should be brighter than its background, but sometimes,
as in Figure 3a, this distinction is not clear. Several factors can cause
this: a background whose brightness is not uniform, clouds or haze that
partially obscure the Sun, or a thin band of rain. In each case, the dullness
of the primary bow compared with its background means that an observer notices
the rainbow’s pattern of colours, rather than its brightness. For the supernumerary
bows, even the coloured patterns will be invisible if the contrast between
them and their background is quite low. One consequence is that if any of
the bows are dim, then their colours will be pastel. But how much brighter
and purer are the rainbow colours in a bright bow, such as the one in Figure
3b, compared with a pastel one? How can we quantify ‘all the colours of
the rainbow’? Thus far, we have used terms like ‘brightness,’ ‘colour,’
and ‘vividness’ only qualitatively. Because researchers cannot yet describe
colour in a purely neurological, quantitative way, visual perception remains
intrinsically subjective. Instead, we assume that if we quantify the responses
of individuals to light of known physical properties, then we have objectively
measured their subjective colour experience. Not surprisingly, we can choose
among many different methods of measuring colour, collectively known as
colorimetry, to describe the rainbow. Although no single system is universally
accepted, one used widely in science and industry, developed by the Commission
Internationale de l’Eclairage (CIE), is especially useful since it relates
light spectra and colour perception mathematically.

Figure 4 illustrates a recent CIE colorimetric standard, the 1976 uniform
chromaticity scale diagram (UCS). Think of this as a colour wheel that has
been reshaped to accommodate the way people see colours. Pure colours lie
on the border of the diagram, and less pure, less vivid, colours are inside
it. The curved border represents the monochromatic colours, each made by
one wavelength of light. These are known as spectrum colours in colorimetry,
and range from the blues with short wavelengths at the bottom corner, through
greens to reds with long wavelengths at the upper right. A straight line
connects these two extremes, and mixtures of monochromatic red and blue
along it generate purples. Purples are not true spectrum colours, because
they are not monochromatic, but they are the purest possible colours that
link the ends of the visible spectrum. One point in the interior of this
diagram is a white, and is known as an achromatic stimulus. We can form
any other colour in the interior of the diagram by mixing a specific amount
of this achromatic stimulus with a spectrum colour (or a purple). Pick the
spectrum colour by drawing a straight line that connects the achromatic
stimulus, the colour of interest, and the curved edge of the diagram. The
resulting spectrum colour defines the dominant wavelength of the arbitrary
colour. Also, the position of the arbitrary colour, between white and its
spectrum colour, expressed as a fraction, gives a mathematical definition
of its purity. A spectrum colour has a purity of 100 per cent, while white’s
purity is 0 per cent. So this quantifies the descriptive terms ‘hue’ or
(‘colour’) and ‘vividness’ as dominant wavelength and purity, respectively.

The colour diagram sits within a rectangular coordinate system. The
coordinates u’ and v’ come from experiments in which people matched the
colours of test lights by mixing red, green and blue reference lights in
varying intensities. Roughly speaking, the u’ coordinate represents the
balance between green and red in a colour, and v’ indicates the relative
amount of blue. In order to convey a sense of what these numbers mean, we
have calculated and displayed the colours that fill the interior of the
UCS diagram (Figure 4). We produced the diagram without markedly varying
brightness, the remaining attribute of colour sensation, but the plot turned
out to show three bright spokes radiating from the white point, the achromatic
stimulus. The CIE quantifies brightness by psychophysical experiments similar
to those for colour matching (the CIE further distinguishes between the
brightness of objects that emit light and the relative lightness of objects
that reflect light). Ironically, the three bright spokes in Figure 4 nicely
illustrate a visual idiosyncrasy that the UCS diagram was not meant to display.
In fact, the spokes are artefacts of the way that our visual systems enhance
contrast. You can demonstrate that the bright lines are not real by covering
one side of a spoke with a piece of paper; the apparent contrast between
the spoke and its surroundings disappears. Far from being mere curiosities,
these illusory spokes hint that visually complicated images like the rainbow
may offer some surprises.

An important step in describing the colours of the rainbow is defining
the term ‘white’. We call sunlight white, even through it is not spectrally
uniform, that is, it does not contain equal amounts of all colours. As the
Sun’s elevation changes during the day, the spectrum of direct sunlight
also changes. If the Sun is more than, say, 30Degree above the horizon,
most of us would readily call the illumination white. As the Sun sets, its
spectrum is increasingly dominated by longer wavelengths and sunlight appears
redder. Yet most people would maintain that sunlight is white until the
Sun is only a few degrees above the horizon. This is called colour constancy.
We need to consider the colour constancy of sunlight as we analyse rainbows.

We analysed the rainbow colours of Figure 3 by digitising the original
slides and slides of a card containing many different samples of colours.
If the same spectrum of sunlight generates the rainbow and illuminates the
colour card, we can use the composition of colours on the card to quantify
the unknown colours of the rainbow. Figures 5 and 6 show some typical results.
In these Figures, we sample colours along a radius to the bow, and draw
a black line through the resulting chromaticities on a 1976 UCS diagram.
At the bottom of the Figures, we use a coloured bar graph to show how rainbow
brightness changes with radius.

The time and location of Figures 3a and 3b specify the elevation of
the Sun, which in turn lets us estimate its spectrum. We simulate colour
constancy in Figures 5 and 6 by making the achromatic stimulus, the perceived
white on the diagram, coincide with the spot that matches the colour of
the sunlight; we mark this site with an ‘x’. Because the Sun was lower in
Figure 3b than in Figure 3a, this point moves toward red in Figure 6. Although
the changing colour of the sunlight shifts the rainbow’s objectively measured
colours, people tend to see them as the same, at least until the Sun is
quite close to the horizon. However, colour constancy does not change the
fact that there are fewer distinct yellows, oranges, and reds near sunset
and sunrise.

This method of colour analysis springs some surprises. The rainbow chromaticities
in Figure 5 span so little of the UCS diagram that the bow scarcely exists,
yet we photographed this rainbow because it was easy to see, not because
it seemed especially pastel. On the brightness scale, the bow hardly differs
from its background, probably a result of hazy sunlight or a thin shower
of rain. Because the brightness of the rainbow is so low, mixing its colours
with the bluish background of clouds makes the bow a sequence of blues modulated
only slightly by red and yellow. The rainbow has purities lying between
14 per cent and 18 per cent, a range of only 4 per cent. Most of this colour
purity is due to the clouds; the bow’s colour contrast with its background
is quite small. Since bows like this are not uncommon, the rainbow begins
to seem a rather sorry colour standard.

The rainbow in Figure 3b looks like a better example of ‘all the colours
of the rainbow’. Its colours seem both bright and pure. Our analysis in
Figure 6 supports this hunch, but not as well as we might imagine. Certainly
this splendid rainbow is markedly brighter than its background, and its
spread of colours is appreciably larger than that in Figure 3a. Colour purities
vary between 7 per cent and 51 per cent, a range of 44 per cent, giving
a much larger spread than the other bow. Nonetheless, even this quite vivid
rainbow spans only a small fraction of the range of colours that we perceive,
and only the yellows of the bow are especially pure.

These results do not mean that we must discard the old saw about the
rainbow. Although the maximum purities in even the most spectacular rainbows
are well under 100 per cent, we need to remember how we defined this maximum
purity. Monochromatic light sources are all but absent from the environment
in which we have evolved. While we can see light of 100 per cent purity,
we almost never find it in nature. Of all the atmospheric colours, only
the Sun’s reddened disc exceeds the rainbow’s purity: the purity of the
best blue sky is only about 40 per cent. Among celestial displays, spectacular
rainbows provide the best range of reasonably pure hues. Although ‘all the
colours of the rainbow’ only hint at the range of our colour vision, they
do sometimes offer a splendid natural colour catalogue.

Raymond Lee is a postdoctoral researcher in the meteorology department
at Pennsylvania State University, looking at the colours of daylight, the
rainbow, and icebergs. Alistair Fraser is professor of meteorology at Pennsylvania
State University. He and Raymond Lee are writing a book about the science,
mythology and art of the rainbow for Penn State Press.

Further reading Rainbows, Halos and Glories by R. Greenler, Cambridge
University Press, 1980; Clouds in a Glass of Beer: Simple Experiments in
Atmospheric Physics by C. F. Bohren, John Wiley and Sons, 1987; The Nature
of Light and Colour in the Open Air by M. Minnaert, Dover Publications,
1954.