Forces of Nature

by Professor Brian Cox

  As far back as the eleventh century, Ibn al-Haytham searched for a physical explanation for rainbows. He correctly surmised that they are caused by light from the Sun interacting with water in the atmosphere before entering the eye – although he was incorrect in that he thought the rainbow was caused by reflections off clouds, which he believed behaved as giant concaved mirrors. The suggestion that the rainbow is reflected sunlight is not a trivial observation. The theory that vision is active in the sense that the eye generates the light that allows the viewer to perceive objects, rather like a radar system, was widespread in the eleventh century, and had the historical authority of Euclid and Ptolemy to support it. Ibn al-Haytham had little time for the authority of the ancients, however, and placed great emphasis on experimentation and observation rather than pure thought and instinct. This approach, which we now recognise as distinctly modern, is one of the reasons why he is regarded by many historians of science as one of the great early scientific minds. As the historian David C. Lindberg writes, he was ‘undoubtedly the most significant figure in the history of optics between antiquity and the seventeenth century.’

  Alhazen (the Latinised version of his name) is not as well known as Newton, Galileo, Kepler or Einstein, but I think he deserves a much more prominent place in the history of science because of the self-awareness and humility that is evident in his writings; essential components of the modern scientific enterprise which echo Sagan’s thoughts on the Pale Blue Dot. All good research scientists understand that no position is unassailable; there are no absolute truths in science; authority counts for nothing when contradicted by Nature; nullius in verba. Here is Alhazen, writing in Basra a thousand years ago:

  ‘Therefore, the seeker after the truth is not one who studies the writings of the ancients and, following his natural disposition, puts his trust in them, but rather the one who suspects his faith in them and questions what he gathers from them, the one who submits to argument and demonstration, and not to the sayings of a human being whose nature is fraught with all kinds of imperfection and deficiency. Thus the duty of the man who investigates the writings of scientists, if learning the truth is his goal, is to make himself an enemy of all that he reads, and, applying his mind to the core and margins of its content, attack it from every side. He should also suspect himself as he performs his critical examination of it, so that he may avoid falling into either prejudice or leniency.’

  Alhazen’s greatest surviving work (half his writings have been lost), The Book of Optics, was the inspiration for many of the subsequent investigations into the origin of the rainbow and the nature of light. There is no irony here; books are to be read critically. They are not sources of ‘truth’, but of inspiration; snapshots of knowledge and experience which should be read with a critical eye. It is a measure of the power of the written word that Alhazen’s book inspired generations of scientists from cultures widely separated in space and time to seek to improve on his work, as he implored them to do, and to find a rational and experimentally testable explanation for the rainbow.

  It is a measure of the power of the written word that Alhazen’s book inspired generations of scientists from cultures widely separated in space and time.

  Kamal al-Din al-Farisi was one of a long line of pioneering scientists who created a vibrant academic culture throughout Persia during the late medieval period. Born in 1265, al-Farisi completed his studies under the tutelage of astronomer Qutb al-Din al-Shirazi at the celebrated Maragheh Observatory near Maragheh, Iran. Al-Farisi became interested in the refraction of light – the bending of light rays when they pass from air into water or glass. Al-Shirazi told Al-Farisi to read The Book of Optics, and he became so engrossed in it that Al-Shirazi encouraged him to write an updated review of its contents. The result was a complete revision of the work, and a step towards a correct explanation for the formation of rainbows. Al-Farisi suggested that a rainbow is formed by light entering water droplets from the air, being refracted twice – once on entering and once on leaving the drop – and undergoing at least one reflection from the back surface. Following Alhazen’s eloquent entreaties, he conducted a series of experiments to test his theoretical approach; a beautiful early example of the controlled exploration of nature under laboratory conditions. Al-Farisi created a model of a rain-laden atmosphere using large spherical glass vessels filled with water. He placed his glass raindrops into the equivalent of a camera obscura, a dark room with a controlled aperture through which to introduce a beam of sunlight, and flat surface on which to project an image. He observed a rainbow, verifying the broad outline of his theory.

  Alhazen’s Book of Optics included the first accurate description of atmospheric refraction and reflection from curved surfaces, leading to him being considered the inspiration for investigations into the origins of rainbows.

  At virtually the same time, but widely separated geographically, the German monk and scholar Theodoric of Freiberg arrived at the same conclusion, documented in De iride et radialibus impressionibus – ‘On the rainbow and the impressions created by irradiance’. Just like al-Farisi, Freiberg used glass spheres filled with water to model raindrops and explored the interaction between sunlight with water. Despite being thousands of miles apart and with no contact or communication, it is not a coincidence that these two early scientists arrived at the same conclusions almost simultaneously. Both were inspired by and built upon The Book of Optics, which was translated from Arabic into Latin in the twelfth century and disseminated around Europe as well as Persia – an early example of an essential principle that we still fight for today; scientific knowledge must be freely available through open publication. There must be no copyright on ideas.

  The anatomy of the eye, as set out by Alhazen in his Book of Optics.

  These were great steps forward, but neither scientist discovered the correct explanation for the origin of the rainbow’s colours or their most striking geometrical property; the universal angle of the circular arc.

  Kamal al-Din al-Farisi’s beautiful manuscript explaining the mathematical explanation of the formation of a rainbow.

  René Descartes was the first to explain the geometry of the rainbow in a 1637 essay entitled L’arc en ciel. Perhaps unsurprisingly from the father of Cartesian geometry, his method was geometrical. L’arc en ciel contains a well-known and beautiful diagram, shown opposite, which marks out all the angles and lines associated with the formation of a rainbow. Here is how Descartes described the diagram.

  ‘I found that if the sunlight came, for example, from the part of the sky which is marked AFZ and my eye was at the point E, when I put the globe in position BCD, its part D appeared all red, and much more brilliant than the rest of it; and that whether I approached it or receded from it, or put it on my right or my left, or even turned it round about my head, provided that the line DE always made an angle of about forty-two degrees with the line EM, which we are to think of as drawn from the centre of the sun to the eye, the part D appeared always similarly red; but that as soon as I made this angle DEM even a little larger, the red colour disappeared; and if I made the angle a little smaller, the colour did not disappear all at once, but divided itself first as if into two parts, less brilliant, and in which I could see yellow, blue, and other colours ... When I examined more particularly, in the globe BCD, what it was which made the part D appear red, I found that it was the rays of the sun which, coming from A to B, bend on entering the water at the point B, and to pass to C, where they are reflected to D, and bending there again as they pass out of the water, proceed to the point.’

  ‘The Steadfast rainbow in the fast-moving, fast hurrying hail-mist. What a congregation of images and feelings, of fantastic Permanence amidst the rapid change of a Tempest – quietness the Daughter of Storm.’

  – Samuel Taylor Coleridge

  Descartes was able to explain the arc, but not the origin of the colours, because he did not know that the white light from the Sun is made up of all the colours of the rainbo
w. Isaac Newton made this discovery forty years later. Our purpose here is to understand the physics of the rainbow, and it is easier to explain both the geometry and the appearance of the colours at once rather than letting the story unfold chronologically, so this is what we’ll do.

  René Descartes’ famous geometrical drawing from L’arc en ciel, showing light from the Sun entering raindrops and being deflected into the eye of an observer, forming the arc of a rainbow.

  The top illustration (below) shows a ray of light from the Sun entering a raindrop, reflecting off the back surface and entering the eye of the observer of the rainbow. The angle through which the ray is deflected relative to the incoming ray is labelled D(θ). In Descartes’ diagram, this is the angle between the lines AB and ED. If you’re mathematically inclined, have a look at the calculation in the caption. If you don’t fancy that, see the graphical representation of the angle D(θ) in the second illustration (middle, left), because this is the key point. The graph shows how the angle of the light that bounces out of a raindrop changes as the angle of the light from the Sun entering the surface of the raindrop changes.

  A ray of light from the Sun enters a raindrop at angle θ. It is refracted into angle ϕ according to Snell’s law, where sin θ / sin ϕ = nf,water / nf,air, and nf,water and nf,air are the refractive indices of water and air respectively. The refractive index is dependent on the colour of the light, which is the reason for the labels f. If you look at the diagram for quite a long time, you should be able to convince yourself that Df(θ) = (θ–ϕ) + (1800 – 2ϕ) + (θ−ϕ). If you want to plot Df(θ) against θ for yourself, then substitute for ϕ using Snell’s law. The refractive index nf,air is approximately equal to 1 for all colours, and nf,water = 1.33 for red light and 1.34 for violet light. The outgoing angle Df(θ) as a function of the angle of light entering the raindrop θ is shown in the illustration below.

  The important thing is that the angle of the outgoing light ray D(θ) has a minimum value of approximately 138 degrees. Immediately, this should ring a bell; 180 – 138 = 42 degrees, the angle Descartes came up with using geometrical methods and observation, and labelled DEM in his diagram.

  To understand why this special minimum angle corresponds to a bright arc in the sky, have a look at the bottom illustration (left). This shows what happens to a whole bundle of rays from the Sun hitting a raindrop across a large section of its curved surface. Visually, you can see that most of the incoming rays come out at the ‘special’ minimum angle, even though many of them hit the raindrop at different incoming angles relative to the surface. This means that the incoming rays are preferentially focused around the minimum angle of 42 degrees, and this is why the rainbow arc across the sky is brighter than anything else. It’s a focusing effect caused by the spherical geometry of raindrops. In slightly more mathematical language, rays that enter the drops over a wide range of angles clustered around 60 degrees will all emerge at a very similar angle, because D(θ) doesn’t change very rapidly close to its minimum value. The special outgoing ray, corresponding to minimum defection and drawn in red on the diagram, is known as the ‘caustic’ or ‘rainbow’ ray.

  The deflection of a ray of light entering a raindrop from the Sun at angle θ.

  This explains why we see a bright arc, but not the value of the angle – 138 degrees – or the spread of colours. The value of the angle depends on how much a ray of light is refracted when it enters and leaves the raindrop; the relationship between angles θ and ϕ in the top illustration. This depends on the properties of air and water, and there is a very simple relationship between the two angles known as Snell’s law, or occasionally the Snell-Descartes law. The law itself has been known to varying degrees of accuracy since classical times, and almost appears in Alhazen’s Book of Optics, although it is not clearly stated. The history is convoluted, but the law is simple. We’ll state it, and if you don’t know any trigonometry then ignore it and skip to the next sentence: sin θ / sin ϕ = nf,water / nf,air, where nf,water and nf,air are the refractive indices of water and air respectively. The refractive index is a number that describes how light travels through a particular substance. To be specific, it describes the ratio of the speed of light in the substance to the speed of light in a vacuum. For water, the refractive index is approximately 1.3, which means light travels around 1.3 times faster in a vacuum than it does in water. For air, the refractive index is very close to 1. This is the number that sets the angle of the rainbow ray, and you can follow the calculation for yourself. In one sentence, the angle of the rainbow is 42 degrees because raindrops are made of water.

  A cluster of parallel rays from the Sun enter the spherical raindrop at different angles relative to the curving surface of the raindrop. On leaving the raindrop, they tend to cluster around the ‘caustic’ or ‘rainbow’ ray, shown in red, which emerges at an angle of approximately 138 degrees relative to the incoming rays from the Sun. The precise angle of the rainbow ray is dependent on the colour of the incoming light, and it is this that is responsible for the colours of the rainbow.

  On Saturn’s moon, Titan, there are raindrops of liquid methane floating delicately downwards like snowflakes in the dense atmosphere. The Sun is weak on this distant world, and the atmosphere often dominated by a thick haze, so rainbows are rare, but they will still be present if viewing conditions are right. The refractive index of liquid methane is 1.29, which leads to rainbows larger than those on Earth with an angle of 49 degrees. That’s quite a big difference for a small change in the refractive index, and this is the reason for the colours of the rainbow. Even in the same substance, the refractive index of light is different for different colours. In water, the refractive index of red light is around 1.33. Blue light has a refractive index of around 1.34. When refracted through Earth’s water raindrops, the rainbow made by red light is slightly larger than that made by blue light. This is why the outer ring of the rainbow is red and the inner ring is blue, with all the other colours in between. Water droplets split the white light of the Sun into the individual colours that make it up because red light travels at a slightly different speed through water than blue.

  The investigation of the rainbow is a beautiful example of the central theme of this book: simple questions about the origin of everyday things often ... lead us down tangled paths through the dense, interconnected undergrowth of physics.

  With this explanation for the origin of the rainbow, we reach the level of understanding achieved by Isaac Newton in the mid 1680s. It’s not the end of the story, though – not by a long shot. There are many subtle features of rainbows that require a more advanced treatment. Double rainbows occur because there can be two reflections inside the raindrops as well as one. There can be faint arcs above and below the primary rainbow called supernumerary arcs, an interference effect first explained by Thomas Young in 1804 by treating light as a wave. The Astronomer Royal of the time, George Biddell Airy, produced a complete theory of the rainbow in 1831, which triggered mathematical research on what is known as the Airy Integral by two of the great mathematicians of the day, Augustus De Morgan and George Gabriel Stokes. This leads to more questions. If light is to be treated as a wave, what’s doing the waving? Nobody knew in 1831; the answer was discovered in the early 1860s by the Scottish theoretical physicist James Clerk Maxwell – we’ll get to this in more detail later – and as we’ve already seen in Chapter Two, Albert Einstein was the first to take Maxwell’s theory of light at face value, and this lead him to jettison Newtonian physics and construct a new theory of space and time. We could go on, and we will, at least down some of the paths that are opening up.

  On Saturn’s moon, Titan, there are raindrops of liquid methane floating delicately downwards like snowflakes in the dense atmosphere. The Sun is weak on this distant world, and the atmosphere often dominated by a thick haze, so rainbows are rare, but they will still be present if the viewing conditions are right.

  Let us pause for a moment, though, and reflect that the investigation of th
e rainbow is a beautiful example of the central theme of this book: simple questions about the origin of everyday things often – more often than not – lead us down tangled paths through the dense, interconnected undergrowth of physics. This shouldn’t be a surprise, although it is only human to be constantly surprised at the deep interconnectedness of nature; I would say this is one of the great joys of physics. It shouldn’t be a surprise because we’ve established, or at least asserted with some supporting examples, that the complex world we perceive is a shadow of simpler forms: the underlying laws of nature.

  Thomas Young, who discovered the undulatory (wave) theory of light and explained the presence of the faint arcs above and below a primary rainbow.

  If this is the case, it must follow that there are common explanations for many of the shadows, and investigating one will inexorably lead us to touch on the deep underpinnings of another. This is why the modern trend for directing scientific research into areas deemed a priori economically or socially useful is not only misguided but positively harmful to the scientific enterprise, and therefore to the goal of the government advisors who dream up such daft, albeit (to be charitable) well-intentioned policies. Serendipity always was and always will be absolutely central to discovery, because the natural world is so intricately interconnected and functions according to a small set of fundamental laws, as far as we know. There are so many ways to discover deep and ultimately useful things that it is futile to imagine that we can predict which investigation of which tiny corner of the natural world will bear undreamt-of fruit. Nature is too complicated. Investigating rainbows might seem whimsical, but it stimulated a great deal of the early research into optics and the nature of light, and ultimately into the nature of space and time.