Great Calculations: A Surprising Look Behind 50 Scientific Inquiries
Page 19
Roemer's work combining observation and calculation was innovative and clever and marks an important advance in science. Calculation 31, light has a finite speed is worthy of a place on the list of important calculations.
9.2 ORIGIN OF THE RAINBOW
There is something particularly delightful about rainbows. The poet William Wordsworth surely spoke for us all when he wrote:
My heart leaps up when I behold
A rainbow in the sky:
So was it when my life began;
So it is now I am a man;
So be it when I shall grow old,
Or let me die!4
There is also something tantalizing and mysterious about rainbows, and they find a place in many myths and stories; for instance, the rainbow serpent plays a major part in Australian aboriginal creation stories. A rainbow is transient and shifting; you cannot go up to it and find a physical form. (Boyer gives a comprehensive history of the rainbow.)
Rainbows obviously are associated with rain, and Aristotle suggested that they are formed by light reflected from rain clouds. Over the centuries, it became apparent that a rainbow is somehow associated with the drops of water in rain itself. A theory was gradually developed to explain how rainbows occur, and it reached a modern form in the works of Descartes and Newton.
In a rainbow, we see circular arcs of colored bands in the sky as shown in figure 9.2, which is Newton's original drawing. The lower, primary bow is always seen, and as we scan it from lowest edge to highest (from E to F in Newton's drawing), it is made up of the familiar bands of color: violet, indigo, blue, green, yellow, orange, and red. The secondary bow appears higher in the sky; it is not as bright as the primary bow and not always easy to see. In the secondary bow, the order of colors is reversed, going from red to violet as we move up from G to H in Newton's figure. Over time, it was also realized that the position of the rainbow in the sky is given by precise rules: the angle between the sun's direction and the observed light from the rainbow is 42º for the primary bow (angle SEO in figure 9.2) and 51º for the secondary bow (angle SGO in figure 9.2).
Any theory for the rainbow must show its origins, explain why the colored bands occur in that particular order, and allow calculation of the characteristic angles of 42º and 51º. This provides an excellent example of how various elements of optical theory may be combined to give the framework for the calculations.
Figure 9.2. Newton's drawing of the rays of light giving rise to a rainbow. In practice, the upper, or secondary, bow is less bright than the primary bow and sometimes may not be evident at all. S indicates light rays from the sun, and O is the observer. From Isaac Newton, Opticks: Or, A Treatise of the Reflexions, Refractions, Inflexions and Colours of Light (1706).
9.2.1 Some Basic Optics
The essential features of the rainbow may be explained by using ray optics. The principal result concerns reflection and Snell's law for refraction. In keeping with Snell's law, if a ray strikes a planar interface between two media, some light is reflected and some light is transmitted, as shown in figure 9.3. The amounts of light reflected or transmitted depend on the index of refraction n for the materials forming the interface. (For air, the refractive index may be taken as 1, and it is around 1.33 for water.) Reflection follows the law that angles of incidence and reflection are equal. Using the angles shown in figure 9.3, the angles for refraction satisfy Snell's law:
The second piece of information needed concerns colors. Newton used his famous prism experiment (see part 1 of book 1 of his Opticks) to show that white light, like that coming from the sun, comprises many different colored components. Furthermore, the index of refraction for materials like glass and water varies a little as the color of the light involved is changed. (This is why the glass prism separates out the colors.) Thus the light coming from the sun S in figure 9.2 is somehow split into its colored components to form the rainbow seen by the observer O.
9.2.2 Rainbow Formation
We assume a spherical raindrop and the path of rainbow-forming light in it is shown in figure 9.4. Rays from the sun refract into the raindrop and reflect at the inner surface according to the rules introduced above (the sphere is assumed to be locally plane at the points of contact so those rules may be applied). The rays then leave the raindrop by another refraction event. Figure 9.4 (a) shows how light forming the primary bow reflects once inside the raindrop, while (b) shows the mechanisms for the secondary bow involving two reflections. Since there is some transmission also at a reflection point, the light is diminished in intensity at each reflection, and thus we find that with two reflections involved, the secondary bow is less intense than the primary. From figure 9.4 (a), we see that it is the angle γ that measures the observed angle between the sun's rays and those seen by the observer.
Figure 9.3. Reflection and refraction at the interface between two materials with refractive indices n1 and n2. The incidence angle is θ1, and the refracted angle is θ2. Figure created by Annabelle Boag.
The ray incident on the raindrop forms the angle α; then the refraction angle β, and the rainbow angle γ follow from the laws of optics and a little geometry as
Finding the appropriate angle α is a technical problem that involves first considering all possible rays and then identifying those that give the dominant contribution to the rainbow. This leads to the caustic ray (details may be readily found in the fine expository article by Casselman and in the paper by Nussenzveig). The result is that we must set
Using equation (9.2) leads to the internal angle β = 40.4º and then to a rainbow angle γ of 42.4º.
Thus it was that Descartes and Newton explained the mechanism behind the rainbow and calculated the observed angle of 42º for the primary bow. Using similar ideas and figure 9.4 (b) allows the rainbow angle for the secondary bow to be calculated as 51º matching that observed.
Figure 9.4. Paths of rays as they encounter a raindrop. (For clarity, only the relevant refracted or reflected rays are shown at the boundaries, and the other reflections or transmissions that can occur there are not shown.) The incident angle is α, and the reflection angle inside the drop is β. The rays in (a) go from the incident sunlight S to the observer O to form the primary bow, and (b) shows how the secondary bow is formed through two reflections. In (a), γ is the rainbow angle—the angle between the direction of the sun's rays and those going from the rainbow to the observer. Figure created by Annabelle Boag.
9.2.3 Rainbow Colors
You may say that is all very well and an impressive calculation of the rainbow angles, but where do the colors come in? Newton had discovered that the refractive index of materials varies according to the color of the light involved. For water, the refractive index decreases as the wavelength of light increases; n = 1.34055 for blue light of wavelength 450 nm, and n = 1.33257 for red light of wavelength 650 nm. Carrying out the calculations translates this information into a spread of rainbow angles γ, and so each color produces its own band in the rainbow. Furthermore, looking at figure 9.4 (b), it is seen that a more extensive path is followed by the rays giving the secondary bow, and this time the direction of the spread of rainbow angles is reversed, so matching the observed phenomena.
These results can be given in precise form for the size of the rainbows. This is how Newton himself reported it in his Opticks:
The breadth of the interior bow EOF [see figure 9.2] measured across the colours shall be 1 Degr. 45 Min. and the breadth of the exterior GOH [the secondary bow as in figure 9.2] shall be 3 Degr. 10 Min. and the distance between them GOF shall be 8 Degr. 15 Min.5
These calculations represent a triumph for the theory of optics, and calculation 32, seeing a rainbow, certainly deserves a place in my list of important calculations. The initial theory of the rainbow has been built on to explain other features such as the faint green and pink bands, known as supernumerary arcs, which may sometimes be seen beneath the primary bow. (The article by Nussenzveig is good place to start for further details.) There are a
lso similar phenomena, like glories and halos, that may be analyzed using rainbow optics.
9.2.4 A Different Response
A scientist reading about the work of Descartes and Newton might say “what a beautiful result,” but others see it in a quite different way. I began this chapter by mentioning the allure of the rainbow and its magical beauty. For some people, the seeming reduction of the rainbow to angles in ray optics and variations in the refractive index of water destroys the magic, the sense of awe, and the beauty. This was most famously expressed by the poet John Keats (1795–1821) in his poem “Lamia”:
Do not all charms fly
At the mere touch of cold philosophy?
There was an awful rainbow once in heaven:
We know her woof, her texture; she is given
In the dull catalogue of common things.
Philosophy will clip an Angel's wings,
Conquer all mysteries by rule and line,
Empty the haunted air, and gnomed mine—
Unweave a rainbow.6
The interaction of science and poetry is complex (interested readers might consult the book by Marjorie Hope Nicolson and the book edited by Heath-Stubbs and Salman). For example, John Donne (1571–1631) wrote:
And the new philosophy puts all in doubt,
The element of fire is quite put out;7
James Thomson (1700–1748) wrote a flattering poem entitled “To the Memory of Sir Isaac Newton,” which includes the lines:
Even now the setting sun and shifting clouds,
Seen, Greenwich, from thy lovely heights, declare
How just, how beauteous the refractive law.8
Personally, Wordsworth's words given at the start of this section, still resonate with me, but, equally, I feel a certain thrill when I see Newton's brilliant calculations.
9.2.5 Light Rays
You may have noticed that when I introduced the ray theory of light in section 9.2.1 I was talking about how light behaves and that was enough to give us the theory of the rainbow. I did not say anything about the nature of a ray of light. Here is how Isaac Newton begins his Opticks:
Definition I. By the Rays of Light I understand its least Parts, and those as well Successive in the same Lines, as Contemporary in several Lines…. The least Light or part of Light, which may be stopped alone without the rest of the Light, or propagated alone, which the rest of the Light doth not or suffers not, I call a Ray of Light.9
I will have more to say on this later.
9.3 WAVE THEORY AND LIMITS ON VISION
The theory of the rainbow provides a good example of how Newton could use his rays of light and spectrum of colors to explain a great variety of optical phenomena. His Opticks is a masterpiece of physics. However, for some optical effects, his explanations seem contorted, and some strange properties must be introduced for his particles, or corpuscles, of light and rays. At times, Newton is almost introducing elements of the main competitor for his theory of optics: wave theory as set out by Christiaan Huygens. Newton's contemporary, Christiaan Huygens (1629–1695) was a brilliant scientist; unfortunately, and perhaps unfairly, he is overshadowed by Newton. Huygens published his Treatise on Light in 1690, setting out a wave theory for light and explaining how light is propagated, reflected, and refracted. This work might be viewed either as a rival to Newton's ray theory or as complementary to it.
Huygens wrote, “I have shown in what manner one may conceive light to spread successively, by spherical waves, and how it is possible that this spreading is accomplished with as great a velocity as that which experiments and celestial observations demand.”10 His central idea is shown in figure 9.5. It shows how various parts of a candle flame emit spherical light waves to form the total candlelight. The second drawing shows how a point source at A emits light in a spherical wave and how Huygens describes light propagation. If there is a wavefront at some place in the spread of light away from a source, each point on that wavefront may be thought of as the source of a new spherical wave; and the waves from all those points combine to give the new wavefront of the propagating light wave. Thus the whole wavefront DCEF is generated by all the points d on the preceding wavefront L. This is known as Huygens's principle, and it is a central part of the theory of optics. If the wavefront is planar, the secondary sources on it give individual spherical waves that combine to form the next propagating plane wavefront. Thus using this principle, Huygens could describe the propagation of light in a great variety of circumstances.
Figure 9.5. Spreading of light waves as described by Huygens. From Christiaan Huygens, Treatise on Light (1690).
Using these ideas, Huygens showed how wave theory explains various phenomena in optical reflection and refraction. This included his chapter 5, “On the Strange Refractions of Iceland Crystal,” an optical effect we now know as double refraction.
Following Newton and Huygens, there was a vigorous investigation of optical phenomena and much debate about the appropriate theory for light (the book by Cantor is recommended). One of the leaders in these endeavors was Thomas Young (1773–1829). (Young was a polymath, and the aptly named book by Robinson gives a very readable account of his life.) In particular, Young investigated the interference and diffraction of light, two optical effects that were difficult to reconcile with Newton's particle theory of light. Young's famous double-slit interference experiment is a classic in optical science (see, for example, chapter 4 in Heavens and Ditchburn).
The proponents of the wave theory of light had the advantage as they could use ideas coming from the theory for sound and water waves. With these theories, it was very clear what was happening, and it was easy to see how waves could spread around obstacles and diffract from apertures. (There was a problem, of course: the media carrying those sound or water waves were well known, whereas if an underlying medium was sought for optical waves, it led back to unresolved questions about an ether, a medium that had to have very strange properties.)
Figure 9.6. Interference of water waves as depicted by Thomas Young. The sources are A and B, and Young pointed out that the interference at C, D, E, and F meant that the water was nearly smooth there. From Wikimedia Commons, user Sakurambo.
Young did not do extensive mathematical calculations, but he explained things using what we might call an analogue computer. Around 1802, Young set up a ripple tank which allowed water waves to be created and observed in a variety of configurations (see Robinson chapter 7). In one particularly valuable case, he demonstrated how waves from two sources could create an interference pattern, as shown in figure 9.6.
Gradually a whole body of experimental work was carried out to support the ideas propounded by Huygens, Young, and others supporting the wave theory. Huygens's principle could be used to understand many of these effects, and Young's analogue demonstrations proved to be very convincing. What was lacking was a solid mathematical foundation for the wave theory of light.
The mathematical theory was developed in the nineteenth century with Augustin-Jean Fresnel (1788–1827) playing a leading part. Essentially Fresnel gave a mathematical form to Huygens's principle. Suppose we have a light field E0 given in the xy-plane as shown in figure 9.7. Then an element in that plane contributes to the light E at observation point P an amount
This is just the mathematical form of Huygens's spherical wave contribution from an element of the initial wave. To find the total light wave at P will require integration over all the elements making up the initial light wave. An aperture has been indicated in figure 9.7, since that is the usual case in practice, and then the integration is carried out over that aperture.
Carrying out the integration is not so simple unless approximations are made appropriate to the physical or experimental arrangement being used. The cases known as Fraunhofer diffraction and Fresnel diffraction are commonly considered. (See Heavens and Ditchburn chapter 6, or Lipson and Lipson chapter 7.) The result is a beautiful set of patterns illustrating the consequences of light behaving like a wave.
Figure 9.7. Geometry for Fresnel's optical wave theory. An element in the source plane contributes to the light wave at point P, which is a distance R from that element. Figure created by Annabelle Boag.
9.3.1 The Airy Disc
The matching of theory and experiment for diffraction and interference effects was central to the proof that light behaves like a wave. However, from a practical point of view, it is the consequences of the wave-like nature of light for the operation of imaging systems (such as our eye, a telescope, or a microscope) that assumes a vital importance. All practical systems (like those shown in figure 9.8) will involve an aperture, and this will limit the extent of the incident wave that the lens delivers to the observation point P in the image. The effects of the aperture are exactly the kind of diffraction effects first examined by those nineteenth-century optical physicists mentioned above.
Figure 9.8. Effects of apertures in imaging systems. (a) The case for long-distance observation with a telescope giving an image in the focal plane. (b) How some waves from an object are lost as it is imaged at P. Figure created by Annabelle Boag.
Viewing stars through telescopes led the Astronomer Royal George Biddell Airy (1801–1892) to calculate the effect of the telescope aperture on the nature of the observed image. In his 1834 paper “On the Diffraction of an Object-Glass with Circular Aperture,” he writes that