Файл: Pye D. Polarised light in science and nature (IOP)(133s).pdf

Figure 2.7. One of the most easily improvised polariscopes for detecting polarisation of light, sometimes called Minnaert’s design. It can be made by adding a strip of Sellotape diagonally across a piece of polaroid at 45◦ to its direction of polarisation. The retardation is generally about half a wave (here about 300 nm) and gives a clear contrast in polarised light, except when the polarisation direction is exactly halfway. It is shown in two orientations over a background polariser.

different colour in its own mirror image. The mirror must be held at 45 ◦ to the direction of polarisation which, as reflected in the mirror, appears to run away at 90◦ to the original direction (figure 2.6). The crossed polarisers now appear to be light in the mirror and any colours produced by retardation films are changed into their complementary colours. A more formal demonstration of this is seen with a graded retardation wedge (taken from colour plate 4) and its reflection, as shown in colour plate 6. Of course the colours are not changed if the mirror is held after the second polariser, or indeed if the mirror is held parallel to or at right angles to the polarisation direction. Even more mystifying at first sight is the fact that a surface-silvered mirror or a polished metal reflector shows different colours from those shown in a standard back-silvered glass mirror. This phenomenon will be explained in chapter 8.

Retardation colours can be exploited in some interesting ways. In chapter 1 two simple polariscopes were described. Instead of producing a brightness contrast by two polarisers side by side, ‘Minneart’s polariscope’ achieves the same result by a single polariser with a diagonal strip of half-wave retarder film. An easily improvised example is a strip of sellotape placed at 45◦ across a single piece of polaroid (figure 2.7). The tape, on the side facing the light source, forms a retarder film with a retardation of around 300 nm, acting almost as a half-wave

plate and thus producing a strong contrast if the light is polarised. But a visual contrast of two colours is often thought to be more sensitive than a contrast of ‘grey’ intensities. So another alternative is to use two polarisers orientated at right angles and to cover them both (on the far side) with a retarder film of say 650 nm. Then polarised light will produce a blue colour alongside the complementary yellow (colour plate 7). Simply reversing the device makes the retarder film ineffective so the colours disappear and are replaced by grey contrasts (as seen earlier in figure 1.4). The user can easily compare each method and choose between them. Other colour pairs, say green and red, may be preferred and can be obtained by using different thickness of retarder film.

A quite magical result is obtained when polarisation colours are used in a kaleidoscope. Three mirrors fixed at 60 ◦ in the normal way produce a repeated pattern with sixfold symmetry. But instead of using coloured materials to produce the initial image, pieces of clear cellophane of random shape and thickness are jumbled together. Two polaroids, one on each side of the ‘specimen chamber’, then produce a variety of polarisation colours. When an attractive pattern is seen, rotating one polaroid changes all the colours without altering the pattern. Any gaps between the ‘coloured’ pieces simply change between light and dark, but if another retarder film is stretched across the whole chamber, these background holes themselves become coloured. Rotation of this film independently of the other elements modulates all the colours in the image, not just the background. A virtually infinite variety of images and colours can be obtained simply by rotating the appropriate supporting collars (colour plate 8).

I once imagined this was an original invention but then discovered that it had been patented in Beijing in 1985. The patent is probably invalid, however, because Sir David Brewster, the inventor of the kaleidoscope, described the method himself in 1858! His book on the kaleidoscope was published in 1819 and the second edition 39 years later had an additional chapter describing just how to use sheets of herapathite and/or a Nicol prism as polarisers and pieces of mica, selenite or other crystals as retarders (all described in chapter 3). He would surely have welcomed a gift of polaroids and cellophane films from the 20th century! Both his and the Beijing instruments placed the second or ‘analyser’ polariser at the eyepiece so that it can be small and consist of a Nicol prism, say. But this alters some of the colours that are seen after multiple reflection as explained earlier. It is better to place both polaroids in front

Figure 2.8. A simple U-shaped piece of perspex is normally invisible between crossed polarisers but when the arms are squeezed together gently, the internal strains so produced are clearly revealed due to their birefringence.

of the mirrors although this needs both of them to be as large as the specimen cell itself.

Some polymers such as polymethyl methacrylate (Perspex, Plexiglas etc) do not show birefringence in normal manufactured sheets. But if mechanical stresses are applied, then the internal strains in the material become birefringent and these areas can be seen as light–dark or coloured fringes if viewed between crossed polarisers (figure 2.8). This effect forms the basis for an industrial technology called photoelastic stress analysis. Any engineering component, from a simple lever or a gear wheel to a railway bridge or a cathedral arch, is first modelled in polymer resin such as methacrylate or epoxy. Then stresses are applied to simulate the loads to be expected in real situations and the distribution of internal strains can be analysed in polarised light (figure 2.9). This allows the design engineers to add strength where necessary and save material where possible. Two-dimensional or threedimensional examinations can be made. In one variant of this technique, some actual components (of steel, say) are coated with a layer of resin and the surface strains are then viewed by reflected light. While in some ways more realistic, this method cannot show internal strains within the material.

Many common objects are made from polymer resins by heat forming or other moulding techniques. In these cases the strains imposed during shaping are retained or ‘frozen in’ and are easily revealed by viewing between crossed polarisers. Examples abound in any domestic environment and some examples are shown in colour plates 9 and 10.

Figure 2.9. An epoxy resin model of part of a large electrical generator viewed in a professional polariscope. The coloured fringes show the strains induced by simulated centrifugal force. (By courtesy of Ken Sharples, Sharples Stress Engineering Ltd, Preston.)

Figure 2.10. Two pieces of worked glass viewed between crossed polarisers. One was allowed to cool immediately and its internal strains show as photelastic fringes; the other was kept overnight in an annealing oven at 565◦C (not quite hot enough to soften the glass) which allowed the strains to dissipate, as shown by the absence of fringes. (Made and kindly loaned by John Cowley, Glass Workshop, Queen Mary, University of London.)

Birefringence also occurs when glass is strained and becomes permanent if the glass is cooled too rapidly after being worked. Such strains make for fragility, so glassblowers often examine their finished work between crossed polarisers and put it in annealing ovens until the strains are relieved. In the example shown in figure 2.10, one specimen was left

overnight in an oven at 565 ◦C, which eliminated all the strains that are still evident years later in the other piece, which had been cooled rather quickly.

Some car windscreens show darkened or coloured patterns when seen through polaroid sunglasses. These screens have been toughened by heat treatment followed by deliberately rapid cooling; the resultant permanent strains ensure that under impact the glass shatters into relatively harmless small granules rather than breaking into sharp shards. The strained regions, however, are birefringent and show up under a variety of circumstances if the driver wears polaroid sunglasses: for example when the incident light is polarised by reflection, say by a wet road (see chapter 7) or comes from the blue sky (see chapter 6). Even light that is not polarised will be partly reflected from the glass and this has a polarising action (see chapter 7), causing the transmitted light to be partly polarised. The patterns may even be seen without polaroid glasses if the windscreen is itself seen by reflection in another window or in the car’s paintwork. Many windscreens are strengthened by being laminated instead of being heat toughened and do not show these effects on polarised light. Laminated screens are therefore preferable if the driver likes to wear polaroid sunglasses.

An extreme example of stressed glass is shown by Prince Rupert’s drops, so named because they were demonstrated to Charles II in 1661 by Prince Rupert of Bavaria. They consist of molten glass, about 1 cm in diameter, that has been dropped into cold water and so cooled very rapidly. Glass shrinks as it solidifies, so after the outer part of each drop has hardened very quickly, the inner parts cannot shrink as they should and a central space, assumed to be a vacuum, is left. The internal strains are so high that coloured polarisation fringes are very close together (colour plate 11). Although the heads of these glass drops are extremely robust, a slight scratch on the long ‘tail’ causes the whole object to disintegrate explosively into tiny fragments. They should therefore be treated with great care.

Chapter 3

Crystals

Crystals act on light in some fascinating ways and show many important influences on polarisation. Indeed the early studies of polarised light depended entirely on crystals and they have continued to be of fundamental significance. Crystals can affect light in several different ways and the result is often quite complex, although the basis is quite straightforward.

Crystals consist of a three-dimensional lattice of atoms or ions, all held together with extreme regularity. For instance common salt, sodium chloride, has equal numbers of sodium and chlorine atoms in a perfect cubic arrangement (figure 3.1). A crystal of pure sodium chloride is itself cubic and is formed of a single lattice of such cubic cells, each with an edge length of 0.562 nm. This spacing is fairly typical of crystal lattices which are all around one-thousandth of the wavelength of visible light. Since the sodium chloride crystal lattice is exactly the same in each direction, light travels through it at the same speed in each direction—the crystal is said to be isotropic and it has only one refractive index or speed of light. But other crystal lattices do not have the same structure in different directions: light travels each way at different speeds, so they are said to be anisotropic and birefringent, having two main values of refractive index, a maximum and a minimum. Just as with birefringent polymer films (chapter 2), an anisotropic crystal divides polarised light into two components vibrating at right angles and with different velocities of propagation. One component has the same velocity in all directions but the other has a velocity that varies with direction, either greater or less than the other, depending on the crystal. But, in essence, the long rows of very regularly arranged atoms in such a

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Crystals 21

Figure 3.1. The lattice structure of a simple cubic crystal of sodium chloride. Positive sodium ions (charged atoms—dark) and negative chloride ions (charged chlorine atoms—pale) are held by electrical forces to form a regular cubical pattern with a repeat distance, as shown by the arrow, of 0.562 nm (one-thousandth of the wavelength of yellow–green light). On an enormously greater scale, such a lattice forms a crystal that is itself cubic.

crystal can act on light just like the long, parallel molecules of polymers.

Ice is crystalline in structure and a spectacular demonstration is produced by rapidly freezing a shallow dish of water (by pouring liquid nitrogen onto it) between crossed polaroids on an overhead projector. Initially the liquid water is isotropic but as the ice crystals grow, they are birefringent and show up in brilliant colours, each according to its own orientation until they all meet within the solid mass. The example shown in colour plate 12 was frozen more slowly in a freezer compartment. Another nice example is salol (phenyl salicylate) which is very strongly birefringent. Crystals can easily be melted (at 43◦) on a glass plate and another warm plate is then pressed onto the melt. As the sandwich cools, the salol recrystallises in a very thin layer that shows splendid colours between crossed polarisers (colour plate 13). As explained in chapter 2, thicker birefringent crystals show no retardation colours because many wavelengths across the spectrum are rotated and all the wavelengths in between are not. So the crystal looks clear between polarisers, whether crossed or parallel, provided that it is properly orientated (figure 3.2).

Rotating thick crystals extinguishes the light every 90 ◦, as happens with

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Figure 3.2. Quite large quartz crystals between crossed polaroids. As the crystals are turned they become transparent four times for each rotation as they turn the direction of polarisation; at the intermediate points they can only be seen by reflected background light. No colours are seen, however, unless the crystals are very small (the same material is seen on a microscopic scale in colour plate 16).

a half-wave plate, but the explanation is rather different.

The term ‘dichroism’ originally referred to crystals that simply looked to be different colours (or clear) when viewed along different axes; indeed the word literally means ‘two coloured’. But the effect is often much clearer when different directions of polarisation are used in viewing the crystals. In some cases. one component of the light is absorbed (the crystal is more or less opaque to it) whereas it may be quite transparent to the other component. A good example of a crystal of this kind is tourmaline. As shown in figure 3.3, two thin pieces of tourmaline act just as polaroid film: they are fairly transparent to green light (the colour varies between specimens) but when they are crossed, the combination is quite opaque. Slices of unflawed tourmaline crystals were often used as polarising components in optical instruments (figure 3.11) as they were cheaper than Nicol prisms (described later) although they were generally of an inferior optical quality and their selfcolours were sometimes undesirable. Another well-known example of a naturally occurring dichroic crystal is epidote. The artificial crystals of herapathite (iodo-quinine sulphate) were described in chapter 1 as they were a component of early kinds of polaroid.

In other crystals, one component may have only some of its wavelengths absorbed so that it emerges coloured, while the other

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Figure 3.3. Tourmaline is a dichroic crystal: left, slices of green tourmaline one parallel to and one crossed with a background polaroid; centre, the same two pieces, both turned by 90◦; right, the same two slices of tourmaline crossed with each other in normal light, with no other polariser.

component is clear. An example is sapphire which is deep blue for one direction of polarisation and clear for the other; since the eye cannot distinguish the different polarisations, they are seen mixed together and the effect is a paler blue. Obviously, any polariser allows one to distinguish immediately between a real sapphire and an isotropic crystal or blue glass. The same relation applies to the red colour of rubies. Alternatively, in some crystals the two components may both be coloured, but of different hues. A good example is copper acetate which is a bluish-green in colour, but when viewed through a polariser the colour changes from deep, royal blue to clear light green as the polariser is turned (this difference is shown in colour plate 14).

Even more variety is added by crystals that are different along three axes rather than two—a property called ‘trichroism’ or ‘pleochroism’ (literally ‘more colours’), associated with three significant values of refractive index in different directions. These crystals are sometimes different in colour when simply viewed along each axis by unpolarised light (tourmaline sometimes shows this property). But again such crystals may also show quite different responses to polarised light in the different directions. Each self-colour may turn out to have two components under a rotated polariser or the transmitted light may be

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absorbed for one direction of polarisation. These effects may be different for each axis of the crystal. In other words each axis may be dichroic in a different way from another axis.

Even when there is no dichroism, most crystals (all except cubic ones) show some degree of birefringence and thus affect polarised light. One of the most birefringent of natural crystals is calcite or Iceland spar (calcium carbonate) whose birefringence was described by the Danish scientist Erasmus Bartolinus in 1670, in what seems to be the first ever observation of an effect due to the polarisation of light. This material is one of the major constituents of the earth’s crust, usually in microcrystalline form in marble, limestone, chalk or coral but sometimes as large, clear, rhombic crystals which are found in Iceland (‘Iceland spar’) and Mexico. The degree of birefringence is expressed by the difference between the two refractive indices, which for calcite are 1.486 and 1.658, giving a large difference of 0.172. Sodium nitrate (‘Chile saltpetre’) has an even larger birefringence of 0.251 but it is much less convenient to experiment with as it readily dissolves in water and so can easily be disfigured by handling (for this reason it only occurs naturally in very dry conditions as in Chilean deserts). In both these cases the two refractive indices are so different that the two refracted rays can be seen to diverge very markedly. A calcite crystal placed over a dot or other mark on a piece of paper shows two images (figure 3.4) and a sodium nitrate crystal does the same (figure 3.5). If the crystal is rotated, one image stays still while the other one moves in a circle around it. It was this observation, first made by Bartolinus in 1670, that eventually led to the discovery of polarisation and, in turn, contributed greatly to our basic understanding of the nature of light itself. We now know that the ‘ordinary ray’, which gives the stationary image, has a lower velocity within the crystal than the ‘extraordinary ray’ that gives the moving image. In some crystals it is the other way round—the ordinary ray is faster.

Viewing the double image through a polariser shows that the two images are polarised at right angles to each other because turning the polariser brightens one image and extinguishes the other in turn. If the dot on the paper is replaced by a small hole in a black card, a polariser can be placed over the hole itself and again the two images seen through the crystal can be extinguished in turn, as shown in figure 3.5 with a crystal of sodium nitrate. A calcite crystal combined with a lens to view the double image of the hole (figure 3.6) makes a kind of polariscope called a dichroscope that is used by jewellers. Any dichroic

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Figure 3.4. Left: a calcite crystal over a single typed word on a sheet of paper, showing a clear double image. This spectacularly large, clear crystal belongs to the Royal Institution, whose help is gratefully acknowledged. Right: a smaller calcite crystal over a small hole in a black card. In both cases, rotating a polariser over or under the crystal would extinguish each image in turn, showing that they are polarised at right angles to each other.

Figure 3.5. Left: a small sodium nitrate crystal over a regular array of holes in a black background. Each hole seen through the crystal creates a double image. Centre: the same seen through a sheet of polaroid that suppresses half the images. Right: the same again but with the polaroid turned by 90◦ to suppress the other set of images instead. Clearly the two sets of images are polarised at right angles to each other.

material placed in front of the hole can give different effects side by side in each image; thus a sapphire gives one blue and one clear image simultaneously, while copper acetate gives one blue image alongside a green image (colour plate 14). Materials with only one refractive index, such as glass, cannot produce such differences in colour between the two images of a single hole.

The impression is often given that calcite crystals also produce a double image of distant objects, but simply looking through the crystal

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Figure 3.6. A calcite crystal can be made into a simple dichroscope that can be used to reveal dichroism in other specimens. A single small hole, seen through a calcite crystal and a small lens, produces two images that are polarised at right angles to each other. An improvised instrument was conveniently housed in a black plastic film can, and to get a neater edge the hole was drilled in a sheet of metal that was then fixed over a larger hole on the end of the case.

does not work. If one watches through a calcite crystal as a dot or hole is steadily moved further away, the spacing between the images appears to diminish with distance just as if they really are a double structure. The reason is that the two polarised rays diverge within the crystal but when they emerge again they become parallel, although separated by a little over 1 mm for each 10 mm of crystal thickness. So a spacing of, say, 2 mm from a fairly large crystal is easily seen when close to the crystal but it becomes insignificant at a distance of more than 1 m and distant objects just do not look double. But if one looks down into the crystal so that the image is seen after it has been reflected in the intermediate face (figure 3.7), then two images can be seen, each polarised at right angles to the other and at 45◦ to the plane of the incident and reflected rays within the crystal. This is because the two emerging rays end up diverging by about 20◦ and produce two well-separated and oppositely polarised images. Figure 3.8 shows how this can be demonstrated and figure 3.9 shows the result.

But this technique is rather inconvenient—for one thing the two images overlap extensively and for another the field of view is very restricted because either the rays entering the crystal or the emerging rays (or both) are close to glancing angles to the respective crystal faces. To make a more practical use of the birefringence of calcite, special ‘double image prisms’ have been invented. In one design, a 60 ◦ prism is cut from a calcite crystal and, instead of producing a spectrum of rainbow colours from each point of the image, it produces two spectra in different directions, each polarised at right angles to the other. A glass prism can then recombine the colours of each spectrum to produce two

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Figure 3.7. A ray of light entering a calcite crystal so that it is reflected at the next face produces a double image because birefringence results in two divergent beams and after refraction they continue to diverge. By looking down into the third face, one can see the two images and simple tests with a piece of polaroid show they are polarised at right angles to each other and at 45◦ to the plane of the diagram (see figure 3.9). As with most figures in this chapter, the angles are not all depicted accurately but have been altered to help clarity.

Figure 3.8. The setup used to photograph the polarised images produced by a calcite crystal as explained in figure 3.7. A light box, consisting of a backlit translucent screen, had two square polaroids mounted on it. Their oblique directions of polarisation were set at right angles as shown by marks in the top corners and by the small central area of overlap.

virtually uncoloured images, polarised at right angles to each other and well separated in space.

In 1828 William Nicol realised that one of the divergent beams within an ordinary calcite crystal could be eliminated by using the principle of total internal reflection. He cut across a crystal at a carefully calculated angle and cemented the two halves together again with Canada balsam (figure 3.10). If the angle of the cut is just right, one beam is reflected out through the side of the crystal while the other one proceeds, giving a complete separation of the two polarised components. This design was capable of several modifications (notably one by Sylvanus P Thompson) which together formed the best polarising