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

Chapter 8

Going circular

So far only linearly polarised light has been considered. It occurs very commonly in both science and nature but it is actually a special case of a more general system which also includes circular and elliptical polarisation. Examples of these are much rarer but are of considerable interest anyway. In most cases the circular polarisation is derived from linearly polarised light which is itself very common. The theory may seem a little abstruse at first, but it is worth following in order to understand the larger picture.

The explanation of ‘changing direction’ given in chapter 2 was rather oversimplified because it was deliberately limited to the case of retardation by half a wave (the half-wave plate). Although thicker and thinner retarders were then discussed briefly, the consideration of vectors in such cases was felt to be an unnecessary distraction at that point. The half-wave plate is simple to understand because the two emergent vectors always recombine to produce another linear vector and, as a result, linear polarisation is preserved on emergence, though the direction of polarisation may be changed. However, if a retardation of only a quarter of a wave occurs, the emerging components combine to form light in which the vector that represents the direction of electrical oscillation actually rotates through one complete revolution during each wave. This is called circular polarisation. It has no single ‘direction’ of polarisation but the vector can rotate either clockwise or anticlockwise (as seen from behind—i.e. from the source) and is called right circular or left circular polarisation respectively.

If all this seems a bit bizarre (and such a wave is certainly difficult to imagine at first), consider the view of a helical spring shown in

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Figure 8.1. A coiled wire spring suspended from a straight wire and photographed from above (lower image) and also seen simultaneously from the side in a mirror inclined at 45◦ (upper image). Both profiles look just like a simple, sinusoidal wave but they are staggered by just a quarter of a wavelength as shown by the points of intersection with the straight wire—at the midpoint below and at the peak in the reflection.

Figure 8.2. A diagrammatic reconstruction of figure 8.1. Two simple sine waves are a quarter of a wave apart and the dotted guide line shows the relative alignment of the two waves.

figure 8.1. From one side, the helix looks just like a simple wave—in fact if perspective is overlooked (i.e. if the spring is viewed from a great distance as drawn in figure 8.2) the wire is exactly the shape of a sine wave. The same spring viewed at right angles to the first view, as seen in the inclined mirror in the figure, is still a sinusoidal wave but the ‘two’

Figure 8.3. Some familiar objects with the form of a right-handed helix. When reflected in a mirror, all would appear to be coiled in a left-handed manner.

waves are just a quarter of a wave ‘out of step’—the peaks and troughs of one view are the nearest and furthest points in the other view. One knows that in three dimensions the wire traces round a circle, once for each ‘wave’ that is seen from the side. Other familiar examples of the same shape include the thread of a screw and a corkscrew (figure 8.3). In the former, the thread generally rotates clockwise as one traces it away from the point of view and it is then called a right-handed thread; in the less common ‘left-handed screws’, the threads rotate anticlockwise away from the viewpoint; left-handed corkscrews are also occasionally seen, being made to help left-handed people. So the description of circular polarisation with a rotating vector as given here also applies to some very familiar objects. It is rather like the schoolboy challenge to ‘describe a spiral [sic] staircase without using your hands’—it always seems unnecessarily complicated when put into words alone.

The fact that two waves vibrating at right angles and one quarter of a wave out of step will generate a circular motion for every wave is also seen in two simple models. If a pendulum is suspended from another pendulum that swings at exactly the same rate but at right angles to its own swing (figure 8.4), the path of the lower bob will depend on the relative timing of the swings. If the two are in step because each was started at the end of its swing at the same moment, then the bob moves in a straight line—say from back left to front right and back again.

Starting one of the pendulums at the other extreme, just half a complete

Figure 8.4. One pendulum suspended from another pendulum. The two swing with the same period but in directions at right angles to each other. Changing the relative timing (phase) of the swings produces a series of different patterns of movement that can be demonstrated by letting sand trickle from the lower bob. When the two swings differ by a quarter of a wave, the sand is deposited around a circular path.

swing later, makes the bob swing in a straight line at right angles to the first path. This change in relative timing by half a wave is exactly analogous to the half-wave plate that retards one wave and so twists the resultant vector through a right angle (as seen in chapter 2). But if one pendulum is delayed by a quarter of its complete swing, then the lower bob swings round in a circle. Imagine it swinging over an upturned clock face. Starting from far left and middle distance (at 9), it moves nearer and to the right; at the nearest point it is only half way to the right (at 6) and continues to the right as it recedes (to 3); starting back to the left, it reaches its furthest point (at 12), and finally completes its leftward swing as it comes forward again (to 9).

The second illustrative model is an electronic version of the same thing. The glowing spot on the screen of an oscilloscope screen can be made to swing left and right by applying a sinusoidal deflecting signal. Another such signal can be applied to make the spot move up and down

Figure 8.5. The bright spot on an oscilloscope screen can be deflected from side to side and also vertically up and down by application of sinusoidal waves. When the two waves are identical but staggered by a quarter of a wavelength, then the spot revolves in a circle. This is a special case of a Lissajou’s figure.

on the screen. When the two waves are in step, the spot moves in a straight line diagonally; when they are a quarter of a wave out of step the spot moves in a circle (figure 8.5). These models can be used to investigate other situations. A perfect circle is only obtained if the two waves are exactly a quarter-wave apart and also of exactly the same size (amplitude), otherwise the circle becomes an ellipse. In the case of light we need not consider unequal waves but it is clear that ‘exactly in step’ and ‘exactly a quarter of a wave apart’ are simply two special cases of a whole range of possible timings. Starting with the two waves in step and delaying one slightly makes the straight line ‘open out’ into a thin ellipse (figure 8.6). Increasing the delay makes the ellipse fatter until at one-quarter of a wave difference it becomes a circle; then it gets thinner ‘the other way’ until at half a wave difference it becomes a straight line again but at right angles to the original one. These patterns are known as Lissajou’s figures for equal frequencies, and they can be used to make very accurate comparisons between two waves.

The point of all this for polarised light is that a birefringent material resolves a wave into two orthogonal components and then retards one relative to the other. The result depends on the thickness of the material and the relative timing of the two emergent components when they recombine. The general case is elliptical polarisation for which circular

Figure 8.6. When the relative timing of the two waves of figures 8.4 or 8.5 is varied, a family of Lissajou’s figures is produced. The figures shown here are the family produced by equal (1:1) frequencies and equal amplitudes; other families of figures are produced when the two waves are of different but simply related frequencies such as 1:2, 2:3 etc, but those patterns do not concern us here.

and linear polarisation are both special, extreme cases.

It is important to emphasise that elliptical polarisation is not the same as partial polarisation, although in both cases a rotating linear polar will show a partial dimming with no complete extinction of the transmitted light. Partial polarisation consists of polarised light mixed in with some unpolarised light, half of which always passes through the analysing polariser. With 100% elliptically polarised light, there is a larger electrical field vector along the long axis of the ellipse and a smaller one along its short axis. With circularly polarised light, of course, the vector is the same in all directions and there is no dimming as an analysing polar is rotated—although in practice there are usually slight colour changes due to retardations not being quite equal for all wavelengths.

Circular polarisers are therefore made by applying a quarter-wave retarder film or crystal to one side of a linear polariser, with its two special optical directions (properly called the privileged directions) at 45◦ to the direction of polarisation. Quite effective circular polarisers can be improvised by selecting quarter-wave cellophane films (see chapter 2) and combining each with a piece of standard, linear polaroid. Rotating the retarder by 90◦ interchanges the two special directions and turns a clockwise circular polariser into an anticlockwise circular polariser or vice versa, but in commercial materials the two films are usually bonded together. For some reason, nearly all the circular polaroid that is easily available is left-handed although the right-handed form can be obtained if needed.

A simple, if rather clumsy, method of changing the handedness or sense of rotation is to add a half-wave plate or film. This is because, if correctly orientated, retarders have a cumulative effect. A quarter-wave

plus a half-wave retardation gives a three-quarter retardation, equivalent to minus a quarter; and conversely, if the extra plate is turned by 90 ◦, a quarter minus a half also gives minus a quarter.

A circular polariser only works in one direction, with the light passing first through the linear polaroid and then through the quarterwave plate. When light passes in the opposite direction, the random, unpolarised light is essentially unchanged by the quarter-wave plate (it is different but still random) and then emerges with linear polarisation from the polaroid. A test piece of linear polariser will immediately show the difference—on rotation it goes black in two orientions as it extinguishes light transmission on the linearly polarised side but it shows (almost) no change on the circularly polarised side.

It is not easy to distinguish between left-handed and right-handed polarisers unless one has a reference piece of known handedness. When two oppositely handed polarisers are held ‘face to face’ the transmission is extinguished (at all points of relative rotation) whereas two similarly handed polarisers transmit freely (even when one is rotated). A simple circular polariscope, called a Cotton polariscope after its originator, can be made from one polariser of each handedness mounted side by side. One will go dark when circularly polarised light of the opposite handedness is examined (figure 8.7). Simply by reversing this device, so that light passes the other way (first through the linear polaroids and then through the retarder films), makes it into an equally effective linear polariscope (as seen in chapter 1) since the retardation then has no effect on the visible result.

Circular polaroid films of this kind find a very useful application in greatly reducing troublesome reflections from aircraft instrument panels and radar screens. The radar screen has an image that glows by phosphorescence and instruments with dials can be illuminated from within. But in both cases lamps and illuminated objects in the room or flight deck will be reflected in the glass screens and will degrade visibility or at least create distraction. If a sheet of circular polaroid is placed over the panel, light from the display passes first through the quarter-wave retarder and then through the linear polariser, suffering only a 50% loss of intensity which can easily be compensated at source. But light from the room is circularly polarised by the filter and when it is reflected by glass the direction of rotation is reversed: left-handed circular becomes right-handed circular. This cannot pass out through the polariser because the quarter-wave retarder turns the circular into linear polarisation with its direction at right angles to the original, and this is

Figure 8.7. A Cotton circular polariscope for detecting whether light is circularly polarised and its handedness, here seen over a circular polaroid background film. Two pieces of circular polaroid, one left-handed and the other right-handed, are simply mounted next to each other. The device must be viewed from the side bearing the linear polaroids, otherwise it acts as a linear polariscope (see figure 1.4). If circular polaroid of only one handedness is available, half of it can be covered with a suitably orientated half-wave film to change its handedness. Another alternative is to mount two pieces of linear polaroid, with their directions at right angles, over a single, properly orientated quarter-wave retarder film.

then blocked by the linear polariser. One can imagine, by analogy, that the mirror image of a corkscrew, reversed in its sense of rotation, cannot be pushed through holes previously made by the real corkscrew. So a sheet of circular polaroid, when inserted the right way round, provides a ‘black screen’ through which self-luminous displays are seen clearly. Such components are sometimes supplied with oscilloscopes too.

A second application for circularly polarising filters is in photography for the removal of intrusive reflections from a scene or to enhance the contrast between clouds and blue sky. As described in chapter 7, the use of a linear polarising filter achieves the wanted effect but might influence the amount of light reflected by mirrors within the camera, which sample the light for metering or for automatic focusing. But if the linear polariser, having done its job of selecting one direction of vibration, is then followed by a quarter-wave retarder, the light becomes circularly polarised and the filter can be rotated to any desired position to enhance the image without affecting the strength of internal reflections. The circular polariser is therefore not actually sensitive to

circular polarisation in the scene itself, as its name might suggest, since it is mounted the ‘wrong way round’.

Another way to make circularly polarised light that, in some ways, is both simpler and better than using a retarder is by the use of total internal reflection. Light that is incident at about 45 ◦ upon a glass–air surface from within the glass itself is completely reflected (in contrast to external reflection where a glass surface reflects only a small part of the light). The principle is used in prisms to make good mirrors, such as those that invert the image in binoculars, where four reflections at 45 ◦ are needed and silvered glass mirrors would degrade the image intolerably. When the light is also linearly polarised at 45◦ both to the surface and to the plane containing the rays, it is divided into two components, one parallel with the surface and one normal to it (in the plane of the rays). These are reflected with a relative difference in timing of about oneeighth of a wave, giving elliptical polarisation. A second reflection, therefore, can give another eighth of a wave difference to make a quarter of a wave and so produce circular polarisation.

To be precise, the angle of incidence within the glass should either be about 48.5◦ or about 54.5◦, but the former is so close to 45◦ that the very convenient combination of two right-angle prisms gives a broadly elliptical polarisation that is a good approximation to truly circular polarisation (figure 8.8). A prism specially designed to give two reflections at the proper angle is called the Fresnel rhomb (figure 8.9) and this gives truly circular polarisation, usually using the larger angle of incidence and reflection. Although these arrangements are not such a handy shape as the quarter-wave plate device, they do not suffer from the slight wavelength dependence that accompanies birefringence and so they avoid spurious colour effects. But they are degraded if the light does not approach from exactly the ‘proper’ direction and so they have a narrow angle of effective view.

Incidentally, circular polarisation is not produced within a pair of binoculars because the prisms happen to be so arranged that the effect of the first pair of reflections is cancelled by the second pair; they do not even add to give a half-wave difference that would rotate the plane between crossed polars. In addition, these considerations do not apply to the stack of plates reflecting polarisers of chapter 7, where the linear polarisation is never slanted at 45◦.

Reflection is also responsible for perhaps the commonest natural occurrence of elliptical polarisation: reflection of linearly polarised light from shiny metals. In chapter 7, reflection was considered from materials

Figure 8.8. Circular polarisation can be produced by the combination of a linear polaroid and two reflecting prisms. With right angle prisms the circularity is not precise but is close enough for many purposes. On the left an oblique linear polariser (p) and two right-angle prisms; on the right the view looking into the upper prism with oblique linear polarised light converted into almost circularly polarised light. Turning the linear polariser by 90◦ would reverse the rotation.

Figure 8.9. Exactly circular polarisation can be produced from oblique linearly polarised light by a Fresnel rhomb on which the angles of internal reflection are around 54◦ for various types of glass. Both the necessary reflections can be arranged within a single piece of glass and the circularly polarised light emerges on a path parallel to the original beam. With the linear polaroid (p) set obliquely as shown, the circular polarisation will be reversed.

that are essentially non-conductors of electricity and from dark coloured metals. It might be supposed that reflection from nice shiny, electrically conductive metal surfaces would be much simpler and it is certainly true that these reflect a much greater proportion of the incident light. But the interaction of the electromagnetic fields of light waves with such conducting materials, which have free charged electrons, is quite complex. Once again incident light is divided into two components, one polarised parallel with the surface and the other normal to it, in

the plane of the light rays. As with internal reflections inside nonmetals such as glass, these two components are given a timing difference on reflection from the surface of shiny metals. If the incident light is already linearly polarised at rather more than 45 ◦ to the surface, then the resultant reflected rays are strongly elliptically polarised. At high angles of incidence, around 70–80 ◦, and the correct angle of initial polarisation (near to but not quite 45◦), the ellipticity becomes circular. Both these specific angles vary somewhat between different metals. The circular polarisation after linearly polarised light has been reflected from some shiny metal objects is shown in figure 8.10. It is also worth pointing out again that back-silvered glass mirrors also produce ellipticity but the effect is very small. Due to refraction in the glass, even light that is incident at a high, grazing angle has a much smaller angle of incidence on the metallic surface itself (figure 8.11). The maximum real angle of incidence with an ordinary glass mirror is about 41 ◦ which produces very little ellipticity.

It is now possible to see why the retardation colours seen in a mirror placed at 45◦ to the direction of polarisation, as described in chapter 2 and shown in colour plates 5 and 6, are different depending whether the mirror is front silvered or back silvered. The front-silvered mirror causes a degree of ellipticity in the reflected light that varies somewhat as the mirror is tilted to change the angles of incidence and reflection. The different ellipticity changes the interaction with the second polariser so that the reflected colours alter quite noticeably. Different metals also produce different combinations of polarisation and timing of retardation so that the colours may be strikingly different in two apparently similar cases. With an ordinary back-silvered mirror, the situation is simpler since little polarisation or ellipticity is produced: the angles of actual incidence and reflection (at the silver layer) vary rather little and cannot approach the high, grazing angles necessary to produce pronounced ellipticity.

The phenomena of dichroicity and birefringence, described earlier for their effects on linearly polarised light, also have their counterparts for circular (and therefore elliptical) polarisation. Circular birefringence, by analogy with its linear counterpart, is where left-handed and righthanded circularly polarised light are propagated at different speeds. This effect occurs in optically active materials: those with an asymmetrical structure or with chiral molecules in solution (see chapter 5). Indeed it provides an explanation for the rotation of the direction of linear polarisation seen in such cases.