Interference

Please note that because of the difficulty of representing a wave on a flat surface, false colours have been used in this page. In all cases the brightness represents the displacement of the wave from the average, black indicating places with no effect at all. Red is used to represent positive excursions: green is used for negative ones. Blue or purple is used for evanescent waves. These colours are not related to the colours of the light they represent, and indeed many of these diagrams could just as well represent sound waves. All the diagrams are snapshots - in reality the waves would be moving along.

Please note also that the relative sizes of objects and wavelengths have been chosen in order to make the effects easily visible. In practice most everyday objects have structural details which are huge compared with wavelengths of light. Most of the diagrams are the result of one-dimensional calculations. In three dimensions the results would be qualitatively different. but the general principles would be the same.

Interference is a property of waves. The name is a poor one - waves generally pass through each other with no effect. If they did affect each other, understanding speech would be very difficult in the presence of other sounds. A band or an orchestra would create chaos, and even a chord on a guitar would probably be a mess. Radio communication would be impossible, bathed as we are in a sea of radio waves. Interference in a radio receiver only happens when two signals, or signal and noise, are at similar frequencies, and the word then has a different meaning from the one described here.

Seeing would be a strange thing too, if all the criss-crossing light rays did not pass with no ill effects.

No - interference of waves actually refers to the idea that waves can go up or down, (or at least positive and negative, in more abstract cases). Where two waves occupy the same space, their effects combine. If both waves are going up or both are going down, the result is a bigger displacement. But if one is up and one is down, the net effect is less than either would give alone.

Each of the next three pictures shows the sum of ten sine waves with randomly chosen amplitudes and wavelengths.

The variable amplitudes are the source of the "every seventh wave" idea. The sea is criss-crossed by waves going in all directions, with many different heights and wavelengths. The result is usually an apparently patternless motion. The chance of three waves being in phase at a given point at a given time is clearly less than for two. The chance of four adding up is less than for three, and so on. Seven is just a convenient way of saying "not very often". If these waves had been continued long enough, much greater amplitudes would have been seen, and eventually one where all ten waves more or less add up.

The fourth picture above is the spectrum of wave heights for 500000 waves. To make this picture, ten sine waves were added, all with the same amplitude. The wavelengths were set at successive multiples of the golden ratio (1.618033989 . . . .), to make all the ratios irrational. A peak was defined as the highest point between two successive zeroes. The width of the graph runs from zero to the maximum possible height, obtained when all ten waves add together. The height of the graph corresponds to 1275 events in a bin. The graph shows that the chance of even seven of the ten waves adding up is very small.

This illustrates the difficulty of building an absolutely unsinkable ship of any size. The larger the wave, the more unlikely it is and the more it costs to design for it.. The designer has to estimate the probability of a ship being struck during its useful life by a wave of a given size.

Given the number of ships at sea, it might be impossibly expensive to design so that no ship is ever overwhelmed. The same idea affects defences against floods - the higher the water level, the less often it happens, and the more expensive it is to resist.

The pictures below represent simple examples of interference between two sources of waves.

In each picture the outer panels represent snapshots of two waves radiating from sources. The centre panel represents the sum of the two. In the first three pictures positive and negative excursions from the average are represented by red and green respectively. The dark areas are places where the displacement is small. There are lines where the amplitudes add to make strong waves, shown by the alternate red and green blobs. Between these there are dark lines where there is little activity because the waves have almost cancelled each other out.

In the second picture the wavelength has been doubled, and the pattern is clearer. One reason that we don't see such effects normally is that the wavelengths of visible light are so very small. They run from about 400 nm for far violet light to about 700 nm for far red light. One nm is 0.000000001 metre, or 10^-9 m. In the third picture the wavelength is a half of the one in the first picture. The fourth picture shows the intensity, which adds the effect of many waves, for green light. In practice this can only be done with tiny light sources very close together. The two sources must be coherent, in other words, derived from the same lamp, otherwise the fringes move around at a tremendous rate and are invisible.

You can see a similar phenomenon on a large scale on a motorway, as you approach a bridge which has railings. The image on the retina of the nearer set of railings looks a little larger than the further set. And so you see "interference" fringes, which are spaced much more widely than the actual railings. These are called Moiré fringes. They can be seen in some kinds of cloth.

A two dimensional version can be seen in containers that have metal sides which are pierced with holes on a hexagonal grid.

Again the fringes are spaced more widely than the holes.

This phenomenon can be used in measuring machines with high resolution, by using a light detector and a scaler to count the fringes.

To see the interference fringes from light, you can use a powerful light source behind two narrow slits in a screen. If the slits are 1 mm apart, there will be narrow fringes at a large distance, for example 1 metre. If there were two minute light sources behind the slits, the fringes might tend to reinforce or cancel, depending on the distance between the sources. Even for an extended sources the net effect will vary with the size.

This effect was used by Michelson to measure the angular diameter of some stars.

He used a long girder in from of the telescope, carrying mirrors which deflected light into the instrument. By varying the separation of the mirrors Michelson could see the fringes come and go. By measuring the mirror positions he could work out the angular sizes of stars . For those whose distance was known, the actual diameter could be calculated.

Using radio telescopes at great distances apart, astronomers can create a similar effect.

The next picture is a photograph a small piece of paper cut from a promotional leaflet, coated with a thin film of metal - Click here for picture. The surface was illuminated by sunlight, and it has created a spectrum of colours. The surface is ruled with minute grooves and ridges. For each wavelength there is a direction in which the reflections from all the ridges are in step. The metal film forms a diffraction grating. A later section will explain how this works. Diffraction of Beams

When light meets an object, it behaves as if it goes off in all directions from every point in the object. But only where all the parts of the wavefront add up properly is there a detectable effect. The behaviour of mirrors, matt surfaces and lenses can all be accounted for by this description. It turns out that the light goes along the path of minimum time. This appears to be a fundamental principle in many aspects of physics. Not only that, but all the light that goes into a perfectly focussed image takes the same time to get there.

$Refract.jpg (34098 bytes)$ These diagrams show simulations of refraction and reflection. In the first picture, light coming from the top right meets the surface of a medium in which it travels more slowly, producing a shorter wavelength. Its direction changes abruptly. The path it takes is not the shortest in distance - it is the shortest in time. Compared with the straight path the light goes further in the fast medium and less far in the slow medium.

The behaviour is rather like that of a driver who has to make a journey partially on fast roads and partially on slow roads - he or she may choose a route to minimise the time rather than the distance. Another example is a person running down the beach to the sea, across pebbles and then across sand. The picture could equally well be read backwards - as light coming from the bottom right and entering a medium in which it travels faster. The second picture shows reflection from a polished metal surface

The next pictures reveal a little more about reflection and refraction.

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In these pictures we see light from the top right in air, reaching the surface of a solid or a liquid medium. A small fraction of the light is reflected, but most goes on into the medium.

It changes direction, because light goes more slowly in the medium, and in order to preserve the continuity of the waves there must be a change of direction.

The amount by which the direction changes is related to a quantity called the refractive index of the medium, which is inversely proportional to the speed of light in the medium. If the surface is curved, focussing may occur.

Some substances have different refractive indices in different directions, while others have indices which depend on the direction of polarization of the light. This happens because the internal structure of the substance is asymmetrical.

We saw that light travels so as to take the path of minimum time, and that the waves in two media must be continuous across the transition. We don't actually need these rules.

Suppose we take each infinitely narrow ray of light, and let it take all possible paths at the transition. When we do this right across a beam of light, we find that all the contributions add up to a negligible intensity, except at the angles that we are already familiar with. It is almost as if light goes in all possible directions, most of which result in no visible effect.

What is going on in these pictures?

Light comes down from the top left, in a piece of glass, and it reaches the edge. The speeds of light in air and in glass are such that there isn't any angle in the air for which the waves match up at the edge. With a perfectly clean smooth surface, the reflection is perfect. Nature so often doesn't seem to be on our side, but if we are designing prism reflectors for binoculars, nature is kind - we can design perfect reflectors without any fancy tricks and without any silvering. The same principle is used to keep light inside fibre optic cables.

The effect happens when sin A > 1 / n, where A is the angle of incidence and N is the refractive index. The critical angle is given by sin A_C = 1/n.

But what is the little glow in the air just by the glass? This is called an evanescent wave. Nature doesn't like discontinuities, and at the glass-air interface the wave amplitude would suddenly go to zero if there were no effects in the air. But it doesn't have to go to zero - the blue glow represents an exponential function that fades gradually with distance from the glass. It doesn't oscillate or travel l like the proper wave, and it doesn't carry any energy. But it does prevent the wave having a discontinuity in it. The exponential in time is a familiar function - it occurs in the calculation of compound interest, and in the fading away of fluorescent light and radioactivity. In this case the exponential is in space. Note that the blue, like the red and green, is not the colour of the light - it is used to distinguish evanescent waves from real ones.

The diagram above represents some waves approaching a surface and receding from it. This can happen if the speed, and thus the refractive index, vary with the distance from the surface. The air above a hot surface such as a tarmac road is heated. Its temperature falls with the height above the road. Consequently the density increases, and so does the refractive index. Light waves that are almost horizontal can be deflected upwards, never hitting the road..

The onlooker does not know the path of the light. She or he only detects the final direction as the light enters the eye. So the eye-brain system thinks there has been a reflection, which in a sense is true.

The shimmering is caused by the many little convection cells and by turbulence..

The vertical scale of the diagram is of course greatly exaggerated.

Let's see if we can work out when a mirage will occur. The refractive index n of air at 20 C and standard pressure is about 1.0003. If n - 1 is roughly inversely proportional to the temperature, then for a 1 degree rise, which is a change of about 0.3 % in absolute temperature, the refractive index becomes about 1.000299, giving a ratio of about 1.000001, or 0.999999, depending which way we look at it

This does not look very hopeful, but let's carry on anyway. We have seen that waves reaching a surface at less than a critical angle, A_C, are reflected.

This happens when n = 1/sin(A_C).

So n=1.000001=1/sin(A_C), or

sin(A_C) = 0.999999. But to get anywhere near this we need 6-figure tables. All we can say is that the answer is near 90 degrees.

It's better to work out the small angle between the light and the ground.

Then we get about 0.0014 radians. If we are driving along a road, and our eyes are 1.5 m above the ground, the start of the mirage will be 1.5/0.0014 metres away, which is about 1000 metres. In a flat desert, that's close enough to make you think there's a pool ahead.

If we take a 2 degree increase in temoerature, the distance drops to about 750 metres. A 4 degree increase gives 500 metres.

Note that the thickness of the hotter layer is not important. Only the initial and final directions of the light matter.

These calculations are not exact.

Since the density of the atmosphere decreases with altitude, light from the sun, moon and stars is slightly refracted downwards. The effect is greatest for an object near the horizon.

A different type of curvature is predicted by general relativity theory. Light passing near a massive object slows down, and so its path is curved. Classical physics would havelight speeding up near an object.

During the solar eclipse of 1919, some stars were photographed "near" the sun.

Their positions relative to those of surrounding stars were measured. These results were compared with ones made when the sun was not near the path of the light. The results appeared to confirm the theory. More recent results using radar reflected from Venus have provided very accurate confirmation.

Light passing near distant galaxies is curved inwards. The lens effect can produce enhanced images of distant stars. The effect can be used to search for dark matter.

These curvature effects are not strictly refraction, as they are the result of the curvature of space-time.

See Black Holes Gravity

Returning to the evanescent waves, how do we know that they "exist"? If we bring up another piece of glass, within a wavelength or two of the first one, something truly miraculous happens.

Some light somehow tunnels through the forbidden gap and appears inside the second piece of glass, as the second picture above shows.

There are practical applications of this tunnel effect. All particles can behave as waves. The electrons in solid state devices do just that. In a tunnel diode the electron waves tunnel across a forbidden region and allow a current to flow when enough voltage difference is provided.

The decay of radioactive atomic nuclei by emission of alpha particles is explicable by the same effect. They are bound inside the nucleus by very strong nuclear forces. If only they could get outside and move away a little they would be repelled by electrical forces and speed away. Between the attractive position and the repulsive one there is a region where they cannot exist.

But if we wait long enough, the particle wave, which is like a standing wave inside the nucleus, will tunnel through, and appear triumphantly on the outside as a travelling alpha particle. This might take minutes, or a thousand million years, depending on the species of nucleus, the mean life being extremely strongly dependent on the energy of the alpha particle. It is as if a boulder, having been lying inside the crater of Mt Fuji for many years, suddenly appeared at the same height on the outside of the cone, and then rolled and bounced to the bottom.

The picture above simulates the potential energy of an alpha particle in relation to an atomic nucleus. Two cases are shown, with alpha particles of two different energies. The blue waves represent the standing wave of a trapped particle. The green curves represent the evanescent wave. The purple waves represent the particle after escaping. The amplitudes of the two green waves represent the probability of escape in the two cases.

The whole diagram is only schematic - the actual ratio of probabilities is so great that no diagram could show it. This is mainly because the decay-rate is hugely sensitive to the energy of the particle, which affects both the height of the energy barrier above the energy of the alpha particle, and the distance through the barrier.

For example, Thorium 232 has a half-life of about 14 billion years, while Francium 204 has a half-life of about 2 seconds . The ratio of the half-lives of these two alpha emitters is about 4.5 X 10¹⁷. The graphs above show the half lives of alpha emitting nuclides of the elements Francium and Polonium, on logarithmic scales. Both trends cover more than a factor of 100, for a range of alpha particle energies differing by only about 10%.. The points do not lie exactly on smooth curves, presumably because other nuclear properties can affect the half life.

The two lines would not lie on each other if the graphs were superimposed, because the electric charges of the two families of nuclei are different, and therefore so are the heights of the potential barriers for a given alpha energy.

This picture shows Mount Fuji. The outline is a graph of the curve of your gravitational potential energy should you climb it straight up that sky-line. A pebble in the crater will never tunnel through.

The next set of diagrams tries to explain a little more about waves.

These pictures simulate the effects of 100 coherent wave sources placed in a vertical line at the left hand edge of the screen. The wavelengths from left to right are 0.5, 0.2, 0.05, 0.02 and 0.006 of the width of the line of sources. As the wavelength decreases the emitted light grows more like a parallel beam, with flattening wave-fronts. But the beam never becomes perfectly parallel and sharp edged. We have the illusion that images and shadows are sharp, because the wavelength of light is so small compared with most everyday objects.

But nature does not like hard edges. If you try to enforce one you pay for it elsewhere - the light just fans out around the edges and goes in directions you didn't ask for. This effect is more noticeable with sound. Low frequencies with long wavelengths go round barriers much better than high frequencies with short wavelengths.

Radio waves behave in the same way. The relatively short waves used for FM radio have rather short range, whereas those in the long wave region can go far past the visible horizon. The fact that some short waves can apparently go far around the globe is anomalous - they are reflected from the ionosphere, a layer of ionised air which has a similar effect to a sheet of metal.

The pictures so far have been snapshots of simulated waves. The next three show the intensity with the waves averaged out, for three different wavelengths. The faint coloured fringes are artefacts of the colour system and computer screen.

The behavior of these beams of light leads to the idea that the product of the width and the divergence of a light beam can never be less than a certain fixed quantity. This result applies to any other wave phenomenon, such as sound. Material objects also behave as waves. The wavelengths are unimaginably small except when the object itself has a tiny momentum, which is attainable only for very light objects, such as elementary particles, atoms and molecules.

Recently some almost spherical molecules of buckminsterfullerene - C₆₀ - were shown to behave like waves on passing through a slit. They fanned out according to patterns like the ones shown above. Please click here for more about diffraction of particles.

If particles such as photons of light are sent through a small hole or a slit one at a time, they fan out according to the wave theory. So light waves are not interfering with each other, but with themselves - even one photon "knows" that it is a wave and behaves accordingly.

Nevertheless, each photon is finally detected at a single place. The diffraction pattern is seen only after many photons have been transmitted. For a single photon, the pattern gives the probability that it will be detected at a given place. Yet during the journey of a photon, it seems to "know" of all the possible paths - if you block some of the paths it could have taken, the diffraction pattern changes.

The sixth picture above has 300 sources instead of 100, resulting in a more geometrical beam . Note that the effects of the edges are seen within the beam as well as outside. Looking through a feather or an umbrella at a small bright light will often show the effects of waves, in the form of extra images around the main one.

The next two pictures use a much larger scale, to show the effects of individual sources more clearly . Some wavefronts have been drawn in white. The direction of travel of the light is at right angles to these wavefronts.

The images above show the near-field region, close to the light sources - further away the beam settles down into a fairly smooth shape. What is happening in the fifth image in the earlier row of six, repeated above? The wavelength is too small for the graphics, and is in fact smaller than the spacing between the 100 light sources.

To understand the two extra beams, or side-lobes, consider a military march-past with soldiers in step - see picture below.

In the picture above, a column if 19 files of soldiers goes from left to right. At the line, six files on each side turn outwards, but instead of temporarily changing the lengths of their strides to make the turn, they continue as before. The blue marks show what happens to one rank of soldiers. The red marks show groups of soldiers who were originally on a diagonal line. At the top of the picture these marchers form a line at right angles to the column, and once those in the green triangle have been forgotten, the column looks perfectly normal.

Light behaves in a similar way, but whereas only complete soldiers exist, light waves can vary in amplitude, and so the lobes of the distribution do not have sharp edges.

The picture above shows diffraction from a small grating embossed on a promotional leaflet.

The differing wavelengths corresponding to different colours cause the side-lobes to fan out according to colour, because the angle is wavelength dependent. You can see this by looking through a pigeon's feather at a small bright light - the colours are spread out. The diffraction grating mentioned earlier works in exactly the same way. Very precisely ruled grooves on a smooth metal surface form a grating that can be used to measure wavelengths of light very accurately. By shaping the cross-section of the grooves, most of the light can be directed into one side-lobe, rather than into the ordinary reflected beam.

These wave effects show that a pinhole camera can never give perfectly sharp images.

A large hole makes a blur - a small hole spreads the light. For any image size there an optimum pinhole size where the two effects are about equal. It is also clear that to transmit or receive a wave directionally, the antenna must be very much bigger than the wavelength. This is why many bats have big noses and ears, and why their sounds have such short wavelengths. Of course, the more directional the antenna, the more accurately it must be aligned. The surface must be accurate to a small fraction of a wavelength. The same principle applies to antennas for radar and for satellite TV. A radar at an airport is often concerned with direction rather than height, and so it is very wide but not very high.

The same effect limits the sharpness of telescopes and microscopes, so that large aperture is needed not just for light gathering, but also for good resolution. Even camera lenses, used at very small apertures, can show the effects of diffraction. The machines that project the shapes of semiconductor components on to silicon wafers are subject to the same limitation. One solution is to use shorter wavelengths in order to pack more parts on a wafer.

Astronomers want telescopes with wide apertures, not only to gather more light, but to obtain better angular resolution. By two or more optical or radio telescopes, spaced well apart, they can achieve some of the benefits of a single very large instrument. In the optical case, the minute wavelengths mean that positioning and alignment must be extremely accurate, so this technique was first used with radio telescopes.

Any kind of wave can be diffracted, and if the sizes of the waves and the objects are suitably related, the diffraction is easily detected. The spacings between atoms in solids and liquids are far too small for light to be diffracted, but the much shorter X-rays are ideal, and have been used to elucidate the structure of a great many substances.

See X-Ray Diffraction

Small particles can behave as waves, and so, in principle, neutrons, which can penetrate most substances easily, can be used in a similar manner to X-rays.

In these simulations a beam from 500 sources is interrupted by an object that covers 160 of them. The light beam gradually fills in the shadow. In the first picture the waves are shown as a snapshot. In the second only the intensity is shown. Because of the symmetry the one line where light is guaranteed, albeit very faint, is along the centre of the shadow. In a real case the wavelength would be far smaller, relative to the object, than the waves in these pictures. Nevertheless, if a 1 mm circular obstacle is placed in a parallel beam of light, there will be a tiny spot of light at the centre of the shadow. In fact, during the 19th century, when some physicists were advocating the wave theory of light, the physicist Poisson pointed out this very fact - that if light were waves, the shadow of a circular object would be a point of light. He was trying to refute the theory, because the spot had never been seen. This was because nobody had ever looked. So they did look, and they did see the spot, which is named after Poisson.

The pattern in the pictures above could have been obtained by subtracting the effects of a narrow beam from the effects of a wide beam.

Since all particles of matter behave like waves, atomic nuclei generate diffraction patterns when bombarded by electrons and other particles. The patterns can be used to learn about the distribution of matter in the nucleus. When a pion or a kaon strikes a proton, it may collide elastically, that is, after the event, the particles retain their identity, and no new ones are created. The collision may also be inelastic, meaning that after the event, different particles may be present. If the kaon or pion turns into a collection of particles whose total properties are the same as those of the original particle, the outgoing jet produces a diffraction pattern just as if the original particle had emerged.

See Diffraction of kaons

The next picture shows the head of a butterfly - Like most insect eyes, the eys of butterflies are composed of many little tubular receptors, called ommatidia. Each one points in a different direction. The large dark spot is the part where the camera is looking more or less straight down the ommatidia. These ommatidia have the camera near the centre of their field of view. Around the main dark spot there are several others which are not so dark.

These correspond to the side-lobes of the diagrams above. They show that the ommatidia take in light from other directions than the axial one. Perhaps evolution has not been able to remove them. Perhaps they convey information that is not in the main lobes. Who knows? At least some dragonflies and mantises do not seem to have side-lobes, so perhaps they interfere with the accurate binocular vision which is needed for locating prey.

The next picture shows sunlight coming off a compact disc - Sunlight has a continuous spectrum of colours, from violet to red, but the slide film, the scanning and the screen of the PC must have modified the results. The spectrum is spread out by the equally spaced concentric rings of pits which contain the digital information. The rings act as a diffraction grating. The spectrum of sunlight is not actually continuous - there are very fine gaps which can only be seen with specialist instruments.

These gaps are the signatures of the numerous chemical elements which are present at the surface of the sun. Atomic emission If you put a compact disc under a yellow street light, you will see only yellow light.. The spectrum of sodium has a dominant yellow band which is very narrow. No optical system can disperse this light into other colours, though a spectrometer will show two very closely spaced yellow lines in the spectrum. Here is a picture of a mercury lamp viewed through a diffraction grating -

The production of colours without pigment is familiar in the colours found in the peacock's tail, in the purple colour of some camera lenses, and in oil films on puddles. These mechanisms differ from the previous examples in the type of colour that can be produced. In the case of early coated lenses, the idea was to reduce the reflections from the glass surfaces. By adding a transparent coating of a thickness that made the front and back reflections out of step, the reflections could be made to cancel. This could only be perfect at one wavelength, which was set at the middle of the spectrum.

Towards the red and blue ends the cancellation was imperfect, and so some reflection occurred. The colours added to make a purple hue. Some later lenses have several layers which enable a more thorough cancellation over a wider range of colours. The coatings are arranged to give a graded range of refractive indices as the glass is approached . Ideally, these refractive indices would start near that of air, and end near that of glass. In practice, no solid substance can have a refractive index anywhere near as low as that of air.

The first picture below shows a comparison between a plain glass plate and a multi-coated lens. Each plain glass surface reflects about 4 % of the light, so a multi-element lens has great potential for flare, as these reflections can bounce around in the lens. This picture shows the 8% reflection from a small piece of glass, and the minute reflection from a coated multi-element lens. It also shows the bad effect of small grains of dust which were not detected by a cursory inspection of the lens and glass..

The second picture is a computer simulation of the effect of a single coating (medium grey strip) on a glass surface. Light waves of six colours, red, orange, yellow, green, blue and purple are shown (thick lines) falling on the glass. The coating has a thickness of one quarter of the mean of the yellow wavelength and the green wavelength.

This second picture shows the two waves (thin lines) which are reflected from the two surfaces of the coating, and the sum of the two. The amplitudes of the reflected waves are greatly exaggerated to make them visible, because they would in reality be only a few per cent of the incident amplitudes. The picture shows that the yellow and green reflections have very small amplitudes, and that the orange and blue are also fairly weak. The red and the purple are the most significant amplitudes. That is why an old single-coated lens looks often looks purplish.

This picture simulates the edge of an oil-film on water, starting from zero thickness at the left hand side. The vertical scale is greatly exaggerated. The fourth spectrum corresponds to the case of the singly coated lens. The crudely drawn spectra give some idea of the way that bands of colour are seen in an oil-film. The height of each part of a spectrum is proportional to the amplitude of that colour. The last picture is similar, but has a more rapid increase in thickness. The numbers at the bottom are the ratio of the thickness to the yellow-green wavelength. when this is 0.25, 0.75, 1.25, etc (0.5(0.5 + N)), the yellow and green are weak. As N gets bigger, the coloured fringes start to fade out, because the fringes of different colours are getting out of step.

Contrasting with the oil film and the multi-coated lens is the production of flashing colours by butterflies such as Morpho and Purple Emperor, Apatura iris. Some butterflies even flash in the ultra-violet part of the spectrum. The scales on the wings have a complicated structure which selects a narrow band of frequencies. In the case of the oil-film and the coated lenses we saw that a single layer can only select a wide wavelength band. To make a narrow band filter needs more structure. So it is with butterflies. Presumably the butterflies don't make the bandwidth too narrow, because not enough light would be reflected.

After all, they can only reflect what is there - they can't create new light, and they can't change the colour of light. Changing the colour, more precisely the frequency and wavelength, of an individual photon of light is not possible by any mechanism accessible to living things.

Although the butterflies and other animals may use complicated structures, the second picture above shows that at certain thicknesses of film, the reflections can add over quite a narrow band of colours, which would produce an iridescent effect.

If you blow a big bubble on the surface of water that has some washing-up liquid in it, you will notice subtle difference between the bubble and the oil-film. As the water drains down and the bubble gets thinner, you will see colours as with the oil-film. But eventually a part of the bubble may disappear completely. What has happened is that the film has become much thinner than the wavelength of light. In the case of the oil-film the very thin film produces a reflection, because both reflections have the same phase.

But in the case of the bubble, one reflection is at an air-water interface, and one is at a water-air interface. One of these inverts the wave and one does not, and so they cancel.

Such a cancellation could conceal a thin crack or delamination in an industrial part being tested using an ultrasonic beam, unless the wavelength is short enough to get an adequate phase shift between the reflections.

The problem of reflection is a universal one with waves. For example, a loudspeaker cone generally radiates both forwards and backwards about equally. There is no way of getting both lots of sound to go the same way and be in phase at all frequencies. But absorbing any wave in a distance that is not a lot larger than the wavelength is tricky. One solution is to make a long transmission line, usually folded or curved, behind the speaker, and to line it with absorbing material. Such a speaker can sound very good indeed.

A small box never sounds as good because it reflects waves back through the cone to the outside, or at least lets these waves change the motion of the cone.

An oil film works in a similar way to the single lens coating. The subtraction colours that you see in an oily puddle are roughly complementary to those of a rainbow or a spectrum. The colours vary because the thickness of the oil film varies. Again there is some variation with the angle of viewing. The thinnest layer that can make a colour subtracts violet light, leaving a reddish tint. As we go towards thicker layers we find green or yellow missing, leaving red and blue to make purplish tints. Thicker still and we lose red, leaving bluish-purple. Even thicker layers repeat the same cycle of colours, so that a tapering film may show several bands of colour.

These pictures show the colours from tiny oil-slicks on water in a black tray. These colours can never be as saturated as those of the normal spectrum, because they are more or less white with a narrow band of colour subtracted. Furthermore, it is difficult to reproduce them using a three colour system, because every colour made by the oil film contains a large part of the visible spectrum. The colour process has to mimic these colours using only the three colour spectra at its disposal.

Colour film seems to work best with a more natural scene, in which individual colours are less noticeable, and we are not so sure what they should really look like. Some people may even prefer a film which makes the colours unnaturally saturated. Colour film may also have trouble with flesh tones, because this is a case where we do know what the result ought to be.

These pictures are of light passing through thin metal films on CD-ROMs.

As before, the bands are lines of equal thickness, showing the non-uniformity of the transparent coatings on the labelled side of the disc, which also exhibits fringes by reflection. The other side of each disc shows no bands by reflection, because the coating is much thicker.

What is the difference between interference and diffraction? When waves from two or more sources produce patterns the phenomenon is usually called interference. When light spreads on passing through a hole the phenomenon is usually called diffraction. In fact both are caused by the effects of the summation of waves. There is no real difference in principle.

Everything about light can in principle be explained in terms of waves, yet the emission and reception of light occurs in the form of minute units called photons. More generally, every other form of matter or energy is in the form of these small units, called quanta. The existence both waves and quanta has provoked decades of debate, calculation and experimentation. In fact the implications of quantum theory are still being worked out, leading to many results which are totally counter-intuitive to minds accustomed to the large scale world.

The quanta and their wavelengths are so small, and their frequencies are so high, that their effects are largely hidden, except when they interact with minute objects like other light quanta and elementary particles. Yet the whole of modern physical science and technology has been transformed by the understanding of quantum mechanics.

The difficulty of reconciling the wave and particle ideas recalls other great debates and discussions in science, which arise when there are different ways of understanding.

In 1900 it was still possible for some people to argue that atoms were only a mathematical tool, and that everything could be explained with methods such as thermodynamics, dealing only with observable quantities. More recently, it might have been possible for some people doing molecular biology to hope that everything could be derived from basic ideas, sidelining fields such as ecology. In practice it appears that for different levels of structure, different approaches are most useful.