Module 12 · physical optics
Module 9 treated light as a ray and got a very long way. This module finds the place where rays fail — and the failure is not a small correction. Light passing through two slits arrives somewhere it could not possibly reach by travelling in straight lines, and that single observation settled a hundred-year argument about what light is. It also, eventually, unsettled it again.
Module 7 gave you the only rule needed here: when two waves meet, their displacements add. Crest on crest builds; crest on trough cancels. Now put two identical sources side by side in a pond and let their ripples overlap.
At any point, what matters is how far that point is from each source. Suppose the two sources rise and fall in step with each other. A wavelength is the distance from one crest to the next, so a wave that has travelled one whole wavelength further than the other is a full cycle behind — which looks exactly the same as not being behind at all. If the two distances differ by a whole number of wavelengths, then, the two waves arrive in step and reinforce. If they differ by half a wavelength — or one and a half, or two and a half — one is at a crest exactly when the other is at a trough, and they cancel. Since those two conditions are met along fixed lines spreading out from the sources, the result is a permanent, motionless pattern of bright and dead directions.
Block one of the two sources. What happens at a point that was sitting in a dark band?
Tick the box on the bench and watch it happen. That result is impossible for particles travelling in straight lines: two streams of bullets cannot add up to fewer bullets. It is entirely ordinary for waves. This is why the experiment settles the argument. Hold on to that sentence — later in this module you will meet a puzzle that puts a single indivisible particle into an apparatus set up for waves, and everything Young thought he had settled comes open again.
One practical point, and it explains why you never see this in daily life. Two ordinary light bulbs side by side produce no interference pattern at all, because each atom in a hot filament emits an independent burst lasting perhaps a hundred-millionth of a second, with no fixed timing relationship to any other. A pattern does still form — but it is rebuilt in a new position about a hundred million times a second, and what your eye adds up is all of those patterns laid on top of one another: a smooth blur. To see fringes you need coherent sources: two sources locked in step and staying locked, which in practice means taking one source and splitting it in two. Realising that in 1801, with no laser and no electricity, was Young's real genius.
Thomas Young let sunlight through a single pinhole, and then let the light from that one hole fall on two closely spaced slits. Because both slits are lit by the same original hole, they are coherent — locked in step. Beyond them he put a screen.
Rays predict two bright lines, one behind each slit. What appears is a whole row of evenly spaced bright and dark bands — fringes, as they are always called — including a bright one dead centre. That central one is bright because light from the two slits reaches it by paths of exactly equal length, so the path difference there is zero.
Each slit has a width of its own, so the fringes sit under the single-slit envelope of Bench 3 — the outer ones are dimmer, and far enough out they vanish.
Young's result was decisive: only waves interfere, so light is a wave, and by 1860 Maxwell would say what kind. But turn the light source down until only one photon at a time is in the apparatus — what happens? The experiment was first done with feeble light in 1909 and with electrons in 1961, and the answer is the subject of Module 21. Do not look it up first; predict it. Whatever you settle on, Feynman called what happens "the only mystery" of quantum mechanics and said the whole subject is contained in it — so everything you learn about fringe spacing here you will need there, unchanged.
Now close one slit and narrow the other. Rays say the beam should simply get thinner.
You narrow a slit further and further. The bright patch on the screen behind it:
What actually happens is the opposite: below a certain width the light starts to spread out, and the narrower the slit, the wider it spreads. That is diffraction, and it happens whenever a wave meets an obstacle or an opening comparable to its own wavelength.
This is not a defect of slits — it is a hard limit on every optical instrument. Any lens or mirror is an opening of a definite size — an aperture, in the language of optics — and a wave arriving at it is diffracted by that opening exactly as it was by the slit. The only difference is the shape: a circle rather than a long slit, which spreads each point of light into a small bright disc rather than a band. Doing the same sum for a circle gives the same λ-over-width answer with a factor of 1.22 in front of it; that number is the one thing the round shape changes, and where it comes from is a piece of mathematics you can take on trust for now. Two stars closer together than that disc cannot be separated, however perfect the glass and however great the magnification.
A film of oil on a puddle shows swirling colours. Light reflects off both the top and the bottom surface of the film, and those two reflections have travelled different distances — the second reflection has gone down through the film and back up again, so it has travelled twice the film's thickness further, and travelled that extra distance inside the film, where light moves more slowly. What counts is 2nt, twice the thickness times the film's refractive index. Where that path difference suits a particular wavelength, that colour reinforces; where it does not, that colour cancels. One extra rule is needed, and it is the whole reason a bubble goes black. A wave reflecting off a denser medium comes back flipped — half a wavelength out of step — while the reflection off the far side, going from dense to thin, is not flipped. So the two reflections start out already half a cycle apart, and a film much thinner than a wavelength cancels for every colour at once. Bright reflection needs 2nt = (m + ½)λ, not mλ. Since the film varies in thickness, different colours are favoured in different places, and you get bands of colour from a colourless oil in colourless water.
That is why a bubble turns black just before it pops. The film thins to far less than a wavelength, so the extra distance is almost nothing — but the flip is still there, the two reflections are still half a wavelength apart, and they cancel for every colour at once. The bubble stops reflecting altogether: you are looking at a hole in the light. An anti-reflection coating works the other way round. The coating is denser than air and the glass is denser than the coating, so both reflections are flipped and the two flips cancel out — which leaves the plain path-difference rule in charge. Make the layer a quarter of a wavelength thick — as measured inside the coating, λ/4n — the round trip down and back is half a wavelength, the two reflections destroy each other, and the light goes through instead of bouncing back. That faint purple sheen on a camera lens or a pair of spectacles is one of these.
Interference and diffraction prove light is a wave. Polarisation tells you what kind. In a light wave, what is doing the waving is an electric field — a push much like the one that makes your hair rise towards a comb you have just rubbed, except that in a light wave it oscillates from one side to the other unimaginably fast and points at right angles to the direction the wave is travelling. Because that push points sideways, it must point some particular way sideways: up and down, or left and right, or anything in between. That direction is the light's polarisation. Sound has no room for it, because sound is longitudinal — the air moves back and forth along the direction of travel, and there is no "sideways" to speak of. Light is transverse, and a transverse wave can be oriented.
Ordinary light from a lamp or the Sun is a jumble of every orientation at once. A polarising filter does not simply hunt through that jumble for the light already pointing its way and burn the rest. It takes whatever arrives and keeps only the part of it that lies along the filter's own direction — and what comes out is polarised along the filter, whatever came in. Light arriving at a small angle to the filter loses only a little. Light arriving at 45° loses half. Light arriving at 90° has no part lying along the filter at all, and is stopped completely. Because ordinary light is an even mixture of every orientation, averaging over all of them leaves exactly half the intensity after the first filter. Pass that through a second filter turned at an angle θ to the first, and what survives is:
Two filters are crossed at 90°, so no light gets through at all. You now slide a third filter, at 45°, into the gap between them. What happens?
Tick the box and watch. About an eighth of the original light emerges from an arrangement that passed exactly nothing a moment ago. The filter is not a sieve that only removes; it is a projection that also rotates what survives. Once you have seen that, the which-slit measurement of Module 21 will feel a good deal less alien, because it is the same mathematics.
| Where polarisation shows up | What is going on |
|---|---|
| Sunglasses that kill glare | light bouncing off a flat horizontal surface like water or a road comes back strongly polarised horizontally, because a surface reflects the part of the field lying along it far better than the part that does not. Polaroid lenses are oriented vertically, so they block precisely the reflected glare while letting ordinary light through. Tilt your head 90° while wearing them and the glare comes back. |
| Every screen you own | an LCD is two crossed polarisers with liquid crystal between them. The crystal is twisted at rest, and in that state it rotates the light's polarisation through 90° so it passes the second filter. A voltage pulls the molecules straight, the rotation stops, and the light is blocked — which is how each pixel is dimmed. Look at your phone through polarised sunglasses and rotate them — it goes black. |
| The blue sky is polarised | the scattering of Module 9 comes out polarised in a band 90° from the Sun. Bees navigate by it, Vikings may have used a crystal to find the Sun through cloud, and a polarising filter is why a photographer's sky comes out dramatically darker. |
| 3D cinema | two images projected with opposite polarisations, and glasses with a different filter over each eye. Each eye receives only its own picture. |
| Stress in plastic | place a clear plastic ruler between crossed filters and bend it: a riot of colour appears along the lines of stress. Engineers used this to find stress concentrations in models of bridges and dams before computers could. |
Take a pair of polarised sunglasses and look at your phone or laptop screen through one lens, then slowly rotate the glasses. At some angle the screen goes completely black — you have crossed your lens with the polariser built into the display. Then look at a puddle, or the sky at right angles to the Sun, and rotate again. Same filter, three different stories.
1. In a double-slit experiment you move the screen twice as far away. The fringes become:
2. You repeat Young's experiment with red light and then with blue. Compared with red, the blue fringes are:
3. Unpolarised light of intensity 100 units passes through a polariser, then a second one at 60° to the first. The intensity emerging is: