The direction of the shaking, and the filter that only asks about it

A wave travelling along a rope can be shaken up and down, or side to side, or at any angle in between, and all of those are the same wave travelling in the same direction at the same speed. The direction of travel does not fix the direction of the shaking. It only rules out one: the shaking cannot be along the direction of travel, because that would be a different kind of wave.

Light has this freedom, and a piece of plastic can interrogate it.

Only transverse waves have the freedom #

The property exists at all because light is a transverse wave, and it is worth being clear that this is a real division rather than a technicality.

A wave on water or a rope displaces the medium across the direction of travel, leaving a whole plane of directions to choose from. A sound wave in air displaces it along the direction of travel: the molecules move forwards and backwards, there is no choice available, and sound cannot be polarised. No filter, however cleverly made, can ask a sound wave which way it is shaking, because the question has one possible answer.

So the existence of polarising filters is direct evidence about what kind of wave light is — as is the angle at which a reflection picks one direction over the other — and historically that is exactly what it was: the phenomenon was known and inexplicable for a century, because the wave theory of the time had light as a longitudinal wave in an ether, like sound. Polarisation was the observation that would not fit, and accommodating it forced the transverse description that Maxwell’s equations later produced without being asked. A crystal that answers a single ray with two was the observation that made it unavoidable.

Malus’s law, which is a projection #

A polarising filter passes the component of the field along its own axis and absorbs the rest. That single sentence gives the whole law, because a component is a projection and a projection is a cosine.

If the light’s field has amplitude $E$ at an angle $\theta$ to the axis, the amplitude that gets through is $E\cos\theta$. Intensity goes as the square of amplitude, so

$$\frac{I}{I_0} = \cos^2\theta.$$

The squaring is the only step that is not geometry, and it is the step that makes the law look like a statement about light rather than about vectors. Resolving a vector into components is the same operation used to put a weight onto a slope; the difference is that here what is measured is the square of the component, because a detector responds to energy.

Three values are worth carrying. At $45°$, half. At $30°$, three quarters. At $10°$, ninety-seven per cent — the law is flat near its maximum, in exactly the way a projectile’s range is flat near 45°, so alignment does not have to be precise to be effective. It is the other end that is sharp: near $90°$ the transmission falls as the square of the angle from crossed, which is why a good extinction is achievable and why polarising filters are specified by the tiny fraction they leak rather than by the large fraction they pass.

The stack, and the third filter #

Two crossed filters pass nothing. That is unremarkable — the first admits only vertical shaking, the second admits only horizontal, and no light is both.

Now slide a third filter between them, at 45°. Light appears.

The arithmetic is two applications of the same cosine. The vertical light meets the middle filter at $45°$ and half of it survives, now shaking at $45°$. It then meets the last filter at $45°$ again, and half of that survives: $0.5 \times 0.5 \times 0.5 = 0.125$ of the original unpolarised beam.

Nothing about the arithmetic is difficult and the result is still surprising, because the intuition it violates is a strong one: filters take things away, and adding one more should not put light where there was none. The resolution is that a polariser does not select from a fixed set of light. It transmits a component, and the light that leaves it is genuinely shaking at the filter’s angle rather than at its previous one — so the middle filter does not pick out light that was already at $45°$, it produces it.

This is the classical shadow of something that becomes central in quantum mechanics, where the same experiment is done with single photons and the same numbers come out. The middle filter is a measurement that changes what it measures, and the two-filter and three-filter results differ for that reason. Very little of the strangeness belongs to the quantum version; most of it is already here, in a stack of plastic.

Reflection does it for free #

Light does not need a filter to become polarised. Any reflection does it partially, and at one particular angle it does it completely.

The angle is Brewster’s, given by $\tan\theta_B = n_2/n_1$, and the reason it is special is visible in the drawing rather than in the formula: at that incidence the reflected ray and the refracted ray are at right angles to each other.

The mechanism follows from that geometry. Reflected light is radiated by charges in the surface, driven into oscillation along the direction of the refracted wave’s field. A charge oscillating along a line radiates nothing along that line — an oscillating dipole has a null on its own axis. When the reflected direction happens to lie along the oscillation direction for one of the two polarisation components, that component cannot be radiated at all, and the reflection consists entirely of the other one. Send that reflection back through the same medium in a magnetic field and the rotation does not undo itself.

So the polarisation of reflected light is not an extra fact about surfaces. It is a consequence of Snell’s law — which fixes where the refracted ray goes — plus the fact that accelerating charges do not radiate along their own line of acceleration.

What decides the polarisation of the reflected ray is the direction of the refracted one. When the two are at right angles the reflected beam is completely polarised, because the oscillation that would have to radiate along the reflected direction is the one direction a dipole cannot radiate in. That is why Brewster’s angle depends on the indices in the same way refraction does, and why it is an angle rather than a property of the surface.

What it is used for #

Polarisation is unusually rich in instruments, because it carries information that intensity and colour do not.

Glare. Light reflected from a horizontal surface — water, a road, a car bonnet — is partially polarised horizontally, so a filter passing only vertical shaking removes most of it while leaving the rest of the scene. Polarising sunglasses are a Brewster-angle device, and they work far better on water than on a vertical shop window because the geometry is what matters.

Stress. Many transparent materials become birefringent when strained: they rotate the polarisation by an amount proportional to the stress. Placed between crossed filters, a plastic model of a bridge or a spanner shows coloured fringes that map the stress field directly. Before finite-element analysis this was how stress distributions in complicated shapes were found, and the pictures are still used to check the software.

Displays. Every liquid-crystal screen is two crossed polarisers with a controllable rotator between them — the three-filter experiment above, with the middle element’s angle set electrically, pixel by pixel. The reason a phone screen goes black at some angles through polarising sunglasses is that its output is polarised, and the reason 3D cinema works is that two projected images can be given orthogonal polarisations and separated by the glasses.

The sky. Scattered light is polarised, most strongly at 90° from the sun, which is why a polarising filter darkens a blue sky selectively and why some insects navigate by a patch of sky rather than by the sun itself. That the sky is polarised at all is the same physics as the reason it is blue, and the two facts come out of one calculation.

The window that started it #

Malus discovered the reflection effect in 1808, by accident, looking through a calcite crystal at sunlight reflected from the windows of the Luxembourg Palace in Paris.

Calcite splits a beam in two — it was already known to do that, and the two images were the standing puzzle of the subject. What Malus noticed was that when he turned the crystal while looking at the reflection, one image faded to nothing at a particular orientation, which sunlight looked at directly never did. The reflection had done to the light what the crystal did, and the effect was in the light rather than in the stone.

The observation is unusually cheap to reproduce and unusually hard to interpret, which is why it took a further fifteen years and Fresnel’s transverse wave theory to explain. It is also a good example of a result that arrived through an instrument being used on the wrong thing: the crystal was the object of study, the window was scenery, and the finding was in the scenery.

What the half costs #

A polariser throws away half of any unpolarised beam before it does anything useful, and that half becomes heat in the filter. For sunglasses it is a benefit. For a display it is the dominant inefficiency in the whole device.

The arithmetic is worth following because it explains an object most people own. A liquid-crystal screen’s backlight is unpolarised. The first polariser removes half of it. The colour filters that make a pixel red, green or blue remove roughly two thirds more, since each subpixel passes only its own third of the spectrum. The liquid crystal and the second polariser take a further share, and the aperture ratio — the fraction of the panel that is not wiring — takes another. The light reaching a viewer is a few per cent of what the backlight emitted, and about half of the loss happened in the first component, for a reason that is pure geometry.

That is why the alternative display technologies that emit light directly are more efficient than any improvement to the filters could make an LCD: the loss is not a manufacturing shortcoming but the cost of asking an unpolarised source a question it has no single answer to. The recycling schemes that exist — reflecting the rejected polarisation back into the backlight to be scrambled and given another chance — are attempts to get some of it back, and they work partially, which is the best available.

The same accounting sets the limit on the three-filter trick. Twelve and a half per cent is the maximum through a crossed pair with one filter between them; the arrangement is not free light but a smaller share of the light that got past the first filter. Adding more intermediate filters improves it — with $N$ filters each turned by $90°/N$, the transmitted fraction tends to a half as $N$ grows, since each small rotation costs $\cos^2(90°/N)$ and there are $N$ of them — which is a genuinely surprising limit and one that a reader can check on the curve above.

The window cut at the angle where nothing reflects #

Brewster’s angle is usually presented as the angle at which a reflection becomes polarised. Turned round, it is the angle at which one polarisation is not reflected at all, and that reading is the one that gets built into hardware.

A gas laser’s discharge tube has to be sealed, and every window in the path costs a reflection at each of its two surfaces — four per round trip, several per cent each, which is more than the gain of a small tube can afford. Set the windows at Brewster’s angle and the loss disappears for one polarisation, exactly, while the other still pays its several per cent on every pass.

The consequence is that only one polarisation ever reaches threshold. A helium–neon laser with Brewster windows emits light that is linearly polarised to better than a part in a thousand, not because anything polarised it but because the other option was quietly prevented from oscillating. It is a filter built out of an absence.

The same zero is used to split rather than to seal. A polarising beamsplitter passes one polarisation and reflects the other, with both outputs usable, so no light is thrown away — which is the fix for the half-the-beam loss described below in any application where both halves can be given a job. Every projector that recycles the rejected polarisation and every interferometer that separates two channels is built on that.

The second channel that costs no bandwidth #

At radio frequencies the same law is measured as received power, and it turns polarisation into a resource.

A receiving dipole responds to the component of the field along its own length, so the power it delivers falls as the cosine squared of the angle between the wave’s polarisation and the antenna — Malus’s law with metres in place of micrometres. Two antennas at right angles, at the same place, therefore receive two signals that do not interfere with one another.

That is worth a great deal, because bandwidth is scarce and polarisation is free. A satellite broadcasting link sends two independent programmes on the same frequency, one horizontally polarised and one vertically, and a receiver rejects the unwanted one by twenty or thirty decibels simply by being aligned. Terrestrial links and cellular systems do the same thing, and the number quoted for how well it works — the cross-polarisation discrimination — is Malus’s law near its steep end, where a small misalignment leaks a squared-small fraction.

And a reader can check that they are not entirely blind to any of this. Stare at a large, uniformly polarised bright field — a white screen on a laptop, or a patch of blue sky ninety degrees from the sun — and after a few seconds a very faint yellowish bow-tie appears near the centre of vision, at right angles to the polarisation, fading if the gaze is held still and returning if the head is tilted. It is Haidinger’s brush, produced by dichroic pigment arranged radially in the retina, and it is the only polarisation detector on this page that needs no equipment at all.

What the picture cannot show #

The figures on this page draw a filter’s axis as a line and the light as a bar of intensity, and both are compressions of something with a time dimension.

The field of a linearly polarised wave is not a fixed arrow. It oscillates along its direction at the frequency of the light — some $5\times10^{14}$ times a second for visible light — passing through zero twice per cycle and reversing. The “direction of shaking” is the line that oscillation happens along, and a still picture can draw the line or an instant, not both. The intensity a detector reports is the average over an enormous number of cycles, which is where the factor of a half in $\cos^2$ averaged over all angles comes from, and where the squaring is hiding.

That averaging is also why the stack figure draws bars rather than waves. What survives each filter is a time-averaged quantity, and the quantity that composes multiplicatively through a stack is the intensity rather than the field. Drawing the fields would show amplitudes multiplying as $\cos\theta$ at each stage, and the reader would have to square at the end to get anything measurable — which is correct, and less legible, and is the reason almost every treatment of polarisation shows intensities and mentions amplitudes only when the phases matter.

They matter exactly once on this page, and it is the case the figures cannot draw at all: when two components are out of step, the tip of the field traces an ellipse, and there is no instant, no bar and no axis line that captures it.

Where the model stops #

Unpolarised light is not a direction. The phrase suggests light shaking in no particular way, and the honest description is statistical: a beam whose polarisation changes randomly, on a timescale of nanoseconds or shorter, so that no direction is preferred on average. There is no instant at which the light is unpolarised. This is why the first filter in a stack always passes half regardless of its angle, and it is why “unpolarised” cannot be drawn as a state of the field but only as an ensemble.

Linear is not the only kind. If the two perpendicular components are out of step rather than in step, the field’s tip traces an ellipse or a circle rather than a line, and the light is circularly polarised. A linear filter passes half of circular light at every angle, so the apparatus on this page cannot distinguish circular light from unpolarised light at all. Telling them apart needs a quarter-wave plate, and the failure is a real limitation of the measurement rather than a subtlety.

Partial polarisation is the usual case. Real reflections at angles other than Brewster’s give a mixture, described by a degree of polarisation between zero and one. Everything on this page treats the two extremes.

The filter is not ideal. A real sheet polariser passes about 40 per cent rather than 50 of an unpolarised beam, and leaks a fraction of a per cent when crossed. The second number is what matters for a display’s black level, and improving it is most of what distinguishes a cheap polariser from an expensive one.

Where the ladder goes next #

The rungs from here: birefringence, and why one crystal produces two rays; the quarter- and half-wave plate, and circular polarisation as a controllable state; optical activity and the sugar solution that rotates the plane; the Fresnel equations, which give the amount reflected at every angle and of which Brewster’s angle is a single zero; scattering polarisation in full; and the quantum version, where the three-filter experiment is run one photon at a time and the classical arithmetic survives unchanged, which is more surprising than any of the numbers on this page.

The idea to carry forward is the projection. A polariser does not sort light into the kinds it likes and the kinds it does not — it takes a component, and the light that emerges has been changed into the thing that gets through.