The light with no direction of shaking

A polarised wave has a direction of shaking, and a polariser is a device that asks about it. Both statements are true, both are useful, and neither describes most of the light there is. Sunlight, lamplight and the light scattered from almost any surface have no direction of shaking, and the usual account of what that means turns out not to be a description of anything measurable.

The usual account is that the direction jitters rapidly and randomly, averaging to nothing. It is a good picture of one way to produce such light. It is not what the light is, because a beam made by combining two independent sources — one horizontal, one vertical, with no correlation between them — has no direction that jitters and is experimentally indistinguishable from the first in every respect.

What can be said about a beam is what a detector can measure, and a detector measures powers. That turns out to be enough.

The measurement that separates them #

The experiment is a polariser and a power meter, and the whole content is in the shape of one curve. A fully polarised beam goes to zero at one angle. A beam with no preferred direction gives a flat line at half. Everything between gives a sinusoid in twice the angle, sitting on a pedestal, and the ratio of the swing to the pedestal is the number that says how polarised the beam is.

That ratio is called the degree of polarisation, and the figure’s assertion is that the modulation depth measured off the curve is that number exactly — not approximately, and not for a particular model of how the light was made. The identity holds because the transmitted power for any beam is a fixed combination of the beam’s own measurable quantities, whatever produced them.

The pedestal is worth a sentence of its own. A polariser passes half of any beam, averaged over a turn — including a fully polarised one, which passes everything at one angle and nothing a quarter turn later. That is why the average carries no information at all and why the whole measurement lives in the modulation.

Four numbers, and why four #

The four quantities are the total power; the excess of horizontal over vertical; the excess of light at 45° over light at 135°; and the excess of right-circular over left-circular. Each is a difference of two powers, so each is measurable with a polariser, a waveplate and a detector, and no phase or field is measured anywhere.

Why exactly four, and why these? Because a beam’s polarisation is carried in the correlations between two field components, and there are four independent real numbers in a two-by-two Hermitian matrix of such correlations: two diagonal entries and the real and imaginary parts of one off-diagonal. The four intensities are those four numbers in a different basis, chosen so that each is a difference of two things a laboratory can actually measure.

That the description is a matrix rather than a vector is the whole difference from the pure-state case. A fully polarised beam is a two-component complex vector with two real degrees of freedom once the overall amplitude and phase are removed. A partially polarised one has three, and the third is exactly how polarised it is. The matrix carries all four; the vector cannot carry the fourth, and no repair of the vector description reaches it.

The ball, and what its inside is for #

The Poincaré sphere is already the map of pure states: linear round the equator, circular at the poles, and antipodes orthogonal to each other. What the Stokes description adds is the interior.

The geometry then makes a statement that is not obvious from the algebra. Any point inside the ball can be written as a weighted sum of the centre and a point on the surface, in exactly one way — so any partially polarised beam splits uniquely into a fully polarised part and a part with no preferred direction, and the fraction in the polarised part is the radius. That decomposition is not a modelling convenience; it is what a polarimeter reports, and it is why “seventy per cent polarised at twenty-five degrees” is a complete description of a beam rather than a summary of one.

The uniqueness matters. There are many ways to build a beam at a given interior point — two uncorrelated beams of different intensities, one beam whose direction wanders, a beam depolarised by scattering — and the decomposition into a polarised part plus an unpolarised one is the same for all of them. The physics has no access to which construction was used, and the geometry says so by giving one answer.

Why the same structure keeps appearing #

The Stokes description is the classical instance of something that recurs whenever a system is not in a definite state: a matrix of correlations rather than a vector, with its trace fixed and the rest describing how definite things are.

The state that has lost its interference is described the same way in quantum mechanics, by a density matrix whose diagonal entries are probabilities and whose off-diagonal entries are the coherences — and the analogy is not loose. A photon’s polarisation is a two-state quantum system, its density matrix is a two-by-two Hermitian matrix, and the Stokes parameters are its expansion in the Pauli matrices. The sphere in the figure is the Bloch sphere under another name, and the interior is mixed states.

That correspondence explains why the classical measurement is complete. Six intensities determine the density matrix of a two-state system, and nothing else about the polarisation exists to be measured. It also explains why the depolarisation is irreversible in the same sense: a beam at the centre of the ball cannot be turned into a beam on the surface by any lossless device, because such devices act as rotations of the ball and rotations preserve the radius. Making light more polarised means throwing some away.

Two polarisers, and what the pedestal does #

Putting the ordinary two-polariser experiment beside the rotating-polariser one settles a confusion that the two figures make obvious and that words do not.

Malus’s law describes the second polariser in a pair, and it goes to zero at ninety degrees for any input whatever — because the first polariser has already thrown away everything except one direction. So the classic demonstration reports nothing about the light that went in. It reports what a polariser does to already-polarised light, which is a property of the polariser.

The single rotating polariser is the measurement that asks about the source, and its curve does not go to zero unless the source was fully polarised. The difference between the two experiments is exactly the pedestal, and confusing them is the reason “unpolarised light gives half” and “crossed polarisers give nothing” sound like statements about the same thing.

There is a third arrangement worth mentioning because it looks impossible and is a straight consequence of the same arithmetic. Two crossed polarisers pass nothing; slide a third between them at forty-five degrees and light appears. Nothing has been added — the middle sheet only absorbs — and what has happened is that the middle polariser produced a beam at forty-five degrees, which the last one is no longer crossed with. The state after each element depends only on that element, not on where the light came from, and the third sheet erases the history the second one would otherwise have been blocked by.

Where the light gets its polarisation, and loses it #

Almost nothing emits polarised light. A thermal source radiates from many independent emitters with no common orientation, and independence is exactly what puts a beam at the centre of the ball — the same independence that stops two lamps interfering, acting on the polarisation rather than on the phase.

What polarises light is a process that treats two directions differently. Reflection at a dielectric surface is one, and it is the reason light from a wet road or a lake is strongly polarised horizontally. Scattering by molecules is another, and it makes the blue sky about eighty per cent polarised at right angles to the Sun. Birefringent crystals are a third, and dichroic sheet a fourth.

Each of those is a partial process and produces a partially polarised beam, which is why the interior of the ball is where most real light sits. The sky’s eighty per cent is not a fully polarised beam contaminated by stray light; it is a beam at radius 0.8, and its polarised fraction is a genuinely well-defined thing that insects and cephalopods navigate by.

Depolarisation runs the other way and is more interesting than it sounds. A beam passing through a scattering medium loses polarisation gradually, and the rate depends on the size of the scatterers — small ones preserve it over many scattering events and large ones destroy it in a few. So the polarisation surviving a passage through fog, tissue or paint is a measurement of what is in the way, and it is one of the few optical measurements that survives multiple scattering at all.

What the interior is measured for #

A number between zero and one that no pure-state description can carry sounds like an abstraction, and it is one of the more heavily used quantities in observational physics.

Astronomers measure it to find magnetic fields. Dust grains in interstellar space align with the field and absorb one polarisation preferentially, so starlight arriving through them is partially polarised — a per cent or two — with the direction mapping the field projected on the sky and the degree measuring how well aligned the grains are. Synchrotron emission from relativistic electrons is intrinsically polarised up to seventy per cent, and the measured degree says how ordered the field is in the emitting region, which no other observation reaches.

The same measurement is made on the microwave background, where a degree of polarisation of a few parts in a million carries information about what the plasma was doing at the moment it became transparent — and the whole of that signal lives in a quantity that a description in terms of a wave with a direction would have no room for.

Closer to hand, the polarisation surviving a passage through tissue is used to see through it: light that has scattered many times is depolarised, so rejecting the unpolarised part rejects the multiply scattered background and leaves the light that went nearly straight through. That is a measurement whose whole content is the radius in the ball, and it works because the radius, unlike an intensity, distinguishes light by its history.

The geometry does the work #

The contrast between the surface and the interior is worth one more paragraph, because it is where the two descriptions stop being alternatives and become different subjects.

On the surface, the sphere is a space of states, and paths on it mean something. A beam taken round a closed circuit by a sequence of waveplates comes back to where it started with an extra phase, and the phase is half the solid angle enclosed — a geometric phase, depending on the shape of the circuit and not on how quickly it was traversed. That is a statement about the wave, and it needs the wave to have a definite state at every moment.

The interior has no such structure. A partially polarised beam has no phase, because it has no single wave; what it has is a set of correlations, and a circuit through the interior means nothing more than a sequence of intensities. Devices that move a beam through the interior are exactly the ones that lose information — depolarisers, scatterers, absorbers with more than one axis — and no arrangement of lossless components enters it.

So the ball divides cleanly into a surface where interference is available and an inside where it is not, and the radius is the measure of how much interference remains. That is the same statement the visibility of a fringe pattern makes about coherence, applied to the polarisation degree of freedom rather than to the phase, and the two visibilities are the same kind of number for the same reason.

Where the model stops #

The four parameters describe one direction of propagation at one wavelength. A real beam is a bundle of directions with a spectrum, and its polarisation can vary across both — light can be horizontally polarised at one wavelength and vertically at another, giving a beam that appears unpolarised to a broadband detector and is fully polarised at every frequency in it. The four numbers are an average, and the averaging is over whatever the instrument accepts.

The description assumes the beam is transverse, which is a statement about a plane wave. A tightly focused beam has a longitudinal field component near the focus, and a two-component description does not cover it. The correction is small for anything below about NA 0.7 and is not negligible at the apertures a microscope works at.

Nothing here is about a single photon. The Stokes parameters are intensities, so they are ensemble quantities by construction, and a beam at the centre of the ball is a statement about a population rather than about any member of it. What a single photon does in a polariser is a separate question with a separate answer.

And the measurement assumes ideal components. A real polariser passes a little of the wrong polarisation and a real waveplate is a quarter wave only at one wavelength; both errors put a fully polarised beam at a radius slightly under one, which is indistinguishable from genuine depolarisation. Calibration of a polarimeter is largely a matter of measuring how far its own components move the answer.

What the pictures cannot show #

The ball is drawn flat and the sphere it represents is three-dimensional, so the picture foreshortens the circular direction badly and the positions of the marked states are more honest about their radii than about their orientations. The radius is the quantity the essay is about, and it is the one the projection preserves.

None of the figures shows time. Every quantity here is an average over an interval long compared with whatever correlations the light has, and a measurement faster than that interval sees something quite different: a beam that is unpolarised over a millisecond may be fully polarised over a femtosecond, with the direction wandering between. The Stokes parameters are exactly the quantities that do not depend on how the wandering happens, which is their strength and the reason they cannot report it.

Where the ladder goes next #

The polarisation ladder began with the direction of the shaking and the filter that asks about it and went through the angle at which reflection picks a side, the crystal that answers twice, the rotation a return trip doubles, the phase that is only a shape and the retarder with no crystal in it. This rung asks what to do about light that has no direction at all. The rungs after it: the Mueller matrix, which describes what a component does to a Stokes vector and can represent depolarisation where a Jones matrix cannot; polarimetric imaging, where the four parameters are measured at every pixel and reveal stress, orientation and surface shape invisible to intensity alone; and the polarisation of a single photon, where the same matrix returns as a density matrix and the interior of the ball becomes entanglement with something else.

The habit worth carrying away is about what a description can represent. A model that cannot express “no definite value” will describe every system as having one, and the repair is almost always the same: replace the state by the correlations, and let the length of what is left say how definite the system actually is.