Illustrated Physics
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Series

Polarisation — the series

11 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Malus's law. The fraction of polarised light passing a filter, against the angle between the light's own direction of shaking and the filter's axis. It is the cosine squared: half at 45 degrees, nothing at 90.

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

    A wave that travels one way can still shake in any direction across that way. Light does, most of it shakes in all of them at once, and a sheet of plastic can ask which.

    part 1 · optics
  2. The two reflectances, and the angle one of them loses. Reflectance against angle of incidence for light going from n = 1 into n = 1.5. The upper curve is light polarised with its electric field along the surface, which reflects more and more strongly until at grazing incidence everything reflects. The lower curve is light polarised in the plane of incidence, and it does something the other cannot: it falls to exactly zero at 56.31°, where tan θ = 1.5000, and then rises again. At normal incidence the two are equal at 4.00% because there is no plane of incidence to tell them apart. The dashed curve is the transmittance, computed from the transmission coefficients and the two media's projected impedances rather than as one minus the reflectance; it agrees with one minus the reflectance to 4.4e-16 across the whole range, which is where the energy accounting can be seen to close.

    The angle at which reflection picks a side

    At one angle of incidence, a water surface reflects no light at all of one polarisation. The Fresnel algebra says so, and says nothing about why. The reason is that the reflected ray would have to leave along the axis of the charges radiating it — and a shaking charge sends nothing along the direction it shakes in.

    part 2 · optics
  3. The index that depends on which way the light is going. The two refractive indices of 3 uniaxial crystals, against the angle between the wave normal and the crystal's optic axis. The flat lines are the ordinary index, which is the same in every direction because the ordinary wave's field is always perpendicular to the axis. The curves are the extraordinary index, which runs from the ordinary value along the axis — where the two waves are identical and the crystal behaves like glass — to its extreme value at right angles to it. calcite (CaCO₃) has n_o = 1.6584 and n_e = 1.4864, so n_e − n_o = -0.1720; quartz (SiO₂) has n_o = 1.5443 and n_e = 1.5534, so n_e − n_o = 0.0091; lithium niobate has n_o = 2.3005 and n_e = 2.2075, so n_e − n_o = -0.0930. The sign of that difference is what makes a crystal positive or negative, and it decides which of the two images in a double-refracting crystal is the one that moves.

    The crystal that answers twice

    Lay a piece of calcite on a printed page and the print appears twice. One image sits still when the crystal is turned and the other goes round it. Nothing has been done to the light except pass it through a material whose response to a field is not a number.

    part 3 · optics
  4. What the return journey does to each of them. The rotation of the plane after a beam has gone through a rotator, been reflected, and come back, against the length of the rotator — divided by the one-way rotation, so the two answers are 0 and 2 and nothing else can happen. A naturally active medium is handed with respect to the beam: reverse the beam and the sense of the rotation reverses with it, and the second pass undoes the first exactly, at every length and every wavelength. A Faraday rotator is handed with respect to the field, which does not care which way the light is going, so the second pass adds to the first and the round trip is twice the single one: 1 mm of it gives 21.7° out and 43.4° back; 2 mm of it gives 43.4° out and 86.8° back; 4 mm of it gives 86.8° out and 173.6° back; 8 mm of it gives 173.6° out and 347.2° back. That is a violation of reciprocity, and it is only available because a magnetic field is odd under time reversal. Everything a passive optical component can do — a lens, a mirror, a waveplate, a piece of quartz — looks the same run backwards, and none of them can be made into a one-way street. This can.

    The rotation a return trip doubles

    Quartz turns the plane of polarisation and so does glass in a magnetic field. The two look identical on the way through and are opposites on the way back — the crystal undoes its own rotation exactly, and the magnet adds to it. That difference is the whole of why a one-way street for light can be built at all, and why nothing passive will ever be one.

    part 4 · optics
  5. A circuit on the sphere of polarisations. The Poincaré sphere, on which every polarisation state is a point: linear states around the equator, circular at the poles, and orthogonal states at opposite ends of a diameter. The triangle is a closed circuit — linear at 0°, then linear at 23°, then left circular, and back — taken along geodesics, which is what a sequence of ideal polarisers does. Bringing a state round it returns it to exactly the state it started in, and multiplied by a phase: 0.3927 radians here, against minus half the enclosed solid angle of -0.7854 steradians. The two agree exactly, and neither calculation knows about the other — the phase is the argument of a product of three overlaps between Jones vectors, and the solid angle is spherical geometry. Nothing about the elements used, their thickness, or the wavelength enters. A phase that depends only on the shape of a path is the signature of a geometric phase, and this is the oldest known example of one.

    The phase that is only a shape

    Take a beam of polarised light through a sequence of elements that returns it to the polarisation it started with, and it comes back with a phase it did not have before. That phase is not an optical path length — it does not depend on the thickness of anything, or on the wavelength, or on how slowly the sequence was carried out. It is minus half the area the path enclosed on the sphere of polarisation states, and nothing else.

    part 5 · optics
  6. Past the critical angle, all that is left of a reflection is its phase. The phase each polarisation acquires on total internal reflection at an index ratio of 1.518, against angle of incidence, together with the difference between them. Below the critical angle of 41.19° there is a transmitted beam and the reflection coefficients are real; above it the transmitted wavenumber is imaginary, the coefficients have modulus exactly one — every photon comes back — and the only thing that distinguishes one angle from another is the phase. The two polarisations acquire different phases, and their difference peaks at 46.533° at an incidence of 51.05°, which the closed form puts at the same place. That difference is a retardation: a wave plate made out of an angle, with no birefringent material anywhere in it, and — because the expression contains only the index ratio — one that barely changes with colour.

    The retarder with no crystal in it

    A wave plate turns linear polarisation into circular by making one component travel a little further than the other, which requires a birefringent crystal cut to a thickness and works properly at one wavelength. Total internal reflection does the same job with a phase that comes from the geometry instead — and because a refractive index barely changes across the visible where a wavelength changes by a factor of two, the same block of ordinary glass is a quarter-wave plate for every colour at once.

    part 6 · optics
  7. One measurement that separates unpolarised from polarised. What a rotating linear polariser passes, against its angle, for beams of the same total intensity and degrees of polarisation 0, 0.35, 0.7, 1. Every curve has the same average — a polariser passes half of any beam over a whole turn, whatever its state — and they differ only in how deeply they modulate. The depth of the modulation is the degree of polarisation, exactly: a fully polarised beam goes to zero at one angle and a beam with no preferred direction gives a flat line at a half. That is the whole measurement, and it is why "unpolarised" is a statement about a modulation depth rather than about what a wave is doing.

    The light with no direction of shaking

    Unpolarised light is not a state of a wave; it is the absence of one, and no description of a single wave can represent it. What can is a set of four numbers, all of them powers a detector reads — and they describe every beam there is, including the ones that are neither polarised nor not.

    part 7 · optics
  8. The pattern the whole sky is written in. The sky as a disc — zenith at the centre, horizon at the rim, equal angles at equal distances — with the sun 30° above the horizon. Each short line is the direction the electric field vibrates in at that point, and its length and darkness are how polarised the light there is. The directions are perpendicular to the plane containing the sun, the observer and the point, which puts them tangent to circles centred on the sun. The heavy arc is the locus 90° from the sun, where the polarisation is strongest — 74 per cent here — and it is a great circle rather than a patch: a band across the sky, not a region near the horizon. This is what a polarising filter on a camera acts on, and it is why turning one darkens a band of sky and leaves the rest almost untouched, and why the effect is strongest when the sun is off to one side and absent when it is behind the photographer.

    The pattern the sky is written in

    Scattered sunlight is polarised, so the whole sky carries a direction of vibration at every point — arranged in circles about the sun, strongest on the great circle ninety degrees away from it, and vanishing at points that were found by looking before anyone could explain them. Bees navigate by it and a camera filter reads one band of it.

    part 8 · optics
  9. A squeezed disc between crossed polarisers. The dark fringes seen in a 50 mm disc of polycarbonate, 6 mm thick, squeezed across a vertical diameter by 500 N and viewed between crossed circular polarisers in light of 550 nm: each dark line is a contour on which the principal stresses differ by a whole number of fringe values, from the stress-optic law with a coefficient of 78 Brewsters. At the centre the fringe order is 3.61; it climbs steeply towards the two loading points, where the fringes crowd together and the stress grows without limit, and falls to zero at the free edges level with the centre. The pattern is the exact stress field of an elastic disc under diametral load; every fringe can be counted from a dark point, and the count times the fringe value is the difference of the principal stresses at that place.

    The stress that polarised light can see

    Squeeze a disc of clear plastic and look at it between two crossed polarising filters, and it fills with dark lines and coloured bands that crowd towards the points where it is pressed. Each line is a contour of stress. Glass, plastic and most transparent solids become doubly refracting when they are loaded, in proportion to the difference between their largest and smallest stresses, and polarised light turns that difference into a pattern that can be counted. David Brewster found it in 1816, the same Brewster whose angle polarises reflected light, and for a century the way to learn the stresses in a bridge or a dam was to cut its shape from plastic, load it and photograph the fringes.

    part 9 · optics
  10. Light that turns with the molecules it passes. Through a layer of liquid crystal whose molecules turn steadily through 90° from the entrance face to the exit (dashed line), the direction of the light's polarisation against depth, for linearly polarised light entering along the molecules, in a thick cell and a thin one; u = 2dΔn/λ measures the layer's birefringence against the wavelength. In the thick cell the polarisation turns with the molecules, never more than a few degrees behind, and leaves rotated by 90°: 100.0 per cent of the light passes an analyser set along the exit molecules. In the thin cell the light barely notices the twist; its polarisation lags further and further behind and leaves turned only part of the way, with 31 per cent passing. Mauguin saw the first case in 1911: a birefringent medium that twists slowly enough drags the light's polarisation round with it.

    The twist that carries light round a corner

    Between two crossed polarisers, a sandwich of liquid crystal a few micrometres thick turns black to white. Its long molecules lie flat, but turned a quarter-turn from one face of the layer to the other, and polarised light entering along the molecules at one face comes out along them at the other — rotated by ninety degrees and able to pass the second polariser. Charles Mauguin found in 1911 that a slowly twisting birefringent layer drags light's polarisation round with it. Sixty years later the effect became the liquid-crystal display. The layers in those displays, though, are too thin to drag anything: the light inside them wanders off and comes back, at exactly the right thickness.

    part 10 · optics
  11. How fine a grid has to be to polarise. The fraction of the power transmitted by a grid of thin parallel wires, against the ratio of the wire spacing d to the wavelength, on logarithmic axes — for the field along the wires with three wire radii (solid), and for the field across the wires with the thickest (dashed), from the quasi-static theory of strip gratings, valid while the spacing is under about half a wavelength. Across the wires nearly everything passes at every spacing. Along them the leak is about four times the square of (d/λ)·ln(d/2πa): 0.1% of the power at a spacing of a hundredth of a wavelength, 10% at a tenth, 64% at four-tenths, for wires of radius 3% of their spacing. Thicker wires block better because the inductance of the grid falls. The diamond marks Hertz's grid of 1888, 4% leaking along the wires; a visible-light polariser needs the same ratio at a wavelength of half a micrometre, which means wires about 140 nanometres apart.

    The grid that stops the light along its wires

    A row of parallel wires looks like a fence, and a fence ought to let through a wave that shakes along its slats. A grid of wires does the opposite. A field along the wires drives electrons up and down them, the currents re-radiate, and the wave is reflected; a field across the wires can move charge only through a wire's width, and passes. Heinrich Hertz used such a grid in 1888 to show that his new waves were polarised. The same grid, shrunk two hundred thousand times to wires 140 nanometres apart, polarises visible light in projectors — and every sheet of polarising plastic is the same idea made from chains of iodine.

    part 11 · optics

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