Illustrated Physics
physics, drawn
Optics

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.
15 min read 5 figures Made into an instrumentApproximations that lie

Assumes: The crystal that answers twice · The direction of the shaking, and the filter that only asks about it

The crystal that answers twice found that a birefringent material splits light into two components polarised along its two axes, travelling at different speeds, and that a slab of the right thickness is a wave plate: it converts one polarisation into another by delaying one component against the other. The direction of the shaking had begun the subject with a pair of crossed polarisers that pass no light at all. The question this essay asks is what happens when a birefringent layer between those polarisers is not a single slab with one axis, but a stack of thin slabs each turned slightly from the last, so that its axis twists steadily through the layer.

The answer turns out to be the basis of the liquid-crystal display — and to come in two versions, only one of which the displays use.

A layer whose axis turns

A nematic liquid crystal is a fluid of long, rod-shaped molecules that tend to lie parallel to one another. Optically it is birefringent, with its slow axis along the molecules, and its birefringence Δn\Delta n is large — typically 0.1 to 0.2, against 0.009 for quartz. Placed between two glass plates whose surfaces have been rubbed in one direction, the molecules at each surface align with the rubbing. If the two plates are rubbed at right angles, the molecules between them turn steadily through ninety degrees across the gap, in a uniform helix with a quarter-turn over the thickness of the layer.

For light, that layer is a stack of thin wave plates whose axes rotate from one to the next. Its effect on any polarisation can be computed by multiplying the Jones matrices of the slices in order, and that is how every figure here was made: four hundred to two thousand slices, each a weak retarder turned a fraction of a degree further than the last, with no closed form assumed.

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.
Fig. 1 The direction of the polarisation through a 90° twisted layer against depth, for light entering along the molecules, beside the molecules’ own direction (dashed): in a thick cell, u = 2dΔn/λ = 12, the light turns with them and leaves rotated by 90°, all of it passing an analyser along the exit molecules; in a thin cell, u = 0.6, it lags and only 31 per cent passes.

In a layer that is thick enough — measured by u=2dΔn/λu = 2d\Delta n/\lambda, twice the number of wavelengths of retardance it holds — linearly polarised light entering along the molecules at the first face stays polarised along the molecules all the way through, turning with them, and leaves polarised along the molecules at the far face. It has been rotated by ninety degrees. Between crossed polarisers aligned with the two rubbing directions, all of it passes. Charles Mauguin observed this in 1911 in a layer of liquid crystal, and called it the guiding of light by the twist.

In a thin layer the light barely notices the twist. Its polarisation turns a little, lags further and further behind the molecules, and leaves turned only part of the way, so that most of it is blocked by the second polariser.

Order matters

The slice-by-slice calculation is worth a moment, because it shows something a single wave plate never does. Each slice changes the polarisation by a small rotation of the light’s state, and those rotations are about different axes, because each slice’s axis is turned from the last. Rotations about different axes do not commute: turning a book a quarter-turn about one edge and then about another does not leave it as the reverse order does. So the stack’s effect depends on the order of its slices, and it is not the effect of one wave plate with the average axis. A layer twisted clockwise and one twisted anticlockwise, with the same slices in opposite order, give the same transmission at normal incidence but different light inside.

That is also why the twisted layer cannot be reduced to ordinary optics of crystals. The crystal that answers twice had one fast axis and one slow, fixed throughout. Here the axes are different at every depth, and the light’s state at the exit is the result of a continuous sequence of small, non-commuting changes — computed here as a product of matrices, and in the uniform helix also solvable exactly, which is what Gooch and Tarry did.

Why a slow twist drags the light

The reason is a competition between two rates. The twist turns the axis by ninety degrees over the thickness dd. The birefringence, meanwhile, makes the two components of the light along and across the local axis drift out of phase, by 2πΔn/λ2\pi\Delta n/\lambda radians per unit length. Light polarised along the axis consists of only one of those components. If the axis turns slowly compared with that dephasing rate, the small amount of the other component that the turning generates at each slice is out of phase with the amount generated at the next, and the contributions cancel instead of accumulating. The light stays in the component that follows the axis. If the axis turns quickly, the contributions add up, and the light leaks out of the guided state.

That is the adiabatic theorem in optical form. The crossing that never happens found its quantum version: a system in one state of a slowly changing Hamiltonian stays in the corresponding state as the Hamiltonian changes, and the probability of a jump falls steeply as the change is slowed. Here the “state” is a polarisation, the “Hamiltonian” is the local birefringence, and slowness is the ratio uu of birefringent dephasing to twist.

The thicknesses that work exactly

The thicknesses at which the twist turns the light exactly. The share of light passing a 90° twisted cell between crossed polarisers, each set along the molecules at its face, against u = 2dΔn/λ: the Gooch–Tarry formula 1 − sin²((π/2)√(1 + u²))/(1 + u²) (line) and the product of 800 thin wave plates computed slice by slice (dots). With no birefringence nothing passes the crossed polarisers. As u grows the transmission climbs and touches one exactly at u = √3 = 1.732, √15 = 3.873 and √35 = 5.916, with shallow dips between; by u = 8 it never falls below 97.9 per cent. A display is made bright where the transmission is one, which for green light at the first maximum means a layer whose dΔn is 476 nm.
Fig. 2 The share of light passing a 90° twisted cell between crossed polarisers aligned with its faces, against u = 2dΔn/λ: the Gooch–Tarry formula (line) and the product of 800 slices (dots). The transmission touches one at u=3u = \sqrt3, 15\sqrt{15} and 35\sqrt{35}; by u = 8 it never falls below 97.9 per cent. For 550 nm light the first maximum needs dΔn = 476 nm.

The guiding is not perfect at any finite thickness, and the leakage does not fall smoothly. In 1975 C. H. Gooch and H. A. Tarry worked out the transmission of a ninety-degree twisted layer between polarisers aligned with its surfaces exactly:

T=1−sin⁡2 ⁣(π21+u2)1+u2.T = 1 - \frac{\sin^2\!\left(\tfrac{\pi}{2}\sqrt{1+u^2}\right)}{1 + u^2}.

It rises from zero for an unbirefringent layer, and it oscillates, touching one exactly whenever 1+u2\sqrt{1+u^2} is an even number: at u=3u = \sqrt3, 15\sqrt{15}, 35\sqrt{35}. Between those maxima it dips, by less and less as uu grows. The slice-by-slice product reproduces the formula to five decimal places, which is the check that the formula — derived for a perfect helix — is the physics of a stack of plates and not of some special assumption.

How much light fails to follow. The share of light that fails to turn with a 90° twist — 1 − T between crossed polarisers — against u = 2dΔn/λ, both on logarithmic axes (solid), with its envelope 1/(1 + u²) (dashed). The leak falls between its zeros as the inverse square of u: 0.99 per cent at most once u is 10, 0.010 per cent at 100. That is the adiabatic theorem in optics. A state carried by a slowly changing system follows the change, and the fraction left behind falls as the square of how slowly the change is made — here the ratio of how fast the birefringence turns the polarisation to how fast the twist turns the axis.
Fig. 3 The share of light that fails to turn with the twist, 1 − T, against u on logarithmic axes, with the envelope 1/(1+u2)1/(1 + u^2). Between its zeros the leakage falls as the inverse square of u: at most 0.99 per cent at u = 10 and 0.010 per cent at 100.

The envelope, 1/(1+u2)1/(1+u^2), is the adiabatic theorem’s quantitative form here: the fraction of light left behind falls as the square of how slowly the twist is made. The zeros between are interference. The small amount of light that leaks out of the guided state at the entrance can, at the right thickness, be returned to it by the time the light reaches the exit, and at those thicknesses nothing is left behind at all.

A coincidence, not a guide

Displays use the first of those zeros, and they do so for good reasons. But the first zero is not in the regime Mauguin described.

A cell that turns the light without guiding it. Inside a 90° twisted cell at the first Gooch–Tarry maximum, u = √3, against depth: the direction of the light's polarisation (solid) and its ellipticity angle (dotted, 0° for linear light, 45° for circular), beside the molecules' direction (dashed); for comparison, the direction in a cell at the third maximum, u = √35 (thin line). In the thin cell the light does not follow the molecules. It lags behind them, becomes elliptical — up to 30° of ellipticity, more than half-way to circular — and swings back, arriving linear and exactly along the exit molecules only because the cell's thickness is just right. The thick cell guides its light round smoothly. Displays use the thin one: the arrival is a coincidence of interference rather than the slow following Mauguin described, and it is a coincidence that holds well across the visible colours.
Fig. 4 Inside a cell at the first maximum, u=3u = \sqrt3: the polarisation’s direction (solid) and its ellipticity (dotted), beside the molecules’ direction (dashed), and the direction in a cell at u=35u = \sqrt{35} (thin line). At u=3u = \sqrt3 the light lags behind the molecules, becomes elliptical by up to 30°, and arrives linear along the exit molecules; at 35\sqrt{35} it follows them closely.

At u=3u = \sqrt3 the light inside the layer does not follow the molecules at all. Its polarisation falls behind them, and the light becomes elliptically polarised, at one point two-thirds of the way to circular. Then, in the second half of the layer, it swings back and arrives at the exit face exactly linear and exactly along the exit molecules. Nothing has guided it. It has taken a detour through elliptical states and returned, because at that particular thickness the phase accumulated by the two components brings it back. A thicker cell at the third maximum, u=35u = \sqrt{35}, does guide its light smoothly, and the comparison shows how different the two mechanisms look from inside even though their transmissions are identical.

The distinction is not academic. A guided cell’s transmission is insensitive to its exact thickness and to the wavelength, because guiding works for any large enough uu; an interference cell’s transmission depends on uu being close to 3\sqrt3, and uu depends on wavelength. That the first-maximum cell nonetheless works across the visible spectrum is a fortunate property of the curve’s shape near its first peak.

How white the bright state is. The share of light a 90° twisted cell passes between crossed polarisers, against wavelength across the visible, for cells designed at the first and second Gooch–Tarry maxima for 550 nm green light (dΔn = 476 and 1065 nm) and one too thin to guide, 250 nm. The first-maximum cell passes at least 90.6 per cent of every colour from 400 to 700 nm and the second-maximum cell at least 91.2: both bright states are close to white. The thin cell falls to 42 per cent at the red end, and its bright state is tinted. Between the first two the choice is made on speed: the first-maximum layer is less than half as thick, and a liquid-crystal layer relaxes in a time that grows as the square of its thickness.
Fig. 5 Transmission across the visible for cells designed at the first and second maxima for 550 nm (dΔn = 476 and 1065 nm) and one at 250 nm. The first two pass at least 90.6 and 91.2 per cent of every colour; the thin one falls to 42 per cent at the red end.

Both designs at the maxima give a nearly white bright state. The thin cell, too thin for either guiding or the first maximum, passes the blue and loses much of the red. Between the first two designs the choice is made on speed. A liquid-crystal layer switches by the molecules rotating against the fluid’s viscosity, restored by its elasticity, and the time it takes grows as the square of the layer’s thickness. The first maximum allows the thinnest layer, about five micrometres for common materials, and it became the standard.

How the layer becomes a display

The twisted cell between crossed polarisers is bright. To turn it dark, a voltage is applied across the layer through transparent electrodes on the glass. Liquid-crystal molecules with a positive dielectric anisotropy tend to line up with an electric field, and above a threshold of a volt or two — the Fréedericksz transition, where the field’s torque overcomes the elastic resistance of the twist — the molecules in the middle of the layer tilt to stand along the field, perpendicular to the plates. Light travelling along the molecules sees no birefringence, the twist is undone in the middle of the layer, and the guiding fails. The light reaches the second polariser with its polarisation unturned, and is blocked.

Martin Schadt and Wolfgang Helfrich at Hoffmann-La Roche in Switzerland, and James Fergason in the United States, made the first such twisted nematic cells in 1971. They needed a few volts and microwatts, and within a decade they were in every watch and calculator. Each pixel of a colour screen built on the same principle is a twisted cell behind a colour filter, its voltage setting a grey level by how far the molecules tilt and the guiding fails.

The twisted cell’s weakness is the angle at which it is viewed. Off the axis the light crosses the layer obliquely, sees a different birefringence and a different twist, and the grey levels shift and sometimes invert. Later display technologies — in-plane switching, in which the molecules rotate in the plane of the panel rather than tilting out of it, and vertically aligned cells — were developed largely to fix that, and have replaced the twisted cell in most large screens. The twisted cell survives where cost and speed matter more than angle.

The light a display throws away

A twisted cell only modulates light that is already polarised, and that costs. Ordinary light from a backlight is unpolarised — the light with no direction of shaking found it to be an incoherent mixture of all directions — and the first polariser absorbs the half that is shaking the wrong way. Colour filters then remove two-thirds of what is left at each subpixel, and the electrodes, the glass and the second polariser take more. A typical liquid-crystal screen delivers about five per cent of its backlight to the viewer.

Much engineering has gone into recovering the lost half. A reflective polariser — a stack of hundreds of alternating birefringent layers, each reflecting a little of one polarisation by the same interference that makes a multilayer mirror — sends the wrong polarisation back into the backlight instead of absorbing it, where scattering scrambles it and gives it another chance to come out right. Each pass recovers part of what the absorbing polariser would have wasted, and such films raise a screen’s brightness by a third or more for the same power.

Under a polarising microscope the same physics turns a liquid crystal into a picture of itself. Between crossed polarisers, a sample whose molecules point in different directions in different places shows dark brushes wherever the molecules lie along either polariser, and bright regions elsewhere, in patterns that map the director field — the liquid-crystal analogue of the fringes the stress that polarised light can see drew in a loaded disc. The textures that identify the different liquid-crystal phases are read in exactly this way.

Two ways to rotate a polarisation

There is an older way of rotating light’s polarisation, and the comparison is instructive. The rotation a return trip doubles described optical activity: a solution of handed molecules, like sugar, rotates the polarisation steadily as light passes through it, because left- and right-circularly polarised light travel at slightly different speeds. That rotation is a few tens of degrees per decimetre of syrup. The twisted cell rotates by ninety degrees in five micrometres, tens of thousands of times faster, by an entirely different mechanism: it is not circular birefringence at all, but the guiding (or the interference) of linearly polarised light by a structure that is handed only in its arrangement, built from molecules that need not be handed themselves.

The two meet in liquid crystals whose molecules are handed, which twist spontaneously into a helix of their own. When that helix’s pitch is many wavelengths long, the material guides light as a twisted cell does; when the pitch is comparable with the wavelength, it reflects one circular polarisation selectively, giving the iridescence of beetle shells; and when it is much shorter than the wavelength, the material behaves as an ordinary optically active liquid with an enormous rotatory power. The same twisted structure gives all three, depending only on how its pitch compares with the light’s wavelength.

Mauguin guiding also has a geometric cousin. When a polarisation is carried round a closed path of directions, it can return rotated by an angle fixed by the path’s shape, the phase the phase that is only a shape found in light sent round a coiled fibre. Both are cases of a polarisation following a slowly changing frame, one in space, one in the material.

A perfect helix, at normal incidence

Every figure treats a perfect, uniform ninety-degree helix of molecules at zero voltage, with light at normal incidence and a single, wavelength-independent birefringence. Real cells have a small pretilt of the molecules at the surfaces, set by the rubbing, which slightly changes the effective birefringence; their birefringence varies with wavelength, by ten per cent or so across the visible, which shifts the colour figure’s curves; and real light arrives over a cone of angles, which is where the twisted cell’s viewing-angle problems come from. The polarisers are taken as perfect, and reflections at the glass and electrode surfaces are ignored. The description of switching is qualitative: the voltage figure a display engineer uses comes from solving the elastic and electric equations for the tilted, twisted profile, which none of these figures does. The domain is 90° twisted layers between crossed polarisers, uu from zero to about a hundred, at normal incidence.

Still open: guiding light where the twist is not uniform

The uniform twisted cell is solved exactly. Real optical structures increasingly are not uniform: liquid-crystal layers patterned so that their axis varies across the plane as well as through the thickness, making flat lenses, beam steerers and vortex beams, and metasurfaces of nanostructures whose local axes are set by lithography. Light passing through them is guided in some places, interferes in others, and is diffracted by the pattern everywhere, and whether such devices should be designed in the adiabatic regime, where the local rules of guiding hold, or at interference points like the first Gooch–Tarry maximum, where they are thinner and faster but more sensitive, is a choice made device by device. How much of a non-uniform twisted structure’s behaviour can be predicted from local guiding, before the full wave calculation is needed, is still being worked out.

The uniform case is short. A birefringent layer whose axis twists through 90° turns light polarised along its entrance axis to its exit axis, transmitting 1−sin⁡2(π21+u2)/(1+u2)1 - \sin^2(\tfrac\pi2\sqrt{1+u^2})/(1+u^2) of it between crossed polarisers, with u=2dΔn/λu = 2d\Delta n/\lambda; for large uu the light follows the twist, its leakage falling as 1/(1+u2)1/(1+u^2), and at u=3u = \sqrt3, 15\sqrt{15}, 35\sqrt{35} it arrives exactly — at the first of these, the one in displays, not by following but by interference after a detour through elliptical states. Mauguin’s guided light made the idea; the display runs on its coincidental cousin.

Part 10 of 10

This essay is one argument about Polarisation. The others:

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 Part 9 — 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.

Where this goes next

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. 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. 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.

The objects named here

The third axis, after the field and the reading path: the things themselves, and every essay that touches each one.

Adiabatic followingBirefringenceDisplayJones calculusLiquid crystalOptical rotationPolarisationWave plate