The grid that stops the light along its wires

The direction of the shaking introduced polarisation with a picture almost every textbook uses: a rope threaded through a picket fence. Shake the rope up and down and the wave passes between the vertical slats; shake it sideways and the slats stop it. The picture explains what polarisation is, and it suggests what a polariser must be — something with a grain, which passes the shaking along its grain and blocks the shaking across it.

Build that fence for an electromagnetic wave, out of parallel metal wires, and it does exactly the reverse. A grid of wires passes the wave whose electric field lies across the wires and reflects the wave whose field lies along them. The rope picture is not merely imprecise here but inverted, and the reason it is inverted is the whole of how most polarisers work — including the plastic sheet in a pair of sunglasses, which turns out to be a grid of wires made from molecules.

What the wires do to each field #

A light wave is not a rope, and the grid does not act on it by obstruction. It acts through the electrons in the wires. When the wave’s electric field lies along a wire, it pushes the wire’s conduction electrons along it, and they are free to move: they surge up and down the wire at the wave’s frequency, a current oscillating in each wire of the grid. An oscillating current radiates — a charge that turns must glow is the general statement — and the wave it radiates forward is, for a good conductor, almost exactly the negative of the incident wave. Beyond the grid the two cancel. Backwards, the radiated wave is a reflection. For this polarisation the grid behaves like a sheet of metal, which reflects light for the reason the frequency below which nothing gets in gives: free charges move to cancel any field that tries to enter.

When the field lies across the wires, it can push electrons only from one side of a wire to the other — through a distance equal to the wire’s thickness, far less than a wavelength. A little charge piles up on each side, the wire becomes a tiny dipole of negligible strength, and almost no current flows. The wave passes as though the wires were barely there.

The two panels above are the argument in one picture. What decides a polariser’s axis is not which way the gaps run but which way the conductors let charge travel. The transmission axis of a wire grid is perpendicular to its wires, and the rope-and-fence analogy, which fixes attention on the gaps, predicts the wrong one.

The same reasoning separates this from the polariser in the angle at which reflection picks a side, where reflection off glass at Brewster’s angle removes one polarisation because the dipoles that would radiate the reflected wave would have to radiate along their own axis, which a dipole cannot do. There, the charges in the glass move freely in every direction, and the selection is made by the geometry of radiation. Here, the selection is made one step earlier, by whether the charges can be set moving at all.

Hertz’s grid #

Heinrich Hertz built such a grid in 1888, a year after he first produced and detected radio waves, to show that the waves were transverse and polarised as Maxwell’s theory required. His transmitter was a spark gap at the focus of a cylindrical mirror, and his receiver a second gap at the focus of another, so that both the emitted and the detected field were polarised along the gaps. Between the two he placed a frame about two metres high strung with copper wires a millimetre thick and three centimetres apart. With the wires perpendicular to the spark gaps, the receiver sparked as before. Turned until the wires were parallel to the gaps, the grid blocked the waves completely; and at forty-five degrees, a grid placed between a transmitter and a receiver set at right angles to each other let some signal through, which neither alone could have done — the effect the direction of the shaking describes for three crossed polarising filters.

Hertz’s waves had a wavelength of about sixty-six centimetres, so his wires were spaced at a twentieth of a wavelength. That ratio is the parameter that governs everything, and the next figure shows why.

The curves come from the theory of thin strip gratings, which treats the grid as a sheet across the path of the wave with an electrical impedance of its own — an inductance for the field along the wires, because the wave’s energy is spent driving currents through them, and a small capacitance for the field across them, because it can only charge their sides. A small inductance across the path of a wave behaves like a short circuit across a transmission line, and reflects. Its size, relative to the impedance of free space, is $(d/\lambda)\ln(d/2\pi a)$ for wires of radius $a$ spaced $d$ apart, and the power that leaks through along the wires is about four times its square while it is small.

Two things follow at once. The leak falls as the square of the spacing in wavelengths, so a grid one-hundredth of a wavelength fine leaks about a thousandth of the power, while one at a tenth of a wavelength leaks ten per cent — still a polariser, but a poor one. And thicker wires help, because their inductance is lower; the logarithm makes the help slow, so a grid cannot compensate for coarse spacing by fattening its wires. Hertz’s grid, marked with the diamond, leaked about four per cent of the power along its wires and passed practically all of the power across them. That was ample to show the effect plainly at a wavelength where wires three centimetres apart could be strung by hand.

The theory also says where it stops. Once the spacing reaches about half a wavelength, the grid starts to diffract: the waves scattered by neighbouring wires add up not only straight ahead and straight back but along other directions, as they do from a grating with a thousand slits. A grid with its spacing near a wavelength is a diffraction grating that happens to treat the two polarisations somewhat differently, not a polariser.

Shrinking the grid to light #

For visible light, the same ratio means wires spaced at about a tenth of half a micrometre’s wavelength or better, and real visible grids are made with a period of about 140 nanometres. That was not possible for most of the twentieth century. The first wire-grid polarisers for infrared light, made around 1960, were produced by evaporating gold at a grazing angle onto a finely ruled grating, so that the metal landed only on one side of each ridge and formed parallel strips; they worked well at wavelengths of several micrometres and above. Only when the lithography developed for integrated circuits could pattern lines at a hundred nanometres did a wire grid for visible light become manufacturable, around the year 2000: lines of aluminium about 70 nanometres wide and 150 tall on a glass plate, with 70-nanometre gaps between them.

At that scale the wires are taller than they are wide, and a second way of seeing the grid takes over. Each gap is a narrow channel between two metal walls — a parallel-plate waveguide — and the pipe that will not carry a low note showed what such a guide does to a wave too long for it. A field lying parallel to the walls must vanish on them, so it can travel through the gap only if the gap is wider than half a wavelength; below that the field does not travel but decays, exponentially, with a decay length close to the gap width divided by $\pi$. A field lying across the gap meets the walls at right angles, which the walls allow, and passes as the lowest mode of a parallel-plate guide, which has no cutoff at all. The polarisation-selection now has an exact waveguide meaning: the grid is a forest of guides that are all below cutoff for one polarisation and all open for the other.

In the perfect-conductor model the figure draws, a 70-nanometre gap passes about two millionths of the power along the wires through 150 nanometres of height, and each extra fifty nanometres of height divides it by roughly another hundred. Narrowing the gaps helps faster still, since the decay length is proportional to the gap. Real aluminium grids do not reach these numbers: they achieve extinction ratios — the transmission of the wanted polarisation divided by that of the unwanted — of about a thousand to ten thousand across the visible. The limit is the metal. Aluminium is the best conductor available for visible light, yet a visible field reaches about 13 nanometres into it, as how far a field gets into metal describes for the general case, and in a wire only 70 nanometres wide that is a sizeable fraction of the conductor. Some of the light along the wires leaks through the metal itself, and some is absorbed.

The grid is also weakest at the blue end of the spectrum, where the ratio of period to wavelength is largest and the leak along the wires is greatest. A visible wire grid is a better polariser for red light than for blue, and a better one again in the infrared, where the same plate is a far finer grid relative to the wavelength.

Polaroid as a grid of molecules #

The polarising sheet in sunglasses, in the camera filters that darken the polarised blue of the sky, and in older liquid-crystal displays works the same way, with molecules as the wires. Edwin Land’s H-sheet, introduced in 1938, is a film of polyvinyl alcohol stretched to several times its length, which aligns its long polymer chains along the stretch direction, and then soaked in an iodine solution. The iodine forms chains of its own, polyiodide ions, strung along the aligned polymer, and electrons can move along these chains but not between them. A field along the chains drives charge along them, exactly as along Hertz’s wires — but the chains are short, lossy and disordered, and instead of re-radiating coherently into a reflection they absorb, turning the light into heat. A field across the chains moves almost nothing and passes.

So a Polaroid sheet’s transmission axis lies perpendicular to its stretch direction, and it absorbs the polarisation that a metal grid reflects. The spacing between chains is a few nanometres, so the sheet is far finer than any wavelength of light and has no diffraction to worry about; its limits come from the chains’ absorption spectrum, which is not perfectly flat across the visible, and from the polymer itself, which yellows and softens with heat. A typical sheet passes about forty per cent of unpolarised light rather than the ideal fifty, and leaks a fraction of a per cent when crossed with another.

The difference between absorbing and reflecting the wrong polarisation is not a detail. An absorbing polariser in the path of an intense beam — a projector lamp or a powerful laser — must dissipate half the light’s power in a thin organic film, and degrades. An aluminium grid on glass sends that half back the way it came, or off to the side if it is tilted, and survives temperatures that would char the sheet. That is why the visible wire grid found its first large market inside projectors, where a tilted grid serves as a polarising beam splitter: it reflects one polarisation to the side and transmits the other, over a much wider range of angles and colours than the multilayer coatings it replaced.

One direction makes a polariser; two make a shield #

The argument that makes a grid a polariser also says what happens when wires are added in the other direction. A square mesh of strips has conductors running both ways, so whichever direction the field lies in, there are wires along it for electrons to move in. Both polarisations drive currents, and both are reflected.

At wavelengths much longer than the period, on the right of the figure, the grid passes the field across its strips and stops the one along them, while the mesh stops either, each exactly as well as the grid stops its blocked polarisation. At wavelengths much shorter than the period, on the left, both are simply windows: light passes through the open area, and the strips cast shadows. In between, where the wavelength is comparable with the period, both structures diffract and neither of the simple descriptions holds.

The door of a microwave oven is a mesh of that kind, with a period of a millimetre or two in front of a wave 12.2 centimetres long. In the thin-sheet model it lets through about a tenth of a per cent of the microwave power; the thickness of a real perforated sheet, which makes each hole a short waveguide below its cutoff, multiplies the attenuation further, and the door’s metal frame and seals do the rest. To visible light, whose wavelength is three thousand times shorter than the period, the same mesh is a window with dark lines across it. The cage of a Faraday shield, a radio telescope’s dish made of wire netting and the screen room of an electronics laboratory are all the same object at different scales: a conductor with holes much smaller than the wavelength it must stop, and arranged so that no direction of field is left without a conductor to move charge along it.

Recovering the half that is thrown away #

Any polariser offered unpolarised light keeps at most half of it, and in a backlit liquid-crystal display that half is the entire budget: the twist that carries light round a corner works only on polarised light, so the rear polariser of every such display discards half the backlight’s output before the image is formed. An absorbing polariser turns that half into heat. A reflective polariser — whether a wire grid or a stack of birefringent layers — sends it back into the backlight, where diffusers and reflectors scatter it, scramble its polarisation, and return it for another attempt.

The output is a geometric series. Half the light passes on the first attempt; the reflected half returns with an efficiency $\eta$ set by the backlight’s losses, and half of that passes; and so on. Summed, the light delivered in the wanted polarisation is $\tfrac12 T/(1 - \tfrac12 R\eta)$, where $T$ and $R$ are the polariser’s transmission of the wanted field and reflection of the other. With a backlight returning seventy per cent, a reflective polariser passing ninety per cent and reflecting ninety-five delivers about two-thirds of the light, against the forty-three per cent an absorbing polariser manages: one and a half times the brightness for the same lamp and the same battery. Brightness-enhancing films of this kind have been standard in laptop and phone displays since the 1990s, and the figure shows what limits them: the curve rises steeply only as the return efficiency approaches one, so every per cent of loss in the backlight is worth more than a per cent of output.

Grids beyond the visible #

At the long-wavelength end the wire grid never went away. Free-standing grids of tungsten wires ten micrometres thick and a few tens of micrometres apart are the standard polarisers and polarisation beam splitters for millimetre and submillimetre waves, used in the spectrometers and polarimeters that measure the faint polarisation of the cosmic microwave background and the dust in star-forming clouds. One design, the Martin–Puplett interferometer, uses wire grids as the beam splitter and the analyser together, dividing light by polarisation rather than by partial reflection, which works over a far wider band than any dielectric splitter could. At those wavelengths the grid is again a hand-made object, strung and tensioned like Hertz’s, and its spacing-to-wavelength ratio sits near a hundredth.

Antennas embody the same physics from the other side. A rod antenna responds to the component of a passing field along the rod, because only that component drives a current in it, which is why a television aerial’s elements must be turned to match the transmitter’s polarisation. A wire grid is an array of such antennas, each receiving the field along it and re-radiating it in antiphase. That a grid passes the perpendicular field is the same fact as that an antenna is deaf to it.

Thin wires, perfect metal and straight-on light #

The transmission curves for thin grids use the quasi-static strip-grating formulas, which hold when the period is well under half a wavelength and the strips are thin and perfectly conducting; a round wire is represented as a strip four times its radius wide, which is the standard equivalence for that regime. The waveguide picture of thick grids treats the metal as perfect and ignores the reflection at the entrance and exit of each gap, so it gives the decay through the gap but not the transmission of the grid as a whole. Neither model includes absorption in real metals, which is what limits real visible grids, nor oblique incidence, where a grid’s extinction ratio changes and the two polarisations no longer stay cleanly separated. The recycling model assumes the backlight fully scrambles the returned polarisation; a backlight that preserves some of it returns light that the polariser rejects again, and the gain is smaller.

Still open: grids that sort circular polarisation #

A grid of straight wires distinguishes linear polarisations because straight wires have a direction. Circular polarisation has a handedness instead, and a structure that sorts it must itself be handed. A layer of wires that is flat cannot be fully handed in three dimensions — viewed from the other side, its pattern is its own mirror image — and a symmetry argument shows that such a layer cannot transmit one circular polarisation while reflecting the other unless it also absorbs light. Arrays of gold helices a few micrometres tall, made by direct laser writing, have been shown to work as broadband circular polarisers in the infrared, transmitting the circular polarisation of opposite handedness to the helix and reflecting the other, exactly as a straight grid does for linear light; how small, how efficient and how cheap such structures can be made for visible light, where the helices would have to be a few hundred nanometres tall, is still being worked out, alongside alternatives built from chiral liquid crystals and from stacks of rotated planar layers.

The habit worth carrying away is to ask which way a conductor lets charge move. A wire carries current only along itself, so it can answer only the field that lies along it — and a grid of such wires, spaced well under a wavelength, reflects that field and passes the one across it, the reverse of the picket fence. That is Hertz’s grid at a twentieth of a wavelength, a visible polariser at 140 nanometres, a sheet of iodine chains at a few nanometres, and, with wires added the other way, an oven door that no polarisation can get through.