The pattern two gratings make that neither contains

When two waves meet began this subject with the rule that waves add point by point, and found beats as one of its first consequences: two notes a little apart in pitch, sounding together, swell and fade at the difference of their frequencies. The silence that fits in a tenth of a wavelength followed addition in the opposite direction, using a second sound to cancel the first, and found the cancellation confined to a small region near the microphone. Both were about waves in time, added. There is a cousin of the beat that lives in space instead of time, and in which the two patterns are not added but multiplied. It can be seen by laying one window screen over another, or by holding two combs against the light, and it was named by French weavers centuries before anyone could explain it: moiré, after the watered silk whose rippling sheen comes from pressing two layers of fabric together.

Two gratings, one coarse pattern #

Two gratings of dark bars, one with ten units between bars and the other with eleven, are laid one over the other. Near the left edge the bars of the two coincide; the gaps line up, and light passes through. Moving right, each bar of the second grating slips a tenth of a period further behind the first’s, and after five bars the second grating’s bars sit exactly in the first’s gaps, blocking almost everything. After ten more they coincide again. The light that gets through the pair rises and falls with a period of 110 units — the length over which ten periods of one grating fit eleven of the other.

The period comes from the same arithmetic as acoustic beats. A grating of period $p$ has a spatial frequency $1/p$, the number of bars per unit length. Two gratings of spatial frequencies $1/p_1$ and $1/p_2$ drift in and out of step at the difference $|1/p_1 - 1/p_2|$, so the moiré period is

$$P = \frac{p_1p_2}{|p_1 - p_2|}.$$

Neither grating contains that period. It belongs to their combination, and in particular to their product: light passes only where both are transparent, so the transmission of the pair is the product of the two transmissions, and the product of two oscillations at nearby frequencies contains an oscillation at their difference.

Why a product, and why the eye sees a sum as one too #

The distinction between adding and multiplying is worth a moment. Two sound waves add: the pressure at the ear is the sum of the two, and the sum of two sinusoids at nearby frequencies contains no component at the difference frequency at all — only the two original tones. The beat is in the envelope, the slowly varying loudness, and it is heard because the ear responds to loudness, which goes as the square of the pressure. Squaring a sum produces a cross term, the product of the two, and that product oscillates at the difference frequency. Beats in sound and moiré in gratings are therefore the same term: one arises from a detector that squares, the other from two transmissions that multiply.

The same mixing is how a radio receiver works. A heterodyne receiver multiplies the incoming signal by a local oscillator at a nearby frequency, and the product contains the difference — a low frequency that is easy to amplify and filter. The speeds a pulsed radar folds into one found a related folding when a signal is sampled at a rate close to its own: the samples show a slow, false oscillation at the difference between the signal frequency and a multiple of the sampling rate. A camera’s grid of pixels photographing a striped shirt does exactly that in space, which is why fine stripes on television break into slowly swimming bands, and why scanners are fitted with filters that blur the image slightly before it is sampled.

The angle and the ratio #

Gratings can differ in two ways: in period, and in direction. Two identical gratings turned by a small angle $\theta$ also make a moiré, because along any line across them the bars of one drift relative to those of the other. In general each grating is described by a wavevector — a vector pointing across its lines, of length $2\pi/p$ — and the moiré is a grating whose wavevector is the difference of the two. Its period is

$$P = \frac{p_1p_2}{\sqrt{p_1^2 + p_2^2 - 2p_1p_2\cos\theta}},$$

which reduces to the beat formula for parallel gratings and to $p/(2\sin\tfrac\theta2) \approx p/\theta$ for identical ones at a small angle. The figure’s numbers are the important part: a twist of one degree gives fringes 57 grating periods apart; a tenth of a degree, 573. The smaller the mismatch, the coarser the moiré, without limit.

The direction of the moiré is the surprising part to anyone who has not drawn it. Two sets of nearly vertical lines, twisted by a few degrees, make fringes that are nearly horizontal. The difference of two nearly equal vectors at a small angle is a short vector nearly perpendicular to both, and the moiré’s lines run perpendicular to that, along the bisector of the two gratings’ normals. Turn one grating further, and the fringes swing round and close up; turn it back, and they spread and turn the other way, until at zero angle the window is filled by a single fringe.

A magnifying glass for mismatch #

That limitless coarsening is the moiré’s usefulness. If a grating is printed on a surface and the surface is stretched by a strain $\varepsilon$, the grating’s period grows by the fraction $\varepsilon$, and viewed through an unstretched copy it makes a moiré of period $p/\varepsilon$. The mismatch has been magnified by $1/\varepsilon$.

A steel bar near its working stress is strained by about a thousandth. A grating of 1,200 lines per millimetre printed on it, viewed through a reference grating, shows a fringe every 0.83 millimetres, and each fringe is a contour along which the surface has moved by one more grating period — a contour map of displacement, drawn by the light, of a deformation far too small to see directly. Moiré interferometry, which uses such fine gratings and laser light, maps strains in welds, composite materials and electronic packages to a few parts in a million. It is the displacement counterpart of the stress that polarised light can see, which maps stress through birefringence. The magnification has a price. The smaller the strain, the coarser its fringes, and a strain of a millionth over a few centimetres produces less than one fringe across the whole field; such strains are read not by counting fringes but by measuring the phase of a single fringe to a small fraction of a cycle, shifting the reference grating in known steps and watching how the light at each point changes.

The same magnification aligns things. Shift one grating against the other by a hundredth of its period, and the moiré moves by a hundredth of its own, much larger, period — a movement easy to see and to measure. Lithography machines that print the successive layers of a microchip use moiré patterns between marks on the mask and on the wafer to align layers to a few nanometres, with marks whose own lines are hundreds of nanometres apart. A vernier scale on a calliper is the same principle with a handful of lines: ten divisions against nine, and the division that lines up reads a tenth of the scale’s spacing.

Contours drawn by a shadow #

A grating does not need a second grating to make a moiré; its own shadow will do. Hold a grating of fine lines a little above a curved surface and light it from one side: the surface carries the grating’s shadow, distorted by its shape, and viewed through the grating from a different direction, shadow and grating make a moiré whose fringes are contours of equal height above the grating. This shadow moiré was developed into a measuring technique around 1970, and it was used for years in schools to screen children for scoliosis: a child stands behind a grating, and a curvature of the spine shows as an asymmetry in the contour fringes on the back, without a single measurement being taken.

The fringes are contours for the same reason a strained grating’s fringes are contours of displacement. Each fringe marks the places where the shadow has shifted, relative to the grating, by one more whole period, and the shift is proportional to the height. A projected fringe pattern, viewed through a reference grating or computed against a stored one by a camera, does the same at a distance, and it is how many optical three-dimensional scanners measure the shape of objects they cannot touch.

Screens that are kept apart on purpose #

Printers have spent a century avoiding moiré. A colour picture in a magazine is printed as four halftone screens — grids of dots in cyan, magenta, yellow and black — whose sizes encode the shade. Laid on top of one another, four grids of the same spacing would beat into visible moiré, and the trade’s solution is to rotate them: the strong colours are printed at angles thirty degrees apart, typically fifteen, forty-five and seventy-five degrees, with the faint yellow at the remaining angle. Thirty degrees is large enough that the moiré between any two screens has a period of about two dots, too fine to see, and the four combine into a small, even rosette. A screen that slips by a degree or two on the press produces moiré at once, with a period of thirty to sixty dots, which is why misregistration is so conspicuous. The figure of the moiré period against angle is the printer’s rule, written as a curve.

Moiré was used as a tool almost as soon as it was understood. Lord Rayleigh proposed in 1874 that the quality of a diffraction grating could be judged by laying it against a copy: any error in the ruling — a period that drifted, a line out of place — would show as a distortion of the moiré fringes, magnified as every mismatch is. Gratings of a few thousand lines per millimetre, whose individual errors were far too small to see, could be inspected by eye. That magnification also explains why the image that is a diffraction pattern twice needed a microscope’s aperture to pass at least two diffracted orders to show a grating’s lines: the image of a periodic object is itself an interference of its diffracted beams, a product of waves of the kind that makes moiré.

The same beat in time #

The wheels of a stagecoach in an old film often seem to turn slowly backwards. The spokes pass a fixed position at a rate set by the wheel’s speed, the camera samples the picture twenty-four times a second, and when the spoke rate is slightly below a multiple of the frame rate, each frame catches the spokes a little short of where the previous frame had them, so they seem to creep the wrong way. The wagon-wheel effect is a moiré in time, between the periodic spokes and the periodic shutter, with a beat period set by the same subtraction. What adding does to the energy found that superposed waves only redistribute energy in space and time without creating it; a moiré is a vivid case, putting the light from two fine patterns into a coarse one without adding a photon. And the grating that photographs itself found a grating’s self-images recurring at fixed distances behind it; a second grating placed at one of those distances makes a moiré with the self-image, which is how X-ray and neutron phase imaging use moiré to read off tiny deflections of the beam.

Sheets of carbon, twisted #

The most striking modern moiré is not optical at all. A sheet of graphene is a single layer of carbon atoms in a hexagonal lattice, with atoms 0.246 nanometres apart along the lattice directions. Lay one sheet on another and twist it by a small angle, and the two lattices make a moiré: regions where the atoms of the two layers sit on top of each other alternate with regions where they are offset, in a pattern that repeats on a scale far larger than the atoms.

At a twist of 1.1 degrees the moiré repeats every 12.8 nanometres, a cell holding about eleven thousand atoms. Electrons in the bilayer move through a landscape that varies on that scale, and the moiré acts on them as a much larger crystal would — with a periodic potential of its own, opening gaps in their energy spectrum as the gap a repeat opens found any periodic structure must. At that particular angle, predicted in 2011 by Rafi Bistritzer and Allan MacDonald, the lowest bands become almost flat: the electrons’ energy barely depends on their momentum, so they hardly move, and their mutual repulsion, which ordinarily is a small correction, dominates their behaviour. In 2018 Pablo Jarillo-Herrero’s group at MIT found that such a bilayer, with electrons added or removed by a gate voltage, becomes an insulator at some fillings and a superconductor at others. The angle had to be set to within a tenth of a degree, and the whole of the new physics came from a pattern neither layer contains.

Overlaid gratings, perfect lines and no diffraction #

The figures treat the gratings as perfectly regular and the moiré as a purely geometric product of two transmissions, seen with light whose wavelength is small compared with the grating periods. When the gratings are fine enough that light diffracts through them — as in moiré interferometry, where gratings of 1,200 lines per millimetre are comparable to the wavelength — the fringes are better described as the interference of diffracted beams, with the same spacing but a contrast that depends on the diffraction efficiencies. The smoothed transmission curve keeps only the lowest moiré component; sharp-edged bars produce higher harmonics that the figure omits. The graphene figure uses the geometric moiré period only; the electronic structure, the flat bands and the superconductivity come from calculations and experiments that no figure here repeats, and real twisted bilayers relax their atoms to enlarge the aligned regions, which changes the pattern’s shape though not its period. The domain is two gratings of nearly equal periods and small mutual angles, viewed with light short compared with both.

Still open: what the electrons in a moiré are doing #

The geometry of moiré patterns is completely understood, and their use in measurement is mature. What is not understood is the physics of electrons living on a moiré. In twisted graphene, and in the many other twisted and stacked layered materials made since, superconductivity, magnetism, correlated insulators and states with fractional charges have been found, depending on the twist, the number of layers and how many electrons are added. Whether the superconductivity in twisted graphene arises from the electrons’ repulsion, from vibrations of the lattice, or from something new, is argued; the samples are hard to make reproducibly, because a twist that varies by a few hundredths of a degree across a sample changes its behaviour. The moiré has given physicists a crystal whose lattice constant can be set with a twist, and they are still finding out what it can be made to do.

The geometry is one subtraction. Two gratings of wavevectors $\mathbf{k}_1$ and $\mathbf{k}_2$, multiplied as overlaid transmissions — or added and then squared by a detector — contain a pattern at $\mathbf{k}_1 - \mathbf{k}_2$, of period $p_1p_2/\sqrt{p_1^2 + p_2^2 - 2p_1p_2\cos\theta}$: 110 units for gratings of 10 and 11, 57 periods for a twist of one degree, $p/\varepsilon$ for a strain $\varepsilon$, and 12.8 nm for two graphene sheets turned 1.1°. A moiré is a beat drawn in space, and like a beat it makes a small difference large enough to see.