Optics · Interference · One equation, printer to superconductor

The Pattern Between the Lines

Lay one set of fine lines over another and turn it a single degree. A huge, slow pattern rolls across the page — and it is in neither set of lines. It looks like a screen glitch. It is one of the most exact things in optics, and the same equation runs from your printer to a superconductor.

The pattern is called a moiré — French for the watered shimmer of certain silks, which is itself two weaves seen through each other. It has a reputation as an artefact: the crawling mess when you photograph a screen, the shirt that strobes on video, the print that muddies. That reputation is exactly backwards. The moiré is not noise added to two patterns; it is a third, precise pattern created by their difference — and its spacing is fixed to the last decimal by the two you started with. Here is the whole law, and then four instruments that let you put your hands on it.

1 · Turn one grating a degree

Two identical rulings of pitch p, one rotated by an angle θ. Drag θ down toward zero and watch the fringe spacing blow up. Tick the box to lay the predicted fringe lines over the pattern — the formula's answer, drawn on top of the emergence it predicts.

At one degree the pattern is 57 times coarser than the lines that make it. That is the first surprise: a tiny angle produces an enormous feature, and the relationship is not vague — it is an exact function of θ.

This is just the subtraction of two directions. Each grating is a wave with a wavevector — a little arrow pointing across its lines, whose length is 2π/p. The moiré you see is the wave whose wavevector is the difference of the two arrows, and the length of a difference of two equal arrows separated by θ is 2·(2π/p)·sin(θ/2). Invert that and you have D. When the two arrows point almost the same way (small θ), their difference is tiny, so its wavelength — the moiré — is huge.

The one law underneath all of it

Rotation is only half of it. Let the two gratings also differ in pitch, and one formula covers every case at once — the length of the difference arrow, written out:

Set the pitches equal and it becomes the rotation law above. Set the angle to zero and it becomes the beat law of the next instrument. It is the spatial cousin of the acoustic beat: two close frequencies, subtracted.

2 · Two rulers, almost the same

Now hold both gratings parallel and make them slightly different in pitch. No rotation at all — yet a fringe still appears, marching wherever the two rulings drift into and out of step. Its period is p₁·p₂ ⁄ |p₁ − p₂|: the closer the two pitches, the wider the fringe, without bound.

This is the same arithmetic your ear does. Two tones a few hertz apart don't blur — they throb, loud-soft-loud, at exactly |f₁ − f₂| times a second. Press the button and hear it; the throb rate is the beat law for sound, the fringe is the beat law for light. (The tones are chosen to make the beat audible; what's shared is the structure — a difference of two near-equal frequencies — not the specific numbers.)

3 · The glitch is a ruler

Here is why the moiré is not an artefact but an instrument. Because the fringe is magnified by M = D/p, a movement of one grating too small to see shifts the fringe by M times as much — a movement you can't miss. Nudge the grating below by a fraction of a line-width and watch the whole pattern lunge across the frame.

4 · The same equation, at 0.246 nanometres

Now shrink the two gratings until the "lines" are rows of atoms. Twist two sheets of graphene — each a honeycomb of carbon with a spacing of 0.246 nm — by a small angle, and the same law makes a moiré superlattice: a new, giant repeat set by a ⁄ (2 sin(θ⁄2)). Drag the twist; at 1.1° the superlattice is about 13 nm — fifty-two atoms wide.

And why your printer already knows this

Colour printing lays four dot screens — cyan, magenta, yellow, black — one over another. Four gratings almost on top of each other is a moiré catastrophe waiting to happen, so printers rotate the screens to fixed angles: black 45°, cyan 15°, magenta 75°. Three of them sit a full 30° apart, and by the rotation law a 30° moiré repeats every 1.93 line-widths — so tight and fine it reads as a clean rosette rather than a crawling pattern. Only three screens can be 30° apart, so the fourth — yellow, the faintest ink — is parked at the leftover 15°, where the moiré is 3.83 line-widths wide. That the trade literature quotes exactly "3.83×" for the 15° screen is the rotation law 1 ⁄ (2 sin 7.5°) arriving from the pressroom instead of the blackboard — same number, checked two ways.