Illustrated Physics
physics, drawn
Concept

Diffraction grating — where it appears

A surface ruled with many parallel lines that sends each wavelength of light in a different direction. It is used to spread light into a spectrum and, run backwards, to combine beams of different colours into one.

Named by 7 essays across 3 fields — each of them below, with the objects they name alongside it.

Colours that leave a grating on different roads. A pair of parallel diffraction gratings with 1,200 lines per millimetre and light arriving at 30°, drawn to scale for three wavelengths, 760, 800 and 840 nm. The first grating sends each colour off at its own angle — 24.3° for 760 nm, 27.4° for 800 nm, 30.5° for 840 nm — and the second turns them all back parallel to the input, side by side. Between the gratings the red travels the longest road: for gratings 10 mm apart the slant paths are 10.97 mm, 11.26 mm, 11.61 mm. Measured to a common wavefront after the second grating, the longer road still wins, so red arrives last: the geometry has given the pulse the opposite ordering to the one glass gives it.

The dispersion made of angles

Glass delays blue light more than red, so a short pulse that passes through glass comes out stretched with its red end first. At the wavelengths where most short-pulse lasers work, no transparent glass does the opposite. Two diffraction gratings do it with no material at all. The first spreads the colours in angle, the second turns them back parallel, and in between the red has travelled the longer road. A spread in angle, however it is made, always adds this kind of dispersion, of the sign glass cannot give. It is how every high-power ultrafast laser squeezes its pulse back after amplifying it.

optics · Dispersion The brightness N lasers can be made into. The greatest radiance — power per unit area per unit solid angle — that N identical single-mode lasers can be combined into by lossless optics, as a multiple of one laser's, against N, for four arrangements. Laid side by side and passed through any lenses and mirrors, they fill N modes and are never brighter than one: the étendue of the combination is at least N times one beam's. Two can share a mode if their polarisations are crossed, so polarisation doubles it once. Beams of N different wavelengths can be overlapped into one spatial mode by a grating, and beams with their phases locked together can be made into one mode outright; either way the radiance grows as N, because the beams are no longer N copies of the same mode.

The beams that add only if they differ

Take two identical lasers and try to merge their beams into one beam twice as bright. No arrangement of lenses, mirrors and prisms will do it: the brightness of light cannot be increased by passive optics, and two copies of the same beam can only ever be put side by side. The escape is to make the beams not copies. Give them crossed polarisations and two will merge. Give each a different colour and a grating will stack dozens into a single beam. Lock their phases so that they stop being separate beams at all, and they add — but only as well as the gaps between them allow and only while their phases are held to a twentieth of a wavelength.

optics · Etendue Three waves, and the points where the light is zero. The overlap of three equal plane waves travelling 120° apart: brightness (shading) and lines of equal phase every 60° (curves). Lines of every phase meet at isolated points where the brightness is exactly zero (dots); elsewhere only branches of a single phase cross, at the bright spots, where the phase is momentarily level. Round each such point the phase changes by one full turn, clockwise (red, charge +1) or anticlockwise (green, charge −1); in this window there are 18 of one and 18 of the other, arranged on a lattice in which every neighbour has the opposite sign. No adjustment of the phases removes them; shifting a wave only slides the lattice.

The dark lines a wave cannot avoid

Overlap three waves of light and there are places where they cancel exactly — not along fringes, as two waves cancel, but at isolated points, round which the phase of the light turns through a full circle. They cannot be removed by any small change, they can only be created or destroyed two at a time, and random light is full of them, about three to every square wavelength. A grating with a fork in it writes one into a beam on purpose, and the beam that comes out has a dark core and carries angular momentum in its twist.

optics · Diffraction A lens with its glass taken out, focusing the same. Two lenses of focal length 200 mm and aperture 100 mm in acrylic (n = 1.49), each with its flat face to a parallel beam and the exact aspheric curve that sends every ray to one focus: solid (top), 12.2 mm thick at the centre, and the same curve cut into 13 rings 1 mm deep with the glass between them removed (bottom), on a backing 0.4 mm thick. Thickness is drawn six times larger than height, and the distance to the focus is not to scale. Every ray leaves each lens at the same angle from the same height, because the angle a ray turns through at a surface depends only on the slope of the surface where it crosses, and every ring keeps the slope it had. What was removed was glass the rays travelled through without turning.

The lens with its glass taken out

Snell's law turns a ray at a surface by an angle set by the slope of the surface where the ray crosses, and the glass behind the surface does not enter it. So a lens can be sliced into rings, the glass between the slopes thrown away, and the rays will not notice. Fresnel did it for lighthouses in 1822. The waves notice — every step cuts a few hundred wavelengths out of them — unless the steps are made exactly one wavelength deep, and then the lens becomes something else: a diffractive element whose colours run backwards, and which can cancel a glass lens's colour with a twentieth of its power.

optics · Refraction Every colour at its own angle, from one charge. The wavelength radiated by a charge passing over a grating, as a multiple of the grating's period d, against the angle from the charge's direction, for speeds of 0.3, 0.776, 0.99 of light's, in the first order: λ = d(1/β − cos θ). The shaded band is visible light for a grating of period 1.67 micrometres, the one Smith and Purcell used in 1953. At 0.776c — 300 keV electrons, their speed — visible light comes out between 0° and 30° from the beam, blue-green straight ahead and red at the wider angles; straight ahead the wavelength is 0.29d, 482 nm, and straight back 2.29d. A slow charge at 0.3c radiates only at wavelengths longer than 2.33d. A fast one at 0.99c sends 1.0 per cent of a period straight ahead, the d/2γ² of an undulator.

The rainbow a passing electron leaves on a grating

An electron moving steadily in a straight line through empty space does not radiate, at any speed. Send it skimming over a ruled metal grating, without touching it, and it lights up — every colour at its own angle, blue-green ahead and red further out, from an electron that is neither turning nor slowing. The grating hands the electron's field the one thing it lacked: a wavenumber that lets part of it escape. It is the same mechanism as the blue glow in a reactor pool, with a ruled surface standing in for the slow light in the water.

astrophysics · Radiating charge Two fine gratings that make a coarse one. Top: two gratings of dark bars laid over each other, one with a period of 10 units and the other of 11, both with bars half a period wide. Bottom: the average transmission of the pair, smoothed over a few periods. Where the bars of the two gratings coincide, light passes through the gaps between them; where the bars of one fill the gaps of the other, nothing passes. The two drift in and out of step once every 110 units — the moiré period, p₁p₂/|p₁ − p₂| — producing broad dark and light bands 11 times coarser than either grating. Neither grating contains a period of 110; their product does.

The pattern two gratings make that neither contains

Lay one fine net curtain over another and broad, swirling bands appear that are in neither curtain. Two sets of lines a tenth of a millimetre apart, differing by one part in a hundred, make bands a centimetre apart. This is moiré: the beat of two gratings in space, as two notes a little out of tune beat in time. It magnifies whatever makes the gratings differ — a twist, a stretch, a shift — by the ratio of the bands to the lines, so it is used to measure strain in bridges and to align the layers of microchips. And two sheets of carbon, twisted by just over one degree, make a moiré on which electrons stop moving and the material superconducts.

waves · Superposition The chirp a staircase returns. The pitch of the echo of a clap from the staircase, heard by a listener 10 m from its foot, against time since the clap: the strongest frequency in successive 24 ms slices of the synthesised echo (dots), and the rate at which the reflections arrive, together with twice that rate (lines). The fundamental falls from about 767 Hz to 476 Hz over a sixth of a second, with its second harmonic an octave above — a descending chirp, which to many listeners sounds like a bird. It is Bragg reflection: the stair is a grating, the first-order echo is the frequency whose half-wavelength equals the extra path from one step to the next, and as the line of sight climbs, that extra path grows and the reflected pitch falls.

The chirp a staircase sings back

A hand clap at the foot of the great stair of El Castillo at Chichén Itzá is answered by a chirp — a falling whistle that some have compared to the call of the quetzal bird. Nothing in the stone is tuned. Each of the ninety-one steps returns its own copy of the clap, the copies arrive as a train of clicks, and the ear hears a regular train of clicks as a pitch. The pitch falls because the steps are not evenly spaced as the listener sees them: low steps are seen almost level, a tread apart, and high ones almost along the slope, a whole step's diagonal apart. It is Bragg reflection from a grating whose period, seen from one spot, changes along its length.

waves · Periodic media

Named alongside it

The objects these essays reach for when they reach for this one.

DispersionAbbe numberAliasingAngular dispersionBeatsBragg reflectionCherenkov radiationChirpChirped-pulse amplificationChromatic aberrationCoherenceCoherent emission

All concepts