Illustrated Physics
physics, drawn
Concept

Reciprocal lattice — where it appears

The set of wavevectors whose plane waves repeat with a given periodic pattern, one for each family of lattice planes. A periodic structure diffracts only into these directions, and a pattern made of crossing waves is periodic when its wavevector differences form one.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

Where the refracted ray comes from, drawn with a compass. Wavevectors in units of the vacuum wavenumber, for light arriving at 30° from a medium of index 1.5 at a medium of index 1. Every direction available in the first medium lies on the circle of radius 1.5 and every direction available in the second on the circle of radius 1; the horizontal axis lies in the interface. The boundary cannot change the component along itself, because the two sides have to agree on the phase at every point of the interface, so the refracted wave is fixed by the vertical line at 0.7500 — and where that line cuts the smaller circle is the refracted direction, 48.59° from the normal. Nothing about least time or about wavefronts enters, and the ratio of sines is what the construction reads when the two radii are written as indices. The normal component is not conserved and is not meant to be: it goes from 1.2990 to 0.6614, which is the whole of the difference between the two rays.

The law that only asks about one component

A boundary between two media cannot change the part of a wave that runs along it, because both sides have to agree on the phase at every point of the surface at every instant. That single restriction produces the refracted ray, the critical angle, the evanescent field and every order of a diffraction grating, out of one drawing made with a compass.

optics · Refraction 3 beams crossing: change the phases and the pattern only slides. The intensity where 3 plane waves of one wavelength cross in a plane, their directions spread evenly round the circle, for three different sets of relative phases, each panel spanning 2.2 wavelengths. Darkest shading is brightest, in five steps of a fifth of the peak. The three patterns are the same lattice of bright spots moved sideways: for each of the altered sets there is a shift that reproduces the first pattern to 2.0 per cent of the peak. With 3 beams in two dimensions there are 2 relative phases and 2 directions to slide in, so every change of phase is a slide.

The lattice no phase can bend

Cross a few laser beams of one colour and they paint a crystal of light in the space where they overlap. The beams' phases drift whenever a mirror trembles, and it would seem the pattern must tremble with them. Whether it does depends on a count. Three beams in a plane, or four in space, give a lattice whose shape no change of phase can alter — the phases can only slide it. One beam more and the phases decide the shape, and must be held still.

waves · Superposition

Named alongside it

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

WavevectorBoundary conditionDispersion relationEvanescent waveGrating equationGuided modeInterferenceInterferometryLaser coolingOptical latticePhasePhase matching

All concepts