Illustrated Physics
physics, drawn
Concept

Wavevector — where it appears

A vector whose direction is the direction of propagation and whose magnitude is 2π over the wavelength. Its component along a boundary is conserved across it, which produces Snell's law, the critical angle and every grating order from one construction.

Named by 5 essays across 2 fields — each of them below, with the objects they name alongside it.

The lowest note a pipe will carry. The dispersion relation of a guided wave for three cutoffs, in units where the free wave speed is one. Each curve leaves the vertical axis at its own cutoff and bends toward the diagonal, which is the free wave. Above the cutoff the phase velocity is the slope of the line from the origin and always exceeds one, while the group velocity is the slope of the curve and never does: at k = 2 their product is 1.0000, 1.0000, 1.0000, which is one to four decimal places in every case and is an identity rather than a coincidence. Below the cutoff there is no curve, because there is no travelling wave to draw.

The pipe that will not carry a low note

A wave squeezed sideways acquires a lowest frequency. Below it nothing travels — the field is there, it is large, and it goes nowhere. Above it the guide is dispersive whether or not anything in it is, and the pattern inside runs faster than light while the signal does not.

waves · Guided waves 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 One slit, four distances, one multiplication. The intensity across the beam behind a slit 5 wavelengths wide, at distances of 0.5, 5, 25, 100 wavelengths, each computed by multiplying the slit's plane-wave spectrum by the phase each wave accumulates and transforming back — no approximation about angles. Close to the slit the pattern is the slit's own shape with ripples at its edges; further out the ripples move inwards and the beam develops a bright centre; far away it spreads into the diffraction pattern. The travelling part of the field keeps its power to 10⁻¹⁰, running it back 100 wavelengths recovers it to 5 × 10⁻¹⁴, and at 100 wavelengths the result matches a direct Fresnel integral to 2.75 per cent rms. Near field and far field are not two theories; they are one multiplication at different distances.

The fan of plane waves inside every beam

Huygens added up wavelets from every point of a front. The same content can be written as a sum over plane waves travelling in every direction, and then propagation stops being an integral and becomes a multiplication: each plane wave picks up a phase in proportion to the distance. One square root in that phase holds all of diffraction, near field and far field alike — and when the square root turns imaginary, it holds the reason no instrument a wavelength away can see detail finer than half a wavelength.

waves · Huygens 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 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

Named alongside it

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

Evanescent wavePlane waveReciprocal latticeSuperpositionAliasingBeatsBoundary conditionBoundary conditionsCutoffDiffractionDiffraction gratingDispersion

All concepts