Illustrated Physics
physics, drawn
Concept

Mode — where it appears

One of the independent patterns in which a wave can exist in a given space, such as a standing wave in a cavity or a beam of one shape and polarisation. Counting modes turns étendue into a number and fixes how much light can be combined.

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

A lens with two flat faces. Rays entering a rod whose refractive index falls parabolically from the axis outward, parallel to the axis and at -2.4, -1.2, 0, 1.2, 2.4 mm from it. The ray equation in such a medium is the harmonic oscillator's, so each path is a cosine of the same period whatever height it started at — which is exactly the condition for a focus, and the reason all of them cross the axis together at 12 mm. Nothing is curved anywhere: the faces are flat and the bending is done by the inside of the glass. Cut the rod at a quarter of the period and it images; cut it at half and it relays the beam parallel again, inverted. The period is a property of the profile alone, which is why a rod like this is specified by a length rather than by a curvature.

The channel with no walls

A pipe will not carry a note below its cutoff, and no length of pipe helps. Replace the walls with nothing but a region where the wave travels slightly slower, and the cutoff disappears — however weak the contrast and however thin the channel, at least one mode is bound. The difference is not a matter of degree; it is the difference between a boundary condition and a potential well.

waves · Guided waves 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 Two kinds of flicker in one heat bath. The variance of the number of photons in one mode of black-body radiation, against hν/kT, both on logarithmic axes, split into its two parts: n̄, the variance a shower of independent particles would have, and n̄², the variance of the intensity of randomly phased waves; their sum is what thermodynamics demands of Planck's law. Where the photons are rare, at hν much more than kT, the particle term wins; where they are crowded, the wave term wins; they are equal at hν/kT = ln 2 = 0.693, where each mode holds one photon on average. At hν = 5kT the wave term is 0.68 per cent of the particle term; at hν = kT/10 the particle term is 10.5 per cent of the wave term.

The flicker that is half wave and half particle

In 1909, sixteen years before quantum mechanics, Einstein asked how much the energy in a small corner of a hot oven should fluctuate, and answered with nothing but thermodynamics and Planck's law. The answer has two terms. One is exactly what a spray of independent particles would give; the other is exactly what randomly overlapping waves would give. Neither theory of light could produce the sum, and the sum is what the oven does. The two terms are equal where each mode of the radiation holds one photon on average — a dividing line that runs through every hot thing, from the microwave sky to the Sun.

quantum · Photon

Named alongside it

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

CoherenceBlackbody radiationBound stateCutoffDiffraction gratingDispersionEtendueEvanescentFluctuationsGuided wavesLaserNumerical aperture

All concepts