Concept

Geometric optics — where it appears

The description of light as rays travelling in straight lines through uniform media and bending at boundaries according to Snell's law. It is the limit of wave optics when every length in the problem is large compared with the wavelength, and it fails near foci, edges and shadow boundaries.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

$A caustic, by rays and by waves. The brightness across a fold caustic, computed two ways. Geometric optics gives the rising curve: on the illuminated side two rays arrive at every point and the intensity goes as the inverse square root of the distance from the caustic, so it becomes infinite exactly at it; on the other side no ray arrives at all and the intensity is zero. The wave answer is the squared Airy function, and it disagrees in three ways that are all observable. It is finite, peaking at 1.0188 in the scaled variable rather than at the caustic itself, so the brightest line is displaced onto the bright side. It oscillates, with maxima at -1.02, -3.25, -4.82, -6.16 — those are the supernumerary fringes, and they are not interference between two separate objects but between the two rays the caustic joins. And it leaks: on the dark side, where geometry forbids any light, the Airy function decays exponentially rather than stopping, which is the same mathematics as tunnelling and is why the edge of a shadow is soft before diffraction from any aperture is considered. The two curves agree far from the caustic, which is where the ray picture is a good approximation and where they have been matched here.$

The fringes below the rainbow

Geometric optics puts the whole rainbow at one angle and predicts an infinite brightness there. What is seen instead is a peak displaced inside that angle, followed by a train of pink and green arcs — and their spacing is a measurement of the raindrops, because a caustic's structure is set by the wavelength to the two-thirds power over the drop radius to the two-thirds.

optics · Diffraction $A lens that cannot focus a wave longer than itself. How much a point mass brightens a wave passing behind it — the factor on the wave's power, on a logarithmic scale — against the dimensionless frequency w = 8πGMf/c³, for a source exactly behind the lens and 0.1, 0.3, 1.0 Einstein radii off it, from the exact diffraction integral. Geometric optics puts the magnification at 10.0, 3.4 and 1.34 for the three offsets and at infinity on the axis, whatever the frequency. The wave disagrees: at w = 0.01 every curve is within 1.6 per cent of 1 — a wave a hundred times longer than the lens passes as if it were not there — and on the axis the power factor is πw/(1 − e^(−πw)): 3.28 at w = 1, 31.4 at w = 10, finite at every frequency. The off-axis curves rise to their geometric values only above w ≈ 1 and then oscillate about them, as the two images begin to interfere.$

The lens a long wave walks round

Geometric optics says that a star passing exactly in front of a source magnifies it infinitely. No real wave agrees, because a wave cannot be focused to a point, and for the gravitational waves the detectors hear the disagreement is the whole story. A star, or a black hole of a few tens of suns, is smaller than one of those waves in the only measure that matters — the delay between paths round it — and the wave flows past it almost untouched. What decides whether a mass lenses a wave is not the mass's size, which is millions of kilometres, but its Schwarzschild radius set against the wavelength.

astrophysics · Light deflection

The rings the full solution finds. The light scattered back by water drops of about 10 µm radius, at 550 nm, from the exact Mie series with around 134 terms for each drop, averaged over a ten per cent spread of sizes so that the fine ripple each single drop has is washed out. Instead of the smooth, weak background of the ray picture, the intensity rises sharply toward 180° through a series of rings about one and a half degrees apart, each brighter than the last, to a first ring 1.5° from straight back. These are what an observer sees round the shadow of an aircraft on a cloud, or round a shadow on a mist at a mountain top: the glory. It is present in the same mathematics that gives the rainbow, and it takes the whole series, with its surface-wave terms, to find.

The rings no ray can reach

Look down from an aircraft at a cloud, and round the aircraft's shadow there may be coloured rings, centred exactly on the point opposite the Sun. Mountain climbers have seen them round their own shadows for centuries, and every drop in that cloud is a sphere of water. Yet ray optics says that a drop of water cannot send light straight back. Its best ray falls fourteen degrees short. The glory lives entirely in those fourteen degrees.

optics · Scattering

Named alongside it

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

Diffraction Airy function Backscatter Caustic Cloud droplet Einstein radius Glory Gravitational lens Gravitational waves Interference Magnification Mie scattering

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