Concept

Stationary phase — where it appears

The principle that a rapidly oscillating integral is dominated by the places where its exponent stops changing, everything else cancelling in pairs. It is why a classical path exists in a sum over all paths, and why a caustic — where the stationary point is degenerate — is bright.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

Stationary, and not always shortest. Three mirrors through the same point, with the same source and the same detector, and the length of the reflected path against where on the mirror the ray strikes. The actual ray is the one at the centre in all three, because the tangent there is horizontal and the two angles are equal whatever the curvature. Its path is a minimum at 0.6× the ellipse's curvature, every path equal at 1× the ellipse's curvature, a maximum at 1.6× the ellipse's curvature. On the most curved mirror every path the light did not take is shorter than the one it did, so the principle cannot be stated as least time and survive; it has to be stated as stationary time, and the difference is not a technicality.

The path that takes the longest time

Light is said to take the quickest route. Put a source and a detector in front of a mirror curved a little more than the ellipse through them, and the route it takes is slower than every route beside it — by construction, not by exception. The principle was never about least; it was about stationary, and the difference is where optics stops being geometry.

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

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

The deviation that has a bottom. The angle by which a 60° and a 90° ice prism bends a ray, against the angle at which the ray arrives, for an index of 1.31. Each curve is traced ray by ray through both faces and stops where it stops: outside the plotted range the ray meets the second face beyond the critical angle and never leaves. Both curves have a minimum, and the minimum is the point of the figure twice over. Its value — 21.84° for the 60° prism and 45.73° for the 90° prism — is where the sky puts a halo. And its flatness is why there is a halo at all: near a minimum the deviation changes only in second order, so a wide band of orientations all deliver light to nearly the same angle, and a cloud of randomly tumbling crystals piles up a bright ring there while sending the rest of the light nowhere in particular. The passage at the minimum comes out symmetric — in at 40.92°, out at 40.92° — which was found by searching the traced curve rather than assumed. The window of incidence that gets through at all is 76.5° wide for the 60° prism against 32.2° for the 90° one, a factor of 2.4, and that is why one of the two halos is common and the other is rare.

The ring at twenty-two degrees

A halo round the sun is a caustic in orientation rather than in space. Most of the ice crystals in a cirrus cloud send light nowhere in particular; the ones near minimum deviation all send it to nearly the same angle, because a minimum is flat — and the angle they pick has a red inner edge, which is the reverse of a rainbow.

optics · Dispersion

The echo hidden in random noise

A seismometer that is not recording an earthquake is not recording silence. It hears the Earth's hum: the pounding of ocean waves on every coast, arriving from all directions as surface waves with periods near ten seconds, indistinguishable from random noise. Two seismometers sixty kilometres apart hear different noise. Yet if their records are correlated and averaged over hours, a pulse appears at a lag of twenty seconds — the time a wave takes to travel from one to the other — exactly as if an earthquake had happened under one station and been recorded at the other. Noise from everywhere contains the echo between any two points, and since 2005 seismologists have mapped the Earth's crust with it.

waves · Guided waves

Named alongside it

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

Caustic Wavefront Airy function Cross correlation Diffraction Diffuse field Dispersion Fermat Focus Geometric optics Greens function Halo

All concepts