The deviation that has a bottom

At its defaults it draws 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.

prism-deviation is one function in lib/figures/optics.js — rays, lenses, mirrors and what light does to a surface. Everything below came out of it during this build, at parameters taken from the essays rather than invented for this page. A figure here is the figure a reader meets in an essay, and if the generator changes, this page changes with it.

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.

Parallel rays entering a spherical drop at different heights, refracting in, reflecting once from the back, and refracting out. The outgoing rays crowd together near one particular direction, and that crowding is the bow.

Total deviation of a ray through a raindrop, plotted against how far off-centre it entered. The curve has a minimum, so rays near it emerge in almost the same direction whatever their entry point — and that stationary point is the angle of the bow.

Total deviation of a ray through a raindrop, plotted against how far off-centre it entered. The curve has a minimum, so rays near it emerge in almost the same direction whatever their entry point — and that stationary point is the angle of the bow.

Total deviation of a ray through a raindrop, plotted against how far off-centre it entered. The curve has a minimum, so rays near it emerge in almost the same direction whatever their entry point — and that stationary point is the angle of the bow.

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.

physicscheck asserts something about prism-deviation that could fail — it draws it and measures the result against a value reached some other way.

Across the library: 100 interrogated, 2 exercised only, 1 untouched, of 103. Read out of the gate's source by the gate's own two patterns — and the gate's last claim fails the build if that read disagrees with what it was handed while running.