The bargain no optical system gets out of

At its defaults it draws the bargain no optical system gets out of. A beam entering an aperture 208 units across, filling a half-angle of 12°, and leaving one 3 times smaller. The angle it fills on the way out is not 12° but 38.6°, because the product of the aperture and the sine of the angle is the same at both ends — that product is the étendue, and no lossless system can reduce it. Measured off the drawing, 43.246 at the entrance against 43.246 at the exit. The consequence is the one that matters: flux divided by étendue is the radiance, so a smaller spot is a larger angle and never a brighter image. No lens, no mirror and no arrangement of either has ever made anything brighter than the thing it was looking at.

concentration-limit 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.

A beam entering an aperture 208 units across, filling a half-angle of 12°, and leaving one 3 times smaller. The angle it fills on the way out is not 12° but 38.6°, because the product of the aperture and the sine of the angle is the same at both ends — that product is the étendue, and no lossless system can reduce it. Measured off the drawing, 43.246 at the entrance against 43.246 at the exit. The consequence is the one that matters: flux divided by étendue is the radiance, so a smaller spot is a larger angle and never a brighter image. No lens, no mirror and no arrangement of either has ever made anything brighter than the thing it was looking at.

The greatest concentration a receiver can be given, against the half-angle it accepts, both logarithmically, for a receiver in a medium of index 1. The upper curve is a three-dimensional concentrator, n²/sin²θ; the lower is a trough, which concentrates in one direction only, n/sin θ. The Sun's angular radius is 0.267°, so sunlight can be concentrated by at most 46165 times in a dish and 215 times in a trough — marked. Nothing about glass, mirrors, wavelength or aperture appears in either expression. The bound is thermodynamic: a receiver that accepts light out to θ also *radiates* out to θ, and the concentration at which what it emits balances what it absorbs is exactly where these curves are.

A beam entering an aperture 208 units across, filling a half-angle of 12°, and leaving one 3 times smaller. The angle it fills on the way out is not 12° but 38.6°, because the product of the aperture and the sine of the angle is the same at both ends — that product is the étendue, and no lossless system can reduce it. Measured off the drawing, 43.246 at the entrance against 43.246 at the exit. The consequence is the one that matters: flux divided by étendue is the radiance, so a smaller spot is a larger angle and never a brighter image. No lens, no mirror and no arrangement of either has ever made anything brighter than the thing it was looking at.

The temperature a perfectly black, perfectly insulated absorber reaches under concentrated sunlight, against the concentration, from one Sun to the 46165 the geometry allows. 1× gives 394 K; 10× gives 700 K; 100× gives 1245 K; 1000× gives 2213 K; 10000× gives 3936 K; 46165× gives 5770 K. The last of those is 5770 K, and the Sun's surface is 5772 K: the greatest concentration geometry permits delivers exactly the temperature of the source and stops. That is not a coincidence and it is not a coincidence twice, because the optical bound was derived with no thermodynamics in it at all. If a lens could beat it, an engine run between the hot spot and the Sun would produce work from a single temperature — which is why the ceiling has the value it has, and why it cannot be moved by a better mirror.

Concentration against rim angle for a paraboloidal mirror aimed at the Sun, against the thermodynamic ceiling of 46165. The curve is sin²φ·cos²(φ+θ)/sin²θ, which is what the Sun's image on the focal plane of a paraboloid is worth. It rises, peaks at 11434 at a rim angle of 44.87° — located by scanning the drawn curve, and displaced from a round 45° by the Sun's own angular radius — and falls back to nothing at 90°. The best of them is 25 per cent of what is allowed. The shortfall is not a manufacturing tolerance and no figuring of the mirror recovers it: a dish that makes an *image* wastes étendue, because the edge of the image is formed by rays arriving at the wrong angle to be useful. Concentrators that reach the ceiling give up imaging altogether, and the compound parabolic funnel is one.

A compound parabolic concentrator for an acceptance half-angle of 18.0°, drawn from the edge-ray construction: each wall is a parabola whose axis is tilted by the acceptance angle and whose focus is the opposite edge of the exit. Measured off the drawn profile, the entry is 3.2361 times the exit against the 1/sin θ = 3.2361 that is the two-dimensional ceiling — the bound reached exactly, to 0.0e+0. It is 2.01 times as long as it is wide, which is the price. What it gives up is the image: a ray entering inside the acceptance angle emerges somewhere on the exit and nobody can say where, and a ray entering outside it is turned round and sent back out of the front. That is the trade the previous figure priced — the dish knows where every ray came from and gets a quarter of the light; this knows nothing and gets all of it.

physicscheck asserts something about concentration-limit 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.