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 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.

A diffraction grating of 1480 lines per millimetre used backwards to combine lasers of different wavelengths: the angle at which each wavelength must arrive, against wavelength from 1000 to 1100 nm, so that all leave the grating in the same direction, 60° from its normal, in the first order. The angle changes by 0.117° per nanometre near 1,050 nm, so lasers spaced a nanometre apart in wavelength are fed in from directions 0.12° apart and leave as one beam. A grating is a lossless device, and it can do this without breaking the étendue bound because the beams are not in the same mode: each is a different colour, and the output beam is one spatial mode holding many.

The far-field intensity of 8 identical beams side by side, each filling 50 per cent of its share of the aperture, against the angle in units of the wavelength over the beams' spacing, as a fraction of the phase-locked peak. With random phases (dashed) the intensities add: the pattern is 8 times one beam's, as broad as a single beam's. With the phases locked (solid) the fields add: the central peak is 8 times higher again, 64 times one beam's, and as narrow as the whole array's width allows rather than one beam's, with side peaks at whole multiples of the wavelength over the spacing. The side peaks are the price of the gaps between the beams: power the array cannot put in the central peak because the beams do not fill the aperture.

For 8 phase-locked beams side by side, the fraction of the total power in the central peak of the far field, against the fill factor — the fraction of the aperture the beams actually cover. With the beams filling half the aperture the central peak holds 46 per cent; at 90 per cent, 82 per cent; completely filled, 90 per cent. In one dimension the share rises nearly in proportion to the fill factor, and for a two-dimensional array it goes roughly as its square. Locked phases do make the beams one mode, but only a mode of the whole aperture, gaps included, and the gaps' share is diffracted away.

The share of the combined power that lands in the single coherent beam, against the RMS error of each laser's phase, in radians, for 2, 8, 64 beams with independent random errors. To keep 90 per cent the phases must agree to about 0.32 radian RMS — 0.052 of a wavelength, some 55 nm at 1,060 nm — and with many beams the share falls exponentially with the square of the error. Lasers drift in phase by whole wavelengths in milliseconds as their fibres warm and vibrate, so coherent combining needs every beam's phase measured and corrected continuously, thousands of times a second.

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.