The Lens

What it does to a point

A lens is judged by what it makes of something that has no size. Every ray of three colours across the whole pupil, landed on the focal plane, at the centre of the field and two-thirds out and right in the corner. The dashed circle is the Airy disc: the blur that diffraction alone would give a perfect lens at this aperture. Anything inside it is as good as glass can be.

The prescription

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Focal length

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Speed

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Clear aperture

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Field

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On axis

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In the corner

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Airy disc

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Spot against it

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Blue against red focus

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Thinnest edge

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Work done

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How a lens gets cut here

1. 01

   Take the order

   The day's number decides who wants a lens and what for, and that decides the form — a singlet behind a stop, a cemented achromat, an air-spaced doublet, a Cooke triplet, a Petzval — along with the focal length, the speed and how much of the world it has to cover.

2. 02

   Split the power

   Glasses are drawn from the catalogue and each element is given its share of the total power by thin-lens arithmetic — for a doublet, the split that makes the red and the blue focus in the same place — and then bent into a starting shape.

3. 03

   Push every number until it stops helping

   Each curvature, each airspace and the focal plane are nudged one at a time, the step halving whenever a whole pass fails to improve anything. Between every trial the system is rescaled so the focal length comes out exactly as ordered, which is why it is never a compromise.

Two things here were not invented on the premises. The first is the glass: sixteen catalogue types, each stored as its index at the yellow helium line and its Abbe number, which is the trade's one-figure summary of how badly a glass splits colour. The dispersion in between is reconstructed as a two-term Cauchy curve fitted to those two figures, which by construction passes exactly through the red, yellow and blue lines a catalogue quotes and tracks the published curve to about a unit in the fifth decimal across the visible. The second is Snell's law, applied as a vector at every surface — rays are intersected with real spheres by solving a quadratic, refracted in three dimensions, and dropped entirely if they miss the glass or turn round in total internal reflection, which is exactly what you will see happen if you push a surface far enough.

What is simplified, said plainly. The stop sits at the front surface of every design, and moving it is one of the strongest tools a real designer has — this bench does not hand you that one. Distortion is measured but taken out of the picture above, which shows blur and not shape. The surfaces are spherical, with no aspherics and no cemented-surface tolerance. The diffraction blur folded into the picture is a Gaussian standing in for an Airy pattern, so there are no rings around a bright point. And the optimiser is coordinate descent, which is a plodding thing next to the damped least squares a real design program uses: it finds a good lens near the form it was given and it does not go looking for a better form.

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The same date integer that drifts the field on the front page also seeds the Garden, writes the Chimes, deals the Lattice, names the town in the Almanac, scatters the Observatory, raises the coast in the Atlas, breeds the Menagerie, lights the star in the Orrery, invents the alphabet in the Codex, founds the countries in the Chronicle, designs the building in the Folly, cuts the gears in the Engine, threads the warp in the Loom, prices the goods in the Exchange, writes the rules in the Gambit, locks the doors in the Warren, hides the thief in the Inquest, cuts the gorge in the Span, writes the timetable in the Junction, enciphers the dispatch in the Bureau, cuts the letters in the Foundry and mixes the glaze in the Kiln — and here it grinds something you are meant to look through rather than at. The Observatory hands you a sky already in focus. This is the instrument that would have had to bring it there.