A converging lens making a real image

At its defaults it draws a converging lens making a real image. An object 2.44 focal lengths from a thin converging lens. The image sits where the construction rays cross, at 1.69 focal lengths, magnified -0.69×.

thin-lens 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.

An object 2.44 focal lengths from a thin converging lens. The image sits where the construction rays cross, at 1.69 focal lengths, magnified -0.69×.

The quantity h divided by the sine of the angle at which the ray converges on the focus, in units of the paraxial focal length, against how far up the aperture the ray entered. Abbe's sine condition says that a system already free of spherical aberration images a small region round the axis faithfully only if this ratio is the same for every ray. A horizontal line means the condition is met. The parabola departs by 12.96 per cent across the aperture; The sphere departs by 7.18 per cent across the aperture. The paraboloid is the interesting case, because it is *exactly* stigmatic on axis — every ray from infinity crosses at one point, which is the definition of the shape — and it still fails this test. Perfection at one point buys nothing at the next one along. What the departure predicts is coma, a blur that grows linearly with the distance off axis and quadratically with the aperture, and the `offaxis` figure measures exactly that blur on the same surfaces. The condition is not a design rule invented for telescopes: it follows from requiring that the same optical path length join object and image for every route, and any instrument that images a field rather than a point has to meet it.

The size of the image blur at the paraxial focal plane, against how far off axis the source is, for parabola and sphere mirrors of the same focal length, traced with the exact law of reflection at every ray. On axis the paraboloid is perfect, by construction, and the sphere is not. A fifth of a degree off axis the paraboloid has a blur of 2.4 units at 1.2° for the parabola, 30.3 units at 1.2° for the sphere. The paraboloid's grows very nearly in proportion to the field angle, which is the signature of coma and is exactly what the offence against the sine condition predicts. This is the trade every reflecting telescope makes and the reason two mirrors are usually better than one: a single paraboloid buys a perfect axis at the price of a field a few minutes of arc wide, and the Ritchey–Chrétien pairing gives up the perfect axis to satisfy the sine condition and gets a usable field in exchange. The blur here is geometric only. Whether it matters depends on the diffraction limit sitting underneath it, which is the previous rung's subject, and on a large instrument the two cross at a field angle worth computing.

An object 2.44 focal lengths from a thin converging lens. The image sits where the construction rays cross, at 1.69 focal lengths, magnified -0.69×.

Parallel rays reflected off a parabolic mirror, each by the exact law of reflection about the local normal. Every ray crosses the axis at the same place, which is the defining property of the shape.

Where a lens of 50 mm focal length brings each part of a flat scene to a focus, against distance from the centre of the field, out to 21.6 mm — the corner of a 35 mm frame. The surface of best focus is a sphere of radius 76 mm curving towards the lens, and the corner of the frame sits 3.13 mm in front of the plane the centre is focused on. The shaded band is the depth of focus at f/8, which is 0.070 mm — 45 times smaller than the sag it has to cover. The curvature is not an error in the lens. It is what Σ1/nf comes to for this stack, and it depends on the powers and the glasses and on nothing else: bending the surfaces, moving the stop or stopping down changes every other aberration and leaves this one exactly where it was.

physicscheck asserts something about thin-lens 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.