Illustrated Physics
physics, drawn
Concept

Thin lens — where it appears

A lens idealised as having no thickness, so that rays bend at a single plane and the image distance obeys 1/u + 1/v = 1/f. It maps planes to planes, which is why a tilted subject images onto a tilted plane meeting it on the lens plane.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

Cancelling a derivative, and what is left over. How far the focus of a 500 mm lens moves with wavelength, for a single crown element and for a cemented pair of crown and flint whose powers satisfy the achromatic condition. The singlet's focus runs over 18.1 mm across the visible — a fifth of a per cent of its focal length, and utterly ruinous at any useful aperture. The doublet's runs over 2.268 mm, some 8× less, and — this is the whole content of the figure — it is not flat. The condition sets the rate of change of power with wavelength to zero, so the curve is stationary rather than constant: it returns to the corrected focus at exactly 2 wavelengths — 486 nm and 656 nm, which are the two Fraunhofer lines the condition was written at — and departs from it everywhere else, most at 400 nm. That residual is the secondary spectrum, it has the same sign at both ends of the visible, and no pair of ordinary glasses removes it, because two conditions cannot be met with one free ratio.

Two glasses that cancel a derivative

A single lens focuses blue light closer than red, and the difference ruins the image. Cementing a second lens of another glass behind it fixes the fault at two wavelengths and at no others, because the condition sets a slope to zero rather than a value.

optics · Dispersion Three planes that meet in one line. A thin lens of focal length 50, drawn as a vertical line, and an object plane tilted at 45° to the axis, crossing it 150 units in front of the lens. Four points on the object plane are traced to their images through the lens's centre and along the axis-parallel ray. The images lie on a straight line too — a thin lens maps planes to planes — but one tilted far less steeply, and when the object plane, the lens plane and the image plane are extended they meet in a single line, here 150 units below the axis. That is the Scheimpflug rule. A sensor laid along the tilted image plane records the whole tilted object plane sharp, which no sensor square to the axis can do.

The plane of focus a tilted lens tilts

An ordinary camera focuses on a plane square to its axis, so a road receding from it, a table top or a field of flowers can be sharp only over the depth an aperture allows. Tilt the lens, or the sensor, and the plane of sharp focus tilts too, by a rule a surveying officer patented in 1904: the plane of the subject, the plane of the lens and the plane of the image always meet in a single line. Turn the focus and the plane of focus does not slide back and forth; it swings about a fixed hinge. The same geometry lets an eye camera see the whole front of the eye in one exposure and a lidar keep a laser beam kilometres long in focus.

optics · Imaging

Named alongside it

The objects these essays reach for when they reach for this one.

Abbe numberChromatic aberrationDepth of fieldDepth of focusDiffraction limitDispersionFocal lengthHinge ruleLidarMagnificationOptical designPlane of focus

All concepts