Illustrated Physics
physics, drawn
Concept

F number — where it appears

The focal length of a lens or camera divided by the diameter of its aperture. It sets how bright the image is, falling as its square, and how deep the zone of acceptable focus is, so a small f-number means a bright, shallow image.

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

Stopping down removes one colour error and not the other. The two chromatic errors of a single 50 mm N-BK7 lens with its stop 20 mm in front, at an image point 10° off axis, against the f-number. Axial colour — blue and red focusing at different distances, seen as a coloured blur at the yellow focus — is about 775/N µm across: 387 µm at f/2, 97 at f/8, 35 at f/22. Lateral colour — blue and red images of the point centred at different heights — is 55.0 µm at every f-number, because it is carried by the ray through the centre of the stop. The two are equal near f/14, and past that stopping down leaves only the error it cannot touch.

The colour fringe no aperture can close

A lens bends blue light more than red, and that one fact produces two different faults. One puts the blue focus nearer the lens than the red, and closing the aperture shrinks it. The other puts the blue image of an off-axis point nearer the centre of the frame than the red one, and closing the aperture does nothing at all, because it is carried by the one ray every aperture keeps. It is why photographs have coloured fringes in their corners and not in their middles, and why a lens designer who cannot remove it with glass can sometimes remove it with symmetry, or with a distance.

optics · Imaging Too small blurs by diffraction, too large by shadow. The size of the image of a distant point through a pinhole 100 mm from the screen, in 550 nm light, against the hole's diameter, both on logarithmic axes. The dashed lines are the two simple estimates: the hole's own shadow, as wide as the hole, and the diffraction spread, 2.44λf/d. The dots are the diameter holding half the light, computed from the Fresnel diffraction integral. They follow the shadow for large holes and the diffraction for small ones, and are least — 0.207 mm — for a hole 0.336 mm across, where the simple sum of the two estimates puts its minimum at 0.366 mm. At the best hole the Fresnel number d²/4λf is 0.51: the sharpest hole, judged by where half the light lands, lets through about half of the first Fresnel zone.

The hole that makes the sharpest picture

A pinhole camera makes a picture without bending any light: a small hole lets through, from each point of the scene, only the rays that reach one small patch of the screen. A smaller hole makes a sharper picture — until it does not. Below a certain size the hole spreads light by diffraction faster than it narrows the shadow, and the picture blurs again. The best hole balances the two, and at that balance it turns out to pass about half of the first Fresnel zone of the incoming wave: the sharpest pinhole is already beginning to behave like a lens.

optics · Imaging

Named alongside it

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

Abbe numberAperture stopChief rayChromatic aberrationDiffractionDispersionFresnel zoneImagingLateral colourLens designPinhole cameraPoint-spread function

All concepts