Illustrated Physics
physics, drawn
Concept

Depth of focus — where it appears

The distance the detector can be moved before a point image blurs by more than an agreed amount, roughly twice the wavelength times the square of the f-number. It is a few tens of micrometres for an ordinary camera lens, which is why a curved field of a few millimetres is a catastrophe rather than a detail.

Named by 5 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 The bowl a lens actually focuses onto. 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.

The flat scene that comes back curved

A lens does not image a plane onto a plane. It images it onto a bowl, and the curvature of that bowl is fixed by the powers and the glasses alone — not by the shapes of the surfaces, not by where the stop is, and not by stopping down. Everything a designer usually plays with leaves it exactly where it was.

optics · Imaging Sturm's conoid: a line, a circle, a line. The cross-section of a parallel beam after a lens whose focal length is 100 mm in its vertical meridian and 125 mm in its horizontal one, traced ray by ray through a round aperture, at seven distances behind the lens: 60.0, 100.0, 105.6, 111.1, 118.1, 125.0, 135.0 mm. There is no point focus anywhere. At 100 mm the vertical meridian has focused and the beam is a horizontal line; at 125 mm the horizontal meridian has, and it is a vertical line. Between them the beam is an ellipse that turns from wide to tall, and at 111.1 mm — the harmonic mean of the two focal lengths — it is a circle, the circle of least confusion, of diameter 22.2 per cent of the aperture's.

The focus that is two lines

A lens curved more steeply in one direction than the other has no focus. It brings light to a short line at one distance, to a line at right angles at another, and in between to an ellipse that turns from wide to tall and is round at exactly one place. Most eyes are built this way to some degree, so do mirrors used at an angle, and the best any ordinary lens can do for them is to put that round patch on the retina. A cylinder, curved in one direction only, can do what no sphere can: move one line onto the other.

optics · Imaging A bright core in a set of rings. The intensity across the beam 29 cm beyond an axicon — a glass cone 10 mm across that tilts every ray by 0.5° towards the axis — lit by a 633 nm laser, computed from the Fresnel integral and scaled to its peak (solid). It is the pattern of a cone of plane waves meeting on the axis: a bright core 56 μm across at its first dark ring, at 2.405/kα from the axis, ringed by rings of slowly falling brightness. Dashed: a Gaussian laser beam focused to a spot of the same size, with no rings. The rings are not a flaw of the axicon; they are where the beam keeps the power that the core is short of.

The beam that does not spread and the price it pays

Pass a laser beam through a shallow cone of glass and it emerges as a thin bright line of light, a core a few tens of micrometres wide that keeps its width for half a metre, where a beam focused to the same width by a lens would have doubled in a few millimetres. It seems to defeat diffraction. It does not. The core is not made of the light that was in it a moment before; every point along it is fed by a fresh ring of the aperture, and the core holds about one per cent of the beam's power. The rest is in the rings, and the rings are what pays for the line.

optics · Diffraction The focus that moves out of the eye. Parallel rays from a distant object entering a simplified eye 24 mm long, in millimetres along its axis: in air the eye's 61 dioptres bring them to a focus 21.9 mm behind its principal plane, on the retina; under water its 18 dioptres would bring them to a focus 73 mm behind, about three times the length of the eye. On the retina itself the underwater rays have converged only a third of the way, and every point of the scene is spread into a disc the size of the pupil's shadow. The eye is not out of focus by a little, as a short-sighted eye is; it is missing most of a lens.

The eye that loses its lens in water

Eyes opened under water see only a blur, and the reason is not water in the eye or salt or murk. The human eye does two-thirds of its focusing not with its lens but with the curved front of its cornea, whose power comes from the jump in refractive index between air and tissue. Water is almost as dense optically as the cornea, and the jump all but disappears: about forty-three of the eye's sixty dioptres vanish, and the focus moves to three times the eye's own length behind the retina. A mask gives the cornea back its air; a small pupil and a hard squint buy back a little; animals that hunt in both worlds have flattened their corneas and given the work to the lens.

optics · Imaging

Named alongside it

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

Focal lengthRefractive indexAberrationAstigmatismDioptreOptical designAbbe numberAccommodationAngular spectrumApertureBessel functionBlur

All concepts