Illustrated Physics
physics, drawn
Concept

Abbe number — where it appears

A single number describing how much a glass's refractive index varies with colour, computed from its index at three standard wavelengths. A high value means little dispersion; lens designers pair glasses of different values so that their colour errors cancel.

Named by 4 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 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 A filter made of glass powder and oil. The fraction of light transmitted straight through a 5 mm layer of N-BK7 grains packed to half its volume in the matching liquid, against wavelength, for grains of radius 5, 10 and 20 μm, from van de Hulst's extinction for large, weakly refracting particles. At the crossing the grains vanish and the layer is clear. A few tens of nanometres away each grain shifts the light passing through it in phase by enough to scatter a little of it out of the beam, and a thousand grains in a row scatter all of it: the passband is 51.7 nm wide at half height for 5 μm grains, 36.3 nm for 10 μm, 25.5 nm for 20 μm. Larger grains make a narrower filter, because the phase a grain imposes grows with its size; the same mismatch of index that a fine powder lets through, a coarse one scatters.

The colour at which glass disappears

Lower a glass rod into an oil of the same refractive index and it vanishes, except for a faint coloured outline. The outline is the clue. Glass and oil disperse light differently, so their indices can agree at only one wavelength; at every other the glass is still there, bending and scattering a little. Grind the glass to powder and pack it in the oil, and the mixture is clear at that one wavelength and opaque at the rest — a colour filter made of nothing but two transparent materials, whose colour moves by a whole hue for a few degrees of warming.

optics · Dispersion A lens with its glass taken out, focusing the same. Two lenses of focal length 200 mm and aperture 100 mm in acrylic (n = 1.49), each with its flat face to a parallel beam and the exact aspheric curve that sends every ray to one focus: solid (top), 12.2 mm thick at the centre, and the same curve cut into 13 rings 1 mm deep with the glass between them removed (bottom), on a backing 0.4 mm thick. Thickness is drawn six times larger than height, and the distance to the focus is not to scale. Every ray leaves each lens at the same angle from the same height, because the angle a ray turns through at a surface depends only on the slope of the surface where it crosses, and every ring keeps the slope it had. What was removed was glass the rays travelled through without turning.

The lens with its glass taken out

Snell's law turns a ray at a surface by an angle set by the slope of the surface where the ray crosses, and the glass behind the surface does not enter it. So a lens can be sliced into rings, the glass between the slopes thrown away, and the rays will not notice. Fresnel did it for lighthouses in 1822. The waves notice — every step cuts a few hundred wavelengths out of them — unless the steps are made exactly one wavelength deep, and then the lens becomes something else: a diffractive element whose colours run backwards, and which can cancel a glass lens's colour with a twentieth of its power.

optics · Refraction

Named alongside it

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

DispersionChromatic aberrationRefractive indexAperture stopChief rayDepth of focusDiffraction gratingDiffraction limitExtinctionF-numberFocal lengthFresnel lens

All concepts