Illustrated Physics
physics, drawn
Concept

Diffraction limit — where it appears

The smallest angular detail an aperture can resolve, set by the wavelength divided by the aperture and by nothing about the instrument's quality. It is 1.22 times that ratio for a circular opening, which for the eye at green light is about a minute of arc.

Named by 8 essays across 2 fields — each of them below, with the objects they name alongside it.

The dip that decides it. Two equally bright points seen through a circular aperture, their Airy patterns added, at 0.7, 1, 1.6 times the Rayleigh separation. Below each is the depth of the dip between the peaks, measured off the drawn sum: 0.0%, 26.5%, 91.6%. At exactly the Rayleigh separation the dip is 26.5% — the criterion is a convention about how much of a dip a detector can see, not a threshold anything crosses. Below it the two peaks merge into one and the pair is gone; above it the answer was never in doubt.

How far apart two things have to be

An instrument's ability to tell two things apart is not set by the quality of its glass. It is set by the width of the hole light comes through, by a factor that is the first zero of a Bessel function — and the threshold everyone quotes is a convention laid over a computed dip of 26.5 per cent.

optics · Diffraction 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 Image distance against object distance. Image distance in focal lengths against object distance in focal lengths. At exactly one focal length the image runs off to infinity; inside it the image distance goes negative, which means virtual.

The focus that is a slab, not a plane

A lens images one plane and no other, which would make every photograph and every micrograph almost entirely out of focus. What rescues them is a tolerance — and there are two of them, one from rays and one from waves, which give different answers and stop being interchangeable exactly where microscopes work.

optics · Imaging The étendue, tiled. The phase space of a one-dimensional optical system: position across a twenty-micrometre aperture, against the optical direction cosine n·sinθ, for a system accepting ±0.1. The shaded rectangle is what the beam occupies, and its area — 4 micrometre-radians — is the one-dimensional étendue, the quantity no arrangement of lenses can reduce. Divided by a wavelength of 500 nanometres it is 8, and the rectangle is tiled here with exactly 8 cells of area one wavelength each. That is the whole content of the count: a cell of phase-space area λ is the smallest patch a field can be confined to, because squeezing it in position spreads it in direction by the same relation that gives a slit its diffraction pattern, so an étendue is a number of modes and its conservation is the conservation of a count. The cells are drawn four wide and two high, which is arbitrary: only a cell's area is fixed, not its shape — a beam may be confined tightly in position and loosely in angle or the other way round, and the trade is what an optical system is for.

The invariant that is a count

Étendue is an area times a solid angle, and ray optics gives it no floor — nothing in a ray has a size. Divide it by the square of the wavelength and it becomes a number of modes, its conservation becomes the conservation of a count, and the count has a least value of one. That is where the ray bound hands over to diffraction, and the handover is the same number written two ways.

optics · Etendue A field that oscillates faster than any of its components. The oscillation of (cos x + i·2·sin x) to the power 10 over half a period, with its changing size divided out so that only the turning of its phase shows, beside the fastest wave it contains, cos 10x (dashed). Multiplied out, the function is a sum of waves e^(inx) with n running from −10 to 10 and no further. Yet in the shaded window, |x| < 0.62, it oscillates faster than cos 10x — near the origin like cos 20x, 2 times faster — completing 3.0 cycles there where its fastest component completes 2.0. Outside the window it slows down, and the next figure shows what happens to its size there.

The wiggle faster than any wave in it

Light passing through a lens carries no detail finer than half a wavelength, because no travelling wave in it varies faster than that. It would seem to follow that the light itself cannot vary faster anywhere. It can. A sum of slow waves can be arranged to wiggle faster than any of them over a stretch of any length chosen — and to focus to a spot as narrow as wanted — provided the stretch sits between side lobes that grow exponentially with every extra wiggle.

waves · Huygens The focus that gets sharper when the wave is scattered. The refocused amplitude against distance across the source, in wavelengths, for the same one-sided mirror and pulse, with the space between empty (dashed) and with a slab of 200 lossless point scatterers, 15 wavelengths thick, filling the space between source and mirror (solid). Through the slab the wave reaching the mirror is a jumble — scattered many times, arriving from every direction the slab can send it — yet time-reversed and sent back through the same slab, it refocuses on the source, in a spot 1.5 wavelengths wide against 4.8 in empty space. The scatterers, bouncing waves in from a wide range of angles, act as a lens much wider than the mirror: disorder that would ruin an ordinary lens makes a time-reversal mirror sharper.

The mirror that sends a wave back to where it began

Record a wave as it arrives at a row of microphones, then play each recording backwards through the loudspeaker beside it, and the sound converges on the place it came from — however twisted the route it took. Huygens said every point on a wavefront acts as a source; a time-reversal mirror takes him literally, making every point of a recorded front re-emit its own history in reverse. The result has a property no lens has. Put a thicket of scatterers between the source and the mirror, so that the arriving wave is an unrecognisable jumble, and the reversed wave still finds its way back — to a sharper focus than it could reach in empty space, because the disorder acts as a lens wider than the mirror.

waves · Huygens A point of light split by an uncoated roof. The image of a distant point across the direction of the roof edge, behind an Amici roof in glass of index 1.517, for unpolarised light, as a fraction of the peak an undivided pupil would give, against position in units of the diffraction width λ/D: with a phase-coated roof, and with bare glass. The coated roof gives the ordinary diffraction pattern of the full aperture. Bare glass leaves the two halves of the pupil only partly able to interfere, and the image's peak falls to 0.70 of its proper height, its central lobe widening across the edge into a shoulder on each side — a slight doubling of every point, which an observer sees as a loss of crispness and contrast.

The roof that splits a binocular's image

A slim, straight binocular turns its image upright with a roof — two glass faces meeting at a right angle, which reflect every ray twice by total internal reflection and lose no light at all. They lose sharpness instead. Total reflection shifts the phases of the two polarisations of light by different amounts, and the half of the beam on one side of the roof's edge meets the two faces in the opposite order from the other half. The halves leave in different polarisations, interfere only partly at the focus, and every star becomes a slightly doubled smudge — unless a coating a fraction of a micrometre thick makes the glass behave like a mirror.

optics · Total internal reflection A facet too wide and a facet too narrow. The blur of an insect's compound eye of radius 1 mm against the width of one facet, in green light. A wide facet looks at a wide patch of sky compared with its neighbours, so the eye's sampling is coarse, and that blur rises with the width. A narrow facet is a small aperture, so diffraction spreads what it collects, and that blur falls with the width. Added in quadrature they make a minimum at D = √(λR), here 22 µm, and that is close to the facets of real insects with an eye of that size. No choice of facet width can get below about 1.8°: that figure is forced by the size of the head. Horace Barlow worked this out in 1952 and compared it with bee eyes.

The eye a fly cannot afford

A compound eye is built from thousands of tiny straws, each looking in one direction. Wide straws look coarsely at the world, and narrow ones blur what they collect by diffraction, so there is a best width, and it depends only on the size of the head. To see as sharply as a person, an eye of this kind would need to be about twelve metres in radius. A single-lens eye gets the same result with a pupil of under two millimetres.

optics · Diffraction

Named alongside it

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

Numerical apertureAngular resolutionApertureFocal lengthPoint-spread functionAbbe numberAiry patternAntennaBandwidthBeam qualityChromatic aberrationCoherence

All concepts