Illustrated Physics
physics, drawn
Concept

Resolution — where it appears

The ability of an instrument to report two nearby things as two, measured by the separation or wavelength ratio at which they stop merging. It is set by diffraction at the aperture rather than by the detector, so improving it means changing the wavelength or the opening and nothing else.

Named by 6 essays across 3 fields — each of them below, with the objects they name alongside it.

Single-slit diffraction at three slit widths. Intensity against angle behind a single slit, evaluated from the integral across the aperture, for slits two, six and twenty wavelengths wide. A wide slit throws a nearly sharp shadow; a narrow one spreads light through a wide angle.

Where rays stop being enough, and a shadow acquires a bright centre

Light going through a narrow gap spreads. No amount of ray tracing predicts it, the size of the spreading is set by one ratio, and taking that ratio to zero is exactly what the ray model is.

optics · Diffraction A grating of 20 slits. Intensity against angle behind a grating of 20 slits spaced 4 wavelengths apart, from I = [sin(Nu)/(N sin u)]² with u = π d sinθ/λ. The principal maxima sit at sinθ = mλ/d — -14.48°, 0.00°, 14.48° for m = -1, 0, 1 — and those angles contain no N at all, so they are exactly where two slits put them. What N changes is the width: the central maximum measures 0.635 degrees between its half-maximum points, measured off the drawn curve, against 0.635 degrees from bisection on the pattern itself. That width goes as 1/N — N times it is 12.692°, 12.691°, 12.691° at N = 64, 256, 1024, and 12.705° here, which is more, because the 1/N law is asymptotic and few slits are the far end of it: two slits give a peak 13 per cent wider than the limit. So a grating's resolving power R = mN is bought with the number of lines illuminated and with nothing else. In first order this grating resolves λ/Δλ = 20; the sodium doublet needs 982. Away from the maxima the pattern stays below 4.8 per cent of one, because there it is bounded by 1/(N sin u)².

What a thousand slits buy that two cannot

The bright directions behind a grating are fixed by its ruling pitch and the wavelength alone, and no count of lines appears in them. What the count changes is the width of each maximum, which falls as 1/N — so resolving power is mN, and 1,200 illuminated lines separate the sodium D lines with a dip of 53.4 per cent where 300 show one line and no dip at all.

optics · Diffraction A current that falls by a decade for every ångström. Tunnelling current against the width of a vacuum gap, on a logarithmic scale, for three work functions covering the range of clean metal surfaces. The curves are straight because the transmission is an exponential in the gap, and their slopes are 0.833, 0.944, 1.044 decades per ångström — fitted to the drawn curves and agreeing with 2κ/ln 10 to a part in a million. At 4.5 electronvolts a change of ten picometres, a tenth of an atomic radius, changes the current by 24 per cent. That is the sensitivity a scanning tunnelling microscope lives on, and it is why the instrument measures height by holding the current fixed and recording what the piezo had to do: the current is far too steep a function of height to be read as one.

The last atom does all the seeing

A tunnelling rate falls by a factor of eight for every tenth of a nanometre of extra barrier, which is normally quoted as the reason nothing ever tunnels anywhere. Read the other way it is a microscope: the second-nearest atom of a blunt metal tip carries a two-hundredth of the current the nearest one does, so a tip nobody sharpened resolves a single atom, and the resolution comes from an exponential rather than from any piece of engineering.

quantum · Tunnelling 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 Two terms, and the distance neither can beat. The smallest distance a probe of a given momentum can resolve, as the sum of two terms. The falling one is the uncertainty relation: more momentum, shorter wavelength, finer resolution. The rising one is gravity: the probe's own energy curves the region it is probing, and past a point it makes a horizon larger than the thing being looked at. With the gravitational term at 1× the Planck area, the least resolvable distance is 2.286e-35 m — each located by scanning the drawn curve over four hundred thousand momenta rather than by substituting into a formula. There is no momentum at which the resolution is better than that, so the ordinary procedure for measuring a distance has a floor, and the floor is the Planck length up to a factor of order one.

The length no experiment can resolve

Measuring a small distance needs a short wavelength, a short wavelength needs a large energy, and a large energy in a small region makes a horizon. Past a point, pushing harder makes the probe bigger — and the distance where that turns round is the Planck length.

astrophysics · Planck scale The rings belong to the edge, not to the size. The far-field intensity of 3 apertures of the same width, against angle in units of the diffraction limit, on a logarithmic intensity axis spanning ten decades. They differ only in how the transmission falls off toward the rim. With a hard edge the first sidelobe is 13.3 decibels down and the core is 0.89 wide. With a Hann taper the first sidelobe is 31.5 decibels down and the core is 1.44 wide. With a Blackman taper the first sidelobe is 58.1 decibels down and the core is 1.64 wide. The hard edge's rings are not a defect of the optics and are not reduced by making it larger — they are the transform of a discontinuity, and the only way to remove them is to remove the discontinuity. What it costs is the width of the core, which is the resolution.

The rings that belong to the edge

Every account of diffraction so far asks what the size of an aperture does. The rings around a star are not about its size: they are the transform of a discontinuity, they do not shrink relative to the core when the telescope grows, and the only way to remove them is to stop the transmission falling to zero abruptly. Softening the edge buys forty-five decibels of contrast and costs eighty per cent of the resolution.

optics · Diffraction

Named alongside it

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

ApertureDiffractionInterferenceRayleigh's criterionWavelengthApodisationBlack holeCoherenceContrastCoronagraphDecay lengthDensity of states

All concepts