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 11 essays across 4 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 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 Echoes from a layer that thins to nothing. Synthetic seismic echoes from a layer between two identical rocks — the top reflecting +R, the base −R — for thicknesses of 0, 5, 10, 15, 20, 25, 30, 40, 50, 60, 75, 100 m, sounded with a 30 Hz pulse in rock of 3000 m/s, whose wavelength in the layer is 100 m. Time runs downward; each trace is one thickness. Thick, the layer gives two separate echoes, one from each face, and its thickness can be read from the time between them. As it thins the two echoes overlap and merge. Near 20 m, about a fifth of a wavelength, they interfere constructively and the combined echo is strongest, 1.45 times one face's echo. Thinner still, the combined echo keeps a fixed shape — the pulse's derivative — and simply weakens in proportion to the thickness; nothing in its timing changes.

The layer too thin to see

A seismic survey sends a pulse of sound into the ground and listens to the echoes from every boundary where the rock changes. A thick layer returns two echoes, one from its top and one from its base, and the time between them gives its thickness. A thin one returns one echo, and below about a quarter of a wavelength the timing says nothing more: every thin layer looks the same, just weaker as it thins. It has not become invisible. It has become unmeasurable in the one way that was being used, and its thickness has moved into the echo's strength and its spectrum.

waves · Impedance A scan into a layered sample, with a broad source and a narrower one. The envelope of the interference signal against depth in a sample with five reflecting boundaries (marked), at 60, 72, 150, 158 and 260 µm, scanned with a 50 nm source (solid) and a 10 nm source (dashed), both centred at 840 nm. The broad source, with a 6.2 µm resolution, separates every boundary, including the two that are only 8 µm apart. The narrow one, with 31 µm, merges the pairs into single blurred echoes. Every boundary in a real eye or skin sample is a small change of refractive index that reflects a tiny fraction of the light; the scan is a map of where those changes are.

The depth a broad spectrum can see

Interference is usually spoiled by a broad spectrum: many colours drift out of step, and fringes wash out a few wavelengths from where two paths are equal. Turned round, that failure is a ruler. Light whose colours stay in step only when two paths match to within a few micrometres can report the depth of every reflecting layer inside something translucent — a retina, a skin, a coat of paint — to that precision, without cutting it open. Optical coherence tomography reads depth off the width of a spectrum, and it has become one of the most used imaging methods in medicine; its resolution is set by how many colours the light contains, and the finest instruments use sources whose spectra span hundreds of nanometres.

optics · Coherence Position written as frequency. Four tubes of water side by side in a 1.5 T magnet, their protons all precessing at 63.87 MHz, until a gradient of 10 mT/m is switched on along the row. Then the field, and so the precession frequency, rises steadily across the row, by 425.8 Hz for every millimetre. The signal the receiving coil picks up is a sum of waves at different frequencies, and its spectrum is a picture: the amount of signal at each frequency is the number of protons at the corresponding position. The axis below is position and the axis above it frequency, and they are the same axis. Nothing has been focused. The radio wave at 64 MHz has a wavelength of 4.7 m in air, and the tubes are centimetres apart.

The image made of frequencies

A hydrogen nucleus in a magnetic field precesses at a frequency proportional to the field — 63.87 megahertz in a 1.5 tesla scanner. Make the field rise steadily across the body and every position precesses at its own frequency, so the radio signal the body gives off is a chord whose spectrum is a map of where the protons are. A magnetic resonance image is never focused. It is computed, as a Fourier transform of a signal recorded one number at a time, and its resolution, a millimetre, is set by how strong the gradient is and how long the signal is listened to — not by the wavelength of the radio waves, which is several metres.

quantum · Spin The long pulse that arrives as a spike. The output of the matched filter for the chirp, in decibels against delay in units of 1/B, beside the same filter's output for an unmodulated pulse of the same duration (dashed). The plain pulse's echo is a broad triangle 100 units wide; the chirp's is a spike 0.878/B wide at its half-power points — compressed by the time-bandwidth product, 50. The compression is not free of shape: the spike has sidelobes, the first at −13.6 dB, from the abrupt edges of the band, and a weak echo sitting in a strong echo's sidelobe can be hidden by it. The energy is all there — the filter has concentrated the whole pulse's echo into one instant of output.

The long pulse that arrives as a spike

A radar that wants to see far must send a lot of energy, and a transmitter limited in peak power can only do that with a long pulse; a radar that wants to tell close targets apart needs a short one. A chirp — a pulse whose pitch rises steadily from start to finish — is both. Received through a filter that delays its low notes more than its high ones, every part of the echo arrives at the same instant, and fifty microseconds of pulse leave the receiver as a spike one microsecond wide. Resolution turns out to belong to bandwidth, not to duration, and the price of the trick is paid in sidelobes and in a target's speed being mistaken for its range.

waves · Wave packets

Named alongside it

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

DiffractionApertureFourier transformInterferenceSpectrumBandwidthCoherenceImagingPoint-spread functionRayleigh's criterionSidelobesWavelength

All concepts