Illustrated Physics
physics, drawn
Concept

Fourier transform — where it appears

The pairing of a signal with its spectrum, under which a narrow feature in one is a wide one in the other and neither can be sharp alone. That reciprocal width is the source of the diffraction limit, the bandwidth of a pulse and the uncertainty relation, which are three statements of one theorem.

Named by 14 essays across 4 fields — each of them below, with the objects they name alongside it.

How far apart the two paths can be. Fringe visibility against the difference between the two path lengths, for light at 550 nm with a bandwidth of 100 nm. Each curve is the modulus of the Fourier transform of its own line shape, summed over the spectrum here rather than taken from a standard result, and the three shapes have the same width at half height. The conventional coherence length λ²/Δλ is 3.02 µm for this light, and what the curves show is that the convention is a rounding of three genuinely different behaviours: a flat band halves at 1.83 µm, a Gaussian line halves at 1.33 µm, a Lorentzian line halves at 0.68 µm. The flat band comes back — a rectangle's transform rings — and the Lorentzian's tails keep a little visibility very much further out than its width suggests. Nothing here is about the apparatus: the fade is the source forgetting its own phase.

How far a wave can remember

Split a beam, delay one half, and put them back together. The fringes are bright while the delay is short and fade as it grows, and the distance at which they die is fixed by nothing but the width of the source's spectral line. Watching them fade is reading the line shape.

optics · Coherence The narrow packet is the one that spreads. Three packets on deep water, ω = √(gk), all built on the same 4 m carrier and differing only in bandwidth — 10%, 18%, 28% of the carrier wavenumber. Each curve is the width of the emitted envelope, measured as the second moment of its intensity about its own centroid in a frame moving at the group velocity, divided by that width at the start. The starting widths are 4.50 m, 2.50 m, 1.61 m, and the order of the curves is the reverse of the order of the widths: the shortest packet, which is the one with the widest spectrum, is the one that comes apart first. The dashed curves are √(1 + (t/τ)²) with τ = σ₀²/|d²ω/dk²| — computed from the dispersion relation, not fitted. They agree with the measured widths to 0.3% at 10% bandwidth, 3.3% at 18% bandwidth, 8.5% at 28% bandwidth, and that ordering is the second thing the figure says: the closed form keeps only the curvature of ω(k), so it is exact for a narrow spectrum and starts to fail for a wide one, by about as much as the cubic term is worth. A packet with no bandwidth would never spread at all, and would also never begin or end.

The packet that will not keep its shape

A group velocity is only the first thing a dispersion relation says. The second is that the packet spreads — at a rate fixed by its own bandwidth and by the curvature of ω(k) — so a short pulse comes apart quickly and a long one hardly at all. It is a trade quantum mechanics is usually given credit for, and classical waves make it too.

waves · Wave packets What arrives in the back focal plane. A grating of pitch 1.2 µm illuminated at 550 nm, and the spectrum that appears in the objective's back focal plane. Each spatial frequency in the object leaves at its own angle, sinθ = mλ/d, and the bar heights are the Fourier coefficients of the object's transmittance. The aperture admits everything inside sinθ = 0.65, which here is orders -1, 0, 1 — 2 of them carrying information about the pitch. Orders outside it are drawn faint and are simply lost: they never reach the image plane, and no amount of magnification afterwards recovers them. This is where a microscope's resolution is decided — not at the image, not by the eyepiece, but by which of these bars the front of the objective is wide enough to catch.

The image that is a diffraction pattern twice

A lens does not project an object onto a screen. It takes the object's spatial frequencies apart, spreading them across its own back focal plane at an angle each, and then puts them back together — so an image is the object's spectrum, filtered by whatever the aperture admits, transformed back. With only the zeroth order through, the image is a uniform grey with no information in it at all.

optics · Imaging Nine planes behind a grating, with no lens anywhere. The intensity across two periods of a 20 µm grating, at nine planes between it and the Talbot distance z_T = 2d²/λ = 1.26 millimetres, illuminated at 633 nm. The top profile is the grating itself and the bottom is the plane at z_T, and they agree to 2.8e-14: free space has reproduced the object with no imaging element of any kind. The middle profile, at half the Talbot distance, is the object shifted sideways by half a period, to 1.1e-14. At a quarter of the way the grating's own period has vanished entirely — its amplitude there is 1.3e-16 — because the odd orders have all turned by the same right angle and the even ones have not. With this grating open for half of each period there are no even orders either, so the plane is uniform: 9.6e-3 at twice the frequency as well, and a screen there shows no grating at all. None of this is interference between two beams; it is the whole spectrum of the object arriving with the phases exp(−iπλzm²/d²), which are all multiples of 2π when z is z_T.

The grating that photographs itself

Put a grating in a beam of light and hold a screen behind it. At one particular distance the screen shows the grating again — sharp, at full contrast, right way up, with no lens anywhere in the apparatus. Half way there it shows the grating shifted sideways by half a period. A quarter of the way there, a half-open grating shows nothing at all.

optics · Diffraction Flat, then exponential, then a power law. Survival probability against time in lifetimes, both logarithmic, for a resonance 20 linewidths above the bottom of its band and 400 below the top. The straight dashed line is exp(−Γt), and the computed curve sits on it through the middle — a fitted rate of 0.9998 per lifetime between one and eight — and leaves it at both ends. Below 1.57e-2 lifetimes the curve is flat, falling as (t/τ_z)² with τ_z = 0.1253 lifetimes; beyond 22.6 lifetimes it is an inverse square, which any exponential eventually loses to. Both departures are forced: the head by the state being normalisable and the tail by the band having a bottom.

The exponential that is only true in the middle

A decay law is not an assumption about nuclei; it is the Fourier transform of an energy distribution. Do that transform honestly and the exponential fails at both ends — flat at the start, because the state is normalisable, and an inverse square at the end, because no system has states of arbitrarily negative energy. Neither departure is a correction that could be made small.

quantum · Decay The same number read off a decay and off a linewidth. A lightly damped oscillator released and left alone, above, and the power spectrum of exactly those samples, below. The decay falls to 1/e of its starting amplitude after 8.0 cycles, which makes the quality factor π times that, or 25.0. The spectrum peaks at 1.0000 radians per second and falls to half its power 0.04001 radians per second wide, which makes the quality factor the peak divided by the width, or 25.0. The two disagree by 0.02 per cent, which is the resolution of the frequency grid rather than a difference in the physics. They cannot disagree by more, because they are the same statement: a resonance is narrow because its ringing is long, and the transform that turns one into the other is not an approximation but an identity. A measurement of either is a measurement of both — which is why a bell can be characterised by hitting it and listening, or by driving it and sweeping, and why the two instruments never argue.

The width that is a lifetime

Hit a bell and time how long it rings; drive it and measure how narrow its response is. The two numbers are the same number, and they cannot disagree — not because the physics conspires but because a decay and a linewidth are one function seen in two coordinate systems.

waves · Resonance Fringe contrast against baseline, for four stellar diameters. The visibility of the fringes an interferometer would obtain at 575 nm, against the separation of its two apertures, for uniform discs of angular diameter 10, 20, 47, 100 milliarcseconds. Each curve is 2J₁(πθB/λ)/(πθB/λ), the transform of a uniform disc, and each first reaches zero at 14.47 m for 10 mas, 7.23 m for 20 mas, 3.08 m for 47 mas, 1.45 m for 100 mas. Dividing each of those by λ/θ returns the same number, 1.2197, which is the 1.22 in every textbook and is the first zero of J₁ divided by π — recovered here from the four curves rather than written into them. The practical content is that a smaller star needs a longer baseline, in exact inverse proportion, and that the measurement is of a contrast rather than of a picture. Michelson and Pease found the fringes from Betelgeuse vanishing at a 3.07 m separation in 1920 at a wavelength of 575 nm, which by the same arithmetic is a disc 47.1 milliarcseconds across — and no telescope resolved that star for another seventy years.

The fringe that measures a star

Set two apertures 3.07 metres apart in 1920 and the fringes from Betelgeuse vanish. That single fact gives the star's angular diameter to two significant figures, without ever forming an image of it — because the contrast of a fringe pattern is a Fourier component of the source's own shape.

optics · Coherence A drive at one frequency, and what comes back at three times it. The Fourier components of the steady motion of an oscillator driven at a single frequency ω = 1.35, for drive strengths of 0.02, 0.05, 0.1. The equation is a harmonic oscillator with a cubic term added, and the components are projected out of the integrated motion rather than assumed. A linear oscillator answers only in the first column. This one answers in the third as well, because x³ of a cosine contains a cosine of three times the angle. The third harmonic grows as the drive to the power 3.00 where the fundamental grows as the power 1.00, so it is negligible at small drive and not at large — which is why nonlinearity in an instrument is a specification rather than a yes or no.

The oscillator that answers at three times the question

Push a spring hard enough that the parabola stops being the whole story, and three things happen that a linear oscillator cannot do: it emits frequencies nobody supplied, its resonance leans over, and its amplitude at one drive frequency depends on where the drive has been.

mechanics · Harmonic approximation The spectrum, and what the interferometer records instead. On the left, a source spectrum: 1 line near 2000 reciprocal centimetres. On the right, what a detector behind a two-beam interferometer reads as the path difference is scanned — the interferogram. It is the cosine transform of the spectrum, so the two panels carry exactly the same information and neither is more fundamental. The fast oscillation is the mean wavenumber; the envelope that decays over about 0.133 centimetres is the reciprocal of the linewidth, which is the coherence length; and where two lines are present, the beat between them is the splitting. Nothing disperses anything anywhere in the instrument.

The fringe and the spectrum are one measurement

An interferometer with no prism and no grating in it measures a spectrum, because what it records as the path difference is scanned is the Fourier transform of the source's spectrum. Coherence length and linewidth are the same fact stated twice, and the resolution is bought in centimetres of travel.

optics · Coherence The floor a state's survival cannot go below. The probability that a quantum state is still found in its initial state, against time measured as its energy spread times time over ħ, for four states with the same spread. The shaded region under cos²(ΔE t/ħ) is forbidden by Mandelstam and Tamm's theorem, and every curve — each a sum of phases over the state's energies — stays out of it. Two equally weighted levels run along its edge and reach an orthogonal state at exactly π/2, the fastest any state with this spread can. The same two levels driven off resonance, with the same spread, never get further than a survival of 0.500. Three equally spaced levels become orthogonal only at 1.7101, 1.0887 times the limit, and a coherent state never does, bottoming out at 0.0183. A spread of energy is permission to change, not an obligation.

The fastest a state can stop being itself

Time has no operator, so the energy–time relation cannot be the commutator inequality it resembles. What stands in its place is sharper: a state whose energy is spread by ΔE cannot become a different, orthogonal state in less than πħ/2ΔE, and cannot do it faster than its mean energy above the ground state allows either. Two equally weighted levels reach both limits exactly. Nothing else does.

quantum · Uncertainty One slit, four distances, one multiplication. The intensity across the beam behind a slit 5 wavelengths wide, at distances of 0.5, 5, 25, 100 wavelengths, each computed by multiplying the slit's plane-wave spectrum by the phase each wave accumulates and transforming back — no approximation about angles. Close to the slit the pattern is the slit's own shape with ripples at its edges; further out the ripples move inwards and the beam develops a bright centre; far away it spreads into the diffraction pattern. The travelling part of the field keeps its power to 10⁻¹⁰, running it back 100 wavelengths recovers it to 5 × 10⁻¹⁴, and at 100 wavelengths the result matches a direct Fresnel integral to 2.75 per cent rms. Near field and far field are not two theories; they are one multiplication at different distances.

The fan of plane waves inside every beam

Huygens added up wavelets from every point of a front. The same content can be written as a sum over plane waves travelling in every direction, and then propagation stops being an integral and becomes a multiplication: each plane wave picks up a phase in proportion to the distance. One square root in that phase holds all of diffraction, near field and far field alike — and when the square root turns imaginary, it holds the reason no instrument a wavelength away can see detail finer than half a wavelength.

waves · Huygens The ripple that sits on top of an absorption edge. The Cu K absorption of copper foil, 293 K through its edge and for seven hundred electronvolts above it, with the smooth atomic background it would have if the absorbing atom were alone drawn beneath it. The difference between the two is the fine structure: a modulation reaching 11 per cent, dying away as the photon energy rises, and entirely absent from a free atom. It is there because the ejected electron is a wave that the neighbouring atoms scatter back onto the atom that emitted it, so the absorption depends on whether the returning wave arrives in step with the outgoing one — which depends on the distance to the neighbour and on nothing else about the sample.

The ripple that counts the neighbours

An absorption edge is drawn as a step and it is a step with a ripple on it — a modulation of eleven per cent in copper, five in a zinc site buried in a protein. The ripple is the ejected electron's own wave, scattered back onto the atom that emitted it, so its period is a distance. It is the only way of measuring where an atom's neighbours are that does not need a crystal.

waves · Attenuation 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 The peak that is lost to a fraction of a wave. The height of the central peak, relative to a perfect pupil of the same size, against the root-mean-square error of the wavefront in waves, for four kinds of error — each computed from the transform and each normalised to the same rms. The dashed curve is the usual approximation, the exponential of minus the square of two pi times the error. What the figure shows is that to a good approximation it does not matter WHAT the error is, only how large it is in the mean square: four quite different shapes of wavefront give nearly the same peak. A fourteenth of a wave leaves 80 per cent of the peak, which is the conventional definition of diffraction-limited, and it corresponds to a quarter of a wave peak-to-valley for a simple defocus — which is where Rayleigh's quarter-wave rule comes from and why it is a convention laid over a computed number rather than a threshold in the physics.

How accurate a mirror has to be

The pupil's amplitude decides the rings; its phase decides the peak. A wavefront error of a fourteenth of a wave root-mean-square leaves eighty per cent of the peak intensity, which is the whole of what 'diffraction-limited' means — a convention laid over a computed number. And the number barely depends on what the error is, only on how large: four quite different aberrations of the same magnitude give nearly the same answer.

optics · Diffraction

Named alongside it

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

CoherenceDiffractionBandwidthInterferenceWavefrontResolving powerResonanceSpectrumSuperpositionWave packetApertureBeats

All concepts