Illustrated Physics
physics, drawn
Concept

Interference — where it appears

The addition of two or more waves at a point, giving a result that depends on their relative phase rather than on their intensities alone. It requires the phases to stay in step for as long as the detection lasts, which is why two independent lamps show none of it.

Named by 57 essays across 7 fields — each of them below, with the objects they name alongside it.

Two sources 3 wavelengths apart. Circular wavefronts from two sources, with the lines along which they arrive in step drawn through the pattern. Those lines are where the path difference is a whole number of wavelengths.

When two waves meet, they simply add

Waves pass through each other unchanged and their displacements add point by point. From that one impoverished-sounding rule comes interference, beats, and the evidence that light is a wave at all.

waves · Superposition 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 An electron's wavelength against the voltage that accelerated it. The de Broglie wavelength of an electron after falling through a potential difference, in picometres. At 100 volts it is 122.6 picometres, at 400 volts it is 61.3 picometres, at 900 volts it is 40.9 picometres. The wavelength goes as the inverse square root of the voltage, so quadrupling the voltage halves it. The calculation is non-relativistic; at a kilovolt that costs a tenth of a per cent. The dashed line is the 215 picometre spacing between atomic planes in nickel, which is what makes an electron beam diffract off a crystal at all.

Everything has a wavelength, and almost nothing shows it

If light with a momentum can behave like a particle, a particle with a momentum can behave like a wave. The wavelength is Planck's constant over the momentum, which for anything larger than a molecule is a number too small to have consequences.

quantum · Matter waves The interference pattern arriving one particle at a time. The same double slit — 100 micrometres apart, slits 40 micrometres wide, lit at 633 nm, screen 1 metre away — recorded after 20, 200, 1000 arrivals, with the intensity that governs them plotted underneath. Each arrival is a single dot in one place, drawn at a position sampled from that intensity. After 20 there is no pattern to see; after 1000 the fringes are unmistakable, with the dark ones exactly 6.33 millimetres apart — the wavelength times the screen distance over the separation. The bright ones are not evenly spaced, because the single-slit envelope pulls each maximum toward the centre; its first zero is at 15.8 millimetres and is set by the width of one slit alone.

One arrival at a time, and the pattern still appears

Send particles through a double slit slowly enough that only one is ever in the apparatus, and each arrives as a single dot in one place. Wait, and the dots assemble into fringes that no dot knew about.

quantum · Matter waves What a passing wave does to a ring. A ring of 8 free masses at 4 phases of a passing gravitational wave, in both polarisations. The upper row is the + mode: one diameter lengthens while the perpendicular one shortens, and half a cycle later they swap. The lower row is the × mode, which is the same pattern rotated by forty-five degrees rather than ninety — the signature of a spin-2 field, and the reason a detector is built as two arms at a right angle. The drawn strain is 0.42; a real one is 10⁻²¹, so the deformation is exaggerated 4.2·10²⁰ times. At that true strain a four-kilometre arm changes length by 4·10⁻¹⁸ m.

The wave that stretches one way and squeezes the other

A gravitational wave passing through a ring of free masses lengthens one diameter while shortening the perpendicular one, then swaps. The two conservation laws that forbid anything simpler are why the effect is a part in a thousand million million million.

astrophysics · Gravitational waves 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 Two sources 4 wavelengths apart. Circular wavefronts from two sources, with the lines along which they arrive in step drawn through the pattern. Those lines are where the path difference is a whole number of wavelengths.

Why two lamps never interfere

Adding amplitudes is unconditional; fringes are not. What decides is whether the phase difference holds still for longer than a detector takes to record it — and a 10 nm slice of white light holds it for 100 femtoseconds, across a path of 30 micrometres.

optics · Coherence The correlation, and the best a shared list of answers can do. The coincidence correlation between two polarisation analysers against the angle between them, over two full turns of the correlation — a polariser turned through 180° is the same polariser, so the picture repeats. The singlet gives −cos 2Δ, drawn through −1.00 at 0°, 1.00 at 90°, −1.00 at 180°, 1.00 at 270°. Beside it is the best correlation any shared list of pre-agreed answers can produce: straight lines between the same four extremes, with corners where the cosine is smooth. The two agree exactly at the multiples of 45° and nowhere else, and they are furthest apart — by 0.2105 — at 19.77° and 70.23°, which is ½ arcsin(2/π) from either end of the quarter turn. The difference is not a matter of degree: it is a curve against a shape with a corner in it, and no list can be bent into the curve.

The correlation no instructions can produce

A pair of gloves in two boxes agrees perfectly and needs no physics, because the answers were settled at packing. What no packing can imitate is the shape that appears as the two analysers are turned relative to each other, and the shape is a number — 2.828 where every list of pre-agreed answers is stuck at 2.

quantum · Entanglement 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 What a real coating leaves behind, and over what range. Reflectance against wavelength for a glass surface of index 1.52 in air, uncoated and with quarter-wave layers of 2 different indices, each a quarter of a wave thick at 550 nm. The ideal index is the geometric mean, 1.2329, and the layer made of it takes the reflectance to zero at the design wavelength exactly. No durable solid has that index: magnesium fluoride at 1.38 is the usual compromise and it leaves 1.26% at the design wavelength against 4.26% bare — a reduction of 3.4× rather than a removal. Both curves rise away from the design wavelength, because the thickness is a quarter of a wave only there, and the useful band is wide but not unlimited: the better coating stays under a quarter of the bare reflectance from 415 to 780 nm. The purple cast of a coated lens is that residual — the ends of the visible reflecting while the middle does not.

The layer that makes a reflection vanish

A wave meeting a step in impedance reflects, and nothing can be done about the step. Put a third medium between the two, a quarter of a wavelength thick and of exactly the intermediate impedance, and the reflection stops existing — not reduced, cancelled.

waves · Impedance The Cornu spiral, and the chords that are amplitudes. Fresnel's two integrals plotted against each other, traced from v = -4.2 to 4.2. Arc length along the curve is v — the distance along the wavefront in units of √(λL/2) — and the amplitude arriving from any stretch of that front is the straight chord between the stretch's two endpoints, not the length of curve between them. The two eyes at ±(½, ½) are where the far parts of an unobstructed front pile up, so the whole open front is the chord between them, of length 1.4142. A straight edge blocks half of it and leaves a chord of 0.7071 — exactly half the amplitude, so a quarter of the intensity, at the geometrical shadow's edge. The chord drawn here runs to v = 1.217, is 1.6556 long, and is the longest chord from the eye that exists: its square over the open chord's square is 1.370, which is why the brightest fringe outside a shadow is brighter than no obstacle at all.

The spiral that says how much light arrives

Huygens' construction says where a wave has got to and refuses to say how bright it is, because an envelope is a locus and a locus has no amplitude. Adding the wavelets with their phases instead of taking their envelope turns the whole subject into one curve, and every near-field pattern there is becomes a chord of it.

waves · Huygens Stationary, and not always shortest. Three mirrors through the same point, with the same source and the same detector, and the length of the reflected path against where on the mirror the ray strikes. The actual ray is the one at the centre in all three, because the tangent there is horizontal and the two angles are equal whatever the curvature. Its path is a minimum at 0.6× the ellipse's curvature, every path equal at 1× the ellipse's curvature, a maximum at 1.6× the ellipse's curvature. On the most curved mirror every path the light did not take is shorter than the one it did, so the principle cannot be stated as least time and survive; it has to be stated as stationary time, and the difference is not a technicality.

The path that takes the longest time

Light is said to take the quickest route. Put a source and a detector in front of a mirror curved a little more than the ellipse through them, and the route it takes is slower than every route beside it — by construction, not by exception. The principle was never about least; it was about stationary, and the difference is where optics stops being geometry.

optics · Fermat The frequencies a repeat will not carry. The band structure of a medium made of quarter-wave layers of index 1 and 2, repeated for ever: frequency against Bloch phase across one cell, in the reduced zone. Inside a band the phase runs from 0 to π and the wave travels. Between bands there is no real phase at all, and the shaded strips are frequencies at which the medium supports nothing — not a weakly transmitted wave, no wave. Gap 1 runs from 0.784 to 1.216; Gap 2 runs from 2.784 to 3.216, in units of the quarter-wave design frequency. The first, measured off the drawn band edges, is 0.4327 wide against the 0.4327 of (4/π)·arcsin|r| — the same number computed from the Fresnel ratio of the two indices alone, agreeing to 2.6e-14 per cent. Every band edge sits where the phase is 0 or π, which is to say where the wave's own period fits the repeat a whole number of times: the gap is a property of the periodicity, and the materials only decide how wide it is.

The gap a repeat opens

Stack two transparent materials in alternating layers and there is a band of frequencies the stack will not carry — not weakly, not with loss, but not at all. Nothing has been absorbed and neither material has a resonance there. What forbids those frequencies is the repeat itself, and the width of the band has a closed form containing only the ratio of the two indices.

waves · Periodic media How fast a spinning electron would have to turn. The equatorial speed of a uniform sphere with the electron's mass and an angular momentum of ħ/2, against the radius it is given, both logarithmically. The expression is 5ħ/4mr and it passes the speed of light at 4.83e-13 m — half a picometre, which is four hundred times larger than a hydrogen nucleus. Giving it a smaller radius only makes the answer worse: at the experimental upper bound, 10⁻¹⁸ m the equator would move at 4.8e+5 times the speed of light; at the classical electron radius the equator would move at 1.7e+2 times the speed of light; at the reduced Compton wavelength the equator would move at 1.3e+0 times the speed of light. No radius the electron is permitted to have gets anywhere near a legal answer, and the experimental bound is off the scale by eight orders of magnitude. So the angular momentum is not the angular momentum of anything going round. It is a property the particle has, in the same way a charge is, and the only thing it shares with a spinning top is the algebra it obeys — which is, admittedly, the whole of what angular momentum means in physics.

The angular momentum that is not a rotation

An electron has angular momentum, and it is not going round anything. A sphere of its mass carrying that much angular momentum would need its equator moving at half a million times the speed of light at any size the electron is allowed to have. What survives of the analogy is the algebra — and the algebra turns out to require that turning the thing all the way round leaves it changed.

quantum · Spin A quarter of the time, an object is found without being touched. The outcomes of sending one photon into a balanced interferometer, with and without an opaque object in one arm, computed from the same amplitudes. With the arm clear, every photon leaves by the bright port and the dark port receives nothing. With the object in place, 50 per cent of photons are absorbed by it, 25 per cent reach the bright port and say nothing, and 25 per cent reach the dark port — which is impossible unless something is in the arm, and which happens with the object still sitting there unabsorbed and unlit. The photon that produced that click did not go through the blocked arm, because a photon that goes through a blocked arm is absorbed. So an object has been located by light that never met it. The price is that 50 per cent of attempts destroy the thing being looked for: an efficiency of 33 per cent, which the Zeno version of the apparatus takes as close to one as one likes.

The measurement that never touched it

A balanced interferometer sends every photon to one output and none at all to the other. Put an object in one arm and the empty port starts clicking — and a click there is caused by a photon that cannot have gone near the object, because a photon that goes near it is absorbed. The object has been found by light that never met it.

quantum · Measurement 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 One expression over 3 decades of area times rate. Sagnac time difference against the product of enclosed area and rotation rate, both logarithmic. The relation Δt = 4AΩ/c² is linear in that product — the fitted slope of the drawn points is 1.0000 — and it contains no refractive index, no shape of the loop and no position of the axis inside it. a 1 km fibre gyroscope on a 10 cm coil: 1.62e-19 s, 3.14e-5 fringes; Sagnac's own ring, 1913: 4.84e-17 s, 0.0666 fringes against a reported 0.07; a laboratory turntable at one revolution a second: 6.99e-17 s, 0.0331 fringes; Michelson and Gale, 1925: 4.50e-16 s, 0.2364 fringes against a reported 0.23. Michelson and Gale's rectangle in Illinois is the one that carries the check: 0.236 fringes predicted from its own dimensions and the vertical component of the Earth's rotation at its latitude, and 0.230 reported.

The ring where the two beams disagree

Send light both ways round a closed loop on a turntable and the two beams come back at different times, by 4AΩ/c² — an expression with no refractive index in it, no shape of the loop and no position of the axis. The same number is what a set of clocks round the rim fails to close by, which is the sharper statement — on a rotating platform there is no global simultaneity to be had.

relativity · Simultaneity A circuit on the sphere of polarisations. The Poincaré sphere, on which every polarisation state is a point: linear states around the equator, circular at the poles, and orthogonal states at opposite ends of a diameter. The triangle is a closed circuit — linear at 0°, then linear at 23°, then left circular, and back — taken along geodesics, which is what a sequence of ideal polarisers does. Bringing a state round it returns it to exactly the state it started in, and multiplied by a phase: 0.3927 radians here, against minus half the enclosed solid angle of -0.7854 steradians. The two agree exactly, and neither calculation knows about the other — the phase is the argument of a product of three overlaps between Jones vectors, and the solid angle is spherical geometry. Nothing about the elements used, their thickness, or the wavelength enters. A phase that depends only on the shape of a path is the signature of a geometric phase, and this is the oldest known example of one.

The phase that is only a shape

Take a beam of polarised light through a sequence of elements that returns it to the polarisation it started with, and it comes back with a phase it did not have before. That phase is not an optical path length — it does not depend on the thickness of anything, or on the wavelength, or on how slowly the sequence was carried out. It is minus half the area the path enclosed on the sphere of polarisation states, and nothing else.

optics · Polarisation A grain that is not on the object. A speckle pattern, computed as the far field of a circular aperture 421 samples in area filled with random phases — which is what a rough surface does to coherent light, and nothing else. The texture is not a picture of the surface: change the phases and the grains move, but their size and their statistics do not. Five shades are drawn here, from the darkest fifth of the range to the brightest. The measured contrast — the standard deviation of the intensity divided by its mean — is 1.0101, against exactly one for a fully developed speckle, and that is a strong statement: it says the most likely intensity anywhere in this pattern is zero, and that the bright grains are as far above the mean as the dark ones are below. Anybody who has pointed a laser at a wall has seen this and most take it for a property of the wall. It is a property of the light and of the aperture looking at it — including, when the aperture is an eye, of the pupil, which is why the pattern swims when the head moves and why its grain size tells an optometrist about the eye rather than about the wall.

The grain that is in the light

Point a laser at a wall and the wall appears to be covered in a fine boiling texture. Nothing on the wall is that size and nothing about the wall decides it: the grain belongs to the aperture looking at it, the statistics are the same for every rough surface there is, and the most likely brightness anywhere in the pattern is zero.

optics · Coherence A step, a wave from the rim, and what they make together. The pattern behind a straight edge, split into the two things Young said it was: the incident wave where the edge does not block it, which is a step, and a wave that appears to come from the rim itself. The step is discontinuous at the shadow boundary by the whole incident amplitude. The edge wave's magnitude is continuous there, to 0.0e+0 — it changes phase by π instead of changing size — and it is exactly half the incident amplitude on both sides, which is why the total intensity on the geometrical shadow boundary is 0.250000 of the unobstructed value rather than a half. Everywhere else the two add, and their sum reproduces the directly computed field to 0.0e+0. The fringes in the lit region are the interference of the two, which is why they are fringes at all: a monotonic decay has nothing to beat against. In the shadow there is only the edge wave, so there is nothing to interfere with and the intensity falls smoothly.

The wave that comes from the rim

A shadow's edge is not a boundary between light and no light, and the fringes on either side of it are not a smudge. The whole pattern is the sum of two things — the light nothing blocked, and a single wave that behaves in every respect as though the rim of the obstacle were radiating it — and splitting it that way is exact rather than a picture.

waves · Huygens Rings whose radii go as the square root of their number. A 20-zone plate for 550 nanometres at 200 millimetres, drawn to scale, beside its zone radii against zone number. The outermost is 1.483 millimetres and the innermost 0.332, and the curve is a square root because the radii come from making each zone's extra path exactly half a wavelength longer than the last. The consequence worth noticing is on the drawing rather than in the formula: every ring has the same area, to 0.003 per cent, so each contributes about equally to what arrives on the axis and the rings get thinner outwards to keep it so. Alternate rings are opaque. Half the light is thrown away and the axis gets brighter, because what is thrown away is the half that would have arrived out of phase with the rest.

The lens that is a set of rings

A lens focuses by delaying the light at its centre until every path takes the same time. A zone plate does the opposite: it changes nothing about the light that gets through, and paints out the light that would have arrived out of step. Half the aperture is thrown away and the axis gets brighter, which sounds like a contradiction and is the whole idea.

optics · Diffraction Two barriers, and the energies at which they stop being barriers. Transmission against energy on a logarithmic scale, for one barrier of width 0.9 and height 20, and for two of them separated by a well of width 2. Below the top of the barrier the single one transmits an exponentially small amount and does so smoothly. The pair does not: at particular energies the transmission climbs by orders of magnitude and reaches 1.000000 — one, to every digit the arithmetic has — even though each barrier on its own passes 3.9e-2 of what arrives. Multiplying the two single-barrier transmissions would give 1.5e-3, which is wrong by 6e+2: amplitudes are what compose, not probabilities, and the region between the barriers is a box whose quasi-bound levels are where the amplitudes reinforce. The resonances sit at 6.438, 13.915 and the levels of an infinite well of the same width are at 2.467, 9.870 — near, and not equal, because a well with leaky walls has states that are not quite bound.

Two walls that let more through than one

Put a second barrier behind the first and the transmission does not fall — at particular energies it rises to exactly one, through a pair of walls each of which stops all but a few per cent. Probabilities cannot do that. Amplitudes can, and the region between the barriers is a box whose levels say where.

quantum · Tunnelling Why only a sideways scattered wave takes anything away. The transmitted amplitude behind a thin scatterer, drawn as a phasor: the incident wave of unit length along the axis, plus a forward-scattered wave of length 0.12 at 0°, 60°, 90°, 150°. What a detector reads is the square of the total length. A scattered wave along the incident one lengthens or shortens the sum in proportion to itself; one at right angles changes the length only in second order, because a small perpendicular addition to a long vector barely alters its length. So a scatterer that removes energy from the beam at first order must scatter forward with a component perpendicular to the incident wave, and the size of that component is the whole extinction — which is the optical theorem.

Everything a scatterer removes, from one direction

How much light a particle takes out of a beam — by scattering it anywhere at all, and by absorbing it — is fixed entirely by what it does in the forward direction, where its scattered wave cannot be told apart from the incident one. The mechanism is interference, and it also gives the refractive index.

optics · Scattering 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 cross term, and the fact that it averages to nothing. The intensity of two waves of amplitude 1 and 0.7 added together, against the phase difference between them, in turns. A detector reads the square of the summed amplitude, which is the sum of the two intensities plus a cross term that swings between plus and minus twice the product. At no phase difference the reading is 2.89 and at half a turn it is 0.09; the flat line is what the two would give with no interference, 1.49, and it is exactly the average of the curve over a whole turn. Interference redistributes and does not create — which answers the question of where the energy goes at a dark fringe by saying that it never left.

What adding does to the energy

Waves add their amplitudes and detectors read squares, so two waves together do not deliver the sum of what each delivers. Where the two get dimmer, the natural question is where the energy went — and the answer depends entirely on whether the sources can feel each other.

waves · Superposition A resonance that goes to zero before it peaks. Fano profiles for asymmetry parameters of 5, 1.5, 0.5, 0, each normalised to its own peak, against detuning in half-widths. A large parameter gives an almost symmetric peak — the resonant path dominates and the shape is nearly Lorentzian. A parameter of zero gives a symmetric dip, a window in which the system transmits nothing on resonance. In between the profile is lopsided, with a zero on one side of the resonance and the maximum on the other, at positions whose product is exactly minus one. The asymmetry is not a defect of the measurement: it is the interference of two ways through the system, and its sign says which side of the resonance the two paths cancel on.

The resonance with a zero in it

Where a resonance is the only way through a system, the response is a symmetric peak. Where there is also a smooth path that does not care about the frequency, the two add before anything is squared — and the result is lopsided, with a frequency at which nothing gets through at all.

waves · Resonance Three ways for a wavelet to be strong, and what each leaves behind. On the left, the strength of a secondary wavelet against the angle from the forward direction, for three candidate rules. Huygens' construction as stated has no such rule: a wavelet is spherical and equally strong in every direction. On the right, what each predicts when the wavelets over a whole plane are added up, on the axis, in front of the plane and behind it. All three reproduce the incident wave in front, which is the part of the construction that has always worked. Only the rule that falls to exactly nothing at a hundred and eighty degrees leaves nothing behind, and that rule is not a repair invented for the purpose — it comes out of solving the wave equation.

The backward wave Huygens had to remove

Every point of a wavefront is a source of a spherical wavelet, and a spherical wavelet goes in every direction — so the construction predicts a wave travelling backwards as well as forwards. Nothing of the kind exists, and the repair is a factor that Huygens' geometry has no room for.

waves · Huygens Minima that are not zeros. The amplitude along a line carrying a wave towards a load and its reflection back, for reflection magnitudes of 0, 0.35, 0.7, 1. With everything reflected the pattern touches zero and is a standing wave in the strict sense. With less than everything it does not: the minima sit at one minus the reflection and the maxima at one plus it, so the pattern is a partial standing wave sitting on a travelling one. The spacing is half a wavelength in every case, and the depth is the only thing that changes — which is why one number, the ratio of the maximum to the minimum, is enough to report the whole pattern.

The node that is not standing still

A wave meeting a perfect reflector makes a standing wave with real nodes. A partial reflection makes something that looks the same and is not: the minima are not zeros, energy flows steadily through them, and the depth of the pattern is a measurement of the load that caused it.

waves · Standing waves Outside the winding, where nothing is supposed to be. The mid-plane of a solenoid 20 radii long, wound at 8 turns per radius, with the field summed turn by turn rather than assumed. The upper curve is the axial field against distance from the axis: it holds up across the winding and collapses outside, reaching 0.47 per cent of its central value at 2 radii. The lower curve is the flux enclosed by a circle of that radius, which is what the vector potential integrates to. The two behave completely differently, and that difference is the whole subject: the field an electron outside can feel has effectively gone, and the flux it encircles has not. The enclosed flux does fall, by 3.29 per cent out to four radii, because the lines that leave the ends come back through the plane outside the coil — that return flux is the leak a real experiment has to defeat, and the reason the definitive versions used a closed toroidal magnet with no ends at all.

The phase a magnet leaves on a path it never touched

An electron beam split around a solenoid comes back with its fringes displaced, although neither path ever entered a magnetic field. What the electron responds to is the flux it went round, and the only local quantity that knows about that flux is the potential.

quantum · Matter waves What comes back after one turn, and what needs two. A spin-½ pointing along z and rotated about the x axis through 720°, with the rotation integrated step by step rather than evaluated from a formula. The direction of the spin — the quantity a Stern–Gerlach magnet, a compass or any other instrument reports — is back where it started after 360°, exactly as the orientation of any other object would be. The state is not: its overlap with the state it began in has reached −1 there, and returns to +1 only after 720°. At 360° the overlap is -1.000 and ⟨σz⟩ is 1.000; At 720° the overlap is 1.000 and ⟨σz⟩ is 1.000. Both curves come off one integration of dψ/dθ = −(i/2)σx ψ whose norm is checked before anything is drawn, so the factor of two between their rates is a property of the propagation rather than of two separate formulae that were chosen to differ.

The turn that has to be made twice

Turn a spin-½ through a full circle and it does not come back. The direction it points in does, and every measurement on it does, but the state itself has changed sign — and a second full turn is needed before anything is where it started. The sign is invisible on one spin and measurable the moment a superposition has one branch turned and the other not.

quantum · Spin The fringes a flow of water moves. The interference fringe shift against the speed of the water, for two tubes 1.5 m long, an index of 1.333, and light of 526 nm — Fizeau's apparatus. The beam is split, each half goes with the flow in one tube and against it in the other, and the two are recombined; the shift is the difference in transit time counted in wavelengths. Three predictions are drawn and they are not close together. If the water did not affect the light at all the shift would be zero, flat along the bottom. If the water carried the light with it completely the shift would be the steepest line. Fresnel's partial drag is the middle one, and at 7 m/s it gives 0.207 of a fringe — which is what was measured, to the accuracy of an eye reading a fringe pattern in 1851. The experiment therefore did not merely detect an effect; it chose between three quantitative possibilities that differ by factors of two, which is why a fraction of a fringe settled something.

The drag that was only an addition

Light in moving water is carried along by it, but only partly — by a fraction of the water's speed that depends on the refractive index in a way nobody could account for. Fresnel invented the coefficient to save a theory, Fizeau measured it in 1851, and it sat unexplained for half a century. It is the first term of the relativistic velocity addition and nothing else.

relativity · Velocity-addition The colours a draining film runs through, and the end of them. The fraction of light a free soap film reflects, against its thickness, at three wavelengths — 450, 550, 620 nm — computed from the sum over multiple reflections rather than from the two-beam approximation. The three curves peak at different thicknesses, which is why a draining film runs through a sequence of colours as it thins. Below about 11 nm every curve is under a tenth of a per cent and the film looks black. At zero thickness the reflectance is exactly zero, checked before the figure is drawn: the two surfaces reflect equally and half a cycle out of step, so a film much thinner than a wavelength cancels itself. That is the whole of why a black film is black. Nothing is absorbing; the film is there and has simply stopped being able to interfere constructively at any visible wavelength — which means the blackness is a measurement, and a film that has gone black is known to be thinner than about a tenth of a wavelength without anything being measured directly.

The film that goes black before it bursts

A soap film drains, runs through every interference colour, and then stops reflecting anything at all. The black patch is not a hole and not a film about to break: it is the thinnest and most stable state the arrangement has, held apart by a pressure between its two surfaces that only exists at distances of nanometres.

fluids · Surface tension What a row of sources does that one cannot. The pattern of 8 identical sources in a row, spaced 0.5 wavelengths apart, all driven in phase. Each source alone radiates the same in every direction drawn here; together they radiate almost entirely along one. The sum being performed is the sum over path differences across the row, which is the sum a diffraction grating performs over its slits — the same function with the same first null, at sin θ = 1/Nd, measured here off the curve. What has been exploited is retardation: the contributions arrive at different times, and the pattern is a map of where they arrive in step.

When the source is not heard all at once

The dipole approximation is not a statement that a source is small. It is a statement that every part of it is heard at the same retarded time, and dropping that assumption turns one source into a sum over a source. The sum is the same one a diffraction grating performs over its slits, with the same first null and the same extra orders — so a phased array and a grating are one piece of arithmetic met twice.

electromagnetism · Retardation A quasi-probability that goes negative. The Wigner function of the oscillator's state at a quarter of its revival time, with 4 quanta on average: two copies of the packet, at x = ±2.83, in equal superposition. It is computed from the wavefunction by the Wigner transform on a lattice, and it integrates to one. The two copies are the two positive blobs. Between them lies a pattern of stripes with no classical counterpart, running from 0.289 down to −0.289, where the shaded warm regions and their outlines mark negative values. A negative probability is not a probability, so this distribution cannot describe a cloud of classical particles. The stripes are also taller than the blobs, so the interference carries more structure than the copies themselves: the highest stripe reaches 0.289 and the centre of a blob 0.159.

The probability that goes below zero

Classical mechanics describes an uncertain state as a cloud of points in the plane of position and momentum. Quantum mechanics has an exact counterpart, the Wigner function, whose shadows are the true position and momentum distributions — and which goes negative. It goes negative for a single photon, for every superposition of two packets, and for every pure state that is not a Gaussian. Where it is negative no classical cloud can imitate the state, and losing energy to the surroundings erases the negative regions first.

quantum · Correspondence Resonances drawn as bands. 60,000 decays of a D⁰ decaying to K⁻π⁺π⁰, accepted from 1,287,365 flat ones in proportion to the square of an amplitude built from 3 short-lived intermediate states, each decaying to two of the three products. A ρ⁺(770) in m²(π⁺π⁰), 67.0 per cent of the rate on its own; a K⁻(892) in m²(K⁻π⁰), 25.5 per cent of the rate on its own; a K⁰(892) in m²(K⁻π⁺), 33.2 per cent of the rate on its own. Each appears as a band at its own mass squared — vertical, horizontal or diagonal according to which pair it decays to — holding 51 per cent, 19 per cent, 23 per cent of the decays within one width of its mass, where phase space alone would put 34, 10, 9. The separate fractions add to 126 per cent, not 100, because the amplitudes interfere where the bands overlap. The magnitudes and phases are a model chosen to make all three visible, not a fit to data.

The plane in which three bodies are flat

A particle breaking into two gives each product a fixed energy; one breaking into three gives none of them one. What it gives instead is a plane of two invariant masses in which a decay with no forces spreads perfectly evenly inside a curved boundary — so every band, dark stripe and bright crossing a real decay draws there is a force, its spin, or a phase between two routes to the same three particles.

relativity · Relativistic dynamics One width that falls to nothing. The decay rates of the two modes of a pair of resonances that leak into one shared channel at rates 0.1 and 0.05 and are coupled to each other with strength 0.2, against the detuning of the first from the second. The two rates always add to 0.15, the trace of the leak matrix, to 10⁻¹². Where the resonances are far apart each mode keeps roughly its own resonance's leak. Near them the leaks interfere, and at a detuning of 0.1414 — κ(γ₁ − γ₂)/√(γ₁γ₂) — the slower mode's decay rate is zero to within 10⁻¹², and the faster carries all 0.15. That mode sits at a frequency where the channel is open and does not leak into it: the two routes by which it could escape cancel.

The resonance that refuses to leak

A resonance that sits at a frequency where waves can escape has a width, because it leaks. Put two such resonances into the same channel and let them talk to each other, and at one precise detuning one of their combinations stops leaking altogether — a mode with no width, surrounded by a continuum it could escape into and does not. It cannot be seen from outside, it traps whatever energy lands in it, and if a symmetry is what forbids the leak, it survives any change that keeps the symmetry.

waves · Resonance Two guides, and the two solutions they have. The transverse field of the two modes a pair of identical slab guides supports, against position across them, for guides half a micrometre wide separated by a gap of 300 nanometres at a wavelength of 1550 nanometres. A single guide has one fundamental mode; two guides side by side have two, and neither of them lives in one guide. The symmetric one is a single hump spanning both, the antisymmetric one has a node exactly between them — checked here to be exactly zero rather than nearly so — and they have slightly different propagation constants because the symmetric one has more of its field in the high-index gap region. That difference, computed from the slab's own dispersion condition, is 4.60e-2 per micrometre. Everything else about a coupler follows from it: a wave launched into one guide alone is the sum of the two supermodes in equal parts, they run at different speeds, and the interference between them moves the power from one guide to the other and back. There is no leakage in the account anywhere — only two solutions beating.

Two tails that swap everything

Bring two guides close enough for their evanescent tails to overlap and they do not leak a little power into each other. They exchange all of it, and then exchange it back, over a length fixed by the splitting between two modes that belong to neither guide — so a coupler is cut to a length rather than tuned to a ratio, and the length depends exponentially on a gap of a few hundred nanometres.

waves · Guided waves 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 With interference kept, transmission falls exponentially; without it, only as one over the thickness. Light through stacks of randomly thick transparent layers, alternating indices 1 and 2.6, with thicknesses scattered by ±50 per cent about a quarter wave, on a logarithmic scale against the number of layers. The falling line is the transmission with the waves' interference kept, computed exactly by multiplying transfer matrices and averaged as the logarithm over 40 random stacks and five wavelengths. It falls in a straight line: the transmission drops by a factor of e every 17 layers, however thick the stack, which is exponential decay — localisation. The upper curve is the same stacks with every surface's reflection and transmission added as intensities, so that no interference survives. It falls only as one over the thickness, checked to grow in exact proportion, which is the diffusion of light through a cloud: at 400 layers it still transmits 1.0 per cent where the coherent stack typically transmits 4.5·10⁻¹¹.

The walk that interference can stop

Light scattered many times walks through a cloud, and a walk always gets through eventually — a slab twice as thick lets through half as much. Keep the waves' interference instead of adding intensities, and in one dimension the same disorder does something a walk cannot: it stops the light exponentially, traps it in modes with nothing special about where they sit, and turns transmission from a number into a spread over powers of ten. Whether the same can happen to light in three dimensions has been claimed, retracted and argued for thirty years.

optics · Scattering Modes added in step: a train of pulses. The intensity of the sum of N waves of equal amplitude at equally spaced frequencies, all starting in phase, against time in units of the round trip — the inverse of the spacing — for N = 5, 10, 20. The sum is a train of pulses, one per round trip, each of peak intensity N² times one wave's and width about 1/N of the round trip. Between the pulses the waves cancel almost completely. The average intensity is N, as it must be — 5.01 for N = 5, 10.04 for N = 10, 20.19 for N = 20 — so the pulses do not create energy; they move it all into a fraction 1/N of the time, where it is N times more intense than it would be if spread evenly.

The phases that turn a glow into pulses

A laser oscillates on many frequencies at once, spaced by the time light takes to go round its cavity. Add those waves with random phases and the output is a steady glow with noise in it. Add exactly the same waves with their phases equal and the output is a train of pulses, each N² times brighter than one wave and a fraction 1/N of the round trip long. Nothing about the frequencies, the amplitudes or the average power changes — only the phases, and they decide everything a detector fast enough would see.

waves · Superposition 3 beams crossing: change the phases and the pattern only slides. The intensity where 3 plane waves of one wavelength cross in a plane, their directions spread evenly round the circle, for three different sets of relative phases, each panel spanning 2.2 wavelengths. Darkest shading is brightest, in five steps of a fifth of the peak. The three patterns are the same lattice of bright spots moved sideways: for each of the altered sets there is a shift that reproduces the first pattern to 2.0 per cent of the peak. With 3 beams in two dimensions there are 2 relative phases and 2 directions to slide in, so every change of phase is a slide.

The lattice no phase can bend

Cross a few laser beams of one colour and they paint a crystal of light in the space where they overlap. The beams' phases drift whenever a mirror trembles, and it would seem the pattern must tremble with them. Whether it does depends on a count. Three beams in a plane, or four in space, give a lattice whose shape no change of phase can alter — the phases can only slide it. One beam more and the phases decide the shape, and must be held still.

waves · Superposition 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 A strong oscillator lifts a picowatt to the quantum limit. The signal-to-noise ratio for detecting a 1 pW signal at 1.55 µm in a 1 MHz bandwidth, with a detector of 80 per cent quantum efficiency whose amplifier adds 2 pA/√Hz of noise, against the power of the local oscillator. Detected directly, the signal is 66 dB below the amplifier's noise. Heterodyned, it rises with the oscillator's power — −3.3 dB with a microwatt, 7.4 dB with a hundred — until the oscillator's own shot noise swamps the amplifier's, above about 12 µW, and then it stops rising at 8.0 dB: ηPₛ/(hνB), the number of detected signal photons per resolution time, which no receiver can beat. A stronger oscillator raises the signal and its own noise together.

The faint light a bright one makes measurable

A photodetector does not respond to a light wave's field; it responds to the square of it. Shine a strong beam onto the detector together with a weak one at a slightly different frequency, and the square contains a term that is the weak field multiplied by the strong one — oscillating at the difference frequency, carrying the weak light's phase, and as large as the strong beam cares to make it. That product lifts a picowatt out of an amplifier's noise to the limit set by counting photons, turns an optical frequency shift into a radio one, and carries with it a quantum price that decides why radio telescopes and optical telescopes are built so differently.

optics · Coherence 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 Light from a charge that crosses into glass at a steady speed. The transition radiation emitted as an ultrarelativistic charge crosses from vacuum into a dense medium, per unit solid angle, against the angle to the path in units of 1/γ, for photons at three frequencies: one tenth, three tenths and one times γωₚ, where ωₚ is the medium's plasma frequency. The scale is αħγ²/π² per unit frequency. Nothing is emitted straight ahead; the light comes out in a narrow hollow cone with its brightest ring at γθ = 0.98, 0.88, 0.68 for the three. At a tenth of γωₚ the peak is about ten times higher than at γωₚ. The charge never accelerates: it is the field's sudden change of shape at the surface, from the flattened disc it carries in vacuum to the one the medium's electrons allow, that is radiated.

The light a charge makes by changing medium

A charge moving at constant speed in a straight line radiates nothing in a vacuum, and nothing in glass either if it is slower than light there. Let it cross from one into the other and it radiates at the boundary, though it neither turns nor slows. The field it carries has one shape in each medium and has to be rebuilt as it crosses, and what is shed in the rebuilding is light. The energy shed grows in proportion to the particle's Lorentz factor, the one property of a fast particle that almost nothing else measures at high energy, and this is why detectors stack hundreds of plastic foils in a particle's path to collect one or two X-rays.

astrophysics · Radiating charge A signal that fades as the fourth power of distance. Received power, in decibels relative to free space at one metre, against distance on a logarithmic scale, for a 900 MHz transmitter 30 m up and a receiver 1.5 m up over flat, perfectly reflecting ground, with the ground's reflection taken as −1; dashed, free space alone, falling 20 dB per decade. Close in, the direct and reflected waves beat as their path difference passes through whole wavelengths. Beyond the break distance 4hₜhᵣ/λ = 540 m, the path difference 2hₜhᵣ/d is less than half a wavelength and shrinking, and the reversed reflection cancels the direct wave more and more completely: the power falls 40 dB per decade, as d⁻⁴, 10 dB below free space at ten times the break and 30 dB below at a hundred times. The ground, which was supposed to add a second path, takes the signal away; raising either antenna, which lengthens the path difference, is the remedy, and is why masts are tall.

The surface that cancels whatever skims it

Glass reflects four per cent of light falling straight on it, and water at radio frequencies two-thirds of the wave's amplitude. Tilt either towards grazing and the difference disappears: every surface, whatever it is made of, reflects a wave that only skims it completely, and turns it upside down. A wave travelling along a surface therefore meets its own reflection reversed and cancels on the surface itself. That one limit explains a dark fringe Humphrey Lloyd saw with a single mirror in 1834, why a mobile signal over flat ground fades with the fourth power of distance rather than the second, why a radar above the sea is blind at the horizon, and how a cliff in Sydney became the first radio interferometer.

optics · Refraction The share of the reflection that stays a mirror. The fraction of the reflected power that goes into the mirror direction rather than being scattered, which falls exponentially with the square of the phase roughness g = 4πσ sin ψ/λ, against the grazing angle on a logarithmic scale, for surfaces whose height varies with a standard deviation σ of 0.05, 0.5 and 5 wavelengths. A surface rough by a twentieth of a wavelength is a good mirror at every angle: 67 per cent at normal incidence. One rough by half a wavelength keeps only 7·10⁻¹⁸ at normal incidence but 74 per cent at 5°. One rough by five wavelengths reflects nothing specularly until the angle is below about 0.9°, where g = 1, and is a mirror below a few tenths of a degree. The angle at which a surface turns from matt to glossy scales as the wavelength over the roughness; no surface is too rough to be a mirror for light that grazes it closely enough.

The rough surface that is a mirror at a glance

A sheet of office paper is matt from above and shines when sighted along its surface. A choppy sea scatters a radar beam from overhead and returns it like a mirror near the horizon. An X-ray mirror polished to a third of a nanometre is still rough to X-rays except at angles below a degree. All three are one rule: two rays reflected from points at different heights differ in path by twice the height times the sine of the grazing angle, so roughness is measured in wavelengths times that sine — and every surface, however rough, is a mirror for light that skims it closely enough.

optics · Refraction Light added to its own reflection. Green light of 550 nm in an emulsion of refractive index 1.5, falling on a metal mirror at the left and adding to its reflection, which comes back inverted. Thin lines: the total electric field across the first 750 nm at three moments; heavy line: its envelope, 2|sin kz|; dashed: the envelope of the magnetic field, 2|cos kz|. The sum does not travel. Its electric field is zero at the mirror at every instant and again every 183 nm, half a wavelength in the emulsion, with maxima between; the magnetic field is largest exactly where the electric field vanishes. A light-sensitive layer anywhere in that space is exposed in sheets parallel to the mirror, and which field it responds to decides whether the first sheet lies at the mirror or a quarter-wavelength above it.

The photograph that keeps the spectrum

Light falling on a mirror adds to its own reflection, and the sum does not travel: it stands, with sheets of darkness every half-wavelength. Otto Wiener photographed those sheets in 1890 and found the film clear where it touched the mirror, which settled which of light's two fields acts on matter. A year later Gabriel Lippmann recorded the sheets through the depth of an emulsion and developed them into stacked layers that, lit with white light, send back exactly the colours that wrote them. His photographs store the spectrum itself, so they can tell apart two yellows that every colour film and every eye records as one.

waves · Superposition Three waves, and the points where the light is zero. The overlap of three equal plane waves travelling 120° apart: brightness (shading) and lines of equal phase every 60° (curves). Lines of every phase meet at isolated points where the brightness is exactly zero (dots); elsewhere only branches of a single phase cross, at the bright spots, where the phase is momentarily level. Round each such point the phase changes by one full turn, clockwise (red, charge +1) or anticlockwise (green, charge −1); in this window there are 18 of one and 18 of the other, arranged on a lattice in which every neighbour has the opposite sign. No adjustment of the phases removes them; shifting a wave only slides the lattice.

The dark lines a wave cannot avoid

Overlap three waves of light and there are places where they cancel exactly — not along fringes, as two waves cancel, but at isolated points, round which the phase of the light turns through a full circle. They cannot be removed by any small change, they can only be created or destroyed two at a time, and random light is full of them, about three to every square wavelength. A grating with a fork in it writes one into a beam on purpose, and the beam that comes out has a dark core and carries angular momentum in its twist.

optics · Diffraction Ramsey fringes: two short pulses and a long wait. The fraction of atoms found in the upper state against the detuning of the drive from the atoms' frequency, in cycles per pulse length, for two π/2 pulses of length τ separated by a free time T = 10τ (solid), computed from the exact two-level evolution. Fringes of period 1/T ride under an envelope set by the short pulses (dotted, the response to one pulse of length 2τ). The central fringe is 0.0444 cycles per τ wide at half height, close to 1/2T. A single long pulse covering the same total time, 12τ (dashed), gives a peak 0.0666 wide: 1.50 times wider, and with no way of telling a large detuning from a small one by its side fringes alone.

The wait that sets the precision

To measure an atom's frequency, the obvious method is to drive it and look for the response, and the sharpness of the answer is set by how long the drive lasts. Norman Ramsey found in 1949 that most of the drive is wasted. Two short pulses with a long wait between them do better than one long pulse, because during the wait the atom keeps its own time undisturbed, and any difference from the drive's time piles up as an angle the second pulse can read. Every primary clock in the world runs on that idea, and its precision is set by two numbers: how long the atoms can be left alone, and how many of them there are.

quantum · Measurement A pile of glass plates, added two ways. The transmission of light at normal incidence through a pile of N glass plates (index 1.5) in air, on a logarithmic axis, against N. Dashed: Stokes's 1862 result for light whose reflections add as intensities, T = (1 − R)/(1 + (2N − 1)R) with R = 0.040 per surface, which falls only as 1/N: 0.107 at 100 plates, 0.029 at 400. Solid: the same pile with the reflections added as waves and every plate and gap of random thickness, as ⟨ln T⟩ over 200 random piles (red) and for one pile (grey). It falls in a straight line, by 0.0843 per plate, an exponential decay with a localisation length of 24 plates: 10⁻³ at 100 plates and 10⁻¹⁴ at 400. Interference with random phases does not average away; it traps.

The pile of glass that stops every colour

A sheet of window glass reflects about eight per cent of the light that falls on it. A pile of a hundred sheets, George Stokes calculated in 1862, should still let through a tenth, because each extra sheet only adds a little more reflection to the total. That is right for sunlight, whose colours are too mixed to interfere. For a single colour, with every sheet and every gap a slightly different thickness, the same pile lets through a thousandth, and a pile of four hundred a hundred-millionth of a millionth. Interference with random phases does not average away into the intensity sum. In one dimension it traps every wave, at every wavelength, however weak each reflection is.

waves · Periodic media Fringes that identical nuclei add. The cross-section for two identical spinless nuclei repelling each other, divided by the classical answer — the Rutherford counts for the nucleus hit and the nucleus thrown, added — against the scattering angle in the centre-of-mass frame, for Sommerfeld parameters η = 1, 4 and 16 (η is half the closest approach in units of the wavelength over 2π). Because the detector cannot tell which nucleus it caught, the two amplitudes add before squaring, and the cross term oscillates in angle: at 90° the two paths are mirror images and the rate is exactly twice the classical one at every energy. Between 30° and 90° the curve completes about 2 oscillations for η = 4 and 7 for η = 16 — their number grows in proportion to η, which is to say in proportion to 1/ħ.

The scattering quantum mechanics returns unchanged

Rutherford worked out how alpha particles bounce off a nucleus by treating them as tiny planets on hyperbolic orbits, two years before Bohr's atom and fifteen before quantum mechanics. He had no right to be exactly correct, and he was: the full quantum calculation returns his formula without a single correction, at every energy. Coulomb's force is the one force for which that happens. Make the two nuclei identical, though, and quantum mechanics adds fringes no orbit can produce — doubling the rate at right angles for one kind of particle and halving it for the other.

quantum · Correspondence A transparent cell, seen four ways. Brightness across the image of a transparent cell 20 μm across, which delays the light passing through its middle by 0.3 radians (and its nucleus by a third as much again) and absorbs none of it, imaged with green light (0.55 μm) through an objective of numerical aperture 0.25; brightness relative to the background. In focus with a plain objective (dashed) the cell is all but invisible: the brightness varies between 0.998 and 1.001. Defocused by 8 μm it shows, 0.96 to 1.06, but only at its edges and nucleus, as bright and dark fringes that do not follow its thickness. With a quarter-wave plate on the undiffracted light the cell appears in focus, its brightness following its thickness: with the plate one way it is brighter where it is thicker, up to 2.02 times the background; with the plate the other way, darker, down to 0.38.

The specimen that is invisible until its light is delayed

A living cell in water is nearly transparent: it delays the light passing through it by a fraction of a wavelength and absorbs almost none, and a perfect microscope shows it as nothing. The delay is there in the light, and the eye cannot see a delay. Frits Zernike's remedy was to delay a different part of the light — the part the cell did not scatter — by a further quarter of a cycle, at the one place in a microscope where it is separated from the rest. The invisible phase becomes brightness, the cell appears, and the price is a bright halo round everything large.

optics · Imaging A bright core in a set of rings. The intensity across the beam 29 cm beyond an axicon — a glass cone 10 mm across that tilts every ray by 0.5° towards the axis — lit by a 633 nm laser, computed from the Fresnel integral and scaled to its peak (solid). It is the pattern of a cone of plane waves meeting on the axis: a bright core 56 μm across at its first dark ring, at 2.405/kα from the axis, ringed by rings of slowly falling brightness. Dashed: a Gaussian laser beam focused to a spot of the same size, with no rings. The rings are not a flaw of the axicon; they are where the beam keeps the power that the core is short of.

The beam that does not spread and the price it pays

Pass a laser beam through a shallow cone of glass and it emerges as a thin bright line of light, a core a few tens of micrometres wide that keeps its width for half a metre, where a beam focused to the same width by a lens would have doubled in a few millimetres. It seems to defeat diffraction. It does not. The core is not made of the light that was in it a moment before; every point along it is fed by a fresh ring of the aperture, and the core holds about one per cent of the beam's power. The rest is in the rings, and the rings are what pays for the line.

optics · Diffraction A lens that cannot focus a wave longer than itself. How much a point mass brightens a wave passing behind it — the factor on the wave's power, on a logarithmic scale — against the dimensionless frequency w = 8πGMf/c³, for a source exactly behind the lens and 0.1, 0.3, 1.0 Einstein radii off it, from the exact diffraction integral. Geometric optics puts the magnification at 10.0, 3.4 and 1.34 for the three offsets and at infinity on the axis, whatever the frequency. The wave disagrees: at w = 0.01 every curve is within 1.6 per cent of 1 — a wave a hundred times longer than the lens passes as if it were not there — and on the axis the power factor is πw/(1 − e^(−πw)): 3.28 at w = 1, 31.4 at w = 10, finite at every frequency. The off-axis curves rise to their geometric values only above w ≈ 1 and then oscillate about them, as the two images begin to interfere.

The lens a long wave walks round

Geometric optics says that a star passing exactly in front of a source magnifies it infinitely. No real wave agrees, because a wave cannot be focused to a point, and for the gravitational waves the detectors hear the disagreement is the whole story. A star, or a black hole of a few tens of suns, is smaller than one of those waves in the only measure that matters — the delay between paths round it — and the wave flows past it almost untouched. What decides whether a mass lenses a wave is not the mass's size, which is millions of kilometres, but its Schwarzschild radius set against the wavelength.

astrophysics · Light deflection The same fringes from cooler and hotter molecules. The rate at which molecules arrive across a few periods of the interference pattern behind the last grating, relative to its mean, for molecules at 1000, 2500, 3000 K, each with the visibility the emission model gives: 46 per cent, 14 per cent, 4 per cent. The cooler molecules draw fringes nearly as strong as cold ones; at 3000 K the pattern is a faint ripple on a flat background. The average number arriving is the same in every case: the heated molecules are not lost, and none of them has been deflected out of the beam. They arrive in the same places on average and no longer interfere.

The molecule that interferes until it glows

A molecule of seventy carbon atoms, sent through a matter-wave interferometer, travels both arms at once and lands in fringes, as an electron or a neutron does. Heat it, and the fringes fade. Nothing has touched it from outside: a hot molecule glows, and each photon it gives off is a messenger that may carry away the information of which arm it was in. Whether it does depends on one comparison — the photon's wavelength against the separation of the arms. Long photons leave the fringes alone; short ones erase them. The loss of interference that is usually described as the onset of the classical world is here a measurement made by the molecule's own light, and its threshold can be computed.

quantum · Matter waves A chaotic rotor heats without limit, and its quantum version stops. The mean kinetic energy of a rotor kicked once a period by a potential of strength K = 5, against the number of kicks, starting at rest. Classically (an ensemble of 4,000 rotors with random angles) the motion is chaotic and the momentum random-walks, so the energy grows in a straight line, 6.3 per kick, near the K²/4 = 6.25 a random walk of kicks K sin θ gives. Quantum mechanically, with an effective Planck constant ƛ = 1 and 0.5, the energy follows the classical line for a while and then stops, fluctuating about 58 and 123: the smaller ƛ lets it follow the classical growth for longer and climb further before it stops. Nothing in the quantum rotor's equations damps it; the growth stops by interference.

The heating a quantum rotor refuses

Kick a freely turning rotor hard, at regular intervals, and classical mechanics says its motion becomes chaotic and its energy grows without limit, the momentum random-walking upward with every kick. Do the same to a quantum rotor and it follows the classical heating for a while, then stops — not because anything damps it, but because the waves of its momentum states interfere so as to cancel further spreading. The correspondence principle, which promises classical behaviour when quantum numbers are large, is broken here in the most direct way: by a quantum system that refuses to get hot.

quantum · Correspondence

Named alongside it

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

SuperpositionDiffractionCoherencePhaseFourier transformPath differenceDecoherenceSpectrumBandwidthImpedanceWavelengthBeats

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