Illustrated Physics
physics, drawn
Concept

Wavefront — where it appears

A surface of constant phase, whose motion and bending carry everything a ray picture summarises as a direction. Rays are the normals to it, so refraction, focusing and diffraction are all statements about how a front changes shape as it advances.

Named by 14 essays across 3 fields — each of them below, with the objects they name alongside it.

Why a straight front stays straight. A plane wavefront, with 9 points on it treated as sources and a wavelet of radius vt drawn about each. The envelope of those circles — the curve touching all of them — is a second straight line, parallel to the first and displaced by exactly vt. That is the whole of straight-line propagation: nothing else has to be assumed, and in particular nothing has to be said about rays, which are afterwards defined as the normals to these fronts. The construction is drawn with the wavelets left in, because they are the part that does the work.

Every front is a source

Treat each point of a wavefront as though it were a little source of its own, and take the envelope of what they produce. That one rule gives straight-line propagation, reflection, Snell's law and diffraction — and its most famous failure is what told everyone what was missing from it.

waves · Huygens Rays that turn round without meeting anything. Rays leaving an eye 1.5 m above a road, at 0.18°, 0.28°, 0.36°, 0.42°, 0.52° below the horizontal, in air whose refractive index is reduced by 3.0e-5 at the hot surface and recovers over 5 cm. Each is traced by integrating the ray equation, with the conserved quantity n cos θ fixed by where and how the ray set out. The shallow ones come back up without touching anything — the lowest gets to 0.6 cm above the surface and turns — and a ray that returns to eye level from below is seen as sky lying on the road. The steep ones run out of gradient first and hit it. The dividing angle is 0.444°, which is what the whole effect is made of: an unremarkable temperature difference and an angle a fiftieth the width of the Moon. Heights are exaggerated 128× against distances; at true scale every ray here would be indistinguishable from the axis. The conserved n cos θ holds to 4.3e-14 over every trace, which is what says the turns are the physics and not the integrator.

The ray that bends without a surface

Snell's law is about a boundary, and light bends in air where there is no boundary anywhere. Let the index vary continuously and the law of angles becomes a differential equation — one that carries a conserved quantity, forbids the ray from reaching certain heights, and turns a hot road into a mirror a hundred metres long.

optics · Fermat 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 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 A source moving faster than its own waves. Circles showing where each crest has reached, centred on where the source was when it emitted them. Ahead of the source the crests are closer together and the frequency heard is higher; behind, they are spread out and it is lower.

The cone the source leaves behind

Take the Doppler construction past the speed of the wave and the wavefronts acquire an envelope. Its half-angle obeys sin θ = 1/M, an expression with no pressure, no density and no shape of the object in it — so a photograph of the cone is a speedometer. And the bang is not an event at the moment of crossing — it is a signature dragged along the ground for the whole of the flight.

waves · Doppler One sharp kick, heard 6 pulse-lengths away. The signal arriving at a fixed distance from a point source that emits a single short pulse, in one, two and three dimensions, each normalised to its own peak. In three dimensions the arriving signal is the emitted pulse, unchanged in shape: it arrives, and then there is silence. In two dimensions the same kick arrives at the same moment and then keeps arriving — a tail falling as one over the time, which is still at a tenth of its peak 12.1 pulse-lengths after the front has passed. In one dimension it never comes back down at all; the medium is left displaced. The wave equation is the same equation in all three, and the source is the same source; what differs is only how many dimensions the disturbance has to spread into. Sharp arrival is the exception rather than the rule — it happens in three dimensions, and in five, and in seven, and in no even number of them — and every argument that treats a wavefront as the whole of the signal is an argument that has quietly used the fact that we live in three.

The arrival that keeps arriving

A clap heard across a field arrives and stops. The same clap in two dimensions arrives at the same instant and then goes on arriving for ever, fading as one over the time — and the difference is not absorption, or echo, or scattering. It is the number of dimensions, and sharp arrival happens in three of them and in no even number at all.

waves · Wave motion A caustic, by rays and by waves. The brightness across a fold caustic, computed two ways. Geometric optics gives the rising curve: on the illuminated side two rays arrive at every point and the intensity goes as the inverse square root of the distance from the caustic, so it becomes infinite exactly at it; on the other side no ray arrives at all and the intensity is zero. The wave answer is the squared Airy function, and it disagrees in three ways that are all observable. It is finite, peaking at 1.0188 in the scaled variable rather than at the caustic itself, so the brightest line is displaced onto the bright side. It oscillates, with maxima at -1.02, -3.25, -4.82, -6.16 — those are the supernumerary fringes, and they are not interference between two separate objects but between the two rays the caustic joins. And it leaks: on the dark side, where geometry forbids any light, the Airy function decays exponentially rather than stopping, which is the same mathematics as tunnelling and is why the edge of a shadow is soft before diffraction from any aperture is considered. The two curves agree far from the caustic, which is where the ray picture is a good approximation and where they have been matched here.

The fringes below the rainbow

Geometric optics puts the whole rainbow at one angle and predicts an infinite brightness there. What is seen instead is a peak displaced inside that angle, followed by a train of pink and green arcs — and their spacing is a measurement of the raindrops, because a caustic's structure is set by the wavelength to the two-thirds power over the drop radius to the two-thirds.

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

The focus that is a slab, not a plane

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

optics · Imaging The 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 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 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 The action at one instant, drawn as a map. Trajectories leaving one point at the same moment, at launch speeds 0.6, 1 and sixteen directions each, in a uniform field pulling downward, drawn up to time 1. Behind them, dashed, are the level curves of the action at that instant, regarded as a function of where a trajectory ends. They are circles, and their common centre is neither the launch point nor anywhere the particles have reached: it is 0.500 above the launch point, while the whole swarm has fallen by the same 0.500. Every arriving velocity points straight out from that centre, so every trajectory crosses the level curves at right angles, and the arriving momentum equals the gradient of the action to one part in ten thousand. The action integrated along each path agrees with the map's value at its end.

The action that knows where every path ends

The action is usually a number attached to one path. Treat it instead as a function of where the true path ends, and a single function of position and time holds every trajectory at once — its slope is the momentum, its rate of change is the energy, and its level curves are wavefronts, drawn about a point that sits above the source while everything falls.

mechanics · Least action

Named alongside it

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

DiffractionHuygens principleInterferenceFourier transformObliquity factorPhaseCausticCoherenceEnvelopeImagingSuperpositionAperture

All concepts