Illustrated Physics
physics, drawn
Concept

Diffraction — where it appears

The spreading of a wave past an obstacle or through an aperture, through an angle set by the ratio of the wavelength to the opening. It is not a small correction to a ray picture but what a ray picture is an approximation to, and it becomes the whole story once the opening approaches a wavelength.

Named by 20 essays across 4 fields — each of them below, with the objects they name alongside it.

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

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

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

optics · Diffraction 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 A packet, and the wavenumbers it is made of. Above: a wave packet built by adding a continuum of plane waves centred on wavenumber 12 with a spread of 1.6. Below: the weight given to each wavenumber. The packet's width, measured as the standard deviation of its probability, is 0.442; the spread of wavenumbers is 1.131; their product is 0.500, which is a half and cannot be less. Narrowing one bracket widens the other by exactly as much. Nothing quantum has been used to draw either panel.

Sharpness has to be paid for

A wave with one exact wavelength has no beginning and no end. Making it short requires adding wavelengths, and the two widths trade against each other exactly — which is a fact about waves, with Planck's constant added only to convert the units.

quantum · Uncertainty 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 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 6 modes of a drum, and their frequency ratios. Nodal-line diagrams for 6 modes of a circular membrane, each labelled with its frequency as a multiple of the lowest mode's. A mode (m, n) has m nodal diameters and n − 1 nodal circles, and the circles are drawn at the radii where the computed radial function J_m(j(m,n)·r/R) crosses zero — not at guessed fractions of the radius. The two tints are the two directions the head is moving in at that instant, and the lines between them are the parts of it that never move. The ratios are 1.000, 1.593, 2.136, 2.295, 2.653, 2.917: each one is a quotient of two zeros of Bessel functions, computed here from the power series and checked against their published values to 4.4e-7. Not one is a whole number, which is why a drum has no harmonic series and no pitch in the sense a string has one — and why these same ratios belong to every circular membrane ever stretched, whatever it is made of and however tightly it is pulled.

The drum that has no harmonics

A string's allowed frequencies are 1, 2, 3, 4 times its lowest, because counting half-wavelengths is arithmetic. Clamp a membrane round a circle and the same reasoning returns 1.000, 1.593, 2.136, 2.295 instead — zeros of Bessel functions, not integers. And those numbers belong to the shape of the boundary alone, which raises a question nobody could answer until 1992.

waves · Standing waves 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 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 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 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 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 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 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 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 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 No beam is narrow enough and wide enough at once. Three lengths at the far end of a Stern-Gerlach magnet 10 centimetres long with a gradient of 1000 tesla a metre, against the width of the beam entering it, for an electron at 100 electronvolts. The spin splitting is a horizontal line at 2.9e-6 metres: it does not depend on the beam's width. The Lorentz blurring rises in proportion to the width, because the field a particle sees depends on where in the beam it is, and the divergence-free condition ties a gradient in one component to a gradient in another. It overtakes the splitting at 1.0e-9 metres. Narrower than that and diffraction has already spread the beam by 3.9e-3 metres, which is larger still. There is no width at which the splitting is the largest of the three, and the magnet's length and gradient cancel out of the comparison entirely — so no magnet helps.

The experiment that defines spin and cannot be done on it

A Stern–Gerlach magnet separates magnetic moments and is how spin was discovered. It cannot be made to work on a free electron, and the obstruction is not the apparatus: the field gradient that splits the beam also deflects the charge by an amount that varies across it, and the ratio of the splitting to that blurring comes out as the de Broglie wavelength over the beam width — with the magnet's length and gradient cancelling exactly.

quantum · Spin

Named alongside it

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

InterferenceApertureWavefrontSuperpositionFourier transformHuygens principleWavelengthCoherencePhaseFresnel zonesObliquity factorRayleigh's criterion

All concepts