Field

Optics

Light, and the small number of rules it obeys.

$Refraction from n = 1 into n = 1.5. A ray crossing a boundary between media of refractive index 1 and 1.5, bending by the amount Snell's law requires.$

The bend at the boundary, and what it is really about

Light changes direction when it changes speed. Snell's law is the geometry of that statement, and it can be derived without knowing anything about light at all.

A converging lens making a real image. An object 2.44 focal lengths from a thin converging lens. The image sits where the construction rays cross, at 1.69 focal lengths, magnified -0.69×.

What a lens is doing, and why three rays are enough

A lens bends every ray that reaches it. The construction uses three, because three are all that can be drawn without calculation — and any three that meet prove all the rest do.

$The critical angle for n = 1.5 into n = 1. Refracted angle against incident angle. It rises faster than the incident angle and reaches 90° at 41.8°, beyond which no refracted ray exists at all.$

The angle past which light cannot leave

Snell's law asks for the sine of an angle greater than one, and no such angle exists. What happens instead is a perfect mirror made out of nothing but a change of speed.

$Rays through a raindrop. Parallel rays entering a spherical drop at different heights, refracting in, reflecting once from the back, and refracting out. The outgoing rays crowd together near one particular direction, and that crowding is the bow.$

The angle the rainbow has to be, and why nobody chose it

A rainbow is at forty-two degrees because a function has a minimum there. Nothing about water, light or weather picks the number — it falls out of running Snell's law three times through a sphere.

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

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.

A sphere does not have a focus. Parallel rays reflected off a spherical mirror. Rays striking further from the axis cross it nearer the mirror, so there is no single point where all of them meet — the blur is spherical aberration, and a perfect sphere has it inherently.

The mirror that cannot focus, and the shape that can

A perfect sphere does not bring parallel light to a point. The blur is not a manufacturing defect — it is what the shape does, and the shape is used anyway, for a reason worth knowing.

$Malus's law. The fraction of polarised light passing a filter, against the angle between the light's own direction of shaking and the filter's axis. It is the cosine squared: half at 45 degrees, nothing at 90.$

The direction of the shaking, and the filter that only asks about it

A wave that travels one way can still shake in any direction across that way. Light does, most of it shakes in all of them at once, and a sheet of plastic can ask which.

How much each colour is scattered. Scattering strength against wavelength, as the inverse fourth power, normalised to one at 550 nanometres. Light at 450 nanometres is scattered 4.35 times as strongly as light at 650 nanometres — which is the whole reason the sky is the colour it is.

Why the sky is blue and the sunset is not, from one exponent

Scattering goes as the inverse fourth power of wavelength, and that single number produces a blue sky and a red sun without any second explanation. The two facts look opposite and are the same arithmetic.

The path that does not change

Reflection, refraction and the angle of the rainbow are not three laws. They are one condition — that the optical path length is stationary — and the word stationary rather than shortest is the whole of what makes an elliptical mirror and a rainbow the same statement.

The dip that decides it. Two equally bright points seen through a circular aperture, their Airy patterns added, at 0.7, 1, 1.6 times the Rayleigh separation. Below each is the depth of the dip between the peaks, measured off the drawn sum: 0.0%, 26.5%, 91.6%. At exactly the Rayleigh separation the dip is 26.5% — the criterion is a convention about how much of a dip a detector can see, not a threshold anything crosses. Below it the two peaks merge into one and the pair is gone; above it the answer was never in doubt.

How far apart two things have to be

An instrument's ability to tell two things apart is not set by the quality of its glass. It is set by the width of the hole light comes through, by a factor that is the first zero of a Bessel function — and the threshold everyone quotes is a convention laid over a computed dip of 26.5 per cent.

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.

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.

The angle at which reflection picks a side

At one angle of incidence, a water surface reflects no light at all of one polarisation. The Fresnel algebra says so, and says nothing about why. The reason is that the reflected ray would have to leave along the axis of the charges radiating it — and a shaking charge sends nothing along the direction it shakes in.

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

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.

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.

Cancelling a derivative, and what is left over. How far the focus of a 500 mm lens moves with wavelength, for a single crown element and for a cemented pair of crown and flint whose powers satisfy the achromatic condition. The singlet's focus runs over 18.1 mm across the visible — a fifth of a per cent of its focal length, and utterly ruinous at any useful aperture. The doublet's runs over 2.268 mm, some 8× less, and — this is the whole content of the figure — it is not flat. The condition sets the rate of change of power with wavelength to zero, so the curve is stationary rather than constant: it returns to the corrected focus at exactly 2 wavelengths — 486 nm and 656 nm, which are the two Fraunhofer lines the condition was written at — and departs from it everywhere else, most at 400 nm. That residual is the secondary spectrum, it has the same sign at both ends of the visible, and no pair of ordinary glasses removes it, because two conditions cannot be met with one free ratio.

Two glasses that cancel a derivative

A single lens focuses blue light closer than red, and the difference ruins the image. Cementing a second lens of another glass behind it fixes the fault at two wavelengths and at no others, because the condition sets a slope to zero rather than a value.

Where the sky's blue goes. How many times more strongly a sphere scatters 450 nm light than 650 nm light, against its radius, with the radius on a logarithmic axis running from a couple of nanometres to twenty micrometres. On the left the ratio sits at 4.35, which is the fourth power of the wavelength ratio and the whole reason the daytime sky is blue. It does not stay there. By a radius of 245 nm the preference has halved, and by a micrometre it has essentially gone: a particle comparable with the wavelength scatters every visible colour within a few per cent of equally, which is why a cloud is white, why fog is white, why milk is white and why the exhaust of a cold diesel is white while the smoke of a cigarette — whose particles are ten times smaller — is blue. Nothing about the material changed between one end of this axis and the other; only the size did.

When the particle is the size of the wave

The sky is blue because small things scatter short wavelengths far more strongly. A cloud is made of the same water and scatters every colour alike. Nothing about the material changed — only the size, and one dimensionless number crossing one.

$The index that depends on which way the light is going. The two refractive indices of 3 uniaxial crystals, against the angle between the wave normal and the crystal's optic axis. The flat lines are the ordinary index, which is the same in every direction because the ordinary wave's field is always perpendicular to the axis. The curves are the extraordinary index, which runs from the ordinary value along the axis — where the two waves are identical and the crystal behaves like glass — to its extreme value at right angles to it. calcite (CaCO₃) has n_o = 1.6584 and n_e = 1.4864, so n_e − n_o = -0.1720; quartz (SiO₂) has n_o = 1.5443 and n_e = 1.5534, so n_e − n_o = 0.0091; lithium niobate has n_o = 2.3005 and n_e = 2.2075, so n_e − n_o = -0.0930. The sign of that difference is what makes a crystal positive or negative, and it decides which of the two images in a double-refracting crystal is the one that moves.$

The crystal that answers twice

Lay a piece of calcite on a printed page and the print appears twice. One image sits still when the crystal is turned and the other goes round it. Nothing has been done to the light except pass it through a material whose response to a field is not a number.

The reflected beam does not leave from where it arrived. The lateral displacement of a totally reflected beam along the surface, for the two polarisations, at 550 nm from n = 1.5 into n = 1. Geometrical optics puts the outgoing ray at the point where the incoming one struck. It is not there: it is displaced forward along the surface by a distance comparable with a wavelength, which was measured by Goos and Hänchen in 1947 by reflecting a beam many times and looking at the accumulated offset. The displacement is different for the two polarisations — at 42°, 1361 nm for s and 2991 nm for p, at 45°, 350 nm for s and 560 nm for p, at 50°, 237 nm for s and 261 nm for p, at 60°, 183 nm for s and 127 nm for p, at 75°, 161 nm for s and 79 nm for p — which is why an unpolarised beam comes back slightly split. A displacement is only possible if the light spent time on the far side of a boundary it never crossed, and it is the most direct evidence there is that the evanescent field is a real field rather than a bookkeeping term.

The reflection that happens where the glass is not

Total internal reflection sends back every photon, which is why it is called total. It does not send them back from where they arrived — the beam re-emerges displaced along the surface, by a fraction of a wavelength, and a displacement is only possible if the light spent time on the far side of a boundary it never crossed.

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.

The most any concentrator is allowed. The greatest concentration a receiver can be given, against the half-angle it accepts, both logarithmically, for a receiver in a medium of index 1. The upper curve is a three-dimensional concentrator, n²/sin²θ; the lower is a trough, which concentrates in one direction only, n/sin θ. The Sun's angular radius is 0.267°, so sunlight can be concentrated by at most 46165 times in a dish and 215 times in a trough — marked. Nothing about glass, mirrors, wavelength or aperture appears in either expression. The bound is thermodynamic: a receiver that accepts light out to θ also radiates out to θ, and the concentration at which what it emits balances what it absorbs is exactly where these curves are.

The brightness no lens can increase

A lens can make an image smaller and therefore hotter, and there is a temperature at which it stops — the temperature of the source. Every arrangement of glass and mirrors ever built obeys a bound that contains no wavelength, no aperture and no material — only the angle the receiver is allowed to accept — and the bound comes from thermodynamics rather than from optics.

What the return journey does to each of them. The rotation of the plane after a beam has gone through a rotator, been reflected, and come back, against the length of the rotator — divided by the one-way rotation, so the two answers are 0 and 2 and nothing else can happen. A naturally active medium is handed with respect to the beam: reverse the beam and the sense of the rotation reverses with it, and the second pass undoes the first exactly, at every length and every wavelength. A Faraday rotator is handed with respect to the field, which does not care which way the light is going, so the second pass adds to the first and the round trip is twice the single one: 1 mm of it gives 21.7° out and 43.4° back; 2 mm of it gives 43.4° out and 86.8° back; 4 mm of it gives 86.8° out and 173.6° back; 8 mm of it gives 173.6° out and 347.2° back. That is a violation of reciprocity, and it is only available because a magnetic field is odd under time reversal. Everything a passive optical component can do — a lens, a mirror, a waveplate, a piece of quartz — looks the same run backwards, and none of them can be made into a one-way street. This can.

The rotation a return trip doubles

Quartz turns the plane of polarisation and so does glass in a magnetic field. The two look identical on the way through and are opposites on the way back — the crystal undoes its own rotation exactly, and the magnet adds to it. That difference is the whole of why a one-way street for light can be built at all, and why nothing passive will ever be one.

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.

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.

N when the phases are random, N² when they are not. Scattered intensity against the number of scatterers, both logarithmic, for two ways of adding the same amplitudes. The lower curve averages 400 draws of N unit amplitudes with independent random phases and grows as N^1.000; the upper one adds them in phase and grows as N². At 3000 scatterers the two differ by a factor of 2921. Nothing about the scatterers is different between the two — same number, same strength, same wavelength. Only the arrangement is, and it is worth three decades here.

Why a litre of water is not blue for the reason the sky is

The same molecules that make the sky blue also make the refractive index of air, and the two numbers agree because the sideways sum has random phases and the forward one does not. Condense those molecules into a liquid and the sideways sum collapses by a factor of sixteen — and what is left is thirty-four times smaller than the absorption that actually colours the water.

Abbe's ratio, measured on the traced rays. The quantity h divided by the sine of the angle at which the ray converges on the focus, in units of the paraxial focal length, against how far up the aperture the ray entered. Abbe's sine condition says that a system already free of spherical aberration images a small region round the axis faithfully only if this ratio is the same for every ray. A horizontal line means the condition is met. The parabola departs by 12.96 per cent across the aperture; The sphere departs by 7.18 per cent across the aperture. The paraboloid is the interesting case, because it is exactly stigmatic on axis — every ray from infinity crosses at one point, which is the definition of the shape — and it still fails this test. Perfection at one point buys nothing at the next one along. What the departure predicts is coma, a blur that grows linearly with the distance off axis and quadratically with the aperture, and the offaxis figure measures exactly that blur on the same surfaces. The condition is not a design rule invented for telescopes: it follows from requiring that the same optical path length join object and image for every route, and any instrument that images a field rather than a point has to meet it.

The condition a lens must meet

A paraboloid brings every parallel ray to exactly one point. Move the source a fifth of a degree off axis and the image is a fan rather than a point, and the reason is a condition Abbe wrote down that has nothing to do with the axis — perfection at one point buys nothing at the next one along.

The fringe that measures a star

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

The wavelength at which a fibre stops smearing a pulse. Group-delay dispersion against wavelength for fused silica, computed by differencing Malitson's Sellmeier fit twice rather than read off a table, in picoseconds of spread per nanometre of source width per kilometre of fibre. The material curve crosses zero at 1273 nm — a property of the glass, fixed by where the ultraviolet and infrared absorptions balance, and not adjustable. Below it a fibre is normally dispersive and above it anomalously so, and at 1550 nm, where silica is most transparent, it is 21.9. The second curve adds a waveguide term, which is negative because a mode confined by a core spreads into the cladding differently at different wavelengths, and which depends on the fibre's geometry rather than on the glass; it is drawn here as a constant offset chosen to put the total zero at 1310 nm, which is what standard single-mode fibre is built to do. The consequence in a system is the thing worth carrying away: a source one nanometre wide sent 100 km through fibre at 1550 nm arrives 1833 picoseconds broader than it left, against 0.0 at the zero. That is the whole reason a network runs at one wavelength rather than another, and the reason the two quantities an engineer wants — lowest loss and zero dispersion — sit at different wavelengths and have to be reconciled by design rather than chosen.

The wavelength a fibre does not smear

Fused silica's index has a second derivative that passes through zero at 1,273 nanometres, and that is not a design choice — it is where the ultraviolet and infrared absorptions balance. A pulse sent at that wavelength arrives the shape it left. A pulse at the wavelength of least loss arrives 1,800 picoseconds wider after a hundred kilometres.

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

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.

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.

The deviation that has a bottom. The angle by which a 60° and a 90° ice prism bends a ray, against the angle at which the ray arrives, for an index of 1.31. Each curve is traced ray by ray through both faces and stops where it stops: outside the plotted range the ray meets the second face beyond the critical angle and never leaves. Both curves have a minimum, and the minimum is the point of the figure twice over. Its value — 21.84° for the 60° prism and 45.73° for the 90° prism — is where the sky puts a halo. And its flatness is why there is a halo at all: near a minimum the deviation changes only in second order, so a wide band of orientations all deliver light to nearly the same angle, and a cloud of randomly tumbling crystals piles up a bright ring there while sending the rest of the light nowhere in particular. The passage at the minimum comes out symmetric — in at 40.92°, out at 40.92° — which was found by searching the traced curve rather than assumed. The window of incidence that gets through at all is 76.5° wide for the 60° prism against 32.2° for the 90° one, a factor of 2.4, and that is why one of the two halos is common and the other is rare.

The ring at twenty-two degrees

A halo round the sun is a caustic in orientation rather than in space. Most of the ice crystals in a cirrus cloud send light nowhere in particular; the ones near minimum deviation all send it to nearly the same angle, because a minimum is flat — and the angle they pick has a red inner edge, which is the reverse of a rainbow.

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.

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.

Past the critical angle, all that is left of a reflection is its phase. The phase each polarisation acquires on total internal reflection at an index ratio of 1.518, against angle of incidence, together with the difference between them. Below the critical angle of 41.19° there is a transmitted beam and the reflection coefficients are real; above it the transmitted wavenumber is imaginary, the coefficients have modulus exactly one — every photon comes back — and the only thing that distinguishes one angle from another is the phase. The two polarisations acquire different phases, and their difference peaks at 46.533° at an incidence of 51.05°, which the closed form puts at the same place. That difference is a retardation: a wave plate made out of an angle, with no birefringent material anywhere in it, and — because the expression contains only the index ratio — one that barely changes with colour.

The retarder with no crystal in it

A wave plate turns linear polarisation into circular by making one component travel a little further than the other, which requires a birefringent crystal cut to a thickness and works properly at one wavelength. Total internal reflection does the same job with a phase that comes from the geometry instead — and because a refractive index barely changes across the visible where a wavelength changes by a factor of two, the same block of ordinary glass is a quarter-wave plate for every colour at once.

$Where the refracted ray comes from, drawn with a compass. Wavevectors in units of the vacuum wavenumber, for light arriving at 30° from a medium of index 1.5 at a medium of index 1. Every direction available in the first medium lies on the circle of radius 1.5 and every direction available in the second on the circle of radius 1; the horizontal axis lies in the interface. The boundary cannot change the component along itself, because the two sides have to agree on the phase at every point of the interface, so the refracted wave is fixed by the vertical line at 0.7500 — and where that line cuts the smaller circle is the refracted direction, 48.59° from the normal. Nothing about least time or about wavefronts enters, and the ratio of sines is what the construction reads when the two radii are written as indices. The normal component is not conserved and is not meant to be: it goes from 1.2990 to 0.6614, which is the whole of the difference between the two rays.$

The law that only asks about one component

A boundary between two media cannot change the part of a wave that runs along it, because both sides have to agree on the phase at every point of the surface at every instant. That single restriction produces the refracted ray, the critical angle, the evanescent field and every order of a diffraction grating, out of one drawing made with a compass.

The same source measured by amplitude and by intensity. The degree of coherence of a 47 milliarcsecond disc at 550 nm, and its square, against the separation of two apertures. A Michelson interferometer measures the upper curve, because fringe contrast is |γ|. Correlating the intensities at the two apertures instead measures the lower one, because the excess correlation of two thermal beams is |γ|² — the same information about the source, since one curve determines the other, and reaching zero at the same baseline of 2.94 m. What is lost is the phase of γ, which the intensity correlation never sees; what is bought is that a path error of many wavelengths does not matter, because the quantity being correlated is a slow fluctuation of brightness rather than a wave. The half-coherence baselines differ — 1.71 m against 1.25 m — which is the practical statement that the squared curve is the steeper one to measure against.

The correlation that survives what the phase does not

Two telescopes can measure a star's diameter by interfering the light, which requires holding two paths equal to a fraction of a wavelength through an atmosphere that will not hold still. Or they can throw the phase away entirely and correlate the brightness fluctuations, which needs the paths equal to a few metres and works.

$A cone of 14.3°, from two indices and nothing else. On the left, the acceptance cone of a step-index fibre with a core index of 1.4677 and a cladding of 1.4624. A ray entering steeper than the cone reaches the wall inside the critical angle and is refracted out at the first bounce; one inside it is trapped. The sine of the half-angle is √(n₁² − n₂²) = 0.125, which is 7.2° in air. On the right, that number against the fractional index difference between core and cladding. Nothing about the core's diameter appears: a fibre a hundred times thicker accepts exactly the same cone, and takes a hundred times the area's worth of light through it.$

The cone a fibre will accept

A fibre takes light from a cone whose half-angle depends on two refractive indices and nothing else — not on how thick it is, not on how long, not on what is shining into it. That single number, squared and multiplied by the core's area, is all the light it will ever carry.

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.

One measurement that separates unpolarised from polarised. What a rotating linear polariser passes, against its angle, for beams of the same total intensity and degrees of polarisation 0, 0.35, 0.7, 1. Every curve has the same average — a polariser passes half of any beam over a whole turn, whatever its state — and they differ only in how deeply they modulate. The depth of the modulation is the degree of polarisation, exactly: a fully polarised beam goes to zero at one angle and a beam with no preferred direction gives a flat line at a half. That is the whole measurement, and it is why "unpolarised" is a statement about a modulation depth rather than about what a wave is doing.

The light with no direction of shaking

Unpolarised light is not a state of a wave; it is the absence of one, and no description of a single wave can represent it. What can is a set of four numbers, all of them powers a detector reads — and they describe every beam there is, including the ones that are neither polarised nor not.

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.

The surface that focuses 1 into 1.52 with no error at all. A Cartesian oval: the locus of points for which 1 times the distance from the object plus 1.52 times the distance to the image is a constant. Every ray drawn takes exactly the same optical path, so every one arrives at the image point — not nearly, and not for small angles, but exactly, for rays at any angle the surface reaches. There is no spherical aberration because there is no approximation: this is what Fermat's principle asks for, solved rather than expanded. The surface is not a sphere, not a conic in general, and not anything a grinding machine makes easily, which is most of why lenses are spherical and aberrated instead.

The surface that images one point exactly

Fermat's principle says every ray from an object to its image must take the same time. Written as an equation, that is a curve Descartes found — a surface with no aberration at all, at any angle. It exists, it is easy to compute, and it images exactly one pair of points and nothing else.

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.

The bowl a lens actually focuses onto. Where a lens of 50 mm focal length brings each part of a flat scene to a focus, against distance from the centre of the field, out to 21.6 mm — the corner of a 35 mm frame. The surface of best focus is a sphere of radius 76 mm curving towards the lens, and the corner of the frame sits 3.13 mm in front of the plane the centre is focused on. The shaded band is the depth of focus at f/8, which is 0.070 mm — 45 times smaller than the sag it has to cover. The curvature is not an error in the lens. It is what Σ1/nf comes to for this stack, and it depends on the powers and the glasses and on nothing else: bending the surfaces, moving the stop or stopping down changes every other aberration and leaves this one exactly where it was.

The flat scene that comes back curved

A lens does not image a plane onto a plane. It images it onto a bowl, and the curvature of that bowl is fixed by the powers and the glasses alone — not by the shapes of the surfaces, not by where the stop is, and not by stopping down. Everything a designer usually plays with leaves it exactly where it was.

The pattern the whole sky is written in. The sky as a disc — zenith at the centre, horizon at the rim, equal angles at equal distances — with the sun 30° above the horizon. Each short line is the direction the electric field vibrates in at that point, and its length and darkness are how polarised the light there is. The directions are perpendicular to the plane containing the sun, the observer and the point, which puts them tangent to circles centred on the sun. The heavy arc is the locus 90° from the sun, where the polarisation is strongest — 74 per cent here — and it is a great circle rather than a patch: a band across the sky, not a region near the horizon. This is what a polarising filter on a camera acts on, and it is why turning one darkens a band of sky and leaves the rest almost untouched, and why the effect is strongest when the sun is off to one side and absent when it is behind the photographer.

The pattern the sky is written in

Scattered sunlight is polarised, so the whole sky carries a direction of vibration at every point — arranged in circles about the sun, strongest on the great circle ninety degrees away from it, and vanishing at points that were found by looking before anyone could explain them. Bees navigate by it and a camera filter reads one band of it.

$A wave with two directions in it. Light at 60° entering silver, 550 nm, whose index is 0.055 + 3.32i. Phase matching along the boundary fixes the transmitted wave's tangential wavenumber and leaves the normal one to the medium, which supplies a complex answer. The real part decides where the phase goes and the imaginary part where the amplitude does, and they are not the same direction: the surfaces of constant amplitude are parallel to the interface, because the decay is entirely into the metal, while the surfaces of constant phase are tilted by 86.5° from the normal. There is therefore no single refracted angle to quote. The familiar picture, in which one set of parallel planes carries both, requires the absorption to be exactly zero.$

The angle that is two angles

Snell's law survives a complex index by giving a complex answer, and a complex angle is not an angle. What the phase-matching argument actually fixes is the tangential wavenumber, and when the medium absorbs, the surfaces of constant phase and the surfaces of constant amplitude stop being parallel. In silver at 550 nanometres the phase fronts run within four degrees of the surface while the amplitude decays straight into it.

Two intersections, and the rule that picks one. The phase-matching construction for light arriving at 40° from a medium of index 1 into one of index -1. The circles are each medium's own relation between wavevector and frequency; the vertical line is the tangential wavenumber, which the boundary conserves. The line crosses the second circle twice, and the construction alone does not say which point is the answer — the rule that does is that energy must travel away from the interface. For a positive index the energy runs along the wavevector and the upper point is taken. For a negative one the energy runs against it, so the lower point is taken, the wavevector points back toward the boundary, and the ray leaves at -40.0° — on the same side of the normal as it arrived. Nothing in the drawing has changed except which intersection is circled.

The ray on the wrong side of the normal

The phase-matching construction draws a circle and a line, and the line crosses the circle twice. Every earlier construction silently took the upper intersection. Which one is physical is decided by where the energy goes rather than by where the wavevector points, and in a medium whose group velocity opposes its phase velocity the answer is the other one — so the refracted ray leaves on the same side of the normal it arrived on, a flat slab focuses, and a lens can beat the diffraction limit until loss stops it.

$A hemisphere outside is a cone of 16.6° inside. Rays leaving a flat face from a material of index 3.5 into air. Every direction in the hemisphere above the face is reached by a ray from inside a cone of half-angle 16.6°, arcsin(1/n), because refraction at the face fans the cone out; rays steeper than that are reflected back, drawn in the warning colour. The squeeze is exact in the way the invariant requires. The projected solid angle of the inside cone is π sin²θc, which is π/12.25, and multiplied by n² it equals π, the projected solid angle of the whole hemisphere outside — checked on the drawn geometry. So the light that does get out has not gained anything: the radiance inside is n² times higher and the directions available are n² times fewer, and the étendue passing the face is the same on both sides.$

The cone light has to find to get out

Inside a dense material the whole hemisphere of directions outside a flat face shrinks to a narrow cone, and light made inside can leave only if it happens to be travelling within it. For gallium arsenide that is two per cent of the light. Turned round, the same cone keeps light in: a slab of silicon with a rough surface holds the light it admits for fifty-one passes. Both numbers are the n² the optical invariant carries, and neither one breaks it.

A delay that grows as the square root of the length. The differential group delay between the fastest and slowest polarisation states of a fibre built from sections 100 m long, each a slightly birefringent waveplate with its axis at a random angle, against length up to 400 km. Three individual fibres are drawn faint, and the root-mean-square delay over 300 of them heavy. The ensemble grows as a power 0.50 of the length — the square root, because each section rotates the polarisation it receives before adding its own delay, so the delays add like the steps of a random walk in three dimensions rather than like lengths laid end to end. At 100 km the mean delay is 5.18 ps against 5.00 ps for a coefficient of 0.5 ps/√km. Had the axes all been aligned, the same sections would have added to 172 ps at that length, growing in proportion, and the drawn dashed line leaves the frame within a few kilometres. The randomness is what keeps the delay small, and it is also what makes it impossible to compensate with a fixed device.

The delay that is a random variable

A fibre's core is very slightly elliptical, so its two polarisations travel at very slightly different speeds — and the ellipse turns, at random, every hundred metres or so. The delays of the pieces do not add. They random-walk, so the total grows as the square root of the length, follows the same distribution as the speeds of gas molecules, differs from one wavelength to the next, and on any particular day may be three times its average.

Light that walks through a cloud. Seven photons entering a slab 8 scattering mean free paths thick from above, straight down, and followed until they leave, with every scattering equally likely to send them in any direction; their paths are projected onto the page. Of 20000 photons followed the same way, 82.5 per cent come back out of the top and 17.5 per cent out of the bottom. Diffusion theory predicts 18.2 per cent through the bottom. Only 10 photons cross without scattering at all, where e⁻⁸ predicts 6.7 — within the counting error of so few: nearly all of what is transmitted has walked, and the direction it arrived from is forgotten on the way.

The cloud light has to walk through

A beam crossing thirty scattering lengths of anything should keep e⁻³⁰ of itself — a ten-millionth of a millionth. A cloud thirty scattering lengths thick lets through nearly a third of the sunlight falling on it. The light has not crossed; it has walked, one scattering at a time, and a walk through a slab obeys a law with the shape of Ohm's rather than of an exponential.

Snell's law with space and time exchanged. Two constructions on the same diagram of frequency against wavenumber, with the light lines of a medium of index 1 and of index 1.5. On the left, a boundary in space: the wave crosses a still surface, the frequency is conserved, and the horizontal line at the incident frequency meets the new medium's line at a wavenumber 1.5 times larger — the familiar shortening of the wavelength. On the right, a boundary in time: the whole medium changes at once, the wavenumber is conserved, and the vertical line at the incident wavenumber meets the new medium's line at a frequency 0.667 times the old one. The vertical line also meets the new line's negative-frequency branch, which is a wave running backwards: a reflection in time. A spatial boundary reflects into the same frequency and a temporal one into the same wavelength.

The reflection that needs no surface

Change the refractive index of a whole medium at one instant and a wave already travelling through it splits in two, one part running on and one running back, though there is no surface anywhere for it to reflect from. A boundary in time is Snell's law with space and time exchanged: the wavelength is kept and the frequency changes, momentum is conserved and energy is not.

$The étendue, tiled. The phase space of a one-dimensional optical system: position across a twenty-micrometre aperture, against the optical direction cosine n·sinθ, for a system accepting ±0.1. The shaded rectangle is what the beam occupies, and its area — 4 micrometre-radians — is the one-dimensional étendue, the quantity no arrangement of lenses can reduce. Divided by a wavelength of 500 nanometres it is 8, and the rectangle is tiled here with exactly 8 cells of area one wavelength each. That is the whole content of the count: a cell of phase-space area λ is the smallest patch a field can be confined to, because squeezing it in position spreads it in direction by the same relation that gives a slit its diffraction pattern, so an étendue is a number of modes and its conservation is the conservation of a count. The cells are drawn four wide and two high, which is arbitrary: only a cell's area is fixed, not its shape — a beam may be confined tightly in position and loosely in angle or the other way round, and the trade is what an optical system is for.$

The invariant that is a count

Étendue is an area times a solid angle, and ray optics gives it no floor — nothing in a ray has a size. Divide it by the square of the wavelength and it becomes a number of modes, its conservation becomes the conservation of a count, and the count has a least value of one. That is where the ray bound hands over to diffraction, and the handover is the same number written two ways.

Two temperatures for the same sunlight. Sunlight described two ways, against how much it has been concentrated. The flat line is the temperature its spectrum belongs to — 5,762 K, the Sun's surface, which concentration does not change because a mirror does not alter a photon's energy. The rising curve is the temperature a blackbody would need in order to radiate the flux actually arriving: 394 K unconcentrated, 2213 K under a parabolic dish, and 5771 K at the geometric limit, where the two meet — computed here and checked against the Sun's own temperature, because a perfect concentrator reproduces the source's radiance and cannot exceed it. The gap between the two curves is the dilution, and dilution is entropy: the same energy spread over a hundred thousand times more directions occupies a hundred thousand times more modes. That is what a converter has to carry, and it is why the ceiling on solar conversion is not the Carnot efficiency between 5,762 K and 300 K.

The work a diluted beam will not do

Sunlight at the top of the atmosphere has the spectrum of a body at 5,762 kelvin and the energy flux of one at 394. The mismatch is not an accident of units: the light has been spread over a hundred thousand times more modes than it left in, and that dilution is entropy. Run it through a heat engine unconcentrated and five per cent of it is available as work.

The same cone, and a different arrival

The invariant fixes what a guide accepts and says nothing about when it arrives, and the two turn out to be nearly independent. Shaping the index so that the rays which travel furthest also travel fastest cuts the spread in arrival times by a factor of five hundred, at the cost of exactly half the light — and the acceptance cone the invariant governs is untouched throughout.

$The rings belong to the edge, not to the size. The far-field intensity of 3 apertures of the same width, against angle in units of the diffraction limit, on a logarithmic intensity axis spanning ten decades. They differ only in how the transmission falls off toward the rim. With a hard edge the first sidelobe is 13.3 decibels down and the core is 0.89 wide. With a Hann taper the first sidelobe is 31.5 decibels down and the core is 1.44 wide. With a Blackman taper the first sidelobe is 58.1 decibels down and the core is 1.64 wide. The hard edge's rings are not a defect of the optics and are not reduced by making it larger — they are the transform of a discontinuity, and the only way to remove them is to remove the discontinuity. What it costs is the width of the core, which is the resolution.$

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.

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

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.

A repeat in space gaps the frequencies; a repeat in time gaps the wavenumbers. Two media with the same modulation depth, 0.2, computed the same way. Left: permittivity repeating in space, a stack of layers. For each frequency the wave equation is integrated across one spatial period, and the Bloch wavenumber drawn against frequency; between frequencies 0.478 and 0.526 (in units of c over the period) there is no real wavenumber, so light of those frequencies cannot travel and is reflected. Right: permittivity repeating in time. For each wavenumber the equation is integrated across one period, and the frequency drawn against wavenumber; between wavenumbers 0.474 and 0.518 there is no real frequency. The axes of the two panels are swapped, and so is everything else: the spatial gap is a band of frequencies that decays in space, the temporal gap is a band of wavenumbers that grows in time.

The crystal made of moments

A stack of layers that repeats in space refuses a band of frequencies and reflects them. A medium that repeats in time — its refractive index swung up and down everywhere at once — refuses a band of wavenumbers instead, and a wave with a wavenumber in that band does not reflect. It grows, exponentially, drawing on whatever is swinging the index. The construction is exact, the gap is computable from one period of the modulation, and the obstacle to building one for light is how fast a material would have to change.

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.

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