Illustrated Physics
physics, drawn
Concept

Refraction — where it appears

The change in a wave's direction at a boundary between media, fixed by the requirement that the component of the wavevector along the boundary be unchanged. That conservation follows from the boundary being uniform along itself, and the ratio of sines is what it looks like when the two wavevector magnitudes are written as indices.

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

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 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.

optics · Dispersion A refraction with no wave in it. Field lines crossing the boundary between two dielectrics whose permittivities differ by a factor of 4, at incidences of 12°, 26°, 40°, 54°, 68°. Nothing is oscillating and nothing is travelling: these are static fields, and the only inputs are that the component of E along the surface is the same on both sides and that the component of D across it is. Dividing one condition by the other gives the ratio of the tangents of the two angles, and it comes out equal to the ratio of the permittivities — measured off the constructed directions as 4.000, 4.000, 4.000, 4.000, 4.000, against 4. The lines bend towards the surface on the side with the larger permittivity, which is the opposite sense from a light ray entering glass, and the chart on the right shows the whole relation: it is a version of Snell's law with the sines replaced by tangents, and there is no speed anywhere in it. Two things follow that are worth carrying. A field line meeting a surface at grazing incidence stays nearly parallel to it whatever the materials, so the ratio matters least where the field is largest along the surface. And in the limit of a very large ratio every line comes out very nearly perpendicular on the low side, which is the electrostatic ancestor of a conductor's boundary condition — a conductor is the ratio taken to infinity.

A refraction with no wave in it

A field line crossing from one dielectric into another bends, by a law that looks exactly like Snell's with the sines replaced by tangents. Nothing is oscillating, nothing is travelling, and no speed appears anywhere in the derivation — only the two conditions that say what a boundary may and may not do to a field.

electromagnetism · Dielectrics 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.

optics · Refraction 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.

optics · Fermat 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.

optics · Refraction 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.

optics · Refraction 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.

optics · Refraction The speed of sound in dry sand rises with the load on it. The speeds of compressional and shear waves in a dry random packing of quartz grains, porosity 0.36, against the pressure on it, from the Hertz–Mindlin effective-medium theory, for coordination numbers from 6 to 9 (the bands). The pressure axis is logarithmic. Both speeds rise as the sixth root of the pressure. At 1 kPa — the load about six centimetres down in dry sand — compressional waves travel at 316–362 m/s; at 100 kPa, about six metres down, 681–780 m/s; at 10 MPa, 1468–1680. The dashed lines are the speed of sound in air, 343 m/s, and in water, 1,480 m/s. Near the surface dry sand carries sound no faster than air; measured shallow speeds are lower still, by up to a factor of two, because real contacts are fewer and weaker than the theory assumes.

The sound sand carries faster the deeper it goes

A solid carries sound at a speed set by what it is made of. A heap of quartz grains is made of quartz, and near its surface it carries sound more slowly than the air above it. The grains are not the solid; their contacts are, and a contact between two spheres grows stiffer the harder it is pressed. So the speed of sound in dry sand rises as the sixth root of the load on it — from about air's speed a few centimetres down to twice that at six metres — the paths of sound in a beach curve back up to the surface, and a water table is a mirror to one kind of wave and almost invisible to the other.

fluids · Granular matter Rays inside a drop never reach the critical angle. Rays of light entering a drop of water (index 1.333) from the left at four heights — 0.3, 0.6, 0.85, 0.97 of the radius — followed through six internal reflections, each chord drawn with a width proportional to the light it carries. Every chord inside a sphere meets the surface at the same angle at both ends, so each ray strikes the inside of the drop again and again at its refraction angle: 13.0°, 26.8°, 39.6°, 46.7°, all below the critical angle of 48.6°. None is ever totally reflected; at each contact most of the light leaves, and what is left inside dwindles — the light that, leaving after one and two reflections, makes the primary and secondary rainbows.

The drop that cannot hold the light it lets in

Light inside water meeting the surface at more than 48.6° is totally reflected, and every raindrop is made of water. Yet no ray of sunlight that enters a raindrop is ever totally reflected inside it. A sphere's geometry forbids it: a ray refracted on the way in meets the inside of the surface at its refraction angle, and keeps meeting it at that same angle every time it bounces, and that angle is always smaller than the critical one. The light leaks out a little at every bounce, which is why there are rainbows at all. Light born inside the drop is another matter, and a sphere can keep it for ever.

optics · Total internal reflection A lens with its glass taken out, focusing the same. Two lenses of focal length 200 mm and aperture 100 mm in acrylic (n = 1.49), each with its flat face to a parallel beam and the exact aspheric curve that sends every ray to one focus: solid (top), 12.2 mm thick at the centre, and the same curve cut into 13 rings 1 mm deep with the glass between them removed (bottom), on a backing 0.4 mm thick. Thickness is drawn six times larger than height, and the distance to the focus is not to scale. Every ray leaves each lens at the same angle from the same height, because the angle a ray turns through at a surface depends only on the slope of the surface where it crosses, and every ring keeps the slope it had. What was removed was glass the rays travelled through without turning.

The lens with its glass taken out

Snell's law turns a ray at a surface by an angle set by the slope of the surface where the ray crosses, and the glass behind the surface does not enter it. So a lens can be sliced into rings, the glass between the slopes thrown away, and the rays will not notice. Fresnel did it for lighthouses in 1822. The waves notice — every step cuts a few hundred wavelengths out of them — unless the steps are made exactly one wavelength deep, and then the lens becomes something else: a diffractive element whose colours run backwards, and which can cancel a glass lens's colour with a twentieth of its power.

optics · Refraction Earthquake waves through the Earth, and the ring they do not reach. Paths of compressional (P) waves from an earthquake at the top of the Earth, traced through the mantle, the liquid outer core (orange) and the solid inner core (darker), drawn in one half of a cross-section: rays that stay in the mantle (blue) and rays that pass through the core (red), with Snell's law applied shell by shell to a reference model of the Earth's speeds. The mantle rays curve back up because the speed rises with depth, and reach the surface no further than 97° from the source. Rays that strike the core are bent sharply inward, because the speed drops from 13.7 to 8.1 km/s at its surface, and those that turn in the liquid outer core come out no closer than 144°. Between the two, marked on both sides, is a ring of the surface that these direct P waves do not reach: the P-wave shadow, which is how the core was found. A few rays that just enter the solid inner core (purple, dashed), where the speed jumps up again, are bent back out early and land inside the shadow, as close as 115° — the faint arrivals from which the inner core was found.

The shadow that shows the core is liquid

Nobody has been more than a few kilometres into the Earth, and its interior was mapped from the timing of earthquake waves alone. The decisive observations were of waves that do not arrive. Seismometers between about 100° and 140° from a large earthquake record no direct compressional wave, because something deep inside bends the rays away; and over two-fifths of the planet beyond 100° no direct shear wave arrives at all, because that something cannot be sheared. A slow, liquid core casts both shadows, and a few faint waves inside the first one revealed a solid core within it.

waves · Wave motion The shadow a thunderclap's sound cannot enter. Sound rays from a point 3 km up a lightning channel, on an afternoon with 25 °C at the ground and the air cooling 6.5 K per kilometre upwards, so that sound travels slower the higher it goes; the vertical scale is stretched 3.5 times. Every ray curves upwards. Rays aimed below the horizontal reach the ground, reflect and climb away; the last ray to touch the ground grazes it 23.1 km from the channel. Beyond that, shaded, no ray arrives at all: a listener's ears 2 m up are inside the shadow from 23.7 km. The flash is visible there and the thunder is not, though the air between is perfectly clear and absorbs the low notes of thunder hardly at all.

The thunder that passes over the listener's head

Lightning twenty-five kilometres away is plainly visible and usually silent, though the air between is clear and absorbs the low notes of thunder hardly at all. The thunder has not faded; it has gone over the listener's head. Daytime air cools upwards, sound slows as it climbs, and Huygens's wavelets, smaller at the top of every front than at the bottom, tilt each front towards the sky. Beyond the ray that just grazes the ground there is a shadow the sound cannot enter — and a hundred and fifty kilometres further out, the stratosphere can bend it back down.

waves · Huygens The focus that moves out of the eye. Parallel rays from a distant object entering a simplified eye 24 mm long, in millimetres along its axis: in air the eye's 61 dioptres bring them to a focus 21.9 mm behind its principal plane, on the retina; under water its 18 dioptres would bring them to a focus 73 mm behind, about three times the length of the eye. On the retina itself the underwater rays have converged only a third of the way, and every point of the scene is spread into a disc the size of the pupil's shadow. The eye is not out of focus by a little, as a short-sighted eye is; it is missing most of a lens.

The eye that loses its lens in water

Eyes opened under water see only a blur, and the reason is not water in the eye or salt or murk. The human eye does two-thirds of its focusing not with its lens but with the curved front of its cornea, whose power comes from the jump in refractive index between air and tissue. Water is almost as dense optically as the cornea, and the jump all but disappears: about forty-three of the eye's sixty dioptres vanish, and the focus moves to three times the eye's own length behind the retina. A mask gives the cornea back its air; a small pupil and a hard squint buy back a little; animals that hunt in both worlds have flattened their corneas and given the work to the lens.

optics · Imaging

Named alongside it

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

Snell's lawBoundary conditionDispersionPhase matchingRefractive indexEvanescent wavePermittivityTotal internal reflectionCausticDispersion relationHuygens principleSeismic waves

All concepts