Illustrated Physics
physics, drawn
Concept

Aberration — where it appears

The change in the apparent direction of light caused by the observer's motion, which remaps the whole sky rather than shifting its colours. The shift is of order the observer's speed divided by that of light, so the Earth's orbit swings every star through 41 arcseconds a year. At speeds approaching c it crowds the entire sky into a forward cone.

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

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.

optics · Imaging Where the light that was sideways ends up. The direction a photon is seen to travel in the laboratory, against the direction it was emitted in the frame of the source, for a source moving at 0.5c, 0.9c, 0.99c. The straight diagonal is what would happen if a boost only changed frequencies; every curve lies well below it, which is aberration. The number that matters is where the emitted right angle lands, because half of everything emitted is on that side of it: 60.0° at 0.5c, 25.8° at 0.9c, 8.1° at 0.99c. The usual shorthand for that angle is 1/γ, which gives 49.6°, 25.0°, 8.1° — good to a few per cent only once the source is genuinely relativistic, and wrong by 17% at 0.5c. Nothing is emitted differently in any of these cases: the source is radiating exactly as it always did, and it is the map from its angles to ours that has changed.

The sky that crowds into a cone

A boost does not only shift frequencies. It remaps directions, so half of everything a fast traveller can see is squeezed into a forward cone of half-angle about 1/γ — and because brightness carries four powers of the Doppler factor, what lies ahead is overwhelming and what lies behind has effectively gone.

relativity · Doppler A cube photographed at four speeds. The outline a camera records of a cube passing at β = 0, β = 0.4, β = 0.7, β = 0.9, moving to the right, seen at the moment it passes. The dashed square is what the usual picture shows: the cube flattened along its motion by the Lorentz factor. It is not what the camera gets. Light from the far face left earlier than light from the near face, by the time it needed to cross the extra distance, and the cube moved in that interval — so the far face appears displaced backwards and the side of the cube comes into view. The two effects together are exactly a rotation: at β = 0.4 the cube photographs as one turned 23.6°, at β = 0.7 the cube photographs as one turned 44.4°, at β = 0.9 the cube photographs as one turned 64.2°. The contraction has not gone anywhere — a ruler laid alongside still reads 43.6 per cent of the proper length at the highest speed drawn — but it is a statement about two positions at one time, and a photograph is not that.

The contraction no photograph shows

Every textbook picture of a relativistically moving object drawn between 1905 and 1959 showed it squashed. A camera would have shown it turned. The contraction is entirely real in the measurement that defines it, and a photograph is not that measurement.

relativity · Length contraction The blow-out band has two edges, not one. The ratio of radiation force to gravity against grain radius, with the radiation-pressure efficiency included: a grain much smaller than the wavelength of the light barely interacts with it — the efficiency falls as the fourth power of the size, which is Rayleigh's law — so the ratio stops rising and turns over. The dashed line is the same ratio with the efficiency taken as one, which is the usual drawing and is right only to the right of the turnover at 115 nm. The consequence is that a grain can be too small to be blown out as well as too large. Taking the threshold at a half — the value at which a grain released from a circular orbit is unbound — the band runs from 48.2 nm to 574 nm, and the largest ratio any grain of this material reaches is 1.87. Everything outside that band stays, and what stays does not stay put: it spirals.

The size the light cannot blow away

Radiation pressure and gravity both fall as the inverse square of distance, so their ratio is a property of the grain and not of where it is. What follows is a band of sizes that get blown out — with a lower edge as well as an upper one — and a drag, on everything else, that is the same pressure read one order further in v/c.

astrophysics · Radiation pressure 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.

optics · Etendue 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.

optics · Imaging The same source, seen coming and seen going. The observed flux of a moving source, relative to the same source at rest, against the angle between its motion and the line of sight, on a logarithmic scale, at β = 0.5, β = 0.9, β = 0.99. The curves span 8.1e+1, 1.3e+5, 1.6e+9 between the approaching and receding directions, which is ((1+β)/(1−β))⁴ exactly. The fourth power comes from the one quantity every observer agrees about: the specific intensity divided by the cube of the frequency. Three powers of the Doppler factor come from that invariance and the fourth from integrating over frequency. Two consequences are worth reading off. A source seen side-on is fainter than the same source at rest, by γ⁴ — a factor of 2.5e+3 at the fastest speed drawn. And half the light arrives inside a cone of 60.0°, 25.8°, 8.1°, which for a fast source is one over γ.

The brightness that is not the same for everyone

A moving source is not merely shifted in colour. Its light is concentrated forwards, so the same lamp is enormously brighter seen coming than seen going — by the fourth power of one number, which for a jet at ninety-nine per cent of light speed is a factor of a thousand million. The reason is that only one combination of intensity and frequency is the same for every observer, and everything else follows from it.

relativity · Doppler 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 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.

optics · Imaging 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 Every ray from one point, meeting again at another. Rays from a point half a radius from the centre of Maxwell's fish-eye, n = n₀/(1 + r²/R²), traced in 11 directions round the full circle. Every one is an arc of a circle, and every one passes through the point on the other side of the centre at distance R²/r from it — 2.00 radii — which is the image. The worst miss is 8.8e-4 radii, the size of the integration's own error. Rays leaving in opposite directions, rays leaving at right angles, rays that go the long way round: all arrive. The faint circles are contours of the index, which falls from n₀ at the centre to n₀/2 at one radius and n₀/5 at two.

The medium that images every point

A single refracting surface can be shaped to image one point perfectly and no other. A medium whose index falls smoothly away from a centre can do better: in Maxwell's fish-eye every ray from any point, leaving in any direction, arrives at one image point, and every such path takes exactly the same time. It is a perfect instrument for all of space at once — and the image it forms is sharp everywhere and a faithful copy nowhere.

optics · Fermat Rays and the surfaces of equal optical path. Rays from a point source 20 mm in front of a curved glass surface (radius 4 mm, index 1.5), traced exactly, and the surfaces on which the optical path from the source — distance times index — takes the same value (curves, every 2.5 mm of path). In air the surfaces are spheres round the source. In the glass the rays from the edge of the surface cross the axis at 12.8 mm, well short of the paraxial focus at 20.0 mm, and the equal-path surfaces are no longer spheres. They are still at right angles to every ray: measured on a fan of 641 rays over 10 surfaces in the glass, the largest departure from 90° is 0.005°. Further in, the surfaces fold into cusps where neighbouring rays cross, and there a surface has no single normal.

The surface every ray is normal to

Light from a point, bent by any number of lenses and mirrors, comes out as a tangle of rays that no longer meet anywhere. Hidden in the tangle is a surface every ray crosses at a right angle: the set of points the light reaches at the same optical time. That a surface always exists is a theorem two centuries old, and it is the reason a modern wavefront sensor works — an array of tiny lenses measures only the direction each patch of light is travelling, and the measurements can be added up into a shape only because they are the slopes of one.

optics · Fermat Sturm's conoid: a line, a circle, a line. The cross-section of a parallel beam after a lens whose focal length is 100 mm in its vertical meridian and 125 mm in its horizontal one, traced ray by ray through a round aperture, at seven distances behind the lens: 60.0, 100.0, 105.6, 111.1, 118.1, 125.0, 135.0 mm. There is no point focus anywhere. At 100 mm the vertical meridian has focused and the beam is a horizontal line; at 125 mm the horizontal meridian has, and it is a vertical line. Between them the beam is an ellipse that turns from wide to tall, and at 111.1 mm — the harmonic mean of the two focal lengths — it is a circle, the circle of least confusion, of diameter 22.2 per cent of the aperture's.

The focus that is two lines

A lens curved more steeply in one direction than the other has no focus. It brings light to a short line at one distance, to a line at right angles at another, and in between to an ellipse that turns from wide to tall and is round at exactly one place. Most eyes are built this way to some degree, so do mirrors used at an angle, and the best any ordinary lens can do for them is to put that round patch on the retina. A cylinder, curved in one direction only, can do what no sphere can: move one line onto the other.

optics · Imaging The bend that depends on when the planet was there. The difference, in microarcseconds, between the quasar's deflection computed from Jupiter's position when the light passed it — its position as seen from the Earth, 53 minutes earlier — and from its position at the moment of observation, over the day of the 2002 pass. In those 53 minutes Jupiter moves 40998 km, 0.57 of its own radius, and the deflection, a vector of 1.14 mas pointing away from the planet, turns and shrinks by up to 46 µas — about one twenty-fifth of itself. That small term was what the experiment was designed to see. The VLBA measured 51 ± 9 µas, the value general relativity predicts for a deflection pointing away from the position light reveals.

The bend a moving planet gives

Jupiter bends starlight by sixteen thousandths of an arcsecond at its edge, a hundredth of what the Sun manages, and in 2002 radio telescopes spread across a continent measured the bend on a quasar three and a half arcminutes from the planet. The experiment was designed to see something smaller still: the fact that Jupiter moves, by half its own width in the time light takes to reach the Earth, so that the bend depends on where the planet was when the light went past. The measured term was announced as the speed of gravity, and the argument over whether it was anything of the kind is the clearest lesson there is in what an experiment's number is a number of.

astrophysics · Light deflection

Named alongside it

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

Optical path lengthFermat's principleFocal lengthAdaptive opticsAstigmatismDepth of focusImagingMagnificationOptical designReference framesRefractive indexRelativistic beaming

All concepts