Illustrated Physics
physics, drawn
Concept

Multiple scattering — where it appears

Light that has been scattered more than once before arriving, and has therefore lost the geometry that a single scattering would have imprinted on it. It is what keeps a real sky's polarisation below a hundred per cent, what makes a hazy sky white, and what makes the neutral points a measurement of turbidity.

Named by 8 essays across 3 fields — each of them below, with the objects they name alongside it.

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.

optics · Scattering 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.

optics · Polarisation 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.

optics · Scattering 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.

optics · Scattering How much darker a surface looks under a film of water. The diffuse reflectance of a surface covered by a thin film of clear water (solid), against its reflectance dry, with the dry value for comparison (dashed): (1 − Rₑₓₜ) a (1 − Rᵢₙₜ)/(1 − a Rᵢₙₜ), where the film traps and returns light that falls outside its escape cone. A bright surface of reflectance 0.9 falls to 0.77; a mid-grey one of 0.5 to 0.32, 64 per cent of its dry value; a dark one of 0.2 to 0.11, 54 per cent. Dark surfaces darken proportionally most, because each return trip costs them most; and a coloured surface grows more saturated, because the colours it absorbs are absorbed again on every return while the colour it reflects survives.

The sand that darkens when it is wet

Wet sand is darker than dry sand, a wet stone darker than a dry one, a splashed pavement darker in patches, and a wet white shirt turns grey and then see-through. Water is clear and absorbs almost nothing, so it cannot be adding darkness. What it does is change where the light goes. A film of water on a surface traps light that has been scattered back up, returning most of it for another chance to be absorbed. Water in the pores between grains replaces air with something closer to the grains' own refractive index, so each grain turns light aside less, and light wanders deeper before it gets out. Both lengthen the path, and on a longer path more light is absorbed by the material that was there all along.

optics · Scattering The focus that gets sharper when the wave is scattered. The refocused amplitude against distance across the source, in wavelengths, for the same one-sided mirror and pulse, with the space between empty (dashed) and with a slab of 200 lossless point scatterers, 15 wavelengths thick, filling the space between source and mirror (solid). Through the slab the wave reaching the mirror is a jumble — scattered many times, arriving from every direction the slab can send it — yet time-reversed and sent back through the same slab, it refocuses on the source, in a spot 1.5 wavelengths wide against 4.8 in empty space. The scatterers, bouncing waves in from a wide range of angles, act as a lens much wider than the mirror: disorder that would ruin an ordinary lens makes a time-reversal mirror sharper.

The mirror that sends a wave back to where it began

Record a wave as it arrives at a row of microphones, then play each recording backwards through the loudspeaker beside it, and the sound converges on the place it came from — however twisted the route it took. Huygens said every point on a wavefront acts as a source; a time-reversal mirror takes him literally, making every point of a recorded front re-emit its own history in reverse. The result has a property no lens has. Put a thicket of scatterers between the source and the mirror, so that the arriving wave is an unrecognisable jumble, and the reversed wave still finds its way back — to a sharper focus than it could reach in empty space, because the disorder acts as a lens wider than the mirror.

waves · Huygens A pile of glass plates, added two ways. The transmission of light at normal incidence through a pile of N glass plates (index 1.5) in air, on a logarithmic axis, against N. Dashed: Stokes's 1862 result for light whose reflections add as intensities, T = (1 − R)/(1 + (2N − 1)R) with R = 0.040 per surface, which falls only as 1/N: 0.107 at 100 plates, 0.029 at 400. Solid: the same pile with the reflections added as waves and every plate and gap of random thickness, as ⟨ln T⟩ over 200 random piles (red) and for one pile (grey). It falls in a straight line, by 0.0843 per plate, an exponential decay with a localisation length of 24 plates: 10⁻³ at 100 plates and 10⁻¹⁴ at 400. Interference with random phases does not average away; it traps.

The pile of glass that stops every colour

A sheet of window glass reflects about eight per cent of the light that falls on it. A pile of a hundred sheets, George Stokes calculated in 1862, should still let through a tenth, because each extra sheet only adds a little more reflection to the total. That is right for sunlight, whose colours are too mixed to interfere. For a single colour, with every sheet and every gap a slightly different thickness, the same pile lets through a thousandth, and a pile of four hundred a hundred-millionth of a millionth. Interference with random phases does not average away into the intensity sum. In one dimension it traps every wave, at every wavelength, however weak each reflection is.

waves · Periodic media A cloud that grows wider instead of denser. The density of rubidium atoms in a magneto-optical trap against distance from its centre, for 10⁵, 10⁶, 10⁷, 10⁸ atoms, computed from the trap's spring (200 Hz), the atoms' temperature (100 μK) and the repulsion of rescattered light, with σᵣ/σₗ = 1.30. The smallest cloud is the trap's Gaussian, 78 μm in radius, with a peak of 0.91 × 10¹⁰ per cm³. Larger clouds stop getting denser: each fills a sphere at the same 2.5 × 10¹⁰ atoms per cm³ (dashed), out to 0.21, 0.46, 0.99 mm, with a soft edge about a Debye length, 45 μm, wide. More atoms make a bigger ball, not a denser one.

The light that makes a cold cloud repel itself

Load more atoms into a magneto-optical trap and the cloud does not get denser. It gets bigger, at a density that stays fixed whatever the number. The cause is the trap's own light. A photon one atom scatters can be absorbed by another, which is pushed away, and the push falls off as the inverse square of the distance, like the force between two charges. A cloud of laser-cooled atoms is therefore a plasma of neutral particles. Each behaves as if it carried a ten-thousandth of an electron's charge, and the trap fills with them as a harmonic well fills with charge: at one density, out to whatever radius the number requires.

astrophysics · Radiation pressure

Named alongside it

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

Random walkAlbedoAnderson localisationAsymmetry parameterCoherent backscatteringDiffusionDipole radiationDisorderInterferenceOptical depthPolarisationRayleigh scattering

All concepts