Illustrated Physics
physics, drawn
Waves

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.
14 min read 5 figures Order out of the randomApproximations that lie

Assumes: The gap a repeat opens · The mode that lives in the mistake

The gap a repeat opens found that a stack of alternating transparent layers refuses a band of frequencies entirely — not absorbing them, but reflecting them, because the small reflections from every interface add in step when the repeat matches half a wavelength. The mode that lives in the mistake broke the repeat once, inserting one layer of the wrong thickness, and found a single frequency let through, trapped at the defect. Both were about order: a periodic arrangement, with or without one flaw.

The obvious next question is what happens when there is no order at all. Pile up glass plates with every plate and every gap a slightly different thickness, chosen at random. Each surface still reflects a few per cent; the reflections still interfere; but now their phases are random. The natural expectation is that random phases cancel on average, leaving the plain sum of intensities — the answer for a pile of plates in sunlight, which George Stokes worked out in 1862. That expectation is wrong, and the way it is wrong is one of the deepest results about waves in disordered media.

Stokes’s pile and its sum

A single glass surface at normal incidence reflects a fraction R=((n−1)/(n+1))2R = \left((n-1)/(n+1)\right)^2 of the light, four per cent for glass of index 1.5. A plate has two surfaces, and between them the light bounces back and forth, losing a little at each reflection. For light whose colours are mixed enough that its reflections do not keep a fixed phase relation — sunlight, a lamp — the intensities of all the bounced beams simply add, and Stokes summed the series for a pile of NN plates:

TN=1−R1+(2N−1)R.T_N = \frac{1 - R}{1 + (2N - 1)R}.

The transmission falls, but only as 1/N1/N. A pile of a hundred plates still transmits a tenth of the light, a pile of four hundred three per cent. This is the behaviour of diffusion: each surface sends some light back, the light that is sent back can be sent forward again, and the net flow through the pile falls in inverse proportion to its thickness, exactly as electrical current through a resistor falls with its length or light through a thick cloud falls with its depth, the result the cloud light has to walk through derived by following photons on random walks. Stokes’s formula is Ohm’s law for light. Stokes had a practical reason for wanting it. A pile of glass plates tilted to the Brewster angle was the standard polariser of his day: at that angle each surface reflects almost none of the light polarised in the plane of incidence and a few per cent of the other, as the angle at which reflection picks a side found, so a pile of a dozen plates strips one polarisation from the transmitted beam by reflecting it away bit by bit. How many plates were needed, and how much light was lost, were exactly what his series answered, and because the light was sunlight or lamplight his intensity sum was exactly right.

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.
Fig. 1 Transmission through a pile of NN glass plates (index 1.5) in air, on a logarithmic axis. Dashed: Stokes’s sum of intensities, falling as 1/N1/N — 0.107 at 100 plates. Solid red: the same pile summed as waves with every plate and gap of random thickness, as ⟨ln⁡T⟩\langle\ln T\rangle over 200 random piles; grey, one pile. It falls in a straight line on this scale, by 0.084 per plate, a localisation length of 24 plates: 10−310^{-3} at 100 plates and 10−1410^{-14} at 400.

The same pile, summed as waves

For a single colour the reflections do keep their phases. Every reflected and transmitted wave in the pile has a definite phase set by the thicknesses it has crossed, and the light that emerges is the sum of their amplitudes, not their intensities. The calculation is done exactly by multiplying the two-by-two matrices that describe how each layer carries the electric and magnetic fields from one face to the other, one per plate and one per gap, and reading the transmission from the product.

With the thicknesses chosen at random, the result is the solid curve in the figure. It does not follow Stokes. It falls in a straight line on a logarithmic scale — exponentially — losing about eight per cent of the remaining light’s logarithm with every plate. After a hundred plates the typical transmission is a thousandth, a hundred times below Stokes; after four hundred it is 10−1410^{-14}, where Stokes still has three per cent. The pile reflects almost all of a single colour back, and nothing is absorbed; the energy is all accounted for in the reflected beam, as what adding does to the energy required of any interference.

The rate of the exponential defines a localisation length: the number of plates over which the logarithm of the transmission falls by two, here 24. Beyond a few localisation lengths a pile is, for one colour, a perfect mirror.

Why random phases trap

The intuition that random phases average out is right about the average of the amplitude, and wrong about what the transmission depends on. Think of the light inside the pile as a forward wave and a backward wave. At each surface a little of each is reflected into the other. In Stokes’s incoherent sum, the backward light is just more light travelling the wrong way, and it can be reflected forward again; nothing prevents light from diffusing through. With coherent waves, light that is reflected back and then forward again rejoins the forward wave with some phase, and the question is whether, over many such loops, those rejoined contributions add constructively or destructively on average.

In one dimension, they do not average to zero. Every path that returns to a point has a partner that traverses the same loop in the opposite order, and the two arrive with the same phase — they have crossed exactly the same layers — so they always add constructively. The backscattered light is systematically enhanced, and the forward light correspondingly reduced, at every step. The enhancement compounds multiplicatively along the pile, which is what makes the decay exponential rather than inverse. This is the same pairing of time-reversed paths that, in three dimensions, produces the narrow cone of enhanced backscattering above a white surface, which the cloud essay found as the first hint of interference in a random walk. In three dimensions the effect is a correction to diffusion; in one, where every path must pass through every layer, it wins outright.

The theorem behind the picture belongs to Philip Anderson, who argued in 1958 that an electron in a sufficiently disordered crystal cannot diffuse at all, because its wavefunction is confined to a region whose size is set by the disorder. Nevill Mott and William Twose showed in 1961 that in one dimension any disorder at all is sufficient: every state is localised, and the only question is how long the localisation length is.

No two piles alike

No two piles transmit the same. How the transmission is spread over 1500 random piles of 25, 100 and 200 plates, as the distribution of log₁₀T. Each is close to a bell curve in the logarithm, widening as the pile lengthens, so the transmission itself spreads over orders of magnitude. At 25 plates the typical value, exp⟨ln T⟩, is 10⁻¹ and the mean ⟨T⟩ is 10⁻¹. At 100 plates the typical value, exp⟨ln T⟩, is 10⁻⁴ and the mean ⟨T⟩ is 10⁻². At 200 plates the typical value, exp⟨ln T⟩, is 10⁻⁷ and the mean ⟨T⟩ is 10⁻³. The mean is carried by rare piles that happen to transmit well; a single pile almost never does what the average says.
Fig. 2 The distribution of log⁡10T\log_{10}T over 1,500 random piles of 25, 100 and 200 plates. Each is close to a bell curve in the logarithm and widens as the pile lengthens. At 100 plates the typical transmission, exp⁡⟨ln⁡T⟩\exp\langle\ln T\rangle, is 10−410^{-4} and the mean ⟨T⟩\langle T\rangle is 10−210^{-2}; at 200, 10−710^{-7} and 10−310^{-3}.

A localised system has a property that diffusing ones lack: its transmission is not self-averaging. Two piles made in the same way, with different random thicknesses, transmit very different amounts, and the spread grows with length. The distribution is close to a bell curve in the logarithm of the transmission — a log-normal distribution — because the logarithm is a sum of many roughly independent contributions, one per plate. The transmission itself then spans orders of magnitude, and its average is dominated by the rare piles that happen to transmit well. For 200 plates the average transmission is about 10−310^{-3}, while the transmission of a typical pile is ten thousand times smaller.

That is why the curves in the first figure plot the average of the logarithm, not the logarithm of the average. The average of the logarithm is what a single pile, chosen at random, almost certainly does. The average transmission is what an experiment over many piles would add up to, and it is carried by a few. Any measurement on a localised system has to say which it means.

Inside a pile that traps

The light inside a pile that traps it. The intensity of the light inside one random pile of 160 glass plates, relative to the light falling on its front, at the boundary after each plate and gap, on a logarithmic axis, computed from the far side back by the same characteristic matrices. It falls on average in a straight line on this scale — an exponential decay into the pile — with large jumps up and down where the random spacings happen to build up a local resonance, and leaves the far side at 10⁻⁴ of what entered. Almost all the light is sent back out of the front; none is absorbed.
Fig. 3 The intensity inside one random pile of 160 plates relative to the light falling on its front, at the boundary after each plate and gap, on a logarithmic axis. It falls exponentially on average, with large jumps where chance spacings build up a local resonance, and leaves the far side at 10−410^{-4} of what entered.

Inside a single pile the light’s intensity does what the average says, but noisily. It falls on a straight line on the logarithmic scale, with large excursions up and down where a stretch of the random spacings happens to form a cavity that resonates at this wavelength and builds up the field inside. Those chance resonances are the same object as the defect mode of the ordered stack: a local arrangement that traps one colour. In a random pile they occur everywhere, at random wavelengths and positions, and transmission through a long pile happens mostly when a chain of them lines up across it, which is rare and grows rarer with length.

Weak reflections, long but finite

Weaker reflections, longer reach. The localisation length of a random pile of plates — the number of plates over which ⟨ln T⟩ falls by two — against the reflectance of one surface of one plate, both on logarithmic axes, for plates of index 1.05, 1.1, 1.2, 1.3, 1.5, 1.8, 2.2, 2.6 in air (dots, from 120 random piles of 200 plates each). Weak reflectors localise light only over long distances: 0.06 per cent per surface, 1921 plates; 0.23 per cent per surface, 488 plates; 0.83 per cent per surface, 120 plates, while index 2.6 glass localises it within 4.5. The length scales close to the inverse of the reflectance (dashed line, slope −1), as perturbation theory predicts: every plate adds a small random scattering, and it is their accumulation, not any one, that traps the wave.
Fig. 4 The localisation length of a random pile against the reflectance of one surface, both on logarithmic axes, for plates of index 1.05 to 2.6 in air, from 120 random piles of 200 plates each. It scales close to the inverse of the reflectance (dashed, slope −1): 1,921 plates at 0.06 per cent per surface, 120 at 0.83 per cent, 4.5 for index 2.6.

Weak reflections lengthen the localisation length but do not remove it. With plates of index 1.05, whose surfaces reflect six hundredths of a per cent, light is localised over nearly two thousand plates; with index 1.2, over 120; with 2.6, within five. The length scales almost exactly as the inverse of the reflectance per surface, which is what a calculation treating each surface as a small random perturbation predicts. A pile of plates of any index whatever, long enough, reflects a single colour completely.

The same is true of any one-dimensional wave in any disordered medium: an electron in a thin wire with impurities, sound in a rod of varying cross-section, light in an optical fibre whose core wanders slightly in index along its length, seismic waves in a layered sediment. Each has a localisation length set by the strength of its disorder, and beyond it, a wave sent in at one end comes back.

Order lets colours through; disorder stops them

Order lets colours through; disorder blocks them all. The transmission of 60 glass plates against wavelength (relative to the wavelength at which each plate and gap is a quarter-wave thick), for a perfectly regular pile (blue) and for one whose plates and gaps vary randomly by ±25 per cent (red). The regular pile has a stop band round its design wavelength, where it reflects almost everything, and pass bands either side where the transmission rings between about 0.77 and 1, averaging 0.92. The disordered pile has no pass band: averaged over the same wavelengths it transmits 0.135, with a few sharp spikes where a chance resonance lets one colour through. Disorder does not blur the bands of a periodic medium; for a long enough pile it removes them.
Fig. 5 Transmission of 60 glass plates against wavelength relative to the quarter-wave design wavelength: a perfectly regular pile (blue), with a stop band at the design wavelength and pass bands either side averaging 0.92, and a pile whose plates and gaps vary randomly by ±25 per cent (red), averaging 0.135, with sharp spikes where a chance resonance lets one colour through.

The comparison with the ordered stack is the clearest way to see what disorder does. A regular pile of quarter-wave plates has a stop band centred on its design wavelength, where it reflects almost everything, and pass bands either side where its transmission rings close to one: the wave’s phase advances by the same amount through every period, and the Bloch waves the gap a repeat opens described carry the light through without loss. Randomise the thicknesses and the pass bands vanish. What is left is a forest of narrow transmission peaks at chance wavelengths, each a resonance through some favourable stretch of the pile, and a background that falls exponentially with the number of plates. It is not a blurred version of the periodic spectrum. The pass band, which depended on order to carry the wave, has gone.

That makes the contrast with the sandbars that reflect the sea and the light that spreads in two beams worth stating directly: in a periodic structure, waves in an allowed band travel as far as the structure goes, and in a disordered one, every wave stops within a localisation length. One-dimensional disorder is a perfect insulator for every frequency.

A silver that nature makes from disorder

The result has a use that evolution found first. Many fish are silver — herring, sardines, the sides of most open-water species — and the silver is not a metal. It is a stack of thin crystals of guanine, a transparent substance of high refractive index, separated by layers of cytoplasm, in the skin and scales. A regular stack of such layers would reflect one colour strongly and look coloured, as the regular multilayers in a peacock’s feather or a beetle’s wing case do, the coherent reflection the mirror that works from every direction designed. The fish’s stacks are irregular: the crystal and cytoplasm thicknesses vary from layer to layer over a wide range.

By the argument here, an irregular stack reflects every wavelength that it is long enough to localise, and with enough layers that is every visible wavelength at once. The fish is a broadband mirror built out of disorder, reflecting white light as a silver surface and taking on the colour of the water around it, which is the best camouflage open water allows. The design choice that a mirror maker would avoid — layers of random thickness — is the one that makes it work for all colours.

Why a laser can see it and a lamp cannot

The figure’s pile shows localisation because each calculation uses a single wavelength. A lamp emits many, and why two lamps never interfere found that light of a spread of wavelengths keeps a fixed phase only over a coherence length inversely proportional to the spread. Within the pile, the waves that loop back and forth travel extra distances of many plate thicknesses. If those extra paths are longer than the coherence length, the loops add as intensities and Stokes’s sum is restored; only if they are shorter does the interference survive and the light localise. Sunlight’s coherence length is about a micrometre, a pile of window glass’s plates are millimetres apart, and every pile of window glass that has ever been stacked has obeyed Stokes.

A laser, with a coherence length of metres or more, sees the pile as the calculation does, and so do radio waves in a layered ground, sound in a layered sediment and electrons in a wire cold enough that their phase survives from one end to the other. That last condition is the one that made localisation a problem in electronics. A thin metal wire at room temperature has an ordinary resistance, because its electrons lose phase by colliding with vibrating atoms every few nanometres; cooled to a fraction of a kelvin, where phase survives over micrometres, a sufficiently thin wire’s resistance begins to rise exponentially with its length instead of linearly, as the electrons’ waves localise in the impurities’ random potential exactly as the light does in the plates.

What the plates leave out

Light is not always single-coloured. A real light source has a spread of wavelengths, and each colour has its own random set of phases. For light whose spectrum is broader than the spacing of the pile’s chance resonances, the colours average, and the transmission returns towards Stokes’s diffusive value. Localisation shows up only for light coherent enough to keep its phases through the whole pile.

The plates do not absorb. Real glass absorbs slightly, and absorption shortens all paths, including the long ones that carry transmission through a localised pile. Separating localisation from absorption in an experiment is notoriously hard, because both make transmission fall exponentially with thickness; the distinction is in how the fluctuations behave, which absorption does not enhance.

One dimension is special. In two dimensions every state is still localised, but the localisation length grows exponentially as the disorder weakens, and for weak disorder it can exceed any sample. In three dimensions there is a threshold: weak disorder leaves waves diffusing and only disorder above a critical strength localises them, the Anderson transition.

Still open: localising light in three dimensions

Anderson’s original prediction was for electrons, and in one and two dimensions localisation of light, sound and matter waves has been demonstrated cleanly: in disordered arrays of optical waveguides, in random photonic structures, and in clouds of cold atoms released into speckled laser light. In three dimensions it has been seen for sound waves in disordered networks of aluminium beads and for matter waves of cold atoms. For light in three dimensions it has not been convincingly demonstrated. Claims made in the 1990s and 2000s in powders of strongly scattering semiconductor particles turned out to be explained by absorption or by fluorescence, and simulations have since shown that in a dense random packing of point-like scatterers, light’s vector nature — the near-field coupling between neighbouring scatterers through the longitudinal part of the field — may prevent the transition altogether.

Whether three-dimensional Anderson localisation of light can be reached in any real material, perhaps with scatterers designed to suppress that coupling, or whether it is ruled out by the vector character of electromagnetic waves, is not settled. The one-dimensional case leaves no such doubt. A pile of glass plates with random spacings, long enough, stops every colour of light that keeps its phase, and the reason is not that the disorder scrambles the waves but that it does not: every reflection is kept, and the reflections that return together always agree.

Part 8 of 8

This essay is one argument about Periodic media. The others:

Light from one guide, spreading through a row of them. The power in each of a row of identical coupled waveguides after light launched into the central one has travelled 1.5, 3, 5 and 7 coupling lengths (1/C) along the row, drawn as stacked bars with the distance increasing upward: |Jₙ(2Cz)|², where n counts guides from the centre. The light does not spread into a single widening hump as it would in a uniform slab. It forms two beams that run outward at a constant rate, carrying most of the power in their leading edges near n = ±2Cz, and leaves the centre nearly dark: after 7 coupling lengths the central guide holds 2.9 per cent and the brightest guides, at ±12, hold 8.1 per cent each. This is discrete diffraction: the spreading of light that can only step between neighbouring sites. The light that spreads in two beams Part 7 — Shine light into one of a row of identical optical fibres laid side by side and close enough to leak into each other, and it does not spread out into a widening glow as it would in a slab of glass. It splits into two beams that run outward at a fixed speed, bright at their leading edges, leaving the middle almost dark. The reason is that light hopping between neighbours has a fastest possible sideways speed, and piles up at it. The same periodicity gives the row two tricks no uniform medium has: a direction in which a beam travels without spreading at all, and a direction in which diffraction runs backwards, so that a spread beam can be gathered up again.

Where this goes next

The frequencies a repeat will not carry. The band structure of a medium made of quarter-wave layers of index 1 and 2, repeated for ever: frequency against Bloch phase across one cell, in the reduced zone. Inside a band the phase runs from 0 to π and the wave travels. Between bands there is no real phase at all, and the shaded strips are frequencies at which the medium supports nothing — not a weakly transmitted wave, no wave. Gap 1 runs from 0.784 to 1.216; Gap 2 runs from 2.784 to 3.216, in units of the quarter-wave design frequency. The first, measured off the drawn band edges, is 0.4327 wide against the 0.4327 of (4/π)·arcsin|r| — the same number computed from the Fresnel ratio of the two indices alone, agreeing to 2.6e-14 per cent. Every band edge sits where the phase is 0 or π, which is to say where the wave's own period fits the repeat a whole number of times: the gap is a property of the periodicity, and the materials only decide how wide it is. The gap a repeat opens Stack two transparent materials in alternating layers and there is a band of frequencies the stack will not carry — not weakly, not with loss, but not at all. Nothing has been absorbed and neither material has a resonance there. What forbids those frequencies is the repeat itself, and the width of the band has a closed form containing only the ratio of the two indices. 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. Light from one guide, spreading through a row of them. The power in each of a row of identical coupled waveguides after light launched into the central one has travelled 1.5, 3, 5 and 7 coupling lengths (1/C) along the row, drawn as stacked bars with the distance increasing upward: |Jₙ(2Cz)|², where n counts guides from the centre. The light does not spread into a single widening hump as it would in a uniform slab. It forms two beams that run outward at a constant rate, carrying most of the power in their leading edges near n = ±2Cz, and leaves the centre nearly dark: after 7 coupling lengths the central guide holds 2.9 per cent and the brightest guides, at ±12, hold 8.1 per cent each. This is discrete diffraction: the spreading of light that can only step between neighbouring sites. The light that spreads in two beams Shine light into one of a row of identical optical fibres laid side by side and close enough to leak into each other, and it does not spread out into a widening glow as it would in a slab of glass. It splits into two beams that run outward at a fixed speed, bright at their leading edges, leaving the middle almost dark. The reason is that light hopping between neighbours has a fastest possible sideways speed, and piles up at it. The same periodicity gives the row two tricks no uniform medium has: a direction in which a beam travels without spreading at all, and a direction in which diffraction runs backwards, so that a spread beam can be gathered up again.

The objects named here

The third axis, after the field and the reading path: the things themselves, and every essay that touches each one.

Anderson localisationDisorderInterferenceLocalisation lengthMultiple scatteringPhotonic crystalRandom walkTransfer matrix