Illustrated Physics
physics, drawn
Waves

The mirror that works from every direction

A stack of alternating transparent layers reflects nearly all the light of one colour arriving straight on, and less as the light tilts, because tilting moves the forbidden band. A structure that repeats in only one direction ought therefore to be a mirror for only a range of directions. It is not, if the layers differ enough: light arriving from air cannot bring enough sideways momentum to escape the forbidden band at any angle, for either polarisation, and a flat stack of plastic and tellurium reflects every angle over a band of frequencies almost half as wide as its centre.
15 min read 5 figures The shape decidesOnly some values fit

Assumes: The gap a repeat opens · The cone light has to find to get out

The gap a repeat opens is a band of frequencies that a stack of alternating transparent layers will not carry. Inside it the stack is a mirror, and a good one: a few dozen layers of two ordinary dielectrics reflect more of the light than polished silver does, and absorb none of it, which is why the mirrors inside lasers are built that way rather than from metal.

The gap was computed for light arriving straight on. Tilt the light and the gap moves. The layers present a longer path to light crossing them obliquely in some respects and a shorter one in others, the condition for the reflections to add up shifts to higher frequencies, and a mirror built for red light at normal incidence reflects orange at an angle. Every multilayer reflector in nature shows this: the colours of a butterfly’s wing or a beetle’s shell change as they turn, because the band they reflect slides along the spectrum with the angle.

That suggests a limit. A structure that repeats in only one direction has a gap that depends on direction, so it should be a mirror for some directions and not others, and a mirror for every direction should need a structure that repeats in all three — a photonic crystal. In 1998 a group at MIT showed that for light arriving from air the suggestion is wrong, and the reason is not in the stack. It is in the air.

A gap that every angle from air falls into. The stop bands of a stack of quarter-wave layers of index 4.6 and 1.6 — tellurium and polystyrene, the pair of the first such mirror — against frequency, in units of the design frequency, and the parallel index β = n₀ sin θ₀ the light brings along the layers; TE polarisation on the right, TM on the left. Shaded regions are gaps. Light from air can only have β between −1 and 1, the vertical lines; within those lines the gaps for both polarisations overlap between f = 0.848 and 1.321, the band marked across the figure, so every angle of incidence and both polarisations are reflected: a relative width of 43.6%. The TM gap narrows as β grows and closes at the internal Brewster index, 1.51, which light from air cannot reach.
Fig. 1 The stop bands of a quarter-wave stack of indices 4.6 and 1.6 — tellurium and polystyrene — against frequency and the parallel index β = n₀ sin θ₀; TE on the right, TM on the left. Light from air has β between −1 and 1. Inside those lines the gaps for both polarisations overlap between 0.848 and 1.321 of the design frequency: every angle and both polarisations are reflected.

The quantity the layers cannot change

A stack of flat layers is the same everywhere along its surface, and that symmetry fixes one property of the light passing through it: the component of its wavevector along the layers. That is Snell’s law in its most useful form. The product nsin⁡θn \sin\theta is the same in every layer, in the medium the light came from and in the medium it leaves into, and it can be written as a single number for the whole stack, the parallel index β=n0sin⁡θ0\beta = n_0 \sin\theta_0.

The cone light has to find to get out used the same conservation from the other side: light inside a dense material with β\beta larger than one has no angle in air to go to, and is trapped by total internal reflection. Here the constraint runs the other way. Light arriving from air has an angle, so its β\beta is at most one, whatever the angle. Grazing incidence, the most oblique light there is, brings β=1\beta = 1 and no more.

That turns a question about every angle into a question about a finite range of one number. The figure plots, for each β\beta, the frequencies the stack forbids. For light arriving from air only the strip between the two vertical lines at β=±1\beta = \pm 1 can ever be reached. Everything outside the strip is real physics — light inside a prism, or in the evanescent tail of a guided wave, can have those values — and none of it matters for a mirror in air.

Two polarisations, two sets of gaps

At normal incidence the two polarisations of light are the same thing, and any state of polarisation behaves alike. At an angle they are not. Light with its electric field parallel to the layers, the TE polarisation, and light with its magnetic field parallel to them, TM, meet each interface with different reflection coefficients, and a stack’s gap depends on how strongly each interface reflects. So each polarisation has its own gap, and the figure draws TE on the right and TM on the left, mirror images at β=0\beta = 0 where they coincide.

Both gaps move up in frequency as β\beta grows, for the reason the butterfly’s colours shift. The stack is the continuous counterpart of the chain of masses in the frequency a lattice cannot carry, with layers in place of masses and a stop band in place of a highest frequency, and tilting the light changes the effective spacing the wave sees. What differs is their width. The TE gap stays wide, and even widens a little, because the reflection at each interface for that polarisation strengthens with angle. The TM gap narrows, because for that polarisation there is an angle at which an interface does not reflect at all.

The omnidirectional window is the set of frequencies that lie inside both gaps for every β\beta from zero to one. In the figure it is the band between 0.848 and 1.321 of the design frequency: the top is set by the gap at normal incidence, where it is highest, and the bottom by the TM gap at grazing incidence, which has moved furthest up. The window’s width relative to its centre is 43.6 per cent. Across that whole range, every direction of incoming light and both polarisations lie in a gap.

Why a high index keeps the gap still

The upward shift has a simple origin, and it says why tellurium matters. What sets a stack’s gap is the phase each layer adds to light crossing it, and that phase depends on the component of the light’s wavevector perpendicular to the layers, which is n2−β2\sqrt{n^2 - \beta^2} times the free-space wavenumber. At normal incidence it is nn; tilted to grazing incidence from air it is n2−1\sqrt{n^2 - 1}. To keep the same phase, and so the same position in the gap, the frequency has to rise by the ratio of the two, n/n2−1n/\sqrt{n^2-1}.

For the polystyrene layers that ratio is 1.281: a 28 per cent rise. For the tellurium layers it is 1.025, barely 2 per cent, because light entering a layer of index 4.6 is bent so close to the normal that it crosses the layer almost straight whatever angle it arrived at. Snell’s law bends light toward the normal in a dense medium, and a very dense layer is nearly blind to the angle outside. The stack’s gap shifts by a weighted compromise between its two layers, so a high-index layer anchors the gap and a low-index one drags it; the more the high index dominates, the less the gap moves and the more easily the gaps at different angles overlap.

Where the TM gap closes

Where the gap for one polarisation closes. The width of the lowest TM stop band against the parallel index β, for two stacks: indices 4.6 and 1.6, whose gap closes at β = 1.511, and 2.3 and 1.45, whose gap closes at β = 1.227. At that index the light meets every interface at Brewster's angle, the two layers present the same admittance, and the interfaces stop reflecting light polarised in the plane of incidence, so no gap can form. Light arriving from air brings at most β = 1, the dashed line, and both closings lie beyond it, which is the necessary condition for an omnidirectional mirror; whether one exists also depends on whether the gaps for all those β overlap.
Fig. 2 The width of the lowest TM stop band against β for two stacks: indices 4.6 and 1.6, whose gap closes at β = 1.511, and 2.3 and 1.45, whose gap closes at β = 1.227. There the light meets every interface at Brewster’s angle, the layers present the same admittance, and no gap forms. Light from air brings at most β = 1, and both closings lie beyond it.

Light polarised in the plane of incidence is not reflected at all when it meets an interface at Brewster’s angle, where the reflected and refracted rays would be at right angles. Inside a stack every interface is between the same two materials, so if the light meets one at Brewster’s angle it meets them all at Brewster’s angle, no interface reflects, and there is nothing from which a gap could form. The figure computes the TM gap’s width against β\beta and finds it closes exactly where that condition holds, at βB=nHnL/nH2+nL2\beta_B = n_H n_L/\sqrt{n_H^2 + n_L^2}.

For tellurium and polystyrene that is 1.511. For a pair like titanium dioxide and silica, 2.3 and 1.45, it is 1.227. Both are larger than one. Light from air would have to arrive at more than grazing incidence to reach them, which it cannot. This is the first condition for an omnidirectional mirror, and it is met by almost any pair of materials with indices above about 1.4, because βB\beta_B exceeds one whenever both indices do by enough.

It is not the only condition, which is why a Brewster closing out of reach does not by itself make a mirror omnidirectional. The gaps must also overlap: the TM gap at grazing incidence must not have moved up past the top of the gap at normal incidence. That depends on how much the two materials differ.

Every angle, and what happens just outside

Every angle reflected, and what happens just outside. The reflectance of a free-standing stack of 10 pairs of the same layers in air, against the angle of incidence, computed by multiplying the layers' characteristic matrices. At f = 1.084, the middle of the omnidirectional window, both polarisations stay above 99.99% at every angle from normal to 89°. At f = 0.764, inside the gap at normal incidence but below the window, both are reflected near normal incidence, but TM falls to 0.0% by 78° while TE stays above 99.9%, because the TM gap moves up in frequency and narrows as the angle grows, and leaves this frequency behind. A mirror that is perfect at normal incidence is not therefore perfect at all angles; one inside the window is.
Fig. 3 The reflectance of ten pairs in air against the angle of incidence. At 1.084 of the design frequency, in the middle of the window, both polarisations stay above 99.99% from normal to 89°. At 0.764, inside the normal-incidence gap but below the window, TE stays above 99.9% while TM falls to zero by 78°.

The band diagram describes an infinite stack. The reflectance of a real one, ten pairs of layers standing in air, is computed by multiplying the layers’ characteristic matrices, the same arithmetic the mode that lives in the mistake used for a stack with a defect. In the middle of the window both polarisations are reflected above 99.99 per cent at every angle the calculation tries, from normal incidence to 89 degrees. Ten pairs is enough because the contrast is large: each pair lets through only a small fraction of what reaches it, and the fractions multiply.

Just below the window the same stack is still inside its gap at normal incidence, and a test at normal incidence would call it a perfect mirror. The TE polarisation is still reflected at every angle. The TM polarisation is reflected near normal incidence and then, as the angle grows, its gap slides up past the frequency and narrows, and by 78 degrees the stack is transparent to it. A mirror specified at one angle is specified at one angle; only inside the window is it specified at all of them.

How much contrast is enough

The contrast a stack needs to reflect everything. The relative width of the omnidirectional window against the ratio of the two indices, with the lower index fixed at 1.6 and light arriving from index 1. Below a contrast of about 1.45 there is no window: the gaps exist at every angle but move too far to overlap. Above it the window opens and widens steadily, reaching 43.1% near the tellurium–polystyrene ratio of 2.87. Each point is the frequency range gapped for both polarisations at forty-one angles.
Fig. 4 The relative width of the omnidirectional window against the ratio of the two indices, with the lower index fixed at 1.6 and light from air. Below a contrast of about 1.45 there is no window. Above it the window opens and widens steadily, reaching 43.1% near the tellurium–polystyrene ratio of 2.87.

The width of the gap at normal incidence grows with the contrast between the two indices, and so does its tolerance for being moved. The figure fixes the lower index at polystyrene’s 1.6 and raises the higher one. Below a ratio of about 1.45 there is no window: every angle has a gap, but the gap at grazing incidence has moved too far to overlap the gap at normal incidence. Above it the window opens and grows, and at the ratio of tellurium to polystyrene it is 43 per cent wide.

That is why the first omnidirectional mirror used tellurium, a semiconductor with an unusually high refractive index in the infrared, and why it worked at wavelengths of about ten micrometres, where tellurium is transparent. At visible wavelengths the materials with high indices and no absorption are fewer, and omnidirectional windows are narrower, but they exist, and multilayer coatings designed to reflect over wide angles are routine.

The comparison with a metal is instructive. A metal reflects every angle at frequencies below its plasma frequency, without any window, because its free electrons screen the field whatever direction it arrives from. But a metal absorbs a few per cent at every reflection, and at high power or after many reflections those few per cent are the whole problem. A dielectric stack reflects every angle only in its window, and absorbs almost nothing there.

A dielectric stack has one more property in common with the chain in the end that knows how the middle was cut. Where a stack meets a different mirror — a metal film, or a second stack with its layers in the other order — light at a frequency inside the gap can be trapped at the interface, decaying into both sides, and whether such a surface state exists depends on how each stack’s repeat is terminated. These optical Tamm states are the photonic cousin of the chain’s end states, and they are one more sign that a one-dimensional repeat is not a simple object: its bulk and its boundary each carry information the other needs.

The same mirror, seen from denser media

The same mirror, seen from denser media. The relative width of the omnidirectional window of the 4.6/1.6 stack against the index of the medium the light arrives from. A denser medium lets light bring a larger parallel index, reaching further across the band diagram towards the places where the gaps move and close, so the window narrows: 43.6% from air, 15.0% from index 1.34, like water, and none at all from index 1.44 upwards. An omnidirectional mirror is omnidirectional for a particular outside medium, not in general.
Fig. 5 The relative width of the same stack’s omnidirectional window against the index of the medium the light arrives from. A denser medium lets light bring a larger parallel index, and the window narrows: 43.6% from air, 15.0% from an index of 1.34, like water, and none from 1.44 upwards.

The argument used one fact about air: that light arriving from it cannot bring a parallel index above one. Light arriving from water can bring up to 1.33, and from glass up to 1.5. In the band diagram the accessible strip widens with the index of the outside medium, and it reaches further towards the region where the TM gap climbs and narrows. The window shrinks from 43.6 per cent in air to 15.0 per cent from a medium like water and closes altogether from an index of 1.44 — before the strip reaches the TM Brewster point at 1.51, because the gaps stop overlapping first.

An omnidirectional mirror is omnidirectional with respect to a particular outside medium.

That is the precise sense in which one dimension of repetition is enough and not enough. A structure that repeats in one direction controls the component of the light’s wavevector along that direction and leaves the other two alone; it can forbid every wave only if something else limits those other two components, and the outside medium is what does. A structure that repeats in two or three directions controls more components itself, and needs no help from outside, which is why a gap for every direction inside a material needs a crystal and a gap for every direction arriving from air does not. Put the same stack in contact with glass and it has directions of transmission again. That is also why the result does not contradict the need for three-dimensional photonic crystals: a light source embedded inside a dense material, which can emit at any parallel index at all, can always find a direction the one-dimensional stack will pass.

The practical use of the idea runs through this condition. A hollow fibre whose core is air and whose wall is a cylindrical omnidirectional mirror guides light by reflection alone, at every angle the light strikes the wall, and fibres built this way deliver the ten-micrometre beam of carbon dioxide lasers for surgery, a wavelength that solid glass fibres absorb. The core is air, which is what makes the mirror omnidirectional; a fibre that relies on total internal reflection has to do the opposite, keeping its light in the denser material.

Where the model stops

The indices do not depend on frequency. Real materials are dispersive, and across a window 43 per cent wide their indices change; the window of a real stack has to be recomputed with measured indices, and it shifts.

Nothing absorbs. The reflectance above 99.99 per cent assumes lossless layers. Any absorption caps it, and in a thick stack light that tunnels partway in and back out samples the absorption many times.

The layers are quarter-wave at normal incidence. That design maximises the normal-incidence gap. Other thickness ratios can widen the omnidirectional window by trading some of the gap at normal incidence for better overlap at grazing incidence, and practical designs are optimised numerically.

And the stack is finite. Near a gap edge the decay per pair is weak, and a few pairs let some light through by tunnelling. Light can also be injected into the stack at a parallel index above one from a prism placed close against it — the frustrated total internal reflection of the reflection that happens where the glass is not — and an omnidirectional mirror offers no protection against that route.

What the pictures cannot show

The band diagram uses one number, the magnitude of the parallel index, for every direction of arrival. That is correct because the stack is the same in every direction along its surface, so light arriving from the north at thirty degrees and from the east at thirty degrees see identical stacks. The drawing’s single axis stands for a whole circle of directions and cannot show it.

Nor can a reflectance curve show where the light goes inside the stack. At a frequency in the window the field decays into the layers within a few pairs and never reaches the far side; just outside it, the field penetrates the whole stack as a travelling wave. The two curves look alike at normal incidence and describe completely different fields.

Still open: a complete gap for visible light, made at scale

A one-dimensional stack is omnidirectional only for light from a low-index medium. A gap for light travelling in every direction inside a material requires a three-dimensional structure, and the structures predicted to give the widest complete gaps — arrangements based on the diamond lattice, and inverted opals of high-index material — are demanding to make with features a few hundred nanometres across and few enough defects to matter.

Some beetles and weevils grow such structures in their scales, with diamond-like networks of chitin that reflect strongly over wide ranges of angle and give colours that barely change as the insect turns — remarkable for a material whose index is only about 1.56. Whether synthetic methods can match them at useful sizes, and whether disordered materials designed to suppress density fluctuations at long wavelengths — which have been shown to have complete gaps without any periodicity at all — can outperform crystals for making light-trapping structures, are active questions rather than settled ones. How much order a gap really requires is not yet known in general.

The habit worth keeping is the one the strip between the two vertical lines teaches. Before asking whether a structure can do something for every direction, ask what the incoming wave is able to bring. The stack cannot reflect every parallel index, and it does not need to: air can only supply a strip of them, and the mirror has to work only there.

Part 5 of 5

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

Two states at zero that only one side of the chain has. Every energy level of a chain of 20 cells, 40 sites, whose couplings alternate between v inside a cell and w = 1 between cells, against the ratio v/w from 0 to 2. The bulk levels fill two bands, bounded by the dashed lines ±|v − w| and ±(v + w), with a gap between them that closes at v = w. For v < w two levels sit at zero energy, in the middle of the gap, drawn in the warning colour: at v/w = 0.5 they are within a millionth of the coupling of zero. For v > w the gap is empty — at v/w = 1.5 the nearest level to zero is 0.527. Nothing about the chain's middle distinguishes the two sides of v = w; the difference is at its ends. The end that knows how the middle was cut Part 4 — A chain whose links alternate, strong and weak, has the same bands whichever kind of link is counted as inside a cell. The infinite chain cannot tell the two choices apart. A finite chain can: cut it so that a weak link is outermost and each end holds a state at exactly zero energy, in the middle of the gap; cut it the other way and it holds none. What decides is not anything at the ends but a whole number counted from the bulk — how many times a loop winds round a point.

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. Two states at zero that only one side of the chain has. Every energy level of a chain of 20 cells, 40 sites, whose couplings alternate between v inside a cell and w = 1 between cells, against the ratio v/w from 0 to 2. The bulk levels fill two bands, bounded by the dashed lines ±|v − w| and ±(v + w), with a gap between them that closes at v = w. For v < w two levels sit at zero energy, in the middle of the gap, drawn in the warning colour: at v/w = 0.5 they are within a millionth of the coupling of zero. For v > w the gap is empty — at v/w = 1.5 the nearest level to zero is 0.527. Nothing about the chain's middle distinguishes the two sides of v = w; the difference is at its ends. The end that knows how the middle was cut A chain whose links alternate, strong and weak, has the same bands whichever kind of link is counted as inside a cell. The infinite chain cannot tell the two choices apart. A finite chain can: cut it so that a weak link is outermost and each end holds a state at exactly zero energy, in the middle of the gap; cut it the other way and it holds none. What decides is not anything at the ends but a whole number counted from the bulk — how many times a loop winds round a point. One line, 21 decades of density, and every mirror on it. Plasma frequency against electron density, both logarithmic, with the horizontal rules at frequencies a reader already has a feel for. A wave is reflected by everything to the right of where its own rule meets the line and passes through everything to the left. a fluorescent tube at 1.0e+17 m⁻³ cuts off at 2.84 GHz; ionosphere, D layer at night at 1.0e+8 m⁻³ cuts off at 90 kHz; ionosphere, E layer at 1.0e+11 m⁻³ cuts off at 2.84 MHz; ionosphere, F2 layer at 1.0e+12 m⁻³ cuts off at 8.98 MHz; aluminium's conduction electrons at 1.8e+29 m⁻³ cuts off at 3819.9 THz. So the F2 layer turns back a 1 MHz broadcast and lets a 100 MHz one straight out, which is why one of them is heard across an ocean at night and the other stops at the horizon; and aluminium's cutoff sits in the far ultraviolet, which is why it is a mirror for everything visible and a window above 78 nm. The dependence is a square root, so the line has slope one half: a hundredfold denser plasma reflects only ten times the frequency. Arriving at an angle helps by a factor of sec θ — 1.00 at 0°, 1.15 at 30°, 2.00 at 60° — because only the component of the motion along the density gradient has to be turned round. The frequency below which nothing gets in Free charges give a medium a permittivity that is negative, and a negative permittivity is not an absorbing medium — it is one in which no wave exists at all. Below that frequency the reflection is total, exactly rather than nearly, because there is no transmitted wave and nothing to absorb. The same expression puts the number at 9 MHz for the ionosphere and 3.8 PHz for aluminium.

What links here

Essays that reach for this one mid-argument — the half of a link its own author cannot write down.

The objects named here

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

Band gapBragg mirrorBrewster's angleLight coneOmnidirectional reflectionPhotonic crystalPolarisationSnell's lawThin film interferenceTransfer matrix