The gap a repeat opens

A mirror made of glass and metal reflects because the metal will not admit the light. A mirror made of two transparent materials, neither of which reflects more than a few per cent on its own, can reflect ninety-nine point nine nine per cent — and it does so over a band of frequencies with edges, outside which the same object is nearly clear.

The shaded strips are the subject. They are not regions of high loss or of strong reflection; they are regions in which the wave equation has no travelling solution, and the reason is the repeat rather than either material.

What a repeat does to a wave equation #

In a uniform medium a wave is $e^{i(kx - \omega t)}$ and any frequency will do.

In a uniform medium nothing distinguishes one wavelength from another: the dispersion relation is a straight line and the spectrum is unbroken. That is the baseline the whole essay departs from, and it is worth stating because the departure is caused by the medium rather than by the wave. Impose a repeat and the straight line acquires breaks — at particular wavelengths, decided by the repeat and by nothing about what is travelling.

In a medium that repeats with period $a$, Bloch’s theorem says a solution is a periodic function multiplied by $e^{iKx}$. All the physics is then in one number per unit cell — the trace of the matrix that propagates the wave across it — and the condition for a travelling solution is

$$\cos(Ka) = \tfrac12\,\mathrm{Tr}\,M(\omega).$$

Where the right-hand side lies between $-1$ and $+1$ there is a real $K$ and the wave travels. Where it does not, $K$ is complex and the solution grows or decays instead of oscillating. That is the whole of the phenomenon, and it needs neither absorption nor resonance to produce it.

Where a gap is allowed to open #

The clearest way to see why gaps appear where they do is to switch the periodicity nearly off.

The branches meet because two different travelling waves — one going right with wavenumber $k$, one going left with $k - 2\pi/a$ — arrive at the same frequency. At such a point the medium can no longer choose between them, and any coupling however weak must mix them. The two mixtures are standing waves.

At a zone boundary there are two standing waves of the same wavelength, differing by a quarter of one in position, so that one puts its antinodes where the other puts its nodes. Those two see different averages of the potential — one concentrates where the potential is low and the other where it is high — so they have different energies, and the gap is that difference. Nothing else is happening.

One of those standing waves concentrates its field in the high-index material and one in the low, and they therefore have different frequencies. That difference is the gap. Nothing else about the medium is disturbed, because everywhere else the two travelling waves have different frequencies and a weak coupling between states of different energy does almost nothing.

This is exactly the argument that produces an energy gap for an electron in a crystal, and it is worth seeing the two side by side.

Two wells make two levels out of one, and N wells make a band, which is the same statement arrived at from the opposite end. Bands can be built up from isolated levels being brought together, or cut out of a free spectrum by imposing a repeat, and the two pictures meet in the middle. Which is more natural depends on whether the electrons are more nearly bound or more nearly free.

How wide, and what decides it #

For a stack of quarter-wave layers the gap has a closed form:

$$\frac{\Delta f}{f_0} = \frac{4}{\pi}\arcsin |r|, \qquad r = \frac{n_2 - n_1}{n_2 + n_1},$$

with $r$ the Fresnel ratio of the two indices — the amplitude reflected at a single interface between them.

Two readings matter. The gap is linear in the single-interface reflectivity for small contrast, which is the perturbation result and is why a weak modulation still forbids a real range of frequencies. And it saturates: an index ratio of 4.5 opens a gap only twice the one a ratio of 2 opens, and no pair of materials whatever can open a gap wider than the mid-gap frequency itself. That ceiling belongs to the arcsine and it is why a stack of this kind is made from many weak reflections rather than a few strong ones.

The individual reflections are worth looking at, because they are the ingredient the whole structure is built from.

A pulse meeting a change of impedance is partly transmitted and partly reflected, with the sign of the reflection depending on which way the impedance steps. Each interface in a stack does that, and the gap is what happens when all the reflections arrive back in phase — so the gap’s width is set by how strong each reflection is, which is the contrast between the two media.

Inside the gap, where there is no wave #

At a frequency in the gap the Bloch phase is $\pi + i\kappa a$, so the field falls by $e^{-\kappa}$ per cell.

The exponential is in the number of cells rather than in the amount of material, and that distinction is the practical one. Doubling the thickness of a stack while halving its period does not improve it; adding cells does. A dielectric mirror is a counting device.

There is a family resemblance here worth naming. A wave in a gap decays exponentially with distance and carries no energy away, which is precisely what a wave does beyond a boundary it cannot cross, and what a quantum particle does inside a barrier. In every case the wavenumber has become imaginary and the same arithmetic follows, and in every case the medium is doing no work. The wall a particle crosses without having the energy to climb it is this calculation with different symbols, and its transmission is the same exponential in the same kind of thickness.

The three numbers a stop band is described by #

It is worth collecting what the arithmetic has produced, because three numbers describe the whole of a one-dimensional stop band and each depends on something different.

Where it is is set by the period alone. A quarter-wave stack has its first gap centred on the frequency whose quarter-wavelength is the optical thickness of each layer, and gaps at odd multiples of that; the even ones are closed, exactly, because at those frequencies the two layers’ reflections cancel. Neither material’s index enters.

How wide it is is set by the index ratio through (4/π)·arcsin|r|, and by nothing else — not by the absolute indices, not by the thicknesses beyond their being quarter-wave, not by the number of layers.

How deep it is is set by the number of cells, through e^(−2Nκ). This is the only one of the three an engineer can improve without changing materials, and it improves exponentially, which is why a good dielectric mirror has thirty layers rather than three hundred.

Separating the three is what the infinite-medium calculation buys. A finite stack’s reflectance curve mixes all three together and it is not obvious from looking at one which feature belongs to which cause. The same separation appears in a two-layer antireflection coating, where the position of the null is a thickness and its depth is an index match, and confusing them produces a coating that is excellent at the wrong wavelength.

The finite stack, which is the object anybody builds #

An infinite crystal has bands; a real mirror has a spectrum.

A finite stack shows the edges arriving before the middle does. Two quarter-wave pairs already put the band edges in place and reflect weakly between them; eight pairs reflect almost perfectly across the whole band. So the gap’s position is a property of the repeat and its depth is a property of how many repeats there are — which is why a thin crystal has the same band structure as a thick one and transmits far more through it.

That is the observable signature of the argument. If the reflectivity came from accumulating small reflections with no band structure behind it, the whole curve would rise together as layers were added. It does not: the edges are fixed from the start.

No length in the problem #

There is one further property, and it is the reason a figure like the band structure is worth drawing in ratios rather than in nanometres.

The same figure therefore describes a dielectric mirror at 550 nanometres, an acoustic filter in a pipe at a metre, a phononic crystal, a fibre Bragg grating and a layered seismic bed at a kilometre. Nothing has to be re-derived to move between them; only the units on the horizontal axis change. Very few results in wave physics are scale-free in this way, and the ones that are tend to be the ones about geometry rather than about materials.

What the edges of a band do to a pulse #

Frequencies just inside a band are allowed, and they travel strangely.

Group velocity is the slope of the dispersion relation, and at a band edge that curve meets the zone boundary flat. So the group velocity goes to zero there: a pulse tuned to a band edge does not travel, it sits and spreads — a packet whose components disagree about speed spreading with nowhere to go. That is the practical consequence of the flatness, and it is exploited deliberately — slow light in a photonic crystal is a pulse parked at a band edge.

Standing still at a band edge is the same statement as a note that fits a boundary — what travels is a shape, and a shape that reflects back and forth is not travelling — and it shades continuously into the standing waves that opened the gap in the first place.

That is where slow light comes from, and where the enhanced emission near a band edge comes from: a mode that is not going anywhere spends a long time in the material. It is also the practical limit on a mirror’s bandwidth, because the group velocity dispersion that goes with a flat band distorts any short pulse the mirror reflects.

The oldest instance, and the newest #

The argument is old enough that it has been discovered independently in at least four subjects, and the names are worth collecting because they conceal how much the four have in common.

Rayleigh, in 1887, worked out the reflection from a stratified medium and found the band — the first appearance of the idea, sixty years before anyone had a use for it. Kronig and Penney, in 1931, solved the same equation for an electron in a periodic potential and got the electronic band gap that decides whether a solid conducts. Brillouin gave the folding construction its name in the 1930s and applied it to lattice vibrations. And the phrase photonic crystal, and with it the deliberate manufacture of structures with gaps, arrived only in 1987.

The gap between Rayleigh and the deliberate use of what he found is the interesting one. Nothing was missing from the physics; what was missing was the habit of treating the structure as the thing to be designed rather than the material. A stack of two ordinary transparent solids does something neither of them does, and the something is a property of the arrangement.

That is the same reversal as the one that made metamaterials a subject — and the same limitation applies to both. The arrangement can only rearrange what the materials already have. It can forbid a frequency, slow a wave and steer it, and it cannot make a lossless material out of a lossy one or a strong index contrast out of two things that are nearly the same.

The colours that do not fade #

The scale-invariance figure said the same band structure describes an optical mirror and a seismic bed. It also describes a great deal of biology, and the consequences are visible without an instrument.

A Morpho butterfly’s wing is blue, and there is no blue pigment in it. The scales carry ridges built of alternating layers of cuticle and air, spaced a couple of hundred nanometres apart, which is a quarter-wave stack of about eight periods with an index contrast of 1.56 to 1. Everything in this essay applies: there is a stop band, its position is set by the spacing, its width by the contrast, and its depth by the number of layers. Blue is reflected because blue is the band, and the rest of the spectrum passes through to a layer of melanin behind, which absorbs it and makes the blue look brighter by removing everything else.

A peacock’s barbules do the same thing in two dimensions, with rods of melanin arranged in a lattice inside keratin; changing the lattice spacing by a few tens of nanometres between one part of a feather and another is what produces the blues, greens and bronzes on one bird from one material. And opal is a three-dimensional version — silica spheres a few hundred nanometres across, packed regularly, with the colour coming from the packing rather than from the silica.

Two properties follow directly from the physics and distinguish structural colour from pigment at a glance.

It is iridescent. The stop band shifts to the blue as the angle of view increases, for exactly the reason a tilted dielectric filter does, so the colour changes with the direction it is seen from. Pigment does not do this: a molecule absorbs the same wavelengths whichever way it is lit.

And it does not fade. A pigment loses its colour when the molecule that absorbs is broken by light, oxygen or heat. A structure has nothing to break — the colour is in the arrangement, and the arrangement is as durable as the material. Museum specimens of Morpho collected in the nineteenth century are as blue as they were, beside pigmented insects from the same drawers that have gone brown.

That distinction is now being copied. Structural colour without pigment is being engineered into paints, textiles and displays for exactly those two reasons — a colour that cannot bleach, and one that can be changed by altering a spacing rather than a chemistry.

The same condition, written by Bragg #

The centre of the first stop band sits where each layer is a quarter of a wavelength, so the optical period of the stack is half a wavelength. Written as a condition on the period rather than on the layer, that is

$$2d = m\lambda,$$

which is Bragg’s law at normal incidence. The stop band is first-order Bragg reflection, and everything in this essay is a statement about it.

Making that identification is worth doing because it puts X-ray crystallography inside the same formula. A crystal is a periodic medium with a period of a few tenths of a nanometre, and X-rays of comparable wavelength meet the Bragg condition at accessible angles. The reflection from a set of atomic planes is the same phenomenon as the reflection from a dielectric mirror, at a scale ten thousand times smaller.

What differs is the width, and the essay’s own formula predicts it. The refractive index of a solid for X-rays differs from one by about a part in a hundred thousand, so the single-interface reflectivity $|r|$ is of that order, and $(4/\pi)\arcsin|r|$ gives a fractional bandwidth of a few parts in a hundred thousand. A crystal’s stop band is a few arcseconds wide where a dielectric mirror’s is twenty per cent.

That extreme narrowness is not a defect; it is what makes a crystal useful. A silicon crystal used as a monochromator on a synchrotron beamline selects a wavelength to a part in $10^4$ or better, from a white beam, using nothing but the same band-gap physics with a very small contrast. The width of the reflection curve has a name — the Darwin width — and it is measured routinely, and it is the arcsine formula evaluated at a very small argument.

The same reading turns a periodic medium into a sensor. A fibre Bragg grating is a weak periodic index modulation written into the core of an optical fibre by ultraviolet light; its contrast is around one part in ten thousand, so its stop band is about a tenth of a nanometre wide at a wavelength of 1550. Stretch the fibre and the period grows, so the reflected wavelength moves — by about a picometre per microstrain, and by ten picometres per kelvin. Thousands of such gratings can be written along one fibre at different periods, each reporting the strain and temperature where it sits, which is how a bridge, a dam or an aircraft wing is now instrumented along its whole length with a single strand of glass.

Where the model stops #

The medium is one-dimensional. A gap in one direction is not a gap; a real three-dimensional structure can have a stop band for waves travelling along one axis and none at all for waves travelling at an angle, and a complete gap — one that forbids a frequency in every direction and polarisation — requires a much larger index contrast than anything drawn here and is hard to arrange. Everything above is the easy case.

Normal incidence is assumed throughout. At an angle, the phase across a layer picks up a cosine and the two polarisations behave differently, so a stop band shifts to the blue and splits as the angle grows. Anybody who has tilted a dielectric filter has seen this.

The layers are lossless and the indices are constant. Real materials absorb a little and disperse, and in the ultraviolet the second effect is not small; the design frequency is then a function of itself and the stack has to be solved iteratively.

Only two materials, and only two thicknesses. A quarter-wave stack is the simplest periodic medium and not the best one for every purpose; a chirped stack, whose period varies slowly along its length, has a much wider band because different depths reflect different frequencies, at the cost of a delay that depends on colour. Nothing in the Bloch argument covers a structure whose period is not constant, because there is then no cell for the theorem to apply to.

And the structure is perfect. A random error in the layer thicknesses fills the gap in from the edges and eventually destroys it, and the sensitivity grows with the number of cells — which is the practical ceiling on how deep a stop band can be made, rather than any limit in the arithmetic.

What the pictures cannot show #

The band structure is drawn as a set of curves, and a curve implies that a state exists at every point on it. In a finite stack it does not: there are as many states in a band as there are cells, and the band is a set of closely spaced levels rather than a continuum. Everything about the drawing is right in the limit and misleading about a mirror eight layers thick.

Nor can any of these figures show what the field looks like inside the gap. The transmission curve reports what emerges; the field inside the stack is a standing pattern decaying into the structure, with most of its energy in the first two or three cells, and a drawing of it would make it much clearer why the number of cells stops mattering after about ten.

Where this ladder goes next #

What has been established is that periodicity alone forbids frequencies. The medium need not resonate, need not absorb and need not be strongly modulated; it need only repeat, and the repeat picks out the frequencies whose half-wavelength fits it a whole number of times.

The habit worth carrying away is the degenerate-perturbation move, which is one of the most reliable in physics. A weak coupling does nothing except where two states already agreed, and there it must split them. That single sentence explains the band gap, the avoided crossing of two coupled oscillators, the splitting of atomic levels in a field, and why two identical pendulums joined by a spring will not stop swapping — in every case the interesting physics lives at a degeneracy and nowhere else.

What is left on this ladder is what happens when the repeat is broken on purpose: a single defective layer in an otherwise perfect stack creates a state inside the forbidden band, localised at the defect, and that state is the basis of every filter, laser cavity and waveguide built out of a periodic medium.