The mode that lives in the mistake

A stack of alternating quarter-wave layers has a band of frequencies it will not carry. The rung below this one computes it: the band’s width has a closed form containing only the ratio of the two indices, no length appears in the problem, and inside the band there is no propagating wave at all.

Now break the repeat. Insert one layer of the wrong thickness — a half-wave instead of a quarter — in the middle of the stack, and change nothing else.

Why exactly one #

The half-wave layer is a resonator. Light entering it, bouncing between the two mirrors either side and coming back, returns in phase with itself — the round trip is a whole wave — so the amplitude inside builds up until what leaks out of the far side equals what is coming in at the near side.

Only some frequencies fit between two ends, and the condition here is the same: a round trip must be a whole number of wavelengths. What makes exactly one mode appear in the gap is that the defect provides exactly one cavity, and the mirrors either side of it admit only frequencies inside the forbidden band — so the count is one because there is one round trip and one band, not because anything has been arranged.

The frequency it settles at is not the defect layer’s own business either. A half-wave layer on its own, in open air, is resonant at the design frequency and at every multiple of it; here it is resonant at only one of those, because the others lie outside the gap where the surrounding stack is transparent and there is nothing to bounce off. The state exists only where the crystal forbids propagation, and the crystal is what makes it a state rather than a passing feature.

The transmittance being exactly one, and not merely large, is a property of the symmetry. Two identical mirrors either side of a resonant cavity are the same arrangement as a Fabry–Pérot etalon, and at resonance the reflected waves from the two halves cancel exactly. Every photon that arrives goes through. It takes a long time about it.

What the two mirrors do to a wave arriving at them is the ordinary business of a boundary where the medium changes, repeated: each interface reflects a little, and the stack’s enormous reflectance is those little reflections arriving in step.

The two rates that turn out to be one #

The agreement in that last figure is the point of the essay, so it is worth stating in both directions.

Why the band is forbidden. Inside a gap, the Bloch phase is complex: a wave entering the medium decays instead of propagating, by a factor $e^{-\kappa}$ per cell, with $\kappa = \operatorname{arccosh}|\tfrac12\operatorname{Tr}M|$. That decay is what makes the stack a mirror.

Why the state is bound. The defect’s field has to fall off away from the defect, or it would carry energy to infinity. It falls off by decaying into the surrounding crystal, and the rate at which it can decay is the rate the crystal offers, which is the same $\kappa$.

So the two are not analogous. They are one evanescent wave, described from either side. A crystal with no gap has $\kappa = 0$, offers no decay, and can bind nothing; a crystal with a wide gap has a large $\kappa$ and binds tightly.

Inside a barrier the wave does not oscillate, it decays — and that is the same evanescent behaviour, met first in a quite different setting. Here it decides both rates: how fast the mode’s field falls away into the mirrors, which is its confinement, and how much leaks through, which is its loss. One decay length answers both questions, which is why they turn out to be the same number.

The arithmetic, which is a product of two-by-two matrices #

The whole calculation is worth showing, because it is the standard tool of the subject and it is elementary.

A layer of index $n$ and phase thickness $\delta$ relates the field and its derivative at one face to their values at the other by a two-by-two matrix whose entries are $\cos\delta$ on the diagonal and $\sin\delta$ divided or multiplied by $n$ off it. A stack is the product of one such matrix per layer, in order, and the transmittance follows from the four entries of the product by one line of algebra.

That is all. There is no approximation anywhere in it, no assumption that the layers are thin or the contrast small, and the same product handles a perfect stack, a stack with one wrong layer, a stack with a hundred wrong layers and a stack of randomly chosen thicknesses. The band structure of the rung below comes out of the same matrix by asking when its trace exceeds two in magnitude; the defect resonance comes out by asking where its transmittance is one.

The economy is worth noticing. One object — the product of a chain of two-by-two matrices — answers which frequencies propagate, how fast they decay when they do not, what a break in the chain holds, and how sharply it holds it. The same structure appears wherever a wave meets a sequence of similar things: a periodic waveguide, a chain of coupled pendulums, an electron in a one-dimensional potential, and a beam of light through a stack of lenses. When the matrices are all the same the answer is a band structure; when one differs it is this essay.

The cavity’s own condition — a round trip of a whole number of wavelengths — is the same one that allows a box only some energies, with a mirror of finite reflectance in place of an infinite wall. A finite wall is what makes the state a resonance rather than a bound state, and the difference is a lifetime.

The same thing in a solid #

The electronic version arrived first, is the more familiar, and is the same calculation.

Replace one silicon atom in a hundred thousand with a phosphorus atom and the extra electron sits in a state whose energy is inside the gap, 45 millielectronvolts below the conduction band, localised on the impurity over a few nanometres. Its wavefunction decays into the surrounding crystal at exactly the rate the gap allows, and the whole of semiconductor electronics is the deliberate placing of such states.

What the narrowing buys #

The quality factor of the resonance rises geometrically with the number of pairs — 400, then 6,410, then 100,000 in the figure — because the light has to leak out through a barrier whose transmission falls exponentially with its thickness.

A resonance’s sharpness is the reciprocal of its loss, and here the loss is a leak through a mirror rather than a friction. So adding mirror pairs sharpens the mode exponentially — every pair multiplies the confinement and divides the linewidth — which is why the quality factors of photonic-crystal cavities run to millions where a mechanical resonator struggles past thousands.

That relation is what every application is about. A vertical-cavity laser is a half-wave defect between two such stacks, and it lases at the defect frequency because that is the only frequency the cavity supports. A narrowband filter is the same object used in transmission. A photonic-crystal cavity is the two-dimensional version, and the reason the field figure’s peak matters is that a high field in a small volume is what a nonlinear interaction or a single-atom coupling needs.

The mirror on its own reflects moderately with two pairs, well with four and almost perfectly with eight. That is the comparison worth making: the mirror’s job is to be uninteresting across the band, and the defect’s job is to be interesting at one frequency inside it. The stack supplies the first and the mistake in it supplies the second.

Where the idea came from twice #

The optical and the electronic versions of this result were obtained independently, thirty years apart, and neither community noticed the other for a long time.

The electronic one came first, out of the semiconductor work of the late 1940s. Impurity levels in the gap were inferred from conductivity that rose with temperature in a way no perfect crystal could produce, and by 1950 the shallow donor and acceptor levels of silicon and germanium had been measured and understood as hydrogen-like states with the crystal’s dielectric constant and effective mass substituted. The transistor is a device for placing and moving such states, and it was working before the theory of them was complete.

The optical one was proposed in 1987, by Eli Yablonovitch and by Sajeev John in the same year and for different reasons — one wanting to suppress spontaneous emission by giving an excited atom no mode to emit into, the other wanting to localise light in a disordered medium. Both realised that a three-dimensional periodic dielectric could have a full gap, and that a defect in one would hold a state exactly as an impurity in a semiconductor does. The vocabulary followed the electronic case directly: donor and acceptor defects, shallow and deep states, the whole terminology imported.

What makes the parallel exact rather than suggestive is the equation. An electron in a periodic potential and a wave in a periodic dielectric obey the same one-dimensional problem with different constants, and every result about one transfers. The transfer matrix in this essay is the same object a solid-state text calls the Kronig–Penney matrix, and the state in the gap is the same state.

An atom with nowhere to emit #

The reason Yablonovitch went looking for a full gap was not to make a mirror or a filter. It was to stop an atom from radiating, and that motivation is worth following because it explains what the defect state is actually for.

An excited atom does not emit at a rate fixed by the atom. It emits at a rate proportional to how many modes of the electromagnetic field are available at its transition frequency and at its position — the local density of states. In free space that number is a smooth function of frequency and gives the familiar lifetime. Put the atom somewhere the density of states is different and the lifetime changes, and nothing about the atom has been touched.

Inside a full three-dimensional gap the density of states is zero. There is no mode for the photon to go into, in any direction, so an excited atom sitting there cannot decay by emitting one. Spontaneous emission, which every textbook presents as an irreducible property of an excited state, is suppressed by the surroundings.

The defect state is the other half of the same statement. Put the atom in the cavity instead, at the frequency the defect holds, and the density of states is not zero but enormous — one mode, in a very small volume, with a very long lifetime. The emission rate is enhanced, by the Purcell factor

$$F_P = \frac{3}{4\pi^2}\left(\frac{\lambda}{n}\right)^3\frac{Q}{V},$$

which is the ratio of quality factor to mode volume in units of a cubic wavelength. The eight-pair stack’s quality factor of $10^5$, in a mode volume of order a wavelength cubed, gives an enhancement of thousands.

So a photonic crystal with a defect is a device for controlling whether and how fast an atom emits, and for controlling where the photon goes when it does — because the one available mode is the cavity’s, and the cavity’s mode leaks out in a known direction. That is the whole basis of single-photon sources, of low-threshold lasers, and of the cavity quantum electrodynamics experiments that couple one emitter to one mode.

The one-dimensional stack computed here does a partial version of it. It suppresses emission along its axis and does nothing to emission sideways, so an atom in a half-wave defect between two stacks has a lifetime altered by a fraction rather than by orders of magnitude. Getting the full effect is why the three-dimensional structures are worth the enormous difficulty of making them.

Many flaws rather than one #

The transfer matrix handles a stack of randomly chosen thicknesses as easily as a periodic one, and it is worth asking what it says about that case, because the answer was the other 1987 motivation.

Make every layer’s thickness random. There is now no periodicity, so there is no band structure and no gap; by the reasoning of this essay there should be nothing to bind a state and nothing to forbid propagation. What actually happens is that transmission through a long enough disordered stack falls exponentially with its length, at every frequency.

This is Anderson localisation, and in one dimension it is not a threshold effect: any amount of disorder, however small, localises every state. A wave in a long random stack is not attenuated by absorption — nothing is absorbing — but by interference between the very many partial reflections, which almost always conspire to send the wave back. The rate of the exponential defines a localisation length, and a sample much longer than it is opaque.

The connection to the single defect is closer than it looks. A disordered stack transmits essentially nothing on average, and at particular frequencies it transmits almost everything — because somewhere inside it a few layers have accidentally arranged themselves into a resonator with mirrors either side, which is exactly the object this essay computes. Those rare transmission resonances are the same defect state arrived at by chance rather than by design, and when several of them line up in frequency and space the transmission is carried by a chain of them.

Sajeev John’s proposal was to take that mechanism into three dimensions with strong enough scattering to localise light in a disordered dielectric, in the same way electrons localise in a disordered solid. It is much harder than the periodic route and it has been achieved in some systems and disputed in others, because separating localisation from absorption in a real material is genuinely difficult.

What the disordered case adds to this essay is a caution about the word defect. The single flaw produces a sharp, predictable, engineerable state because everything around it is perfect. Many flaws produce a dense forest of states at unpredictable frequencies and positions, each one localised, and their collective effect is opacity rather than a filter. The value of a defect is inseparable from the perfection of its surroundings, and that is not a manufacturing preference — it is the same statement as the decay rate being set by the crystal’s own band structure.

What it costs #

The defect has to be the only one. Two defects at a distance couple, their states split into two, and the arrangement stops being a single resonance. That is a nuisance when a filter is wanted and is the mechanism when a waveguide is wanted: a line of defects makes a chain of coupled cavities along which light can travel, at frequencies the crystal around them forbids entirely.

Loss has been ignored throughout. Every index above is real. A real material absorbs a little, and absorption limits the quality factor independently of the mirrors — so past some number of pairs the line stops narrowing and the peak transmittance starts falling below one. Where that happens is what decides the design.

And the calculation is one-dimensional. A stack confines light in one direction and does nothing at all in the other two. A defect in a one-dimensional stack is not a trap: light leaks out sideways, and every real device has to confine the other directions some other way — by total internal reflection, by a two- or three-dimensional crystal, or by accepting the loss.

Where the model stops #

Nothing here says how a state gets in or out. The transmission calculation is for a steady state, and a real cavity is filled and emptied on a timescale of the quality factor over the frequency — which for the eight-pair stack is a hundred thousand optical cycles. Everything about the dynamics of such a cavity is outside a transmittance curve.

The defect’s frequency has been placed at the middle of the gap by construction. A half-wave layer is resonant at the design frequency, which is the gap’s centre; a different defect thickness puts the state elsewhere in the gap, more weakly bound, and eventually pushes it out of the gap altogether, at which point it stops being a state and becomes a feature of a transmission spectrum.

One dimension is a special case in a second way. The modes of a two-dimensional resonator are not a harmonic series and are not simply related to each other at all — a drum has no pitch — so a planar photonic cavity’s spectrum has to be computed rather than counted.

And a “state” here is a resonance rather than a bound state. A genuinely bound state has no leakage and an infinite lifetime; this one always leaks, however many pairs are added, and its lifetime is finite at every mirror thickness. The distinction is invisible in a transmission measurement and is the whole difference between a filter and a trap.

A finite structure has a discrete set of modes, and the defect state is one of them — a mode of the whole stack that happens to have most of its energy in one place. That is where the idealisation stops: an infinite crystal has a genuine gap and a finite one has closely spaced modes that are merely faint, so “forbidden” is a statement about a limit and “localised” a statement about degree.

What the pictures cannot show #

The transmission figure plots a resonance whose width at eight pairs is one part in a hundred thousand of the frequency. Drawn on the axis at true width it would be thinner than the stroke that draws it, and what is on the page at that thickness is the sampling of the curve rather than the curve.

Nor can the field figure show a field. It draws an amplitude at each interface, which is an envelope: the actual field oscillates within every layer, many times over the width of the drawing, and what is plotted is the size of that oscillation. The peak at the defect is a statement about how large the oscillation is there, and the oscillation itself — the thing that is resonating — has been averaged away to make the localisation visible.

Where this ladder goes next #

Two rungs stand on periodic-media. The first found that a repeat forbids a band of frequencies, that the band’s width has a closed form containing only the ratio of the two indices, and that no length appears in the problem. This one breaks the repeat once and finds that the gap’s real value is the state a single flaw can hold in it.

The habit worth carrying away is about where to look in a periodic system. The interesting physics of a crystal is nearly always at the place where it stops being one. A perfect lattice has bands and nothing else; the states that carry current, emit light, trap a photon, pin a vortex or scatter an electron all live at an impurity, a surface, a dislocation or an edge — and the perfection of the surroundings is what makes those states sharp.

What is left on this ladder is the boundary rather than the interior. A semi-infinite crystal has states bound to its surface, decaying into the crystal at the gap’s rate and into the vacuum at another, and the conditions under which they exist depend on how the repeat is cut — which turns out to be a question about a topological invariant of the band structure rather than about the surface’s chemistry.