Illustrated Physics
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The resonance that refuses to leak

A resonance that sits at a frequency where waves can escape has a width, because it leaks. Put two such resonances into the same channel and let them talk to each other, and at one precise detuning one of their combinations stops leaking altogether — a mode with no width, surrounded by a continuum it could escape into and does not. It cannot be seen from outside, it traps whatever energy lands in it, and if a symmetry is what forbids the leak, it survives any change that keeps the symmetry.
15 min read 5 figures The shape decidesWhat stays the same

Assumes: The resonance with a zero in it · The width that is a lifetime

A resonance that can lose its energy has a width, and the width is its lifetime seen in another coordinate. A bell rings down because its vibration radiates sound; a cavity’s light leaks through its mirrors; an atom’s excited state decays by emitting a photon. Whenever a resonance sits at a frequency where waves can travel away from it — inside a continuum of travelling states — it is expected to leak into that continuum, and the question is only how fast.

The resonance with a zero in it found what happens when a resonance shares a channel with a smooth background: the two routes through the system interfere, and the spectrum acquires an exact zero. It closed by naming the limit at which interference does something stranger — makes a resonance’s width go to zero while it stays inside the continuum, so that a state which ought to decay cannot. That limit has a name, a bound state in the continuum, and it can be computed from a pair of resonances and a single channel.

Two leaks into one channel

Take two resonances with frequencies ω1\omega_1 and ω2\omega_2, each leaking into the same outgoing channel — the same waveguide, the same free space, the same radiation — at rates γ1\gamma_1 and γ2\gamma_2, and coupled to each other with a strength κ\kappa. Because they leak into one channel, their leaks are not independent: the wave one sends out can interfere with the wave the other sends out. The pair is described by a two-by-two matrix whose real part holds the frequencies and the coupling, and whose imaginary part holds the leaks, and the leak part has a peculiar structure. It is built from a single vector, (γ1,γ2)(\sqrt{\gamma_1}, \sqrt{\gamma_2}) — one channel, one direction in which leaking happens.

A matrix built from one vector annihilates everything perpendicular to it. The combination of the two resonances proportional to (γ2,−γ1)(\sqrt{\gamma_2}, -\sqrt{\gamma_1}) radiates into the channel with two amplitudes that cancel exactly, whatever the frequencies. It does not leak. It is not, in general, a mode of the pair, because the coupling and the detuning mix it with the other combination, and the mixture leaks. But when it happens to be a mode as well — which requires the detuning to take the value

ω1−ω2=κ(γ1−γ2)γ1γ2,\omega_1 - \omega_2 = \frac{\kappa(\gamma_1 - \gamma_2)}{\sqrt{\gamma_1\gamma_2}},

a condition Friedrich and Wintgen found in 1985 — the pair has an eigenmode with a real frequency and no width at all.

A width that falls to nothing

One width that falls to nothing. The decay rates of the two modes of a pair of resonances that leak into one shared channel at rates 0.1 and 0.05 and are coupled to each other with strength 0.2, against the detuning of the first from the second. The two rates always add to 0.15, the trace of the leak matrix, to 10⁻¹². Where the resonances are far apart each mode keeps roughly its own resonance's leak. Near them the leaks interfere, and at a detuning of 0.1414 — κ(γ₁ − γ₂)/√(γ₁γ₂) — the slower mode's decay rate is zero to within 10⁻¹², and the faster carries all 0.15. That mode sits at a frequency where the channel is open and does not leak into it: the two routes by which it could escape cancel.
Fig. 1 Decay rates of the two modes of a pair of resonances leaking into one channel at 0.1 and 0.05, coupled with strength 0.2, against their detuning. The two rates add to 0.15 to 10⁻¹². Near a detuning of 0.1414 the leaks interfere, and there the slower mode’s decay rate is zero to within 10⁻¹² while the faster carries all 0.15.

The figure diagonalises the pair numerically at each detuning. Far apart, each mode keeps roughly the leak of the resonance it mostly is. As the resonances approach, the modes mix, and their decay rates move in opposite directions — always adding to γ1+γ2\gamma_1 + \gamma_2, because the sum of the eigenvalues of a matrix is its trace, and the trace of the leak part does not depend on anything but the two leak rates. At the predicted detuning of 0.1414, one decay rate touches zero and the other takes the whole of the leak.

This is related to, and not the same as, the crossing that never happens. There, two coupled levels repel in energy and exchange character. Here the repulsion happens in the widths: the two modes share a fixed total leak, and the interference that mixes them can push all of it into one. The real frequencies go on repelling as usual; what reaches an extreme is the division of the loss.

A budget of leaking

The rule that the two decay rates always add to the same total is worth dwelling on, because it is what makes the effect possible and what it costs. Energy leaves the pair only through the channel, and what reaches the channel from the pair is a sum of two amplitudes, one from each resonator. A combination of the resonators radiates the square of that sum. The combination that is in phase with the leak — proportional to (γ1,γ2)(\sqrt{\gamma_1}, \sqrt{\gamma_2}) — radiates at the full rate γ1+γ2\gamma_1 + \gamma_2; the one perpendicular to it radiates nothing; and any mode of the pair is some mixture of the two, radiating in proportion to how much of the first it contains.

The two modes of the pair together contain exactly one of each combination, so their decay rates add to γ1+γ2\gamma_1 + \gamma_2 whatever the coupling and detuning do. Interference does not remove loss; it decides which mode carries it. Making one mode perfectly dark makes the other perfectly bright, carrying the leak of both resonators at once, and the broad dip in every spectrum below is that bright mode.

The same structure appears in the mass that makes another stand still, with a crucial difference. There, a tuned absorber delivers a force that cancels the drive on a machine, and the machine’s response goes to zero at one frequency; the cancellation is of a response to something pushing from outside. Here the cancellation is of the system’s own loss, and what goes to zero is not a response but a width. The absorber makes a mass stand still while it is being pushed; the bound state makes a vibration last for ever while nothing pushes it.

Sharper than anything it is made of

A mode with zero width is an idealisation that no real structure reaches exactly, and what matters in practice is how sharp a resonance becomes near it.

The quality factor near a state that cannot leak. The quality factor of the slower mode — its frequency over its full width — against how far the detuning misses the bound state, on logarithmic axes. Missing by 10⁻⁴ gives 260,839,853; by 10⁻³, 2,612,597; by 10⁻², 26,554. The slope is −1.9961: the width vanishes as the square of the error, so every factor of ten closer buys a factor of a hundred in quality. A structure tuned near such a state has a resonance far sharper than either resonance it was made from, and the sharpness is limited in practice by whatever the model leaves out — absorption, roughness, a second channel — rather than by the leak the interference removed.
Fig. 2 The quality factor of the slower mode — its frequency over its full width — against how far the detuning misses the bound state, on logarithmic axes. Missing by 10⁻² gives 26,554; by 10⁻³, 2,612,597; by 10⁻⁴, 260,839,853. The slope is −1.9961: the width vanishes as the square of the error.

Near the bound state the width falls as the square of the detuning error, so the quality factor rises as its inverse square: every factor of ten closer buys a factor of a hundred. The two resonances in the figure have quality factors of about five and ten on their own — they are broad, leaky resonances — and a detuning within a thousandth of the right value gives their combination a quality factor of more than two million. A bound state in the continuum makes a sharp resonance out of blunt parts, not by reducing any loss but by arranging for losses to cancel.

The square law is the reason the effect is usable. A width that vanished linearly would demand linear precision; one that vanishes quadratically forgives small errors in the first place and rewards precision steeply beyond it, so that the practical limit is set by whatever the model leaves out — absorption in the material, scattering from roughness, a second channel the cancellation does not cover — rather than by the tuning.

Invisible from outside

The obvious way to look for a sharp resonance is to send a wave past it and watch for a narrow feature in what comes out. For this resonance that fails completely.

A resonance that disappears from its own spectrum. The fraction of a wave transmitted along a channel past the pair of coupled resonances, against frequency, for detunings of 0.6, 0.35, 0.22 and at the bound-state detuning, 0.1414; each spectrum is drawn on its own row. Transmission and reflection add to one to 10⁻¹⁰. The slower mode's full width is 0.029 at detuning 0.6, 0.0099 at detuning 0.35, and 0.0018 at detuning 0.22, and its narrow feature in the spectrum narrows with it. At the bound state the feature is gone: the mode still exists, at frequency 0.8586, but a mode that cannot leak into the channel also cannot be reached from it, and the spectrum shows only the broad dip of the other mode. The sharpest resonance of the system is the one no transmission measurement can see.
Fig. 3 Transmission along a channel past the coupled pair, against frequency, for detunings of 0.6, 0.35 and 0.22 and at the bound state, each on its own row; transmission and reflection add to one to 10⁻¹⁰. The slower mode’s width is 0.029, 0.0099 and 0.0018 at the three detunings and its narrow feature narrows with it. At the bound state the feature is gone, though the mode still exists at frequency 0.8586.

Away from the bound state the spectrum shows two features, a broad dip from the leaky mode and a narrow one from the slow mode, and the narrow one has the interference-shaped profile of a resonance on a background. As the detuning approaches the bound state, the narrow feature narrows exactly as its mode’s width does — and at the bound state it is not there. The broad dip of the other mode is all that remains.

The disappearance is not a coincidence of the model; it is reciprocity. A mode’s coupling to a channel works both ways, because the equations are the same run backwards: the coupling that is the same both ways is the circuit version of the same statement. A mode that radiates nothing into the channel receives nothing from it. The sharpest resonance of the system is the one no transmission measurement can see, and experiments that find bound states in the continuum find them indirectly — by watching a feature narrow and vanish as a structure is tuned, or by breaking the cancellation slightly so the mode becomes visible and very sharp.

Energy that stays

What can be done with a mode that cannot be reached from outside is to put energy into it from inside, and watch it not come out.

Energy that stays. The energy remaining in the pair after the first resonator alone is set ringing, against time, at the bound-state detuning and at detunings 0.01 and 0.03 past it. At the bound state the energy falls quickly as the leaking part escapes and then stops falling, at 0.3333 — exactly γ₂/(γ₁ + γ₂) = 0.3333, the share of the initial excitation that lies in the combination of the two resonators whose leaks cancel — and stays there. Missing the bound state by 0.01 or 0.03 turns the plateau into a slow decline. Whatever part of an excitation projects onto a bound state in the continuum is trapped for as long as the model holds.
Fig. 4 Energy remaining after the first resonator alone is set ringing, against time, at the bound state and missing it by 0.01 and 0.03. At the bound state the energy falls as the leaking part escapes and then stops, at 0.3333 — exactly γ2/(γ1+γ2)\gamma_2/(\gamma_1 + \gamma_2), the share of the excitation lying in the non-leaking combination — and stays. Off it, the plateau becomes a slow decline.

Striking one resonator excites both modes of the pair. The leaky mode’s share escapes in a few units of time, and the bound state’s share does not escape at all. The energy that stays is the square of the initial excitation’s projection onto the non-leaking combination, γ2/(γ1+γ2)\gamma_2/(\gamma_1 + \gamma_2) — a third, for the rates in the figure — and the calculation reproduces it to ten decimal places. Missing the bound state by a hundredth of a unit turns the plateau into a slow leak whose rate is the slow mode’s small width.

Trapping of this kind is what a defect in a periodic structure does by a different route. A defect mode sits in a band gap, at a frequency where no wave can travel, so it is confined because there is nowhere to go — the mechanism of the gap a repeat opens. A bound state in the continuum sits at a frequency where waves can travel freely, and is confined only because every route out cancels. The first is a wall; the second is an argument.

Protected by a mirror

The tuned condition is fragile by construction: it holds at one detuning. There is a second way for the leaks to cancel that needs no tuning at all.

A state protected by a mirror. Two identical resonators, each leaking at 0.08, coupled with strengths of 0.05, 0.2, 0.5. With no tuning at all, the combination in which they oscillate oppositely does not leak, for every coupling: its two leaks are equal and opposite by symmetry, and its decay rate is zero to 10⁻¹⁴. The figure breaks the symmetry by making the second resonator leak ε more, and plots that mode's quality factor against ε on logarithmic axes: between ε = 10⁻³ and 10⁻² the slopes are −1.995, −1.998, −1.998, so the protection is lost as ε² while the asymmetry is small. At ε = 0.01 the quality factors are 1,711,696, 466,961, 257,745. A bound state that a symmetry forbids from leaking survives any change that keeps the symmetry, and one that is tuned into existence survives nothing.
Fig. 5 Two identical resonators, each leaking at 0.08, coupled with strengths 0.05, 0.2 and 0.5. Their antisymmetric combination does not leak at any coupling, to 10⁻¹⁴. Making the second resonator leak ε more restores a leak, and between ε = 10⁻³ and 10⁻² the quality factor falls as ε to the −1.995, −1.998 and −1.998. At ε = 0.01 the quality factors are 1,711,696, 466,961 and 257,745.

Two identical resonators leaking identically into the same channel have a symmetric combination and an antisymmetric one. By symmetry the antisymmetric combination’s two leaks are equal and opposite, so it does not radiate, whatever the coupling between the resonators and whatever their common frequency. The figure confirms it at three couplings. Only an asymmetry can restore the leak, and it does so as the square of the asymmetry: a one per cent difference in the leak rates still leaves quality factors between a quarter of a million and 1.7 million.

This is the distinction that matters in any real structure. A bound state protected by a symmetry survives every change that keeps the symmetry; one that exists by tuning survives nothing. Photonic crystal slabs have modes at the centre of their band that cannot radiate perpendicular to the slab because of the lattice’s symmetry, and those modes stay dark as the slab’s thickness and hole size are varied, while an accidental bound state elsewhere in the band appears only at particular values and moves when anything is changed.

Counting what has to be tuned

The Friedrich–Wintgen condition is one equation among the pair’s parameters, and it is a real equation: a relation between the detuning, the coupling and the two leak rates. Satisfying it takes one adjustment — a detuning, in the figures — and that is why a bound state in the continuum formed from two resonances in one channel appears as a single point on a line of detunings rather than needing a lucky coincidence of several numbers.

The counting is the same kind that the crossing that never happens used to explain why two levels of the same symmetry do not meet when one parameter is varied, and it comes from the same place: von Neumann and Wigner’s paper of 1929 contained both the rule about crossings and the first example of a bound state in the continuum. More channels need more conditions. A mode that must not leak into two independent channels needs two cancellations, and generically two parameters to tune; a mode in a continuum of many channels needs as many.

Symmetry changes the count. If a symmetry forbids the mode from coupling to a channel at all — the antisymmetric combination of two identical resonators cannot radiate into a symmetric channel, just as a symmetry hands over a conserved quantity that no symmetric process can change — the condition is satisfied identically, and nothing needs tuning. That is the whole difference between the fragile, tuned bound states and the robust, protected ones, and it is why the structures that have made them useful are built around symmetries.

Where they have been found

The idea is older than the mechanism. In 1929 von Neumann and Wigner constructed a quantum potential, oscillating and decaying in a finely arranged way, that held an electron in a bound state at an energy above the top of the potential, where it should have been free to leave; the construction was a mathematical curiosity for decades. In 1951 Ursell showed that water waves can be trapped over a submerged obstacle at frequencies where waves propagate freely to infinity, and in 1966 Parker found acoustic resonances of flat plates in ducts — the plates of a jet engine’s compressor among them — that sing at frequencies the duct ought to carry away.

Friedrich and Wintgen’s interference picture of 1985 unified these as the same effect, and optics made it an engineering subject. In 2013 a bound state in the continuum was observed in a photonic crystal slab, its quality factor rising towards the fabrication limit as the angle of incidence approached the protected point, and in 2017 one was used as the cavity of a laser whose threshold was lowest where the mode was darkest.

Where the perfect trap leaks

One channel was assumed. The cancellation removes the leak into one channel. A structure that can radiate into two — two polarisations, two directions, a second diffraction order — needs the leaks into both to cancel, which takes more than one tuned parameter or a symmetry that covers both.

The resonators had no loss of their own. Absorption in the material is not a leak into the channel and is not cancelled. A bound state in the continuum in a real material has a quality factor limited by absorption, which for good dielectrics is very high and for metals is low.

The structure was infinite and perfect. A photonic crystal of finite size lets its protected mode leak sideways out of the edges, and fabrication disorder breaks the symmetry at random; the ε2\varepsilon^2 law says how forgiving the mode is, and the measured quality factors of such devices are set by exactly these two effects.

The coupling was constant. Real resonators have couplings and leak rates that depend on frequency, and the bound-state condition then has to be satisfied self-consistently. The two-mode model is the simplest version, and it captures the mechanism rather than any particular device.

What the figures leave out

Every figure is in units where the second resonance’s frequency is one, and the leak rates of 0.1 and 0.05 make the resonances very broad — quality factors of five and ten. That was chosen so the mixing is visible on one axis. Real resonators are usually much sharper to begin with, and the bound state’s region of influence is correspondingly narrower, but the shapes of every curve are the same.

The spectrum figure shows transmission along a channel, which is one of several ways the pair could be probed. Probing through a second, weakly coupled channel — which a real experiment often has — would reveal the bound state as an extremely sharp peak, because that channel’s leak is not cancelled. The invisibility is a statement about the channel whose leaks cancel, not about every possible measurement.

Still open: how dark a real dark mode can be

The quality factor of a bound state in the continuum is infinite in the model and finite in every device, and the question of what limits it has become a subject of its own. In photonic crystal slabs the dominant limits are scattering from fabrication roughness and leakage at the edges of a finite sample. Since 2019 structures have been designed in which several bound states in the continuum, each a point in the space of directions where radiation vanishes, are brought together so that radiation is suppressed over a whole neighbourhood of directions, making the mode far less sensitive to disorder.

How far that protection extends — whether it can make a mode robust to three-dimensional disorder rather than only to disorder that preserves some symmetry, and whether devices built on it can outperform conventional high-quality cavities once fabrication is taken into account — has not been settled.

The habit worth carrying away is to ask whether a loss is a property of a part or of an arrangement. A width is a statement about where energy goes, and when two parts send it to the same place, the arrangement decides whether it goes at all.

Part 6 of 6

This essay is one argument about Resonance. The others:

A resonance that goes to zero before it peaks. Fano profiles for asymmetry parameters of 5, 1.5, 0.5, 0, each normalised to its own peak, against detuning in half-widths. A large parameter gives an almost symmetric peak — the resonant path dominates and the shape is nearly Lorentzian. A parameter of zero gives a symmetric dip, a window in which the system transmits nothing on resonance. In between the profile is lopsided, with a zero on one side of the resonance and the maximum on the other, at positions whose product is exactly minus one. The asymmetry is not a defect of the measurement: it is the interference of two ways through the system, and its sign says which side of the resonance the two paths cancel on. The resonance with a zero in it Part 5 — Where a resonance is the only way through a system, the response is a symmetric peak. Where there is also a smooth path that does not care about the frequency, the two add before anything is squared — and the result is lopsided, with a frequency at which nothing gets through at all.

Where this goes next

A resonance that goes to zero before it peaks. Fano profiles for asymmetry parameters of 5, 1.5, 0.5, 0, each normalised to its own peak, against detuning in half-widths. A large parameter gives an almost symmetric peak — the resonant path dominates and the shape is nearly Lorentzian. A parameter of zero gives a symmetric dip, a window in which the system transmits nothing on resonance. In between the profile is lopsided, with a zero on one side of the resonance and the maximum on the other, at positions whose product is exactly minus one. The asymmetry is not a defect of the measurement: it is the interference of two ways through the system, and its sign says which side of the resonance the two paths cancel on. The resonance with a zero in it Where a resonance is the only way through a system, the response is a symmetric peak. Where there is also a smooth path that does not care about the frequency, the two add before anything is squared — and the result is lopsided, with a frequency at which nothing gets through at all. The same number read off a decay and off a linewidth. A lightly damped oscillator released and left alone, above, and the power spectrum of exactly those samples, below. The decay falls to 1/e of its starting amplitude after 8.0 cycles, which makes the quality factor π times that, or 25.0. The spectrum peaks at 1.0000 radians per second and falls to half its power 0.04001 radians per second wide, which makes the quality factor the peak divided by the width, or 25.0. The two disagree by 0.02 per cent, which is the resolution of the frequency grid rather than a difference in the physics. They cannot disagree by more, because they are the same statement: a resonance is narrow because its ringing is long, and the transform that turns one into the other is not an approximation but an identity. A measurement of either is a measurement of both — which is why a bell can be characterised by hitting it and listening, or by driving it and sweeping, and why the two instruments never argue. The width that is a lifetime Hit a bell and time how long it rings; drive it and measure how narrow its response is. The two numbers are the same number, and they cannot disagree — not because the physics conspires but because a decay and a linewidth are one function seen in two coordinate systems. Two levels that refuse to cross. On the left, the energies of a two-state system as one state is swept past the other, for couplings of 0, 0.05, 0.15, 0.35 electronvolts. With no coupling at all the two levels cross, which is the dotted pair. With any coupling whatever they do not: the eigenvalues are plus and minus half the square root of the squared detuning plus four times the squared coupling, so the closest approach is exactly twice the coupling — measured off the drawn curves as 0.000, 0.100, 0.300, 0.700 eV against 2V, agreeing to 0.0e+0. Far from the crossing the curves rejoin the uncoupled lines, so the repulsion is local: it is largest where the two states are degenerate and dies away as the square of the coupling divided by the detuning. On the right is what makes this more than a picture of two hyperbolas — the character of the upper state, meaning how much of the first basis state is in it. It swaps completely across a region whose width is set by the coupling, so the level that arrives as one thing leaves as the other. That exchange of identity is why the phrase avoided crossing is misleading: nothing is avoided, the labels are. The crossing that never happens Two energy levels swept past one another do not cross. Any coupling between them, however small, opens a gap of exactly twice the coupling — and the two levels exchange their identities across it, so the state that arrives as one thing leaves as the other while the labels are what avoided anything.

The objects named here

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

DampingDecayInterferenceNormal modesQuality factorReciprocityResonanceSymmetry