The resonance with a zero in it

A driven oscillator’s response is a peak: it rises toward the natural frequency, reaches a maximum, and falls away symmetrically. That shape is so familiar that an asymmetric one is usually read as a defect — an instrument that has broadened one side, or two lines too close to separate.

It is neither. A lopsided resonance with a zero in it is what a resonance looks like when it is not the only way through.

The four curves are one function with one parameter. What that parameter is, physically, is the ratio of how strongly the resonant path couples to how strongly the smooth path does — and every feature of the shape follows from it.

Where the zero comes from #

The mechanism is entirely in the phase sweep, which is the part of a resonance that gets less attention than the peak. A driven oscillator does not merely respond most strongly on resonance; it responds in quadrature there, lagging the drive by a quarter cycle, and its phase runs from zero far below resonance to half a turn far above. That sweep is what the quarter cycle nobody mentions is about, and here it is the whole story.

Add a fixed amplitude that does not sweep, and somewhere on one side of the resonance the resonant amplitude points exactly opposite to it. If the two are also equal in size there, the total vanishes — and they always are equal somewhere, because the resonant amplitude grows without bound as the damping falls and passes through every size on its way. So a Fano profile has an exact zero, at a frequency displaced from the resonance by an amount set by the coupling ratio.

Two things about the zero are worth stating because both are commonly got wrong. Nothing is being absorbed there: a completely lossless system has the zero in exactly the same place, and what happens to the energy is that it is returned rather than taken. And the zero is exact — not a deep minimum, but a genuine null, at one frequency, however weak the resonance is. A weak resonance moves the zero far out into the wings, where nothing much is happening; it does not soften it.

One parameter, two features #

The reciprocal relation is the most useful thing to remember about the shape, because it means the profile has no freedom left once one feature is located. Measure where a spectrum goes to zero and the peak’s position follows; measure the two and the parameter is over-determined, which is what makes a fit to a Fano profile a real test rather than a curve-matching exercise.

It also explains the two familiar shapes as ends of one family. A large parameter — a strong resonance and a weak background — is a peak, and the zero is so far out that nobody notices. A parameter of zero — no direct coupling at all between the resonance and the outgoing channel — is a symmetric window, a frequency at which the system transmits nothing, sitting in an otherwise flat response. That case has its own name, the antiresonance, and it is what a tuned mass absorber does to a structure: a small oscillator attached to a large one holds it still at one frequency, and holds it exactly still rather than nearly.

The intermediate cases are the general one, and the fact that both familiar shapes are limits of it is the reason the asymmetric profile is so widespread and so often misread.

The ordinary resonance, for comparison #

Putting the two beside each other says what the extra path costs and what it buys.

The symmetric peak has two numbers in it — where it is and how wide — and the width is the damping. The response never vanishes; far from resonance it falls off as the inverse square of the detuning and reaches zero only at infinity. And the shape carries no information about anything except the oscillator.

The asymmetric profile has a third number, and it is not about the oscillator at all: it is about how the oscillator is connected to everything else. That is why the profile is so useful diagnostically. A symmetric peak measures a resonator; an asymmetric one measures a coupling.

There is also a practical difference that matters in instruments. The symmetric peak’s steepest slope is at the half-power points, where the response is changing at a rate set by the width. The Fano profile’s steepest slope is between the zero and the peak, and if the parameter is small those are very close together — so the response there changes much faster with frequency than any symmetric resonance of the same width can. That steepness is the reason such features are used as sensors, and it is bought with the interference rather than with a higher quality factor.

Where it turns up #

Fano wrote the profile down in 1961 for the autoionising states of an atom, and the setting is the clearest instance of the mechanism.

An atom absorbing a photon of enough energy can eject an electron in two ways. It can be ionised directly, which is a smooth process available at every energy above the threshold. Or it can be excited into a bound state above the ionisation energy, which then falls apart — a resonance, at a definite energy with a definite width. Both routes end in the same place: an ion and a free electron with a given energy. Because the final states are identical, the two amplitudes add, and the absorption spectrum is asymmetric with a window in it.

The same structure appears wherever a discrete state sits inside a continuum, which turns out to be almost everywhere.

In a solid, a sharp lattice vibration inside a continuum of electronic excitations gives asymmetric phonon lines whose asymmetry measures the coupling between the two — the standard diagnostic in doped semiconductors and in metals.

In optics, a narrow resonance in a structure that also transmits directly gives sharp asymmetric features: a whispering-gallery mode beside a waveguide, a metal grating with a surface wave on it, a photonic crystal with a defect. The sharpness of the transition from zero to peak is what makes these attractive as sensors — a shift of the resonance moves a very steep edge past a fixed wavelength, and the change in transmission is much larger than the shift.

In mechanics, a resonator attached to a transmission path gives a notch in the response, and this is how mechanical filters and vibration absorbers are designed.

The common feature in all of them is not a mechanism but a structure: two routes from the same start to the same end, one of them resonant. That is a much weaker condition than sharing any physics, which is why the same three-parameter curve fits data from atoms, crystals, waveguides and bridges.

What the shape is evidence for #

The profile’s real value is as evidence, and it is worth separating what it establishes from what it does not.

An asymmetric line establishes that there are at least two indistinguishable routes. That is a strong statement: it rules out the resonance being the only thing happening, and it rules out an explanation in terms of two separate populations, because populations add intensities and cannot produce a zero.

The sign of the asymmetry says which side the destructive interference falls on, which is a statement about the relative phase of the two couplings and is often the quantity a theory predicts.

What it does not establish is the mechanism of either route. Two very different systems with the same coupling ratio give the same curve, and fitting one says nothing about what the continuum is. That is the ordinary situation with a universal lineshape and it is worth being explicit about, because a good fit to a distinctive curve is persuasive out of proportion to what it settles.

How the parameter is read from data #

Fitting the profile is a three-parameter business — the resonance position, its width and the asymmetry — and it is worth knowing what each is determined by, because the three are not equally well constrained.

The position and the width come mostly from the peak’s neighbourhood, as they would for a symmetric line. The asymmetry comes from the zero, which is often far from the peak and in a part of the spectrum where the signal is weak and the background poorly known. So the asymmetry is the parameter that a fit determines worst, and reported values of it are frequently much less certain than they look.

There is a way round that, and it is the reciprocal relation. If the zero can be found at all — and a zero is a very distinctive feature, being the one place the signal genuinely vanishes — then it fixes the parameter directly, and the peak’s position becomes a prediction rather than a fit. Measuring both and checking that their product is minus one, about the resonance, is a test that a two-peak explanation fails and a Fano profile passes.

The other trap is normalisation. The profile as written has an overall scale, and a spectrum measured as a ratio to a background has already divided that out — so what is being fitted is the shape and not the amplitude. A fit that treats the scale as free will trade it against the asymmetry, since a large parameter and a small amplitude look much alike near the peak, and the two are separated only by the wings.

Where the model stops #

The background is assumed flat across the resonance. In practice it varies, and slowly-varying continua give profiles that are asymmetric for a second reason — the background’s own slope — which is not the same as the interference. Distinguishing the two needs the resonance to be narrow compared with the scale on which the background moves, and near a threshold it is not.

There is one resonance. Two overlapping ones interfere with the continuum and with each other, and the result is not a sum of Fano profiles: the coupled system has its own eigenmodes and the observed lineshapes belong to those. The two-resonance case is where the parameter loses its simple meaning.

Damping is treated as a width and nothing else. A resonance that decays into the same continuum it is being excited through is the case in the essay; one that also decays somewhere else — into a channel the measurement does not see — has a profile with a floor rather than a zero, and the depth of the minimum then measures the branching between the two.

And the profile is a statement about amplitudes, so it needs the two routes to be indistinguishable. If anything in the apparatus could in principle tell which route was taken, the amplitudes do not add and neither does the interference. That condition is the same one that decides whether interference happens at all, and in a solid at finite temperature it is what limits how sharp a Fano feature can be.

Where the sharpness is bought #

It is worth being quantitative about the sensor claim, because it is the reason the profile has had a second life since about 2000.

A symmetric resonance of width $\Gamma$ has its steepest slope at the half-power points, and the fractional change in transmission per unit change in frequency there is of order one over $\Gamma$. That is the whole basis of resonant sensing: make the resonance narrow, sit on the flank, and a small shift produces a large change.

A Fano feature does better, and by a factor that is not the quality factor. Between the zero and the peak the response goes from nothing to everything, and if the asymmetry parameter is small those two points are separated by much less than the width — by the width times the parameter, in fact. So the maximum slope is of order one over $\Gamma q$, and a system with a modest resonance and a small asymmetry outperforms a much sharper symmetric one.

The cost is that the feature is fragile in a way a peak is not. The zero depends on the two amplitudes cancelling exactly, so anything that adds an uncorrelated contribution — stray light, a second route with a different phase, absorption in the resonator — fills the zero in and destroys the steepness while leaving the peak nearly untouched. A Fano sensor is therefore much more sensitive to the quality of the fabrication than a resonant one of the same width, and the depth of the measured minimum is the honest figure of merit rather than the width.

What the pictures cannot show #

The phasor figure draws the two amplitudes at a few detunings and cannot show how quickly the sweep happens. The whole phase change takes place over a width, so a resonance with a quality factor of a thousand does all of this within a thousandth of its frequency — and the zero and the peak are that close together too, which is why the feature is easy to miss entirely at low resolution and easy to mistake for noise at moderate resolution.

The profile figures draw a response and say nothing about what is being responded to. The same curve describes an absorption, a transmission, a scattering cross-section and a mechanical amplitude, and which one it is decides what the zero means physically — a window, a notch, a dark state or a standstill. The mathematics is indifferent and the interpretation is not.

The name, and who else found it #

The profile is called Fano’s and it was not first written down by Fano, which is worth a sentence because the priority tells something about how such a shape gets noticed.

Beutler measured the asymmetric absorption lines in the rare gases in 1935 and reported them as a puzzle. Fano gave the theory in 1935 and the general form in 1961, and the 1961 paper is the one everybody cites because it is the one that states the shape as a one-parameter family and identifies the parameter with a ratio of matrix elements.

The same profile had already been derived twice elsewhere under other names. Rice had it in 1933 for predissociation in molecules, where a bound state above a dissociation limit interferes with the direct dissociation. And in nuclear physics the same interference between resonant and potential scattering was standard by the 1950s and is written as a sum of amplitudes with no lineshape name attached at all.

Three subjects, three derivations, one curve — and it took until 1961 for anyone to state that the shape has a single parameter and that the parameter means something. That is the usual pattern with a universal lineshape: the mechanism is found repeatedly in specific settings, and the general statement, which is the useful one, arrives long afterwards and from whichever subject happens to need it stated.

Where the ladder goes next #

The resonance ladder began with the frequency that gets an answer, went through the swing that is pumped rather than pushed, the mass that makes another stand still and the width that is a lifetime. This rung asks what a resonance looks like when it is not alone. The rungs after it: two coupled resonances, where the interference between them produces a transparency window rather than a zero; the connection between a lineshape and the analytic structure of the response, in which the zero and the pole are two features of one function; and bound states in the continuum, which are the limit in which the resonance’s width goes to zero while it sits inside a continuum — a state that cannot decay and has no width at all.

The habit worth carrying away is to read an asymmetry as information. A lopsided line is not a spoiled symmetric one; it is the signature of a second route, its shape has one parameter, and that parameter is a ratio of couplings that no symmetric measurement gives.