The taper that matches every note

A wave meeting a change of impedance reflects, and how much reflects is fixed by the ratio of the two impedances. The standard cure is a layer a quarter of a wavelength thick with an impedance the geometric mean of the two, so that the echo from its front face and the echo from its back face arrive half a wavelength out of step and cancel.

That cure has a design wavelength in it, and everything about its behaviour follows from that. There is another cure with no design wavelength at all.

Two different shapes, not two settings of one #

The two curves are worth reading against each other before any explanation, because the difference is one of kind.

The quarter-wave layer is a band-pass device. It has a centre, it is perfect there, and it degrades in both directions. Widening its band means adding more layers, and a multilayer transformer is a filter-design problem with the usual trade between bandwidth and how flat the passband is.

The taper is a high-pass device. It has no centre and it is never exactly perfect anywhere; instead it is negligible everywhere above a cutoff, and it stays negligible for every shorter wavelength, without limit. Making it longer moves the cutoff down and changes nothing above it.

That is not a difference in degree. Nothing that could be done to a quarter-wave layer would give it the taper’s flat top, and nothing that could be done to a taper would give it the layer’s exact zero.

Where the cutoff comes from #

The mechanism is a cancellation, and it is worth being precise about what cancels against what.

Divide the graded region into thin slices. Each slice presents a small step of impedance, and each small step sends back a small reflection. The total reflection is the sum of all of them, each with the phase it acquired on the round trip to its own depth.

If the grading is long compared with the wavelength, the depths cover many wavelengths, the phases are spread over many turns, and the sum of a smoothly varying amplitude over uniformly spread phases is very nearly nothing. If the grading is short compared with the wavelength, all the slices are at essentially the same phase, they add coherently, and the sum is the full step — the wave cannot tell a gradual change from an abrupt one because both happen within a fraction of a cycle.

So the criterion is a comparison of lengths: the grading works when it is longer than about half a wavelength. The computed cutoffs bear this out — 398 nm for a 150 nm taper, 1062 nm for a 400 nm one, a ratio of 2.65 in both cases, which after allowing for the mean index of the graded material is about twice the optical length.

The same reasoning explains why the change has to be smooth and not merely gradual. A grading with a kink in it has a slice whose step is much larger than its neighbours’, and that slice’s reflection has nothing to cancel against; the classic result is that the reflection falls off as a power of the wavelength whose exponent is set by how many derivatives of the profile are continuous.

How the curves were obtained #

Everything above is computed by the same piece of arithmetic, and it is worth a paragraph because it is what makes the comparison fair.

Each layer of a stack is represented by a two-by-two matrix relating the field and its slope at one face to the field and slope at the other. The matrix for a whole stack is the product of the individual matrices, in the order the wave meets them, and the reflectance follows from the product’s four entries and the impedances on either side. The method is exact for any number of layers of any thicknesses, and there is nothing in it about quarter waves or gradings: it is handed a list and it multiplies.

The taper is therefore not a special case with its own theory. It is one hundred and twenty entries in the same list, with thicknesses that happen to be small and indices that happen to climb geometrically, and it goes through the identical product as the single layer beside it. That is the discipline this collection tries to keep — a figure that evaluates a formula its own caption quotes demonstrates nothing — and it is the reason the two behaviours can be laid on one axis and believed.

The one thing the method assumes is that the wave is a single frequency travelling in one direction with a well-defined impedance at each depth. That holds for a plane wave at normal incidence in a lossless isotropic medium and needs amending for every departure from it.

The same device in six trades #

Once stated as “make the impedance change slowly on the scale of a wavelength”, the idea turns out to have been invented independently a great many times.

The horn on a loudspeaker or a trumpet. A driver radiating into open air is a small piston meeting a very low acoustic impedance, and almost all its energy reflects. A flare that widens smoothly over a length comparable with the longest wavelength of interest matches the two, and the cutoff of an exponential horn is exactly the effect above: it works above a frequency set by its own flare rate and does nothing below. That is why a horn loudspeaker is large if it is to reach low notes, and why a small horn is a treble device.

The moth’s eye. An array of bumps much smaller than the wavelength of light, tapering from full height to nothing over a few hundred nanometres, behaves optically as a medium whose effective index rises smoothly from air’s to the material’s. The cutoff sits in the near infrared, so it is antireflective across the whole visible band at every angle — which a quarter-wave coating is not — and it works by having no surface rather than by having a clever one. It is also self-cleaning, which is presumably the reason it evolved.

The waveguide transition. Joining two waveguides of different cross-section with a step reflects; joining them with a flare over several wavelengths does not. The same figure, the same cutoff, the same conclusion that the flare’s length is set by the lowest frequency to be carried.

The ultrasound probe. Piezoelectric ceramic has an acoustic impedance about twenty times that of tissue, so a bare transducer couples almost nothing. A single quarter-wave matching layer is the standard solution and gives a bandwidth of perhaps sixty per cent; a graded stack gives more, at the price of thickness, and thickness costs resolution.

The gradient-index rod. Grade the index across a rod rather than along it and the same idea bends rays instead of transmitting them, which is a different use of one profile.

The tapered optical fibre. Drawing a fibre down to a fraction of its diameter over a length of centimetres carries the light from the core into a mode guided by the outside surface without reflecting any of it, which is how a fibre is coupled to something that is not another fibre. Do it over a millimetre instead and most of the light is lost — the same length criterion, with the wavelength being the beat length between the two modes rather than the free-space one.

And the seismic transition zone. A wave arriving at a boundary in the Earth reflects if the boundary is sharp on the scale of its own wavelength and passes if it is not — so long-period waves see the mantle’s gradual changes as no boundary at all and short-period ones see them clearly. Which frequencies reflect is therefore a measurement of how thick the transition is, and that inversion is how the sharpness of the 410-kilometre discontinuity is known.

The stack that goes the other way #

The transfer-matrix arithmetic behind the figures is indifferent to what it is asked to compute, and asking it for the opposite of a taper produces the opposite behaviour.

A quarter-wave stack has the maximum number of steps arranged so their echoes add rather than cancel, and enough of them produce a mirror that is better than metal over its band and transparent outside it. That the same machinery produces a perfect reflector and a perfect transmitter, depending only on whether the impedance is made to vary smoothly or in steps of a quarter wavelength, is the strongest statement of what this subject is about: it is the arrangement of the change that decides, not the amount of it.

The number that decides, and what it is made of #

Everything above turns on one dimensionless quantity: the length of the transition divided by the wavelength in it. That is the only parameter, and it explains why the same device works at scales eleven orders of magnitude apart.

It also says what a designer actually has to decide. Not the profile, which matters only at the level of ripple; not the material, beyond the endpoints; only the length. Fix the longest wavelength the device must handle, multiply by about a half, and that is the taper — everything else is refinement.

The corollary is the honest limitation. A transition that must be short compared with a wavelength cannot be made non-reflecting by shaping, at all, by anybody. Where that constraint binds — a probe that must be thin, a coating that must be a monolayer, an antenna that must fit in a handset — the answer has to come from somewhere other than geometry, and it usually comes from resonance, with a resonance’s bandwidth. The speed at which a wave travels in a medium fixes the wavelength, so the only remaining freedom is how fast the change is made.

What it costs #

Length. The taper’s whole advantage is bought with distance, and the distance is set by the longest wavelength to be handled. A horn for forty hertz is metres long; an antireflection grading for the visible is a few hundred nanometres and is cheap; a graded matching layer for one-megahertz ultrasound would be a millimetre and would blur the image.

Manufacture. A smoothly varying index is harder to deposit than two discrete materials, and the moth’s eye is made by etching rather than coating for exactly this reason.

And nothing below the cutoff. The taper is not a compromise below its cutoff; it is simply absent. A device that must work over a range including very long wavelengths cannot use one, and this is the point at which the quarter-wave design, with its band around a centre, is the better answer.

There is also a limit that has nothing to do with the design. A gradual transition removes the reflection at normal incidence and does less well at a glancing one, because the relevant length is the depth measured along the ray rather than along the normal. And where a wave is below the cutoff of the guide carrying it no matching helps at all, because there is nothing propagating to match into.

The room with spikes on the walls #

The most conspicuous use of a taper is in a room designed to have no echoes, and the spikes are the length criterion made visible.

An absorbing material only absorbs what enters it. Present a flat slab of acoustic foam to a sound wave and a good deal of the wave never gets in: the foam’s impedance is not air’s, so the surface reflects, and the absorption behind it never gets the chance. Shaping the same foam into long pyramids fixes it. Near the tips the material occupies a small fraction of each cross-section, so the average impedance is close to air’s; deeper down the fraction rises and so does the impedance; at the base it is the foam’s. The wave meets a graded transition rather than a face.

Which means the pyramids’ height is the taper length, and everything in this essay applies to it directly. An anechoic chamber is quiet above a cutoff set by that height and stops being anechoic below it, abruptly. A metre-long wedge works down to a hundred hertz or so and does nothing useful at thirty, which is why the chambers are so large: the room has to be big enough to hold spikes as long as a fraction of the longest wavelength it is meant to absorb, and then big enough again to leave a usable volume between them.

The electromagnetic version is identical in every respect but the material. A radar test chamber is lined with carbon-loaded foam pyramids, whose height is set by the lowest frequency to be measured.

And the fix for the low-frequency end is exactly the alternative this essay has been contrasting with the taper. Ferrite tiles a few millimetres thick absorb well at low frequencies by resonance rather than by grading — narrowband, thin, and effective precisely where the pyramids give out. A modern chamber uses both, which is a room whose walls carry one broadband high-pass device and one narrowband tuned one, chosen for the two halves of the spectrum on the grounds this page sets out.

Where a taper will not fit #

The essay’s honest limitation — that a transition forced to be short compared with a wavelength cannot be fixed by shaping — has a worked example that everybody is carrying.

Sound arriving at an ear has to get from air into the fluid of the inner ear, and the acoustic impedances differ by a factor of about three and a half thousand. Left to itself that boundary reflects 99.9 per cent of the incident power: an ear consisting of a membrane on a fluid-filled tube would be some thirty decibels deaf.

A taper is not available. The wavelengths concerned are between a couple of centimetres and several metres, and there is no room in a head for a graded transition metres long. So the problem is solved the other way, by a transformer: the eardrum is about seventeen times the area of the stapes footplate that drives the fluid, and the chain of three small bones between them adds a lever ratio of about a third again. Force collected over a large area and delivered over a small one is a pressure gain of roughly twenty, which recovers the great majority of what the mismatch would have cost.

Two features of that solution are the ones this page predicts. It is compact, occupying a few millimetres where a graded match would need metres. And it is not flat: a lever-and-area transformer has its own frequency response, set by the masses and stiffnesses of the bones, so the middle ear works best over a band of a few kilohertz and less well outside it — which is a large part of why human hearing is shaped the way it is.

Where geometry cannot supply the length, the answer comes from a resonance, and a resonance brings a bandwidth with it. That is the trade the whole essay is about, arrived at by evolution rather than by design.

What the picture cannot show #

The taper is drawn as a hundred and twenty discrete layers. That is a numerical approximation to a continuous profile, and it has its own artefacts: a stack of identical thin layers is itself periodic, and at a wavelength comparable with twice the slice thickness it would open a stop band. The slices here are a few nanometres and the artefact sits far into the ultraviolet, but the reader should know the curve is a limit being approached rather than an exact continuum result.

The profile drawn is geometric, and it is not optimal. An exponential grading is the simplest choice and gives a reflection that falls off as the inverse square of the frequency with visible ripple. Profiles that taper the derivative smoothly to zero at both ends — the Klopfenstein and the raised-cosine among them — do measurably better for the same length, and designing them is a filter-synthesis problem rather than a physics one.

Absorption is absent. Every curve here is for lossless media. A real graded layer absorbs, and past a certain thickness the absorption costs more than the reflection saved, which puts a practical ceiling on length that the physics above does not.

The medium is treated as non-dispersive. Every index in these figures is a fixed number, and a real material’s index depends on wavelength, so a taper’s effective profile is slightly different at every frequency it is asked about. Over the visible band in glass the variation is a per cent or two and changes nothing; across an octave in a polymer it is not negligible, and the same dependence that makes a prism work reappears here as a slow drift of the cutoff.

And nothing here is about angle. All the reflectances are at normal incidence. At other angles the two polarisations behave differently, one of them vanishing entirely at a particular angle, and a coating designed for one incidence is a different device at another.

The ladder from here #

Later rungs on this anchor: the optimal taper profile, and the trade between length and passband ripple that makes it a filter-design problem; the Bode–Fano limit, which puts a hard bound on how well any finite network can match a reactive load over a band and says the taper’s success has a price paid elsewhere; matching at oblique incidence and the two polarisations; the impedance of an antenna, where the load is radiation itself and the matching problem is the whole of the design; and the acoustic case in a duct with flow, where the impedance is no longer a property of the medium alone.

The neighbouring ladders are what happens at a change of medium, which is the rung this one stands on, and the single quarter-wave layer, whose exact zero at one wavelength is the thing a taper trades away for a flat top everywhere else.