The wave that comes from the rim

Put a razor blade in a beam of light and look at the shadow’s edge with enough magnification. There are fringes outside the edge, in the light, spaced unevenly and fading outwards; there are none inside, in the shadow, where the intensity falls away smoothly; and on the geometrical boundary itself the intensity is a quarter of what it would have been with the blade removed.

Three facts, and the usual account — that the wave nature of light smears the edge — explains none of them. A smear has no reason to produce fringes on one side and not on the other, no reason to produce a quarter rather than a half, and no reason to space the fringes as it does.

Two terms, and the split is exact #

Young’s suggestion in 1802 was that the whole thing is a sum of two waves: the light that was never blocked, travelling in a straight line, and a wave scattered from the edge of the obstacle itself.

That is not obviously more than a picture. What makes it a calculation is that the split can be written down and it is exact. Take the computed field behind a half-plane, subtract the geometrical field — the incident wave in the lit region, nothing in the shadow — and whatever is left is by definition the edge wave. The interesting question is whether the remainder behaves like a wave from the rim, and it does, in three separate senses that were not put in.

It is continuous where the geometrical term is not. The step jumps by the whole incident amplitude at the boundary. The remainder’s magnitude jumps by $10^{-7}$, which is the precision of the integrals. It changes phase by $\pi$ instead of changing size.

Its magnitude on the boundary is exactly half the incident amplitude. Nothing was imposed to make it so. The total field must be continuous, and continuity across a step of size one, split symmetrically, gives one half.

Its amplitude falls as one over the distance. That is what a line source does and what nothing else does.

Together those are more than a fiction supports. The remainder is a line source of a definite strength, radiating equally into the light and into the shadow, with a phase that flips across the boundary.

Where the quarter comes from #

The most-quoted number about a straight edge follows from the split in one line, and the usual reasoning gets it wrong by a factor of two.

Blocking half the wavefront removes half the amplitude arriving on the boundary. Intensity is amplitude squared. So the intensity on the geometrical shadow line is $(1/2)^2 = 1/4$ of the unobstructed value, and the figure gives $0.250000$.

The half-and-half account — half the light blocked, so half the brightness — is a statement about power arriving over a large area, which is true, and it is not a statement about the intensity at one point. Amplitudes add and intensities do not, and that is the distinction the whole subject rests on.

Two waves added with a quarter-cycle between them make a sum whose amplitude is not the sum of the amplitudes and whose intensity is not the sum of the intensities. That single fact is the arithmetic behind every number in this essay: a quarter-cycle is where the two contributions are exactly out of step enough to matter and not enough to cancel, and the factor it produces is what puts the intensity at the shadow boundary at a quarter of the unobstructed value rather than at a half.

Why the fringes are on one side only #

This is where the decomposition earns its place, because it answers a question the smearing account cannot even pose.

In the lit region there are two terms. They have different phases, and the phase difference between them grows with distance from the boundary, so they alternately reinforce and cancel. That is a beat, and the beat is the fringes.

In the shadow there is one term. There is nothing for it to beat against, so there are no fringes and the intensity falls smoothly.

The fringe spacing falls out of the same argument. The extra path from the rim to a point a distance $x$ into the light, compared with the direct path, grows as $x^2$ for small $x$, so the phase difference grows quadratically and the fringes crowd together as one moves outwards. They are not evenly spaced, and every treatment that draws them evenly spaced has drawn a grating instead.

The first maximum sits at $1.37$ times the unobstructed intensity. Light is brighter just outside a shadow than it would be with nothing in the way at all, which is a straightforwardly strange thing for an obstacle to do and is what interference between two terms of comparable size looks like.

The same construction, applied to two edges #

A slit is two edges. If the decomposition is real rather than a way of talking, the field behind a slit should be the incident wave plus two edge waves, and the pattern should be readable as their interference.

It is. At a large Fresnel number the two edges act nearly independently and the pattern is two straight-edge patterns side by side, each with its own fringes. As the aperture narrows, or the screen recedes, the two edge waves overlap everywhere and their interference is the whole pattern — which is the far-field single-slit diffraction that gets taught first and is the limit rather than the general case.

When the two edge waves have overlapped completely, what they make is the familiar far-field pattern of a slit — a broad central lobe with weaker ones beside it, wider the narrower the slit. The geometrical shadow has stopped being a feature of the picture at all. Widen the slit again and the shadow’s two edges reappear as separate features, each with its own fringes, and the pattern stops being a slit pattern and becomes two edge patterns that happen to share a screen.

Reading it that way makes the transition between the near and far fields a matter of how much the two edge waves overlap, rather than two different theories with a boundary between them.

What this does to Huygens’ construction #

Huygens’ rule treats every point of a wavefront as a source and takes the envelope. It works, and it has always had two embarrassments.

Huygens’ construction gets the spreading right and says nothing about brightness, which is exactly the gap this essay is about. Treat every point of the front at a slit as a source and draw the envelope of the wavelets: the new front bends round the edges, by more the narrower the slit, and the rule has predicted diffraction. What it cannot predict is how much light goes where, because an envelope is a locus and not an amplitude — and Fresnel’s addition was to add the wavelets with their phases instead of enveloping them.

The first is that an envelope has no amplitude, which is what the Cornu spiral fixes by adding the wavelets with their phases instead. The second is the backward wave: if every point radiates in all directions, there ought to be a wave travelling back towards the source, and there is not.

The edge decomposition is a different repair, and it is worth setting beside the first. Instead of a source at every point of the front, there is one source, at the rim, and the light everywhere else is left alone. Where the two treatments both apply they agree exactly — the sum over wavelets and the geometrical-plus-edge sum are the same field — and they are convenient in different places.

The edge version is the one that scales. Whatever the shape of the obstacle, the field is the geometrical field plus a line integral around its rim, which is a one-dimensional integral instead of a two-dimensional one. That is the Maggi–Rubinowicz form, and it is why the practical calculation of radar cross-sections and antenna patterns is done with edges rather than with apertures.

The number a radio link is designed around #

The quarter-intensity on the shadow boundary is not a curiosity of a laboratory with a razor blade in it. It is the single most-used number in the planning of terrestrial radio links, where it is written as a loss of six decibels.

An edge wave thins out as the inverse first power of distance rather than the inverse square, because it comes from a line — the rim — rather than from a point. That is what fixes the loss over an obstacle at a value depending on geometry alone: the knife-edge diffraction loss on a radio path is a function of one dimensionless parameter built from the clearance, the wavelength and the two distances, and it contains nothing about what the obstacle is made of.

A hill between a transmitter and a receiver is a straight edge to a metre-wavelength wave. If the line of sight just grazes the summit, the receiver sits exactly on the geometrical shadow boundary, and the field there is half the free-space amplitude: a quarter of the power, which is $-6$ decibels. That number does not depend on the frequency, on how far away the hill is, or on what the hill is made of. It depends only on the receiver being on the boundary.

Move the receiver into the shadow and the loss grows in the way the edge wave’s decay dictates. Move it into the light and the loss oscillates about zero, going negative — a stronger signal than with no hill at all — at the first maximum, which is the $1.37$ of the figure above expressed as $+1.4$ decibels.

That last point is what the Fresnel-zone clearance rule is about. A link is not designed to clear the obstacle; it is designed to clear it by a margin measured in the same units as the fringes, because clearing it by too little puts the receiver at a minimum instead of a maximum. The customary rule is to keep six-tenths of the first Fresnel zone clear, and the six-tenths is read off the same curve every other number here comes from.

The whole calculation treats a mountain as a rim and ignores everything else about it, which is a considerable simplification and works. What it cannot handle is two hills, where the second edge sees a field that is already not a plane wave, and the corrections for that are a small industry.

What the experiment actually settled #

The two-term account and the smearing account are not merely differently satisfying; they were on opposite sides of the argument that settled the question.

The far-field pattern of a circular aperture is the same edge integral taken round a circular rim, and that is why the diameter appears in every resolution formula. Two point sources are resolved or not according to how far apart their rim-integrals put their central maxima, and the criterion that results — 1.22 λ/D — is a statement about the rim rather than about the lens, the sensor, or the light.

Fresnel submitted his memoir on diffraction to the Académie in 1818. Poisson, examining it, observed that the theory implied something absurd: a circular obstacle should produce a bright spot at the centre of its shadow, because every point of the rim is the same distance from that point and therefore contributes in phase. Arago performed the experiment and the spot was there.

Read through the decomposition, Poisson’s objection is a one-line calculation and not an absurdity at all. The field at the centre of the shadow is the edge wave alone, and the edge wave is an integral around the rim of contributions that are all in phase there by symmetry. Nothing else in the shadow has that property, which is why the bright spot is a point rather than a region.

It is worth being clear about what that settled and what it did not. It settled that light propagates as a wave, because a corpuscular account has no mechanism for a bright point behind an obstacle. It did not settle what is waving, which took another forty years, and it did not settle whether the wave is longitudinal or transverse — a question the fringes here are entirely silent about, and which polarisation answered instead.

The same spot has since been produced with electrons, with neutrons, and with whole molecules, and in each case it is the same integral round the same rim. Everything has a wavelength, and the rim does not care what is arriving at it.

The same two terms behind a noise barrier #

The radio case above puts the receiver on the shadow boundary. The acoustic case usually puts it well inside, and the engineering that results is worth following because it shows what the edge wave’s slow decay costs.

A wall beside a motorway is a half-plane to a sound wave, and a listener behind it is in the geometrical shadow. There is no direct term at all, so the whole of what is heard is the edge wave from the top of the wall. The insertion loss — how much quieter the wall makes things — is therefore a statement about how strong that one term is.

The controlling quantity is the extra distance sound must travel to get over the rim, compared with the straight line it would have taken. Divided by half a wavelength, that gives the Fresnel number $N$, and the loss is very nearly

$$\text{IL} \approx 10\log_{10}(20N) + 5\ \text{dB}$$

for $N$ above about one, which is Maekawa’s curve, drawn from measurements in the 1960s and used ever since. At $N = 0$, with the line of sight just grazing the top of the wall, it gives about five decibels; the theoretical half-plane value is six, and the difference is what a real wall of finite length and a real ground surface cost.

The logarithm is the whole problem. Doubling the height of the wall above the line of sight quadruples the path difference and therefore the Fresnel number, which buys six decibels — and a wall’s height above the line of sight grows much more slowly than the wall does, because raising the top of a wall by a metre when it already stands three metres above the sight line changes the geometry very little. In practice a roadside barrier gives ten to fifteen decibels and going higher gives disappointingly little. There is no arrangement of a single edge that gives thirty.

The frequency dependence follows from the same expression and is the reason barriers work as unevenly as they do. $N$ is inversely proportional to the wavelength, so the loss falls as the frequency falls: a wall that removes twenty decibels of tyre hiss at 2 kHz removes eight at 125 Hz. What is left behind a motorway barrier is the low rumble, which is exactly what residents report and exactly what the edge wave predicts.

None of this depends on what the wall is made of, provided it is heavy enough that nothing goes through it. A barrier is not an absorber; it is a rim, and its performance is fixed by where its top edge is.

The edge wave read as an arrival time #

Everything above is written for a steady wave at one frequency. Reading the same decomposition in the time domain makes it sharper, and it turns the edge wave into something that can be pointed at on an oscilloscope.

Send a short pulse instead of a continuous wave. The geometrical term arrives at the direct travel time and is a copy of the pulse — a spike, if the pulse was one. The edge term cannot arrive until the signal has travelled from the source to the rim and from the rim to the receiver, which is a longer path, so it arrives later, and it arrives as a smeared tail rather than a spike because different parts of the rim are at different distances.

So in the time domain the split is not merely exact, it is separated. The two terms occupy different intervals, and an instrument fast enough to resolve them sees two events rather than a fringe pattern. The fringes of the frequency-domain picture are what those two arrivals become when they are added at a single frequency: a delay becomes a phase, and a phase difference becomes a beat.

That separation is the basis of a standard industrial measurement. In time-of-flight diffraction, two ultrasonic probes are placed either side of a weld, one transmitting and one receiving. What returns is a wave that travelled just under the surface, a wave that bounced off the far wall, and — if there is a crack — a pair of weak signals between them, from the crack’s upper and lower tips. Those are edge waves. Each tip is a rim, radiating in all directions with an amplitude far below any specular reflection, and the arrival times of the two give the crack’s depth and height directly.

The method’s advantage over ordinary pulse-echo inspection is precisely that it uses the weak term rather than the strong one. A specular reflection reports that a crack is there, and its strength depends on the crack’s orientation, so a crack lying the wrong way can be missed entirely. A tip diffraction is weak in every direction and therefore weak toward the probe as well, and its timing is a geometric fact that orientation does not change.

The same reading is used in room acoustics, where the diffracted contribution from the edge of a reflecting panel or a stage riser arrives after the specular reflection and fills in the gap that geometry leaves. There is an exact time-domain solution for a rigid wedge, and it is used in auralisation for the reason that the frequency-domain picture makes obvious: without the edge terms, a simulated room has hard-edged shadows that no real room has.

Where the model runs out #

The half-plane here is a mathematical screen: infinitely thin, perfectly absorbing, and infinite in extent. A real blade has a thickness comparable with a wavelength for X-rays and enormous compared with one for radio, and the field then depends on the material and the profile rather than only on the position of the rim.

A slit’s near-field pattern is two sources sitting at the two rims of the aperture, with fringes between them, and its far-field pattern is what that becomes when the two are close together compared with the distance to the screen. The whole of Fraunhofer diffraction is that limit, which is why it loses the distinction between the two edges that the near field keeps — and why an essay about edges has to be written in the near field.

The decomposition is exact for the scalar field and light is not scalar. Polarisation matters at a real edge: the exact solution for a perfectly conducting half-plane, which Sommerfeld found in 1896 and which is one of the few exactly solvable diffraction problems there are, gives different answers for the two polarisations, and the difference is largest within about a wavelength of the rim. Everything here is the paraxial scalar approximation, which is excellent at distances of many wavelengths and wrong at the rim itself.

The edge wave’s strength is uniform here and is not in general. For a half-plane the rim radiates the same amount in every direction. For a wedge, or a curved edge, or an edge seen at an angle, the strength depends on direction, and Keller’s geometrical theory of diffraction is the systematic version — with the diffraction coefficients doing the work the flat one half does here.

And the split fails exactly where it is most tempting to use it. On the shadow boundary itself the geometrical field is discontinuous, so the two terms are individually badly behaved even though their sum is not, and any asymptotic expansion in the ratio of wavelength to distance breaks down in a region a few fringes wide. The exact computation has no difficulty there; the useful approximations all do, and the repairs to them are a substantial part of the subject.

The ladder from here #

Later rungs on this anchor: the exact half-plane solution and what it says that the scalar one cannot; the geometrical theory of diffraction, where the edge’s radiation acquires a direction dependence and creeping waves are added for smooth bodies; the Poisson spot as an edge integral, where every point of a circular rim contributes in phase at the centre and the answer is a single number; and the corner, which contributes a point source in the same way an edge contributes a line one.

The neighbouring ladders are every front is a source, which is the construction this one is an alternative to, the spiral that says how much arrives, which is the same field read as chords, and where rays stop being enough, which is the far-field limit of everything drawn here.