When the source is not heard all at once

Everything on the rung below this one was computed for a point. The field of an oscillating dipole has three terms, they cross at $\lambda/2\pi$, and none of it says anything about the source having a size.

That looks like an approximation about smallness and it is not. Writing the retarded field of an extended source means integrating over the source with each element evaluated at its own retarded time, and the dipole formula is what is left when those retarded times are treated as equal. So the condition is not that the source is small in metres. It is that the difference in arrival time across it is negligible compared with the period — which is to say that the source is small compared with a wavelength, at whatever distance the observer happens to be.

Drop the assumption and the integral does not go away. It becomes a sum over the source of contributions that arrive at different times, and the interesting part is that the sum has already appeared elsewhere in this collection.

The sum, and where it has been met before #

Put $N$ sources in a row, spaced $d$ apart, all driven identically. An observer at angle $\theta$ from broadside receives a contribution from each, and the path from the $n$th is longer than from its neighbour by $d\sin\theta$. That path difference is a delay, and a delay at a fixed frequency is a phase, so the received field is a geometric series:

$$\text{AF} = \frac{1}{N}\left|\frac{\sin(N\psi/2)}{\sin(\psi/2)}\right|, \qquad \psi = kd\sin\theta.$$

That expression is what a thousand slits buy, letter for letter. The grating’s slits are the array’s elements, its orders are the array’s grating lobes, and its resolving power is the array’s beam width. Nothing was carried over from optics; the two arrive at the same function because both are sums over retarded times across a periodic source.

The habit that makes them the same is worth naming, because it is more general than either. Whenever a source has parts, the field is a sum of what each part did at the time it had to act in order to be heard now. In optics that reads as interference and in radio as an array factor, and the reason the vocabulary differs is historical rather than physical.

One equation under two vocabularies #

The correspondence is worth spelling out, because the two subjects use different words for every single quantity and it hides how complete the identity is.

A grating’s period is an array’s element spacing. A grating’s order is an array’s grating lobe. The grating equation $d\sin\theta = m\lambda$ is the condition that neighbouring contributions differ by a whole number of wavelengths, which is exactly the condition for an array’s extra beams. A grating’s resolving power, $mN$, is the array’s beam width inverted. A blazed grating, whose grooves are tilted to throw energy into one chosen order, is a phased array with a progressive phase applied to steer into one chosen direction — the same trick, invented independently, and named after a fire in one field and after electronics in the other.

Even the failures correspond. A grating with too coarse a period produces overlapping orders that a spectrometer must filter out; an array with too coarse a spacing produces grating lobes that a radar must avoid pointing at. A grating illuminated over only part of its width loses resolving power; an array whose outer elements fail loses beam width. The list runs out only when the physical realisations differ — a grating’s grooves are passive and an array’s elements are driven, so an array can steer and a grating cannot.

What makes the identity total rather than an analogy is that neither derivation contains anything about light or about radio. Both are the sum of a geometric series whose ratio is a phase, and the phase came from a path difference divided by a wavelength. Anything periodic that radiates does this: a row of loudspeakers, a line of ultrasound transducers in a medical probe, an acoustic array on a submarine, the atoms in a crystal scattering X-rays. The last of those is Bragg’s law, which is this sum in three dimensions with the period supplied by the lattice.

That is also why the constraint on element spacing appears in each of them with a different name and the same number. In crystallography it is the statement that reflections exist only for lattice spacings above half the wavelength, which is why X-rays rather than light are used to see atoms. In ultrasound it is why a probe’s elements are 200 micrometres apart at 5 megahertz. In every case the quantity being compared is a path difference against a wavelength, and the arithmetic does not care what is doing the radiating.

The width, and what does not set it #

The first null of the array pattern sits where the total extra path across the whole row is one wavelength, which is $\sin\theta = 1/Nd$ — the aperture in wavelengths, inverted. The main beam’s half-power width lands on $0.886\,\lambda/L$ for a uniformly driven aperture of length $L$.

The thing that is absent from that expression is the count. Eight elements over a metre and eighty over the same metre give the same beam, because the beam is decided by the two ends of the aperture and by the sum in between only through its being filled. What the extra elements buy is elsewhere, and the next figure is where.

This is the whole reason a radio telescope is a large object. At 21 centimetres, the wavelength of neutral hydrogen, a one-degree beam needs an aperture of twelve metres and an arcsecond beam needs forty-three kilometres. No dish is forty-three kilometres across, which is why very long baseline interferometry exists: an array’s resolution depends on the separation of its ends and not on the area between them, so the ends can be put on different continents and the middle left empty. The sensitivity falls with the missing area and the resolution does not — the two are independent, which is exactly what the two expressions above say.

The beams nobody asked for #

The sum repeats. Advance $\psi$ by a full turn and every term returns to where it was, so a second full-strength beam appears wherever the path difference between neighbours is another whole wavelength.

Whether those extra beams land in real angles is decided by the spacing alone. For $d < \lambda/2$ the required $\sin\theta$ exceeds one for every extra order and none of them exists; for $d > \lambda/2$ they arrive.

That is why phased-array elements are half a wavelength apart, a constraint that decides the cost of every such system built. A radar array a metre across at 10 gigahertz needs elements 15 millimetres apart, which is 4,400 of them for a square metre, each with its own phase shifter and amplifier. Spacing them at a wavelength would quarter the count and would also put a second beam at 90 degrees, so a target ahead and a target beside would be indistinguishable. The economics of the whole field sit on that one inequality.

The same statement in optics is that a grating with a period longer than the wavelength produces several orders, which is a nuisance in a spectrometer and the entire purpose of a grating in a diffractometer. In neither case is it a defect: it is a periodic structure doing what a periodic structure does, and whether it is wanted depends on what is being built.

Aiming without moving #

Because the pattern depends on the relative phase between elements and not on any absolute one, adding a progressive phase across the row tilts the direction in which the contributions arrive in step.

The broadening is the part usually left out and it is not small. An aperture steered to sixty degrees presents half its length to that direction, so the beam is twice as wide and the gain is halved, in the same way two things have to be further apart to be told apart when the aperture shrinks. That is why a radar wanting hemispheric coverage uses several flat faces rather than one — an aircraft-nose array typically has a mechanical tilt as well as electronic steering, and a shipboard system has four faces.

The invention is older than its electronics. Karl Ferdinand Braun demonstrated directional transmission from three phased antennas in 1905, sharing a Nobel prize with Marconi four years later for it. What changed the subject was not the idea but the ability to control thousands of phases at once, and that arrived with digital electronics — which is also why the modern version steers by time delay rather than by phase when the signal is broadband, since a phase shift is the right delay at only one frequency and a wideband beam steered by phase points in slightly different directions at its two edges.

The same pattern, run backwards #

An array’s receiving pattern is identical to its transmitting one, and the reason is reciprocity: the coupling between two circuits is the same whichever of them is driven, so the sensitivity of an array to a wave arriving from a direction equals the strength it would send in that direction.

That equality is doing more work than it looks. On transmit, the phases are set by hardware and one beam is produced at a time. On receive, each element’s signal can be recorded separately and combined afterwards — and combined more than once, with different phase sets, from the same recording. A digital array can therefore look in a hundred directions at once, which its transmitting half cannot, and the asymmetry is not a violation of reciprocity but a consequence of what can be stored.

The whole of modern radio astronomy sits on that. An interferometer records the signal at each dish, and the beams are formed later in software from correlations between pairs — which is what a fringe knows about the shape of a source applied to an array rather than to a pair. The same recording, reprocessed, gives a different pointing, a different resolution, or a map assembled from every pair at once.

One element, moved, is an array #

The strangest consequence of the sum is that the elements need not exist at the same time.

An array’s pattern comes from combining contributions with the right relative phases. Nothing in that requires the contributions to have been collected simultaneously: a single element carried along a line, transmitting and recording at intervals, produces the same set of phases as a row of elements would have, provided the geometry is known well enough to reconstruct them. Combining them afterwards synthesises an aperture as long as the path travelled.

That is synthetic-aperture radar, and its result is the one worth being surprised by. The along-track resolution of a synthetic aperture turns out to be half the physical antenna’s length, and it does not depend on the range, on the wavelength, or on how long the synthesis was. A shorter real antenna has a wider beam, so it illuminates each patch of ground for longer, so the synthetic aperture is longer — and the two effects cancel exactly. A satellite four hundred kilometres up with a ten-metre antenna resolves five metres, which no real aperture of ten metres could approach by four orders of magnitude.

Radio astronomy uses the Earth’s rotation the same way, filling in an aperture over hours as the baselines between fixed dishes sweep across the sky. In both cases the sum is the array factor of this essay, performed over positions a source visited rather than over positions it occupied at once.

Why nobody makes a tiny array with a huge aperture’s beam #

The array factor has no lower bound on element spacing, and that has an odd consequence. Alternating the sign of the excitation on closely spaced elements produces a pattern as narrow as one likes from an array as short as one likes — a result derived by Schelkunoff in 1943 and called superdirectivity.

It is real, and it has essentially never been used, and the reason is the rung below this one. Making the pattern narrow with alternating excitations means the elements are almost cancelling each other, so the currents are enormous compared with the radiated field, and the energy stored in the near field grows without bound relative to the energy radiated. That ratio is the antenna’s $Q$, and a high-Q antenna is a narrowband and lossy one: a modest amount of superdirectivity costs a bandwidth of parts per million and requires conductor losses that no metal has.

So the array factor is not the whole story, and the missing constraint is stored energy rather than geometry. It is a good example of a limit that lives outside the equation being solved — nothing in the sum forbids superdirectivity, and everything about the field the sum ignores does.

Where treating the source as a point actually fails #

The figures so far are arrays of separate elements. A continuous source is the same sum with the spacing gone to zero, and it is the case that says what the dipole approximation was assuming.

The progression is the answer to when the dipole formula may be used, and it is quantitative rather than a matter of taste. Below about a tenth of a wavelength the error is under two per cent and the source is a dipole for any practical purpose. By half a wavelength the error is tens of per cent, and a half-wave dipole — the commonest antenna there is — is emphatically not described by the dipole approximation despite the name. Past a wavelength the pattern has structure the point formula cannot produce at all, because nulls are not a small correction to a circle.

What is happening is retardation across the source rather than out from it. The far end of the antenna is heard later than the near end by the extra path over $c$, and when that delay approaches half a period the two ends are pulling in opposite directions and their contributions subtract.

The same arithmetic explains a fact about long wire antennas that looks like a defect. A wire several wavelengths long has a pattern of many narrow lobes and a null straight out to the side, so it is a poor omnidirectional antenna and a good directional one — and which of those it is was decided by the length in wavelengths rather than by anything about its construction. The rhombic antennas used for long-distance shortwave in the 1930s were several wavelengths on a side for exactly this reason: the lobe structure was the point.

Where this stops being right #

The elements have been treated as identical and independent, and they are neither. A real array element sits in the near field of its neighbours, so it is loaded by them: its impedance changes with the steering angle, and at certain angles the array can refuse to radiate altogether. Scan blindness is that effect, it is not visible in the array factor at all, and it is found by modelling the whole aperture rather than by multiplying a pattern by a factor.

The array factor multiplies the element pattern rather than replacing it. Every figure here assumes the individual source radiates equally in the plane drawn, which no real antenna does. The full pattern is the product of the two, so an element with a null in a direction gives the array a null there whatever the phases.

The excitation has been uniform. Feeding the ends less than the middle — tapering — lowers the sidelobes and widens the main beam, and the trade between the two is what most array design consists of. A uniform aperture has the narrowest possible main beam and the worst sidelobes, at −13 decibels, and softening the edges is the same manoeuvre as a taper that matches every note.

And everything is monochromatic and far field. Close in, the path differences are not a linear function of angle and the pattern has not formed yet; the distance at which it has is conventionally $2L^2/\lambda$, which for a large array is a considerable way off.

What the drawings leave out #

Each polar figure draws the magnitude of a sum and discards its phase, which is exactly what a power measurement does and exactly what makes the pictures misleading about interferometry. Two arrays whose patterns are drawn identically here can differ entirely in what they do when their outputs are combined, because the combination is sensitive to the phase the drawing threw away.

And nothing here shows the time. The whole subject is that different parts of a source are heard at different moments, and every figure is a steady-state pattern in which all of that has already been summed and forgotten. A picture of an array factor is a picture of the answer to a question about timing, with the timing removed.

What the ladder has and has not covered #

Three rungs stand on retardation. The first split a charge’s field into a piece that stays and a piece that leaves; the second found the three terms of an oscillating source and the single radius where they cross; this one drops the assumption that the source is heard all at once and finds that the resulting sum is the one optics has been performing since Young.

The habit worth carrying away is about approximations named for the wrong quantity. The dipole approximation is named for the size of the source and is a statement about the spread of its retarded times. Once that is seen, the condition writes itself — compare the spread with the period — and the cases where a “small” source fails the approximation stop being surprising. The same renaming clarifies the paraxial approximation, which is about the spread of angles rather than about being near an axis, and the sudden approximation in quantum mechanics, which is about a duration rather than about a magnitude.

What is left on this ladder is the assumption underneath all three rungs, which has never been argued for at any of them. Every field written here has been the retarded solution. Maxwell’s equations have another one, exactly as good, in which the field converges on the source instead of leaving it — and nothing in the equations chooses between them.