The rainbow a passing electron leaves on a grating

In 1953 Steve Smith, a graduate student at Harvard, sent a beam of 300 keV electrons from a Van de Graaff accelerator skimming over the surface of an optical diffraction grating — a mirror ruled with six hundred grooves to the millimetre — and saw light. Not a glow at the point where electrons struck the metal, but a spread of colours coming off the whole length of the grating the beam had passed over, changing from blue-green to red as the angle of viewing changed. He and Edward Purcell published it as “visible light from localized surface charges moving across a grating”, and the effect has been called Smith–Purcell radiation since.

What is surprising is the electron. It was not turned, not slowed, not shaken: it moved in a straight line at constant speed over a surface it never touched. A charge that turns must glow, and a charge moving uniformly in empty space must not — its field travels with it, a flattened disc that the charge that passes as a flash of light found looking like a pulse of light to anyone it passes, and yet carrying no energy away. Two neighbouring arguments in this collection already found exceptions, both involving matter: why the glow of a fast charge is blue followed a charge faster than light in water, and the light a charge makes by changing medium one crossing a boundary. The grating is the third exception, and it explains the other two.

A clock with a period of one groove #

The simplest way to find the colours is to treat each groove as a source. The passing electron induces charges in the metal beneath it — its image, following it along the surface — and each time the image crosses a groove, the surface current it drives is disturbed and the groove radiates a small pulse. The electron crosses one groove every $d/v$ seconds, where $d$ is the groove spacing. Light leaving at an angle $\theta$ from the beam’s direction, from two neighbouring grooves, starts $d/v$ seconds apart, but the light from the later groove has a head start of $d\cos\theta$ along the direction of travel. The pulses add in step when the difference is a whole number of periods of the light:

$$\frac{d}{v} - \frac{d\cos\theta}{c} = \frac{n\lambda}{c}, \qquad\text{so}\qquad \lambda = \frac{d}{n}\left(\frac1\beta - \cos\theta\right).$$

That is the Smith–Purcell relation. It has the form of the grating equation that what a thousand slits buy used for light falling on a grating, with $1/\beta$ in place of the sine of an angle of incidence — which is possible because $1/\beta$ is larger than one, so the “incident wave” here is not a wave that light could be.

Each angle has its own wavelength, and the whole range is swept as the angle goes from forward to backward: from $d(1/\beta - 1)$ straight ahead to $d(1/\beta + 1)$ straight back. For Smith and Purcell’s electrons that is 482 nanometres ahead — blue-green — and visible light only out to 30°; at larger angles the light is infrared. A single electron passing a single grating produces a rainbow laid out by angle, exactly as a grating does with white light, except that there was no white light.

The shortest wavelength a grating can make, straight ahead, depends on the speed. A slow electron at 0.3 of light’s speed makes nothing shorter than 2.33 periods; a fast one at 0.99c makes 1 per cent of a period. In the relativistic limit $1/\beta - 1 \approx 1/2\gamma^2$, so the forward wavelength is $d/2\gamma^2$ — the same factor that the magnet period that comes out as an X-ray found for light from a row of magnets. The grating and the undulator are both periodic structures that an electron sees rushing past, and their forward light is compressed by the same Doppler factor.

The wavenumber the field was missing #

The groove-counting argument gets the colours, but not the reason anything is radiated at all. That reason is clearest in terms of waves. The field of a charge moving at speed $v$ along a line is a pattern that moves with it, so every Fourier component of it has a frequency $\omega$ and a wavenumber along the line $k$ related by $\omega = vk$. Light in empty space travelling at an angle to the line has $\omega = ck/\cos\theta$, so its wavenumber along the line is at most $\omega/c$. Since $v < c$, the charge’s components all have $k = \omega/v$, larger than $\omega/c$ — too much wavenumber along the line for any light wave to match. They are bound to the charge, decaying away from it, and they cannot escape. That is why a uniformly moving charge does not radiate, stated in one line.

A periodic surface changes the accounting. A field reflected from a grating of period $d$ acquires wavenumbers shifted by whole multiples of $2\pi/d$ — the same shift that sends light into the diffracted orders of a grating. Shift the charge’s components by $-2\pi n/d$, and some of them now have a wavenumber small enough for light:

$$k\cos\theta = \frac{\omega}{v} - \frac{2\pi n}{d},$$

which is the Smith–Purcell relation again, read as conservation of wavenumber along the grating.

In that diagram the three exceptions to “a uniformly moving charge does not radiate” are three ways of getting the charge’s line into the cone. In a medium where light is slower than the charge, the cone itself opens wider — light’s line is $\omega = ck/n$, flatter than the charge’s — and the charge’s own components fall inside it without any help: that is Cherenkov radiation, and its angle $\cos\theta = 1/n\beta$ is the condition that a component match a light wave. At a boundary between media the field must change abruptly in space, and an abrupt change contains every wavenumber, some of which fall inside the cone: that is transition radiation, one flash at one place. On a grating the shift is periodic, the matching repeats every groove, and the radiation comes out as a sharp relation between angle and wavelength.

Smith–Purcell radiation is Cherenkov radiation off a spatial harmonic of the grating. The grating’s periodic field, seen by the electron, is a wave whose phase moves along the surface slower than light, and an electron faster than that phase velocity radiates into it as it would into slow light in water. That is why no threshold appears: the grating’s harmonic can always be made slow enough, by making its period small enough.

Every order at every angle, and why a magnet row is brighter #

The relation has an $n$ in it, and the higher orders are always present. At any angle where the first order sends wavelength $\lambda$, the second sends $\lambda/2$ and the third $\lambda/3$, exactly as a grating illuminated with white light sends overlapping orders to the same angle. A spectrometer pointed at the grating from 20° would see a line at the first-order wavelength and fainter lines at its half and third. The orders are weaker in a way the groove-counting picture makes plain: the $n$-th order needs the field’s component at $n$ times the frequency, whose reach is $n$ times shorter, so at a fixed beam height each higher order is suppressed exponentially more. For a beam grazing the grating, the second order is comparable to the first; for a beam passing at the edge of the first order’s reach, it is almost absent.

The comparison with the undulator is worth making in the same terms, since both structures make forward light at $d/2\gamma^2$. In an undulator the electron is deflected sideways by the magnets, and the light comes from the electron’s own acceleration — the Larmor radiation of a charge that turns, organised into a narrow line by the periodicity. On a grating the electron is not deflected at all. The light is made by the metal, from currents the electron’s field drives in it, and only the fraction of the field that reaches the metal takes part. That is why an undulator of a given length and period is far brighter than a grating: it converts the electron’s whole field, through its motion, where the grating converts only the evanescent tail that touches it.

What the grating buys in exchange is the period. Magnets cannot usefully be made shorter than a few millimetres, because the field of an alternating row decays from its surface over a distance comparable to its period, and a millimetre-period undulator has almost no field at the beam. Gratings can be ruled or etched with periods of a few hundred nanometres, and their period sets the wavelength directly. For a beam of tens of keV, which no undulator can use, a grating is the only periodic structure that makes visible light at all.

How close the electron has to pass #

The figures so far say what colours come out. How much light comes out depends on how much of the charge’s field reaches the grating, and that is where the effect gets difficult. The bound components that the grating converts decay away from the charge’s line, and at wavelength $\lambda$ they reach a distance

$$h_{\text{int}} = \frac{\beta\gamma\lambda}{4\pi}$$

before falling by a factor of $e$. The emission, which goes as the square of the field at the grating, falls as $e^{-2h/h_{\text{int}}}$ with the beam’s height $h$ above it.

For Smith and Purcell’s green light the reach is 49 nanometres. Their electrons that produced visible light were the ones skimming within a fraction of a wavelength of the metal — most of the beam passed too high to contribute, and the light was faint. The reach grows with $\gamma$ because the field of a fast charge is flattened into a thin disc, and a thin disc has Fourier components at higher frequencies at the same distance; the charge that passes as a flash of light drew that flattening. A 30 MeV electron reaches 2.4 micrometres for green light. At a millimetre wavelength, a 15 MeV beam reaches 2.4 millimetres, comfortably larger than any practical beam size.

That is why the effect’s main uses are at long wavelengths with relativistic beams. The bound field is the same evanescent field that the mirror that lights a tenth of a micrometre found at a totally reflecting surface, with the electron playing the part of the light that does not get out; the grating is what lets it out, as a prism pressed against the surface let the evanescent light out in the reflection that happens where the glass is not. In both cases the coupling falls exponentially with the gap, and the gap is fixed by the wavelength.

Tuning by speed and by angle #

A source whose colour depends on angle is a monochromator built in. At a fixed angle the wavelength is fixed by the grating and the electron’s speed together, and the two parts behave differently as the speed rises.

Sideways and backwards, once the electron is relativistic, $1/\beta$ is one and the wavelength depends on the grating alone: $d$ at 90°, $1.87d$ at 150°. A relativistic beam over a grating therefore gives a spectrum that hardly depends on its energy except straight ahead, which is a convenient property for a source — the beam energy can drift without moving the colour — and an inconvenient one for reaching short wavelengths, which needs both high energy and observation close to the forward direction, where the emission is weak.

A slow beam is different. At 10 keV, the energy of a cathode-ray tube’s beam, the wavelength is at least 4.1 periods at any angle and moves with the beam voltage everywhere. Low-energy versions have been made on gratings with periods of a few hundred nanometres, by running the beam of an electron microscope over them; the light is visible and tunable by the microscope’s voltage, though faint, because a slow electron’s field reaches only nanometres.

A bunch that radiates as one charge #

One electron radiates a little. A bunch of $N$ electrons radiates $N$ times as much if their emissions add with random phases — and $N^2$ times as much at wavelengths long compared with the bunch, where they all radiate in step. The enhancement is $1 + N|F(\lambda)|^2$, where $F$ is the Fourier transform of the bunch’s shape at the radiated wavelength: for a Gaussian bunch of length $\sigma$, the coherence survives down to wavelengths of about $2\pi\sigma c$.

At $10^8$ electrons the coherent part is a hundred million times brighter than the incoherent, and where it turns over is a measurement of the bunch. Because each angle carries its own wavelength, a ring of detectors around a grating reads the coherent spectrum all at once, angle by angle, and the turnover gives the bunch’s length without touching it — a grating beside a beam used as a ruler for bunches a few hundred femtoseconds long. Systems of this kind have been tested on linear accelerators, including the bunches driving plasma-wakefield experiments, where no other method was fast enough.

The same coherence is how Smith–Purcell devices make power. If the radiated field acts back on the bunch, it can modulate it at the radiated wavelength, and a bunch modulated at the wavelength it radiates radiates more — the feedback that makes a free-electron laser. Millimetre-wave oscillators working this way, with an open mirror cavity above a grating, were built in the 1960s, and terahertz versions remain an active subject.

The process run backwards #

Every radiation process has an inverse. If an electron passing a grating can give energy to a light wave whose wavenumber matches its field’s shifted component, then a light wave shone onto a grating can give energy to an electron passing it, if the same matching holds. Inverse Smith–Purcell acceleration was demonstrated in the 1970s, and in 2013 two groups, at Stanford and at Erlangen, accelerated electrons with laser light on glass gratings with periods under a micrometre — the dielectric laser accelerator, now built on chips.

It is one of the ways round the theorem the light that moves an electron and pays it nothing found for an electron in a plane wave in empty space. A plane wave cannot give an electron net energy, because its phase moves faster than the electron and the electron’s response averages to nothing. Near a grating the light has a component whose phase moves along the surface at the electron’s speed, and the electron can ride it. The structure that let the electron’s field out is the structure that lets the light’s field in.

What the figures leave out #

The figures give the wavelengths, the reach and the coherence, all of which follow from the geometry. They do not give the intensity at each angle, which depends on the grating’s groove profile — a sawtooth, a sinusoid, a set of rectangular slots — and on the metal’s conductivity at the radiated frequency. Several theories of the intensity exist, from treating each groove as an independent radiator to solving the full boundary problem for a perfectly conducting grating, and they differ by factors of several for real profiles; the measured angular distributions have been used to choose between them. The figures also take the beam to be thin and straight, the grating infinite in width, and the electrons non-interacting except through coherence. The domain of what is drawn is a charge in uniform motion parallel to a periodic conducting surface, at a height comparable to $h_{\text{int}}$ or less.

Still open: how bright a grating can be made to shine #

The open questions are about power. A grating extracts energy only from the part of the beam within the evanescent reach, so most of a beam passes uselessly; tightly focused beams, cylindrical gratings surrounding the beam, and structures that turn the grating into a cavity all try to raise the fraction of the beam that couples. Whether a Smith–Purcell free-electron laser can be made to oscillate reliably at terahertz frequencies from a compact, low-energy beam — whether the gain can exceed the losses in a structure small enough to be practical — has been claimed and disputed since the early 2000s, and the debate turns on the mechanism by which the bunching starts.

The radiation itself is understood exactly. A charge moving uniformly at $\beta c$ over a grating of period $d$ radiates $\lambda = (d/n)(1/\beta - \cos\theta)$ at angle $\theta$ — 482 nanometres straight ahead from 300 keV electrons over a 1.67-micrometre grating, $d/2\gamma^2$ ahead for a fast one — because the grating shifts the charge’s bound field by $2\pi/d$ into the light cone; the field reaches only $\beta\gamma\lambda/4\pi$ from the charge, 49 nanometres for that green light. A uniformly moving charge radiates whenever the world around it has a period, a boundary or a slower light to lend its field the wavenumber it lacks.