The magnet period that comes out as an X-ray

The length that depends on when introduced length contraction as a disagreement about simultaneity: a moving rod is measured shorter because marking both ends “at the same time” means different things to different observers. The pole that fits and does not fit and the string that breaks between two rockets pushed it into paradox, the contraction no photograph shows found that a camera sees a rotation rather than a squash, and the contraction the forces work out for themselves derived it from the forces inside an atom instead of from geometry. Every one of those arguments was about establishing that contraction is real and what exactly it is.

This essay is about what it is for. There is a class of machines in which length contraction is not a correction but the working principle, and they are among the most productive scientific instruments ever built: the undulators that line the storage rings of every synchrotron light source and form the long straight sections of every X-ray free-electron laser. Each is a row of magnets a few centimetres apart. Each makes X-rays with wavelengths a hundred million times shorter than that spacing. The factor between them is $2\gamma^2$, and half of it is contraction.

A row of magnets #

An undulator is simple. Two rows of permanent magnets face each other across a gap of a centimetre or so, arranged so that the magnetic field between them points up, then down, then up again, alternating every half of a period $\lambda_u$, typically two to five centimetres. A beam of electrons travels along the gap. Each time the field reverses, the force on the electrons reverses, and they wiggle from side to side in a gentle sine wave, a few micrometres across, as they go.

A wiggling charge radiates, as a charge that turns must glow established. A slow electron wiggling once every three centimetres would radiate at the frequency of its wiggle — a wave whose wavelength is three centimetres divided by its speed over the speed of light, which for a slow electron is long, a microwave. The electrons in a light source are not slow. At six gigaelectronvolts their Lorentz factor $\gamma$ is 11,742, and the light comes out at an ångström and a half. To see why, it is easiest to ride with the electron.

The view from the electron #

In the frame moving with the electron’s average forward velocity, the electron is nearly at rest, wiggling only sideways, and the undulator is the thing that moves. It rushes towards the electron at almost the speed of light. A moving object is contracted along its motion by $\gamma$, so the row of magnets, three centimetres per period in the laboratory, is in this frame a row with a period of $\lambda_u/\gamma$ — 2.55 micrometres for a six-gigaelectronvolt beam.

A row of magnets moving at nearly the speed of light is not, in this frame, a purely magnetic thing. The charge that passes as a flash of light found that a field moving at high speed becomes an electric and magnetic field at right angles to each other and to the motion, of nearly equal strength in the right units: a passing electromagnetic wave. So the electron, sitting nearly still, is struck by a wave of wavelength 2.55 micrometres and frequency $\gamma c/\lambda_u$, coming from straight ahead. It does what any electron struck by a light wave does. It oscillates at the wave’s frequency and re-radiates it — the scattering of the cross-section that forgets the colour — in a dipole pattern, at 2.55 micrometres, in the infrared.

That is the first factor of $\gamma$, and it is length contraction, in its plainest form. The undulator has been made $\gamma$ times shorter by being moved, and the radiation the electron makes is set by the shortened length.

The view from the laboratory, again #

The infrared light the electron emits in its own frame goes out in all directions, but the part that matters is the part going forward, in the direction the laboratory sees the electron moving. Transformed back to the laboratory, light emitted forward by a source moving towards the observer at nearly $c$ is Doppler-shifted, by the factor $\sqrt{(1+\beta)/(1-\beta)}$, which for $\beta$ close to one is very nearly $2\gamma$. A 2.55-micrometre wave is shortened by a further 23,500, to 1.09 ångströms.

Two factors of $\gamma$, one from each frame: the first because the magnets are moving towards the electron, the second because the electron is moving towards the observer. The result is

$$\lambda = \frac{\lambda_u}{2\gamma^2}.$$

The shift a mirror gives twice found the same structure for light reflected from a moving mirror: one Doppler factor on the way in, one on the way out, and a frequency multiplied by roughly $4\gamma^2$. An undulator is a mirror made of magnets: its static field, seen from the electron, is incoming light; the electron reflects it; and the reflected light comes out multiplied by $\gamma$ squared. In the trade, undulator radiation is described as Compton scattering of the undulator’s “virtual photons”, and the description is not a metaphor.

The same number without changing frame #

A result that comes from switching frames ought to be visible without switching, and it is, in a form that reveals why the light is coherent.

In the laboratory, the electron moves forward at slightly less than the speed of light, and the light it emits moves forward at exactly the speed of light. Pick one point in each wiggle — the same phase each time — and follow the wavefront emitted there. By the time the electron has travelled one more period and reaches the same phase again, the first wavefront has got ahead of it by the distance light travels in the time the electron took, minus the period: $\lambda_u(1/\bar\beta - 1)$, where $\bar\beta c$ is the electron’s average forward speed. That distance is the spacing between successive wavefronts, which is the wavelength. For $\gamma$ large, $1/\bar\beta - 1$ is $1/2\gamma^2$ to first order, and the slip is $\lambda_u/2\gamma^2$.

The figure draws it for an electron at a Lorentz factor of only three, where the slip is a tenth of a period and can be seen. The computed slip, from integrating the wiggling path, is 0.102 of a period; the large-$\gamma$ formula gives 0.096, already within seven per cent. At $\gamma$ in the thousands the slip is a hundred-millionth of the period and the formula is exact for all practical purposes.

The laboratory picture adds something the frame-hopping argument hides. The wavefront emitted in every period is exactly one slip behind the one before, so the light from all the periods arrives in step at that wavelength, and adds. An undulator of a hundred periods produces a train of a hundred waves, with a spread of wavelengths about a hundredth of the central one. A bending magnet, which makes the electron turn once instead of wiggling many times, produces a single flash and a broad spectrum. That is why an undulator’s light is concentrated into a narrow band of colour and is, at that colour, many orders of magnitude brighter than a bending magnet’s.

A count that every frame agrees on #

There is a check on all this that needs no formula at all, and it is the best evidence that the contraction is doing real work rather than being a manner of speaking. An undulator of $N$ periods makes the electron wiggle $N$ times. That number is a count of events — the electron passing the centre of a magnet — and a count cannot depend on who is counting. In the laboratory the electron crosses $N$ magnets of length $\lambda_u$ each. In its own frame it must also meet $N$ magnets, and since they rush past it at nearly $c$, the time it spends among them is the undulator’s length in that frame divided by $c$. If the undulator were not contracted, the electron would spend $N\lambda_u/c$ there and oscillate at a frequency of $c/\lambda_u$, radiating a long-wavelength wave. It oscillates $N$ times in a time $N\lambda_u/\gamma c$, because the undulator is $\gamma$ times shorter.

So the electron’s own light is a train of $N$ waves, each 2.55 micrometres long in the example. The laboratory sees the same $N$ waves, since they are the same crests counted again, packed into a train $N$ times 1.6 ångströms long. The light’s relative bandwidth is about $1/N$ in both frames, because the train has the same number of waves in both. A hundred-period undulator, three metres long, gives light with a spread of about one per cent in wavelength, in whichever frame it is measured.

The same bookkeeping in time is just as tidy. The electron takes about ten nanoseconds to cross three metres of undulator. In its own frame it takes $\gamma$ times less, under a picosecond. And the X-ray pulse it emits forward, $N$ waves of 1.6 ångströms, lasts about fifty attoseconds: the electron spends ten nanoseconds in the undulator and the light it makes there arrives in a twenty-millionth of that time, because the electron chases its own light almost all the way. That compression is the second $\gamma$ seen as time instead of length, the same compression everything from an exchange of pulses built a whole account of relativity on. Every factor in the problem is one of these two, applied to a length or to a time, and the number of waves is the one quantity that both frames hold fixed.

Why the wiggle reddens the light #

The formula needs one correction, and it comes from the wiggle itself. An electron moving at nearly the speed of light along a sinuous path is moving more slowly forward than it would along a straight one, because part of its velocity is sideways. A slower forward speed means a larger slip per period, and so a longer wavelength. The size of the wiggle is set by the strength of the field: the deflection parameter $K = eB_0\lambda_u/2\pi m c$, which in practical units is 0.934 times the peak field in tesla times the period in centimetres, is the electron’s maximum angle to the axis in units of $1/\gamma$. Averaging the forward speed over a wiggle gives

$$\lambda = \frac{\lambda_u}{2\gamma^2}\left(1 + \frac{K^2}{2}\right).$$

That factor is the knob. Open the gap between the two rows of magnets and the field on the axis weakens, $K$ falls and the light gets bluer; close the gap and $K$ rises and the light reddens. At six gigaelectronvolts with a three-centimetre period, the forward photon energy runs from 11.4 kiloelectronvolts with the gap wide open to 7.6 at $K = 1$ and 3.8 at $K = 2$. Above $K$ of about one the electron’s motion, seen in its own frame, is no longer a simple sideways oscillation — the strong wiggle drags it back and forth along the axis as well, in a figure of eight — and it radiates harmonics, odd multiples of the fundamental, on the axis. Using the third and fifth harmonics, the same magnets reach tens of kiloelectronvolts.

A colour that changes with angle #

The Doppler factor that shortened the forward light is largest straight ahead and falls off at an angle. Light leaving the electron at an angle $\theta$ to its motion is shortened by less, and its wavelength picks up a further term: $\lambda = (\lambda_u/2\gamma^2)(1 + K^2/2 + \gamma^2\theta^2)$. At an angle of $1/\gamma$, eighty-five microradians for a six-gigaelectronvolt beam, the wavelength has grown by a factor of between 1.3 and 1.9, depending on $K$. An undulator’s light is therefore a narrow cone, a few tens of microradians across, bluest at its centre and reddening outward, as the sky that crowds into a cone would predict for any source moving this fast: the electron’s dipole pattern, broad in its own frame, is squeezed by aberration into a forward cone of half-angle about $1/\gamma$. A small aperture on the axis passes a narrow band of colour; a wide one passes a rainbow.

Centimetres in, ångströms out #

Because the wavelength goes as $1/\gamma^2$, the same magnets cover the whole spectrum from the infrared to hard X-rays as the electron energy is raised over two decades. A beam of a hundred megaelectronvolts, from a modest linear accelerator, makes visible light from a three-centimetre undulator. A beam of one gigaelectronvolt makes extreme ultraviolet at six nanometres. A storage ring at six gigaelectronvolts makes X-rays at an ångström and a half, the size of an atom and the wavelength crystallographers want. An X-ray free-electron laser at fourteen gigaelectronvolts reaches a third of an ångström with these magnets. No other way of making light can be tuned across that range by turning one knob, the energy of the electrons, while keeping the same hardware.

The comparison with a bending magnet sharpens the role of contraction. The flash a circling charge sends once a turn found that an electron bent round a circle radiates a flash whose characteristic frequency goes as $\gamma^3$: one factor from the circling and two from the compression of the flash in time, because the electron nearly keeps up with the light it emits. In an undulator the electron’s direction never changes by more than $K/\gamma$, the circling is replaced by a fixed spatial period, and the frequency goes as $\gamma^2$: contraction in one frame, Doppler in the other. The extra $\gamma$ of the bending magnet buys reach in photon energy; the undulator gives it up for coherence across many periods.

When the light talks back to the electrons #

In an ordinary undulator each electron radiates independently, and the light from a bunch of a billion electrons is the sum of a billion independent trains, adding in intensity. In a long enough undulator, with a bright enough beam, something else happens. The light the electrons have already emitted travels along with them, slipping ahead by one wavelength per period, and its field acts back on them, speeding up some and slowing others. Over many periods this bunches the electrons into slices exactly one wavelength apart, and electrons in step radiate in step: their fields add, so the power grows as the square of the number of electrons in each slice rather than in proportion to it, and the light grows exponentially along the undulator until the bunching saturates.

That is the free-electron laser, and the slip of one wavelength per period is what makes it possible: it is the condition that keeps the light and the electrons’ bunching in step over hundreds of periods. The X-ray free-electron lasers built since 2009 use undulators a hundred metres long, electron beams of up to seventeen gigaelectronvolts, and produce pulses of femtoseconds with a billion times the peak brightness of a storage ring. They are used to take snapshots of molecules in the middle of chemical reactions, and every one of their photons began as a centimetre of magnet, contracted.

What the pictures cannot show #

The formulas describe a single electron on the axis of a perfect undulator. A real beam has a spread of energies, which spreads the wavelength by twice the relative energy spread; a spread of angles, which mixes in the redder off-axis light; and a finite number of periods, which sets a minimum bandwidth. The frames figure is schematic: the three periods it draws differ by factors of twelve thousand and of a hundred million and cannot be shown to one scale. The slippage figure is drawn at $\gamma = 3$, where the effect is visible but the formula’s large-$\gamma$ approximation is off by seven per cent. And the harmonics in the tuning figure are drawn at their positions only; their strengths depend on $K$ and vanish for small $K$, which the figure does not show.

Still open: how short a pulse, how hard a photon #

The frontier of these machines is now in time rather than in wavelength. X-ray free-electron lasers produce pulses of a few femtoseconds routinely and of a few hundred attoseconds by shaping the electron bunch so that only a short slice of it lases; how short the pulses can be made, how stable their timing can be held against a laser that triggers the experiment, and whether two colours can be produced at once with controlled delay are all being pushed now. At the other end, the photon energy of the fundamental is limited by $\gamma^2$ and the shortest practical period; undulators with periods of millimetres, made of superconducting coils or with laser light itself standing in for the magnets, could reach the same X-rays with electrons ten times less energetic, and with accelerators small enough for a laboratory rather than a national facility. Several are being built.

The habit worth carrying away is to look for the frame in which a device is simple. In the electron’s frame an undulator is a row of magnets contracted by γ and rushing past as a wave, which the electron scatters; seen back in the laboratory, that forward light is shortened by another 2γ, so a 3 cm period makes 1.6 Å X-rays at 6 GeV — $\lambda = (\lambda_u/2\gamma^2)(1 + K^2/2 + \gamma^2\theta^2)$. Length contraction is the first of the two factors, and the slip of one wavelength per period is what lets the light from every period add.