The wave an electron rides to a gigaelectronvolt

The spring that becomes a light-clock followed an electron oscillating in a plasma and found that a strong enough oscillation stops being harmonic: the electron’s speed saturates at the speed of light while its momentum keeps growing, and the period stretches with amplitude. It ended on the use that oscillation has been put to. Drive a plasma’s electrons with an intense laser pulse and they oscillate collectively behind it, in a wave that travels with the pulse at nearly the speed of light, and electrons riding that wave are accelerated with fields no conventional accelerator can reach.

The idea was proposed by Toshiki Tajima and John Dawson in 1979, and it took twenty-five years and the invention of lasers producing pulses tens of femtoseconds long and terawatts strong to make it work. In 2004 three groups reported, in the same issue of one journal, electron beams with narrow spreads of energy accelerated to a hundred million electronvolts in a few millimetres; within a decade, beams of several gigaelectronvolts were being produced in plasma a few centimetres long. The physics that decides how far this can go is the physics of an electron riding a wave that moves almost as fast as it does.

A field that cannot break down #

A conventional accelerator pushes particles through a chain of metal cavities in which a radio wave builds up an oscillating electric field. The field cannot exceed about a hundred million volts per metre, because above that the metal surfaces break down — electrons are torn out of them and arcs form. So reaching high energies takes length: the Stanford linear accelerator took three kilometres to reach fifty gigaelectronvolts.

A plasma has no such limit, because it is already broken down. Its field is the field of its own separated charges: electrons displaced from the positive ions behind them, pulled back by the charge imbalance, overshooting, oscillating at the plasma frequency. The largest field such an oscillation can have is set by how far the electrons can be displaced before the wave destroys itself, and for a cold plasma that is about $E_0 = mc\omega_p/e$ — ninety-six billion volts per metre at a density of $10^{18}$ electrons per cubic centimetre, a twentieth of the number of molecules in air. That is a thousand times the field of the best cavities, and it scales as the square root of the density.

Riding a wave that moves at nearly c #

A field that oscillates in time does nothing on average to a particle that stays in one place. The trick is that the plasma wave is not standing still: it travels behind the laser pulse at the pulse’s group velocity, which in a thin plasma is only slightly less than the speed of light. An electron moving at nearly the same speed stays at nearly the same phase of the wave for a long time, and if that phase is one where the field accelerates it, it is accelerated for as long as it stays there. That is surfing, and the drawing shows it as orbits.

In the frame of the wave the electron’s motion conserves a single quantity, a combination of its energy, its momentum and the wave’s potential at its position. Its possible motions are the level curves of that quantity, drawn here as energy against position over two wavelengths. They fall into two families separated by a curve called the separatrix. Outside it, an electron is either too slow, and the wave overtakes it, gaining and losing energy in turn as crests pass; or it is fast enough to outrun the wave, and similarly gains nothing on average. Inside it, the electron is trapped. It oscillates back and forth through one trough of the wave, and along the top of the separatrix it reaches an energy far above where it started.

The separatrix’s height is the whole promise of the method. For a wave whose phase moves with Lorentz factor $\gamma_p$ and whose field is a fraction ε of $E_0$, its top reaches about $4\gamma_p^2\varepsilon$ times the electron’s rest energy. The factor $\gamma_p^2$ is large because the wave moves so nearly with the electron: an electron slightly faster than the wave takes a very long time to slip through it, and all that time it is being pushed.

An electron that outruns its wave #

The same surfing sets the limit. The electron starts at the wave’s speed at the front of the accelerating half of a trough. The field pushes it; it speeds up, and a relativistic electron speeding up changes its velocity hardly at all — its speed saturates while its energy keeps rising — but it does change it, and it is now faster than the wave. It begins to slip forward through the trough. Half a plasma wavelength later it reaches the point where the field reverses, and from there on it is decelerated. This is dephasing, and the drawing follows it: energy rising to a peak and falling away.

How far the electron rides before it dephases is set by how much faster than the wave it becomes. The relative speed between an electron at nearly $c$ and a wave with Lorentz factor $\gamma_p$ is about $c/2\gamma_p^2$, so slipping half a plasma wavelength takes a distance of order $\gamma_p^2$ plasma wavelengths. The energy gained is the field times that distance. Doubling $\gamma_p$ roughly quadruples both the distance and the energy, which the drawing checks by tracing the motion through the wave step by step while holding the conserved quantity fixed.

The phase velocity of the wave equals the group velocity of the laser pulse that drives it, which in a plasma is less than $c$ by an amount that grows with the density. That is the speed that carries the signal, not the phase velocity of the light itself, which in a plasma exceeds $c$. A thinner plasma lets the pulse travel faster, raises $\gamma_p$, and lengthens the ride.

The factor four gamma squared, seen from the wave #

The factor $4\gamma_p^2$ has a simpler reading in the frame that moves with the wave. There the wave stands still, a row of potential hills, and a trapped electron is a ball rolling in one of the valleys: it arrives from one side, climbs, turns and leaves the other way. In the wave’s frame nothing changes its energy — the potential is static — so the electron simply reverses its direction of motion. Transforming back to the laboratory, an electron that was nearly at rest there — moving backwards through the wave at the wave’s own speed — ends up moving forwards at that speed, and the laboratory sees its energy raised to about $2\gamma_p^2$ times its rest energy: the same kind of factor as the $4\gamma^2$ by which a mirror moving at nearly the speed of light multiplies the frequency of light reflected from it. A wave too weak to turn round an electron at rest can only turn round electrons already moving nearly with it, and then the gain is the smaller $4\gamma_p^2\varepsilon$ of the separatrix.

That is the reason a factor of $\gamma_p^2$ appears, and it is the same mechanism that makes a ball bouncing off a wall moving towards it come back faster, the mechanism behind the acceleration of cosmic rays at shock fronts. A trapped electron is reflected once, by a moving potential; its energy gain is set by how fast the reflector moves, and the field strength enters only through how deep the valley is, which decides whether the electron is trapped at all and how quickly it turns round.

Where the riders come from #

An electron must start at the wave’s speed, or nearly, to be trapped, and a cold plasma’s electrons are nowhere near it: they oscillate back and forth about their places. Getting some of them into the accelerating trough is injection, and there are several ways to do it. A strong enough wave breaks at its crest and throws electrons into the trough behind, which is how the first narrow-energy beams were made and why their charge and energy varied from shot to shot. Mixing a little of a heavier gas into the plasma lets the peak of the laser pulse strip electrons from inner shells of its atoms at exactly the right place in the wave, where they are born inside the trough. A sudden step down in the plasma’s density slows the wave for an instant and lets electrons catch it. Each method trades charge against energy spread and stability, and the choice decides what the accelerator is good for.

Thinner, longer, higher #

The two scalings together give a counter-intuitive design rule. The field grows as the square root of the density, so a denser plasma pushes harder. But $\gamma_p^2$ is the ratio of the laser’s frequency to the plasma frequency, squared, which falls as one over the density, and the plasma wavelength falls as its square root. The dephasing length therefore falls as the density to the power −3/2, faster than the field grows, and the energy gained per stage falls as one over the density.

At $10^{19}$ electrons per cubic centimetre a stage is under two millimetres long and gives about a hundred million electronvolts. At $10^{18}$ it is six centimetres and gives a gigaelectronvolt. At $10^{17}$ it is nearly two metres and would give ten. A thinner plasma is gentler, longer and better, at the cost of needing a laser pulse that stays intense over the whole length — which it does not do on its own, because a focused beam spreads. Real experiments guide the pulse through a plasma channel, a tube of plasma denser at its walls than on its axis, which acts as an optical fibre. Guided that way, a pulse drove electrons to 7.8 gigaelectronvolts over twenty centimetres of plasma in 2019.

A wave that holds more when it moves faster #

The limit on the wave’s field has a relativistic twist of its own. A plasma wave is made of electrons oscillating back and forth along its direction of travel, and as the oscillation grows the electrons at its crests are carried forward faster and faster. When they are carried forward as fast as the wave itself travels, they overtake the crest: the wave breaks, like a water wave curling over, and its orderly structure dissolves into a turbulent spray of electrons. For a slow wave this happens at about $E_0$.

For a wave travelling at nearly the speed of light the electrons have much further to go. Their speed is capped below $c$ however strong the field, so to overtake a wave moving at $\beta_p c$ they must be driven to the point where their speed matches $\beta_p$, which for a fast wave means a very large momentum. Aleksandr Akhiezer and Roman Polovin showed in 1956 that the largest field a cold relativistic wave can carry is $\sqrt{2(\gamma_p - 1)}\,E_0$. At $\gamma_p = 10$ the wave can hold four times $E_0$, and at 100, fourteen times. The same saturation of speed that stretched the period of a relativistic spring raises the ceiling of the wave, and wave breaking, far from being only a limit, is also one of the ways electrons are injected: electrons swept out of the breaking crest land in the trough behind it with just the speed to be trapped.

The other side of Landau damping #

The trapped electron takes its energy from the wave. The wave must lose it, and in a plasma this is an old and precisely understood effect seen from the other side. A wave that dies with nothing to rub against is a plasma wave damped by the electrons moving at nearly its own speed: those slightly slower than the wave are pushed forward and take energy, those slightly faster push on the wave and give it, and because a thermal plasma has more slow electrons than fast ones near any speed, the wave loses on balance. Landau damping and wakefield acceleration are the same exchange between a wave and the particles riding it. In one the riders are a thermal crowd and the wave is the loser. In the other the riders are a small injected bunch and the wave is a reservoir built by a laser, and the bunch is the point.

The exchange also sets a limit on how much charge can be accelerated. A bunch that takes energy from the wave changes the wave, reducing its field behind the bunch; a bunch with too much charge flattens the field it is riding and spreads its own energies. Shaping the bunch so that its own wake exactly cancels the variation of the driving wake along it, and all its electrons feel the same field, is called beam loading, and it is how the energy spread of the best plasma-accelerated beams has been brought down to around a per cent.

Where the one-dimensional picture stops #

Three dimensions. The drawings follow a one-dimensional wave. A real wake is three-dimensional: a laser pulse intense enough expels electrons entirely from a region behind it, leaving a bubble of bare ions whose field accelerates electrons along its axis and focuses them towards it, a pinch of the kind a partly neutralised beam feels at its most extreme. The bubble regime is where most modern experiments work; its scalings are close to the one-dimensional ones with different numerical factors.

The laser’s own fate. The laser pulse loses energy as it drives the wave, and after a depletion length it can drive no more. It also spreads by diffraction and can be focused or broken up by its own effect on the plasma. For a stage to reach its dephasing length, the pulse must survive at least that far, and for thin plasmas that requires a guiding channel and a large laser.

Injection and quality. An electron must be put into the wave at the right place with the right speed. Achieving that reproducibly, for enough electrons, with a small spread of energies and angles, is most of what separates a plasma accelerator from a useful machine; the orbits drawn assume an electron already at the wave’s speed.

What the orbits do not show #

The orbits are those of a single electron in a given wave. They do not show the laser, which is in front, or the ions, which barely move, or the radiation the electrons emit as they oscillate sideways in the bubble’s focusing field — radiation that turns out to be a bright source of X-rays in its own right, because the oscillating electrons form a miniature version of the undulators used at synchrotron facilities. Nor do they show the wake left behind after the accelerated bunch, which in a real plasma rings on and heats the plasma, a waste the next stage must deal with.

Still open: many stages in a row #

A single stage gives a few gigaelectronvolts in centimetres. A collider for particle physics needs thousands of gigaelectronvolts, and so dozens or hundreds of stages in series, each with its own laser pulse, with the beam passed from one to the next without losing its quality. The first two-stage experiments have been done, with the second stage adding energy to electrons from the first, and the difficulty is exactly what the dephasing scaling suggests: the beam leaving one stage must be captured by the next with a precision of micrometres and femtoseconds. Whether a staged plasma accelerator can deliver beams of the quality a collider needs, at the repetition rates and efficiencies that would make it affordable, is not yet known, and proton-driven wakes, in which a single proton bunch drives a wave over hundreds of metres without staging, are being explored as an alternative.

The habit worth carrying away is to ask how long a push lasts, not only how strong it is. A particle can be accelerated by a travelling field only for as long as it keeps pace with the field, and near the speed of light the time it keeps pace grows as the square of the field’s Lorentz factor — so the energy of a plasma accelerator is set as much by how nearly its wave keeps up with the electron as by how hard the wave pushes.