The light that moves an electron and pays it nothing

The most intense lasers now focus light to more than $10^{22}$ watts on a square centimetre. In that light the electric field is several hundred times the field that holds an electron in a hydrogen atom, and an electron caught in it is shaken so hard that its speed comes within a fraction of a per cent of the speed of light within a quarter of a cycle. The obvious hope, from the first days of such lasers, was that this was an accelerator — that an electron struck by such a pulse would come out of it moving at nearly the speed of light, with a gain of energy that no metal cavity could match in so short a distance.

For the simplest pulse it is not. An electron overtaken by a plane wave of any strength and any shape is left, when the wave has passed, exactly at rest. It has been thrown about violently and carried forward with the light; it has none of the light’s energy at the end. The statement is called the Lawson–Woodward theorem, after P. M. Woodward, who stated it in the 1940s, and J. D. Lawson, who stated it again in 1979 when laser accelerators were first proposed seriously. It follows from two quantities the electron cannot change while a plane wave is passing over it, and those two quantities fix everything else about the motion as well.

Two things the electron keeps #

Take a wave travelling along $z$ with its electric field along $x$, and describe its strength by the dimensionless number

$$a = \frac{eA}{mc},$$

where $A$ is the vector potential. Its peak value $a_0$ measures the field in natural units: $a_0 = 1$ is the strength at which an electron’s quiver momentum equals $mc$, so that its quiver becomes relativistic. For light of 0.8 micrometres that takes $2.1 \times 10^{18}$ watts per square centimetre, and for light of 1 micrometre $1.4 \times 10^{18}$.

The wave depends on position and time only through the combination $\phi = \omega(t - z/c)$. Two consequences follow from that and from nothing else. Because the wave is the same everywhere across the beam, the electron’s transverse canonical momentum — its ordinary momentum minus $e$ times the vector potential — cannot change, and for an electron that started at rest it stays zero. So its sideways momentum is, at every instant,

$$\frac{p_x}{mc} = a(\phi).$$

Because the wave depends on $t$ and $z$ only through $t - z/c$, a second quantity is conserved: $\gamma - p_z/mc$, the electron’s energy minus its momentum along the beam, both in units of $mc^2$. For an electron that started at rest it is exactly one. Combine that with $\gamma^2 = 1 + (p_x/mc)^2 + (p_z/mc)^2$ and the motion is solved without a differential equation:

$$\frac{p_z}{mc} = \frac{a^2}{2}, \qquad \gamma = 1 + \frac{a^2}{2}.$$

The electron’s energy at any moment is set by the wave’s field at the electron at that moment, and by nothing in its history. When the pulse has passed, $a = 0$: so $p_x = 0$, $p_z = 0$, $\gamma = 1$. The electron is at rest, whatever the pulse did in between.

The paths in the figure are drawn from those two conservation laws alone: the sideways position is the integral of $a$ over the phase, the forward position the integral of $a^2/2$. To check that nothing has been assumed that the force law does not deliver, the same electrons were also integrated step by step under the full Lorentz force, electric and magnetic, and the two methods agree to better than a part in five hundred on where the electron ends and how fast it went.

Pushed forward by a force that does no work #

Where does the forward push come from, if the electric field points across the beam? From the magnetic field of the light, which points along $y$. The electric field drives the electron along $x$; the magnetic force on that motion, $e\mathbf v \times \mathbf B$, points along the beam. The force that does no work found that a magnetic force can turn a velocity and never change its size, and that is exactly what happens here: the magnetic force turns sideways motion into forward motion, while the electric field supplies the energy.

At small $a_0$ the sideways speed is small, the magnetic force is smaller still — a product of two small quantities — and the electron simply quivers across the beam, the textbook picture. At $a_0 = 1$ the sideways speed is relativistic and the magnetic force is as large as the electric one. The electron is then carried along the beam as much as it is shaken across it, and at $a_0 = 2$ the forward motion dominates: in the path above, the electron at $a_0 = 2$ goes forward two and a quarter wavelengths while swinging only a third of one sideways.

The push is not a time average of anything, the way the force that held a pendulum upside down was; it is exact at every instant. But its time average is what is usually called the ponderomotive force, and in a beam of uniform strength the average comes out as a push along the beam, proportional to the intensity. The usual statement — that light pushes on whatever it shines on, as light has a pressure found — is a statement about light that is absorbed or reflected. A free electron in a plane wave does neither, to the accuracy of these equations: it is pushed while the light is on it and the push is taken back as the light leaves.

Energy lent for the length of a pulse #

The energy curve is the most direct statement of the theorem. At $a_0 = 2$ the electron’s Lorentz factor reaches 3 twice in the middle cycle: its kinetic energy is then twice its rest energy, a megaelectronvolt, gained in a fraction of a femtosecond from light that had started out with no particular direction for it. Half a cycle later it is back near rest, and when the pulse has passed it is at rest, with $\gamma = 1$ to the accuracy of the integration.

It is a loan, repaid in full. The electron takes energy from the light on the way in and gives every joule back on the way out, because its energy at each instant is set by the light’s field at that instant, and the field at the trailing edge of the pulse is the same as at the leading edge — zero. The same structure turned up in the bill that arrives when the pushing stops, where a uniformly accelerated charge radiates while the radiation reaction on it is zero and the account is settled only at the ends of the push. Here there is no account to settle. The light is not depleted, to the approximation that one electron does not change a pulse of $10^{19}$ photons.

There is a second way to say why. The theorem’s conditions are a wave of infinite extent across the beam, no boundaries, no static fields and no other matter, so the only things present are the electron and a light wave that depends on $t - z/c$. In the frame of the electron at the end, a plane wave has passed over it with nothing to absorb the momentum a net gain would need. Every scheme that accelerates electrons with light is a way of breaking one of those conditions.

The figure eight #

Inside a long pulse of steady strength, the forward momentum averages $a_0^2/4$ (in units of $mc$) and the energy $1 + a_0^2/4$, so the electron drifts along the beam at

$$\frac{v_d}{c} = \frac{a_0^2}{4 + a_0^2}.$$

At $a_0 = 1$ that is a fifth of the speed of light; at $a_0 = 10$, 96 per cent of it. The drift speed is also, exactly, the share of the electron’s average energy that is motion rather than rest mass — the two are one number, because every bit of energy the electron takes from the wave arrives together with the same amount of forward momentum, which is what $\gamma - p_z/mc = 1$ says.

Watched from a frame moving with that drift, the electron goes nowhere on average, and its path closes once a cycle. The sideways motion oscillates at the light’s frequency, the forward-and-back motion at twice it — the magnetic force is the product of the field and the velocity, both oscillating at the light’s frequency, so their product oscillates at twice. A motion at frequency $\omega$ across and $2\omega$ along is a figure eight, exactly as a pendulum swinging in one direction at one rate and another at twice the rate traces a figure eight, the shape the spring that becomes a light-clock mentioned for an electron driven straight by a strong laser.

The figure eight matters because it is what makes the electron radiate harmonics. A charge moving in a straight line across a beam re-radiates the light’s own frequency, which is Thomson scattering. A charge moving in a figure eight also re-radiates twice and three times the frequency, and as $a_0$ grows the harmonic spectrum broadens. The second harmonic comes out of the forward-and-back motion along the beam and is polarised differently from the first, so its light can be told apart by a polariser as well as by its colour. Harmonics with this pattern were seen in 1998 from electrons in a plasma struck by a pulse at $a_0$ close to one, in an experiment at the University of Michigan: the angular pattern of each harmonic, with the second strongest off the axis rather than on it, matched what a figure-eight orbit radiates.

The eight is also a clock. It closes once per cycle of the light in the drifting frame, so the scattered light is periodic at the light’s frequency as that frame sees it, which is lower than in the laboratory because the frame recedes from the source. Looking back along the beam from the laboratory, the scattered fundamental therefore comes out at a lower frequency than the light that made it, by a factor $1/(1 + a_0^2/2)$ — a redshift by the drift, and a signature of $a_0$ that can be read without knowing the size of the focal spot. The same motion seen by an electron that is already fast is how a fast electron throws a laser photon back as a gamma ray, and when $a_0$ approaches one the scattered gamma rays acquire harmonics of their own.

What a focus changes #

A real laser beam is not a plane wave. It is focused, and its intensity falls away from the axis over a few micrometres. That gradient breaks the first conservation law — the wave is no longer the same everywhere across the beam — and gives the electron a way out. An electron in the focus is pushed down the intensity gradient, away from the axis, by the same averaged force that holds a charge in a Paul trap. If it leaves the beam sideways before the pulse has finished passing, it leaves with whatever energy it had at the moment it left, and keeps it.

The light-front quantity survives this much better than the transverse momentum, because it depends on the wave’s dependence on $t - z/c$ rather than on its uniformity across the beam. So an electron that started at rest still has $\gamma - p_z/mc = 1$ when it emerges, and with $\gamma^2 = 1 + p_\perp^2 + p_z^2$ that fixes $p_\perp^2 = 2(\gamma - 1)$ in units of $mc$. The angle it leaves at, measured from the beam’s direction, is therefore

$$\tan\theta = \sqrt{\frac{2}{\gamma - 1}}.$$

The relation has no property of the light in it: not the intensity, the wavelength or the shape of the focus. Slow electrons leave almost sideways and fast ones almost along the beam, at an angle that is a pure function of their energy. It has been measured for electrons freed from atoms inside an intense focus, which leave at the angles it predicts, and it is used the other way round, as an energy spectrometer: an electron detected at a given angle has a known energy.

The energies reached this way are modest beside the peak quiver energy, because an electron can leave the focus only while the field is still large if the focus is narrow compared with the distance the electron moves in a cycle. And the electrons go out in a ring of angles rather than a beam. As an accelerator a focus is a poor one. As a demonstration that it is the plane-wave geometry and nothing about the strength of the field that forbids the gain, it is decisive.

Every laser accelerator breaks one condition #

The theorem’s conditions are a list, and each item on it has been broken on purpose. A plasma breaks the “no other matter” condition: the wave an electron rides to a gigaelectronvolt is a plasma wave the laser pulse leaves behind, travelling at nearly the speed of light with a longitudinal field, and an electron riding it is accelerated in the direction the light is going by a field the light has handed to the plasma. A grating or a dielectric structure breaks “no boundaries”: a structure with a period near the light’s wavelength can arrange for a longitudinal field component whose phase travels with the electron, and dielectric laser accelerators on chips do exactly that. A magnet row breaks “no static fields”: in an inverse free-electron laser the electron’s sideways wiggle in an undulator, matched to the light’s frequency, keeps its velocity in phase with the light’s field so that the field does work on it in every cycle — the undulator of the magnet period that comes out as an X-ray run backwards.

The theorem’s other condition is that the electron is not already fast. An electron already travelling with the light at nearly its speed still conserves $\gamma - p_z/mc$, but its value is now small rather than one, and the energy it can borrow from a given field grows as the inverse of that value. A fast electron in a plane wave gains and returns enormously more than a slow one; it still returns it all.

What all these schemes have in common is that the electron’s velocity must have a component along the electric field, in phase with it, averaged over many cycles. A plane wave in empty space makes that impossible because the electron’s own response — the quiver it is given — is a quarter-cycle out of phase with what would be needed, and the light-front conservation law is the exact statement of that.

What the figures leave out #

Every figure here treats one electron in a prescribed wave. Three things are missing. The electron radiates as it is shaken, and at $a_0$ of a hundred or more that radiation carries away a significant fraction of the energy it borrows, so the loan is not fully repaid to the electron — radiation reaction, the self-force the force a charge exerts on itself found unavoidable, breaks the theorem at the highest intensities now reached. The wave is taken to be unaffected by the electron, which is true of one electron and false of a dense plasma, where the electrons’ collective response is the whole story. And the electron is classical: the Compton recoil of individual photon emissions, negligible at these energies, begins to matter when the field in the electron’s frame approaches the field at which the vacuum itself can make pairs.

The domain in which the drawn paths are exact is therefore a plane wave of any shape and strength in empty space, an electron starting at rest, and fields well below the strengths at which radiation reaction matters. Inside it, the two conservation laws are not approximations: they are the whole solution.

Still open: where radiation reaction takes over #

At $a_0$ of a few hundred, reached by the most intense lasers now built, an electron colliding head-on with the pulse radiates so much in a single cycle that its motion is set by its own radiation rather than by the light’s field, and the classical equations for that regime are known to be unsatisfactory in their simplest form. Experiments in 2017 and 2018 at the Gemini laser in the United Kingdom collided electron beams from a plasma accelerator with an intense pulse and saw the electrons lose energy in a way consistent with radiation reaction, at the edge of statistical significance, and the question of which equation describes the electron — classical, semiclassical with quantum corrections, or a fully quantum treatment — is being decided by experiments at laser facilities able to reach $a_0$ of a thousand.

The plane-wave result is settled, and short. While a plane wave passes, an electron that started at rest keeps $p_x/mc = a$ and $\gamma - p_z/mc = 1$, so $\gamma = 1 + a^2/2$ and $p_z/mc = a^2/2$ at every instant: it is thrown forward, drifts at $a_0^2/(4 + a_0^2)$ of $c$, traces a figure eight in its drifting frame, and is left at rest when the light has gone — 2.25 wavelengths further on at $a_0 = 2$, having borrowed and returned a megaelectronvolt. Light can move an electron as far as it likes. It cannot pay it, unless something other than the electron and the wave takes part.