Text from the interactive explanation

Kilojoule-scale laser acceleration enabling efficient generation of electron-positron and muon beams

R. Babjak, M. Pouyez, C. Badiali, T. Grismayer, M. Vranic

A laser recipe for antimatter plasma · simulation study

Pulsars and black holes are wrapped in a pair plasma: electrons and their antimatter twins, positrons, moving together. Physicists from Instituto Superior Técnico and Sorbonne Université have simulated a way to make one in the lab. Fire a powerful laser through a cell of gas, send the electrons it accelerates into a slab of lead, and out comes an electron–positron beam dense enough to act as one and tear itself into filaments.

electron

positron

gamma-ray photon

laser light

muon

Numbers: the paper's Geant4 simulations of the lead converter, read approximately from its Fig. 3. Skin depths are counted two ways: with the standard skin depth in the lab, and with the relativistic skin depth in the beam's own frame (Fig. 3e); chapters 2 and 6 explain both. Picture: an illustrative time-lapse, not to scale, in which the laser, the electrons and the pair beam all fly at nearly the speed of light. The gas cell follows the paper's particle-in-cell density map (its Fig. 2b): the laser clears a bubble of electrons inside a guiding channel, and the accelerated bunch rides at the back of it. The shower in the lead is an illustrative cascade with its sideways spread compressed. The pair bunch's width, density and length follow Fig. 3; its filaments come from the illustrative model of chapter 2, with their strength set by the chapter 2 estimate.

1 · Light into matter

A photon that becomes two particles

A positron is the electron's antimatter twin: same mass, opposite charge. You can make one out of light. When a gamma-ray photon (very high-energy light) passes close to a heavy nucleus (lead has 82 protons), its energy can turn into an electron and a positron. This is the Bethe–Heitler process : E = mc² run in reverse.

Mass costs energy, so the photon must carry at least the rest energy of both particles: 1.022 million electron-volts (MeV). Muons , the electron's 207-times-heavier cousins, need photons of at least about 211 MeV. Even well above that, muon pairs are rare. As the paper notes, the chance is smaller by the square of the mass ratio, roughly 40,000 times, and near the threshold it is smaller still.

Try it: drop the photon energy below 1 MeV. No pairs at all: the photons only scatter or are absorbed. Then raise it to a few GeV and fire a burst of a million. How many muon pairs do you get?

Each photon crosses 1 cm of lead, the converter used in the paper's filament simulation. Its chances of making an electron–positron pair, of scattering or being absorbed instead, or of making a muon pair come from standard tabulated cross-sections for lead (NIST XCOM photon data, and the Geant4 muon-pair formula that the paper's simulations use). The single nucleus is only a picture: in reality each conversion happens at one of trillions of lead nuclei.

One pair is easy. A plasma needs trillions of them, packed closely together. Why does the packing matter so much?

2 · From a cloud to a plasma

When does a cloud of particles start acting together?

A plasma is a gas of charged particles that responds as a whole. In a beam rushing forward, every particle is a tiny electric current. Currents flowing the same way attract and opposite currents repel, so positrons gather with positrons, electrons with electrons, and the beam tears itself into alternating filaments. This is the current filamentation instability . Instabilities like it are thought to help accelerate the particles behind the light we see from pulsars and black holes.

Collective behaviour has a natural length scale, the skin depth ; filaments are typically on the order of a skin depth wide. The more tightly the particles are packed, the shorter the skin depth. So a beam forms filaments only if it is dense enough to span many skin depths, and collimated enough (low divergence ) that its particles don't spray apart before the filaments can grow.

Predict first: two beams, both 0.5 mm wide and barely spraying. One is 100 times denser than the other. Which tears into filaments?

Skin depths here are the standard ones, measured in the lab. That is the count the paper uses for sideways filaments: the beam must be much wider than one skin depth, and its spray small enough for the filaments to grow before it thins out. The verdict is a rough estimate built on these two conditions, taking "much wider" as about ten skin depths or more. (Whether the beam is a plasma in every direction is judged in its own frame; see chapter 6.) The animation is an illustrative model made for this page, drawn to resemble the paper's simulation (chapter 8) but not computed like it: a Gaussian beam crosses hydrogen gas at the paper's density and spreads with its divergence, while red and blue filaments on the order of a skin depth wide grow out of noise where they stand and slowly merge. The view always spans the same 3 mm across the beam, so a wider beam looks bigger. The bracket marks the beam's starting width; beams wider than 3 mm overflow the view.

So a lab pair plasma needs a beam that is both dense and collimated. That is exactly where earlier laser recipes have got stuck.

3 · The recipe problem

Plenty of particles, or a collimated beam — why not both?

Every laser recipe for pair beams does two things: it accelerates electrons, and it sends them through a heavy-metal converter , where they create pairs. Until now the electron sources have forced a trade-off, and each runs into one of the problems you just met. Divergence is measured in milliradians (mrad): 1 mrad means spreading by 1 mm for every metre travelled.

Illustrative: all three beams have the same particle density, and so the same skin depth (gold bar), and are drawn on one scale. They differ in width (how much charge they carry) and in spray, and run through the same illustrative filament model as chapter 2.

How does a laser pulse in gas produce so much charge in such a collimated beam?

4 · Surfing the laser

How a laser pulse hands its energy to electrons

Light is an electric field that flips direction hundreds of trillions of times a second. A petawatt pulse (1015 watts, lasting a fraction of a trillionth of a second) is so intense that, in gas, it shoves electrons out of its path and leaves behind a channel of positive ions. The ions pull the electrons back, so they swing from side to side as they race along.

If each swing stays in step with the laser's flipping field, the electron is pushed forward a little every time, like a playground swing pushed at just the right moment. The plasma wave trailing the pulse gives the electrons at the back of it an extra kick. This is direct laser acceleration (DLA) . To get the most out of it, the authors matched the laser power, gas density, focal spot size and acceleration distance. How to choose these to optimise the acceleration was worked out in an earlier paper by members of the team (Babjak et al., Phys. Rev. Lett. 132, 125001, 2024).

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Illustrative and slowed down enormously. Shading and arrows show the laser's electric field: red where it points up, blue where it points down. Being negatively charged, the electron is pushed against the field (arrow beside it). The arrow is gold when that push follows its swing, so it gains energy, and blue when the push opposes it. The trail is gold while gaining and blue while losing. In step, the electron stays locked to the phase that pushes it along.

What the paper's laser simulations found

Fig. 2 (a, b, d) from Babjak et al. 2026, CC BY 4.0; cropped into separate panels.

Now send this beam into the lead.

5 · Inside the lead

A cascade that multiplies particles and spreads them out

Inside the lead, each fast electron swerves near nuclei and sheds gamma rays (so-called braking radiation). Those gamma rays turn into electron–positron pairs, which shed more gamma rays, and so on: a particle shower .

A thicker slab lets the shower multiply further, until the particles have lost so much energy that they get stuck in the lead. The paper puts this shower length at about 3.6 × ln γ0 mm, a few centimetres for GeV electrons. Every particle is also nudged sideways on the way, so the beam widens and thins out. The thickness is a trade-off you can tune.

Find the thickness that makes the most positrons.

Now find where the beam is densest.

Finally, find where the beam spans the most skin depths across, measured in its own frame.

The shower drawing is an illustrative cascade started by 8 electrons, with the sideways spread exaggerated. The charts and numbers are the paper's Geant4 results, read approximately from its Fig. 3.

10 PW laser 1 PW laser · dashed line: your thickness

Thicker lead also shifts the positrons toward lower energies, because the shower keeps sharing out the energy (Fig. 3f). To count as a plasma, the beam must be wider than the skin depth measured in its own frame. For the 10 PW laser that holds from roughly 0.6 cm of lead upward, peaking at a little over three skin depths; counted with the standard lab skin depth, the beam reaches up to 30. The supplementary material shows how much the driver's initial spread matters. An electron beam with the same energies but no sideways spread would give positron densities more than ten times higher: up to 3 × 1017 per cm³ for the 1 PW case. Even a spread of around 10 mrad costs a factor of ten in density over centimetres of lead.

"Skin depths in the beam's own frame" keeps coming up. Whose frame is that, and why does it matter?

6 · Riding with the beam

How big is the beam? It depends on who is measuring

In chapter 2 you counted skin depths in the lab. By that count the paper's pair beam (10 PW laser, 1 cm of lead) is wide: about 25 skin depths across, and up to about 30 for thicker lead. That is ample room for sideways filaments. But the beam rushes along at more than 99% of the speed of light, and relativity says that lengths, densities and energies depend on how fast the person measuring them is moving.

To decide whether the beam is a plasma in every direction, the paper judges it in the beam's own frame : the viewpoint of an observer moving along with it, so that its particles, taken together, have no forward momentum at all (physicists call this the centre-of-momentum frame). The authors found this frame by transforming every simulated positron from their Geant4 run (10 PW laser, 1 cm of lead). It moves with a Lorentz factor of about 9, and in it the average positron energy drops from γ ≈ 290 to γ ≈ 32.

Try it: speed up the observer. Only one speed makes the gold "total momentum" arrow vanish. Watch the bunch stretch and thin out on the way, then compare the skin-depth grids in the lab and in the beam's own frame. In the lab, the fast positrons travel together at almost light speed, while the few slow ones fall behind and leave at wider angles. In the beam's own frame, most positrons sit almost still near the middle of the momentum plot, and the rest drift slowly forward and backward: a warm cloud at rest.

Dots: an illustrative sample of 600 positrons shaped after the paper's beam (its supplementary Fig. 9). In the lab, their momenta follow a power law reaching about 5 GeV/c, faster positrons fly closer to the axis, and about 1% leave at wide angles. The sample is tuned to the paper's averages: γ ≈ 290 in the lab, a Lorentz factor of 9 and γ ≈ 32 in the beam's own frame, where, as in Fig. 9, most positrons have almost no forward momentum, a few drift backward at up to a few hundred MeV/c and none move forward faster than about 300 MeV/c. Both momentum axes are linear close to zero and logarithmic further out, and each change of observer is an exact relativistic transformation. On the right, every positron starts inside the bunch and flies straight on at the velocity its dot on the left gives it (forces between the particles are not drawn). Every observer watches the same stretch of the bunch's life: the time the bunch ages while light crosses the panel in the lab. Moving with the beam, that is only a ninth as long, so there the cloud spreads only slightly. The faint oval and grid mark the bunch at the start of each flight. The width (≈ 0.5 mm, the same for every observer moving along the beam), the lab density and the 38 µm length are the paper's 10 PW, 1 cm values, read approximately from its figures. Lab grid: the standard skin depth; beam-frame grid: the relativistic skin depth the paper uses.

So the beam's own frame is where to judge a plasma. Is the beam big enough there in every direction?

7 · Wide enough, long enough

One slab of lead, two directions pulling against each other

A plasma should be bigger than a skin depth in every direction, measured in the beam's own frame. The lead thickness pulls the two directions in opposite ways.

Across: thicker lead spreads the beam sideways, so it spans more skin depths across. Along: thicker lead also leaves the beam slower and more dilute, so the bunch is stretched less in its own frame while the skin depth grows. The result: fewer skin depths along.

Try it: drag the dashed line across the chart (or use the slider) and watch the bunch on the right change shape. Find the thickness where it is about one grid square both ways.

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Across: the paper's Fig. 3e. Along: its supplementary Fig. 11 (bunch length in the beam's frame divided by the relativistic skin depth). Both are read approximately from the plots; the balance point is where the smaller of the two is largest.

Read off the two curves, the 10 PW beam reaches about one skin depth both across and along near 0.6 cm of lead, but only just. The paper's own conclusion is that the beam does not meet the plasma criteria along its direction of travel: it is long enough only behind thin converters, and there the beam still carries many extra driver electrons, so it is not yet neutral. Size is not the only condition either, and real experiments usually need more. With the 1 PW laser the beam becomes wide enough but stays far shorter than a skin depth. To make longer bunches, the authors suggest accelerating the electrons with longer, picosecond-scale laser pulses. Sideways filaments, however, only need room across, and the beam already has it.

The beam is wide enough for sideways filaments. Does it really form them?

8 · The evidence

In a full simulation, the beam tears into filaments

The team took the pair beam made by the 10 PW laser and 1 cm of lead and followed it through ionised hydrogen gas (4 × 1016 particles per cm³) in a 2D particle-in-cell simulation. Such a simulation tracks huge numbers of particles together with the electric and magnetic fields they create.

Colour shows positrons minus electrons: red where positrons dominate, blue where electrons do. The curves are the electron (blue) and positron (red) densities across the beam, along the dotted line. The view travels with the beam, which covers about 0.3 mm every picosecond.

Look for: one smooth bunch at 0 ps; the first stripes at 152 ps; at 304 ps, red and blue filaments side by side, with the two density curves zig-zagging in opposite directions.

0 ps

152 ps

304 ps

As the electrons and positrons separate, the filament currents generate magnetic (Bz) and electric (Ey) fields, shown here at 304 ps. Because extra driver electrons leave the lead together with the pairs, the bunch starts out with fields of its own, so the authors could not cleanly pick out the instability's early growth phase. The evidence is the filaments themselves and the strong fields they produce. The beam's spread of tens of mrad is small enough that these sideways modes are not suppressed, which the authors call a rough estimate.

The same electron beam can make something else that is useful too.

9 · A bonus beam

The same electrons also make muons

Muons are 207 times heavier than electrons, so they travel much farther through rock and metal. Muon imaging revealed a hidden chamber in the Great Pyramid of Giza, and muons have been used to inspect cargo containers and to look inside the damaged Fukushima Daiichi reactor. Accelerated muon beams are also needed for a future muon collider.

In the lead, gamma rays above about 211 MeV can make muon pairs. The authors derived a formula for how many, checked it against Geant4 simulations, and found a surprise.

Predict: with the same 100 J in the electron beam, which makes more muons?

Curve: the authors' fitted formula (their Eq. 8, valid up to 100 GeV) for a thick lead target and a beam in which every electron has the same energy. Muon rain and pyramid: illustrative.

Muons from the simulated DLA beams (Geant4)

Back to the pulsar

What this brings closer

Making a pair plasma in the lab needs a beam with lots of charge, good collimation and high density, all at once. In these simulations, an electron beam from optimised direct laser acceleration provides all three. Even a 1 PW, roughly 100 J laser yields 5 × 1011 positrons in a beam about one skin depth across in its own frame, cutting the laser energy needed by about a factor of ten. The 10 PW case is a conservative upper limit for the near-term ELI-L4 ATON laser, and its pair beam tore into filaments in the simulation.

The authors point to next steps: converters such as iridium or platinum, which scatter the particles less and so give denser pair bunches; collimating the electron beam before it hits the metal; and shaping the electron energies for each experiment. High-charge sources like DLA suit sideways instabilities, and perhaps even reconnection of magnetic field lines; sources with higher energy but less charge may suit lengthwise effects such as shocks. If experiments follow, a laboratory could watch, up close, a piece of the physics thought to power the light from pulsars and black holes.

Source

R. Babjak, M. Pouyez, C. Badiali, T. Grismayer, M. Vranic. Kilojoule-scale laser acceleration enabling efficient generation of electron-positron and muon beams. arXiv:2609.18775v1 [physics.plasm-ph], 2026. arxiv.org/abs/2609.18775 · DOI 10.48550/arXiv.2609.18775

Credits & notes

Figures 2 and 6 are reproduced from the paper under the CC BY 4.0 licence, cropped or split into panels.

All results in the paper come from simulations: quasi-3D particle-in-cell runs of the laser acceleration, Geant4 Monte Carlo of the lead converter, and a 2D particle-in-cell run of the pair beam.

The animations, the filament model and the rough filament estimate, the photon gun, the swing picture of laser acceleration, the shower drawings, the moving-observer sample and the bunch-to-scale drawing are illustrative re-creations made for this page. Numerical values are from the paper; some are read approximately from its figures.