Text from the interactive explanation

Collisionless shock acceleration of protons in a plasma slab produced in a gas jet by the collision of two laser-driven hydrodynamic shockwaves

J.-R. Marquès, L. Lancia, P. Loiseau, P. Forestier-Colleoni, M. Tarisien, E. Atukpor, V. Bagnoud, C. Brabetz, F. Consoli, J. Domange, F. Hannachi, Philippe Nicolaï, Martina Salvadori, B. Zielbauer

Laser–plasma physics · experiment + simulation

Shock waves around exploding stars fling particles across space. Physicists asked whether a lab-made shock, inside a puff of hydrogen, could throw protons forward in a clean, single-energy beam — the kind of compact, fire-again-and-again source that could one day serve medicine and imaging. The catch: the gas has to be sculpted first.

Two blasts: the collision forms a thin dense slab; forward protons get more energetic and more peaked (up to 3.2 MeV measured at LULI, 5.5 MeV at GSI) — a sideways spray remains.

Heating beams (nanosecond, fire 2.5 ns early)

Main pulse (picosecond, ~10¹⁹–10²⁰ W/cm²)

Plasma density

Neutral gas

Protons

Gas sculpting: simplified simulation calibrated to the paper. Proton burst: illustration of the measured trends. Proton flight shown about a thousand times slower than the gas motion.

01 Chapter 1 · The shock

A wall that throws protons

Heat a gas enough and its electrons tear loose from their atoms. What remains is a plasma : a soup of light, nimble electrons and heavy positive ions — in hydrogen, bare protons.

In space, plasma is so thin that its particles almost never bump into one another, yet shock waves still form, like the bow shock where the solar wind meets Earth’s magnetic field or the blast fronts of supernova remnants. They are held together by electric and magnetic fields instead of collisions, so they are called collisionless shocks — and they are known particle accelerators.

The version in this paper is electrostatic. When a region whose electrons are much hotter than its ions pushes into cooler, lower-pressure plasma, it drives a shock: a moving step in electric potential, an electric wall . Protons waiting ahead of it see the wall coming. If the wall’s potential is larger than their energy of motion relative to the wall, they bounce off.

Picture a ball lying on the rails as a train arrives. Seen from the train, the ball comes in at the train’s speed and bounces back at that same speed; seen from the platform, it leaves at twice the train’s speed. Reflected protons do exactly this. The process is called collisionless shock acceleration (CSA). The shock’s speed is counted in units of the plasma’s sound speed, set by the electron temperature Te: shock speed = M × sound speed, where M is the Mach number .

Surf the electric wall

Predict first: double the shock’s speed. The bounced protons’ energy becomes…

×4. A bounced proton leaves at twice the shock speed, and energy of motion grows with the square of speed: E = ½ m (2 vs)² = 2 m vs². Since vs = M × cs and m cs² = Te, this is the paper’s rule of thumb E ≈ 2 M² Te. Try it: move M from 1.3 to 2.6 and watch the marker.

Check the paper’s estimate

The authors estimate electron temperatures between 0.35 and 1.4 MeV in their experiments and take the smallest Mach number that still reflects protons, M ≈ 1.6. Load either end:

That gives 1.8 to 7.2 MeV. The highest measured energies, 3.2 MeV at the LULI laser and 5.5 MeV at GSI, sit inside this window — the authors call it “in good agreement” with CSA.

In an ordinary shock — a sonic boom, a blast wave in air — molecules collide constantly and that is how the shock front passes its push along. In the hot, thin plasmas here, the particles fly much farther than the shock is thick before they ever collide. The front is held together instead by the electric field that appears when the light electrons run ahead of the heavy ions.

For protons to bounce, the wall’s electric potential must exceed their kinetic energy measured in the wall’s frame. A slow wall is too low: protons climb over it and end up behind. The paper uses Mcr ≈ 1.6 as the minimum Mach number for reflection; in the toy above, the soft threshold comes from the protons’ small thermal jiggle.

02 Chapter 2 · The target

The shock needs a very particular target

A laser makes the hot electrons, the hot electrons make the shock, and the shock throws the protons. For that chain to give one clean beam, the paper lists three requirements:

What does “critical density” mean? Light can only travel through a plasma whose electrons are not too crowded. Below the critical density nc ≈ 1.1 × 10²¹ cm⁻³ (for light of 1 µm wavelength), light passes; above it, the plasma acts like a mirror. Right around it, the light is soaked up most strongly — exactly what heats the electrons.

Light meets plasma

Illustrative behaviour. At these laser intensities electrons move close to light speed and get heavier, which pushes the effective threshold up by a factor γ — the paper estimates about 3 nc at LULI and 5.7 nc at GSI.

The trouble with a gas jet

The target here is a puff of hydrogen from a 250 µm nozzle, probed 400 µm below its exit. The team measured its profile: about 270 µm wide at half height, with long faint wings. Plasma made from it has the same shape once the gas is ionised.

Fit the jet to the target

Turning up the pressure makes the jet denser (800 bar reaches 0.8 nc at the centre), but never thinner — and the faint wings spread further. The shape has to be sculpted.

Why not simply use a wider beam of longer-wavelength light? Earlier CSA experiments used CO₂ lasers with 10 µm light, whose critical density is 100 times lower, so ordinary millimetre gas jets worked. Today’s common solid-state lasers emit near 1 µm and need about 10²¹ cm⁻³ in a thin layer. Gas jets are still worth the effort: no debris to spoil the optics, no target to replace between shots, and pure hydrogen gives pure proton beams.

03 Chapter 3 · The lab

Sculpt the gas with two blasts

The team’s answer, proposed in their earlier simulations and tested here: carve the jet with heat. About 2.5 ns before the main pulse, two weaker nanosecond heating beams (about 1 ns long, a few joules each) are focused on either side of the jet, 300 µm from its centre. They travel perpendicular to the main laser — straight into the screen in the views below.

Each hot spot explodes as a cylindrical blast wave . Its outer half pushes the jet’s faint wings away into vacuum; its inner half ionises and compresses the gas. Where the two waves meet in the middle they pile up a thin, dense slab with steep edges — the authors expect a compression of eight times or more.

Before you press play: where do the two waves meet, and how wide is the result?

They meet at the centre of the jet , where the gas is densest, and pile it into a sheet only a few tens of micrometres thick. In the model below this happens about 0.3 ns before the main pulse would arrive; the paper’s measured streak shows the collision at about −0.1 ns.

Gas-sculpting lab

Time is given relative to the main pulse, which arrives at 0 ns. The heating beams fire at −2.50 ns.

Shot log — fire the main pulse at any moment to record what it would meet.

Each heating spot has a faint halo carrying about 25% of its energy. Switch the halo off. Do the waves meet sooner, later, or not before the main pulse?

Not before the main pulse. Without the halo, each blast starts smaller and farther out, and the waves are still about 200 µm apart at 0 ns. This was a real lesson of the paper: the authors’ 3D simulations only reproduced the measured streak when the low-intensity wing of the heating-beam focal spot was included (their Figs. 10 and 11, below).

Compare with what was measured

The lab runs on a simplified two-dimensional blast-wave model made for this page — not the authors’ 3D TROLL radiation-hydrodynamics code. It treats hydrogen as an ideal gas on a 5 µm grid, in the plane containing the main laser (x) and the gas flow (z).

The starting gas is the paper’s measured jet profile at 200 bar (centre density nc/5), widening slightly away from the nozzle. Each heating beam heats a zone sized from the authors’ simulated temperature maps (their Fig. 12b) for 1 ns: 120 µm radius with the halo, 60 µm without. The heating strength was calibrated so that, with the halo, the waves collide at about 2.2–2.3 ns, like the authors’ Fig. 11. Gas counts as plasma (visible to the probe) once heated above about 1 eV.

Results: with the halo the waves collide at ≈ 2.2 ns and the slab peaks at ≈ 1.7 nc near 2.5 ns, about 15 µm wide (the grid limit); without the halo the fronts are still about ±100 µm from the centre at 2.5 ns. The Probe camera view turns the density into a shadow using a simple deflection rule (light bent more than the lens accepts is lost). Treat all of this as a qualitative guide.

04 Chapter 4 · Reading shadows

The shadow is wider than the wall

Nobody can photograph electrons directly. Instead a weak green probe beam (0.5 µm wavelength) shines through the plasma into a camera. Plasma bends light wherever its density changes quickly; rays bent more than the collecting lens can accept never reach the camera and leave a dark band. So the camera records where the density is steep — and that shadow can be far wider than the dense layer itself.

Bend the probe light

The authors’ 3D simulations show the same effect: at the moment of collision a slab under 20 µm thick casts a shadow about 150 µm wide. They used that ratio to read their own measurements backwards.

Read a measured shadow backwards

Thickness = shadow × (20/150). Density follows from particle conservation: the same gas spread over a thicker slab is less dense, starting from the simulated ≈ 3 nc for 20 µm. The measured shadows were 200–300 µm wide, so the paper’s estimate is a slab ~30–40 µm thick at ~1.5 nc — close to the ~40 µm optimum, but with lower density than ideal. These are estimates, not direct measurements.

05 Chapter 5 · Measured results

Fire! Read the proton detectors

Now the main event. At the LULI PICO2000 laser in France, a 1 ps pulse at 2 × 10¹⁹ W/cm² is focused to a 10 µm spot in the plasma. Three magnetic spectrometers, at 0°, 30° and 70° from the laser direction, sort the outgoing protons by energy onto image plates. Each configuration below replays real shots: the spray directions and the detector numbers follow the paper’s Table I.

Shot replay

Detector spectrum

A peaked group appears at 0° on top of the thermal fall-off — the first sign of shock acceleration — while the sideways spray weakens.

The shock-accelerated part separates clearly from the thermal part, and the maximum energy “more than doubled”.

Cycle the four two-sided shots: the forward signal swings from 0.44 to 65 ×10⁸ per sr at 0°. Shot 4, the best forward shot, also has the least sideways spray.

A second laser: GSI PHELIX

Without shaping, the main pulse ionises the whole millimetre-wide jet. Its light pressure shoves electrons out of the laser channel; the protons left behind are all positively charged and repel one another, flying apart mostly sideways. That gives many protons at 70° with an exponential, “thermal-looking” spectrum below about 2 MeV — and essentially nothing straight ahead. The paper attributes the untailored results to this mechanism. A forward, peaked group is the fingerprint of the shock instead.

06 Chapter 6 · Outlook

Closer — and what’s still missing

The strong shot-to-shot fluctuations tell the authors that the ideal conditions for shock acceleration were not fully reached. Comparing their estimates with what the shock needs:

Estimated ~30–40 µm vs ~40 µm optimum.

Estimated ~1.5 nc vs ≈ 3 nc (LULI) to 5.7 nc (GSI) needed at these intensities.

10 µm vs ≳ 25 µm: in a small spot the hot plasma escapes sideways before the shock can act.

Heating-spot shape and position vary, and the heating and main pulses jitter by 100–200 ps.

Proposed by the authors — not yet demonstrated

So what changed? A puff of hydrogen, sculpted by two blast waves, produced forward, peaked proton beams consistent with collisionless shock acceleration — from a target that can in principle be refreshed for every shot, without debris.

Reaching stable, high-energy, narrow-spread beams is the next step: with better heating spots, higher pressure and a larger main spot, the authors expect more stable acceleration, more protons, higher energies, narrower spectra and lower emittance.

Replay the forge

Source & credits

Source & credits

J.-R. Marquès, L. Lancia, P. Loiseau, P. Forestier-Colleoni, M. Tarisien, E. Atukpor, V. Bagnoud, C. Brabetz, F. Consoli, J. Domange, F. Hannachi, P. Nicolaï, M. Salvadori, B. Zielbauer , “Collisionless shock acceleration of protons in a plasma slab produced in a gas jet by the collision of two laser-driven hydrodynamic shockwaves.”

Version read: arXiv:2309.16277v1 (submitted version, 2023), CC BY 4.0 — https://arxiv.org/abs/2309.16277

Published version: Matter and Radiation at Extremes 9, 024001 (2024) — https://doi.org/10.1063/5.0178253

Facilities: LULI PICO2000 laser (France) and GSI PHELIX laser (Germany).

Figures

Figures 1, 3, 5, 6, 7, 8, 9, 10 and 11 and Table I are reproduced from Marquès et al., arXiv:2309.16277, under the Creative Commons Attribution 4.0 licence (CC BY 4.0). They have been resized; in Chapter 5 a highlight overlay marks the selected detector panel. No data in the figures were altered.

What is measured, simulated, modelled or illustrated

Measured Gas-jet profile, shadowgraphs, streak images, proton spectra and Table I counts are the paper’s experimental results.

Authors’ simulation Figures 10–11 and the slab thickness/density estimates come from the authors’ 3D TROLL radiation-hydrodynamics simulations with ray-traced synthetic shadowgraphy.

Simplified model The gas-sculpting animations (hero and lab) use a simplified 2D blast-wave model made for this page — not the authors’ code — calibrated to their Figs. 10–12.

Illustration Proton sprays: particle numbers are schematic; directions and energies follow the measured Table I and spectra. The shock “surf”, light-meets-plasma and ray-bending demos are teaching illustrations.

Typefaces: Fraunces, Inter and IBM Plex Mono (SIL Open Font License).