Mini WorldLaplace Energy, Flight & Sensing / Plate F67 — Frontier devices

Plasma Wakefield Acceleration

A laser pulse crossing a plasma leaves a wave of charge behind it. Electrons ride it to 7.8 GeV in 20 cm; the best copper cavities would need 78 m.

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Chapters · 15
  1. 017.8 GeV in 20 cm
  2. 02A Slope Hundreds of Times Steeper
  3. 03Pushed Aside, Pulled Back
  4. 04Density Sets the Wavelength
  5. 05The Bubble
  6. 06Catching the Wave
  7. 07Outrunning the Wave
  8. 08Cut Open: a Channel in Sapphire
  9. 09Not a Tabletop Laser
  10. 10A Magnet Reads the Energy
  11. 11Three Ways to Drive It
  12. 12What Is Still Hard
  13. 13What Comes Next
  14. 14Taken Apart
  15. 15The Laboratory

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01

7.8 GeV in 20 cm

In 2019 the BELLA team at Berkeley Lab reported in Physical Review Letters that a laser pulse had accelerated electrons to 7.8 GeV in a plasma 20 cm long. That shot had 5 pC of electrons at 7.8 GeV and several lower peaks besides; the whole shot carried about 420 pC.

Over the 20 cm that is an average of about 39 GeV per metre (this page’s arithmetic). The same team reached 1 GeV in 3.3 cm in 2006 and 4.2 GeV in 9 cm in 2014; in 2024, with 30 cm of plasma, single peaks went up to 9.2 GeV, with some charge beyond 10 GeV.

7.8 GeV · 20 cm · about 39 GV/m on average · 2019

02

A Slope Hundreds of Times Steeper

Conventional accelerators use copper cavities: radio waves fill them, and an electron gets a push in each cell it crosses. Push the field too high and the walls spark. In working machines and beam tests, copper cavities have reached about 100 MV/m, superconducting ones 31.5 MV/m.

To reach 7.8 GeV takes 78 m at 100 MV/m, 248 m at 31.5 MV/m, and 371 m at the roughly 21 MV/m of SLAC’s 3 km linac (accelerating length only; this page’s arithmetic). The plasma’s average of about 39 GV/m is some 400 times the best copper and nearly 2,000 times SLAC.

copper ≈ 100 MV/m · superconducting 31.5 MV/m · plasma ≈ 39 GV/m on average · 78 m is about 16 times the base’s length

03

Pushed Aside, Pulled Back

A boat leaves a wave trailing behind it (see the plate “The V Behind a Ship”). This wave also trails behind something, but what rises and falls is not a water surface: it is how densely charge is packed. A plasma is an ionised gas: its light electrons can move, its heavy ions barely do.

The intense laser pulse’s light pressure (the ponderomotive force) shoves the electrons in its path aside; the ions stay. Once the charges are apart the ions pull the electrons back; they overshoot, are pulled back again, and oscillate. As the pulse moves on, the oscillation follows it: a wave travelling at nearly the speed of light. What accelerates electrons is this wave’s field along the axis, not the laser itself.

the ponderomotive force pushes electrons aside · the ions stay · the wave follows the pulse at nearly c

04

Density Sets the Wavelength

How fast the electrons swing back depends only on how dense they are: the denser, the stronger the pull and the faster the swing. The wave follows the pulse at nearly the speed of light, so one swing stretches behind it into one wavelength: λₚ ≈ 3.3 × 10¹⁰ / √n µm, with n in electrons per cm³.

In BELLA’s 2019 shot the density was 2.7 × 10¹⁷ cm⁻³, giving a wavelength of about 64 µm; the field scale such a wave can hold, E₀ ≈ 96 √n V/m, is about 50 GV/m (this page’s arithmetic). At lower density the wave is longer and its field weaker, but electrons can ride it further.

λₚ ≈ 64 µm · E₀ ≈ 50 GV/m · n = 2.7 × 10¹⁷ cm⁻³

05

The Bubble

When the pulse is strong enough it pushes nearly every electron out of its way, and behind it is a cavity of ions alone, close to a sphere: the bubble. The expelled electrons flow round its wall, meet at its back, overshoot and close in again round the next.

In the analysis of Lu et al. (2007) the bubble’s radius is about 2√a₀ / kₚ, where a₀ is the laser’s dimensionless strength: BELLA’s 2019 pulse had a₀ ≈ 2.2 at its vacuum focus, giving a radius of about 30 µm (this page’s arithmetic). The field along the axis inside is close to a straight line: strongest at the back, pushing electrons forward, zero in the middle, slowing them in the front half.

radius ≈ 2√a₀/kₚ ≈ 30 µm · peak field ≈ √a₀ E₀

06

Catching the Wave

To be accelerated an electron must first fall into the back of the bubble, moving fast enough to keep up. In the 2019 shot electrons fell in on their own (self-injection): in the team’s simulation, about 5 cm into the channel the pulse focused itself, a₀ rose above 3 and the bubble began to trap electrons, starting and stopping several times, which is why several energy peaks came out.

In 2024 the same team added a little nitrogen (up to 5 %) to the first part of the gas only, to trigger injection: with no nitrogen no electrons came out, and with it they got single energy peaks, up to 9.2 GeV. The trapped bunch is shorter than one plasma wavelength.

self-injection from about 5 cm (simulation) · 2024: nitrogen-triggered · single peaks up to 9.2 GeV

07

Outrunning the Wave

A trapped electron moves at almost the speed of light; the wave moves at the laser’s group velocity in the plasma, a little slower. So the electron slowly gains on it, moving from the strongest field at the back of the bubble to the middle, where the field is zero. That distance is the dephasing length; beyond it the field starts to slow the electron down.

In Lu et al.’s formulas the dephasing length is about (2/3)(ω₀/ωₚ)² R and the energy gain about (2/3)mₑc²(ω₀/ωₚ)² a₀: halve the density and the gain doubles while the dephasing length grows about 2.8 times. For more energy: a thinner plasma, a longer channel, and a laser guided all that way.

L_d ≈ (2/3)(ω₀/ωₚ)² R · gain ∝ 1/n · L_d ∝ n^−3/2

08

Cut Open: a Channel in Sapphire

Focused to about 60 µm, the laser starts to spread after about 1.4 cm in vacuum (this page’s arithmetic); unguided, it is about 2.4 mm across after 20 cm. BELLA holds it in a capillary: a channel 20 cm long and 800 µm wide in a block of sapphire, filled with hydrogen, with an electrode at each end and gas inlets 6 mm from each end.

A discharge of about 450 A ionises and heats the hydrogen. The plasma cools at the wall, so it is hotter and thinner on the axis; the refractive index peaks in the middle and the channel holds the light like a graded-index fibre. In 2019 a 532 nm, 8 ns heater pulse deepened the channel: the spot it can hold shrank from about 100 µm to about 60 µm, and the 850 TW pulse was guided the full 20 cm.

20 cm × 800 µm · hydrogen · 450 A · heater 532 nm, 8 ns

09

Not a Tabletop Laser

The plasma is 20 cm long; the laser that drives it is not small. The BELLA petawatt laser is a titanium-sapphire laser at 815 nm: over 40 J squeezed into about 35 fs, a peak power over 1 PW (10¹⁵ W), once a second. Twelve frequency-doubled Nd:YAG lasers of 15 J each pump it, and their rate sets the one shot a second.

It arrived in over 80 crates weighing over 40 t, and fills its own clean laser bay, with its power supplies in another room upstairs and a radiation-shielded target area. The last mirror is an off-axis parabola with a 13.5 m focal length, focusing the light onto the capillary’s entrance; the model shortens those 13.5 m. The 2019 shot brought 31 J in 35 fs, 850 TW, to the focus.

815 nm · about 35 fs · > 1 PW · one shot a second · 13.5 m focal length

10

A Magnet Reads the Energy

Leaving the capillary, the electrons pass through a hole in a wedge-shaped mirror: the wedge sends the laser to an absorber and lets the electrons through. A phosphor screen behind it shows the beam’s direction and spread, about 0.2 mrad; then comes the magnetic spectrometer.

The magnet bends each electron on a radius ρ = p/(eB) that grows with its momentum: low energies bend more, high ones less, so the electrons spread out by energy along a screen; where it glows gives the energy and how bright gives the charge. The beam’s direction jumped by about 1 mrad from shot to shot, so in 2019 only 10 to 20 % of the beams fell within the spectrometer’s acceptance. That paper gives no size for the magnet; the model’s magnet and screen are drawn.

ρ = p/(eB) · 7.8 GeV at 1.5 T: ρ ≈ 17 m · spread about 0.2 mrad

11

Three Ways to Drive It

The wake need not be driven by a laser. In 1985 Chen, Dawson and colleagues proposed a bunch of electrons. At SLAC in 2007, 42 GeV electrons crossed 85 cm of plasma: the front of the bunch drove a wake and a small fraction at its back gained over 42 GeV more, at about 52 GV/m.

At FACET in 2014 two bunches followed each other, one to drive and one to ride: 74 pC gained 1.6 GeV in 36 cm, taking up to half the wake’s energy. CERN’s AWAKE uses 400 GeV protons, 19 kJ in each bunch, and in 2018 accelerated electrons to 2 GeV in 10 m of rubidium plasma. Protons carry so much energy that one stage might go very far; that run’s average gradient was about 0.2 GV/m.

laser: BELLA · electron bunch: SLAC 2007, FACET 2014 · proton bunch: AWAKE 2018

12

What Is Still Hard

The energy is there; the beam quality is not yet. The 2019 electrons came in several peaks; the 2014 beam at 4.2 GeV had a 6 % rms energy spread. Beams change from shot to shot: in 2019 the pointing jumped by about 1 mrad, and in 2024 stability was limited by where the laser hit the channel’s entrance and by pulse lengths varying by 20 % or more.

BELLA fires once a second. Higher energies mean linking stages: in 2016 BELLA joined two separately powered stages, and the second gave part of the beam 100 MeV more. A collider would also need positrons, which are much harder to accelerate in plasma.

several peaks · pointing jitter ≈ 1 mrad · one shot a second · two stages: +100 MeV

13

What Comes Next

Such electrons have already driven free-electron lasers: in 2021 a team at the Shanghai Institute of Optics and Fine Mechanics (CAS) used electrons from a laser wakefield accelerator to amplify light at 27 nm with exponential gain (for free-electron lasers see the plate “Free-Electron X-ray Lasers”). In 2022 a team at Frascati, Italy, made an infrared free-electron laser with a 3 cm beam-driven plasma.

Europe’s EuPRAXIA plans a 1 to 5 GeV plasma accelerator, driven by lasers and by beams, with a compact free-electron laser among its uses; its 2020 design report foresaw full operation within 8 to 10 years. Higgs factories built on plasma acceleration (HALHF, for one) are proposals: their authors estimate key technologies must improve by 10 to 1,000 times.

2021: a 27 nm free-electron laser · EuPRAXIA: planned · HALHF: a proposal

14

Taken Apart

The target chamber, the capillary, the parabola, the dipole magnet and the screen rise one by one; the copper RF structure slides forward.

Point at a part (tap it on a phone) and it lifts on its own, with its name; the rest fade.

15

The Laboratory

The sliders set the plasma density and the laser’s peak power. The readouts are the wake’s wavelength, the peak field in the bubble, the dephasing length and the energy reached there, estimated with Lu et al.’s formulas for a matched bubble (a₀ ≈ 2(P/P_c)^1/3) at 815 nm. It is a model, not a simulation; it assumes the pulse is guided and lasts.

With the 2019 settings (850 TW, 2.7 × 10¹⁷ cm⁻³) the model gives a₀ ≈ 4, about 8.5 GeV and a dephasing length of about 17 cm; the highest measured peak was 7.8 GeV, from a 20 cm channel.

Try thisIn the lab, set the laser to 850 TW and lower the plasma density from 2.7 × 10¹⁷ to 1 × 10¹⁷ cm⁻³: what happens to the wake’s wavelength, the dephasing length and the energy estimate?

What the model leaves out

The set is at 1 : 5.5. The capillary has its published sizes: a channel 20 cm long and 800 µm across; the sapphire block’s outer size is not published and is drawn. The 13.5 m from the parabola to the capillary is shortened with a break. The chambers, the magnet, the screen and the cameras are drawn from general practice, not BELLA’s drawings; the magnet bends the electrons as 1.5 T over 0.8 m would, and the bright spots on the screen stand for the 2019 shot’s several peaks. The copper RF structure is drawn as an S-band disk-loaded structure (35 mm cells), and only a piece of it, about 2 m, broken off at both ends: the same energy would take 78 m, about 16 times the base’s length. The infrared laser is drawn red; the laser and electron beams are drawn wider than they are; a shot takes nanoseconds and is slowed here to seconds. The wake in the magnifier is a sketch, not a simulation: the number of particles, the pulse’s fringes and the shape behind the bubble are simplified.

Common misconceptions

The laser does not push the accelerated electrons directly: the field comes from the separated charge in the plasma; the laser (or a particle bunch) only pushes the plasma’s electrons aside. The accelerated electrons ride at the back of the bubble, not in front of the pulse. “Tabletop” describes the plasma, a few to tens of centimetres long; the petawatt laser that drives it fills a room. Only part of the beam (5 pC) reached 7.8 GeV, not the whole bunch. The 2024 single peaks went to 9.2 GeV, with some charge beyond 10 GeV; that is not “10 GeV reached”. The ratio to copper depends on what is compared: about 400 times the best copper cavities, about 2,000 times SLAC. Plasma-based colliders are proposals; EuPRAXIA is planned, not built.

Common questions · 5
How high an energy has plasma wakefield acceleration reached?

In 2019 the BELLA team at Berkeley Lab reported in Physical Review Letters that a laser pulse had accelerated electrons to 7.8 GeV in a plasma 20 cm long.

017.8 GeV in 20 cm
How much stronger is plasma acceleration than a conventional accelerator?

The plasma’s average of about 39 GV/m is some 400 times the best copper and nearly 2,000 times SLAC.

02A Slope Hundreds of Times Steeper
How does the wake form?

The intense laser pulse’s light pressure (the ponderomotive force) shoves the electrons in its path aside; the ions stay. Once the charges are apart the ions pull the electrons back; they overshoot, are pulled back again, and oscillate.

03Pushed Aside, Pulled Back
Why can’t the electrons keep gaining energy?

A trapped electron moves at almost the speed of light; the wave moves at the laser’s group velocity in the plasma, a little slower. So the electron slowly gains on it, moving from the strongest field at the back of the bubble to the middle, where the field is zero.

07Outrunning the Wave
Is a laser wakefield accelerator really a tabletop machine?

The plasma is 20 cm long; the laser that drives it is not small. The BELLA petawatt laser is a titanium-sapphire laser at 815 nm: over 40 J squeezed into about 35 fs, a peak power over 1 PW (10¹⁵ W), once a second.

09Not a Tabletop Laser

Sources

  1. Gonsalves et al. (2019): Petawatt Laser Guiding and Electron Beam Acceleration to 8 GeV in a Laser-Heated Capillary Discharge Waveguide, Phys. Rev. Lett. 122, 084801 (7.8 GeV; 20 cm × 800 µm; 2.7 × 10¹⁷ cm⁻³; 31 J, 35 fs, 850 TW; the 13.5 m parabola; the heater; several peaks; pointing jitter)
  2. Picksley et al. (2024): Matched Guiding and Controlled Injection in Dark-Current-Free, 10-GeV-Class, Channel-Guided Laser-Plasma Accelerators, Phys. Rev. Lett. 133, 255001 (single peaks up to 9.2 GeV; 30 cm; nitrogen-triggered injection; stability)
  3. Leemans et al. (2014): Multi-GeV Electron Beams from Capillary-Discharge-Guided Subpetawatt Laser Pulses in the Self-Trapping Regime, Phys. Rev. Lett. 113, 245002 (4.2 GeV, 9 cm, 6 % rms spread)
  4. Leemans et al. (2006): GeV electron beams from a centimetre-scale accelerator, Nature Physics 2, 696 (1 GeV, 3.3 cm)
  5. Leemans et al. (2013): BELLA Laser and Operations, Proc. PAC2013, THYAA1 (over 1 PW at 1 Hz; twelve pump lasers; over 80 crates, over 40 t)
  6. Turner et al. (2021): Radial density profile and stability of capillary discharge plasma waveguides of lengths up to 40 cm, High Power Laser Sci. Eng. 9, e17 (20 cm sapphire capillaries; gas inlets 6 mm from the ends)
  7. Tajima and Dawson (1979): Laser Electron Accelerator, Phys. Rev. Lett. 43, 267 (a laser pulse leaves a wake in a plasma)
  8. Chen, Dawson, Huff and Katsouleas (1985): Acceleration of Electrons by the Interaction of a Bunched Electron Beam with a Plasma, Phys. Rev. Lett. 54, 693 (a bunch of electrons drives the wake)
  9. Esarey and Schroeder (2003): Physics of Laser-Driven Plasma-Based Accelerators, LBNL-53510 (the formulas for λₚ and E₀; RF at about 100 MV/m; the ponderomotive force)
  10. Esarey, Schroeder and Leemans (2009): Physics of laser-driven plasma-based electron accelerators, Rev. Mod. Phys. 81, 1229 (review)
  11. Lu et al. (2007): Generating multi-GeV electron bunches using single stage laser wakefield acceleration in a 3D nonlinear regime, Phys. Rev. ST Accel. Beams 10, 061301 (the bubble’s radius, the dephasing length, the energy gain)
  12. Blumenfeld et al. (2007): Energy doubling of 42 GeV electrons in a metre-scale plasma wakefield accelerator, Nature 445, 741 (85 cm, about 52 GV/m)
  13. Litos et al. (2014): High-efficiency acceleration of an electron beam in a plasma wakefield accelerator, Nature 515, 92 (1.6 GeV, 4.4 GV/m, 74 pC, up to 50 % efficiency)
  14. AWAKE Collaboration (2018): Acceleration of electrons in the plasma wakefield of a proton bunch, Nature 561, 363 (400 GeV protons, 10 m, 2 GeV)
  15. Steinke et al. (2016): Multistage coupling of independent laser-plasma accelerators, Nature 530, 190 (two stages; 100 MeV in the second)
  16. Aniculaesei et al. (2024): The acceleration of a high-charge electron bunch to 10 GeV in a 10-cm nanoparticle-assisted wakefield accelerator, Matter Radiat. Extremes 9, 014001 (arXiv:2207.11492)
  17. Wang et al. (2021): Free-electron lasing at 27 nanometres based on a laser wakefield accelerator, Nature 595, 516
  18. Pompili et al. (2022): Free-electron lasing with compact beam-driven plasma wakefield accelerator, Nature 605, 659
  19. Assmann et al. (2020): EuPRAXIA Conceptual Design Report, Eur. Phys. J. Spec. Top. 229, 3675 (planned: 1⁠–⁠5 GeV)
  20. Foster, D’Arcy and Lindstrøm (2023): A hybrid, asymmetric, linear Higgs factory based on plasma-wakefield and radio-frequency acceleration, New J. Phys. 25, 093037 (a proposal)
  21. Shiltsev and Zimmermann (2021): Modern and future colliders, arXiv:2003.09084 (copper about 100 MV/m, superconducting 31.5 MV/m; SLAC’s 3 km, 50 GeV)
  22. Faure et al. (2004): A laser–plasma accelerator producing monoenergetic electron beams, Nature 431, 541 (with Mangles et al. and Geddes et al. in the same issue)

Data

The set follows the layout of the BELLA laser-plasma accelerator described in Gonsalves et al., Phys. Rev. Lett. 122, 084801 (2019) and Leemans et al., Proc. PAC2013 THYAA1 (the focusing parabola, the capillary, the wedge with a hole, the screens and the magnetic spectrometer). The capillary’s published sizes are used (20 cm long, 800 µm across; sapphire as in Turner et al., High Power Laser Sci. Eng. 9, e17 (2021)); the chambers, the magnet, the screen and the copper structure are drawn from general practice, not from BELLA’s drawings, and the notes list what is simplified. The wake’s wavelength and field, the bubble’s size, the dephasing length, the energy estimates, the guided and unguided spot sizes, the bending radii and the linac lengths are computed by this page (science.js, tests/wakefield.test.js) from the quoted formulas; the laboratory is a model, not a simulation. Material presets, parts and the softbox studio of the site’s model kit are ported from AirsupHQ/machine-skill (MIT, © 2026 Air Supply Technologies Inc.).

The text, pictures and models on this page are licensed CC BY-NC 4.0: free to use and adapt for teaching and other non-commercial purposes, with credit to world.lapla.si. Third-party material listed above keeps its own licence.

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Laplace’s Mini World, Plate F67, “Plasma Wakefield Acceleration”, https://world.lapla.si/en/wakefield/, CC BY-NC 4.0, updated 2026-11-05.

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Plate 35The V Behind a ShipA big ship, a small boat, even a swimming duck: the V-shaped wake behind them opens at almost exactly the same angle, about 19.5° on each side. Plate F45Free-Electron X-ray LasersA kilometres-long straight accelerator and long rows of magnets turn electrons into X-ray flashes a billion times brighter at peak than the best synchrotrons, a few to about 100 femtoseconds long.

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