Plasma-based wakefield accelerators can produce high-brightness relativistic electron beams (e beams) of compact size1–3 with an accelerating gradient of 100 GV/m, which is almost three orders of magnitude stronger than those provided by conventional radio-frequency (RF) accelerators.4,5 Over the past decade, great progress has been made in developing laser-driven wakefield accelerators (LWFAs), in terms of both electron energy gain and e-beam quality, with the maximum energy being extended up to 10 GeV6,7 and the divergence and energy spread being reduced to 0.1 mrad and a per-mille level.8,9 These high-energy bright e beams can find promising applications in medicine,10,11 high-energy physics,5 and photon science,12,13 especially for developing compact free-electron lasers (FELs)14,15 and novel light sources.16–23 Nowadays, most interest is focused on high-energy LWFAs beyond 100 MeV, which mainly rely on 100 TW class or even PW laser systems and are limited to a low repetition rate with current technology.

However, using a much smaller laser system with 1–10 TW to develop low-energy LWFAs for generating MeV-class e beams with femtosecond durations have now attracted increasing interest24–28 because of the potential applications in ultrafast electron diffraction and electron probes, gamma-ray flashes,29 and magnetic reconnection in laboratory astrophysics.30 However, in comparison with e beams in the 100 MeV to 1 GeV energy range from a high-energy LWFA, the generated MeV-class e beams from a low-energy LWFA usually have large divergence and energy spread. On the one hand, for these TW-class laser systems driving an efficient LWFA, a high plasma density of n_e ≥ 10²⁰ cm⁻³ is required, making the dephasing length and pump depletion length very short, and usually inducing significant laser–plasma instability to degrade the e-beam quality. On the other hand, the carrier-envelope-phase (CEP) effect of the few-cycle driving laser pulses31,32 also becomes more important for the injection and acceleration and will greatly affect the e-beam stability and quality. Recently, several approaches have been reported to improve the e-beam qualities for few-cycle laser wakefield acceleration, such as by modifying the nozzle structure to increase the beam charge,33 employing circularly polarized light to reduce the divergence and mitigate CEP slip effects,34 and enhancing the beam stability with a specific density structure.35 Although some parameters of the MeV-class e-beam quality have been improved, the absolute energy spread (AES) and divergence are still relatively large, and these need to be optimized further to increase beam brightness.

In this work, we propose a scheme to generate high-quality e beams at around 50 MeV by trapping longitudinally low-divergence electrons in a weakly nonlinear wake driven by a TW-level laser system. Usually, for wave breaking mechanisms,36 longitudinal injection provides better beam quality and stability compared with transverse injection. By designing a plasma with a steep density down-ramp, a moderately intense laser pulse with a normalized laser intensity of a = 1.3 can drive a weakly nonlinear plasma wave with several periods of buckets. Some paraxial electrons can slip backward axially to be trapped into the second wake, which undergoes a much faster decrease in phase velocity in the down-ramp region and therefore has a much lower injection threshold. It has been found that these injected electrons have a very small emittance of ∼10 nm rad and a low absolute energy spread (AES) of ∼50 keV, and the growth of emittance and energy spread are effectively restrained to maintain beam brightness during the subsequent acceleration process. As a result, output e beams with a peak energy in the 5–70 MeV range can be produced with a very small AES and a minimum emittance of ∼28 nm rad, making them suitable for practical applications in ultrafast electron diffraction,37,38 in coherent radiation source generation,39–41 and as injectors for GeV particle accelerators.42

Figure 1 illustrates the overall generation of high-brightness e beams from a weakly nonlinear LWFA with a typical density-tailored plasma that has three down-slope distributions. As shown in Fig. 1(a), a steep density down-ramp, dropping from a peak density n_p = 1.5 × 10¹⁹ cm⁻³ at z = 400 µm to n₀ = 0.9 × 10¹⁹ cm⁻³ within a steep down-ramp length L_d1 = 13.5 µm, which is longer than the plasma wavelength λ_p = 8.6 µm, is introduced to manipulate the wake phase velocity of the second bucket and significantly reduce the electron velocity required for injection via the density down-ramp injection.8,43–46 This kind of sharp density down-ramp can actually be produced by inserting a specific blade placed below a supersonic gas nozzle43 or by the expansion of a pre-ionized plasma.44 The phase velocity of the excited plasma waves47 can be expressed as ${\beta }_p={[1+(\xi /2n_e)d{n}_e/dz]}^{-1}$, where n_e is the density of the plasma and ξ = z − ct < 0 is the longitudinal coordinate in the comoving frame (ξ < 0 behind the peak of the laser pulse). At the steep density down-ramp, the local phase velocity of the plasma wave, which is expressed as ${\beta }_{p,m}={[1+md{\lambda }_p/dz]}^{-1}$ with ξ = −mλ_p (where m is the serial number of the wake structure behind the laser pulse, i.e., m = 2 for the second bucket), will decrease rapidly and the electrons converging into the tail of the second wake will more easily exceed the local phase velocity than the first wake to realize electron injection. To obtain insight into the whole process of electron injection, acceleration, and the evolution of the plasma wave, we have conducted three-dimensional (3D) simulations using the Fourier–Bessel particle-in-cell (FBPIC) code.48 The simulation window is set to a size of 60 × 40 µm² with a 2400 × 800 grid cell along the propagation direction z and the transverse azimuthal direction r, and 16 macroparticles per cell. The plasma is pre-ionized with an open boundary. A linearly polarized laser pulse of 120 mJ and with a central wavelength λ_L = 800 nm enters the plasma from the left boundary into the simulation box. The laser pulse with an on-target peak power of 8 TW has a spatiotemporal Gaussian profile with a spot radius r₀ = 12 µm and a full width at half maximum (FWHM) duration of 15 fs. The peak intensity of the input driving laser pulse is 3.6 × 10¹⁸ W/cm², corresponding to a normalized laser intensity a₀ = 1.3.

Figure 1(b) shows the evolution of the normalized laser intensity a and the phase velocity of the second wake β_p,2 along the propagation. The wake phase velocity β_p,2 decreases rapidly to 0.76 at the density down-ramp at around z = 400 µm. During the propagation in the density-tailored plasma, the driving laser only self-focuses smoothly to reach its maximum normalized laser intensity of a_max ≈ 1.6, ensuring that a weakly nonlinear plasma wave is excited instead of a strongly nonlinear bubble. Figures 1(c)–1(e) show 3D views of the TW laser-driven wakefield acceleration from FBPIC simulations for three segments at t = 1.46, 1.93, and 3.11 ps, respectively. Electrons are injected in the second wake at the density down-ramp around t = 1.46 ps. Owing to the moderate laser intensity, no electrons can be trapped in the first wake.

To understand the mechanism of longitudinal electron injection in this regime, the trajectories of electrons of interest have been carefully traced and analyzed. As presented in Figs. 2(a)–2(c), since the ponderomotive force F_p ∝ ∇a² for a moderately intense laser beam is not large enough to expel all electrons transversely, a thin tabular electron sheath appears mainly at the tail of the plasma wave49 instead of a circular bubble structure. It is observed that from t = 1.40 to 1.51 ps, some backward-moving paraxial electrons with a very small transverse oscillation amplitude slip backward through the center of the first periodic wake and are directly injected into the rear of the enlarging second bucket. Before the injection, these paraxial electrons have undergone deceleration and acceleration twice during their backward movement from the first wake to the second one. Accordingly, the longitudinal momentum u_z first decreases but then increases until the electrons’ longitudinal velocity is larger than the second wakefield phase velocity to realize the longitudinal injection, where the wake expansion is clearly observed at the downward density transition L_d1. The minimum energy for the electrons’ longitudinal capture into the wake can be estimated as Ref. 50γ_min ≈ β_p/(2Δϕ) + Δϕ/(1 + β_p) in the limit of γ_pΔφ ≫ 1, where ${\gamma }_p={(1-{\beta }_p^2)}^{-1}$ is the normalized Lorentz factor of the plasma wave and Δϕ = ϕ_max − ϕ_min is the difference in the wake potential ϕ = eΦ/m_ec² in a single wake period. With an undifferentiated wake strength of Δϕ₁ ≈ Δϕ₂, the electrons can be injected more easily into the second wakefield for a smaller phase velocity β_p,2 < β_p,1 as compared with the first wake. It is found that this injected e beam has a very short duration of <5 fs, accompanied by an ultralow AES of ∼50 keV, as shown in Fig. 2(d).

Apart from the ultralow AES electron injection, these longitudinally injected electrons have an ultralow emittance with only tens of nm rad. As presented in Figs. 2(e) and 3(a)–3(c), the injected e beam exhibits an ultranarrow transverse momentum distribution of ∆u_x ≈ 0.03m_ec and a very small transverse size of ∆x ≈ 0.4 µm at z = 420 µm, corresponding to a normalized emittance of 10 nm rad. It then increases to 32 nm rad at z = 450 µm and then is eventually saturated around 28 nm rad. Meanwhile, the trapped electrons undergo small betatron oscillations with a decreasing transverse size of ∼0.1 µm during the following acceleration. By contrast, if the intensity of the driving laser pulse is increased to a = 2.5, the plasma wave evolves into the bubble regime, where electrons are mostly expelled outward to form a circular bubble sheath, and the typical transverse injection is then induced at the tail of the wake. In this case, the transverse oscillation amplitude will increase by nearly an order of magnitude to be larger than ∼1.5 µm, and the transverse momentum u_x also increases to ∼2m_ec, corresponding to an emittance larger than 200 nm rad, as shown in Figs. 3(d)–3(f). It is verified that this longitudinally localized electron injection in a tailored plasma using a moderately intense driving laser favors high-brightness MeV-class e-beam generation along with ultralow emittance.

After injection, the e beam undergoes natural chirp self-compensation to achieve a low energy spread. As shown in Fig. 4(c), when the steep density down-ramp is terminated, the injected e beam has an ultralow AES of ∼50 keV with a slight negative energy chirp of 0.05 MeV/μm, since the head of the beam was injected earlier and gained more energy than the tail. Over a very short distance from z = 435–580 µm, the electrons still remain in the negative-slope accelerating field at the tail of the second wake, and the negative energy chirp accumulates further as shown in Figs. 2(f) and 4(a). A local chirping strength of ∼20 MeV mm⁻¹µm⁻¹ is estimated for the early acceleration, and its negative energy chirp increases to acquire the maximum energy spread of 3.1 MeV at t = 1.93 ps in Fig. 4(d). Subsequently, a much slower plasma down-ramp that decreases from 9 × 10¹⁸ to 6 × 10¹⁸ cm⁻³ over the acceleration stage L_d2 is introduced to shift the e beam forward into the positive-slope field zone. The accumulated negative chirp will then be mostly compensated via dechirping, before exiting the plasma at z = 0.95 mm with an AES of ∼0.49 MeV at t = 3.33 ps, as shown in Figs. 4(e) and 4(f). In this way, the initial low AES is retained, the energy chirp can be minimized by adjusting the plasma density, and the desired energy gain can be achieved. After passing through the transport stage L_d3, the final output e beam with a peak energy of 51 MeV and beam charge of 4.4 pC is produced with an ultralow emittance of ∼28 nm rad, energy spread of ∼1%, and duration less than 5 fs. Its six-dimensional beam brightness $B_{6D,n}=I/({\epsilon }_n^2\cdot {\sigma }_{\delta })$ is estimated here to be as high as 2 × 10¹⁷ A/m²/0.1%, where I is the peak current, ɛ_n is the normalized emittance, and σ_δ is the relative energy spread in units of 0.1%.

In addition, the e-beam parameters can be tuned and controlled by adjusting the density gradient and density ratio for the down-ramp at L_d1. As illustrated in Figs. 5(a) and 5(c), by decreasing (or increasing) the density gradient dn_e/dz (i.e., lengthening or shortening L_d1) for a given density ratio n_peak/n₀, the energy spread and emittance of the beam can be decreased (or increased), and a higher (or lower) energy gain can be achieved, but the beam charge decreases (or increases) accordingly under these injection conditions. The reason is that the steeper the down-ramp L_d1, the smaller is the phase velocity required to satisfy the conditions for fast local injection, and more electrons will be injected to increase the peak current, and a beam loading effect occurs to reduce the energy gain, leading to an increase in the emittance and energy spread as well. However, in such a weakly nonlinear plasma wake, high-brightness e beams ranging from 5 to 70 MeV with a small emittance of ≤100 nm rad and low energy spread at a few-percent level can be produced after optimization. With a larger density ratio n_peak/n₀, the e-beam charge will be increased while the energy gain E_e will be decreased, as shown in Figs. 5(b) and 5(d), and a relatively large emittance of ∼100 nm rad and energy spread will be introduced accordingly. However, for a steep density down-ramp L_d1 (shown by the blue shaded region), the density gradient and difference can be carefully chosen to generate an ultrahigh-brightness e beam.

In conclusion, we have proposed a simple but promising scheme to generate ultrabright e beams at around 50 MeV by exciting a weakly nonlinear laser wakefield, which, when crossing the sharp density gradient, facilitates longitudinal injection. This approach has the advantage over the bubble regime in that it produces a brighter e beam with a moderate laser driver power. Besides, longitudinal injection can also be more easily induced to obtain low-emittance e beams. By adjusting the steep density down-ramp, low-divergence electrons with initial emittance of <30 nm rad can be trapped longitudinally and then undergo chirp self-compensation during the subsequent acceleration, with the energy spread being compressed down to less than 1% when the e-beam energy reaches several tens of MeV. An ultrahigh six-dimensional beam brightness B_6D,n of as high as 2 × 10¹⁷ A/m²/0.1% is obtained here, which is much higher than the values previously reported. It is anticipated that such bright e beams will be useful for a variety of applications, for example, as ultrafast electron probes for diffraction and imaging, in laboratory astrophysics, in coherent radiation source generation, and as high-quality electron injectors for high-energy particle acceleration.

Acknowledgment. This work was supported by the National Natural Science Foundation of China (Grant Nos. 11974251, 12105180, 12074397, 11904377, and 12005137), the Innovation Program of Shanghai Municipal Education Commission (Grant No. 2021-01-07-00-02-E00118), and the National Key Research and Development Program (Grant No. 2023YFA1406804).