2 Experimental setup

The experiment was conducted using the SECUF-PW laser facility, a Ti:sapphire laser system at the ultrafast X-ray dynamics experimental station of the Synergetic Extreme Condition User Facility (SECUF), Institute of Physics, Chinese Academy of Sciences. The experimental setup is illustrated in Figure 1. A laser pulse with a central wavelength of 800 nm, full width at half maximum (FWHM) duration of 25 fs and energy of 9 J was focused onto the gas target by using an f/4.4 off-axis parabolic mirror. The laser pulse was focused to a spot with radius w ₀ ≈ 4.7 μm, containing approximately 47% of the total pulse energy, yielding a peak intensity of approximately 5.2 × 10²⁰ W/cm² and the normalized vector potential a ₀ ≈ 15.6. Two types of gas nozzles were connected to a fast-operating valve respectively in the experiment. The first was a convergent–divergent Laval nozzle with an exit diameter of 300 μm, while the second featured a 2 mm diameter (see Appendix A). Pure nitrogen was used as the working gas to exploit ionization-induced injection and thereby enhance the electron injection rate. Offline density measurements revealed the nitrogen density profile. The 2 mm nozzle consists of a 1000 μm density up-ramp, followed by an 800 μm plateau and then a 1200 μm density down-ramp, while the 0.3 mm nozzle yielded a Gaussian-like density distribution. Plasma density was varied by adjusting the gas backing pressure. The electron spectrometer consists of an 8 cm × 8 cm permanent magnet with a maximum magnetic field of 0.98 T. The accelerated electrons were deflected by the magnet onto the phosphor screen, coupled with a 16-bit visible charge-coupled device (CCD) camera. The spectrometer covered an energy range from 7.5 to 77.5 MeV. A top-view imaging system was used to monitor the plasma channel via the scattered light. The electron beam charge was measured using 20 cm × 20 cm absolutely calibrated image plates^[ Reference Tanaka, Yabuuchi, Sato, Kodama, Kitagawa, Takahashi, Ikeda, Honda and Okuda ^{21 –} Reference Ingenito, Andreoli, Batani, Boutoux, Cipriani, Consoli, Cristofari, Curcio, Angelis, Giorgio, Ducret, Forestier-Colleoni, Hulin, Jakubowska and Rabhi ^{23 ]} (IPs, Fuji Film SR series) positioned 24.5 cm away from the target along the beam axis inside the chamber. A series of energy cut-off filters allowed charge measurements across a range of energy thresholds, from more than 0.6 MeV up to more than 6 MeV.

Figure 1 Schematic of the experimental setup.

3 Experimental results

The single-shot electron energy distributions obtained by using the 0.3-mm-diameter nozzle (shot #1) and the 2-mm-diameter nozzle (shots #2–#4) are presented in Figure 2(a). For both nozzles, the laser was focused 300 μm upstream of the nozzle center. For the 0.3 mm nozzle, the electron energy spectrum exhibits a two-temperature distribution of the form ${N}_\mathrm{e}\propto {e}^{\frac{E}{T_{\mathrm{e}1}}}+{e}^{\frac{E}{T_{\mathrm{e}2}}}$ where T _e1 = 7.3 MeV and T _e2 = 19.9 MeV (Figure 2(b)), a typical signature of LWFA in high-density plasma conditions^[ Reference Feng, Li, Tan, Wang, Li, Zhang, Meng, Ge, Liu, Yan, Fu, Chen and Zhang ^{24 ]}. The measured electron beam charge above 6 MeV was 7.3 nC, with an FWHM divergence angle of approximately 15° at a nitrogen density of approximately 1.7 × 10¹⁸ cm^–3 (Figure 2(c)), and the laser-to-electron energy conversion efficiency was approximately 1.4%. In contrast, the energy spectra obtained from the 2 mm nozzle show significant enhancements within the nitrogen density range from 1.2 × 10¹⁸ to 2.4 × 10¹⁸ cm^–3 (Figure 2(b)). Notably, at a density of 1.2 × 10¹⁸ cm^–3, a prominent plateau emerges around 30 MeV. The integrated charges above 6, 4 and 2 MeV are 31, 63 and 116 nC, respectively, significantly exceeding the prediction from the nonlinear bubble regime scaling law^[ Reference Gordienko and Pukhov ^{6 ,} Reference Pukhov and Meyer-ter-Vehn ^{25 ]} (see Appendix B). The electron spectrum for energies below 7.5 MeV was extrapolated by fitting the measured charges across different energy thresholds. The laser-to-electron energy conversion efficiency reaches 6.1%, 10.7% and 16.4% for electron energy above 6, 4 and 2 MeV, respectively (see Appendix C). Here, the energy conversion efficiency is defined as the ratio of the total kinetic energy of accelerated electrons to the incident laser energy. The divergence of the electron beam was approximately 22° (Figure 2(d)).

Figure 2 Experiment results of plasma wakefield acceleration. (a) Representative single-shot electron energy distributions for the 0.3-mm-diameter nozzle (shot #1) and the 2-mm-diameter nozzle (shots #2–#4) at different nitrogen densities: shot #1 (1.7 × 10¹⁸ cm^–3), shot #2 (1.2 × 10¹⁸ cm^–3), shot #3 (1.8 × 10¹⁸ cm^–3), shot #4 (2.4 × 10¹⁸ cm^–3). (b) Corresponding electron energy spectra for shots #1–#4. (c), (d) Angular distribution of electron beams for the 0.3 and 2 mm nozzles, respectively. (e), (f) Top-view images of side Thomson-scattered light for the 0.3 and 2 mm nozzles, respectively. White dashed lines indicate nozzle boundaries, while white solid arrows denote laser propagation directions.

The top-view imaging of scattered light was used to estimate the laser propagation distance, offering insights into the physical mechanisms behind the spectral enhancement. The laser propagation direction is indicated by white arrows in Figures 2(e) and 2(f). For the 0.3 mm nozzle, the laser propagates entirely through the gas target (Figure 2(e)). In contrast, for the 2 mm nozzle, the scattered light extends only half of the nozzle length (Figure 2(f)). This suggests that in the longer nozzle, a transition to PWFA may occur after significant laser energy depletion. In addition, we rotated the magnetic spectrometer to deflect electrons in the plane perpendicular to the laser polarization to assess the presence of direct laser acceleration (DLA). If DLA mechanisms were present, electrons would gain preferential momentum along the laser polarization. The spectrum would split into a forked structure at the high-energy tail^[ Reference Shaw, Lemos, Amorim, Vafaei-Najafabadi, Marsh, Tsung, Mori and Joshi ^{26 ,} Reference Shaw, Lemos, Marsh, Froula and Joshi ^{27 ]}. However, no forked structure was observed in our spectrum, confirming that the contributions from DLA are negligible.

4 Simulation results

To investigate the physical mechanism behind the enhanced electron beam conversion efficiency, we performed simulations using the quasi-3D PIC code FBPIC^[ Reference Lehe, Kirchen, Andriyash, Godfrey and Vay ^{28 ]}. The simulation parameters were set to closely match the experimental conditions. A linearly polarized laser pulse along the x-axis with wavelength of 800 nm, laser intensity a ₀ = 14, pulse duration of τ₀ = 25 fs (FWHM) and spot size of w ₀ = 4.72 μm was focused into a pure nitrogen plasma. For the 2 mm nozzle, the plasma density profile consisted of a 1000 μm up-ramp, an 800 μm density plateau and a 1200 μm down-ramp. In comparison, the 0.3 mm nozzle provided a Gaussian density profile with an FWHM of 300 μm and a total length of 600 μm. The simulations were performed in a cylindrical geometry using two azimuthal modes. The simulation window size was 160 μm (longitudinal, x) × 60 μm (radial, r), with grid resolutions of ∆x = λ/20 and ∆r = λ/4, respectively.

Figure 3 presents the evolution of the electron density distribution, transverse electric field, phase space (x, E _k) of the electron beam and electron energy spectrum above 6 MeV for two distinct nozzles. The nitrogen plasma densities were set to 1.7 × 10¹⁸ cm^–3 for the 0.3 mm nozzle and 1.2 × 10¹⁸ cm^–3 for the 2 mm nozzle. For the 0.3 mm nozzle, the laser ponderomotive force rapidly expels electrons to form an ion cavity structure, also known as plasma bubble (Figures 3(a) and 3(e)). This structure captures and accelerates electrons, generating an electron beam of approximately 6 nC, in agreement with experimental measurements. For the 2 mm nozzle, the acceleration process evolves through three distinct stages, namely LWFA, LWFA–PWFA hybrid and PWFA. In Figures 3(b) and 3(f), the early stage is dominated by LWFA, accompanied by minor contributions from ponderomotive acceleration and DLA^[ Reference Martelli, Kononenko, Andriyash, Wheeler, Gautier, Goddet, Tafzi, Lahaye, Giaccaglia, Flacco, Tomkus, Mackevičiūtė, Dudutis, Stankevic, Gečys, Račiukaitis, Kraft, Dinh and Thaury ^{29 ,} Reference Yang, Brunetti, Gil, Welsh, Li, Cipiccia, Ersfeld, Grant, Grant, Islam, Tooley, Vieux, Wiggins, Sheng and Jaroszynski ^{30 ]}. In the hybrid LWFA–PWFA stage (Figures 3(c) and 3(g)), the wakefield is co-driven by both the laser and the electron beam, which can be described by $ \left(\frac{\partial^2}{\partial {t}^2}+{\omega_\mathrm{p}}^2\right)\frac{\delta {n}}{n_0}=-{\omega_\mathrm{p}}^2\frac{n_\mathrm{b}}{n_0}+{c}^2\varDelta \frac{\overset{\rightharpoonup }{a}}{2} $ ^[ Reference Götzfried, Döpp, Gilljohann, Foerster, Ding, Schindler, Schilling, Buck, Veisz and Karsch ^{31 ]}. Here, ω_p denotes the plasma frequency, n _b is the electron beam density and δn = n _e − n ₀ represents the plasma density perturbation, where n _e and n ₀ are the local and background plasma densities, respectively. As the laser is gradually depleted, the electrons accelerated by the laser cease gaining energy and instead begin to drive the plasma wakefield, which in turn accelerates the tail of the electron beam. At this point, the laser pulse energy is nearly fully depleted in the plasma, and the laser-to-electron energy conversion efficiency reaches its maximum. In the pure PWFA stage, as shown in Figures 3(d) and 3(h), a small fraction of the high-energy drive beam transfers its energy to multiple low-energy witness beams, leading to a significant increase in the charge of the low-energy electrons. In this case, the electron charge above 6 MeV is approximately 26 nC, which is consistent with the charge levels in the experiment.

Figure 3 Simulation results of plasma wakefield acceleration. (a)–(d) Simulated snapshots of the electron density distribution and transverse electric field. (e)–(h) The corresponding longitudinal phase space of the electron beam and electron energy spectrum above 6 MeV.

Figure 4(a) presents the evolution of the laser a ₀, the total charge of electrons with energy above 6 MeV and the corresponding laser-to-electron energy conversion efficiency for the case of the 2 mm nozzle. The initial rapid rise in both energy conversion efficiency and electron charge is primarily due to the rapid transfer of energy from the laser to the plasma wakefield, which traps and accelerates a large number of electrons, yielding nearly 18 nC of charge. As the laser gradually depletes, the accelerated electron beam begins to drive the wakefield and transfers energy to multiple low-energy witness electron beams, sustaining continued charge growth, leading to a continued growth of charge up to approximately 26 nC. The conversion efficiency peaks at 15% for electrons above 6 MeV at t ≈ 6 ps. Thereafter, as soon as a ₀ drops, the LWFA–PWFA hybrid stage gradually transitions to pure PWFA, resulting in a decrease of the overall laser-to-electron efficiency. The electron energy spectrum evolution for the 2 mm nozzle is shown in Figure 4(b). During the LWFA stage, electrons are accelerated to the maximum energy of approximately 300 MeV by a highly nonlinear plasma wakefield. In the LWFA–PWFA hybrid and pure PWFA stages, the high-energy electron beam decelerates, while multiple electron beams are injected and accelerated to tens of MeV.

Figure 4 PIC simulation results for the 2-mm-diameter nozzle. (a) Temporal evolution of the laser intensity, total charge of electrons with energies of more than 6 MeV and the corresponding laser-to-electron energy conversion efficiency. (b) Evolution of the electron energy spectrum. (c) Laser-to-electron energy conversion efficiency and total electron charge as a function of nitrogen density. (d) Simulated energy spectra for the 0.3 and 2 mm nozzles. (e) Correlation between electron energy and divergence angle.

Figure 4(c) presents the dependence of laser-to-electron energy conversion efficiency and total electron charge on nitrogen plasma density for the 2 mm nozzle. For nitrogen densities below 0.9 × 10¹⁸ cm⁻³, lower densities lead to slower laser energy depletion and stronger diffraction, resulting in inefficient energy transfer to electrons. At densities above 0.9 × 10¹⁸ cm⁻³, higher densities cause faster laser depletion, prompting a rapid transition from the LWFA stage through the hybrid stage to the PWFA stage, leading to enhanced electron charge and conversion efficiency. However, rapid laser depletion prolongs the PWFA stage, ultimately causing a decline in both the average electron energy and conversion efficiency. Figure 4(d) compares the simulated energy spectra for the cases of the 0.3 and 2 mm nozzles. The case of the 2 mm nozzle yields a significantly stronger energy spectrum than the case of the 0.3 mm nozzle, in qualitative agreement with experimental observations. Notably, the energy spectrum of the 2 mm nozzle exhibits a distinct plateau around 30 MeV, closely matching the experimentally observed feature. However, the simulations predict more high-energy electrons than those experimentally observed for the 2 mm nozzle. This discrepancy may arise because higher-energy electrons, which typically exhibit smaller divergence angles (see Figure 4(e)), do not easily pass through the limited collimation aperture of the magnetic spectrometer.