There has been tremendous progress in laser wakefield accelerators (LWFAs) in recent decades, and they have become one of the most important candidates for the next generation of accelerator-based light sources and colliders.1–7 The present energy record in LWFA experiments is 7.8 GeV,8,9 and the typical energy spread is in the range of a few percent, with the lowest reported value being slightly below 1%, with beam charges of tens of picocoulombs.10–12 However, for challenging applications such as plasma-based colliders and free-electron lasers, orders of magnitude better beam qualities are required, and need to be achieved simultaneously, and this has motivated an ongoing search for methods of generating high-quality beams.13–16

Among these methods, all-optical injection schemes are very attractive owing to their simple setups and their potential for precise controllability.17–19 Experimental demonstrations of these schemes and modified versions typically produce electron beams with energy spread $>$1% and charges of tens of picocoulombs.20–25 Further optimization of these schemes, based on particle-in-cell (PIC) simulations,26–30 and optimizations of other injection schemes relying on assistant lasers such as plasma torches31 and Trojan Horse injection,32–34 as well as of methods that rely on higher laser modes,35 have suggested that better beam qualities are attainable. However, none of these studies has produced a beam with simultaneously small energy spread of a few permille, small emittance ($<$0.5 mm mrad), and large amount of charge ($>$100 pC).

In this work, we propose a novel all-optical injection scheme that can generate high-quality electron beams with parameters beyond the above-mentioned values, by utilizing the interference of two coaxial laser pulses. In this scheme, a driver laser pulse, containing the majority of the total energy, is relatively loosely focused, with its Rayleigh range covering the injection region, while a trigger pulse, containing only a small portion of the total energy, is tightly focused shortly before the injection region, as illustrated in Fig. 1(a). Because of the different wavefront curvatures of the two lasers, interference rings are formed in the cross-section of the lasers, pinching the plasma electrons in the traps of their ponderomotive force.36 As a result, onion-like multiple subcavities are formed in the plasma wake, as shown in Fig. 1(b). With the propagation of the two lasers, the innermost subcavity expands, which slows down the effective phase velocity of the wake. Such an injection process is similar to density down-ramp injection. The one-to-one mapping between the injection phase and the initial phase greatly limits phase mixing and thus guarantees a small slice energy spread, and the defocusing force near the rear of the bubble kicks the electron trajectories from larger ellipses to smaller ones in the transverse space–momentum phase space, ensuring a small emittance of the trapped beam.37,38 The small slice energy spread is transformed to a small energy spread of the entire beam by the self-dechirping effect, which is to be discussed later. The injection length is a few hundred micrometers, which is significantly longer than those in previous all-optical injection studies, allowing large amount of beam charge.

Owing to its close-to-cylindrical feature, this scheme is suitable for simulation using the spectral quasicylindrical algorithm, which decomposes the Maxwell equations in k space through a Hankel–Fourier transform with only a few azimuthal modes and is more computationally efficient than a fully 3D algorithm.39 An example of a simulation using the code WarpX with the spectral quasicylindrical algorithm and the PSATD (Pseudo-Spectral Analytical Time Domain) solver40 is shown in Fig. 2. In the plots, the longitudinal coordinate is transformed to the comoving frame ζ = z − ct, where z is the longitudinal position, with its zero point at the beginning of the flattop of the plasma density profile, c is the speed of light in vacuum, and t is the time, with its zero point at the instant when the driver laser center reaches z = 0. In this simulation, the unperturbed plasma density profile has a linear up-ramp from z = −50 µm to 0 and a flattop from z = 0 to +∞, with the density of the flattop n₀ = 1 × 10¹⁸ cm⁻³. The driver laser has a peak power of 227.13 TW and is focused at z = z_f0 = 450 µm, with a focal waist of w₀₀ = 30 µm. The trigger laser has a peak power of 1.88 TW and is focused at z = z_f1 = 40 µm, with a focal waist of w₀₁ = 2 µm. Both lasers have wavelength λ = 0.8 µm and pulse duration of 25 fs and are linearly polarized in the y direction (perpendicular to the paper), but the trigger laser is 3.75λ ahead of the driver laser. The longitudinal size of the simulation box is 65 µm, with 4096 cells, the transverse size is 500 µm, with 1024 cells, the number of azimuthal modes is 2, and the time step interval is dz/c, where dz is the longitudinal grid size. The number of macroparticles per cell is 192 in the injection region and 24 elsewhere. Owing to the interference of the two lasers, rings with ponderomotive traps are formed, creating subcavities in the plasma wake. The expansion of the innermost subcavity sharply slows down the effective phase velocity of the wakefield from t = 0.6 to 1.7 ps, as shown in Fig. 3(a), and thus injection of background plasma electrons occurs, forming a beam with ∼0.5 MeV slice energy spread. The longitudinal electric field exerts positive and negative chirping on the injected beam respectively before and after t = 18 ps, where the blowout regime is transformed to a partial-blowout regime, as shown in Figs. 3(b) and 3(c), owing to the pump-depletion effect of the driver pulse.41 The energy spread has two minima before and after the self-dechirping point, as shown in Figs. 3(d)–3(g). It can be seen that a 170 pC electron beam with energy spread no more than 0.4% and emittance no more than 0.5 mm mrad is generated. It is worth noting that the two local minima of the energy spread do not persist for other acceleration distances, owing to the non-uniform acceleration field.

A full 3D simulation with the same grid size and parameters as above has also been performed (but only until injection has finished, owing to limited computational resources). The injected beam has a normalized emittance of 0.14 mm mrad in both transverse directions, a charge of 150 pC, and a slice energy spread of ∼0.5 MeV, which confirms the validity of the quasicylindrical simulation. This simulation at t = 1 ps is plotted in Fig. 1(b).

To explain the physics of the multisheath creation and injection, we write down the expression for the electric field of a Gaussian laser pulse in vacuum:$$E(r,z,t)=E_0\frac{w_0}{w(z)}T(z-ct)\exp \left[-\frac{r^2}{w^2(z)}+i\varphi \right],$$(1)where E₀ is the peak electric field strength at the focus, w₀ is the laser waist radius, $r=\sqrt{x^2+y^2}$ is the transverse position, $w(z)=w_0\sqrt{1+{(z-z_f)}^2/z_R^2}$ is the laser radius, z_f is the focal position, $z_R=\pi w_0^2/\lambda$ is the Rayleigh length, $T(z-ct)=\exp [-{(z-ct-z_0)}^2/{(c\tau )}^2]$ is the temporal profile of the pulse, z₀ is the pulse center at t = 0, τ is the pulse duration,$$\varphi =-k\left[\frac{r^2}{2R(z)}+z-ct-z_0\right]+{\varphi }_{\text{CEP}}+{\varphi }_{\text{G}}$$(2)is the phase, k = 2π/λ is the wavenumber, ϕ_CEP is the carrier envelope phase (CPE),$$R(z)=z-z_f+\frac{z_R^2}{z-z_f}$$(3)is the radius of the wavefront curvature, and$${\varphi }_{\text{G}}=\arctan \left(\frac{z-z_f}{z_R}\right)$$(4)is the Gouy phase. We neglect the plasma reaction to the laser pulses, because the interference region is short compared with the self-focusing length,42 and the phase velocity changes due to the plasma reaction are the same for the two lasers. We use extra subscripts 0 and 1 for the driver and trigger lasers, respectively, and the axial phase difference of the two lasers is then$$\mathrm{\Delta }{\varphi }_a\equiv {\left.{\varphi }_1\right|}_{r=0}-{\left.{\varphi }_0\right|}_{r=0}=\mathrm{\Delta }{\varphi }_{-\infty }+\mathrm{\Delta }{\varphi }_{\text{G}},$$(5)where Δϕ_−∞ = kΔz₀ + Δϕ_CEP is the phase difference far before focus, Δz₀ = z₀₁ − z₀₀ is the difference of the centers of the two pulses, Δϕ_CEP = ϕ_CEP1 − ϕ_CEP0 is the difference of the carrier envelope phases, and Δϕ_G = ϕ_G1 − ϕ_G0. The phase difference for arbitrary r can be written as$$\begin{array}{ll}\hfill \mathrm{\Delta }\varphi ={\varphi }_1-{\varphi }_0& =\frac{k{r}^2}{2}\left(\frac{1}{R_0}-\frac{1}{R_1}\right)+\mathrm{\Delta }{\varphi }_a\hfill \\ \hfill & \approx -\frac{k{r}^2}{2}\frac{1}{R_1}+\mathrm{\Delta }{\varphi }_a,\hfill \end{array}$$(6)where only z > z_f1 (and thus R₁ > 0) is considered, and the approximation is taken because |R₀| ≫ R₁ in the injection region. The destructive inference rings have Δϕ = −(2m + 1)π, where m is an integer. These rings are ponderomotive traps, pinching the stream of background electrons and forming subcavity sheaths. Thus, the radii of the subcavities are$$r_c=\sqrt{\frac{2R_1}{k}[(2m+1)\pi +\mathrm{\Delta }{\varphi }_a]}.$$(7)Without loss of generality, we shift Δϕ_a to the range (−π, π] by adding a multiple of 2π. Then, m should be a non-negative integer, and the radius of the innermost subcavity is$$r_c^{*}\equiv {\left.r_c\right|}_{m=0}=\sqrt{\frac{2R_1}{k}(\pi +\mathrm{\Delta }{\varphi }_a)}.$$(8)On the basis of simulation observations, we assume that the longitudinal extent of the innermost subcavity can be estimated by $L_c\sim 2r_c^{*}$. Because the front half of the main wakefield bubble has a decelerating electric field, and the subcavity cannot be longer than the main bubble, a necessary (but not sufficient) condition for injection is$$L_b/2<L_c<L_b,$$(9)where L_b is the longitudinal extent of the main bubble. This implies |z − z_f1| ≫ z_R1 and |z − z_f0| ≪ z_R0 in the injection region, which lead to Δϕ_G ≈ π/2 and R₁ ≈ z − z_f1. Thus, Eq. (8) can be simplified in the injection region as follows:$$r_c^{*}\approx \sqrt{\frac{3\pi +2\mathrm{\Delta }{\varphi }_{-\infty }}{k}}\sqrt{z-z_{f1}},$$(10)where ϕ_−∞ is shifted to the range (−3π/2, π/2] by adding a multiple of 2π. The increasing $r_c^{*}$ leads to an increasing L_c and a slow-down of the wakefield phase velocity, which triggers injection of electrons from the subcavity sheath.

To investigate the influence of the initial phase difference Δϕ_−∞, we vary Δz₀ from 3.7 λ to 4.7 λ while keeping Δϕ_CEP = 0. The theoretical evolution of the peak axial electric field E_p, obtained by adding the electric field of the two lasers using Eq. (1), is shown in Fig. 4(a) as solid lines. The simulation results are plotted as scatter points and show reasonably good agreement with theory. The injected beam charge has a strong correlation with E_p at z = 150 µm, which is a typical injection location, as shown in Fig. 4(b). Injection is suppressed if Δϕ_−∞ ≈ π/2 (thus, Δϕ_a ≈ π with the same assumption as previously, Δϕ_G ≈ π/2), because the interference is destructive on axis and the innermost subcavity does not have a clear sheath. It is worth noting that Δz₀ = 3.75 λ was used for the simulation shown in Figs. 2 and 3, which maximized the injection quantity.

To test the influence of laser pointing jitter, we have also performed one full 3D simulation with the trigger pulse transversely offset by 2 µm, which shows that the emittance of the trapped beam in the offset direction increases to 2 mm mrad, the emittance in the vertical direction and the slice energy spread do not change significantly, and the amount of charge decreases to 115 pC. Thus, this injection scheme is robust under micrometer-scale jitter of the transverse focal position.

In summary, we have proposed a new injection scheme for laser wakefield accelerators that generates electron beams with simultaneously small energy spread, small emittance, and large amount of charge. In this scheme, one laser pulse is relatively loosely focused to drive the plasma wakefield, and another laser is tightly focused in the Rayleigh range of the former. The energy of the latter laser is only a small portion of the former, but the peak intensities are similar owing to their focal spot size difference. Within a certain range, the tightly focused laser has a convex wavefront, while the wavefront of the loosely focused laser is flat. Consequently, interference rings are formed, pinching the background electron stream and creating subcavities in the wakefield. owing to the rapidly varying wavefront curvature of the tightly focused laser, the innermost subcavity expands, slows down the effective phase velocity of the wakefield, and triggers the injection of an electron beam with a few permille energy spread and no more than 0.5 mm mrad emittance. The charge of the injected beam can be modulated by the initial phase difference of the two lasers, with the maximum exceeding 100 pC for a moderate total laser power ∼200 TW. This mechanism is sensitive to the phase difference of the two lasers, which can be either time delay or carrier envelope phase (CEP) difference. In practice, the CEP difference can be eliminated by obtaining the two laser beams by splitting one laser beam, and the time delay should be precisely controlled to maximize the injection quantity.

Acknowledgment. This work is supported by the Research Foundation of the Institute of High Energy Physics, Chinese Academy of Sciences (Grant Nos. E05153U1, E15453U2, Y9545160U2, and Y9291305U2) and the Key Research Program of Frontier Sciences of the Chinese Academy of Sciences (Grant No. QYZDJ-SSW-SLH004).