The laser-plasma accelerator (LPA) has shown extraordinary potential in the compact particle acceleration field since it was first put forward in 1979.1–3 In the past few decades, a great deal of work on electron acceleration mechanisms has resulted in inspiring progress from the perspectives of theory, simulation, and experiment.4,5 In principle, high-charge electrons can be accelerated in a laser field directly (direct laser acceleration, DLA) and by a laser wake field (laser wake-field acceleration, LWFA).2 DLA, which occurs in an ion channel formed by the self-channeling effect, was first described by Pukhov et al.6 It has been validated that this approach can produce an overcritical electron beam because it can be applied to near-critical-density (NCD) plasmas or even at the surface of solid-density plasmas.7–9 However, it is challenging to maintain the electron beam quality owing to the tremendous impact of the laser field.10 Unlike in DLA, electrons are accelerated longitudinally and confined transversely in LWFA. In addition, the accelerating gradient of the wake field is independent of the radial position, while the transverse confinement force increases with radius. These two features are capable of conserving electron beam qualities such as the energy spread, emittance, and divergence.11 Meanwhile, the population of the witness bunch is restricted by the low fraction of electron self-trapping.2,12 Various schemes have been proposed to resolve the population limitation of self-injection, including ionization injection in high-Z gas,13,14 density transitions,15 irradiation of the wake field with another laser beam propagating in the vertical direction,16 and bow-wave injection.17 The electron charge is significantly enhanced from tens to hundreds of pC.18 When increasing the power of the incident laser to hundreds of TW or even to the PW scale, the maximal charge can reach up to several nC.19–21 Despite this, the LWFA total charge performance, particularly for high-energy electrons, is still not comparable to those of conventional radio-frequency sources.

The tailored plasma offers the possibility of improving beam quality through its remarkable contribution to LPA.22–24 The mechanism behind the tailored plasma is the rapid changing of the wavelength of the plasma wave through introduction of a density transition, as the electron plasma frequency is proportional to the electron density, i.e., ω_pe ∝ n_e. Bulanov et al. first studied the excitation of a plasma wave in an inhomogeneous plasma in 1998.25 In their regime, the phase velocity of the plasma decreased with a moderate decline of the plasma density in the laser propagating path. When the plasma phase velocity dropped to the quiver velocity of the background electrons, the Langmuir wave was broken. As a result, the electrons were trapped in the acceleration phase. The condition was totally different when the longitudinal density ramp was sharpened, i.e., k_pL_s < 1. Suk et al. showed that enormous numbers of electrons that were initially at the tail of a high-density wake field could enter the accelerating phase of the wake field as the plasma wavelength rose with a rapid density reduction.26 Numerous efforts to optimize the gradient, scale length, and density profile have been invested to obtain high-charge electron beams in LWFA.27–29 Unlike in a longitudinal density transition, a transverse density gradient is employed in LWFA to tilt the propagation axis of the laser pulse and induce wake-field asymmetry. As a result, it can considerably boost betatron oscillation and increase both the yield and energy of the produced photons.30,31 In order to increase the amount of charge in an accelerated electron bunch, further design improvements of the plasma density profile are still required.

In this paper, we propose a new scheme that utilizes a bubble-like structure formed by electron reflux to accelerate a high-charge electron beam. This is achieved via tailoring the plasma density transversely, which results in a curvature of the plasma wavefronts and thus controls the electron re-injection along the maximal plasma density. The electrons inside the bubble shape feel a strong longitudinal field that acts as the acceleration field and a varying transverse field that confines the electron oscillation in the vertical direction. Simulation results show that an over-critical-density electron beam is injected into the bubble and efficiently gains an energy of 500 MeV. Furthermore, the frequency of betatron oscillation ω_β is adjusted by the laser beam to ten times that in LWFA. This significantly raises the cut-off energy of the emitted photons to the range of MeV.

In a uniform plasma medium, the group velocity of a relativistic laser beam (a₀ ≫ 1) is estimated to be $v_g(r)/c\approx 1-{\omega }_p^2/\left.\sqrt{2}{\omega }_0^{2}a(r)\right]$.32 In this expression, c is the speed of light in vacuum, ω_p and ω₀ denote the frequencies of the plasma wave and the laser, and a(r) is the dimensionless laser intensity. Depending on the spatial profile of the incident laser beam, the plasma wavefronts in a uniform plasma show phase differences in the transverse direction. For example, the plasma wavefronts driven by a Gaussian laser beam look like a cone33 [see the black line in Fig. 1(a)]. When introducing a transverse density gradient, ω_p in the group velocity formula is not constant, which helps further bend the plasma wavefront. Assuming a plasma with an initial density profile of the form $n_e(r)=n_{e0}\exp \left\{-(r-1(\mu \mathrm{m}))/(\sigma /2)\right\}$, where n_e0 is the maximum electron density and σ denotes the waist radius of the laser beam, the red solid line in Fig. 1(a) shows the phase difference after traveling for a period of 20T₀. Here, the situation is simplified to a 2D model, which omits the profile of the laser pulse and the plasma cell in the third dimension. The computational parameters of the laser and the plasma are given in Table I. It is worth pointing out that there are obvious phase delays of the plasma wave at places with the peak plasma density. The laser rays arriving at the curved wavefronts are efficiently refracted. The corresponding refractive index is written as $\eta (r)={\left\{1-{\omega }_p^2(r)/{\omega }^2{[1+a^2(r)/2]}^{1/2}\right\}}^{1/2}$. It is apparently influenced by the laser profile following a function of −exp(r²/σ²) for a standard Gaussian laser beam, while the dependence on the electron density can be described as −exp[−(r − 1(μm))/(σ/2)]. The resulting refractive index follows the gradient of the initial electron density. In other words, the laser rays are bent along the density transition.

We simulate the laser irradiating the transversely tailored plasma with the particle-in-cell (PIC) code EPOCH2D. To facilitate comparisons, the simulation parameters are kept the same as those used in the aforementioned theoretical calculation, as listed in Table I. The sampled window is X × Y = 40 × 30 µm² with cell numbers of N_x × N_y = 2000 × 1500. A Gaussian laser beam is incident from the left boundary of the simulation box. According to the expression for the group velocity in the previous paragraph, the impact of the laser wavelength is on the term ω, which is independent of the density profile. Therefore, the curvature at the plasma front can be observed under various laser wavelength conditions. The laser wavelength used in the simulations is 1 µm, corresponding to a critical plasma density of n_c = 1.1 × 10²⁷ m⁻³.34 The plasma slab is located in the domain of 5 ≤ x ≤ 40 μm. We employ the same plasma density profile discussed in the last paragraph. The characteristic term n_e0 = 2n_c. It is worth noting that such a typical density distribution could be obtained by ablating a channel built of an array of nanotubes in an experiment.35 In fact, such a density profile has already been fabricated with hydrogen gas in a laboratory with the hydrodynamical expansion of an optical-field-ionized plasma.36

Figure 1(b) shows the magnetic field at 20T₀ and the visible curvature of the wavefront. The influence of the bent phase front has two aspects. On the one hand, it gives rise to the refraction of laser rays and shapes the laser profile into a particular distribution in the transverse direction. As displayed in Fig. 1(b), the light field is particularly weak in the region of the maximal electron density. On the other hand, the reflected laser field travels with a small angle with respect to the axis and then produces several off-axis peaks of the laser field. This causes the direction of the ponderomotive force to vary in the vertical direction. From Fig. 1(b), it is seen that there are zones close to the density peaks where the transverse ponderomotive force and the magnetic field share the same direction. For example, if they are both positive, the electrons in these areas will move up. Hence, the force v × B acts on the electrons in the negative x direction. Together with the space-charge force caused by the plasma charge separation, the electrons could be accelerated backward. However, it is relatively difficult to pull electrons back within the channel between the two density peaks. In view of the spatial distribution of the reflux electrons, the enhanced electron re-injection at the two maximal densities comprises a bubble-like structure with a maximum density on the bubble wall of over 20n_c [see Fig. 1(c)]. Figure 1(d) presents several representative trajectories of the backward electrons. As expected, the re-injected electrons finally flow into the tail of the laser pulse after experiencing continuous acceleration and deceleration processes owing to the intense ponderomotive force of the laser.

The self-generated electromagnetic field in the formed bubble resembles that in the laser wake field. It is beneficial to the electron acceleration in two ways. First, an acceleration phase for the electrons appears in the spatial distribution of the longitudinal field as a result of the charge separation [see Fig. 2(a)]. Second, the strong transverse field due to the over-dense electron reflux contributes to confining the witness electrons [see Fig. 2(b)]. It is worth mentioning that the re-injected electrons keep colliding with the laser field and oscillate in the transverse direction as they stream back with respect to the laser pulse. Hence, the bubble structure shakes with the propagation of the laser beam [see Fig. 2(d)]. In spite of this, the acceleration and focusing fields are ideal for electron acceleration.

In the rear of the bubble, a fraction of the re-injected electrons are trapped in the acceleration phase because the ponderomotive force of the laser pulse promotes them from fluid orbits into trapped orbits. The electron dynamics of self-trapping can be described as37,38$$\frac{\text{d}\gamma {\text{m}}_e{v}_y}{dt}=-e[(E_L+E_{Sy})+v_x\times (B_L+B_{Sz})],$$(1)$$m_e{c}^2\frac{\text{d}\gamma }{\mathrm{d}t}=-e(v_y{E}_L\cos \psi +v_x{E}_{Sx}),$$(2)where γ, v_x, and v_y are the Lorentz factor, the longitudinal velocity, and the transverse velocity of the trapped electron, respectively, and m_e is the electron mass at rest. The electrons see both the laser field and the quasi-static self-generated field. E_L and B_L are the electric and magnetic fields of the driving laser. Similarly, E_Sy, E_Sz, and B_Sz are the components of the static electric field and the self-generated magnetic field in the plasma. The angle ψ = φ − ϕ represents the relative angle between the electron motion and the laser field. φ and ϕ denote the polar angles of the electron momentum and the laser electric vector, respectively. Since the electrons move in the laser propagation direction, the terms for the electric and magnetic fields from the laser in (1) are approximately counterbalanced. Under this circumstance, the electrons undergo transverse oscillation owing to the impact of the quasi-static fields. Following previous work, the self-generated fields can be simplified to be E_Sy = k_Ey and B_Sz = −k_By, where k_E and k_B are constants.6 The angular frequency of the electron motion is therefore expressed as$$\frac{\text{d}\phi }{\mathrm{d}t}={\omega }_{\beta }=\sqrt{\frac{e}{\gamma m_e}(v_x{k}_B+k_E)}.$$(3)The laser frequency witnessed by the trapped electrons is written as$$\frac{\text{d}\varphi }{\mathrm{d}t}=(1-v_x/v_{ph}){\omega }_0,$$(4)where v_ph is the phase velocity of the laser in the plasma. It follows that the time derivative of the relative phase between the electron momentum and the laser field is derived as38$$\frac{\text{d}\psi }{\mathrm{d}t}=\frac{\text{d}\phi }{\mathrm{d}t}-\frac{\text{d}\varphi }{\mathrm{d}t}=\sqrt{\frac{e}{\gamma m_e}(v_x{k}_B+k_E)}-(1-v_x/v_{ph}){\omega }_0.$$(5)A Hamiltonian in phase space (ψ, γ) is then given by$$\begin{array}{ll}\hfill H& =\frac{e}{m_e{c}^2}{v}_y{E}_L\sin \psi +\frac{e}{m_e{c}^2}{v}_x{E}_{Sx}\psi +2\sqrt{\frac{e}{m_e}(v_x{k}_B+k_E)\gamma }\hfill \\ \hfill & -(1-v_x/v_{ph}){\omega }_0\gamma .\hfill \end{array}$$(6)The separatrix and fixed point are solved by letting the right-hand side in (2) and (5) equal zero. That is, ψ_c = arccos(−v_xE_Sx/v_yE_L) + 2nπ and ${\gamma }_c=e(v_x{k}_B+k_E)/m_e{(1-v_x/v_{ph})}^2{\omega }_0^2$. Figure 3(a) shows the Hamiltonian in phase space (ψ, γ). It is apparent that electrons that initially reside around the fixed point feel slow phase movements and can stay in the acceleration phase for a long time. The electron density is plotted in the same phase space (ψ, γ) [see Fig. 3(b)], and is consistent with the above theoretical discussion.

Figure 4(a) shows that electrons with a maximum density of $\sim 22n_c$ are self-injected into the bubble. The longitudinal and transverse fields on-axis are drawn in Figs. 4(b) and 4(c). The longitudinal electrostatic field has a maximum of $\sim 4\times 10^{13}\mathrm{V}/\mathrm{m}$, which enables an extremely high acceleration gradient of 4 × 10⁴ GV/m. Since the laser depletion is non-negligible in an NCD plasma, the group velocity of the laser is lower than that given before. For the parameter set in the simulated case, the phase velocity of the plasma wave v_p is approximately equal to the group velocity of the laser v_g, which is calculated as $\sim 0.86c$. Applying the relation (1 − v_p/c)L_d = L_s, the dephasing length L_d of the electrons is then derived as 28 μm. Here, the length of the acceleration phase L_s is around 4 μm, as shown in Fig. 2. At this point, it is possible to accelerate electrons to several hundreds of MeV. We see that the strength of the fields oscillates along the x-axis. Beyond the reasons discussed in Sec. II, including the presence of the laser field and the shaking bubble wall, the existence of the electrons inside the bubble creates a substantial disturbance. The underlying sources of the noise electrons streaming into the bubble are twofold. First, a small fraction of re-injected electrons penetrating near the peak plasma density collapse into the bubble on experiencing a strong enough transverse field [see Fig. 1(d)]. In addition, there are electrons re-injected from the bubble head, although their density is not comparable to those of the two plasma density peaks [see Fig. 1(c)]. In spite of the oscillating fields, the electrons remain in the accelerating and focusing region. One typical trajectory of an injected electron [see Fig. 4(d)] indicates that the electrons remain in the longitudinal acceleration field for an acceleration distance of $\sim 28 \mu \text{m}$ and gain energy reaching up to $\sim 500\mathrm{M}\mathrm{e}\mathrm{V}$. The acceleration process is described by the evolutions of the phase space x–p_x, x–p_y and the work done by forces characterized by two terms, ɛ_‖ = −∫2v_xE_xdt and ɛ_⊥ = −∫2v_yE_ydt [see Figs. 4(e) and 4(f)]. It is not surprising that the acceleration is mainly from the longitudinal electric field. In the transverse direction, the fields that the electrons meet, including the laser field and the self-generated electromagnetic field, shake strongly with the fluctuation of the electron bubble due to the existence of the laser pulse. Under this circumstance, the trapped electrons quiver in the bubble with a relatively short wavelength. This can be estimated by the distance that the electron travels before it slips by one laser wavelength, i.e., λ_β = λ₀/(v_ph/v_x − 1) ≈ 7 µm. As the oscillation frequency of an electron in the perpendicular direction is close to the laser frequency in the electron rest frame, the electron gains energy from the transverse field directly. However, this situation only occurs when the electron betatron oscillation matches the resonance condition ω_β ≈ (1 − v_x/v_ph)ω₀. Therefore, DLA mainly contributes in two cases. One is when the electron oscillation is dominated by the laser field, which happens at the beginning of the injection. The other comes with a high oscillation frequency caused by a strong transverse field [see Figs. 4(e) and 4(f)].

The phase space x–p_x of trapped electrons is shown in Fig. 5(a). It shows that the accelerated electrons are bunched at the laser frequency [see Fig. 5(a)]. This results from the oscillation of the self-generated electromagnetic field in the bubble because the bubble wall is quivering in the laser field. The density of the accelerated electron beams is overcritical [see Fig. 5(b)] and comparable with the electron density that could be obtained in DLA. The total charge of the electron bunches that are accelerated is as high as 0.26 nC. The angular energy distribution of the electrons, displayed in Fig. 5(c), shows great collimation in the high energy band. As expected, the spectrum of the electron bunch [sketched in Fig. 5(d)] is quasi-monoenergetic with a peak at $\sim 440\mathrm{M}\mathrm{e}\mathrm{V}$. It seems that these beam properties are much better than those achieved in DLA occurring in near-critical density plasmas. Overall, the electron beam obtained in the proposed scheme has the advantage of high density and great collimation and is an ideal combination of the DLA and LWFA acceleration regimes.

The transverse oscillation of high-charge electrons in a bubble-like structure could efficiently radiate a large number of photons in comparison with other radiation regimes involving bremsstrahlung and non-linear Compton scattering. As seen from Fig. 6(a), the peak density of a γ-ray is greater than 15n_c. It is worth noting that the betatron oscillation period is extremely short because the quivering is affected by the laser field. Figure 6(b) shows that the transverse oscillation period is shorter than 20 µm, which is normally greater than 100 µm in LWFA.39 As a result, it is possible to improve the energy of a radiated photon to the MeV level. Thanks to the high-quality electrons, the created photons are well collimated with a small divergence of $\sim 10^{◦}$ [see Fig. 6(c)]. In principle, the energy of an emitted photon is related to the amplitude and frequency of electron oscillation. The critical frequency is expressed as ${\omega }_c=\frac{3}{2}K{\gamma }_e^{2}2\pi c/{\lambda }_{\beta }$, where K = γ_eφ is the dimensionless strength of the electron oscillation and is proportional to the Lorentz factor of the electron γ_e and the maximum tangential angle φ of the orbit. Applying the parameters of electron beam oscillation [see Figs. 5(d) and 6(b)], it is reasonable to attain a critical energy of 60 MeV in the photon spectrum [see Fig. 6(d)]. Taking the first photon pulse into consideration, the mean energy is calculated as 6.17 MeV. The number of photons at this average energy is 2.15 × 10⁵ photons (sr 0.1%  BW)⁻¹. Hence, a striking brightness of 4.3 × 10²² photons s⁻¹ mm⁻² mrad⁻² (0.1%  BW)⁻¹ at an energy of 6.17 MeV is obtained. Such a bright MeV γ-ray source provides a new approach for compact light sources. In addition, thanks to the extremely short electron oscillation wavelength, the energy conversion efficiency from laser to γ-rays is improved to up to 0.2%, which further opens a path toward high-efficiency future laser-driven γ-ray sources.40

In conclusion, we have reported a new electron acceleration regime in a self-established bubble resulting from the special electron reflux in a laser interacting with a transversely tailored plasma. With a laser intensity of 5 × 10²¹ W/cm², simulations observe overcritical electron beams with a mean energy of 440 MeV and a total charge of up to 0.26 nC. Furthermore, the electron oscillation wavelength is extremely short — typically hundreds of μm in the classical betatron radiation occurring in a plasma wake field. A large number of photons in the γ-ray range are consequently radiated. This is a significant step in the investigation of betatron radiation. The described acceleration scheme is important to elucidate the physics underlying a laser irradiating a transversely inhomogeneous plasma. The generation of high-quality electrons and γ-rays should inspire the further development of LPA and compact radiation sources.

Acknowledgment. This work was supported by the China Postdoctoral Science Foundation (Grant No. 2021M692204), the National Natural Science Foundation of China (Grant No. 11805278), the Fundamental Research Program of Shenzhen (Grant No. SZWD2021007), the Science and Technology on Plasma Physics Laboratory, the Guangdong Basic and Applied Basic Research Foundation (Grant No. 2022A1515010326), and the Shenzhen Technology University.