Enhancing the flux, brightness, and density of energetic electron beams is crucial for applications such as ultrafast electron diffraction, fast ignition in confined fusion, and free-electron lasers. Laser Wakefield Acceleration (LWFA) has demonstrated potential for accelerating collimated electrons up to 10 Giga-electron volts in 'bubble-like' plasma channels. However, its reliance on the plasma environment constrains the enhancement of acceleration stability and gradients. In contrast, Direct Laser Acceleration (DLA) does not depend on plasma and can achieve efficient acceleration with traditional Gaussian lasers. Nonetheless, traditional DLA often results in uncertain and divergent electron beams due to the ponderomotive force of the Gaussian laser. To overcome these limitations, our proof-of-principle experiments achieved collimated acceleration using a left circularly polarized Laguerre Gaussian (LG) laser in a DLA mechanism. Studies revealed that a novel vacuum bubble field formed by the LG laser is critical in simultaneously concentrating and accelerating electrons. This vacuum bubble field mechanism integrates the advantages of both traditional DLA and LWFA, offering significant benefits for applications such as particle collimation, high-flux particle sources, and coherent radiation sources in new relativistic regimes.
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Electron acceleration via vacuum bubble field in Laguerre Gaussian laser
https://doi.org/10.21203/rs.3.rs-5051722/v1
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As ultra-intense and ultra-short-wavelength laser technologies advance, femtosecond lasers have achieved relativistic characteristics in terms of intensity (> 1018 W/cm2), with some reports recording intensities in the 1023 W/cm2 order1,2. Such extremely intense lasers can ionize matter to the high valence state, thereby forming plasma composed of ions and electrons3. Subsequently, free electrons with light mass, accelerated by the lasers’ electric fields with up to approximately 10 TV/m accelerating gradient, can gain velocities approaching the speed of light4. Notably, the collective free electrons exhibit macroscopic movement against the gradient of the laser field, resembling a propulsion effect by the laser itself5. This phenomenon belongs to the area of direct laser acceleration (DLA) and has found applications in particle accelerators6,7, radiation sources8,9, fast ignition10, and ultrafast attosecond science11,12. DLA driven by the longitudinal ponderomotive force of the traditional Gaussian laser has been commonly employed to accelerate electrons in theoretical and experimental works13–15. However, because of the transverse Gaussian-distributed ponderomotive force, the electrons in the front of the laser are gradually pushed around13–16, which is disadvantageous for stable electron acceleration and applications.
To address this challenge, a DLA mechanism was proposed for application in underdense plasmas (such as argon and nitrogen)4,17,18, where the laser initiates gas ionization and piles electrons at the front of the laser pulse. The heavier ions remain static, generating charged-separated fields within a 'bubble-like' plasma channel. Subsequently, electrons can undergo longitudinal acceleration by the accelerating fields at the bubble's rear, while being simultaneously focused in the transverse direction19. This process, known as laser Wakefield acceleration (LWFA), has achieved a maximum electron energy gain of 8 GeV to date20. Comparatively, LWFA offers more stable electron acceleration than traditional DLA20–22. However, the acceleration gradient in LWFA is only on the order of gigavolts per metre4,20,23, significantly lower than the gradient (TV/m) achieved in DLA. To enhance the acceleration gradient in LWFA, either laser intensity or plasma density must be increased, thereby amplifying the charge separation fields within the Wakefield distribution24. Nonetheless, these two values cannot be indefinitely increased because excessively large values lead to intensified laser self-focusing effects due to their nonlinear interaction processes25. This situation exacerbates Wakefield instability, causing electron dephasing and deceleration25. Consequently, the acceleration gradient in LWFA can be maintained only on the order of gigavolts per metre. Therefore, both DLA and LWFA present inherent advantages and challenges for electron acceleration. The revolutionary development of electron acceleration may be achievable by synthesizing their respective advantages through innovative approaches.
Fortunately, the Laguerre Gaussian (LG) laser, endowed with orbital angular momentum and a phase singularity at its beam centre, realizes a rotating force for manipulating atoms and molecules within weak fields26–29. Advances in laser technology, notably the development of chirped pulse amplification (CPA) technology, has motivated researchers to extend LG lasers into the relativistic realm30. Recent experiments have pushed the intensity of LG lasers to unprecedented levels, reaching up to 1019 W/cm2 by using reflected phase plates31. Unlike conventional weak LG lasers, these high-intensity LG lasers can ionize matter into plasma, comprising charged particles such as electrons and ions30–32. The nonlinear effects of electromagnetic fields in LG lasers are key to the manipulation of these charged particles33. Free electrons exhibit motion against the gradient of the laser field, converging around the beam axis where an inward ponderomotive force arises from the phase singularity along the LG laser axis. Consequently, the collimation and orientation of particle beams may be enhanced by applying intense LG lasers within the DLA mechanism32,34,35 Remarkably, in the case of the special mode of left-handed circularly polarized LG lasers (𝜎z = − 1, l = 1), an accelerating electric field in the longitudinal direction, along with concentrating fields in the transverse directions, intrinsically manifests during each laser duration within the LG laser32, analogous to the bubble field structure observed in traditional LWFA. Here, 𝜎z represents the value of spin angular momentum, while l denotes the value of orbital angular momentum. The acceleration gradient of such a bubble field is independent of the intricate plasma environment, thus rendering its acceleration gradient infinitely proportional to the laser intensity25. For instance, a 10 PW or 100 PW laser could yield an acceleration gradient on the order of petavolts per metre, surpassing the current acceleration gradient in LWFA by four orders of magnitude. Such intense LG modes (𝜎z = − 1, l = 1) may hold considerable potential for generating vacuum bubbles at the micrometre scale, combining the benefits of high acceleration gradients in DLA with the stable acceleration bubble regime in LWFA, thereby facilitating the production of high-energy, collimated, and ultra-short electron beams.
In this work, we successfully conducted a proof-of-principle experiment on the vacuum bubble field acceleration (VBFA) scheme utilizing LCP LG lasers (𝜎z = − 1, l = 1). The objective was to achieve a stable, highly collimated, and lower emittance ultrafast electron beam in the DLA region. Our results demonstrate that the electron beam exhibits a narrow full width at half maximum (FWHM) size of 80 mrad precisely along the reflected laser direction, a significant improvement over Gaussian lasers. This achievement addresses both the collimation and directional issues simultaneously. The longitudinal accelerating electric field and transverse concentrating fields generated by LG lasers construct a stable vacuum bubble, enabling the stable and acceleration of electrons in the DLA region.
A schematic depicting the VBLA-based electron acceleration is illustrated in Fig. 1a. In experiments, a reflected phase plate was used to transform the Gaussian mode to LG mode and an off-axis parabolic mirror further focused the laser into a vacuum spot size of 7.8 µm (FWHM) and an energy concentration of approximately 32% (see Methods). In general, when the target is irradiated by such an intense femtosecond laser, the plasma is ionized and expands outside of the target with the surface density distribution as an exponential expression. The target surface density gradient can be extremely high and ascend to approximately 1023 electron cm− 3 within one laser wavelength, where the relativistic laser is unable to penetrate and can be reflected. Thus, the target behaves as a high-reflective mirror for the intense laser beam36. Meanwhile, the electrons on the target surface are pushed back to the plasma or dragged out of the target surface into the vacuum under the influence of the incident and reflected electric fields12. Those electrons that have gained higher kinetic energy toward the vacuum are extracted from the surface and enter the vacuum, which is continuously modulated by the reflected field in the DLA mechanism.
Figure 2 illustrates the experimental results of the electron angular distribution and spectra. The electron beam exhibited a deviation of approximately 100 mrad towards the normal direction, accompanied by a divergence angle of 200 mrad, induced by the conventional circularly polarized (CP) Gaussian laser with an energy of approximately 12 J (Fig. 2a). This deviation occurred because the CP Gaussian laser field applied Gaussian distribution ponderomotive force in the transverse direction, leading to the uncertainty of electron beams and consequently resulting in a broader electron beam divergence angle. This characteristic is disadvantageous for practical applications, such as electron injection sources37, fast ignition10, and electronic imaging38.
LG lasers featuring hollow phase singularities offer a solution for accelerating electrons with minimal divergence by generating a doughnut-shaped distribution of ponderomotive force in the transverse direction. In the case of an LCP LG laser with a 12 J energy, a distinct electron peak emerged in the reflection direction (45°) with a divergence angle of 80 mrad (Fig. 2b). This favourable result was caused by the constraining influence of the transverse ponderomotive force exerted by the LG laser. Moreover, increasing the laser energy to 15 J maintained the electron beam's alignment with the reflection direction (Extended Data Fig. 1), indicating the stability of the electron direction in relation to laser energy. The charge of emitted electrons increased from 200 pC in the 12 J case to 350 pC in the 15 J case, demonstrating the LG laser's capability to yield higher charge in our experiments.
Furthermore, electron beams propelled by the RCP LG laser with 12 J energy also maintained their concentration around the reflection direction (Fig. 2c). However, the electron divergence angle (approximately 160 mrad) was larger than that of the LCP LG cases (Fig. 2d), and the electron charge decreased to 150 pC. This difference resulted from the distinct field structures in the LCP LG laser, where electrons can be locked in an accelerating phase in the longitudinal electric field along the laser axis and concentrated by the transverse focusing fields. This phenomenon resembles traditional LWFA and is called VBFA in this paper. Subsequently, electrons can sustainably gain energy and achieve collimation within the LCP LG laser environment. Conversely, the absence of a longitudinal acceleration field along the laser axis in the RCP LG laser impedes electron energy acquisition and directional control (Fig. 2d). Analysis of the energy spectrum shown in Fig. 2e confirmed that the LCP LG laser yielded higher energy and charge for the electron beam compared with the RCP LG laser. Meanwhile, the electron angular distributions (Fig. 2d) and energy spectra (Fig. 2e) in 3D particle-in-cell (PIC) simulation agree with the experimental results. Therefore, it can be concluded that the VBFA mechanism driven by the LCP LG laser is better suited for obtaining a collimated, directional, and high-charge electron beam. This consideration is critical for applications relying on the DLA mechanism in fields with stringent requirements for electron-beam quality, such as free-electron lasers and ultrafast electron diffraction.
To illustrate the VBFA mechanism in experiments, three-dimensional (3D) PIC simulations were carried out to study the dynamics between plasma and the LG laser field39. The 3D PIC simulation parameters are detailed in the Methods. At t = 16 T, the laser arrived at the front surface of the target at an incident angle of 45° and was reflected by the plasma with a critical density of nccos2θ (nc = 1.6×1021 cm− 3 is the critical plasma density and θ is the incident angle of 45°)40,41, as shown in Fig. 3a. Simultaneously, the incident and reflected laser fields interfered to form an electric field En along the target normal direction (Fig. 3c), which dragged the electrons out of the target at one half of the laser cycle (negative amplitude) and pushed them into the target at the next half of the laser cycle (positive amplitude)13. The outgoing electrons were initially confined in a singularity area in the beam centre (Fig. 3c) because the ponderomotive force potential wells of the incident and reflected LG lasers interfaced near x = 0 µm and y = -1 µm (Extended Data Fig. 3b). However, these electrons could not be infinitely confined in the singularity because of the Coulomb repulsive force42. Some electrons were emitted outside and were further accelerated by the interfering field in the normal direction En (Fig. 3c)13. When the reflected laser left the target after t = 24 T, the emitted electrons entered the VBFA stage driven by the reflected LG laser (Fig. 3b, d).
Here, the electrons can be phase-locked accelerated along the laser propagation (+ x) direction if the initial electron velocity matches the phase velocity of the longitudinal electric field (Ex) in the centre of the reflected LG laser. Meanwhile, the vertical electric field provided a concentrated force for electrons, together with Ex (Fig. 3d), further collimating and accelerating the electrons up to approximately 80 MeV (Fig. 2e) with a divergence angle of approximately 80 mrad (Fig. 3e) in the VBFA mechanism, which is consistent with the experimental results shown in Fig. 2.
By contrast, the electron density distribution driven by the traditional Gaussian laser deviates from the laser reflection direction (Extended Data Fig. 2a) because the ponderomotive force of the Gaussian laser usually pushes the electrons away from the laser axis, which has been proven in several studies13,11,14,43. Extended Data Fig. 2c shows that the electrons also emit along the laser reflection direction by the RCP LG laser, similar to the case of the LCP LG laser, resulting from the concentrated ponderomotive force around the axis of the LG lasers. However, the divergence angle (approximately 160 mrad) in the former case is twice that in the latter case. The main reason is that no longitudinal electric fields (Ex) appear on the x-axis in the reflected RCP LG laser, resulting in inefficient acceleration and collimation in the subsequent progress. Therefore, only the LCP LG laser provides a VBFA acceleration mechanism to stably accelerate electrons to higher energy with lower divergent angles in our case. Such a new VBFA acceleration mechanism is completely independent of the plasma environment so that its acceleration gradient (on the order or teravolts per metre) can be proportional to the laser intensity, which may be larger than that for the present LWFA mechanism (gigavolts per metre) and has various applications.
Subsequently, a particle test was conducted to demonstrate how electrons are confined within the ponderomotive force potential wells and are further accelerated during the subsequent VBFA process (see Methods). As shown in Fig. 3c, most electrons were emitted around the target normal direction; thus, the initial incident angle of the test electrons was approximately set at 45° (Extended Data Fig. 4a). The model considers different initial electron velocities (Extended Data Fig. 4b). As the initial velocity of the electrons increased from 0.1c (3 keV) to 0.7c (205 keV), the energy gain of the electrons increased from 77 MeV to 104 MeV. However, as the initial velocity continued to increase, the energy gain of the electrons significantly decreased (Extended Data Fig. 4b). This phenomenon occurs because an increase in the longitudinal component of the electron velocity causes the electron to enter the phase-locked acceleration process more quickly. However, when the velocity is excessively high, the transverse component of the electron velocity also increases, leading to the electron escaping from the vertical electric field, thus preventing confinement of electrons within the longitudinal accelerating electric field on the axis. Consequently, the optimal injection velocity was determined to be 0.7c, allowing the electron to be confined by the transverse electric field while achieving stable phase acceleration in the longitudinal direction.
As shown in Fig. 4a, a test electron with the optimum initial velocity of 0.7c was pulled back by the transverse electric field within approximately 1 µm distances (Fig. 4a) and then entered the VBFA acceleration for phase-locked acceleration to achieve a linear increase in energy gain (Fig. 4b). By contrast, the energy gain of external electrons rose slightly and then decreased to approximately 20 MeV because they were pushed out by the transverse outward ponderomotive force at the outside; thus, it was not accelerated by the axial longitudinal electric field in the centre. Therefore, only the electrons with optimum velocity and angle confined in the laser centre can achieve the stable VBFA stage.
In conclusion, we demonstrated electron acceleration under the VBFA scheme using LCP LG lasers (𝜎z = − 1, l = 1) for the first time. Our experimental results showed that the electron beam exhibits a narrow FWHM of 80 mrad precisely along the reflected laser direction, addressing the collimation and directional issues present in the cases of GS and RCP LG lasers. Additionally, 3D PIC simulation results aligned well with the experimental data, revealing that the ponderomotive force confines and accelerates electrons ejected along the target's normal direction into the reflected LG electric field. This process creates a stable vacuum bubble with a high accelerating gradient in the DLA region, enabling stable electron acceleration. This novel VBFA mechanism effectively solves the beam uncertainty and wide divergence issues in traditional DLA, benefiting various applications such as particle collimation accelerations, high-flux particle source generation, and coherent radiation sources in new relativistic regimes. Our results represent the initial proof-of-principle experiment for the VBFA scheme. Currently, the laser intensity is insufficient to achieve a super-high accelerating gradient under existing laboratory conditions. However, this limitation can be overcome with the advent of 10 PW and upcoming 100 PW lasers, which are expected to generate acceleration gradients on the order of petavolts per meter, thus broadening future VBFA applications.
Relativistic LG laser generation. The experiments were conducted utilizing a 1 PW laser system (Fig. 1), capable of delivering energy up to 26 J with a duration of 32 fs at 795 nm. Initially, a quarter-wave plate was inserted into the incident path to manipulate the polarization of the incident Gaussian laser. Then, a phase plate with 32 steps was employed at a 45° angle to generate a vortex hollow laser. Finally, the laser was focused by an off-axis parabolic mirror, resulting in a vacuum spot size of 7.8 µm (FWHM) and an energy concentration of approximately 32% at 1/e². The peak intensity was estimated to reach 3.8×1020 W/cm2, corresponding to a normalized amplitude of a0 = 9.5. Notably, a singularity point emerged at the centre of the laser intensity distribution at the focal point, significantly unlike that of a Gaussian laser. Fig. 1b shows a singularity point at the centre of the laser intensity distribution at the focus, distinct from the Gaussian laser distribution in Fig. 1c. The transverse doughnut distribution of the LG laser generates a ponderomotive force that concentrates electrons transversely, which is advantageous for various applications (Fig. 1e), including accelerators44, high-flux particle generation32, and coherent radiation sources21.
Electron beam diagnostic. The laser pulse irradiated a flat fused silica target (plasma mirror) at a 45° incident angle relative to the target normal direction, driving electrons outside. The spatial characteristics of these electron beams were analysed using an imaging plate positioned 10.5 cm away from the target, with aluminium foil employed to shield against ions and low-energy photons. Additionally, an electron spectrometer was utilized to detect the energetic spectra of the electron beam.
PIC simulation. 3D PIC simulations were conducted to study the injection and acceleration process. The simulations used a circularly polarized LG laser pulse with a central wavelength of 795 nm, pulse duration (FWHM) of 32 fs, laser waist of 3.2 μm, and a normalized laser vector potential a0 = 9.5. The laser intensity contrast ratio was better than 2.5×10−11 at 80 ps before the main pulse45,46. The overdense plasma density was ne = 20nc, where nc = 1.6×1021 cm−3 is the critical plasma density. The plasma scale length was varied by changing the density profile, given by an exponential ramp n(r) = neexp(−(r − r0)/L), where r is the direction normal to the target surface and L = 0.1 λ is the plasma scale length. Simulation parameters included a spatial step of Δx = Δy = Δz = λ/60, 50 particles per cell, and a simulation box size of 22 μm (x) × 22 μm (y) × 22 μm (z). To track the electron trajectories far from the target, a moving simulation box was used: after the laser pulse reflected from the target, the box moved at the speed of light, following the reflected pulse and allowing the dynamics of energetic electrons to be observed over many wavelengths (within the laser Rayleigh length). Here, the PIC simulation parameters are consistent with those in experiments.
Laguerre-Gaussian (LG) laser field. The transverse electric field of an LG laser can be expressed as30
where E0 = a0meωLc/e is the peak amplitude of the electric field, a0 = 10 is the normalization amplitude of the laser pulse, me is the mass of the electron, ωL is the laser frequency, c is the speed of light in vacuum, and e is the charge of the electron. The parameters l and p + 1 represent the number of azimuthal phase cycles and number of radial nodes, respectively. w(x) = w0(1 + x2/xR2)1/2 is the diameter of the focus spot (FWHM), where w0 ∼ 3.2 µm is the beam waist, and xR = πw02/λ is the Rayleigh length. The radial distance is r = (y2 + z2)1/2, is the generalized Laguerre polynomial, ϕ is the azimuthal angle, k is the wave number, and (l + 2p + 1)arctan(x/xR) is the Gouy phase of the mode; σz = -1 is the spin angular momentum. The longitudinal electric field can be defined by Poisson's equation, 
.
3D particle test.
The single-particle model, a simplified numerical approach compared to PIC simulation, is illustrated in Fig. 2. The particle acceleration can be expressed as32
where relativistic parameter γ = (1-v2/c2)-1/2, E is the electric field, B is the magnetic field, c is the light speed and v is the electron speed. The energy gain can be expressed as
The particle model involves solving the relativistic equations of motion for electrons within a laser pulse that propagates along the +x direction.
Data availability statement
The data that support the findings of this study are available from the corresponding authors on reasonable request
Acknowledgements
This study is supported by the Natural Science Foundation of China (Grant No. 12075306), Natural Science Foundation of Shanghai (Grant No. 22ZR1470900), Key Research Programs in Frontier Science (Grant No. ZDBSLY-SLH006), Strategic Priority Research Program of the Chinese Academy of Sciences (Grant No. XDA0380000), Shanghai Science and Technology Committee Program (Grant No. 22DZ1100300).
Competing interest declaration
The authors declare no competing interests.
Author contributions
R. L., Y. L, and Z. X. conceived the project. W. W. designed the experiments. W. W., Z. L., F. S., and Z. S. performed the experiments and collected the data. X. L., X. L., J. L. and R. X. constructed and ran the Ti:sapphire laser system. Z. L. and F. S. conducted the simulations and analysed the experimental data. W. W., Z. L., F. S. and R. L. co-wrote the paper. All authors contributed to the experiments and discussions.
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There is NO Competing Interest.