Attosecond electron bunches from a nanofiber driven by Laguerre-Gaussian laser pulses

Generation of attosecond bunches of energetic electrons offers significant potential from ultrafast physics to novel radiation sources. However, it is still a great challenge to stably produce such electron beams with lasers, since the typical subfemtosecond electron bunches from laser-plasma interactions either carry low beam charge, or propagate for only several tens of femtoseconds. Here we propose an all-optical scheme for generating dense attosecond electron bunches via the interaction of an intense Laguerre-Gaussian (LG) laser pulse with a nanofiber. The dense bunch train results from the unique field structure of a circularly polarized LG laser pulse, enabling each bunch to be phase-locked and accelerated forward with low divergence, high beam charge and large beam-angular-momentum. This paves the way for wide applications in various fields, e.g., ultrabrilliant attosecond x/γ-ray emission.

In this paper, we report an innovative scheme to generate a dense attosecond electron bunch train efficiently from a laser-driven nanofiber target. In the scheme, a circularly polarized LG laser pulse with OAM but without total AM is employed. The attosecond electron bunches are first dragged out by the radial laser electric fields and are quickly phase-locked and accelerated forward by the longitudinal electric fields. The dense attosecond bunch train results from the unique field structure of the circularly polarized LG laser pulse, enabling each bunch to converge into a dense disk and propagate with a narrow energy spread, high beam charge and large beam-angular-momentum (BAM). Full three-dimensional (3D) particle-in-cell (PIC) simulations indicate that the stable attosecond structure remains intact for several hundreds of femtoseconds, paving the way for many potential applications in various fields.

Results
In initial simulations, a hydrogen nanofiber with plasma density of n e = 20n c is used. Here, n c = 1.12 × 10 21 cm −3 is the critical density for a λ 0 = 1 µm laser. The target could be solid-hydrogen, hydrocarbon or, carbon and silicon materials in experiments 33 . The fiber radius and length are R 0 = 0.5 µm and L 0 = 15 µm, respectively. A left-hand circularly-polarized LG laser pulse with mode ( p , ) is focused on the left side of the fiber and propagates along the x-axis. Here the integer  denotes the number of azimuthal phase cycles and (p + 1) represents the number of radial nodes. The laser electric field is given by where E L0 = m e0 cω 0 a 0 /q e is the peak amplitude of the laser electric field with m e0 the electron rest mass, c the speed of light in vacuum, ω 0 the laser angular frequency, q e the charge unit, and a 0 the laser dimensionless parameter, r = (y 2 + z 2 ) 1/2 , σ = σ 0 (1 + x 2 /f 2 ) 1/2 the focal spot radius with σ 0 = 3λ 0 the beam waist radius, and the laser temporal profile for −cτ ≤ x − ct ≤ cτ with the duration τ = 5T 0 and T 0 = λ 0 /c, and  L p is the generalized Laguerre polynomial. Here, ψ= denotes the phase term with k the wave number, R = x + f 2 /x the curvature radius of the wave-front, + +  p xf ( 2 1)arctan( / ) the Guoy phase of the mode, and ϕ the azimuthal angle. We use the (1, 0) mode of the LG laser, which implies an azimuthal phase term e iϕ and a spiral wave-front of the laser pulse. Figure 1 presents the typical simulation results obtained with a 0 = 30. At t = 10T 0 , a series of short electron bunches with a 1 λ 0 spacing are generated along the nanofiber. Each bunch is isolated and doughnut-like, which is markedly different from the sawtooth-like or helical structure in a plasma channel, a hollow cone, a wire, or in the case of micro-droplets using normal Gaussian laser pulses 8,9,[11][12][13][14][15] . These electron bunches move almost at the speed of light with a longitudinal thickness corresponding to a fraction of the laser wavelength ( λ ∼ . 0 2 0 , i.e., ~660 as in our simulations at t = 10T 0 ). The inner ring diameter is only 1.0λ 0 , with the maximal density of 30n c . When the annular electron bunches leave the nanofiber, they are further accelerated and each bunch converges into a dense disk with beam charge up to 0.3 nC, instead of being dispersed in space as in the case with normal Gaussian laser pulses 8,9,[11][12][13][14][15] . As time progresses, the thickness of the disk decreases to, ~200 as, while its peak density is still up to 2n c at t = 50T 0 . The attosecond structure is very stable and remains intact for a duration >300 fs (see Supplementary Fig. S1), which is about one order of magnitude longer than in other cases 8,9,[11][12][13][14][15]18 .
The formation of annular attosecond electron bunches in our scheme results from the unique field structure of the circularly polarized LG laser pulse. Here we considered a left-hand circularly-polarized LG-mode laser pulse with an OAM per photon but without total AM, which is characterized by simultaneously cylindrically-symmetric radially-polarized and azimuthally-polarized components. For convenience, we first convert the laser transverse electric fields into radial and azimuthal equivalents in the cylindrical coordinate system, owing to E r = cosϕ ⋅ E y + sinϕ ⋅ E z and E ϕ = −sinϕ ⋅ E y + cosϕ ⋅ E z : where ψ′ = −k[x + r 2 /(2R)] + ω 0 t − 2 arctan (x/f) is the transformed phase term. One can see that both the electric field E r and E ϕ are circularly-symmetric in the transverse cross-section and have nothing to do with the azimuthal angle ϕ. It is distinct from the normal radially polarized laser pulses 10,[34][35][36] , where only the radial component exists for the electric field. This unique optical field structure would benefit the dense electron bunch acceleration and propagation greatly. When E r is positive, the electrons in the skin layer of the fiber are compressed, forming a rippled structure on the fiber surface; as it becomes negative, the electrons are subject to a radial outward force and are dragged out of the nanofiber, so that an attosecond bunch forms. Since E r is also periodic along the x direction with a period of 1T 0 , the spacing of the attosecond electron bunches is exactly 1λ 0 , as seen in Fig. 1(a-c). However, when we change the laser handedness, the electric field structure varies accordingly. As a result, the attosecond structure disappears and two separate electron helices surrounding the nanofiber are observed. Thus the field structure of the LG laser pulse plays a key role in the electron dynamics.
Once the attosecond electron bunches are dragged out, they experience acceleration by the longitudinal electric fields, which exist for a tightly focused circularly-polarized LG laser pulse. Here one can obtain longitudinal electric field from the Poisson equation, i.e., E x (r, ϕ, x) = −(i/k)(∂E y /∂y + ∂E z /∂z): 0 at the center axis and vanishes at r = σ 0 ; For r < σ 0 , E x < 0, which is able to accelerate electrons. For each single electron dragged out from the fiber, the electron relativistic equations of motion in the LG laser fields together with the energy equation can be written as Here β is the normalized electron velocity by the speed of light, γ is the electron relativistic factor, E and B are the laser electromagnetic fields. In cylindrical coordinates ϕ r x ( , , ), the equations are rewritten as Since the azimuthal electromagnetic fields exist for a circularly-polarized LG laser pulse, it becomes a challenge to find an analytical solution to Eq. (4) above for each electron in the bunches. In order to explore the detailed dynamics, we track some typical electrons in the PIC simulations and present their energy evolution and distribution in Fig. 1(a). Here we separate the two driving terms in the energy equation and thus the energy gain due to the longitudinal and transverse electric fields are the contribution of the transverse electric fields is smaller and most electrons are located within the region ), indicating the dominance of the longitudinal electric field acceleration over the transverse acceleration. Considering 0.5λ 0 ≤ r ≤ λ 0 , we see |E x |/|E x | max changes from 0.94 to 0.80, implying quasi-monoenergetic electron acceleration in our configuration.
Dense attosecond electron bunch formation requires simultaneous longitudinal acceleration and transverse focusing, i.e., E x < 0 and E r > 0. This demands (2r 2 /σ 2 − 2)cosψ′ < 0 (note kr 2 /R → 0) and sinψ′ > 0. As seen in Fig. 2(b), each attosecond electron bunch in our configuration are exactly positioned in the acceleration phase of the longitudinal electric field E x . These bunches are relativistic and co-move with the laser pulse, so that they are quickly phase-locked longitudinally 37 . Figure 2(c) presents the radial field E r experienced by an attosecond bunch as marked in Fig. 1. Once phase-locked, the bunch always suffers positive electric field E r , making the doughnut bunches close up after leaving the nanofiber. Figure 2(d) shows the electron phase-locking in a single laser period. For r < σ 0 , |E x | is much stronger than for σ  r 0 , which is sufficient to accelerate electrons to relativistic velocities in less than one laser cycle. Once entering the matched phase with positive E r , these electrons are subject to a strong focusing force and are confined transversely. By contrast, as r > σ 0 , |E x | gets much smaller so that the electron acceleration is weak and the phase-slip occurs, resulting in an increasing energy spread, even though the electrons are initially in the acceleration phase of the laser pulse. Thus, the appropriate phase for dense attosecond electron bunch formation satisfies ψ′ ∈ [0, π/2] in one laser cycle.
When the energetic electrons are well confined near the laser axis, namely , we obtain the condition for the electron trapping, x . This requires  ψ′ sin 0 for each attosecond bunch, which is exactly the case here for the bunches in phase, so that the transverse electric and magnetic forces cancel each other out. However, in order to compensate the phase-slip during the acceleration (note v x < c), the electron velocity should be high enough. Here the acceleration length approximates to the Rayleigh length, This implies an energy threshold of γ th ≥ 7.5 in order to keep the electrons phase-locked in the LG laser fields. For relativistic LG laser pulses, this is naturally satisfied, because such a laser pulse is capable of accelerating electrons to relativistic energies in less than one laser cycle. Once locked by the LG laser pulse, an electron acquires a velocity of v t = qE x /m e × T 0 in the first laser cycle, which should be such that . As seen in Eq. (2) and Fig. 2(b) above, we take π × E E e a / (1/6) 2 /(3 ) x 0 0 to make sure most electrons in each bunch phase-locked. Therefore, we find that the laser amplitude threshold required for electron trapping in the LG laser field can be written as γ γ > − = . a e 9( 1) /( 2 ) 3 82 , if we ignore the multi-dimensional effects. By a series of 3D PIC simulations, we find the actual threshold is around a 0 = 6. Note that the LG laser pulses with such intensity are already available in several laboratories around the world 32,38 . Figure 3(a) presents the electron trajectories up to t = 35T 0 , which illustrates clearly that most attosecond electrons originate from the left tip of the nanofiber. Since the laser electric field components E r , E ϕ , and E x are circularly-symmetric in the yz plane, these electrons with the same r experience similar acceleration. All these factors contribute to the quasi-monoenergetic electron acceleration as indicated in Fig. 3(b), where a peak energy of 375 MeV and a cutoff energy of 465 MeV are found at t = 60T 0 . The energy peak gradually increases in width, while the density is still above the critical density at t = 120T 0 (400 fs). The final electron energy can be estimated MeV, in excellent agreement with the simulations. Figure 3(c) shows the electron emis- sion angle distribution at t = 60T 0 . The energetic bunches obtained are distinctive with a small divergence angle of <2°, a peak energy of ~400 MeV and nC charges, which have diverse applications in various fields. The corresponding transverse beam root-mean-square emittance ε y and ε z are 1.4 × 10 −2 mm ⋅ mrad and 1.5 × 10 −2 mm ⋅ mrad, respectively (see Supplementary Fig. S2). For example, by applying the electron bunches obtained via our scheme in the recent Thomson scattering experiment 39 , ultrabrilliant attosecond γ-ray bursts at the critical photon energy of 34 MeV are expected, since the beam charge here is enhanced by 10 3 times while the beam energy spreads in both cases are comparable. For a circularly-polarized LG laser pulse, its spin angular momentum (SAM) makes a charged particle spin with respect to its own axis, while its OAM is able to drive the particle rotate with respect to the laser axis. Under appropriate conditions, both can be transferred to particles, which is a fundamental question from a relativistic plasma optics perspective 28 . Here we employ a circularly polarized LG laser pulse with OAM but without total AM. Figure 4(a) exhibits the velocity field of electrons with AM larger than 2.73 × 10 −28 kg⋅m 2 /s in the yz plane. Apparently, most electrons rotate around the origin point and form a vortex. Figure 4 , with y 0 ,z 0 the target axis position in the yz plane. A clear transfer of the laser OAM to the electron BAM is observed because of the circularly-symmetric non-zero azimuthal components of the LG laser field. Once in the phase-locking regime, these bunch electrons also suffer a fixed azimuthal electric field, tend to rotate with respect to the laser axis, and thus get AM from the laser pulse. As expected, almost all AM are carried away by the attosecond electron bunches. At t = 60T 0 ,  is > 10 −17 kg ⋅ m 2 /s. This is a unique property of the bunches, which is otherwise absent in other cases using radially polarized normal Gaussian laser pulses 10,34-36 , making the bunches even more attractive for potential applications, e.g., generating x/γ-ray vortex via nonlinear Thomson scattering 40 .

Discussion
The space charge effect plays an important role in determining the final electron number of each bunch. Here the transverse Coulomb field can be estimated as where ε 0 is the permittivity. Considering the relativistic cancellation of the magnetic and electric forces, the net transverse repulsion force approximates to 1/ s is much smaller than the Lorentz force as γ  1, the repulsion force is suppressed significantly, so that we can take into account only the laser force here. The electron number in each bunch is thus limited to N e,max < (σ 0 /r e )[p ⊥ /(m e0 c)] 2 , according to the impulse theorem ∆ γσ < Here, r e = (4πε 0 ) −1 e 2 /(m e c 2 ) is the classic electron radius. As electrons are dragged out, ⊥ f s increases accordingly. Once ⊥ f s approaches the maximum, electrons are no longer dragged out. Finally, we find the total electron number is up to 10 10 in our simulations, about ~61.3% of which originate from the left tip. This implies that we may replace the nanofibre with micro-droplets to facilitate experiments. Note that it is easier to produce micro-droplets in large numbers and consistent with high repetition rate lasers 41 . One can thus test the set-up in experiments with the current and forthcoming LG-mode lasers in the near future. Figure 5(a) shows the 3D isosurface of the longitudinal current density at t = 10T 0 . Since abundant electrons are dragged out of the nanofiber, strong charge separation fields also form in the nanofiber, resulting in a strong return current to satisfy the plasma neutralization condition 42 . The return current tends to pump electrons to the left tip, some of which can be distinguished from the trajectories above. During the process, quasi-static electromagnetic fields around the nanofiber are also induced, as seen in Fig. 5(b) and (c), though their amplitudes are much smaller as compared to the laser fields.
Finally, we explore the influence of the laser and target parameters on the bunch generation, as summarized in Fig. 6. Here, we vary a single parameter in each case and keep all other parameters fixed. By increasing the fiber radius, the longitudinal electric field E x decreases and the phase-slip occurs more quickly, so that the electron energy decreases. Correspondingly, the radial electric field E r increases and more surface electrons are dragged out, and finally the total beam charge increases. In experiments, the pre-pulse may expand the front end of the nanofiber so that the real diameter increases. However, once the diameter becomes larger than a critical value (r th = σ 0 ), the longitudinal electric field E x in Eq. (3) becomes positive, which accelerates surface electrons along the -x direction (see Supplementary Fig. S3). As a result, these electrons move backward and are separated from the forward bunches. As discussed above, since most bunch electrons come from the front tip, the beam charge and energy weakly depend on the fiber length, which is demonstrated by simulations in Fig. 6(b). Meanwhile, the laser intensity plays an important role in the final beam energy and flux. Figure 6(c) indicates that, by increasing the laser dimensionless parameter a 0 , the longitudinal electric field E x rises linearly, leading to higher electron energies, larger beam flux, and longer pulse lifetime (>500 fs). The stable process is also valid for a much lower laser intensity and an example simulation with a 0 = 10 is given in Supplement. On the other hand, using a larger laser focal spot, the acceleration length of the beam increases and much higher electron energies are achieved. Other experimental factors on attosecond bunch formation are also discussed (see Supplementary Figs S4 and S5). Because there exists a laser intensity threshold for the short bunch formation in our scheme, one may also tailor the laser pulse to realize a single attosecond electron bunch generation.
In summary, we report a novel efficient scheme for generating a dense attosecond electron bunch train by irradiating a nanofiber target with an circularly polarized, intense LG-mode laser pulse. The generated electron bunch train results from the unique field structure of the circularly polarized LG laser pulse, and is distinctive with ultrashort bunch duration, high beam charge and density, small divergence angle, and high BAM. These features are extremely attractive for generating ultrabrilliant attosecond x/γ-ray bursts 43 , driving plasma-based electron acceleration, and generating high-energy-density states of matters. We also observe a clear transfer of the laser OAM to the attosecond electron BAM, offering a new avenue towards the generation of bright x/γ-ray vortex via nonlinear Thomson scattering 40 .

Methods
Numerical modeling. We carry out a series of 3D simulations to investigate the physics of the LG-mode laser interaction with a nanofiber target. The PIC code Virtual Laser Plasma Lab (VLPL 44,45 ) is employed, which allows direct fully electromagnetic simulations of relativistic laser plasma interactions. The particle number per cell (PPC) is 27 for both electrons and ions. Absorbing boundary conditions for the fields and particles are employed in the x direction (the laser propagation direction), while periodic ones for the fields and absorbing ones for the particles are used in the y and z directions. The initial electron and ion temperatures are assumed to be 1 keV and 0.01 keV, respectively. Data availability. Data associated with research published in this paper can be accessed at http://dx.doi. org/10.15129/f9d7cc35-8f3c-461c-9bbe-ed6a9c8a667b. Figure 6. Parametric influence of the fiber radius R 0 (a) and length L 0 (b), the laser dimensionless parameter a 0 (c) and focal size σ 0 (d) on the achieved bunch energy and charge. The blue square denotes the electron charge, while the red circle shows the electron energy.