Manipulating the topological structure of ultrarelativistic electron beams using Laguerre–Gaussian laser pulse

A method of using intense Laguerre–Gaussian (LG) laser pulse is proposed to generate ultrarelativistic (multi-GeV) electron beams with controllable helical structures based on a hybrid electron acceleration regime in underdense plasmas, where both the longitudinal charge-separation electric field and transverse laser electric field play the role of accelerating the electrons. By directly interacting with the LG laser pulse, the topological structure of the accelerated electron beam is manipulated and it is spatially separated into multi-slice helical bunches. These results are clearly demonstrated by our three-dimensional particle-in-cell simulations and explained by a theoretical model based on electron phase-space dynamics. This novel regime offers a new degree of freedom for manipulating ultrashort and ultrarelativistic electrons, and it provides an efficient way for generating high-energy high-angular-momentum helical electron beams, which may find applications in wide-ranging areas.


Introduction
Vortex light beams, which carry a finite orbital angular momentum (OAM), are characterized by an angulardependant phase Φ=lf with helical wavefront and strictly zero intensity on the propagation axis [1], where l is the topological charge that indicates the OAM carried by each photon and f is the azimuthal angle. Such vortex light beams are described by Laguerre-Gaussian (LG) modes, which can be implemented by several techniques such as mode conversion from Hermite-Gaussian to LG, using spiral phase plates, computer holograms, etc [2][3][4][5]. They are widely used in applications such as optical microscopy, micromanipulation, quantum information, astronomy, etc [6][7][8][9][10]. Recently, vortex laser pulses with relativistic intensities are also achieved based on the techniques of plasma holograms [11], Raman amplification [12], and the light fan [13], which have stimulated much research interest in intense laser-matter interactions and extended the study of OAM into a relativistic regime [14][15][16][17][18].
In addition to the vortex laser beams, particle beams such as electrons [19][20][21], neutrons [22], and x/ gamma-ray photons [23][24][25][26][27][28], can also carry OAM. In particular, electron vortex beams carrying OAM have extensive applications ranging from condensed-matter spectroscopy and new electron microscopes, to particle physics [19-24, 29, 30]. Electron vortex beams can be generated through the spiral phase plate and computergenerated holograms in a transmission electron microscope [19][20][21]. However, so far the achieved electron vortex beams are limited to a nonrelativistic energy range. Relativistic electron vortex beams are still unavailable based on the existing techniques. Recently, laser-based plasma accelerators have attracted much research attention, which can produce monoenergetic electron beams with energies up to several GeV in the so-called laser wake field acceleration (LWFA) regime [31][32][33][34][35][36][37][38][39]. In addition, when the intense laser pulse interacts with the plasma, its angular momentum can be transferred into electrons when the conservativeness of photon number is Original content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence.
Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. broken during nonlinear laser-plasma interactions [40][41][42]. Thus, it would be possible to manipulate the topological structure of relativistic electron beams and generate high-energy high-OAM electron beams based on laser-plasma interactions.
In this paper, we demonstrate for the first time that ultrarelativistic screw-like electrons beams with high-OAM can be produced by exploiting a hybrid electron acceleration regime driven by an intense LG laser pulse in undersense plasmas. In this regime, electrons are accelerated by the longitudinal charge-separation electric field and also the transverse laser electric field. Meanwhile, the LG laser pulse effectively transfers its angular momentum to the accelerated electron beam and manipulates the topological structure of the electron beam. Our three-dimensional (3D) particle-in-cell (PIC) simulations show that in this regime, the accelerated electron beam is spatially separated and consists of multi-slice helical bunches. A theoretical model based on the electron phase-space dynamics is also proposed to explain the emergence of multi-slice helical electron bunches. It is clarified that the topological charge of the LG laser pulse determines the number of fixed points in the system, which in turn determines the number of electron bunches that are accelerated in different phases of the laser electric field. In this way, the topological structure of the electron beam can be well controlled by adjusting the laser's topological charge.

Simulation and results
In order to investigate the interaction process of intense LG laser pulse with underdense plasmas, 3D PIC simulations were performed using EPOCH code [43]. A LG laser pulse with central wavelength λ=800 nm and laser period T L =λ/c is normally incident from the left boundary (x=0), where c is the speed of light in vacuum and x is the laser propagation axis. The transverse profile of a LG laser with zero radial index at x=0 is where a 0 is the normalized amplitude of laser electric field, r y z 2 2 = + , r 0 is the laser beam radius, θ is the azimuthal angle in transverse space, l is the topological charge of LG laser pulse, e e , y z   are the unit vectors in y and z direction, and σ represents the laser polarization state. σ is equal to 1 or -1 for right-hand and left-hand circular polarization, respectively. For linear polarization, σ is equal to 0. Here a right-hand circularly polarized (CP) LG laser is considered. In the simulations, LG laser pulses with different topological charges (l=0, 1, 2) are employed. In particular, for l=0, the LG laser pulse is reduced into the normal Gaussian laser pulse. In these simulations, the value of a 0 is set as 70, the initial laser beam radius is set as r 0 =7λ, and the duration (τ) of the laser pulse is set as T 20 L . The plasma is located in the region of 5λ<x<100λ with a uniform density of n e =2.4×10 20 cm −3 . The initial electron temperature is assumed to be 1 keV and ion temperature is 30 eV. The simulation box is 100λ(x)×25λ(y)×25λ(z) with a grid of 1000×250×250 and four particles per cell. In the current setup, the laser pulse length (cτ) is comparable with the size of the plasma bubble ( a n n 2 l c e l p ) created by the ultraintense laser pulse, were n c ≈1.74×10 21 cm −3 refers to the critical plasma density and a a l l exp 2 )is the amplitude of the LG laser pulse. In this case, the self-injected electrons can interact with the laser field directly when they are accelerated by the longitudinal charge-separation electric field [44,45].
When the intense LG laser pulse propagates into the underdense plasma, its ponderomotive force expels the electrons from the laser region. For the LG laser pulse, because there is no component of laser electric field on the central axis, as shown in figure 1(a), the electrons are expelled both outward and inward around the LG laser pulse in transverse space. As a result, a donut-like plasma bubble with a compressed plasma pillar along the central axis is formed, as shown in figure 1(d). In the plasma bubble, quasistatic electromagnetic fields are also generated, as shown in figures 1(b) and (c). The transverse focusing field, which consists of the transverse chargeseparation electric field E Sr and azimuthal magnetic field B Sθ , can be expressed as (E Sr −cB Sθ ) and is shown in figure 1(b). Under these intense fields, the self-injected electrons are trapped and confined in the donut-like plasma bubble, and experience betatron oscillations, as shown in figure 1(d). Meanwhile, these electrons are accelerated longitudinally at the rear of the donut bubble by the negative longitudinal charge-separation electric field E Sx , as shown in figure 1(c). For a LG laser pulse with l a a l l 2, exp 2 103 , the bubble size can be estimated as a n n 2 17.4 l c e l p l » , which is in good agreement with the simulation results shown in figure 1(d). For these parameters, the bubble size is comparable with the laser pulse length (20λ), which indicates a significant overlap between the trapped electrons and the transverse laser field, as can be seen in figures 1(a) and (d). In this case, the electrons can be directly accelerated by the transverse laser field, once the betatron oscillation frequency of the electrons is close to the Doppler-shifted laser frequency [46]. In this process, the trapped electrons are microbunched with the laser wavelength, as shown figure 1(d), which is indicative of direct laser acceleration (DLA) [47]. By directly interacting with the laser field, the electrons gain energy and angular momentum (both spin and orbit angular momentum) simultaneously from the LG laser pulse [17], as a result, they are manipulated by the laser pulse and distributed as helical structures, as indicated in figure 1(d).
The corresponding 3D distribution of the high-energy electrons in the plasma bubble is shown in figure 2 for LG laser pulses with different topological charges l. It is clearly demonstrated that the high-energy electrons are helically distributed in space due to the interaction with the LG laser pulse, which efficiently transfers its angular momentum into these electrons. In particular, the electron beam displays as multi-slice helical bunches and the number of the electron bunches is determined by the topological charge of the LG laser pulse. For example, there is only one slice for l=0, as shown in figure 2(a).When l=0, the LG laser pulse is reduced into a CP Gaussian laser pulse. In this case, the resonant electrons in the plasma bubble are locked in the acceleration phase of CP laser field and they experience circular motions collectively together with the rotating CP laser field. As a result, the energy density of the accelerated electron beam exhibits an annular distribution, as shown in figure 2(a), which also agrees well with previous simulation results using CP Gaussian laser pulse [17,45]. However, for l 0 ¹ , the results are quite different from the case with l=0. For l=1, there are two helical electron bunches intertwined each other, as shown in figure 2(b), and the number of the intertwined electrons bunches becomes 3 for l=2, as shown in figure 2(c).
The energy spectra of the helical electron beams are plotted in figure 3(a), which shows that the accelerated electrons display as peaked energy spectra for different cases. The electron energy at the peaks is about 1 GeV and the maximum energy can reach up to 2.0 GeV. The corresponding OAM of the accelerated electrons is calculated as L yp zp z y = -, where p y , p z are the linear momentum of particles in y and z direction, respectively. = is the peak laser intensity, and ω 0 refers to the central laser frequency. For the right-hand CP laser pulse, σ=1. When l=0, the OAM of the LG laser is zero, but the CP laser pulse itself still carries spin angular momentum. In this case, the accelerated electrons can still gain angular momentum from the laser pulse [17]. In the simulations, the conversion rate of the angular momentum from laser to electrons is about 30% for these three cases, which is comparable with the energy conversion rate. This indicates that the OAM of the electron beam is proportional to the value of l+1. In order to clarify the acceleration mechanism of the high-energy electrons, particle tracing methods were carried out here. The energy gain of electrons can be separated into two parts, i.e., W=W ⊥ +W x , where W ev E t d ) refers to the energy gained from the transverse laser electric field. Figure 3(c) shows that for the high-energy electrons, both the transverse laser field and the longitudinal wake field make great contribution on electron energy gain. In this hybrid regime, both DLA and LWFA play significant roles. In particular, during the interaction with the LG laser field, the electrons gain OAM efficiently from the laser pulse and the topological structure of the accelerated electron beam is manipulated.

Theoretical analysis
In this section, we build a theoretical model to explain the emergence of multi-slice helical electron bunches. When the relativistic LG laser pulse propagates into the underdense plasmas, the electrons will be expelled out of the laser region by the laser ponderomotive force to form a stable donut-like plasma bubble, as shown in figure 1(d). Especially when the laser pulse length is comparable with or longer than the plasma bubble length, the electrons in the plasma bubble will interact with both the laser fields and the quasistatic self-generated fields. Taking these fields into consideration in the electron acceleration process, the electron dynamics inside the (c) l=2.The (y, z) projection plane of the electron energy density on the left is taken at x=62.5λ, the (x, y) projection plane at the bottom is taken at z=0, and the (x, z) projection plane at the rear is taken at y=0. The colorbar represents the energy density of the electron beams ranging from 0 to n 1000 MeV c . The number of the sliced electron bunches can be clearly seen in the (y, z) projection plane. plasma bubble can be expressed as are the electromagnetic fields of the laser pulse, E S  and B S  are the quasistatic electric and magnetic fields, respectively. Here a right-hand CP LG laser beam with zero radial index is described as , where E 0 is the amplitude of the laser field, t kx l The laser frequency in the electron co-moving frame can be written as is the electron velocity along the laser propagation axis, and v ph is the phase velocity of laser in plasma. The first term ω L =κω 0 in equation (5) refers to the Doppler-shifted laser propagation frequency and the second term ω θ is the angular frequency of laser pulse witnessed by the electron, which is equal to the angular frequency of the circular motion of the electron, i.e., ω θ =ω B . The electrons can be accelerated by the laser field continuously when the resonance condition ω B ≈ω f is satisfied. To describe the electron motion in the plasma bubble, we further define a relative phase between the electron motion and the laser field as ψ and its time derivative can be written as The time derivative of electron energy in equation ( where v ⊥ is the electron velocity in transverse space and E L is the amplitude of transverse laser field. In this Hamiltonian system, by letting the right-hand side of equations (6) and (7) become zero, one can get the fixed points that satisfy the following equations For equation (10), its solution can be written as n arccos 2 where n is integer. Then one can obtain the angular phase of electron motion as is the laser propagation phase. For simplicity, when the longitudinal charge-separation electric field E Sx is ignored, indicates that the number of fixed points in this Hamiltonian system is l+1, which is determined by the topological charge of the LG laser pulse. Figure 4 shows the Hamiltonian and trajectories of electrons in phase-space (θ, γ) for a fixed laser propagation phase of f L =π. It is shown from figure 4(a) that there is only one fixed point and one separatrix for the case of l=0, which is consistent with the previous work [45]. While for the LG laser pulse with l 0 ¹ , the number of fixed points is related to the topological charge of the laser pulse and is equal to l+1, as shown figures 4(b) and (c), which agrees well with the above theoretical estimates. If the electrons are initially located around the fixed points, the acceleration phase of these electrons change slowly so that the electrons can stay in the acceleration phase for a relatively long time and then are efficiently accelerated. However, if the electrons are initially located below the separatrix and away from the fixed points [48], they cannot be accelerated to a highenergy because their orbits are open, as can be clearly seen in figure 4(c). In this case, the electrons would  experience dephasing rapidly and the acceleration process is terminated soon. Thus, the number of fixed points actually determines the acceleration phases for the resonant electrons that can be accelerated to a high-energy. The high-energy electrons would be distributed around the fixed points, and display as multi-slice helical bunches, as demonstrated in figure 2. Figure 5 presents the typical trajectories of high-energy electrons in (θ, γ) phase-space from 3D PIC simulations. It is clearly shown that the high-energy electrons are concentrated around the fixed points described in figure 4. This indicates that the topological charge of the LG laser pulse determines the number of electron bunches that are accelerated in different acceleration phases of the laser electric field. In this way, the topological structure of the accelerated electron beam is controlled by the LG laser pulse.

Conclusion
In summary, we propose an approach for producing high-energy helical electron beams based on a hybrid electron acceleration regime by launching an intense LG laser pulse into an underdense plasma. It is shown that due to the special helical phase of the LG laser pulse, when the trapped electrons in the plasma bubble directly interact with the LG laser pulse, the acceleration phase of these electrons is controlled by the topological charge of the LG laser pulse, leading to the generation of multi-slice helical electron bunches. The generated energetic helical electron bunches can be applied to many applications, such as the generation of x/γ-rays with angular momentum [24][25][26][27][28]. The annular beams like these may also be of interest to free-electron-laser applications [23].