Polarized electron-beam acceleration driven by vortex laser pulses

We propose a new approach based on an all-optical set-up for generating relativistic polarized electron beams via vortex Laguerre-Gaussian (LG) laser-driven wakefield acceleration. Using a pre-polarized gas target, we find that the topology of the vortex wakefield resolves the depolarization issue of the injected electrons. In full three-dimensional particle-in-cell simulations, incorporating the spin dynamics via the Thomas-Bargmann Michel Telegdi equation, the LG laser preserves the electron spin polarization by more than 80% at high beam charge and flux. The method releases the limit on beam flux for polarized electron acceleration and promises more than an order of magnitude boost in peak flux, as compared to Gaussian beams. These results suggest a promising table-top method to produce energetic polarized electron beams.


Introduction
Spin is an intrinsic form of angular momentum carried by elementary particles [1]. Numerous studies in particle physics and material science have been carried out using spin-polarized electron beams [2][3][4][5][6]. Generally, generating polarized electrons requires conventional accelerators (Storage ring or Linac) that are typically very large in scale and budget [2,7,8]. In some cases, it also needs sufficient long time to attain high polarization (couple of hours for storage rings [9]). Thanks to the rapid development of laser technology, the focal light intensities are now well beyond 10 20 W cm −2 [10][11][12], which paves a new path to obtain high energy electron beams based on the concept of laser-driven wakefield acceleration (LWFA) [13][14][15]. The latter, due to the extremely high acceleration gradient, promises a more compact and cost-efficient approach for electron acceleration. However, for the realization of a laser-driven accelerator for polarized electron beams several challenges need to be addressed: (i) since a significant build-up of electron polarization from an initially unpolarized target during laser acceleration does not happen [16,17], it requires the use of a gas target where the electron spins are already aligned before laser irradiation. (ii) Polarization losses during the injection of as many as possible electrons into a bubble structure and (iii) subsequent acceleration in the wake field must be kept under control and minimized.
With regard to polarized electron targets suitable for LWFA, we have witnessed promising developments in recent years. Electron spin polarization in strong-field ionization of atoms has been widely studied both theoretically [18][19][20][21][22][23] and experimentally [18][19][20]. Gas targets with spin polarization ∼40% have been achieved Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
The perspective of all-optical laser-driven polarized electron acceleration therefore relies on addressing the key issues (ii) and (iii), i.e. electron depolarization during LWFA. As shown below, spin depolarization mainly happens in the injection phase, which is in line with previous studies [29]. Recently, a method is proposed to mitigate this issue by fine tuning the focal position of the Gaussian laser beam to weaken the electron injection [30]. Here we propose that compared to Gaussian lasers, vortex laser beams are capable of creating a unique topology in LWFA that significantly suppresses the beam depolarization without sacrifice of the injected electron charge. Based on the proposed all-optical experiment set-up available for present techniques, we demonstrate in full three-dimensional particle-in-cell simulations the generation of high polarization electrons at very high beam charge. Such laser beams are readily accessible, for example, with the Laguerre-Gaussian (LG) mode [31,32].
The paper is organized as follows: in section 2 we give a detailed description of the method to generate fully polarized electron target. In section 3 we introduce the spin dynamics and the parameters chosen for our PIC simulations. In section 4 we present the results of simulations and analysis. Finally, in section 5, we give a brief summary.

Description of the all-optical set-up
As stated above, preparation a pre-polarized electron target is crucial to our scheme. Known from previous literatures, the photodissociation of hydrogen halides with circularly polarized UV light yields highly spinpolarized hydrogen and halogen atoms [26,27], at gas densities of at least 10 19 cm −3 [28]. The polarization can approach 100% for specific photodissociation wavelengths (for HCl this is near 213 nm), and if the molecular bonds are aligned prior to photodissociation [25]; otherwise, if the bonds are isotropic, the polarization is reduced to 40%. However, the electronic polarization of the halogen atoms is very low; specifically, the core electrons are completely unpolarized. Therefore, to maximize the polarization of all accelerated electrons, the halogen atoms can be ionized with a resonant laser (e.g. via 2+1 resonance-enhanced multiphoton ionization, for Cl atoms at 234.62 nm [26]) and removed from the target volume before irradiation by the accelerating laser pulse.
With this in mind, the all-optical scheme is sketched in figure 1. A spin-polarized hydrogen-atom target is prepared [16] from the photodissociation of HCl gas emitted from a nozzle. The bolds of HCl molecule are aligned by an 1064 nm infrared (IR) light (laser) first. Then polarized electrons are obtained via photodissociation with a circularly polarized UV light (213 nm, right-handed, perpendicular to IR) so that all the electron spins are oriented along the UV light propagation direction. After that Cl atoms are ionized by another UV light (235 nm, not shown here) and removed by either external electric field or thermal expansion. This is followed by the driving vortex LG pulse with a well-controlled time synchronization. The latter, propagating coaxially with the IR light, stimulates a plasma wakefield in the gas target and accelerates pre-polarized electrons to high energies. Figure 1. Sketch of the all-optical laser-driven polarized electron acceleration scheme with (a) the proposed experimental configuration and (b) the procedure to generate fully polarized electron target. The 1064 nm IR laser propagates along the x axis to align the bonds of the HCl molecules, then a UV light propagates along the z axis with a wavelength of 213 nm is used to photodissociate the HCl molecules. A 234.62 nm UV light is used to ionize the Cl atoms. Thermal expansion of the electrons creates large Coulomb field that expels the Cl ions. A fully polarized electron target is therefore produced for sequential acceleration by the LG laser pulse propagating along the x axis. A density ramp of length scale L is employed to ensure efficient electron injection.
The all-optical set-up requires good time synchronization between the multiple laser pulses. While aligning and photodissociation can be done simultaneously, the key issue is to control the time delay of the driving laser with respect to the previous light sources. As a matter of fact, the electron spins of hydrogen-atoms are aligned after photodissociation. For electron spin of s=+1/2, there exist two combinations of spin states |s=+1/2, n=+1/2> and |s=+1/2, n=−1/2> in the atoms (n is the nuclei spin). The first hyperfine state is an eigenstate and remains unchanged but the latter oscillates between |s=+1/2, n=−1/2> and |s=−1/2, n=+1/2> at a period of 350 ps [16,26,27]. To maximize the polarization of electrons for LWFA, the time delay between the driving pulse and the photodissociation UV light should be controlled, for example, at a precision of one-tenths of the oscillation period. State-of-art technique offers synchronization precision at ps level [33,34], therefore is sufficient for our purpose. It should be mentioned that the driving laser should also be delayed in such a way that the Cl ions are completely removed.

Spin dynamics and simulation parameters
Three relevant mechanisms may have influence on beam polarization, i.e. spin precession in electric and magnetic fields according to the Thomas-Bargmann-Michel-Telegdi (T-BMT) equation [35], the Sokolov-Ternov (S-T) effect (spin flip) and the Stern-Gerlach force (gradient forces). In reality, for LWFA schemes, spin precession according to T-BMT is the main influence while other effects are negligibly small [16,17,36] : Here e is the fundamental charge; m is the electron mass; v is the electron velocity; γ=1/(1−v 2 /c 2 ) −1/2 is the relativistic factor; a e =(g−2)/2≈1.16×10 −3 (g is the gyromagnetic factor); and the vector s is the electron spin in its rest frame [2], respectively. For simplicity, the spin vector is normalized to |s|=1 instead of ÿ/2 in our simulations. We have adopted the rotation matrix method in our moving particle module of the PIC code to minimize the numerical error in solving the T-BMT equation.
The LG laser propagates along the x axis from left side of simulations window of 48 μm(x)×48 μm (y)×48 μm(z) in size and 1200×600×600 in cell numbers, and passes through the fully ionized cold plasma where the electron spins are initially aligned to +z axis. For the sake of tracking the bubble wake during whole simulations, the window moves with the velocity of c before the laser front reaches the right boundary. We employ the LG 01 mode laser [41][42][43]: x R =πw 0 2 /λ and the focusing position x f , respectively. The laser pulse is linearly polarized along the y axis. We also give results for a Gaussian laser for comparison. We use the density ramp injection method [44][45][46] with following profile: similar to the one used in [30], for better comparison. Here Θ(x) is the step function, ξ=x−x 0 , x 0 =20 μm, L=16 μm, ratio between the peak density of the ramp and the background density κ=n p /n 0 =4, n p is the peak density while n 0 =10 18 cm −3 is the background density, respectively.

Results and theoretical analysis
The acceleration processes for both laser modes are illustrated in figure 2, at the same peak intensity of 8.6×10 18 W cm −2 and pulse duration of 21.4 fs. The ordinary Gaussian beam drives a bubble wakefield that traps local electrons due to the decreasing phase velocity in the plasma density bump [47]. A cylindrical electron bunch is formed in the bubble center, as depicted in figures 2(c) and (d). The spins of the injected electrons are well aligned in the same direction at 75T 0 However, at a later stage of 150T 0 the spin orientations are strongly diverged at different positions. Averaging the spin projections onto the z axis, one finds that the overall polarization is vanishing, i.e. the electron beam is depolarized.
To better understand the spin dynamics of electrons, we count all electrons injected into the bubble and calculate the polarization with P P P P / at each interaction time. As illustrated in figure 3(a), the beam polarization barely changes when interacting with the laser field (50-65T 0 ) since the initial spin directions are parallel/anti-parallel to laser magnetic field (along the +/−z axis). The polarization decreases significantly with γ varying slowly and then followed by steady acceleration of electrons where the polarization is almost constant. The former, corresponding to the injection phase of LWFA, is when the major depolarization happens. Along the simulation, one finds that the polarization is maintained at a very high level (∼88%) for the LG case, while the one for the Gaussian beam is only about 30%.
To further illustrate the polarization variation depending on the driving laser geometry, the evolution of the transverse spin component s z is shown in figure 3(b). During the interaction, the spins dilute towards the s z =-1 end for the Gaussian pulse, while most trapped electrons accumulate at s z =1 for the LG laser. From the energy spectra in figure 3(c) we see that, while the cutoff and peak energies are marginally smaller for the LG case, the total number of electrons is significantly higher than that for the Gaussian beam. This generates an enormous electron-beam flux. For instance, the peak current of the LG case reaches 20 kA (polarization ∼88%), about 4 times larger than in the Gaussian case (5 kA, polarization <30%). The boosted bunch charge or the peak flux benefits from the new geometry of the LG-laser-driven wakefield. The vortex beam produces a donut-shaped electron bunch in the vicinity of r 0 ±!r/2, as compared to a cylinder-like beam of radius !r for the Gaussian driver, corresponding to a cross-section area of 2πr 0 !r and π!r 2 , respectively. For a simple estimation, we use !r∼a 1/2 λ p (x p )/π∼4 μm [30,[48][49][50] as the bunch radius of the trapped electrons and r 0 =w 0 / 2 ≈7 μm as the center of the trapped region for the LG case [48,49]. Accordingly, the peak-current ratio between the LG and the Gaussian case is about 2r 0 /!r∼3.5, a factor that is well reproduced in simulations. The black arrows denote the electron spin directions. The electron density is projected onto the x-y (at z=0) and y-z planes, while the laser electric field is projected onto the x-z plane (at y=0). For the LG laser, we choose w 0 =12.5λ. The Gaussian beam also has a beam radius of 12.5λ. Therefore at the same laser amplitude, the pulse energy of the former is 2.7 times of that for the latter.
In LWFA, one usually has B∼B f , E r ∼−B f and v∼v x [47,49], where B f is the azimuthal magnetic field within the bubble. Considering γ∼1 ?a e during the injection (see figure 3(a)), the spin precession frequency from equation (2) takes the simplified form Ω≈eB f (2+β x )/2me f . To solve the equation for each particle, we separate the spin vectors s into the component parallel to Ω with s // =(s·e f )e f and perpendicular to Ω with s ⊥ =(s·e r )e r . From the initial condition s 0 =e z , one acquires evolution of the spin vector as s=(e z ·e f )e f +cos(Δθ s )(e z ·e r )e r +sin(Δθ s )(e z ·e r )e x . Here e x , e r and e f are normalized base vectors in cylindrical coordinates, the rotation angle Δθ s ≈〈Ω〉Δt depends on the time averaged precession frequency and the precession duration Δt. Then the beam polarization in each direction follows P x =1/N∑sin(Δθ s )(e z ·e r )=0, P y =1/N∑(e z ·e f )(e y ·e f )+cos(Δθ s )(e z ·e r )(e y ·e r )=0 and P z =1/N∑(e z ·e f ) 2 +cos(Δθ s ) (e z ·e r ) 2 =[1+∑cos(Δθ s )/N]/2, i.e. the polarization component is mainly along the z direction.
The above analysis shows that the spin procession is strongly related to the azimuthal magnetic field. We averaged B f over the trapped electron region and display its values in the y-z plane during injection in figure 4. For the LG case ( figure 4(a)), B f is anticlockwise in the near axis region (−5 μm<r<5 μm), decays to zero as the radius increases, and changes to clockwise in the outer region. Differently, the B f field for the Gaussian laser is clockwise in the whole displayed area and gradually declines ( figure 4(c)). One sees that the peak magnetic field  in the former is less than half of that in the latter. As a matter of fact, the azimuthal magnetic field B f in the cavity satisfies [45,46]: We assume ∂E x /∂r∼0 since in the blowout regime of LWFA the longitudinal electric field is slowly varying in the trapping area. Therefore, the field strength depends on two quantities: the longitudinal current density (J x ) and the electric field (E x ) along the x axis. Since it is known for the blow-out regime that |j x /ε 0 c 2 |/|∂E x /c∂x|∼n p /n 0 ∼4 [49][50][51], the difference is mainly due to the self-generated magnetic field of the beam current. For a Gaussian laser, the light intensity peaks on-axis such that trapped electrons are concentrated at the center of the bubble, leading to a well-directed current in the counter-propagation direction, as shown in figure 4(d). One finds the highest field strength near the symmetry axis. On the contrary, the intensity of the LG laser beam is maximized off-axis, leaving a hollow space in the propagation center. Two consequences immediately arise: first, trapped electrons are distributed in a circular ring with their density peaking at around r=7 μm. The new topology significantly lowers down the electron areal density and the current density for certain amount of beam charge. Second, electrons located near the symmetry axis leak through the beam center and become the source of a counter-propagating return flux in the region of r<5 μm. These effects lead to the unique current density profile for the LG driver in figure 4(b), where the peak current density in the trapping region is only one-third of that for a Gaussian beam. The already weakened magnetic field is further effectively compensated by the anti-clockwise field resulting from the counter-propagating flux in the center. Together, they strongly reduce B f while maintaining the total beam charge or flux, as lined out along the radial distance in figure 4(e). We also note that the longitudinal accelerating field E x , linearly dependent on the phase [50][51][52], is notably smaller in the LG case as seen from figure 4(f). It leads to less acceleration compared to the Gaussian case ( figure 3(c)). A possible explanation is that the beam loading effect is more significant as compared to the Gaussian case, due to the larger amounts of injected electrons caused by donut-shaped vortex distribution, which weakens the longitudinal electric field E x formed by the background ions.
The essence of beam depolarization is that electron spins precess at difference frequencies. Eventually the unsynchronized spins are oriented in various directions such that the averaged spin, i.e. the beam polarization, vanishes. A direct observation of the depolarization process will be tracking the specific electrons during injection and acceleration. We chose three electrons for each case, at injection radii of 0, Δr/2, Δr for the Gaussian laser and r 0 , r 0 +Δr/2, r 0 −Δr/2 for the LG laser, and present the precession frequencies in figures 5(a) and (b). In the Gaussian case, the electron spins oscillate at much higher frequencies due to larger B f (see figure 4). The precession frequencies, strongly determined by the magnetic field, are diverged for different injection positions. Electrons then lose their initial spin orientations in varied paces, leading to beam depolarization. The same analysis also applies to the LG case, but the field is so much weaker that the spin evolves much slower and their directions for off-axis electrons remain well-aligned during the whole interaction process.
Considering 〈β x 〉∼1/2 during the injection phase, the average precesion frequencies can be written as 〈Ω〉≈5e〈B f 〉/4m. As we noted from figures 5(a) and (b), the electrons with smaller injected radius seems trapped faster (Ω declines faster). Taking this into account, we consider that electrons with injection radius r i undergo 2π|r i −r 0 | for LG case and πr i for gaussian case during injection phase. With this in mind, we treat the injection time for Δt∼π|r i −r 0 |/〈β x 〉∼4π|r i −r 0 |/c LG case and Δt∼2πr i /c Gaussian case, which is close to previous studies where considering trapped electrons as a whole with Δt≈4a 1/2 λ p /πc [30,53,54]. Assuming the injection density is homogenous among whole injection region, the whole polarization after injection is given as: where P LG and P G is the whole polarization of the beam for LG and Gaussian case respectively. We calculate a polarization of about 0.91 for the LG case and 0.31 for the Gaussian case from the B f profiles in figure 4(e), in good agreement with the simulation results from figure 3(a).
After electrons gaining sufficiently high energies the spin directions remain almost unchanged (see figure 3(a)). In fact, the spin precession angle Δθ s can be roughly estimated during this phase. In steady acceleration, one has E r ∼cB f , v∼v x =β x c [45] (see also figure 4), therefore the precession frequency can be written as Ω≈eB f /mγ(γ+1), for β x ≈1 and γ?1, suggesting the electron spin precession is slowed down due to γ?1. Substituting the equation of motion for electrons γ/Δt∼dγ/dt∼eE x /mc into the precession frequency, we finally obtain Δθ s ∼cB f /E x (γ+1)∼1/γ=1, i.e. the spin change is negligible during this phase.
Spin depolarization imposes strong restrictions on the charge or the current of the electron beam from LWFA. One can find out the criteria for the Gaussian laser beam by taking B f ∼B 0 r/Δr∼en p r/8ε 0 c [55,56] and the radius of the injection volume Δr=(I peak /πen p c) 1/2 . The polarization is obtained from the statistic average of spins in the injection volume: where sinc(x)=sin(x)/x and α=5πe/16mε 0 c 3 . Hence to retain polarization >80% the criteria αI peak <1.8 applies, corresponding to the restriction I peak <2.5 kA. In figure 5(c) we show systematic scans at various laser amplitudes a, background densities n 0 , density ratio κ and focal position x f . In figure 5(c), the red circle with polarization 88% and I peak =18.9 kA corresponds to the LG laser, while the blue diamond with polarization 30% and I peak =4.8 kA is for the Gaussian case in figure 3(a). In all cases the beam current is limited to I peak <2.2 kA for the Gaussian laser to preserve polarizations over 80% (pink dotted line in figure 5(c)), consistent with the prediction in equation (7). However, the limitation on the beam flux is released because of the vortex beam structure. The peak current for the LG case I peak reaches ∼20 kA where the polarization is ∼90%, an order of magnitudes higher. We find interesting oscillations of the beam polarization for the Gaussian laser in figure 5(c). This can be seen from equation (7), where the second term in the numerator periodically changes with the peak current, describing the oscillation very well. It is known that only the component perpendicular to the magnetic field in the rest frame of the electron s ⊥ processes. For larger peak current, the precession frequencies increase such that the transverse beam polarization, averaged over accelerated electrons, vanishes for Δθ sm ?2π. In other words, P ⊥ = N cos s å q D » ( )/ 0, leaving the non-changing parallel to the magnetic field in the rest frame of the electron. Therefore the oscillation converges to P=P // =1/2 for growing peak current.
Our simulations are carried out for initial electron polarization along the laser magnetic field. Nevertheless, our scenario is valid for all starting spin directions. For initial polarizations s 0 =e x (parallel to propagation direction) and s 0 =e y (parallel to electric direction) the results are illustrated in figure 6(a). The polarizations of both cases are 81.2% for s 0 =e x and 88.6% for s 0 =e y , validating the novel effect of the vortex laser geometry. For the longitudinal pre-polarized case (s 0 =e x ), the polarization oscillation still happens as shown in figure 6(b). Unlike in the transverse polarization case, one has P=P x =∑cos(Δθ s )/N for s 0 =e x , converging to 0 polarization at sufficiently large currents. The restriction on peak current is also stronger, e.g. to preserve polarization >80%, I peak <1.5 kA.

Conclusions
In conclusion, we proposed a promising scheme to generate polarized electron beams via LWFA driven by a vortex LG Laser. According to our 3D-PIC simulations involving particle spin dynamics, electron beam is of polarization over 80% is achieved with high beam charge and peak flux. Compared to the Gaussian laser driven acceleration, the restriction on the electron beam current density is released, thanks to the novel topology of the vortex LG laser. By releasing the limitation on the beam loading, the averaged electron energy is naturally lower for the LG case. This could be compensated, for example, by enlarging the laser pulse energy as compared to the Gaussian case (at no loss of beam polarization). The scheme relies on an all optical set-up that is accessible at state-of-the-art facilities.