Particle accelerators are widely used in materials science,1 biology,2 medicine,3 fusion research,4 and industry,5 and as sources of intense and energetic photons.6–12 In science, one of the most important roles of accelerators is to probe the properties of fundamental forces as well as particle structure in searches for possible physics beyond the Standard Model.13 Conventional accelerators have an accelerating gradient limit around 100 MV/m owing to the electrical breakdown of radio-frequency cavities. By contrast, laser-plasma-based accelerators can support accelerating fields above 100 GV/m,14,15 enabling acceleration of electron beams to several GeV energy on centimeter scales.16 Several experiments have demonstrated the efficacy and robustness of the laser-wakefield acceleration (LWFA) mechanism.16–23 Compared with conventional large-scale accelerators, plasma-based accelerators generally have advantages in costs, size, and achievable peak current. Thus, LWFA is regarded as a promising route to realizing compact lepton colliders.24–26

To enable LWFA-based spin-dependent process investigations, which could also benefit high-energy lepton colliders,27,28 it is crucial to develop all-optical methods for the controlled and reliable generation of highly polarized electron beams. Recently, theoretical schemes based on the collision of an ultrarelativistic electron beam with a tailored laser pulse have been proposed as a possible source of spin-polarized electron beams induced by hard-photon emissions in the strong-field QED regime29–34 or by helicity transfer.35 However, the above methods require high-power and high-intensity laser pulses and are unsuitable for operating at a high repetition rate.

To generate a high-current spin-polarized electron beam, Wen et al.36 have put forward a scheme based on the LWFA of pre-polarized plasma electrons with a density down ramp for injection.36 With 3D particle-in-cell (PIC) simulations, this method has been shown to deliver 0.31 kA electron beams with 90.6% spin polarization by using a 2.1 × 10¹⁸ W/cm² tens of femtoseconds laser pulse with ∼2.2 TW power.36 In the scheme of Wen et al., a pre-polarized plasma is first produced via laser-induced molecular photodissociation, a method successfully employed in experiments to generate a high-density electron-spin-polarized gas with densities from 10¹⁶–10¹⁹ cm⁻³.37–39 Note that, in practice, it is not the entire plasma source that must be pre-polarized, but rather only the restricted injection volume itself. Although the pre-polarized plasma lifetime is of the order of 10 ns, hyperfine coupling results in a periodic electron-to-nucleus spin transfer with ∼100 ps period.38 Thus, the driving laser pulse can arrive tens of picoseconds after plasma pre-polarization, which is easily achievable with existing laser technology. Furthermore, while the alignment of laser pulses inside a plasma source with micrometer precision in both space and time has been demonstrated (see, e.g., Ref. 40), the laser pulses employed for plasma pre-polarization have low intensity requirements, such that their focal size can be large enough to easily enable spatial overlap. Remarkably, with the advent of 100 nm lasers,41 the molecular photodissociation technique might be applied to pure hydrogen, potentially enabling 100% plasma pre-polarization. The method of plasma pre-polarization via laser-induced molecular photodissociation was initially proposed by Hützen et al. and applied to polarized proton beam generation in laser-plasma interaction in Ref. 42. Following these seminal works, other schemes utilizing a pre-polarized plasma were put forward to generate energetic spin-polarized electron (or proton) beams43–55 or to investigate polarization effects in inertial confinement fusion.56 More recently, Nie et al.57 proposed to exploit the spin-dependent ionization cross section of xenon atoms to generate up to ∼31% spin-polarized and 0.8 kA current electron beams in a beam-driven plasma wakefield accelerator. However, the above methods have limitations on the attainable charge or spin polarization of the beam, and no simple route exists to control the generated beam features.

The colliding-pulse injection (CPI) scheme58,59 has produced high-quality electron beams of low divergence and energy spread,60–63 which are stable and reproducible.60,61,64–66 CPI provides many degrees of freedom that permit control over the generated beam features. For instance, the produced electron beam energy, charge, and energy spread are tunable by adjusting the position of the collision point in the plasma source,60 or the relative polarization between the driving and colliding laser pulses.62 This renders CPI a robust and versatile alternative to single-pulse LWFA to reliably generate high-quality, high-current spin-polarized electron bunches from a pre-polarized plasma, as shown below (see also Ref. 67). In CPI, a driving laser pulse with relativistic intensity induces a wakefield, while a sub-relativistic-intensity colliding pulse enables injection in the wake. As schematically illustrated in Fig. 1, the interaction process consists of two stages: (i) stochastic collisionless heating of plasma electrons by the two colliding laser pulses; (ii) trapping and acceleration of some energized electrons by the wakefield excited by the driving laser pulse.

In this article, we develop a model with an effective Hamiltonian that characterizes the electron dynamics, validate its predictions against quasi-3D PIC simulations, and show that CPI enables the generation of high-current and highly spin-polarized electron beams with controllable average spin polarization. The optimization of the beam features, including its charge and spin polarization, is studied in more detail in Ref. 67. Our work is arranged in four sections. In Sec. II, we present 2D PIC simulation results showing the plasma electron dynamics with and without the colliding laser pulse. In Sec. III, we discuss electron heating and the injection criterion. In Sec. IV, quasi-3D PIC simulations under the assumption of near cylindrical symmetry are performed to validate the theoretical model and further elucidate the electron injection dynamics and its influence on the charge and polarization of the electron beam. Our results are summarized in Sec. V, while the details of the particle spin pusher that we implemented in the spectral numerical-dispersion-free quasi-3D PIC code FBPIC68 are detailed in the Appendix.

Our 2D simulations are performed using the PIC code EPOCH,69 where we implemented the electron spin dynamics. Following Ref. 36, we exploit Ehrenfest’s theorem70 to describe the spin of an electron in a quasiclassical state with a vector s, where |s| = 1. The evolution of s is determined by the Thomas–Bargmann–Michel–Telegdi (TBMT) equation,71,72 and in our EPOCH simulations is implemented via the Boris pusher method73–76 (see below and the Appendix for an alternative implementation). Given the relatively low laser pulse intensities considered here, radiation reaction effects77,78 as well as other spin effects such as the Stern–Gerlach force79,80 and the Sokolov–Ternov effect48,81 are negligible. In simulations, an underdense plasma is irradiated by a relativistic-intensity driving laser pulse and a subrelativistic-intensity colliding laser pulse. The driving pulse is linearly polarized along the y axis, incoming from the left boundary of the computational box, and has a Gaussian transverse and longitudinal profile with w₀ = 8 µm waist radius, I₀ = 8.7 × 10¹⁸ W/cm² peak intensity, and τ₀ = 25 fs duration full width at half maximum (FWHM) of the intensity. Its wavelength is λ₀ = 0.8 µm, and the corresponding normalized field amplitude is a₀ ≈ 0.85λ₀[μm]I0[1018W/cm2]≈2. The colliding pulse has the same parameters as those of the driving pulse, except for the peak intensity, which is I₁ = 5.4 × 10¹⁷ W/cm², corresponding to a normalized field amplitude a₁ ≈ 0.5.

The computational box size is 120λ₀(x) × 70λ₀(y) and is uniformly divided in cells with a size of λ₀/20 (x) × λ₀/20 (y). The pre-polarized plasma has a plateau electron density profile n_e(x) = n_e,0 = 10¹⁸ cm⁻³ for x > x₁ and a linear up-ramp density profile for x₀ < x < x₁, where x₀ = 0 and x₁ = 20λ₀ [see Fig. 1(a)]. In the simulation, 32 particles per cell are used for electrons, and ions are treated as immobile. The focus of both the driving and the colliding laser pulses is set to be located at x = 35λ₀ in vacuum. The simulation box moves at the speed of light c, and open boundary conditions are adopted for both fields and particles. The group velocity of the driving laser pulse propagating inside the underdense plasma is vg=c1−ωpe2/(γ̃ω02), where ωpe=4πnee2/me is the plasma frequency, ω₀ = 2πc/λ₀ is the laser angular frequency, and γ̃≈1+a02/2 is the cycle-averaged Lorentz factor of the plasma electrons. Here, m_e and e are the electron mass and charge, respectively.

To illustrate the effect of the colliding laser pulse on the electron dynamics, we compare the 2D PIC simulation results obtained with and without the colliding pulse. As displayed in Fig. 2, a plasma cavity with length c/ω_pe ≈ 30 µm is sustained behind the driving pulse. In the presence of the colliding pulse, an electron bunch is stably injected at the rear of the cavity, resulting in ∼15 MeV energy gain over 200 µm propagation distance, whereas essentially no electron injection is observed without the colliding pulse (see Fig. 2). The corresponding particle tracking results from 2D PIC simulations are displayed in Fig. 3. For the case without a colliding pulse, the background plasma electrons merely experience the smooth oscillation excited by the driving laser ponderomotive force, and no background electrons are injected into the wakefield cavity [Figs. 3(a)–3(d)]. These electrons do not have a net energy gain, and their growing depolarization over time is attributed to the spin precession induced by the magnetic field while traversing the plasma cavity [see Fig. 3(d)]. By contrast, with a colliding pulse, a fraction of the electrons originating from the central region are injected into the first wakefield cavity and subsequently undergo acceleration [Figs. 3(e)–3(h)]. Electron injection occurs owing to the electrons’ residual longitudinal momentum p_x > 0 after interacting with the colliding laser fields [see Fig. 3(g) and below]. The electron beam spin polarization is primarily determined by the transient chaotic dynamics induced during the driving- and colliding-pulse interaction, while it is almost unchanged during the acceleration phase [see Fig. 3(h)]. In contrast to the spin dynamics observed in the down-ramp injection scheme,36 in CPI no strong correlation between the accelerated electrons’ longitudinal spin polarization loss 1 − s_x and their initial transverse coordinate y is observed [see Fig. 3(h) and Sec. IV].

In the following, we employ a two-stage model to characterize the electron dynamics and elucidate the injection process.

It is known that, in vacuum, an electron initially at rest remains at rest after interacting with a laser pulse, if the pulse can be approximated as a plane wave. Thus, in a one-dimensional model where plasma fields are small compared with the laser fields, the electron longitudinal residual momentum δp_x mainly stems from the interaction with the fields of the two colliding laser pulses. The residual momentum δp_x is of critical importance in determining the electron injection into the forward moving plasma cavity. If the plane-wave fields are derived from a vector potential expressed as A_0,1, then the corresponding electric and magnetic fields are E_0,1 = −∂A_0,1/∂ct and B_0,1 = ∇ × A_0,1, where subscripts 0 and 1 denote the driving and colliding laser pulses, respectively. By considering for simplicity the vector potentials of monochromatic plane waves A0=a0(mec2/|e|)sin⁡ϕêy and A1=a1(mec2/|e|)sin(ϕ+2k0x+ϕ1)êy, with ϕ = ω₀t − k₀x being the light front time, ϕ₁ the initial phase, êy the unit vector along the y direction, and k₀ = ω₀/c the wavenumber, the electron dynamics inside the two colliding laser pulse fields are determined bydpydϕ=|e|cddϕ(A0,y+A1,y),(1)dpxdϕ=−|e|ω0pyp−(B0,z+B1,z),(2)where p₋ ≡ γ_em_ec − p_x, dϕ/dt = p₋ω₀/γ_em_ec, B_0,z = −a₀(m_eω₀c/|e|) cos ϕ, and B_1,z = a₁(m_eω₀c/|e|) cos(ϕ + 2k₀x + ϕ₁). From Eq. (1), one immediately derives an integral of motion for the transverse momentum p_y = |e|(A_0,y + A_1,y)/c, such that Eq. (2) becomesdpxdϕ=me2c2p−a02⁡cos⁡ϕ⁡sin⁡ϕ+a0a1⁡sin(2k0x+ϕ1)−a12⁡cos(ϕ+2k0x+ϕ1)sin(ϕ+2k0x+ϕ1).(3)The terms containing 2k₀x in Eq. (3) hint at a strong dependence on initial conditions. In fact, previous studies have already shown that the resulting dynamics are chaotic,82 and that plasma heating due to stochastic acceleration can occur inside the counterpropagating laser pulses.83

To obtain the dependence of the residual momentum δp_x and spin depolarization δs_x on the laser parameters, we therefore resort to test-particle simulations.85–86 In our test-particle simulations, the two laser pulses are modeled as plane waves with electric fields|e|Ey,0meω0c=a0⁡exp−ϕ−ϕ0ω0τ0/2⁡ln⁡22cos⁡ϕ,(4)|e|Ey,1meω0c=a1⁡exp−ϕ+2k0x−ϕ1ω0τ0/2⁡ln⁡22cos(ϕ+2k0x),(5)and magnetic fields B_0,z = E_0,y and B_1,z = −E_1,y. The laser wavelength and period are denoted by λ₀ = 0.8 µm and T₀ = λ₀/c ≈ 2.67 fs, respectively, while τ₀ is the FWHM of the intensity. Here, ϕ₀ = 0 and ϕ₁ = 100π determine the initial positions of the peaks of the driving and colliding laser pulses, which correspond to 0 and 50λ₀, respectively. The initially at rest and uniformly distributed electrons are located in the region 20λ₀ ⩽ x ⩽ 30λ₀. In the calculations of the electron momentum p and spin s, an explicit Boris pusher method is utilized, where the timestep is Δt = 5 × 10⁻⁴T₀. This timestep satisfies the stringent temporal criteria for electron acceleration.87

For τ₀ = 25 fs, the test-particle simulation results for the residual longitudinal momentum δp_x and spin variation δs_x are shown in Fig. 4. Both δp_x and δs_x are calculated by averaging over the forward-moving electrons after they have separated from the two colliding pulses. By numerically fitting the results over the range 1 ⩽ a₀ ⩽ 3 and 10⁻² ⩽ a₁ ⩽ 1, we obtain the scalings δpx≈0.29a02a1mec and δs_x ≈ 0.25a₀a₁. As shown in Fig. 4, the curves obtained from the above simple scaling model agree fairly well with the test-particle simulation results.

It is worth emphasizing that, in general, stochastic heating, and consequently δp_x and δs_x, are also expected to depend on the laser pulse duration τ₀. To examine the impact of τ₀, for each τ₀ in the range 6.2 fs ⩽ τ₀ ⩽ 43.9 fs, we assume scalings of the form δpx=κpa0n0a1n1mec and δsx=κsa0m0a1m1, where κ_p,s, n_0,1, and m_0,1 are constants obtained by numerically fitting the residual longitudinal momentum and electron spin. Tables I and II report the obtained coefficients. Table I highlights a pronounced dependence of the exponent n₀ on the laser pulse duration, which originates from an increased stochastic heating and longitudinal momentum gain of electrons for longer-duration laser pulses. Given the relative simplicity of the obtained scaling, this is employed for quantitative predictions of the electron injection threshold and of the final beam polarization, which are validated against PIC simulations (see below).

The second stage of electron injection corresponds to electron trapping into the subluminal wakefield, which is investigated through Hamiltonian analysis.

In LWFA, electrons gain energy from the longitudinal electric field of the Langmuir wave excited by the ponderomotive force of the laser pulse. This is modeled, for simplicity, by considering the 1D dynamics of electrons in the moving frame of the first cavity in the wake of the laser pulse. The drifting velocity of the plasma cavity v_d equals the group velocity of the laser pulse inside the underdense plasma v_g, i.e., vd=c1−ωpe2/(γ̃ω02). In the cavity-frame coordinate ξ ≡ x − v_dt, the longitudinal electric field E_x(ξ) depends only on ξ. The electron dynamics are characterized by the equations88dpxdt=−|e|Ex(ξ),(6)dξdt=pxme1+(px/mec)2−vd,(7)where v_d is independent of time. The electric potential of the longitudinal wakefield can be derived as φ(ξ) = −∫E_x(ξ) dξ such that E_x(ξ) = −∂φ(ξ)/∂ξ. This allows us to determine the electron motion in the moving wakefield from the conserved HamiltonianH(ξ,px)=−|e|φ(ξ)+cme2c2+px2−vdpx.(8)Indeed, Eqs. (6) and (7) can be obtained from Hamilton’s equations dpx/dt=−∂H(ξ,px)/∂ξ and dξ/dt=∂H(ξ,px)/∂px. By setting dp_x/dt = 0 and dξ/dt = 0, we find a fixed point (ξ*,px*) in (ξ, p_x) phase space where ξ* satisfies the condition E(ξ*) = 0 and px*=vdme/1−vd2/c2. This fixed point corresponds to a scenario in which an electron with velocity v_x = v_d is comoving with the wakefield and does not exchange energy with the longitudinal electric field.

We are interested in the electron dynamics inside the first cavity of the laser-driven wake. The corresponding longitudinal electric field can be approximated as [see Fig. 5(a)]Ex(ξ)=0ifξ⩽ξ4,−E0ξ−ξ4ξ3−ξ4ifξ4<ξ⩽ξ3,−E0ξ0−ξξ0−ξ3ifξ3<ξ⩽ξ0,E0ξ−ξ0ξ1−ξ0ifξ0<ξ⩽ξ1,E0ξ2−ξξ2−ξ1ifξ1<ξ⩽ξ2,0ifξ2<ξ,(9)where E₀ is the peak value of |E_x(ξ)| and is reached at ξ_1,3, while ξ_2,4 denote the boundaries of the cavity. As shown in Fig. 5(a), ξ₄ = −ξ₂ and ξ₃ = −ξ₁ as a result of the symmetry of the field with respect to ξ₀ = 0. Accordingly, the potential is calculated through φ(ξ)=−∫−∞ξE(ξ)dξ with φ(−∞) = 0, which givesφ(ξ)=0ifξ⩽ξ4,E0(ξ−ξ4)22(ξ3−ξ4)ifξ4<ξ⩽ξ3,E0ξ0−ξ42−E0(ξ0−ξ)22(ξ0−ξ3)ifξ3<ξ⩽ξ0,E0ξ2−ξ02−E0(ξ−ξ0)22(ξ1−ξ0)ifξ0<ξ⩽ξ1,E0(ξ−ξ2)22(ξ2−ξ1)ifξ1<ξ⩽ξ2,0ifξ2<ξ.(10)For definiteness and without loss of generality, we consider the following parameters: v_d/c ≈ 0.9997, ξ₀ = 0, ξ₁ = −ξ₃ = 10.4 µm, ξ₂ = −ξ₄ = 13.0 µm, and E₀ = 0.022m_ecω₀/|e| ≈ 96 GV/m. These parameters are similar to those identified in LWFA experiments where GeV electron beams are produced from a centimeter-scale underdense plasma.89 The corresponding minimum electron potential energy is −|e|φ(ξ₀)/(m_ec²) ≈ 0.9.

For the above-mentioned parameters, Fig. 5(a) displays the profiles of the electric field E_x(ξ) and potential φ(ξ) obtained from Eqs. (9) and (10), respectively. Figure 5(b) displays the corresponding values of the Hamiltonian H(ξ,px), as well as the phase space evolution of a group of electrons initially located at ξ = 10 µm with momentum m_ec ⩽ p_x|_t=0 ⩽ 10m_ec. Their evolution in (ξ, p_x) clearly shows that there exists a longitudinal momentum threshold p_th for the occurrence of electron trapping in the wake. This allows us to determine whether an energized electron gets trapped by the wakefield E_x(ξ) or slides away from the potential cavity [see Fig. 5(b)]. The electrons with p_x|_t=0 < p_th are not sufficiently fast to be trapped by the forward-moving wake. Thus, they slide away from the wake cavity and are not injected. These electrons are termed untrapped electrons. By contrast, the electrons with p_x|_t=0 > p_th are trapped by the potential well φ(ξ) and subsequently efficiently accelerated to an energy of ∼400 MeV in the region of the cavity where the field E_x(ξ) is negative and therefore accelerating for electrons. These electrons are termed trapped electrons. To determine the threshold p_th, we consider the contour of H(ξ,px) between the separatrix point (ξ₄, p_d) and the threshold (ξ, p_th), which is given by H(ξ,pth)=H(ξ4,pd). This gives(1−βd2)pthmec2−2βdApthmec+1−A2=0,(11)where β_d = v_d/c,A=|e|φ(ξ)mec2+1γd,γd=11−βd2,and p_d = γ_dv_dm_ec. The two solutions pth± of Eq. (11) arepth±mec=βdA±βd2+A2−11−βd2.(12)By employing the parameters listed below Eq. (10), we obtain |e|φ(ξ)/(m_ec²) ≈ −0.24 at ξ = 10 µm and γ_d ≈ 41.6. Thus, the momentum threshold for electron trapping is pth−≈1.7mec, which agrees well with the numerically calculated electron trajectories in Fig. 5(b). The conjugate root pth+≈917mec corresponds to the attainable energy of an electron trapped with momentum near the threshold after it undergoes acceleration in the cavity and returns to ξ = 10 µm [see Fig. 5(b)]. In this description, the maximum and minimum longitudinal momenta are reached at ξ₀ = 0. Note that while the simplified longitudinal electric field profile in Eq. (9) does not precisely match that obtained in PIC simulations [compare Figs. 5(a) and 6(a)], our analysis and model are not sensitive to the exact form of the longitudinal electric field. In fact, the injection criterion pth− and the maximum attainable energy γemax∼pth+ in Eq. (12) are determined once the potential φ(ξ) around the peak of the longitudinal electric field and the drifting velocity v_d have been given (see Sec. IV for details).

To validate the model presented in Sec. III and the injection condition δpx>pth−, with pth− defined in Eq. (12), we track the evolution of an electron with initial position y ≈ 0 in the 2D PIC simulations of Sec. II and investigate its evolution both with and without the colliding pulse. The corresponding results are displayed in Fig. 6, where the magenta and green lines correspond to the cases respectively with and without the colliding pulse. In both cases, the electron trajectory in (ξ, p_x) space shows that its evolution nearly follows the contour of the Hamiltonian90 after the electron interaction with the pulses ends [see Fig. 6(b)]. In the case without the colliding pulse, the electron trajectory in (ξ, p_x) space always remains below the separatrix, and the electron is not trapped and readily slides away from the plasma cavity. In the case with the colliding pulse, however, the electron has a residual longitudinal momentum δp_x > 0. The residual momentum satisfies the injection criterion, namely, δpx>pth−, and the electron gets trapped in the cavity [see Fig. 6(b)]. For the traced electron, the longitudinal spin polarization s_x is modulated by the colliding laser fields, but returns nearly to its original value after the passage of the laser pulses. Moreover, s_x does not significantly change during the subsequent acceleration stage inside the cavity.

For a homogeneous plasma, one can infer pth± in Eq. (12) with the following estimates: βd=1−S, 1/γd=S, and A=ρφ̃+1/γd, where S≡ne/γ̃nc, nc=meω02/4πe2, φ̃=|e|φ0/mec2, φ0=4π|e|ne(c/ωpe)2, and γ̃=1+a02/2. The coefficient 0 < ρ ≲ 1 accounts for the unknown position of the electron inside the cavity when the electron–laser pulses interaction ends [see the magenta line in Fig. 6] and for the minimum of the actual potential, which is simply estimated as φ₀. As will become clear below, in practice ρ is extracted from quasi-3D PIC simulations. Now, pth− in Eq. (12) can be recast aspth−mec≈1−Sρφ̃+S−ρ2φ̃2+2ρφ̃SS.(13)By combining Eq. (13) with the scaling of the residual longitudinal momentum δpx≈κpa0n0a1n1mec (see Table I), the injection criterion δpx>pth− becomes a1>a1*, wherea1*=1−Sρφ̃+S−ρ2φ̃2+2ρφ̃Sκpa0n0S1/n1.(14)

To test the validity of the scaling predicted by the injection criterion in Eq. (14) in a more realistic 3D scenario, we perform parametric scans with the spectral quasi-3D PIC code FBPIC68 with N_m = 2 azimuthal modes, where N_m > 1 accounts for departures from cylindrical symmetry in the fields. The simulation domain is a cylinder with a size of x × r = 50 × 25 µm², with cell size Δx = 1/80 µm and Δr = 1/40 µm. Similarly to the 2D EPOCH simulations presented in Sec. II, the two colliding laser pulses are linearly polarized along the y axis with λ₀ = 0.8 µm wavelength, and have a Gaussian profile both in space and in time with τ₀ = 25 fs FWHM of the intensity duration and w₀ = 8 µm waist radius. The driving laser pulse normalized amplitude is 1 ⩽ a₀ ⩽ 2.5, while the colliding laser pulse normalized amplitude is 0.01 ⩽ a₁ ⩽ 0.2. As in Sec. II, the plasma is underdense, with a linearly growing density profile followed by a plateau with density n_e,0 = 10¹⁸ cm⁻³ for x > x₁ [see Fig. 1(a)]. The numbers of macroparticles per cell per species along the cylindrical coordinate axes z, r, and θ are p_n,z = 4, p_n,r = 4, and p_n,θ = 4, respectively. Note that the spectral cylindrical representation implemented in FBPIC prevents numerical Cherenkov radiation (NCR), thus permitting both fast and accurate simulations. To mitigate NCR, a finite difference Cartesian 3D code would require very small spatial and temporal steps, greatly increasing the computational cost. Simulations with N_m = 3 did not show significant differences with respect to N_m = 2, thus indicating that possible effects beyond cylindrical symmetry are properly accounted for. For simulations with laser pulses carrying a high orbital angular momentum, not considered in the present work, the inclusion of several azimuthal modes would be required. In post-processing, the electron beam charge is self-consistently calculated from the weight of each macro-electron. As the FBPIC simulation domain has a cylindrical geometry, the macroparticle weight w is calculated as w = nr dθ dr dz, where n is the plasma number density, r is the radial coordinate, and dθ, dr, and dz are the grid steps along the azimuthal, radial, and longitudinal directions, respectively (see Ref. 91 for details of the FBPIC algorithm and features).

Figure 7 displays the model and FBPIC simulation results obtained with the above-mentioned parameters. In particular, Figs. 7(a) and 7(b) show the generated electron beam charge Q and average longitudinal spin polarization ⟨s_x⟩, respectively, while Fig. 7(c) shows the average beam polarization as predicted by the model, ⟨s_x⟩ = 1 − κ_sa₀a₁ (see Table II). The black dashed line in Figs. 7(a)–7(c) plots a1*, which is the injection threshold according to Eq. (14), where ρ is used as a fitting parameter to the quasi-3D FBPIC simulations, which gives ρ = 0.55. Remarkably, the scaling obtained from the model is in good agreement with the simulation results. Indeed, both the predicted scaling for electron injection as obtained in Eq. (14), which is set once ρ has been fixed, and the longitudinal beam polarization as estimated by assuming the simple scaling δsx≈κsa0m0a1m1 with the coefficients extracted from test-particle simulations (see Table II), are in good agreement with simulations (see Fig. 7).

Figure 7 shows that while highly polarized beams are generated with laser field amplitudes chosen around the injection threshold a1≈a1*, the electron beam charge Q is smaller for these lower field amplitudes. In fact, while the charge Q can be increased with higher a₁, this also results in a decrease in the average spin polarization ⟨s_x⟩ because of the stronger depolarization induced by the colliding laser pulses. By tuning the laser intensities a₀ and a₁, one can prioritize the electron charge Q or the spin polarization of the beam. However, one of the key advantages of the CPI scheme is the availability of more degrees of freedom than LWFA with a single laser pulse. Such flexibility naturally lends itself to multiparameter space optimization, as shown in Ref. 67, where Bayesian optimization is employed to conceptually demonstrate high-charge, highly polarized, and low-emittance electron beam generation. In addition, the average charge delivered per unit time can be simply increased by employing relatively low-power laser systems operating at high repetition rate.

Figures 8(a)–8(d) display the electron density distribution, the transverse focusing force, the electron energy spectrum, and the average spin dependence on the energy of an electron beam obtained with a₀ = 2 and a₁ = 0.05. The corresponding laser intensity and power are 8.6 × 10¹⁸ W/cm² and 8.7 TW, and 5.4 × 10¹⁵ W/cm² and 5.4 GW for the driving and colliding pulses, respectively. Figure 8(a) shows that a bunch of electrons is injected at the rear of the first cavity. The transverse focusing force −E_y + cB_z stabilizes electron acceleration in the cavity during the acceleration stage [Fig. 8(b)]. The injected electron bunch has ∼2.27 pC charge, 90% average longitudinal spin polarization, and about 33 MeV average energy after undergoing acceleration over ∼450 µm, with an accelerating gradient of ∼90 GeV/m. The electron energy spectrum dN_e/dɛ_e as a function of time clearly shows that the beam is highly monochromatic and that the energy spread is preserved over the acceleration stage [see Fig. 8(c)]. The average spin distribution ⟨s_x⟩ as a function of the electron energy ɛ_e exhibits a correlation between electron energy and electron degree of polarization, with the higher-energy electrons having predominantly a lower average spin [see Fig. 8(d)].

Further insights can be gained by analyzing the temporal evolution of the injected electrons [Fig. 8(e)] and by closely examining their distribution in (ξ, p_x) phase space [Fig. 8(f)]. By comparing the electron distribution in (ξ, p_x) space at different times [see Fig. 8(e) and the zoom of the distribution of injected electrons at t = 100T₀ in Fig. 8(f)], we find that the electrons with lower final spin polarization are injected with lower energy, initially, and with a longitudinal momentum near the injection threshold pth− [see Fig. 8(f)]. These electrons are later accelerated to higher energy inside the cavity, as predicted by the Hamiltonian model [see the contours of the Hamiltonian in Figs. 8(e) and 8(f)]. This explains the correlation between electron energy and the average spin of electrons in the generated beam. The zoom of the electron (ξ, p_x) distribution in Fig. 8(f) suggests the existence of three distinct electron populations, which are labeled A, B, and C. This is corroborated by tracking the injected electrons back to their initial positions and examining their evolution. Figure 8(g) plots the initial positions of injected electrons in (x, y) space, while Fig. 8(h) displays their detailed evolution in longitudinal phase space (x, p_x) for t < 100T₀. Figures 8(g) and 8(h) show that the electrons belonging to population A interact with the driving laser pulse significantly before interacting with the colliding pulse, thus suggesting that the colliding pulse merely plays a perturbative role for population A electrons. These electrons do not show a violent stochastic spin precession, and the spin polarization loss induced by the colliding pulses is insignificant. Figure 8(g) shows that the electrons belonging to populations B and C first interact with the colliding laser pulse before experiencing the stronger fields of the driving pulse. Population B and population C electrons occupy nearly the same spatial region around x ≈ 40λ₀, initially, and form transverse “stripes” that are longitudinally shifted by λ₀/2 [see Fig. 8(g)]. Thus, population B and population C electrons have a phase difference ϕ_1,B − ϕ_1,C = π, which implies a different longitudinal momentum according to Eq. (3). As a result, while electrons belonging to population B quickly slide away from the driving laser pulse and remain in the same region around x ≈ 40λ₀, the electrons of population C follow the driving pulse when they experience the combined fields of the colliding pulses, and drift up to x ≈ 45λ₀ longitudinally [see the orange and green dots in Fig. 8(h) corresponding to t = 70T₀ and t = 90T₀, respectively]. This relatively long interaction with the laser fields results in significant spin precession and depolarization.

To confirm the above analysis, we have tracked the dynamics of two representative electrons extracted from each of the three populations A, B, and C, with the two electrons having nearly the same initial conditions. Figures 9(a-1) and 9(a-2) show the evolution of the three components of the momentum p_x, p_y, p_z and of the spin s_x, s_y, s_z of the representative electrons of population A. Figure 9(a-3) shows the evolution in (ct − x, y) space of the representative electrons of population A. Figures 9(b-1)–9(b-3) and 9(c-1)–9(c-3) show the same quantities for electrons extracted from populations B and C, respectively. Figure 9 shows that the momentum and spin dynamics of electrons belonging to populations A and B are weakly dependent on initial conditions and that, in particular, the electron spins nearly return to their initial values after interaction [see Figs. 9(a-2) and 9(b-2)]. Notably, for populations A and B, the electron transverse momentum p_y,z closely follows the evolution predicted by Eq. (1) in the collision of two plane waves in vacuum, the anharmonic dynamics being manifested in the longitudinal momentum p_x, as expected from Eq. (2). By contrast, electrons from population C show signatures of premature and significant transverse momentum gain and a strong dependence on initial conditions. Unlike populations A and B, for population C the electron transverse momentum p_y,z does not follow the evolution predicted by Eq. (1), especially for p_z [see Fig. 9(c-1)]. This suggests that finite-transverse-size effects are important for this population. Note that the theoretical analysis of Sec. III utilizes test particle simulation results where the two colliding pulses are assumed to be plane waves in vacuum. Thus, deviations are expected if the waist radius of even one of the two laser pulses is not much larger than the laser wavelength.

A detailed study of how transverse effects and plasma self-generated fields influence the transverse momentum gain of the injected electrons is beyond the scope of the current work, and parameter optimization is carried out in a separate work.67 Nevertheless, it is worth noting that the contribution of electrons from population C can be controlled by tuning the colliding laser pulse intensity and waist radius. Naturally, this affects both the total beam charge Q and its average spin polarization ⟨s_x⟩. For instance, when the intensity of the colliding pulse is increased from a₁ = 0.05 [see Fig. 10(a)] to a₁ = 0.2 [see Fig. 10(b)], with all other parameters remaining unchanged, the number of electrons from population C greatly increases, and correspondingly the total charge of the electron beam rises from Q≈2.27 to 20.6 pC. Alternatively, the injection of electrons from population C can be strongly suppressed by reducing the waist radius of the colliding laser pulse, w₁. For instance, by reducing w₁ from w₁ = 8 µm to w₁ = 4 µm [see Fig. 10(c)] and w₁ = 2 µm [see Fig. 10(d)] and keeping all other parameters fixed, electrons from population C are suppressed, while the average beam polarization is enhanced from ⟨s_x⟩ ≈ 90.1% to ⟨s_x⟩ ≈ 95.3% and ⟨s_x⟩ ≈ 97.6%, respectively.

We have studied the dynamics of spin-polarized electron injection in the colliding-pulse scheme. By simple analytical modeling and multidimensional PIC simulations, we have shown that the electron injection process can be divided into a first stage of plasma electron collisionless heating and spin precession followed by a second stage of electron trapping and acceleration in the plasma wake. Using test-particle simulations and Hamiltonian analysis, we have obtained a simple scaling for determining the electron injection threshold and the beam polarization as functions of the laser and plasma parameters. Further study is required to show the dependence on additional parameters, such as relative laser polarization. Model estimates are in good agreement with quasi-3D FBPIC simulations over a broad range of experimentally relevant laser parameters. While it was already shown that the colliding pulse injection scheme reliably provides electron beams with excellent quality,60–63 here we have shown that this scheme also enables control of the spin-polarization degree of the generated beam. Remarkably, the required relatively low laser power of this scheme, a₀ = 2 (a₁ = 0.05) and w₀ = 8 µm, corresponding to 8.7 TW (5.4 GW), enable stable, reliable, and highly controllable operations even at high repetition rates, which is particularly relevant for applications such as precision measurements in fundamental physics.92,93

Acknowledgment. The original version of the PIC code EPOCH adapted here is funded by UK EPSRC Grant Nos. EP/G054950/1, EP/G056803/1, EP/G055165/1m, and EP/M022463/1. Z.G. would like to thank Rong-Hao Hu for useful discussions. The authors gratefully acknowledge the Gauss Centre for Supercomputing e.V. (https://www.gauss-centre.eu/) for providing computing time used for the FBPIC simulations through the John von Neumann Institute for Computing (NIC) on the GCS Supercomputer JUWELS at the Jülich Supercomputing Centre (JSC).