Fig. 1. Schematic of colliding pulse injection. (a) Two colliding laser pulses irradiate a pre-polarized underdense plasma with longitudinal density profile n_e(x) shown by the black dashed line. (b) Some plasma electrons (blue) undergo collisionless heating and gain residual energy and longitudinal momentum (red). (c) The electrons that have gained sufficient longitudinal momentum (red) to satisfy the injection criterion are trapped and subsequently accelerated in the wakefield.

Fig. 2. 2D PIC simulation results. The driving and colliding laser pulse intensities are a₀ = 2 and a₁ = 0.5, respectively. Both pulses have w₀ = 8 µm waist radius and τ₀ = 25 fs duration. (a) Snapshot of the electron plasma density n_e and the laser electric field E_y at time t = 100T₀. (b) Same as (a), but at time t = 340T₀. In (a) and (b), the upper and lower half-panels correspond to the cases respectively with and without the colliding laser pulse.

Fig. 3. Particle tracking results from 2D PIC simulations with the same parameters as those in Fig. 2. The rainbow color map shows the initial electron’s transverse position |y_t=0|. The black dashed line indicates the value obtained by averaging over the displayed trajectories. (a) and (e) Electron trajectories in the wake-frame coordinates (ξ, y). (b) and (f) Temporal evolution of the electron energy γ_e. (c) and (g) Longitudinal electron momentum p_x. (d) and (h) Longitudinal spin component s_x of the electron. (a)–(d) correspond to the case without the colliding laser pulse, and (e)–(h) to the case with the colliding pulse.

Fig. 4. Test-particle simulation results. Each color corresponds to a different driving laser pulse amplitude a₀, and the horizontal axis gives the colliding laser pulse amplitude a₁. (a) Residual longitudinal momentum δp_x after the collision of the two plane-wave pulses. (b) Spin polarization loss δs_x ≡ 1 − s_x. In both panels, dashed lines display the prediction obtained by numerical fitting the simulation data as δpx=0.29a02a1mec and δs_x = 0.25a₀a₁.

Fig. 5. Illustration of the Hamiltonian model. (a) Electron potential energy −|e|φ (black dashed line) and longitudinal electric field E_x (blue solid line) as functions of the wake-frame coordinate ξ. (b) Value of the Hamiltonian H(ξ,px) in Eq. (8) in units of electron rest energy m_ec² (brown color map) and its contour levels (black dashed lines). The rainbow color lines display the evolution in the (ξ, p_x) phase space of the electrons initially located at ξ = 10 µm. (a) and (b) share the same horizontal axis.

Fig. 6. Particle tracking results from 2D PIC simulations. The driving- and colliding-laser pulse intensities are a₀ = 2 and a₁ = 0.5, respectively. Both laser pulses have w₀ = 8 µm waist radius and τ₀ = 25 fs duration. The magenta and green lines correspond to the cases respectively with and without the colliding laser pulse. (a) Electron trajectories in (ξ, y) space. The blue–red color map displays the longitudinal electric field. The black dashed line plots E_x at y = 0. (b) Electron trajectories in (ξ, p_x) space. The brown color map shows the normalized value of the Hamiltonian H from Eq. (8), where the potential φ(ξ) is obtained from the E_x at y = 0 of the simulation [see the black dashed line in (a)]. (c) Evolution of the longitudinal spin s_x.

Fig. 7. Parameter scans over the normalized amplitudes a₀ and a₁ of the driving and colliding laser pulses, respectively, performed with the spectral quasi-3D PIC code FBPIC. (a) Injected electron charge Q. (b) Electron beam average spin polarization ⟨s_x⟩. The cross marks in (a) and (b) denote the cases in which no significant electron injection was observed. The black dashed line in (a)–(c) plots the injection threshold according to Eq. (14). (c) Average longitudinal spin polarization ⟨s_x⟩ = 1 − κ_sa₀a₁ as predicted from the scaling obtained with the test-particle simulations (see Table II).

Fig. 8. FBPIC simulation results with a₀ = 2 and a₁ = 0.05 driving and colliding laser pulses, respectively. (a) and (b) Snapshots of electron density distribution n_e and transverse focusing force −E_y + cB_z, respectively, at t = 500T₀. (c) Electron energy spectrum dN_e/dɛ_e. (d) Average spin polarization ⟨s_x⟩ as a function of electron energy ɛ_e. In (c) and (d), each color corresponds to a specific time. (e) Evolution of injected electrons (rainbow color map) in (ξ, p_x) space and the corresponding Hamiltonian distribution H(ξ,px) (brown color map). (f) Zoom of (e) at t = 100T₀ showing the three electron populations labeled A, B, and C. In (e) and (f), the white dashed ellipse marks the electrons near the Hamiltonian separatrix. (g) Initial position in (x, y) space of the injected electrons that eventually constitute the three populations A, B, and C whose evolution is shown in (e) and (f). The rainbow color map in (e)–(g) indicates the spin polarization at time t = 500T₀. (h) Evolution of injected electron populations in longitudinal phase space (x, p_x), where each color corresponds to a different time, namely, t = 50T₀, 70T₀, and 90T₀.

Fig. 9. FBPIC particle tracking results with a₀ = 2 and a₁ = 0.05 driving and colliding laser pulses, respectively, as functions of ct − x (i.e., the time evolution is from left to right). (a1)–(c1) Evolution of momentum components p_x (green), p_y (red), p_z (blue). (a2)–(c2) Evolution of spin components s_x (green), s_y (red), s_z (blue). (a3)–(c3) Evolution of transverse coordinate y of two representative electrons. (a1)–(a3), (b1)–(b3), and (c1)–(c3) are for electrons from populations A, B, and C, respectively.

Fig. 10. FBPIC simulation results showing the initial distribution in (x, y) space of injected electrons for the same driving laser and plasma parameters as in Figs. 8 and 9, but for different colliding pulse parameters: (a) a₁ = 0.05 and w₁ = 8 µm; (b) a₁ = 0.2 and w₁ = 8 µm; (c) a₁ = 0.05 and w₁ = w₀ = 4 µm; (d) a₁ = 0.05 and w₁ = 2 µm. The rainbow color map indicates the electron longitudinal spin polarization s_x at t = 500T₀.

Table 1. Parameters of the scaling $\delta p_x\approx {\kappa }_p{a}_0^{n_0}{a}_1^{n_1}{m}_{e}c$ calculated by numerical fitting of the results of test-particle simulations.

Table 2. Parameters of the scaling $\delta s_x\approx {\kappa }_s{a}_0^{m_0}{a}_1^{m_1}$ calculated by numerical fitting of the results of test-particle simulations.