Polarized electron acceleration in beam-driven plasma wakefield based on density down-ramp injection

We investigate the precession of electron spins during beam-driven plasma-wakefield acceleration based on density down-ramp injection by means of full three-dimensional (3D) particle-in-cell (PIC) simulations. A relativistic electron beam generated via, e.g., laser wakefield acceleration, serves as the driving source. It traverses the prepolarized gas target and accelerates polarized electrons via the excited wakefield. We derive the criteria for the driving beam parameters and the limitation on the injected beam flux to preserve a high degree of polarization for the accelerated electrons, which are confirmed by our 3D PIC simulations and single-particle modeling. The electron-beam driver is free of the prepulse issue associated with a laser driver, thus eliminating possible depolarization of the prepolarized gas due to ionization by the prepulse. These results provide guidance for future experiments towards generating a source of polarized electrons based on wakefield acceleration.

Figure 1

Sketch of the electron beam-driven wakefield acceleration: the driving beam has a longitudinal size ${\sigma }_l$ and transverse size ${\sigma }_r$; plasma target comprises a density ramp and background density; electron is injected at the location defined by the transverse radius of $r_i$ and azimuthal angle $\varphi$; the electron spin vector orientation is defined by the azimuthal angle $\varsigma$ and polar angle $\psi$.

Figure 2

Isosurface of the bubble (green, outside) and the driving beam (blue, inside) are shown in (a) and (b), respectively, where the electron density is projected onto the x-y (at $z=0$) plane, while the longitudinal electric field $E_x$ is projected onto the x-z plane (at $y=0$). The black arrows denote the spin directions for electrons. The azimuthal magnetic field $\left(B_{\varphi }\right)$ distributions at (c)130 f s and 260 fs (d). Data are taken at the beginning (130 fs) and end (260 fs) of injection for the driving electron beam ${\sigma }_r=2\mu \mathrm{m}, {\sigma }_l=2.5\mu \mathrm{m}, n_{b0}=1\times {10}^{19}\mathrm{c}{\mathrm{m}}^{-3}$ with $\alpha =4, n_0={10}^{18}\mathrm{c}{\mathrm{m}}^{-3}$, and ${\mathit{s}}_0={\mathit{e}}_x$.

Figure 3

Simulation results for trapped electrons located in the bubble. Displayed are the evolution for (a) beam polarization and the averaged relativistic factor of electron (dashed), (b) the spin $s_i$ (${\mathit{s}}_i={\mathit{s}}_0$) distributions at 800 fs. In both figures, the blue solid lines and red dotted lines represent ${\sigma }_r=2\mu \mathrm{m}$ and ${\sigma }_r=1\mu \mathrm{m}$ with ${\mathit{s}}_0={\mathbf{e}}_x$ while cyan dashed lines and pink dashed-dotted lines represent ${\mathit{s}}_0={\mathit{e}}_y$, respectively. Other parameters are the same where the ${\sigma }_l=2.5\mu \mathrm{m}, n_{b0}=1\times {10}^{19}\mathrm{c}{\mathrm{m}}^{-3}, \alpha =4$, and $n_0={10}^{18}\mathrm{c}{\mathrm{m}}^{-3}$.

Figure 4

(a) Evolution of the spin rotation angles $\mathrm{\Delta }{\theta }_s$ based on single particle modeling for $r_i=0\sim 3\mu \mathrm{m}$ (lines from bottom to top) with the same parameters of the ${\sigma }_r=2$-μm case in Fig. 2. The polarization as a function of (b) the dimensionless parameter $m$ [Eq. (9)] for initial spin ${\mathit{s}}_0={\mathbf{e}}_x$ and (c) $\psi$ (initial spin directions parameters). In (b), the blue dashed line is the analytical results from Eq. (8); case 1 ∼ 3 corresponds to adjusting ${\sigma }_r$ at ${\sigma }_l=2.5\mu \mathrm{m}, n_{b0}=1\times {10}^{19}\mathrm{c}{\mathrm{m}}^{-3}, {\sigma }_l$ at ${\sigma }_r=1\mu \mathrm{m}, n_{b0}=1\times {10}^{19}\mathrm{c}{\mathrm{m}}^{-3}$, and $n_{b0}$ at ${\sigma }_l=2.5\mu \mathrm{m}, {\sigma }_r=1\mu \mathrm{m}$, respectively, with $\alpha =4, n_0={10}^{18}\mathrm{c}{\mathrm{m}}^{-3}$; case 4 denotes varying plasma density $n_0$ and $\alpha$ with ${\sigma }_l=2.5\mu \mathrm{m}, {\sigma }_r=1\mu \mathrm{m}, n_{b0}=1\times {10}^{19}\mathrm{c}{\mathrm{m}}^{-3}$; case 0 represents the single-particle modeling for parameters in the region: ${\sigma }_l=0.5\sim 5\mu \mathrm{m}, {\sigma }_r=0.5\sim 5\mu \mathrm{m}, n_{b0}=0.2\sim 2\times {10}^{19}\mathrm{c}{\mathrm{m}}^{-3}, \alpha =2\sim 8$ and $n_0=0.2\sim 2\times {10}^{18}\mathrm{c}{\mathrm{m}}^{-3}$. In (c), the blue dashed line is the analytical results from Eq. (4); Case 1 and 2 are for simulations with $n_{b0}=1\times {10}^{19}\mathrm{c}{\mathrm{m}}^{-3}$ and $1.5\times {10}^{19}\mathrm{c}{\mathrm{m}}^{-3}$, respectively, with ${\sigma }_l=2.5\mu \mathrm{m}, {\sigma }_r=1\mu \mathrm{m}, \alpha =4$, and $n_0={10}^{18}\mathrm{c}{\mathrm{m}}^{-3}$. Case 0 represents the single-particle modeling with the same parameters of case 1 and 2. (d) The $s_y$ evolution $({\mathit{s}}_0={\mathbf{e}}_y)$ for different $\varphi =0\sim \pi /2$ (lines from bottom to top) of $r_i=3\mu \mathrm{m}$ with the same parameters of (a) based on single-particle modeling.

Figure 5

Restrictions on the total particle number of the driving beam $N_d$ to obtain beam polarization $>80\%$ for (a) longitudinal $({\mathit{s}}_0={\mathit{e}}_x)$ and (b) transverse polarization $({\mathit{s}}_0={\mathit{e}}_y)$, under certain $k^2$ and ${\sigma }_r$ with $\alpha =4$ based on Eq. (10). The color bar denotes for the maximum driven-beam total charge $Q=e{N}_d$ to obtain polarization $>80\%$.

Figure 6

Polarization as a function of the peak current $I_{\mathrm{peak}}$ under certain plasma densities with $\alpha =4$ and $n_0={10}^{18}\mathrm{c}{\mathrm{m}}^{-3}$ for (a) longitudinal $({\mathit{s}}_0={\mathit{e}}_x)$ and (b) transverse $({\mathit{s}}_0={\mathit{e}}_y)$ polarization. The red lines represent results predicted by Eq. (11) while case 1 ∼ 3 (blue circles, pink squares, and black stars) denote simulation results adjusting $N_d({\sigma }_r=1\mu \mathrm{m},k^2=n_{b0}/n_0=5), {\sigma }_r(N_d=2\times {10}^8,k^2=n_{b0}/n_0=5)$ and $n_{b0}(N_d=2\times {10}^8,{\sigma }_r=1\mu \mathrm{m})$ respectively. The beam flux as a function of α at fixed polarization values for (c) longitudinal and (d) transverse polarization, with fixed drive-beam parameters ($N_d=2\times {10}^8, {\sigma }_r=1\mu \mathrm{m}, n_{b0}=5\times {10}^{18}\mathrm{c}{\mathrm{m}}^{-3}$).

Figure 7

(a) Radii of the beam-driven bubble under different parameters at $x=x_p$ (red, lower) and $x=x_0=x_p+L$ (blue, upper), from Eq. (5) (dashed) and simulations (square and circle). (b) Azimuthal magnetic field $B_{\varphi }$ as a function of the radius. The $B$ field is averaged among the trapping region during the injection process. Analytical results from Eq. (7) are shown by the blue dotted-dashed line. Simulation results are given for case 1 (red circles): ${\sigma }_l=2.5\mu \mathrm{m}, {\sigma }_r=1\mu \mathrm{m}, n_{b0}=1\times {10}^{19}\mathrm{c}{\mathrm{m}}^{-3}$; case 2 (purple squares): ${\sigma }_l=2.5\mu \mathrm{m}, {\sigma }_r=2\mu \mathrm{m}, n_{b0}=1\times {10}^{19}\mathrm{c}{\mathrm{m}}^{-3}$, and case 3 (yellow stars): ${\sigma }_l=2.5\mu \mathrm{m}, {\sigma }_r=1\mu \mathrm{m}, n_{b0}=5\times {10}^{19}\mathrm{c}{\mathrm{m}}^{-3}$, respectively. For all cases $\alpha =4$ and $n_0={10}^{18}\mathrm{c}{\mathrm{m}}^{-3}$.