Fig. 1. (a) Schematic representation of the initially pre-polarized plasma. The longitudinal profile of the electron density is marked by the yellow dashed line, including an up-ramp from to with length and a plateau with . The initial polarization direction is aligned along the -direction, as denoted by the arrows. The laser is focused at the left-hand boundary of the plasma . For the case of longitudinal injection (Case A), (b)–(d) show the density distribution of electron longitudinal polarization at three different times, that is, is the product of electron density (normalized by and the average of polarization in the -direction per cell. Here, , , , and . For the case of transverse injection (Case B), (e)–(g) present the corresponding distributions of at different times, where , , and the other parameters are the same as in Case A. The electrons with kinetic energy in Case A (or in Case B) are chosen as the accelerated electrons, which are marked by a green box in (d) and (g), respectively.

Fig. 2. The history of particle properties, (a), (b) and the average kinetic energy (c) about accelerated electrons in the case of the longitudinal scheme. The distribution of (or ) in the x-direction is shown in the insert of (a) (or (b)). (d)–(f) The corresponding quantities in the case of the transverse scheme. The accelerated electrons are marked in the Figures 1(d) and 1(g), respectively.

Fig. 3. (a) Trajectory of a typical tracked electron for the longitudinal self-injection scheme (Case A) at the wakefield frame, where is the phase velocity of the wakefield calculated using the plateau density. The electron is initially located at the front of the wakefield. (b) The history of (blue solid line) and (red dashed line) for the tracked electron. (c) The evolution of the spin direction with time. (d) The evolution of (green solid line), the term (blue dashed line) and the term (magenta dashed line) of (black solid line) caused by , and the term (red solid line) of caused by for the tracked electron.

Fig. 4. (a) The green dots denote the positions of the chosen accelerated electrons, which are marked in Figure 1(d). The magnetic field at the laser axis is presented as a black solid line. (b) The spectra of (black line) and the longitudinal position (magenta line) for the accelerated electrons at . (c) The profiles of (red line) and (blue line) at the laser axis. (d) The spectra of the longitudinal velocity (red line) and transverse velocity (blue line).