2 Simulation method

In this study, 2D particle-in-cell (PIC) simulations were performed with a modified version of the EPOCH code^[ Reference Arber, Bennett, Brady, Lawrence-Douglas, Ramsay, Sircombe, Gillies, Evans, Schmitz, Bell and Ridgers ^{43 ]}, which includes the spin evolution module based on the Thomas–Bargmann–Michel–Telegdi (TBMT)^[ Reference Mane, Shatunov and Yokoya ^{25 ,} Reference Thomas ^{44 ]} equation via the Boris pusher method^[ Reference Li, Gibbon, Hützen, Büscher, Weng, Chen and Sheng ^{45 ]}. The electron spin is regarded as a quasi-classical quantity with a vector $\boldsymbol{s}$ , which has an absolute value of $1$ and a direction calculated from the TBMT equation $\mathrm{d}\boldsymbol{s}/\mathrm{d}t=\boldsymbol{\varOmega} \times \boldsymbol{s}$ with the following:

(1) $$\begin{align}\boldsymbol{\varOmega} =\frac{e}{m_{\mathrm{e}}}\left[\left({a}_{\mathrm{e}}+\frac{1}{\gamma}\right)\boldsymbol{B}-\frac{{\boldsymbol{a}}_{\boldsymbol{e}}\gamma }{\gamma +1}\boldsymbol{v}\cdot \boldsymbol{B}\frac{\boldsymbol{v}}{c^2}-\left({a}_{\mathrm{e}}+\frac{1}{\gamma +1}\right)\frac{\boldsymbol{v}}{c^2}\times \boldsymbol{E}\right],\end{align}$$

where ${m}_{\mathrm{e}}$ , $e$ and ${a}_{\mathrm{e}}\approx 1.16\times {10}^{-3}$ are the electron mass, charge and dimensionless anomalous magnetic moment, respectively, $\gamma$ is the Lorentz factor of the electron, $c$ is the light speed in vacuum, $\boldsymbol{B}$ is the magnetic field and $\boldsymbol{E}$ is the electric field in the laboratory frame. The effects of the radiation reaction, Stern–Gerlach effect and Sokolov–Ternov effect, can be ignored during the study of LWFAs, based on the work of Thomas et al. ^[ Reference Thomas, Hützen, Lehrach, Pukhov, Ji, Wu, Geng and Büscher ^{46 ]}.

In the simulation, the laser propagates in the $x$ -direction with linear polarization and a Gaussian envelope in the $y$ -direction:

(2) $$\begin{align}E=\frac{E_0{w}_0}{w(x)}\exp \left[-\frac{y^2+{z}^2}{w{(x)}^2}-\frac{{\left(t-\tau \right)}^2}{{\left(0.5\tau \right)}^2}\right]\cos \left(\varphi \right),\end{align}$$

with the laser wavelength $\lambda =800\ \mathrm{nm}$ , the initial laser waist ${w}_0=20\lambda$ , $w(x)={w}_0{\left[1+{\left(x-{x}_0\right)}^2/{z}_{\mathrm{R}}^2\right]}^{0.5}$ , ${z}_{\mathrm{R}}=\pi {w}_0^2/\lambda$ , the pulse duration $\tau =17\ \mathrm{fs}$ and the normalized laser amplitude ${a}_0={eE}_0/{m}_{\mathrm{e}}\omega c=6$ , corresponding to a peak intensity of ${I}_0=7.71\times {10}^{19}\ \mathrm{W}/{\mathrm{cm}}^2$ . The phase is $\varphi = kx-\omega t+\phi (x)+k\left({y}^2+{z}^2\right)/2R(x)$ , where $R(x)=x\left[1+{\left({z}_{\mathrm{R}}/x\right)}^2\right]$ and ${\phi (x)=\arctan \left(x/{z}_{\mathrm{R}}\right)}$ are the radius of curvature of the beam’s wavefronts and the Gouy phase at $x$ , respectively. The simulation box is $200\lambda (x)\times 120\lambda (y)$ with resolution $\mathrm{d}x=0.02\lambda$ and $\mathrm{d}y=0.08\lambda$ . Open boundary conditions are used in each direction and there are four pseudo-particles per cell for each particle species.

The initial longitudinal profile of the pre-polarized plasma is an up-ramp followed by a plateau with constant density ${n}_0=0.04{n}_{\mathrm{c}}$ , as shown in Figure 1(a), marked with a yellow dashed line. Such a density profile makes longitudinal electron injection possible, as first introduced in Ref. [Reference Li, Sheng, Liu, Meyer-ter-Vehn, Mori, Lu and Zhang41]. Here, the length of the up-ramp transition is ${L}_1=45\lambda$ and the laser pulse is focused on the left-hand edge of the plasma target at ${x}_0=30\lambda$ . The pre-polarized plasma could be realized by using the ultra-violet polarization method^[ Reference Sofikitis, Glodic, Koumarianou, Jiang, Bougas, Samartzis, Andreev and Rakitzis ^{26 ]}. For simplicity, the initial polarization rate of the plasma is assumed as $100\%$ , where we are interested in the evolution of the polarization during the electron self-injection scheme. The net polarization of a particle beam is defined as ${P=\sqrt{{\left\langle {s}_x\right\rangle}^2+{\langle {s}_y\rangle}^2+{\left\langle {s}_z\right\rangle}^2}}$ , where ${s}_i$ is the component of spin polarization in each direction and $\left\langle {s}_i\right\rangle$ is the corresponding average value.

Figure 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 $0$ to ${n}_0$ with length ${L}_1$ and a plateau with ${n}_0$ . The initial polarization direction is aligned along the $x$ -direction, as denoted by the arrows. The laser is focused at the left-hand boundary of the plasma $\left({x}_0=30\lambda \right)$ . For the case of longitudinal injection (Case A), (b)–(d) show the density distribution of electron longitudinal polarization $\left({n}_{\left\langle {s}_x\right\rangle}\right)$ at three different times, that is, ${n}_{\left\langle {s}_x\right\rangle }$ is the product of electron density (normalized by ${n}_0)$ and the average of polarization in the $x$ -direction $\left(\left\langle {s}_x\right\rangle \right)$ per cell. Here, ${a}_0=6$ , $\tau =17\ \mathrm{fs}$ , ${w}_0=20\lambda$ , ${n}_0=0.04{n}_{\mathrm{c}}$ and ${L}_1=45\lambda$ . For the case of transverse injection (Case B), (e)–(g) present the corresponding distributions of ${n}_{\left\langle {s}_x\right\rangle }$ at different times, where ${w}_0=10\lambda$ , ${n}_0=0.01{n}_{\mathrm{c}}$ , ${L}_1=10\lambda$ and the other parameters are the same as in Case A. The electrons with kinetic energy ${E}_{\mathrm{k}}>13\ \mathrm{MeV}$ in Case A (or ${E}_{\mathrm{k}}>30\ \mathrm{MeV}$ in Case B) are chosen as the accelerated electrons, which are marked by a green box in (d) and (g), respectively.

3 Results and discussion

Generally, the profile of the wakefield depends on the parameters of the laser and plasma, which inevitably causes the variation of the self-injection scheme, further leading to different evolution of spin polarization during the self-injection process. When the laser spot size is larger than the plasma wavelength (Case A), the wakefield is a quasi-1D regime, as shown in Figure 1(b). At this time, the wakefield propagates in the up-ramp density. When the wakefield reaches the uniform density regime $\left(x=75\lambda \right)$ , several electrons located at the tail of the wakefield $\left(x=88\lambda \right)$ , can be captured and accelerated due to the breaking-wave effect, as presented in Figure 1(c). After that, owing to the effect of laser self-focusing, the laser intensity increases, the wakefield develops into a 3D nonlinear bubble regime and the electrons are accelerated continually, as revealed in Figure 1(d). On the other hand, when the laser spot size is equal to the plasma wavelength and the laser intensity ${a}_0$ is larger than 4, that is, Case B, the bubble regime can be formed directly as the laser propagates into the plasma. In order to avoid the electron at the left-hand boundary being injected into the bubble, an up-ramp density with a short length is also used, as shown in Figure 1(f). Different from the case of longitudinal injection, the bubble regime is a 3D nonlinear regime initially and the corresponding phase velocity slows down. As the bubble geometry changes following the laser evolution, several electrons can be injected into the bubble and achieve acceleration, as shown in Figure 1(g).

Not surprisingly, the distribution of the electron polarization is different in the two cases and the influence of the laser on the electron polarization cannot be ignored. In the quasi-1D regime (Case A), the values of ${s}_x$ for injection electrons mostly are positive, as plotted in Figure 1(c). Meanwhile, in the bubble regime (Case B), the values of ${s}_x$ about the electrons located at the sheath are negative, as revealed in Figure 1(f). For further analysis, the accelerated electrons are chosen to analyze the evolution of polarization, as marked in the green region in Figures 1(d) and 1(g), respectively. For Case A, $6056$ electrons are chosen with a polarization ${P=0.99}$ . For Case B, there are $32,078$ electrons chosen with a polarization $P=0.18$ . Moreover, as presented in Figure 1(d), another bunch of electrons $\left(x\simeq 150\lambda \right)$ was injected in the wakefield owing to the nonlinear evolution of the laser pulse and bubble, which has been analyzed in the work of Kalmykov et al. ^[ Reference Kalmykov, Yi, Khudik and Shvets ^{39 ,} Reference Kalmykov, Beck, Yi, Khudik, Downer, Lefebvre, Shadwick and Umstadter ^{40 ]}. The trajectories of these electrons are similar to those in the case of transverse injection (Case B). Their polarization is nearly 0.8, which is lower than that of the first bunch of electrons; this will be discussed in this paper.

Figure 2 The history of particle properties, $\left\langle {s}_x\right\rangle$ (a), $\langle {s}_y\rangle$ (b) and the average kinetic energy $\left\langle {E}_{\mathrm{k}}\right\rangle$ (c) about accelerated electrons in the case of the longitudinal scheme. The distribution of ${s}_x$ (or ${s}_y$ ) 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.

The histories of $\left\langle {s}_x\right\rangle$ , $\langle {s}_y\rangle$ and the average energy $\left\langle {E}_{\mathrm{k}}\right\rangle$ for these two cases are plotted in Figure 2. As presented in Figures 2(a)–2(c), the evolution of the polarization for the longitudinal injection can be divided into three stages. (i) Here, $t<{t}_{\mathrm{I}}=198\ \mathrm{fs}$ , with the electron fixed. (ii) Here, ${t}_{\mathrm{I}}<t<{t}_{\mathrm{II}}=234\ \mathrm{fs}$ , where the value of $\left\langle {s}_x\right\rangle$ decreases firstly and then returns nearly to the initial value during a short time, which is nearly the duration of the laser $\left(\sim 36\ \mathrm{fs}\right)$ . As presented in the insert of Figure 2(a), the distribution of ${s}_x$ shows that the laser can affect the electron spin directly at $216\ \mathrm{fs}$ . The oscillation period is nearly $0.5\lambda$ for $\left\langle {s}_x\right\rangle$ and $1\lambda$ for $\langle {s}_y\rangle$ , which means that the electron spin is affected by the laser field directly. Meanwhile, the average energy increases firstly and then decreases. (iii) Here, $t>{t}_{\mathrm{II}}$ , where the value of $\left\langle {s}_x\right\rangle$ does not change significantly. The value of $\langle {s}_y\rangle$ is nearly 0 owing to the azimuthal symmetry of the wakefield. At this stage, the electron energy increases with time, which means the electrons are continuously accelerated in the wakefield.

For the case of the transverse injection scheme (Case B), as analyzed in Ref. [Reference Fan, Liu, Li, Qu, Yu, Kong, Weng, Chen, Büscher, Gibbon, Kawata and Sheng30], the evolution of polarization can be divided into four stages. (i) Here, $t<{t}_{\mathrm{I}}$ , where the electrons do not feel the wakefield. (ii) Here, ${t}_{\mathrm{I}}<t<{t}_{\mathrm{II}}$ , where the electrons are located on the bubble shell. Further, $\left\langle {s}_x\right\rangle$ decreases and $\langle {s}_y\rangle$ oscillates in the laser field and stays at nearly $0$ due to the azimuthal symmetry of the bubble field, as shown in Figure 2(d). (iii) Here, ${t}_{\mathrm{II}}<t<{t}_{\mathrm{III}}$ , where the electrons reach the tail of the bubble and $\left\langle {s}_x\right\rangle$ increases, as revealed in Figure 2(d). (iv) Here, $t>{t}_{\mathrm{III}}$ , where the electrons are captured in the bubble and their spin precession slows down.

The electron spin evolution during the transverse injection has been studied through single particle dynamics in the work of Fan et al. ^[ Reference Fan, Liu, Li, Qu, Yu, Kong, Weng, Chen, Büscher, Gibbon, Kawata and Sheng ^{30 ]}. It is found that the electron spin is mainly affected by the magnetic field of the bubble during the second stage and affected by the electric field of the bubble in the third stage. In the fourth stage, the electron moves along with the laser axis, so its spin does not change obviously. For the longitudinal injection, a typical accelerated electron is also analyzed, as shown in Figure 3. The trajectory of the electron in the wakefield coordinate system between 195 and $235\ \mathrm{fs}$ is presented in Figure 3(a). At $195\ \mathrm{fs}$ , the electron is located at the head of the wakefield, and then it slips backward. When reaching the tail of the wakefield, it is captured. Although it vibrates in the transverse direction, the transverse position does not change obviously in the wakefield.

Figure 3 (a) Trajectory of a typical tracked electron for the longitudinal self-injection scheme (Case A) at the wakefield frame, where ${v}_{\mathrm{b}}$ 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 ${s}_x$ (blue solid line) and ${s}_y$ (red dashed line) for the tracked electron. (c) The evolution of the spin direction $\left(\theta =\mathrm{arctan}\left({s}_y/{s}_x\right)\right)$ with time. (d) The evolution of ${\boldsymbol{\varOmega}}_z$ (green solid line), the term ${\boldsymbol{\varOmega}}_{B_z}$ (blue dashed line) and the term ${\boldsymbol{\varOmega}}_{v_x{E}_y}$ (magenta dashed line) of ${\boldsymbol{\varOmega}}_{\boldsymbol{v}\times \boldsymbol{E}}$ (black solid line) caused by ${v}_x{E}_y$ , and the term ${\boldsymbol{\varOmega}}_{-{v}_y{E}_x}$ (red solid line) of ${\boldsymbol{\varOmega}}_{\boldsymbol{v}\times \boldsymbol{E}}$ caused by $-{v}_y{E}_x$ for the tracked electron.

Moreover, the evolutions of ${s}_x$ and ${s}_y$ are plotted in Figure 3(b). Based on the TBMT equation, the electron spin precesses in the XY plane with the laser field. Its spin changes rapidly in the laser field and the ${s}_x$ returns to its initial value at each cycle. The amplitude of oscillation coincides with the laser intensity, and the profile of ${s}_x$ is similar to the laser duration. However, the period of oscillation increases firstly and then decreases. If we defined the spin angle ${\theta =\mathrm{arctan}\left({s}_y/{s}_x\right)}$ , the quiver of electron spin is presented more clearly in Figure 3(c). The electron spin oscillates around the x-direction in the laser field and it causes the oscillation period of ${s}_y$ to be twice as that of ${s}_x$ . In addition, the period of oscillation decreases with time.

In order to investigate the dynamics of the electron spin, the contribution of the electromagnetic field to the precession frequency $\boldsymbol{\varOmega}$ is analyzed in detail. Equation (1) can be rewritten as $\boldsymbol{\varOmega} ={\boldsymbol{\varOmega}}_{\mathrm{a}}+{\boldsymbol{\varOmega}}_{\mathrm{T}}$ , where the following applies:

(3) $$\begin{align}{\boldsymbol{\varOmega}}_{\mathrm{a}}={a}_{\mathrm{e}}\frac{e}{m_{\mathrm{e}}}\left(\boldsymbol{B}-\frac{\gamma }{\gamma +1}\boldsymbol{v}\cdot \boldsymbol{B}\frac{\boldsymbol{v}}{c^2}-\frac{\boldsymbol{v}}{c^2}\times \boldsymbol{E}\right),\end{align}$$

and

(4) $$\begin{align}{\boldsymbol{\varOmega}}_{\mathrm{T}}=\frac{e}{m_{\mathrm{e}}}\left(\frac{1}{\gamma}\boldsymbol{B}-\frac{1}{\gamma +1}\frac{\boldsymbol{v}}{c^2}\times \boldsymbol{E}\right).\end{align}$$

Considering ${a}_{\mathrm{e}}\approx 1.16\times {10}^{-3}$ for electrons, the value of ${\boldsymbol{\varOmega}}_{\mathrm{a}}$ is much smaller than ${\boldsymbol{\varOmega}}_{\mathrm{T}}$ at the initial stage of acceleration, that is, ${a}_{\mathrm{e}}\ll 1/\gamma$ . With the increase of electron energy, the contribution of ${\boldsymbol{\varOmega}}_{\mathrm{a}}$ gradually increases and cannot be ignored, especially when $\gamma \simeq 1/{a}_{\mathrm{e}}\simeq 862$ . As presented in Figure 2(c), the electron energy is smaller than 10 MeV during the process of injection. Therefore, the contribution of ${\boldsymbol{\varOmega}}_{\mathrm{a}}$ can be ignored and $\boldsymbol{\varOmega} \simeq {\boldsymbol{\varOmega}}_{\mathrm{T}}$ can be used in the next analysis.

As shown in Figure 3(c), the electron precesses in the XY plane, which is caused by the part of ${\boldsymbol{\varOmega}}_z$ . It can be divided into three terms, ${\boldsymbol{\varOmega}}_{B_z}$ , ${\boldsymbol{\varOmega}}_{-{v}_y{E}_x}$ and ${\boldsymbol{\varOmega}}_{v_x{E}_y}$ . Figure 3(d) presents the history of ${\boldsymbol{\varOmega}}_z$ , and the contributions of the three different terms have been compared. It is found that the contribution of ${\boldsymbol{\varOmega}}_{-{v}_y{E}_x}$ can be ignored because the motion of electrons is nearly along the laser axis, where ${E}_x$ is nearly zero. The contribution of ${\boldsymbol{\varOmega}}_{v_x{E}_y}$ is comparable with ${\boldsymbol{\varOmega}}_{\boldsymbol{v}\times \boldsymbol{E}}$ . Following with time, the contribution of ${\boldsymbol{\varOmega}}_{\boldsymbol{v}\times \boldsymbol{E}}$ increases firstly and then decreases, which is consistent with the evolution of electron velocity. More importantly, the net contribution of ${\boldsymbol{\varOmega}}_z$ is nearly zero at one cycle, such that the effect of the laser field on electron spin can be ignored.

Figure 4 (a) The green dots denote the positions of the chosen accelerated electrons, which are marked in Figure 1(d). The magnetic field ${B}_z$ at the laser axis is presented as a black solid line. (b) The spectra of ${s}_x$ (black line) and the longitudinal position $x$ (magenta line) for the accelerated electrons at $330\ \mathrm{fs}$ . (c) The profiles of ${E}_y$ (red line) and ${E}_x$ (blue line) at the laser axis. (d) The spectra of the longitudinal velocity ${v}_x$ (red line) and transverse velocity ${v}_y$ (blue line).

As revealed in Figure 2(a), in the third stage, the electron spin does not change obviously. The distribution of the magnetic field at $330$ fs is shown in Figure 4(a). The accelerated electrons are denoted as green dots and they are located at the tail of the wakefield. The spectrum of ${s}_x$ indicates that the spin does not change obviously compared with the initial value, as presented in Figure 4(b), which means that the effect of the wakefield can be ignored. Furthermore, the width of the accelerated electron beam is $0.039\lambda$ (or $103.20$ as), where the full width at half maximum (FWHM) of the energy spectrum was used. The distribution of the electromagnetic field at the laser axis is presented in Figure 4(c) and the spectra of ${v}_x$ and ${v}_y$ are plotted in Figure 4(d). At the tail of the wakefield, ${B}_z$ and ${E}_y$ are nearly zero. Then their contribution to ${\boldsymbol{\varOmega}}_z$ can be ignored. Because of the small ${v}_y$ , the contribution of ${\boldsymbol{\varOmega}}_{-{v}_y{E}_x}$ can also be ignored, which is an essential difference compared with transverse injection. In the case of longitudinal injection, due to the motion of electrons close to the laser axis, the electron spin is affected by the laser field only. Since the laser field cannot depolarize the electrons, it is advantageous to obtain a polarized electron beam with a high energy in LWFAs. Finally, a nearly $115\ \mathrm{as}$ electron beam with $99\%$ polarization and average kinetic energy of $64\ \mathrm{MeV}$ is obtained at $400\ \mathrm{fs}$ .

In this work, the spin dynamics of injection electrons and the related physical mechanism have been briefly addressed based on a series of 2D simulations. Since the evolutions of the laser pulse and bubble regime are 3D nonlinear phenomena, it is necessary to use 3D simulation to analyze more accurately the characteristics of the electron beam, such as electron charge and transverse emittance^[ Reference Kalmykov, Beck, Yi, Khudik, Downer, Lefebvre, Shadwick and Umstadter ^{40 ,} Reference Tsung, Lu, Tzoufras, Mori, Joshi, Vieira, Silva and Fonseca ^{47 ]}.The polarization of electron beams could also be influenced by other effects, that is, beam loading and laser polarization. Besides, a plasma with 100% of the initial polarization rate was assumed since the state of pre-polarization has not been measured in the experiment. The polarization of the acceleration electron could be smaller than the result of simulation, even using the longitudinal injection mechanism. The details will be studied in our future work.

4 Summary

We have studied the generation of an electron beam, including its polarization properties in the bubble regime of an LWFA. By using a series of 2D PIC simulations, it is found that the depolarization process depends on the self-injection scheme. Compared with transverse self-injection, longitudinal self-injection is more suitable to generate an electron beam with higher polarization. The accelerated electrons move around the laser axis in the case of longitudinal injection. It causes the motion and the spin of the electrons to oscillate in the laser field, and the net influence of the laser field can be ignored. The contribution of the bubble field to the spin precession is also negligible, since the transverse electromagnetic field and the transverse velocity of the electrons are both very small. Ultimately, an attosecond electron beam with polarization of $99\%$ is obtained in the simulation. Our work helps to generate a polarized electron beam using the longitudinal self-injection scheme in a pre-polarized plasma and guide future experiments for producing ultra-short electron beams with high polarization.