Control of electron beam polarization in the bubble regime of laser-wakefield acceleration

Laser wakefield acceleration (LWFA) has made remarkable progress since it was first proposed by Tajima and Dawson in 1979 [1] and experimentally realized through the rapid advancement of laser technology via chirped-pulse amplification [2]. Since then, improved understanding of various schemes such as plasma beat wave acceleration [3], multiple laser pulses [4, 5], and self-modulated LWFA [6] have contributed to a series of milestones. Of particular note is the generation of quasimonoenergetic electron beams in the bubble regime [7], which triggered significant experimental progress and widespread interest [8–10]. In recent years, applications of wakefield acceleration have been actively pursued, such as synchrotron radiation sources [11, 12] and polarized particle beams [13–15].

Spin-polarized particle beams are widely used in nuclear and particle physics to study the interaction and structure of matter, and to test the standard model of particle physics [16, 17]. In particular, the structure of subatomic particles like protons or neutrons can be explored to get further insights into quantum chromodynamics [18] or to probe the nuclear spin structure [19]. Additionally, polarized particle beams are advantageous to achieve a deeper understanding of nuclear reactions [20], to investigate symmetry violations, to measure quantum numbers of new particles [16, 21–23], or to investigate molecular dynamics [24, 25]. In 2019, An et al proposed to map electromagnetic field structures of plasmas by using a spin-polarized relativistic electron beam [26]. In contrast, a first polarization measurement of few-MeV laser accelerated protons reported a negative result, i.e. no polarization build-up during the acceleration process with an unpolarized foil target [27]. Recently, polarized multi-GeV proton beams produced by ultra-intense laser interaction with pre-polarized target were studied via simulations [28]. At present, the preparation of polarized electron beams mainly relies on spontaneous polarization in the magnetic fields of storage rings due to the emission of spin-flip synchrotron radiation, i.e. the well-known Sokolov–Ternov effect [29, 30]. This technique requires conventional particle accelerators that are typically very large in scale and budget [29].

The acceleration of polarized electron beams by means of laser-driven acceleration promises to be cost-efficient and highly effective. Despite many advances mostly on the theoretical side, several principal issues need to be addressed, for example: (i) is it possible to alter the polarization of an initially unpolarized target through interaction with relativistic laser pulses [31–35]? or (ii) are the spins so inert during the short acceleration period that a pre-polarized target is required [13, 14, 36–38]? Following the work by Hützen et al [39], Wen et al [13] have proposed to generate high-current polarized electron beams in the interaction of an ultra-intense laser pulse with a pre-polarized gas plasma, which is produced through photo-dissociation by a circularly polarized ultra-violet (UV) laser pulse [40]. The work of Vieira et al showed that spin is depolarized mainly in the injection phase [41].

Previous works show that the distribution of the electromagnetic fields is affected by the accelerating bubble geometry [42–44]. It is thus likely that the self-injection process can be affected by the bubble shape [45, 46]. Moreover, the work of Qu et al [47] indicates that the frequency of THz radiation generated by the shell electrons also depends on the bubble shape. In this paper, the evolution of the electron beam polarization injected into various shapes of ellipsoidal bubbles is discussed in detail, and it is found that the polarization of electron beams can be controlled by adjusting the bubble geometry. The results of our analysis are highly relevant to experimental implementations of polarized electron beams, such as those planned at the European EuPRAXIA facility [48].

By choosing appropriate laser and plasma parameters, different shaped wakefield bubbles can be achieved [45]. A series of 2.5D particle-in-cell (PIC) simulation with the code EPOCH [49] was carried out to analyse the bubble geometry. The laser propagates in the x-direction with linear polarization in the y-direction and a Gaussian envelope

$$\begin{equation}E={E}_{0}\frac{{w}_{0}}{w(x)}\mathrm{exp}\left(-\frac{{y}^{2}+{z}^{2}}{{w}^{2}(x)}\right)\mathrm{exp}\left(-\frac{{(kx-\omega t)}^{2}}{{(0.5\tau )}^{2}}\right)\mathrm{cos}(\varphi ),\end{equation} \tag{ 1 }$$

where, $w(x)={w}_{0}{[1+{(x-{x}_{0})}^{2}/{z}_{\mathrm{R}}^{2}]}^{0.5}$, with laser waist w₀ = 10λ, pulse duration τ = 21 fs, laser intensity a₀ = eE₀/m_e ωc = 20 and wavelength λ = 800 nm. The x₀ = 30λ is the position of the laser waist, and ${z}_{\mathrm{R}}=\pi {w}_{0}^{2}/\lambda$ is the Rayleigh length. The vacuum length was 30λ and the laser beam was focused at the left edge of plasma. The simulation box is 140λ(x) × 100λ(y) with resolution dx = λ/32 and dy = 5 dx. There were 16 pseudo-particles per cell. We define the aspect ratio η = R_⊥/R_|| to describe the shape of bubble, where R_|| and R_⊥ are the longitudinal and transverse radii respectively. Different R_|| and R_⊥ can be obtained by changing the plasma density for the same laser system [45]. The simulation result shows that the bubble shape changes from prolate to oblate spheroid with increasing plasma density. Following this definition, η < 1 indicates a prolate spheroid, η = 1 indicates a sphere, and η > 1 indicates an oblate spheroid. For an initial plasma density of n₀ = 0.011n_c, we find that R_|| = 14.38λ and R_⊥ = 13.13λ. Consequently, the aspect ratio η = 0.91 < 1, represents a prolate spheroidal bubble.

The electromagnetic field distribution in a spherical bubble has already been theoretically and numerically studied [50–52]. Based on the work of Li et al [44], the electromagnetic field of an ellipsoidal bubble can be written as

$$\begin{equation} {E}_{x}=\frac{{\eta }^{2}}{{\eta }^{2}(1-{v}_{\text{b}}^{2})+2}\xi ,\end{equation} \tag{ 2a }$$

$$\begin{equation}{E}_{y}=\frac{2-{\eta }^{2}{v}_{\text{b}}^{2}}{2{\eta }^{2}(1-{v}_{\text{b}}^{2})+4}y,\end{equation} \tag{ 2b }$$

$$\begin{equation}{E}_{z}=\frac{2-{\eta }^{2}{v}_{\text{b}}^{2}}{2{\eta }^{2}(1-{v}_{\text{b}}^{2})+4}z,\end{equation} \tag{ 2c }$$

$$\begin{equation} {B}_{x}=0,\end{equation} \tag{ 2d }$$

$$\begin{equation}{B}_{y}=\frac{{v}_{\text{b}}{\eta }^{2}}{2{\eta }^{2}(1-{v}_{\text{b}}^{2})+4}z,\end{equation} \tag{ 2e }$$

$$\begin{equation}\enspace {B}_{z}=-\frac{{v}_{\text{b}}{\eta }^{2}}{2{\eta }^{2}(1-{v}_{\text{b}}^{2})+4}y,\end{equation} \tag{ 2f }$$

where ξ = x − v_b t, ${v}_{\text{b}}=\sqrt{1-{\gamma }_{\text{b}}^{-2}}$ is the bubble phase velocity and ${\gamma }_{\text{b}}=0.45\sqrt{{n}_{\mathrm{c}}/{n}_{0}}$ [50, 53]. To confine the field distribution inside the bubble, a modified factor $f(r)=\left[\mathrm{tanh}({R}_{{\Vert}}/d-r/d)+1\right]/2$ was used, $r=\sqrt{{\xi }^{2}+({y}^{2}+{z}^{2})/{\eta }^{2}\left.\!\right)}$ and d is the width of the electron sheath. In this work, d = 0.5 was used. Here, we used dimensionless units, by normalizing the length to k_p, the velocity to c, the electron density to n₀, the electric field to m_e cω_p/e, and the magnetic field to m_e ω_p/e.

Initially, the electron is at rest in front of the bubble with a position (x₀, y₀). Considering a fully polarized plasma, the electron spin at initial time is aligned with in the bubble propagation direction (x). To follow the trajectory of an electron in the bubble, a fourth-order Runge–Kutta method was adopted to numerically solve the relativistic Newton–Lorentz equation, $\mathrm{d}\boldsymbol{P}/\mathrm{d}t=-e\left[\boldsymbol{E}+(\boldsymbol{P}/\gamma )\times \boldsymbol{B}\right]$, where $\gamma =1/\sqrt{1-{v}^{2}/{c}^{2}}$ is the relativistic factor. Meanwhile, the spin precession of an electron in the electromagnetic field was calculated according to the Thomas–Bargmann–Michel–Telegdi (TBMT) equation [29] $\mathrm{d}\boldsymbol{s}/\mathrm{d}t=\left({\mathbf{\Omega }}_{T}+{\mathbf{\Omega }}_{a}\right)\times \boldsymbol{s}$ with

$$\begin{equation}{\hspace{-48pt}\mathbf{\Omega }}_{T}=\frac{e}{{m}_{\mathrm{e}}}\left(\frac{1}{\gamma }\boldsymbol{B}-\frac{1}{\gamma +1}\frac{\boldsymbol{v}}{{c}^{2}}\times \boldsymbol{E}\right),\end{equation} \tag{ 3a }$$

$$\begin{equation}{\mathbf{\Omega }}_{a}={\boldsymbol{a}}_{\mathrm{e}}\frac{e}{{m}_{\mathrm{e}}}\left(\boldsymbol{B}-\frac{\gamma }{\gamma +1}\frac{\boldsymbol{v}}{{c}^{2}}(\boldsymbol{v}\cdot \boldsymbol{B})-\frac{\boldsymbol{v}}{{c}^{2}}\times \boldsymbol{E}\right),\end{equation} \tag{ 3b }$$

where a_e ≈ 1.16 × 10⁻³ is the anomalous magnetic moment of the electron and the spin is normalized to | s | = 1. The Boris-rotation method was adopted to numerically solve the TBMT equation [54].

Owing to the azimuthal symmetry of the wake magnetic field, the electron orbit and spin can be placed in the same plane [13, 50], so for simplicity the XY plane was adopted for this calculation. The simulation results of three typical cases are displayed in figure 1. The aspect ratio of these bubbles were η = 0.91 in case 1, η = 1.00 in case 2 and η = 1.09 in case 3, representing prolate spheroidal, spherical and oblate spheroidal bubbles, respectively. The trajectory and the spin orientation of a typical electron are presented in figures 1(a1)–(a3). The motion of electrons can be divided into four stages: (i) t < t_I, the electron does not feel the bubble field; (ii) t_I < t < t_II, the electron is located on the bubble shell and its spin rotates clockwise, such that s_y decreases from 0 to almost −1 as shown in figure 1(b1); (iii) t_II < t < t_III, the electron reaches the tail of the bubble and its spin rotates counter-clockwise, and s_y increases as revealed in figure 1(b1); (iv) t > t_III, the electron is captured in the bubble and its spin procession slows down.

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The degree of electron spin precession differs for the three cases. The electrons can stay longer in the second stage with increasing η, which means that the degree of clockwise rotation is largest for case 3 (η > 1), where s_x approaches −1 as shown as figure 1(b3). During the third stage, the degree of spin precession in the three cases is also different. In case 1 (η < 1) and case 2 (η = 1), s_x increases to 1 (its initial value) and decreases afterwards, while s_y changes from negative to positive. In case 3 (η > 1), the counter-clockwise spin precession is significantly smaller. Thus, s_x increases from −1 to a value near 0 and s_y remains negative.

During the fourth stage t > t_III the electrons remain in the acceleration phase and oscillate around the laser axis. The s_x and s_y values oscillate around their values at t = t_III, as shown in the inset of figures 1(b1)–(b3). Compared to the strong precession during the earlier stages, this spin variation can be ignored. This means that the spin procession for electrons mainly occurs during self-injection, and does not change significantly thereafter in the acceleration stage, and this result is consistent with the study of Wen et al [13]. When the electron arrives at the centre of bubble, the acceleration process is terminated and the spin direction is similar to its value at t = t_III, as revealed as insets of figures 1(b1)–(b3). At this time, the final spin orientation is forward in case 1 (η < 1), case 2 (η = 1) and backward in case 3 (η > 1). We can thus conclude that the spin precession is strongly affected by the bubble geometry. Moreover, during the electron injection, the clockwise spin rotation during the second stage is partly balanced by the counter-clockwise rotation of stage three. As a consequence, a particular ellipsoidal bubble shape can be chosen for which the electron spins can be restored to their initial orientation at t = t_III and maintain dynamic stability over the acceleration phase.

The net polarization of a particle beam is defined as $P=\sqrt{{\langle {s}_{x}\rangle }^{2}+{\langle {s}_{y}\rangle }^{2}+{\langle {s}_{z}\rangle }^{2}}$, where ⟨s_i ⟩ is the average value in each direction. This definition is a statistical average for an electron bunch. An accelerated electron beam can be mimicked by changing the initial position y₀ for a set of test particles. The trapping cross-section has been studied using the same method in a previous study [45]. The distributions of s_x and s_y as a function of initial position y₀ at the end of acceleration stage are displayed in the figures 1(c1)–(c3). The electrons are also distinguished according to their final energy, denoted by the color scale. In 3D geometry, the injected electrons (γ > 50) are distributed in a ring. The accelerated electrons (red dots) are distributed from r_min to r_max in these three cases, which means that the electron charge is affected by the bubble geometry. More importantly, the spin direction of the accelerated electrons also depends on the bubble shape. The value of ⟨s_x ⟩ of the electron beam can be calculated as,

$$\begin{equation}\langle {s}_{x}\rangle =\frac{{\sum }_{i}^{N}{s}_{xi}}{N}=\frac{{\sum }_{{r}_{\mathrm{min}}}^{{r}_{\mathrm{max}}}{s}_{xi}\cdot 2\pi {y}_{0}\delta {y}_{0}}{{\sum }_{{r}_{\mathrm{min}}}^{{r}_{\mathrm{max}}}2\pi {y}_{0}\delta {y}_{0}},\end{equation} \tag{ 4 }$$

with δy₀ = 0.01 in our simulation. Considering the azimuthal symmetry of the bubble field, we have ⟨s_y ⟩ = ⟨s_z ⟩ = 0 and $P=\left\vert \langle {s}_{x}\rangle \right\vert$. The polarization amounts to P = 0.55, 0.70 and 0.09 for cases 1 (η < 1), 2 (η = 1) and 3 (η > 1) from figure 1, respectively.

To verify these findings we present results of 3D PIC simulations with a modified version of the EPOCH code [49], in which the TBMT equation for the study of electron and ion spins was implemented [28]. The laser parameters are the same as in figure 1. The initial plasma density is n₀ = 0.006n_c and n₀ = 0.015n_c, respectively. The size of the moving window is 60λ(x) × 40λ(y) × 40λ(z) with resolutions dx = λ/32 and dy = dz = 5 dx, and 8 pseudo-particles per cell. The pre-polarized plasma can be produced using UV polarization method [40]. While the initial rate of electron polarization in the laser–plasma acceleration structure is not yet totally understood, we assume for simplicity that this is 100%, since we are interested in the evolution of the polarization during the acceleration phase.

As shown as in figures 2(a) and (b), the simulated bubble shapes are prolate (n₀ = 0.006n_c) and oblate (n₀ = 0.015n_c), respectively. The distribution of $\left\langle {s}_{x}\right\rangle$ for the two cases are presented in figures 2(c) and (d). The polarization of electrons, located at the tail of bubble, is positive (green arrows) for the prolate bubble (η = 0.92), while it is negative for the oblate case (η = 1.21). Thus, the results of our 3D PIC simulations verify the results of the test-particle simulations. Here, the electrons located at the shell of the bubble were analysed because of depolarization process mainly occurs before the electron arrives at the rear wall of the bubble. Moreover, the injection process and the evolution of bubble are also affected by the laser, effects which are not included in the earlier test-particle analysis. To further illustrate how the electron-beam polarization variation depends on the bubble geometry, we did a series of 2.5D PIC simulations with initial plasma densities ranging from n₀ = 0.005n_c to n₀ = 0.025n_c and fixed laser intensity a₀ = 20. Other parameters were the same as in figure 1. The data of bubble geometry were substituted in the single electron dynamic simulations. The evolution curves of ⟨s_x ⟩ with time are shown in figure 3(a). Here, bubbles with six typical aspect ratios (η) were selected. We found that the ⟨s_x ⟩ of the electron bunch first decreases then increases in every case, which is similar to the spin dynamics of a single electron—figure 1. When η increases from 0.86 to 0.91, ⟨s_x ⟩ decreases. When η > 1, ⟨s_x ⟩ also decreases with increasing η and even goes negative in an oblate spheroidal bubble. Its value is maximized in a spherical bubble (η = 1) compared with the two cases with η = 0.91 and η = 1.09. This means that the polarization of an accelerated electron bunch can be preserved when the bubble shape is nearly spherical.

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In order to check the persistence of this phenomenon, further 2.5D PIC simulations were carried out for different laser parameters. The dependence of ⟨s_x ⟩ on the various aspect ratios η after acceleration with laser intensities a₀ = 15, 20, 30 are presented in figure 3(b). It is found that when the bubble is nearly spherical, the electron-beam depolarization is always minimal. It can even be zero as shown in figure 3(c), where n₀ = 0.20n_c, a₀ = 30 and η = 1.04, respectively. In the case of a₀ ⩾ 20, the value of ⟨s_x ⟩ is negatively correlated with η when η < 0.9, which arises from the smaller number of electrons injected into the bubble.

Finally, the mechanisms governing the spin dynamics in the bubble fields are considered. In figure 4 the evolution of the rotation frequency is analysed for three cases: (a) η = 0.91 (case 1 in figure 1), (b) η = 1.04 (case in figure 3(c)), where ⟨s_x ⟩ = 0.99; (c) η = 1.09 (case 3 in figure 1). The terms of equation (3) are separated into Ω_E and Ω_B for studying their individual contributions to the rotation frequency. Note that |Ω_a | ≪ |Ω_T |, and that Ω_T (solid black line) can be divided to Ω_B = e B _z /m_e γ (solid blue line) and Ω_E = −e( v × E )/[m_e(γ + 1)c²] (solid red line).

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During the second stage (t_I < t < t_II), the electrons located near (ξ ≈ 0, y ≈ r_⊥) mainly feel E_y and B_z as given in equation (2). Then, ${\mathbf{\Omega }}_{{B}_{z}}$ and the v_x ⋅ E_y term in Ω_E make the largest contribution to Ω_T . Since the electrons cannot obtain enough energy, this results in $\left\vert {{\Omega}}_{{B}_{z}}\right\vert > \left\vert {{\Omega}}_{E}\right\vert$ and the spins rotate clockwise. Assuming a bubble velocity v_b = 1, we obtain B_z ∝ (η² y)/4 and E_y ∝ [(2 − η²)y]/4, based on equation (2). With increasing of η, B_z increases and E_y decreases, then Ω_T increases. Meanwhile, the electrons stay longer in this phase, which allows the contribution of B_z to dominate with increasing η, and the degree of clockwise rotation for electron spin is positively correlated with η.

During the third stage (t_II < t < t_III), electrons arrive at the tail of bubble (ξ ≈ R_⊥, y ≈ 0), where E_y ≈ 0 and B_z ≈ 0. Here the electrons mainly feel E_x , then the part of v_y ⋅ E_x in Ω_E becomes the dominant contribution, which results in counter-clockwise spin rotation. At the tail of bubble E_x ≈ η² ξ/2 under the assumption v_b = 1. Considering R_⊥ is similar for bubbles with different shapes and ξ = R_||, we obtain ${E}_{x}\approx {R}_{\perp }^{2}/(2{R}_{{\Vert}})$. With increasing η, R_|| decreases and the part of v_y ⋅ E_x in Ω_E increases. Moreover, considering that the times when electron reach the tail of bubble are different, then the contribution of ${{\Omega}}_{{B}_{z}}$ is different for these three cases, which results in similar values of Ω_T .

For the overall process, the spin rotation is the sum of the clockwise rotation during the second stage and the counter-clockwise rotation during the third stage. With increasing η, the degree of clockwise rotation increases and the degree of counter-clockwise rotation stays roughly constant. Clockwise rotation dominates in the prolate bubble, whereas the counter-clockwise rotation is prevalent in an oblate bubble. More importantly, the two precessions can cancel each other in a bubble with a certain value of η. As shown as figure 4(b), an accelerated electron beam with no net depolarization can be produced when η = 1.04.

In this work, we investigated the motion of electrons in a pre-polarized plasma, where the initial polarized direction is aligned with the propagation direction of the laser. In this schematic, the s_y and s_z stay zero because of the symmetric precession owing to the azimuthal symmetry of the electromagnetic field in bubble. Although the study of s_x is not a generalized case, the polarization is always preserved for nearly spherical bubble shapes with changing the initial pre-polarized direction.

Previous work has shown that the bubble shape strongly affects the self-injection process, and that this is maximized for near-spherical geometries [45]. The shape can furthermore change over time due to nonlinear propagation beam-loading effects, but fortunately it appears that a spherical bubble will maximize both charge and spin preservation, since the latter is most influence by the period prior to injection. Moreover, the electron polarization also depends on the injection mechanism. Different trajectories of injected electrons cause the disparity of polarization. The further work is necessary to determine which kind of self-injection mechanism is the best choice for a highly polarized electron beam.

In summary, the depolarization of accelerated electron bunches has been studied through test-particle dynamics during the interaction of a high-intensity laser with a longitudinally pre-polarized plasma. Spin rotation occurs mainly during the self-injection process. As a consequence, the bubble geometry has a strong influence during this phase, since it determines the electromagnetic field distributions. It is also found that polarization is preserved for nearly spherical bubble shapes. In contrast, non-spherical bubble shapes lead to strong depolarization. These findings should help to choose suitable laser–plasma parameters for producing a polarized electron beam using a pre-polarized plasma. Moreover, the case of a transversal polarization plasma, which is better accessible experimentally, will be investigated in the near future.

The 3D PIC simulations were carried out on the JURECA supercomputer at Jülich Supercomputing Centre, in particular through the computing time projects JZAM04 and LAPIPE. This work was supported by Germany Postdoctoral Council and the Helmholtz Centre (Grant No. 20191016) and China Postdoctoral Science Foundation (Grant No. 2018M641993). The work of MB was carried out in the framework of the Jülich Short-Pulse Particle and Radiation Center [55] and was supported by the Accelerator Technology Helmholtz Infrastructure consortium ATHENA. This work was also supported by the Strategic Priority Research Program of Chinese Academy of Sciences (Grant No. XDA25050100), the National Natural Science Foundation of China (Nos. 11804348, 11775056, 11975154, and 11991074), and the Science Challenge Project (No. TZ2018005).

The data generated and/or analysed during the current study are not publicly available for legal/ethical reasons but are available from the corresponding author on reasonable request.