The concept of laser plasma wakefield accelerators (LWFAs) was first proposed by Tajima and Dawson^[1^]. Over the past few decades, LWFAs have become increasingly mature and have recently exhibited stable^[2^], low divergence (milliradians)^[3^] and energy tunable^[4^] electron bunches with a charge at the picocoulomb level^[5^]. An electron beam is most efficiently produced in the ‘bubble’ regime^[6^], which requires laser pulses that are both intense (normalized vector potential a₀ > 1) and short (pulse duration $\tau \le 2\pi c/{\omega}_{\mathrm{p}}$, where ω_p is the plasma frequency). The ponderomotive force of these laser pulses propagating in an underdense plasma pushes the background electrons away from the high-intensity regions and drives a relativistic plasma wave. The wave consists of a string of ion cavities (also referred to as ‘bubbles’), and the electrons trapped inside can be accelerated by the electrostatic field set up by the separation of electrons and ions. Moreover, these accelerated electrons will also oscillate in the plasma wakefield with betatron frequency ${\omega}_{\beta }={\omega}_{\mathrm{p}}/\sqrt{2\gamma }$ and emit synchrotron radiation^[7^]. There are several methods of electron capture, including ponderomotive force injection^[8^], colliding laser pulse injection^[9^], plasma density gradient injection^[10^] and transverse self-injection^[11–13^]. With these methods, the injected direction of electrons is hard to control, and these injection processes are not easy to achieve in experiment. In contrast, another method is ionization-induced injection^[14–16^], which is used in this study. Owing to the different ionization potential levels of high Z atoms^[15^][17–19^] (such as nitrogen), the outer shell electrons can be ionized instantaneously by the rising edge of the laser pulses (98 eV for N⁺⁵ requires an intensity of ~2×10¹⁶ W/cm²) and pushed away. The inner shell electrons (552 eV for N⁺⁶ requires an intensity of ~1×10¹⁹ W/cm²) are ionized close to the peak of the laser intensity envelope. These ionized electrons will appear at rest and slip backward relative to the laser pulses and the wake. The electrons are trapped after gaining enough energy from the longitudinal electric field of the first period of the wake to move at the phase velocity of the wake and will gain additional energy^[15^]. Ionization injection is a more controllable method, regarded particularly for its stability^[14^][20^][21^]. Moreover, these trapped electrons mainly oscillate along the direction of laser polarization in the ion cavity.

Owing to the fact that the plasma wakefield has a transverse electric field of tens of gigavolts per metre and the radius of a plasma bubble is limited to several micrometres^[22^], it is difficult to find a strong enough external electric field or magnetic field for controlling the transverse motion of an electron beam in a bubble, especially helical motion. Moreover, Luo et al.^[23^] simulated and acquired the helical motion of an electron beam and elliptically polarized radiation by laser pulses incident at a skew angle to the axis of the plasma waveguide, but this method was hard to achieve in experiment. Thaury et al.^[24^] used one laser pulse to drive an asymmetrical plasma wakefield and another pulse of colliding injection to achieve the helical motion of an electron beam, but the process of colliding injection is not easy to control and has low repetition probability. In addition, Chang et al.^[25^] also generated the helical motion of an electron beam and circularly polarized radiation by using a petawatt-class circularly polarized laser pulse interaction with near-critical density plasma, but the divergences of the electron beam and radiation were very large.

In this paper, we propose a simple all-optical method to control the transverse motion of the ionization injected electron beam by changing the evolution of the plasma wakefield transversal distribution. We also use three-dimensional particle-in-cell (3D-PIC) simulation to verify our experimental results and analyze the dynamics of electron transverse motion.

The 3D-PIC simulations were carried out using KLAPS code^[31^{32], and the tunnel-ionization model was adopted for field ionization. The simulation box size was 50μm×60μm×60μm with 1500×450×450 cells in the x-, y- and z-directions, respectively, and one cell contained two macro particles. In addition, a third-order time interpolation for the magnetic field was used in the simulation. P-polarized (y-direction) 800 nm laser pulses with a₀ = 1.7 were focused to a radius of 15 μm at x = 50 μm behind the front edge of the nitrogen gas. The pulse had a Gaussian transverse profile and sine-squared longitudinal shape with pulse duration of 30 fs (FWHM). The neutral nitrogen longitudinal profile had a 100 μm up-ramp followed by a 2 mm long plateau with uniform density of 6×1017 cm−3.}

If a laser pulse with the focus situation shown in Figure 1(b) is propagating and self-focusing in the plasma, the shape of the laser spot will change from a slanted 45° ellipse to a circle and then to a slanted 135° ellipse. The process of laser spot shape change will continue until the laser pulses cannot sustain self-focusing in the plasma. Therefore, in order to study the process of this laser pulse propagation and self-focusing in the plasma, and the influence of the asymmetrical laser focus on the plasma wakefield acceleration, the asymmetrical laser intensity distribution in front of the focal plane was set according to the intensity distribution measured in the experiments (as shown in Figure 1(b)), and the electric-field intensity distribution is expressed as

$$\begin{align}E\left(x,y,z\right)&={E}_0\cdot \sqrt{w_0/ \text{rs}(x)}\cdot\exp \left\{\!-\left[\!\frac{{\left(y\cdot \cos \theta -z\cdot \sin \theta\! \right)}^2}{2}\right.\right.\notag\\&\quad\qquad\;\;\left.\left.+\frac{{\left(y\cdot \cos \theta +z\cdot \sin \theta \right)}^2}{0.5}\!\right] \middle/ \text{rs}{(x)}^2\!\right\},\end{align}$$ (1)

To compare the influence of the asymmetrical plasma wakefield on the transverse motion of the accelerated electron beam in the plasma bubble, the phase spaces of P_y-z and P_z-y corresponding to the electrons of Figures 5(c) and 5(d) are shown in Figures 5(e) and 5(f) and Figures 5(g) and 5(h), respectively. Electrons in the symmetrical plasma wakefield have more momentum in the y-direction than in the z-direction, as shown in Figures 5(g) and 5(h), resulting in the shape of the electron beam tending to be an ellipse, as shown in Figure 5(d). However, for the asymmetrical plasma wakefield, the maximum momentum in the y-direction is approximately equal to that in the z-direction, as shown in Figures 5(e) and 5(f), and a majority of electrons have momentum in the z-direction larger than that in the case of the symmetrical plasma wakefield, as show in Figures 5(e) and 5(g). Therefore, a majority of electrons have experienced a strong force in the z-direction in the asymmetrical transversal wakefield.

In conclusion, a simple method of controlling the transverse motion of an electron beam in a plasma bubble is presented. Laser pulses with a power of 100 TW drive a plasma wakefield and accelerate an electron beam in the regime of ionization injection. The transverse motion of the accelerated electron beam can be controlled by changing the intensity distribution of the laser focus by adjusting the posture of the OAP mirror. When the shape of the laser focus changes from circular to slanted elliptical, the geometrical symmetry of the transverse force in the plasma bubble is changed, resulting in the motion of the electron beam changing from undulating to helical. Moreover, the profile of the electron beam also changes with the laser focal spot’s shape. The experimental results were verified by 3D-PIC simulations.

This method is expected to conveniently control the transverse motion of an electron beam in a plasma wakefield and to generate synchrotron radiation with orbit angular momentum.