## 3 Results and Discussion  
### Generation of Attosecond Micro-Bunched Electron Beams from LWFA  
Nonlinear trapping in laser wakefield acceleration significantly influences the quality of the final electron beam. Ionization-induced injection [34] is favored in LWFA due to its controllability, enabling the generation of a high-brightness, stable, and tunable electron beam. According to the ionization injection dynamics [35], the relative longitudinal position \(\xi\equiv x-v_{\phi}t\) of the trapped electron is primarily determined by the initial \(\xi_{i}\) as follows:  
\[\xi=-\sqrt{4+\xi_{i}^{2}+r_{i}^{2}-r^{2}-4[\gamma-(v_{\phi}/c)p_{x}]} \tag{2}\]  
where \(v_{\phi}\) is the phase velocity of the wakefield, \(p_{x}\) and \(r\) are the longitudinal momentum and the transverse position of the trapped electrons, and \(r_{i}\) and \(\xi_{i}\) are the initial transverse and relative longitudinal position of the trapped electrons where they are ionized, respectively. Here, parameters of positions are normalized to \(c/\omega_{p}\), velocities to \(c\), and the momenta to \(m_{e}c\), with \(\omega_{p}\) is the plasma frequency, and \(m_{e}\) is the electron rest mass. In general, if \(r_{i},r\ll 1\) and \(\gamma-(v_{\phi}/c)p_{x}\ll 1\), the longitudinal position \(\xi_{f}\) of trapped electron is mainly determined by the initial \(\xi_{i}\), as follows:  
\[\xi_{f}\approx-\sqrt{4+\xi_{i}^{2}} \tag{3}\]  
which means the initial modulation of \(\xi_{i}\) can be nonlinearly mapped to \(\xi_{f}\). Due to the phase-dependent ionization within a linear polarized laser field, the initial distribution of the electrons \(\xi_{i}\) will have a strong modulation at \(2k_{0}\) during the sub-cycle interaction with the laser field, where \(k_{0}\) is the wavenumber of the laser pulse. As a result, a bunched beam with a periodically modulated current profile can be produced. Any spreads in \(r_{i}\), \(r\) and \(\gamma-(v_{\phi}/c)p_{x}\) will broaden the \(\xi\) distribution, called phase mixing during the injection. Previous simulations have shown that it is crucial to suppress phase mixing during the injection and acceleration process for maintaining the initial micro-bunched features [36]. Figure 2 shows the injection and acceleration of an attosecond pre-bunched electron beam. The snapshots of the trapped charge are shown in Figure 2a, where the injected electrons exit the gas mixture at \(x=0.3\) mm. It indicates that electrons are immediately micro-bunched after injection. Fine bunch structures and the on-axis density profiles show strong periodic modulation in the \((\xi_{f},y)\) space and a comb-like density bunch profile is pronounced. Figure 2b displays the initial phase space distribution \((\xi_{i},p_{x})\) of the injected electrons when they are ionized in the electric field of the driving laser. Since the laser intensity is just above the ionization threshold of \(N^{6+}\), electrons are ionized off-peak of the laser electric field and born with the initial coordinates \(\xi_{i}<0\), as shown in Figure 2b. Furthermore, by using a linearly polarized laser pulse, the longitudinal momentum distribution oscillates twice per laser period. Electrons ionized at different  
times with the same initial \(\xi_{i}\) will be nonlinearly mapped to the same position \(\xi_{f}\). When they are trapped by the wake, the comb-like on-axis density bunch profile is evident in Figure 2a. The average interval between the peaks of the beam density profile in Figure 2a is approximately 0.97 fs, and the average bunch length(rms) of micro bunches is about 118 as. Once the electron beam exits the gas mixture with the density-modulated structure, it continues to be accelerated. In Figure 2c, we show the energy spectra of the entire beam energy during the acceleration. Initially, the entire beam shows increasing absolute energy spread as electrons are trapped at the rear of the bubble, transitioning from the negative slope region to the positive slope of the wakefield. Electrons at the front experience a larger longitudinal accelerating field.