## 3 Results
### Spatial and Angular Distribution  
The laser delay time was set at 5 fs to maximize the electron radiation power. To investigate the characteristics of the angular and the spatial distribution of electron radiation, we have plotted the normalized radiation power distribution of the interaction between the electron and the pulse at a 5 fs delay time. As obtained from Figure 3, the radiation spectrum exhibits a comet-like shape, and the peak of the radiation power is located at the head of the comet. Figure 3: The angular distribution of the electron radiation power when the delay time is 5 fs. It can be observed from Figure 3 that with the increase in \(b_{0}\), the peak power of electron radiation will increase. Since the extent of the increase in the peak value of the electron radiation power with the increase in the beam waist radius is significantly reduced when \(b_{0}\geq 6\), especially when \(b_{0}\geq 10\), we have provided Figure 4 to clearly demonstrate the variation of the peak power value and its corresponding polar angle value with \(b_{0}\). The relationship of radiation power is as follows [26]:  
\[P(t)\propto\left[\gamma^{8}|d_{t}u_{z}|^{2}\right]_{\tau} \tag{13}\]  
Among them, \(\gamma\) represents the electron energy, and \(d_{t}u_{z}\) denotes the acceleration of the electron in the z-axis direction. Then, the peak of spatial radiation can be expressed as [25]:  
\[\left[\frac{dP(t)}{d\Omega}\right]_{peak}\propto\left[\gamma^{8}|d_{t}u_{z}|^ {2}\right]_{max} \tag{14}\]  
When the z-axis velocity of the electrons attains its maximum, namely when the z-axis acceleration assumes a minimum value, the spatial radiation attains its peak. By observing Figure 5, the peak value of the electrons' velocity in the z-axis direction does not undergo a substantial change. However, as the \(b_{0}\) increases, the variation period of the electrons' z-axis velocity, namely the number of action cycles, keeps on increasing. The electron and the laser pulse will interact for multiple cycles, and the radiation will be superimposed multiple times throughout the entire process. As \(b_{0}\) increases, the number of superimposed radiation cycles keeps rising, and the total radiation peak keeps escalating. When \(b_{0}\) increases to the non-tightly focused state, the total radiation peak remains nearly invariant. Figure 4: The variation of peak radiation and radiation polar angle \(\theta\) under different waist radii. The radiation distribution characteristics of electrons can be contemplated from two aspects: the polar angle \(\theta\) and the azimuth angle \(\phi\). It can be deduced from Figure 3 that the power radiation of electrons is concentrated in the part where \(\phi=0\). When \(\phi\) deviates from 0, regardless of the value of the polar angle \(\theta\), the electron power radiation will rapidly decay to 0. This is because the oscillation direction of the linearly polarized pulsed laser is in the XOZ plane, and the oscillation of electrons after colliding with it also occurs in the XOZ plane. Consequently, the electron radiation is concentrated in the XOZ plane. Simultaneously, observe \(\theta\) in Figure 3. The radiation of electron power is primarily concentrated within the polar angle \(\theta\) range from 60\({}^{\circ}\) to 90\({}^{\circ}\). Under the circumstance of tight focusing, as \(b_{0}\) increases, the \(\theta\) at which the electron power radiation reaches its maximum gradually diminishes. The variation rule of the peak of electron power radiation with respect to \(\theta\) can be derived from Figure 4. When the laser pulse is in a non-tightly focused situation, the \(\theta\) at which the maximum power occurs remains essentially invariant. Examine Figures 3 and 5 and conduct an analysis of the peak shift phenomenon in terms of the velocity and acceleration of electron movement. It can be derived from Figure 5 that there exists a phenomenon of periodic and sudden reduction in the velocity of the electron in the x-direction.