## 3 Results
### The Electron Deflection Trajectory in a Gaussian Pulse  
Theoretical analysis indicates that in order to investigate the influence of the beam waist radius of the Gaussian pulse on the electron collision trajectory, it is necessary to determine the relative maximum delay time of electron radiation. It can be concluded from Figure 2 that the radiation generated by the electron is the maximum when \(t_{0}=5\) fs. To study the electron trajectory, it is necessary to divide and decompose the electron movement stage. In the incident stage, the electron moves uniformly and linearly from -x-axis to +x-axis, and the Gaussian pulse moves uniformly and linearly from -z-axis to +z-axis. The two cross and collide around the origin of the Cartesian coordinate system. In the oscillation and distortion stage, the electron is subjected to the action of the linearly polarized Gaussian pulse and continuously performs oscillation motion in a single plane. Within the oscillation plane, the amplitude of the electron's movement initially rises and subsequently decreases. In the exit stage, the electron exits the influence range of the Gaussian pulse and is no longer affected by the electromagnetic field. The exit direction is biased towards the positive half-axis of the x-axis, and there is also a certain deviation in the positive and negative half-axes of the z-axis. Even when electrons interact with Gaussian pulses of the identical beam waist radius, variations in the delay time of laser emission will give rise to distinct trajectory characteristics. This is because the velocity of electrons is extremely high, approximating the speed of light. The movement of electrons within the Gaussian pulse only undergoes a very brief period and is only influenced at a particular section of the Gaussian pulse. Regulating the emission delay time and the beam waist radius of the Gaussian pulse enables the acquisition of some distinct patterns. When the delay time of the Gaussian pulse is set at \(t_{0}\) = 5 fs, the electron radiation reaches its maximum, and the interaction between the electron and the laser is the most vigorous. Overall, the increase in beam waist radius \(b_{0}\) will cause the movement time of electrons within the Gaussian pulse to increase, leading to a reduction in the electron emission angle. For the convenience of researching the impact of \(b_{0}\) on the electron trajectory, the delay time \(t_{0}\) of the laser is supposed be set initially. Subsequently, the influence of \(b_{0}\) is investigated under various delay times. Figure 2: Schematic diagrams of electron trajectories corresponding to various beam waist radii and delay times. As depicted in Figure 2a-c, when a relatively large delay time of \(t_{0}\) = 6.74 fs is set, the moving electron will collide with the falling edge of the Gaussian pulse and interact therewith. Irrespective of \(b_{0}\), the electron always exits in the direction of +z-axis. As \(b_{0}\) increases, the oscillation trajectory of the electron within the pulse becomes longer. Within the tight focusing regime, the increase in \(b_{0}\) gives rise to a more pronounced increase in the oscillation trajectory. Nevertheless, in the non-tight focusing case, the length of the electron's oscillation trajectory undergoes hardly any change. Concurrently, in the tight focusing scenario, the deflection angle of the electron upon exit is greater. This is because a larger beam waist radius leads to a longer interaction duration of the electron with the pulse, thereby generating a longer oscillation trajectory. Additionally, it can be discerned that in the non-tight focusing case, the deflection angle of the electron's exit is smaller, and the incident and exit directions of the electron are relatively proximate. This is because, in the non-tight focusing situation, the Gaussian pulse has a larger interaction range with the electron, and the electron moves within the pulse for a longer period, experiencing a more balanced positive and negative deflection from the laser pulse.