## 3 Results
### Motion Trajectory Analysis
#### 3.1.1 The Coupling Analysis of Motion Trajectory and Radiation Power
Based on this condition, the axial motion process of the electron is further analyzed:  
Phase 1: During this stage (\(tspan=0\)\(\sim\)58), the electron, with an initial velocity, comes into contact with the rising edge of the laser pulse. According to Phase 1, shown in Figure 3, the electron's velocity direction is along the negative \(z\)-axis, while the axial component of the ponderomotive potential of the laser's rising edge points in the positive \(z\)-axis direction. From the color bar of the radiation distribution, we can see that in Phase 1, the electron's radiation power increases as the interaction process progresses. In Figure 4, the acceleration in the \(z\)-axis increases negatively in Phase 1. The axial component of the ponderomotive potential acts to decelerate the electron, reducing its axial velocity. Figure 4: Velocities and accelerations of the electron in different directions. Phase 2: According to Figure 4, during this stage (\(tspan\) = 58\(\sim\)74), under the condition \(gm_{0}=2\), the electron's axial velocity reaches a point where it decreases to 0. However, at this moment, the electron does not cross the central axis of the laser pulse and continues to interact with the rising edge of the laser pulse. Therefore, the axial component of the ponderomotive potential remains in the positive \(z\)-axis direction, and the electron starts to accelerate in the positive \(z\)-axis direction due to this effect. In addition, from the electron's motion trajectory and radiation power distribution in Figure 3, we can see that the radiation power of the electron in this stage is still monotonically increasing, with no decrease observed. This is consistent with the above conclusion, and at this point, the effect of the laser on the electron is similar to that in Phase 1. Phase 3: In the third stage, as the electron passes the central position, the acceleration curve begins to decrease, which can be seen in Figure 4. During this stage (\(tspan\) = 74\(\sim\)96), the electron undergoes reverse acceleration, but its velocity remains lower than the speed of the laser pulse. As a result, the laser pulse continues to move relative to the electron along the positive \(z\)-axis, effectively passing through the electron. At this point, the central axis of the laser pulse surpasses the electron, causing the electron to interact with the falling edge of the laser pulse. The axial component of the ponderomotive potential of the laser pulse's falling edge reverses to the negative \(z\)-axis direction. However, due to the influence of the second phase, the electron's axial velocity component remains in the positive \(z\)-axis direction. Consequently, the electron begins to decelerate for the second time under the influence of the laser pulse. In Figure 3 of Phase 3, based on the mapping relationship between radiation and motion trajectory, it can be observed that the radiation power of the electron begins to decline. The peak radiation power appears at the transition between Phase 2 and Phase 3, which occurs when the central position of the laser interacts with the electron. At this moment, the electron experiences the maximum influence of the electric field, resulting in the highest radiation power. Phase 4: In Phase 4 of Figure 3, we can see that the electron's radiation power does not increase. After the electron returns to the same state as its initial motion direction, the radiation gradually decreases, with a peak present. Based on Phase 4 of Figure 3, we observe that the peak of the radiation power occurs at the position where the electron's velocity reverses, meaning the radiation peak appears in Phase 3. During this stage (\(tspan\) from 96 to the end of the interaction between the electron and the laser pulse), the electron's velocity decreases to 0 once again. At this point, the electron and the laser pulse have not yet separated, and the electron continues to interact with the falling edge of the laser pulse, undergoing a second phase of reverse acceleration.