## 3 Results
### Motion Trajectory Analysis
#### 3.1.1 The Coupling Analysis of Motion Trajectory and Radiation Power  
To facilitate the analysis of the relationship between the electron's position at a specific moment and its corresponding radiation, the \(z\)-axis is transformed into \(\eta=z-t\), representing the entire interaction process between the electron and the laser pulse. The radiation values during the interaction process are mapped to the electron's trajectory using color coding. From the color bar of the radiation curve and the motion trajectory, the energy distribution of the electron in different states can be observed. It is important to note that, due to the significant variation in radiation values released throughout the electron's motion, the radiation values and color bar distribution in each figure are normalized to the maximum radiation value released up to that point in time. After the complete evolution process, the formation of the hourglass-spindle-shaped trajectory can be divided into four stages. The motion trajectories and radiation power distributions for these four stages are shown in Figure 3.  
The axial and radial acceleration, as well as the velocity changes during the motion process, are calculated separately. The magnitude and direction of the axial and radial acceleration are plotted as a function of \(\eta\) in Figure 4. The orange vertical lines or orange dots marked in Figure 4 indicate the positions where the electron exactly reaches the central axis of the laser pulse. These markers are used to distinguish between the rising edge and the falling edge interaction regions. Hereafter, this position is referred to as the central position of the laser. By combining Figures 3 and 4, we can conduct a more accurate analysis of the electron's motion at different stages. During the forward collision between the electron and the laser pulse, the electron undergoes two reversals in its axial motion direction along the \(z\)-axis. The axial acceleration exerted by the laser pulse on the electron  
Figure 2: Hourglass\(-\)spindle\(-\)shaped trajectory. +++ ==ERROR: No output for this page==
initially increases in the negative \(z\)-axis direction, then decreases to 0, and subsequently transitions to the positive \(z\)-axis direction, where it again increases and then decreases to 0, marking the end of its effect on the electron. By conducting a time-evolution analysis of the velocity and acceleration, the reasons for the axial reversals in the electron's trajectory are further explained here:  
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Under the influence of a strong laser field, the relativistic motion of electrons is affected by the electric field and the ponderomotive potential. The electric field primarily causes high-frequency oscillations of the electron in the radial direction (within the x-y plane), while the ponderomotive potential dominates the electron's axial motion (along the \(z\)-axis). The ponderomotive potential can be regarded as the gradient of the normalized Lorentz factor, reflecting changes in the electron's velocity and energy. Gaussian laser pulses follow a Rayleigh distribution in the radial direction, with the beam waist radius first increasing and then decreasing along the axial direction. At \(z=0\), the beam waist radius reaches its maximum, where the laser pulse's focusing effect is strongest, meaning the laser pulse's influence on the electron trajectory is most significant. The electric field generated by the laser pulse causes the electron's radial motion radius to first expand and then contract. The main reason for the change in the electron's radial motion radius is the Rayleigh distribution of the laser pulse, where the radial component of the laser pulse's acceleration on the electron resembles the laser pulse distribution. This causes the electron to undergo radial oscillations under the influence of the electric field. In the axial direction, the laser pulse's axial component exerts alternating acceleration and deceleration on the electron, causing velocity changes due to the influence of the ponderomotive force.