## 3 Results  
### Motion Trajectory Analysis  
An electron is fixed to the origin of the coordinate. The incident circularly polarized laser-chirped pulse is an electromagnetic wave whose transverse electric field and irradiance distribution approximately meet the Gaussian envelope, propagating along the \(+z\)-axis. Its wavelength is \(\lambda_{0}=1\)\(\upmu\)m and its optical intensity parameter is \(a_{0}=5\) (_the laser pulse intensity \(I=3.45\times 10^{19}\) W/cm\({}^{2}\)_), and the chirped pulse width \(L\) is taken as three sets of values, which are \(2\lambda_{0},4\lambda_{0},\) and \(6\lambda_{0}\), respectively. The corresponding pulse durations are \(6.7\)\(fs,13.3\)\(fs,\) and \(20\)\(fs\), and the waist radius \(w_{0}\) is determined as \(4\)\(\upmu\)m. The chirped parameter \(c_{0}\) changes from \(-0.05\) to \(+0.05\), and the electron motion, full-angle, and spectral characteristics of the electron radiation during the transition from compactness laser to non-compact-focused laser and even to the plane wave are studied. The environmental parameters are kept constant without being described otherwise and will not be described below. Based on classical electrodynamics, we employed the fourth-order Runge-Kutta algorithm to solve the corresponding partial differential equations, obtaining the electron trajectory as shown in the figure. It can be seen from Figure 2 that the static electron, driven by the Gaussian circularly polarized laser-chirped pulse, first carries out a spiral motion similar to the isogonal spiral, and the change process of its pitch is basically consistent with the change process of the Gaussian function. According to the change, we can see that when the electron lags behind the end of the pulse-falling edge, the electron begins to perform a uniform linear motion. We found that after the chirp parameter was added, the range of electron motion trajectory changed to a large extent. In order to visually display the change in motion trajectory and facilitate a comparison, we placed the axis of motion trajectory under the same axis in the comparison chart. Due to the effect of the chirp parameter, the electron motion trajectory contracted too much. In Figure 2I(_a_1-_c_1,\(a\)3-_c_3), a clearer motion trajectory is attached to the upper right corner of the six pictures, which is convenient when showing the change in motion trajectory of the chirped pulse under the same pulse width. It can be seen from the motion trajectory in Figure 2I that the radial motion radius of the electron without chirp influence is larger than that of the trajectory with chirp influence. The radial trajectory envelope of electron motion is very similar to that of the pulse envelope; the radial motion radius increases first and then decreases, and the radial displacement reaches the maximum when the electron and pulse peak act instantaneously. On the other hand, in the process of an interaction between the Gaussian pulse and electrons, the pulse width of the pulse will affect the trajectory of the electron. A pulse with a narrower pulse width is able to provide a higher peak power, resulting in a stronger field strength in the interaction, resulting in greater acceleration and energy transfer of the electrons. It can be seen from the trajectory diagram that the pulse width is more curved and complex than that of the motion trajectory of the group, and at \(L=2\lambda_{0}\), the electrons eventually deviate from the z-axis and move in a uniform off-axis straight line. From Equation (4), it can be derived that under constant conditions, \(E_{0}\) is inversely proportional to \(L^{2}\). As \(L\) increases, \(E_{0}\) decreases, which reduces the binding effect of the electric field component on the electron. Therefore, during the interaction between the laser pulse with a small pulse width and the electron, as shown in Figure 2 with \(L=2,c_{0}=0\), the electron does not fully interact with the laser and deviates from the laser's influence, moving off-axis.