## 3 Results  
### Electron Motion Characteristics  
In this section, we will discuss the effects of nonlinear Thomson scattering on electron motion trajectory, spatial radiation, and spectrum for various parameters of the incident tightly focused laser pulse in the presence of an applied magnetic field. Based on the analysis of available data, we conclude that the electron trajectory, spatial radiation, and spectrum calculated at the peak laser amplitude \(\mathrm{a}_{0}=5\times 0.85\times 10^{-9}\lambda\sqrt{\mathrm{I}}\), beam waist radius \(\mathrm{b}_{0}=4\lambda_{0}\), and magnetic field strength \(\mathrm{B}_{0}=0.3(\mathrm{kT})\) are more representative. \(\lambda_{0}\) denotes the initial wavelength of action of the laser pulse, and \(\lambda_{0}=1\)\(\mathrm{\mu m}\). The trajectory, spatial radiation, and spectrum of the electron in the above data are determined by writing a MATLAB program. Figure 2 shows the trajectory of the electron at \(\mathrm{L}=1\lambda_{0},\mathrm{L}=2\lambda_{0},\mathrm{L}=3\lambda_{0}, \mathrm{L}=4\lambda_{0}\), \(\mathrm{L}=5\lambda_{0},\mathrm{L}=6\lambda_{0},\mathrm{L}=10\lambda_{0}, \mathrm{L}=15\lambda_{0},\mathrm{B}=0.3\) under the combined effect of the laser pulse and magnetic field, and \(\mathrm{L}=6\lambda_{0}\) without the effect of the magnetic field. The radial radii of the electron trajectories are 1, 1, 1.5, 4, 5, 5.5, 6,6, and 1.5, respectively in each image. The axes labeled in the figure are the same as in the schematic; the direction of electron motion is the z-axis, the longitudinal axis is the x-axis, and the diagonal axis is the y-axis. From Figure 2a-h, we can find that with laser and magnetic fields both acting, the electrons end up making a stable spiral motion with a specific amount of energy. When \(\mathrm{L}=\lambda_{0},2\lambda_{0},3\lambda_{0}\), Figure 2a-c, the electron motion trajectory is a spiral that first becomes larger and then smaller, taking on an olive-like shape. Its final minimal value is a stable value. Specifically, the radial radii of the electrons are less than \(0.4\lambda_{0}\). This is due to the fact that the gradient between the rising and falling edges of the laser field is larger when the pulse width is smaller. As a result, the electron is subjected to significant acceleration and deceleration. The effect of deceleration on the electron is greater than the stabilizing effect of the magnetic field on the electron motion. Therefore, the electron in the smaller pulse width will appear in the first half of the olive-shaped and the second half of the spiral-shaped trajectory. When \(3\lambda_{0}<\mathrm{L}<6\lambda_{0}\), Figure 2d-h, the electron trajectory is a column-like helix, and its final value is a stable value. This is reflected in the increase of radial radius of the electron, which reaches saturation after \(\mathrm{L}=6\lambda_{0}\), and basically becomes stable at \(6\lambda_{0}\). This is due to the fact that the gradient between the rising and falling edges of the laser field is smaller at larger pulse widths. The acceleration and deceleration of the electrons are weaker. The effect of deceleration on the electrons is less than the stabilizing effect of the magnetic field on the electron motion. Therefore, the trajectory of the electron shows a stable spiral. The radial maximum displacement also differs between the two. When \(\mathrm{L}=\lambda_{0},2\lambda_{0},3\lambda_{0}\), the radial maximum displacement of electrons is less than \(\lambda_{0}\), while after \(\mathrm{L}>5\lambda_{0}\), the radial maximum displacement of electrons increases with the increase of pulse width \(\mathrm{L}\), and both are greater than \(5\lambda_{0}\). This indicates that the magnetic field changes the trajectory of electron motion significantly.