# Analysis of the Effect of Pulse Length and Magnetic Field Strength on Nonlinear Thomson Scattering
## 3 Results
### Electron Motion and Corresponding Radiation Properties 1  
Figure 2 shows the trajectory of electrons under the combined effect of laser pulse and magnetic field. The maximal coordinates of the corresponding \(x\)-axis and \(y\)-axis in the plots are both \(\pm 10\lambda_{0}\).  
The color of the line in Figure 2 indicates the magnitude of the radiation at each point on the trajectory, while the direction of the radiation is a section of the cone in the direction of the electron velocity. The blue line in the background of each figure indicates the magnitude of the relativistic factor \(\gamma\) of the electron at the corresponding \(z\)-coordinate position. Please note that the magnitude of the relativistic factor \(\gamma\) is here normalized by the maximum relativistic factor that is under the corresponding parameter. Therefore, in this study, we can only compare the relative magnitude under the same parameter, not the absolute magnitude in different plots, i.e., the magnitude under different parameters.  
Observing the differences between columns, we begin by analyzing the plots where pulse length \(L\leq 6\lambda_{0}\) in Figure 2. As the magnetic field strength B\({}_{0}\) increases, the maximum energy radiated by the electron gradually increases, and the ratio of the magnitude of the relativistic factor \(\gamma\) at the maximum radiation to the magnitude of the relativistic factor \(\gamma\) after smoothing becomes smaller; while after pulse length \(L>6\lambda_{0}\), the ratio of the magnitude of the relativistic factor \(\gamma\) at the maximum radiation to the magnitude of the relativistic factor \(\gamma\) after smoothing becomes larger. From the trajectory diagram, we also find that the electron trajectory changes when L \(>6\lambda_{0}\). The radial radius of the electron begins to contract after reaching its maximum, corresponding to the trend of the relativistic factor \(\gamma\).  
Observing from the difference between rows, we can also find that as the pulse length L increases, the electron trajectory changes from the original olive shape to a regular cylindrical shape. The ratio of the magnitude of the relativistic factor \(\gamma\) at the maximum radiation to the magnitude of the relativistic factor \(\gamma\) after smoothing becomes larger; while for the portion of the pulse length \(L>6\lambda_{0}\), the radiation becomes smaller relative to \(L=\)6\(\lambda_{0}\). This indicates that there is an optimal pulse length parameter \(L=\)6\(\lambda_{0}\) for the tightly focused laser pulse under an applied magnetic field.  
Figure 2: Trajectories of electrons and coincidence plots normalized to the maximum radiated power of all data.  
We also find that when the pulse length \(L>6\lambda_{0}\), the trend of the radial radius is different from the previous one. At pulse length \(L\leq 6\lambda_{0}\), the maximum radial radius increases with the increasing magnetic field strength B\({}_{0}\) until the maximum radial radius equals the final radial radius. In contrast, for pulse lengths \(L>6\lambda_{0}\), the maximum radial radius decreases with the increasing magnetic field strength B\({}_{0}\).