# Analysis of the Effect of Pulse Length and Magnetic Field Strength on Nonlinear Thomson Scattering
## 3 Results
### Electron Motion and Corresponding Radiation Properties 2  
The parameters in Figure 3 are the same as those in Figure 2. The difference is that the radiated power in Figure 2 is normalized by the maximum radiated power of all parameters, while the radiated power in Figure 3 is normalized by the maximum radiated power of itself. In this way, from Figure 3, the magnitude of the radiated power and its difference between the front end and the back end of the motion can be observed and compared. This enables us to study the causes of radiation generation more successfully. First, we can see that several plots in the upper left corner of Figure 3 have a different trajectory compared to the rest of the plots. Their trajectory takes on the shape of an olive, while the trajectory of the other plots has a near-cylindrical shape. Figure 3: Trajectories of electrons and coincidence diagrams normalized to their own maximum radiated power. This is because when the pulse length is small and the magnetic field strength is low, the contact time between the electrons and the laser pulse is very short. This also leads to the situation where, in disengaging from the laser field, the magnetic field does not have sufficient effect on the electrons to maintain large energy in the final motion. Ultimately, the electrons move in a nearly straight line along the \(z\)-axis. This also leads to the fact that when the electrons disengage from the laser field, the magnetic field does not have a large enough effect on the electrons to give them a large amount of energy in their final spiral motion. Therefore, the electrons end up moving in an almost-straight line along the \(z\)-axis. As the magnetic field increases, more energy can be conserved by the magnetic field, and the eventual spiral motion becomes more pronounced as a result. As the pulse length increases, both the rising and falling edges of the laser field become flat, thus lengthening the contact time of the electrons with the laser field. In this way, the duration of action of the magnetic field increases so that ultimately, more energy can be conserved, leading to a helical motion. By observing the opposition of radiation, we notice that the maximum radiated power occurs near the end of the rising edge of the laser pulse. According to Equations (9) and (10), we know that the radiated power of an electron is proportional to the eighth power of the relativistic factor of the electron \(\gamma^{8}\) and also proportional to the square of the magnitude of the acceleration in the x-direction \(d_{t}u_{x}{}^{2}\). Therefore, we can calculate that the maximum radiated power of the electron occurs at the rising edge of the laser pulse, where \(\gamma^{8}d_{t}u_{x}{}^{2}\) is maximized. This occurs because for a laser field of a given laser intensity, beam waist radius, and magnetic field strength, the time of contact of the laser with the electrons becomes longer as the pulse length increases. As a result, the energy acquired by the electrons in the rising-edge portion of the laser field increases until the electrons are in complete contact with the laser field. After that, the maximum velocity of the electrons remains essentially constant as the pulse length increases. The situation is different when the pulse length is too large; i.e., pulse length \(L>6\lambda_{0}\). The excessive pulse length results in a reduced gradient between the rising and falling edges of the laser field, which means that the acceleration of the electrons is diminished. The maximum velocity is the same since the peak amplitude a\({}_{0}\) of the laser field is constant and the energy given to the electrons by the laser field is also constant; however, due to the decrease in the gradient, this leads to a decrease in \(\gamma^{8}d_{t}u_{x}{}^{2}\) such that the maximum radiated power of the electrons decreases instead as the pulse length increasing. The contact time of the laser field with the electrons is shorter at pulse lengths \(L=2\lambda_{0}\).