# Analysis of the Effect of Pulse Length and Magnetic Field Strength on Nonlinear Thomson Scattering
## 3 Results
### Electron Motion and Corresponding Radiation Properties 3  
The parameters of Figure 4 are consistent with those above, with the difference being that the radiated power in Figure 4 is normalized by the maximum value of the radiated power in each set of corresponding pulse lengths L. Figure 4 shows that the magnitude of radiated power in the same set of electron radiation varies with the pulse length L.  
By varying the parameters of the normalisation in this way, it is possible to study the causes of the generation of radiation more successfully. Since the maximum radiated power produced by the same set of pulse lengths L differs by a maximum of about 7 orders of magnitude, in order to observe and represent the variation of the radiated power magnitude with the pulse length L more clearly, we apply some linear transformations to the normalization. Figure 4: Recombination plot of electron trajectories and maximum radiated power normalized to the same pulse length. The specific formula is as follows:  
\[P_{fix}=\left\{ \begin{array}{ll}1&,P=P_{max}\\ 1-\frac{log_{10}(\frac{p_{max}}{c})}{c}&,\frac{p_{max}}{p}>10^{c}\\ 0&,\frac{p_{max}}{p}\leq 10^{c}.\end{array} \right. \tag{12}\]  
\(P_{fix}\) is the value corresponding to the linear transformation and normalized; and c is the transformation parameter, which, in Figure 4, is c = 7. It can be seen from the figure that for the same set of pulse lengths L, there is a significant increase in the radiated power of the electrons as the magnetic field strength B\({}_{0}\) increases. Moreover, the distribution of the radiated power of the electrons varies as the pulse length L increases. When the pulse length L is large, the ratio of the radiated power of the electrons in the smaller magnetic field strength to the maximum radiated power of the whole group also increases significantly. By looking at the \(L=4\lambda_{0},6\lambda_{0}\) sets in Figures 3 and 4, we find that the power radiated by the electrons at the front end of the electron motion and at the back end of the helix increases as the magnetic field strength B\({}_{0}\) increases. However, as can be seen from the figure, the radiated power at the back end of the spiral where the electrons are moving increases to a significantly lesser extent than that at the front end of the movement. This is because at the back end of the helix, the electrons are moving; the laser field is already out of contact with the electrons, and at this point, the only thing controlling the trajectory of the electrons is the magnetic field. Since the magnetic field is nonuniformly strong, the electrons still exchange energy as they move. Thus, the greater the strength of the applied magnetic field, the greater the radiated power emitted from the back end of the helix where the electrons are moving; while at the maximum radiation power at the rising edge of the laser field, the electrons are subjected to the combined action of the laser and magnetic fields. Since the magnitude of the radiated power of the electron is related to the velocity and acceleration of the electron, and the applied nonuniform magnetic field has a periodic acceleration and deceleration effect on the electron, both the velocity and acceleration at the maximum radiated power increase with the applied magnetic field, and the increase is greater than that in the part of the helix at the back end where only the magnetic field is applied. Comparing the \(L=20\lambda_{0}\) groups of Figures 3 and 4, we find the phenomenon of double peaks. This is because there is a process of acceleration to deceleration with respect to the electron's velocity towards the end of the rising edge of the laser field. At this point, there is a brief moment when the acceleration is equal to 0 and the magnitude of the radiated power is 0. As a result, there is a process in which the radiated power decreases from a larger position to zero and increases again to a larger value--the so-called bimodal phenomenon.