## 4 Conclusions  
From the perspective of electron dynamics analysis, electrons in the smaller pulse width of the impact of the trajectory are relatively small because of the smaller role of the magnetic field, so the trajectory for the larger and then the smaller spiral is finally stabilized at a stable value. The larger pulse width tends to stabilize the spiral column. The spatial radiation properties of the electron also correspond to the trajectory of the electron. Collimation is poor at smaller pulse widths, while as the pulse width increases, the maximum radiated power angle stabilizes and collimation becomes better. When the pulse width is greater than \(6\lambda_{0}\), the spatial radiation of the electrons appears to be asymmetric. The spectral characteristics of the electron radiation are also roughly the same as the first two characteristics and are optimal when the pulse width is equal to \(6\lambda_{0}\). When the pulse width is small, there is less influence of the magnetic field, and the spectrum is approximately the same as that without the magnetic field, while its repeatability increases as the pulse width increases due to the predominance of the magnetic field and less influence of the tightly focused laser field. The maximum value of the spectral energy is greatest when the pulse width is equal to \(6\lambda_{0}\) because the collimation is best and the corresponding radiation is available in all directions. When the pulse width is greater than \(6\lambda_{0}\), the asymmetry increases and the maximum value of the corresponding spectral energy decreases more quickly.  
Therefore, in order to obtain high-energy, highly collimated X-rays with a given laser pulse beam radius and peak laser amplitude, modulation can be performed by applying a magnetic field and increasing the pulse width. The resulting X-rays not only increase tremendously in energy but also become highly collimated. However, it should be noted that although the collimation of the radiation and the magnitude of the radiated power are optimal at a pulse width of \(6\lambda_{0}\), some small sources of interference can be found in the X-rays corresponding to \(\mathrm{L}=6\lambda_{0}\) in Figure 6.This leads to the fact that although the maximum power angle of the X-rays at \(\mathrm{L}=6\lambda_{0}\) is extremely small, the actual maximum value of \(\theta\) is still not the minimum. Therefore, if we need X-rays with excellent collimation, it is better to use X-rays with a pulse width \(\mathrm{L}=15\lambda_{0}\) modulation. The maximum value of the corresponding spatial power angle at this point is the smallest, i.e., the best collimation.  
This gives us a good way to use nonlinear Thomson scattering for scientific experiments and some potential applications of modulated X-rays.  
Conceptualization, Y.T.; Methodology, H.W. and Y.W.; Software, F.G., Y.Z. and Y.T.; Validation, H.W., Y.Z. and Y.T.; Formal analysis, F.G.; Data curation, H.W.; Writing--original draft, H.W.; Writing--review & editing, F.G. and Y.W.; Visualization, H.W., Y.Z. and Y.T.; Funding acquisition, Y.T. All authors have read and agreed to the published version of the manuscript.  
This work has been supported by the National Natural Sciences Foundation of China under grants No. 10947170/A05 and No. 11104291, the Natural Science Fund for Colleges and Universities in Jiangsu Province under grant No. 10 KJB140006, the Natural Sciences Foundation of Shanghai under grant No. 11 ZR1441300 and Colleges and Universities in Jiangsu Province under grant No. 10 KJB140006, and the Natural Science Foundation of Nanjing University of Posts and Telecommunications under grant No. 202310293146Y.  
Not applicable.  
The authors declare no conflict of interest.