## 4 Conclusions  
By means of numerical simulation, this paper investigates the impact of laser amplitude on the spatial distribution, temporal characteristics and frequency-domain charac  
Figure 8: The diagram depicting the relationship between laser amplitude and cutoff frequency. Figure 9: Three-dimensional spectrum graphs of the peak radiation power direction under diverse laser amplitudes and polar angles. teristics of radiation power under the circumstance of cross-collision between the tightly focused linearly polarized Gaussian laser pulse and the relativistic electron. In terms of the electron trajectories, an augmentation of the laser amplitude can magnify the z-axis ponderomotive force exerted by a linearly polarized intense laser pulse on high-energy electrons, inducing a certain displacement of the electrons toward the z-axis and eventually elongating the oscillation trajectory of the electrons along the z-axis. Meanwhile, the radiation power distribution diagram produced by the interaction of the linearly polarized intense laser and the high-energy electron will undergo certain distribution changes in the \(\theta\) and \(\Phi\) directions as the laser amplitude increases. When the laser amplitude is very low, the electron radiation power is concentrated at \(\Phi=0^{\circ}\) and \(\theta=90^{\circ}\), and the image presents a \"hill-like\" distribution. As the laser intensity increases, the distribution of the image changes in the \(\theta\) and \(\Phi\) directions, presenting a \"comet-like\" distribution. Generally speaking, the peak of the radiation power spectrum at the comet head shifts toward the direction of smaller \(\theta\), and the power peak continuously increases. This indicates that the radiation power can be enhanced by increasing the laser amplitude. This paper also examines the temporal distribution characteristics in the direction of the maximal radiation power. The temporal distribution is mainly a symmetrical multi-peak structure. The number of peaks is reduced as the laser amplitude increases, eventually evolving into a symmetrical three-peak structure. The height of the secondary peaks relative to the main peak keeps shrinking, and the temporal distribution becomes more concentrated. It can be speculated that when the laser amplitude is large enough, the energy can be concentrated on the main peak, which can be regarded as a single attosecond laser pulse. Ultimately, the morphology of the spectrum in the direction of the maximum radiation power will shift to a multi-peak state spectrum as the laser amplitude increases. When the laser amplitude is relatively intense, radiation with a more concentrated frequency range and superior quality can be output. The research findings of this paper enrich the experimental experience of the nonlinear Thomson scattering theory and contribute to the study of cross-collision in the tight focusing state. Based on the above-mentioned rules, the laser amplitude can be regulated to generate X-rays that are more concentrated and more stable and can eventually be applied in fields such as biomedicine and atomic physics. Conceptualization, Y.T. ; Methodology, Y.T. ; Software, Y.T. ; Validation, J.X. and Y.L. ; Formal analysis, [PERSON], J.X. and Y.L. ; Investigation, [PERSON]; Resources, [PERSON]; Data curation, J.S. ; Writing--original draft, J.S. ; Writing--review & editing, [PERSON]; Visualization, J.S. ; Supervision, Y.L. ; Project administration, G.Y. ; Funding acquisition, G.Y. All authors have read and agreed to the published version of the manuscript. This work was supported by the National Natural Sciences Foundation of China under Grant Nos. 10947170/A05 and 11104291; Natural science fund for colleges and universities in Jiangsu Province under Grant No. 10 KJB140006; Natural Sciences Foundation of Shanghai under Grant No. 11 ZR1441300; and Natural Science Foundation of Nanjing University of Posts and Telecommunications under Grant No. NY221098, and it was sponsored by Jiangsu Qing Lan Project and STITP Project under Grant No. CXXZD2024158.