## 4 Conclusions  
Through numerical simulation, this paper investigates the impact of the beam waist radius of a linearly polarized Gaussian laser pulse on the spatial distribution, temporal characteristics, and frequency domain characteristics of radiation under the condition of  
Figure 8: Two-dimensional spectrum of peak radiation power under different beam waist radii. relativistic electron-laser cross-collision. Concurrently, we also compare the dissimilarities and similarities of the outcomes produced by tightly focused and non-tightly focused laser beams. It has been discovered through experimental studies that the interaction duration between the relativistic electron and the linearly polarized Gaussian laser pulse can be extended by increasing the beam waist radius. Under diverse delay emission times, it induces the electron to exhibit distinct oscillation motion trajectories and exit angles. Simultaneously, the spatial distribution of NCTS in this experimentation presents a comet-like shape, with the peak of electron radiation power concentrated at the head of the comet. The larger the beam waist radius, the closer the direction of the peak radiation power approaches the z-axis. This implies that increasing the beam waist radius can control the principal radiation direction of the rays generated by the collision, concentrating them in the z-axis direction of the laser emission and reducing waste by making the energy more concentrated. Subsequently, this paper also examines the temporal distribution characteristics in the direction of the maximum radiation power. The temporal distribution manifests a symmetrical three-peak structure, and the two secondary peaks escalate with the increase of the beam waist radius, ultimately attaining approximately half the height of the main peak. Ultimately, regarding the temporal distribution characteristics in the direction of the maximum radiation power, the spectrum transitions from a supercontinuum to a multi-peak state as the beam waist radius increases. We innovatively identified a transformation mode of the spectrum: under the tight focusing condition, as the beam waist radius expands, the main peak and the secondary peak of the spectrum interchange. Currently, the primary means to increase the radiation power of the rays is to increase \(a_{0}\) and \(\gamma\), which leads to energy dissipation. By appropriately increasing the beam waist radius through the method presented in this article, rays with higher frequencies and greater energies can be obtained, and energy waste can be mitigated. The aforementioned research achievements enrich the experimental experience of nonlinear Thomson scattering theory and offer novel concepts for the study of the cross-collision between high-energy electrons and tightly focused linearly polarized laser pulses. This research contributes to generating stable and high-quality X-rays under relativistic conditions and facilitating the design of the next generation of ultra-short high-energy ray generators. Conceptualization, Y.T. ; Methodology, J.S. and Y.T. ; Software, J.S. and Y.T. ; Validation, J.L., Y.L. and Y.T. ; Formal analysis, J.L. and Y.L. ; Investigation, J.S. and Z.Y. ; Resources, J.S. and J.L. ; Data curation, J.S. ; Writing--original draft, J.S. ; Writing--review & editing, J.S., Y.L. and Y.T. ; Visualization, J.S. and J.L. ; Supervision, Y.L. and Y.T. ; Project administration, J.S., Z.Y. and G.Y. ; Funding acquisition, G.Y. 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 Grant Nos. 10947170/A05 and 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, the Natural Science Foundation of Nanjing University of Posts and Telecommunications under Grant No. NY221098, and sponsored by the Jiangsu Qing Lan Project and STITP Project under Grant No. CXXZD2024158. Not applicable.