## 3 Results
### The Coupling Effect of Optical Characteristic Parameters on Radiative Characteristics  
To investigate the influence of laser intensity (\(\mathbf{a_{0}}\)) and beam waist radius (\(\omega_{0}\)) on nonlinear Thomson scattering (NTS), we systematically varied \(\omega_{0}\) across a range of \(\mathbf{a_{0}}\) values (\(1\leq\mathbf{a_{0}}\leq 9\), \(1\leq\mathbf{\omega_{0}}\leq 9\)). For the convenience of research, it is stated again that \(\mathbf{a_{0}}\) has undergone normalization processing, so it has no unit. \(\omega_{0}\) is at the micron level, and this will not be repeated later. Numerical results reveal that both increasing \(\mathbf{a_{0}}\) and \(\mathbf{\omega_{0}}\) enhances the electron radiation power. Notably, when \(\mathbf{\omega_{0}}\) is not excessively small, its impact on radiation power exhibits a linear enhancement within approximately one order of magnitude. In contrast, the dependence on \(\mathbf{a_{0}}\) follows a pronounced exponential growth trend. This demonstrates that while both parameters positively modulate radiation power, \(\mathbf{a_{0}}\) dominates the scaling behavior, and a detailed decoupled analysis of their individual effects will follow. Below, we focus on their synergistic coupling mechanisms. In Figure 2, when the laser intensity is relatively low (\(\mathbf{a_{0}}<3\)), variations in the beam waist radius (\(\omega_{0}\)) exhibit negligible effects on the electron radiation power. For instance, as \(\mathbf{\omega_{0}}\) increases from 1 to 9, the radiation power rises by merely 0.5 t/\(\Omega\). Notably, a critical transition occurs at \(\mathbf{a_{0}}=3\), and scaling \(\omega_{0}\) within the same range (1-9) amplifies the radiation power by one order of magnitude. This enhancement escalates progressively with higher \(\mathbf{a_{0}}\)--reaching two, three, and four orders of magnitude at \(\mathbf{a_{0}}=5,7\), and 9, respectively. These observations collectively demonstrate that while \(\mathbf{a_{0}}\) and \(\mathbf{\omega_{0}}\) synergistically enhance radiation power, their coupling strength grows nonlinearly with increasing \(\mathbf{a_{0}}\). Specifically, higher \(\mathbf{a_{0}}\) values amplify the sensitivity of radiation power to \(\mathbf{\omega_{0}}\), revealing a dominant role of laser intensity in mediating the beam waist-radius-dependent energy scaling. Our observations reveal that increasing the beam waist radius (\(\mathbf{\omega_{0}}\)) significantly enhances the spatial symmetry of electron radiation patterns. Initially, the radiation spectrum may exhibit multiple spectral peaks; however, while the laser intensity (\(\mathbf{a_{0}}\)) exerts limited control over the number or intensity of these peaks (failing to effectively concentrate radiation power), enlarging \(\mathbf{\omega_{0}}\) drastically reduces the number of minor peaks, leading to a more centralized energy distribution. By integrating the data of the radiation power and through data fitting, we have drawn a three-dimensional image (Figure 3). Meanwhile, in order to increase the contrast, we have taken the logarithm of the radiation power with base 10. This further supports the above research content. In summary, higher laser intensities (\(\mathbf{a_{0}}\)) amplify the sensitivity of radiation symmetry to variations in \(\mathbf{\omega_{0}}\). Under fixed values of \(\mathbf{a_{0}}\), larger \(\mathbf{\omega_{0}}\) values not only boost the total radiation power but also improve its collimation, with the power scaling approximately being quadratically dependent on \(\mathbf{\omega_{0}}\). These findings highlight the critical role of beam waist radius modulation in tailoring radiation coherence and efficiency under relativistic laser-electron interactions. Given the negligible influence of the beam waist radius (\(\omega_{0}\)) on radiation power at low laser intensities (\(a_{0}<3\)), we selected \(a_{0}=7\) to maximize the sensitivity of radiation dynamics to \(\omega_{0}\), thereby amplifying its effect and enhancing the observability of the phenomenon.