## 2 Materials and Methods  
We have learned that paraxial approximation can no longer meet the requirements of tightly focused lasers and that higher-order field effects [27] need to be considered. The process we simulated is shown in Figure 1. By modeling and performing linear polarization vector decomposition of the light beam, the following electric field components can be obtained:  
\[E_{x}=E\Bigg{\{}S_{0}+\epsilon^{2}\bigg{[}\xi^{2}S_{2}-\frac{\rho^{4}S_{3}}{4} \bigg{]}+\epsilon^{4}\bigg{[}\frac{S_{2}}{8}-\frac{\rho^{2}S_{3}}{4}-\frac{ \rho^{2}\big{(}\rho^{2}-16\xi^{2}\big{)}S_{4}}{16}-\frac{\rho^{4}\big{(}\rho^{2 }+2\xi^{2}\big{)}S_{5}}{8}+\frac{\rho^{8}S_{6}}{32}\bigg{]}\Bigg{\}} \tag{1}\]  
\[E_{y}=E\xi v\bigg{\{}\epsilon^{2}S_{2}+\epsilon^{4}\bigg{[}\rho^{2}S_{4}-\frac {\rho^{4}S_{5}}{4}\bigg{]}\bigg{\}} \tag{2}\]  
\[E_{z}=E\xi\bigg{\{}\epsilon C_{1}+\epsilon^{3}\bigg{[}-\frac{C_{2}}{2}+\rho^{2 }C_{3}-\frac{\rho^{4}C_{4}}{4}\bigg{]}+\epsilon^{5}\bigg{[}-\frac{3C_{3}}{8}- \frac{3\rho^{2}C_{4}}{8}+\frac{17\rho^{4}C_{5}}{16}-\frac{3\rho^{6}C_{5}}{16}- \frac{3\rho^{6}C_{6}}{8}-\frac{\rho^{8}C_{7}}{32}\bigg{]}\bigg{\}} \tag{3}\]Similarly, the magnetic field part can also be decomposed into the following parts:  
\[B_{x}=0 \tag{4}\]  
\[B_{y}=E\bigg{\{}S_{0}+\epsilon^{2}\bigg{[}\frac{\rho^{2}S_{2}}{2}-\frac{\rho^{4}S _{3}}{4}\bigg{]}+\epsilon^{4}\bigg{[}-\frac{S_{2}}{8}+\frac{\rho^{2}S_{3}}{4}+ \frac{5\rho^{4}S_{4}}{16}-\frac{\rho^{6}S_{5}}{4}+\frac{\rho^{8}S_{6}}{32} \bigg{]}\bigg{\}} \tag{5}\]  
\[B_{z}=E\
u\bigg{\{}eC_{1}+\epsilon^{3}\bigg{[}\frac{C_{2}}{2}+\frac{\rho^{2}C_{ 3}}{2}-\frac{\rho^{4}C_{4}}{4}\bigg{]}+\epsilon^{5}\bigg{[}\frac{3C_{3}}{8}+ \frac{3\rho^{2}C_{4}}{8}+\frac{3\rho^{4}C_{5}}{16}-\frac{8\rho^{6}C_{6}}{4}+ \frac{\rho^{8}C_{7}}{32}\bigg{]}\bigg{\}} \tag{6}\]  
In Equations (1)-(6), it is stated that \(\mathbf{\xi}=\mathbf{x}/\mathbf{b_{0}}\), \(\mathbf{v}=\frac{\mathbf{y}}{\mathbf{b_{0}}}\mathbf{E}=\mathbf{E_{0}}\frac{\mathbf{b_{0}}}{\mathbf{E}}\mathbf{ exp}\bigg{[}-\frac{\mathbf{y}^{2}}{\mathbf{b}^{2}}-\frac{\mathbf{r}^{2}}{\mathbf{b}^{2}}\bigg{]}\), and \(\mathbf{E_{0}}=\mathbf{k}\mathbf{A_{0}}\). The cross-section of the tightly focused beam has a circular form with a radius of \(\mathbf{b_{0}}\), since we approximate it as a long cylinder. This indicates that the cross-section across every point \(z\) on the axis is also circular, with a radius of \(\mathbf{b}(\mathbf{z})=\mathbf{b_{0}}\sqrt{1+\bigg{(}\frac{\mathbf{z}}{\mathbf{z}_{r}}\bigg{)}^{2}}\), determined by \(\mathbf{k}=\mathbf{2\pi}/\mathbf{\lambda}\) and the Rayleigh length \(\mathbf{z}_{r}=\mathbf{k}\mathbf{b_{0}}^{2}/\mathbf{2}\). For laser electron collision, we select the z-axis as the trajectory. In the absence of any further requirements, the laser's focus \(z\) is 0. The focus spot is the only location where the laser can be maintained at its strongest since it has the lowest laser radius and the maximum laser intensity. \(\mathbf{\epsilon}=\mathbf{b_{0}}/\mathbf{z}_{r}\) is the diffraction angle. \(L\) represents the pulse width, and it is related to the beam duration. \(\mathbf{L}=\mathbf{\tau}\mathbf{c}\), where \(\mathbf{\tau}\) represents the duration of the laser. \[\mathbf{a_{0}}=\frac{e\mathbf{A_{0}}}{mc^{2}} \tag{7}\]  
\[S_{n}=\bigg{(}\frac{\mathbf{b_{0}}}{\mathbf{b}}\bigg{)}^{n}sin(\mathbf{\xi}+n\mathbf{\xi}_{G}),\mathbf{n}=\mathbf{0},\ \mathbf{1},\ \mathbf{2}\dots \tag{8}\]  
Figure 1: Schematic illustration of the interaction between a chirped laser pulse and a relativistic electron in a tightly focused beam geometry. The diagram outlines the key physical elements considered in the simulation, including laser polarization, beam waist radius (\(w_{0}\)), and the electron trajectory along the \(z\)-axis.