## 2 Materials and Methods  
It can be inferred from Figure 1 that the schematic diagram of the cross-collision between a tightly focused linearly polarized laser pulse and an electron is obtained. The linearly polarized Gaussian laser pulse is incident along the positive z-axis after a certain delay time and collides with the high-energy electron moving in the positive direction on the negative half of the x-axis. Due to the Thomson scattering effect of the linearly polarized strong laser pulse in the plasma, the paraxial approximation of the tightly focused laser is no longer applicable, and the high-order field effects need to be taken into account when establishing the model [16, 17, 18, 19]. A mathematical model is established, and the beam is regarded as a linearly polarized vector along the +x direction. The electric field components can be represented by the following formula [20]:  
\[E_{x}=E\Bigg{\{}S_{0}+\epsilon^{2}\bigg{[}\varphi^{2}S_{2}-\frac{r^{4}S_{3}}{4} \bigg{]}+\epsilon^{4}\bigg{[}\frac{S_{2}}{8}-\frac{r^{2}S_{3}}{4}-\frac{r^{2} \big{(}r^{2}-16\varphi^{2}\big{)}S_{4}}{16}-\frac{r^{4}\big{(}r^{2}+2\varphi^{2} \big{)}S_{5}}{8}+\frac{r^{8}S_{6}}{32}\bigg{]}\Bigg{\}} \tag{1}\]  
\[E_{y}=E\varphi w\bigg{\{}\epsilon^{2}S_{2}+\epsilon^{4}\bigg{[}r^{2}S_{4}- \frac{r^{4}S_{5}}{4}\bigg{]}\bigg{\}} \tag{2}\]  
\[E_{z}=E\varphi\bigg{\{}\epsilon C_{1}+\epsilon^{3}\bigg{[}-\frac{C_{2}}{2}+r^ {2}C_{3}-\frac{r^{4}C_{4}}{4}\bigg{]}+\epsilon^{5}\bigg{[}-\frac{3C_{3}}{8}- \frac{3r^{2}C_{4}}{8}+\frac{17r^{4}C_{5}}{16}-\frac{3r^{6}C_{5}}{16}-\frac{3 r^{6}C_{6}}{8}+\frac{r^{8}C_{7}}{32}\bigg{]}\bigg{\}} \tag{3}\]  
Likewise, the magnetic field components can also be derived and expressed as  
\[B_{x}=0 \tag{4}\]  
\[B_{y}=E\bigg{\{}S_{0}+\epsilon^{2}\bigg{[}\frac{r^{2}S_{2}}{2}-\frac{r^{4}S_{ 3}}{4}\bigg{]}+\epsilon^{4}\bigg{[}-\frac{S_{2}}{8}+\frac{r^{2}S_{3}}{4}+ \frac{5r^{4}S_{4}}{16}-\frac{r^{6}S_{5}}{4}+\frac{r^{8}S_{6}}{32}\bigg{]} \bigg{\}} \tag{5}\]  
\[B_{z}=Ew\bigg{\{}\epsilon C_{1}+\epsilon^{3}\bigg{[}\frac{C_{2}}{2}+\frac{r^{ 2}C_{3}}{2}-\frac{r^{4}C_{4}}{4}\bigg{]}+\epsilon^{5}\bigg{[}\frac{3C_{3}}{8} +\frac{3r^{2}C_{4}}{8}+\frac{3r^{4}C_{5}}{16}-\frac{8r^{6}C_{6}}{4}+\frac{r^{ 8}C_{7}}{32}\bigg{]}\bigg{\}} \tag{6}\]  
The expression for the light field distribution can be expressed as \(\varphi=x/b_{0}\), \(w=y/b_{0}\), and the field intensity \(E=E_{0}\frac{b_{0}}{b}exp\left[-\frac{r^{2}}{l^{2}}-\frac{r^{2}}{b^{2}}\right]\), where \(E_{0}\) is equal to \(kA_{0}\). Formulas (1)-(6) are presented in approximate forms, and the precision of the approximation is related to the orders of \"\(r\)\" and \"\(\epsilon\)\". By constructing a fifth-order high-order field of the diffraction angle \(\epsilon\), the beam can be focused to a size of several micrometers. The orders presented in the formulae can both calculate the structure clearly and reduce unnecessary computational efforts. This model considers the light beam as a cylindrical structure, with each cross-section along the optical axis exhibiting a circular distribution. The variation in the beam radius along the propagation direction is described by \(b(z)=b_{0}\sqrt{1+\left(z/z_{r}\right)^{2}}\), where the Rayleigh length \(z_{r}={kb_{0}}^{2}/2\), and the wave number \(\mathrm{k}=2\pi/\lambda\). The beam radius is the smallest, and the intensity reaches its peak at the beam focus, which is the origin of the coordinate system. The diffraction effect is quantified by the parameter \(\epsilon=b_{0}/z_{r}\), characterizing the divergence property of the beam. The pulse duration \(\tau\) is related to the spatial pulse width \(L\) by \(L=\tau c\) (c being the speed of light), and this parameter  
Figure 1: Schematic illustration of the cross-collision between a tightly focused linearly polarized intense laser pulse and a relativistic electron. simultaneously appears in the transverse distribution of the exponential term, determining the attenuation characteristics of the longitudinal distribution of the light field.