## 2 Materials and Methods  
The cross-collision scenario of a linearly polarized laser pulse with an electron under relativistic conditions is depicted in Figure 1. A linearly polarized high-energy intense laser with a wavelength \(\lambda\) of 1 \(\upmu\)m and parallel to the z-axis is incident along the +z direction. Considering the existence of the NTS effect of the linearly polarized intense laser pulse in the plasma, the paraxial approximation is no longer valid for tightly focused laser beams. Hence, it is necessary to take into account the higher-order field effects [18; 19; 20; 21; 22]. By establishing a mathematical model, the beam is modeled as a linearly polarized vector potential along the +x axis. The specific electric field components can be expressed as [23]:  
\[E_{x}=E\Bigg{\{}S_{0}+e^{2}\bigg{[}q^{2}S_{2}-\frac{r^{4}S_{3}}{4}\bigg{]}+e^{ 4}\Bigg{[}\frac{S_{2}}{8}-\frac{r^{2}S_{3}}{4}-\frac{r^{2}\big{(}r^{2}-16\varphi ^{2}\big{)}S_{4}}{16}-\frac{r^{4}\big{(}r^{2}+2\varphi^{2}\big{)}S_{5}}{8}+ \frac{r^{8}S_{6}}{32}\bigg{]}\Bigg{\}} \tag{1}\]  
\[E_{y}=E\varphi w\bigg{\{}e^{2}S_{2}+e^{4}\bigg{[}r^{2}S_{4}-\frac{r^{4}S_{5}}{ 4}\bigg{]}\bigg{\}} \tag{2}\]  
\[E_{z}=E\varphi\bigg{\{}eC_{1}+e^{3}\bigg{[}-\frac{C_{2}}{2}+r^{2}C_{3}-\frac{ r^{4}C_{4}}{4}\bigg{]}+e^{5}\bigg{[}-\frac{3C_{3}}{8}-\frac{3r^{2}C_{4}}{8}+ \frac{17r^{4}C_{5}}{16}-\frac{3r^{6}C_{5}}{16}-\frac{3r^{6}C_{6}}{8}+\frac{r^ {8}C_{7}}{32}\bigg{]}\bigg{\}} \tag{3}\]  
Likewise, the magnetic field components can also be derived and expressed as:  
\[B_{x}=0 \tag{4}\]  
\[B_{y}=E\bigg{\{}S_{0}+e^{2}\bigg{[}\frac{r^{2}S_{2}}{2}-\frac{r^{4}S_{3}}{4} \bigg{]}+e^{4}\bigg{[}-\frac{S_{2}}{8}+\frac{r^{2}S_{3}}{4}+\frac{5r^{4}S_{4} }{16}-\frac{r^{6}S_{5}}{4}+\frac{r^{8}S_{6}}{32}\bigg{]}\bigg{\}} \tag{5}\]  
\[B_{z}=Ew\bigg{\{}eC_{1}+e^{3}\bigg{[}\frac{C_{2}}{2}+\frac{r^{2}C_{3}}{2}- \frac{r^{4}C_{4}}{4}\bigg{]}+e^{5}\bigg{[}\frac{3C_{3}}{8}+\frac{3r^{2}C_{4}} {8}+\frac{3r^{4}C_{5}}{16}-\frac{8r^{6}C_{6}}{4}+\frac{r^{8}C_{7}}{32}\bigg{]} \bigg{\}} \tag{6}\]  
\(\varphi=x/b_{0}\), \(w=y/b_{0}\), and \(E=E_{0}\frac{b_{0}}{E}exp\bigg{[}-\frac{r^{2}}{L^{2}}-\frac{r^{2}}{b^{2}}\bigg{]}\), where \(E_{0}\) is equal to \(kA_{0}\). In this model, the beam is approximated as a cylinder, where each cross-section along the optical  
Figure 1: A schematic diagram of the collision between a linearly polarized intense laser pulse and a relativistic electron. axis is a circle, and the radius \(b(z)=b_{0}\sqrt{1+\left(z/z_{r}\right)^{2}}\) is determined by \(\mathrm{k}=2\pi/\lambda\) and the Rayleigh length \(z_{r}=kb_{0}{}^{2}/2\). The sole position where the laser can remain at its strongest state is at the focal point. At the focal point, the radius of the laser is the minimum and its intensity is the maximum. \(\epsilon=b_{0}/z_{r}\) represents the diffraction angle, \(L\) represents the pulse width, \(L=\tau c\), and the duration of the laser is equal to \(\tau\). \[a_{0}=\frac{eA_{0}}{mc^{2}} \tag{7}\]  
\[S_{n}=\left(\frac{b_{0}}{b}\right)^{n}sin(\varphi+n\varphi_{G}),\ \ \ \ \ n=0,1,2\ldots \tag{8}\]  
\[E_{x}=E\Bigg{\{}S_{0}+e^{2}\bigg{[}q^{2}S_{2}-\frac{r^{4}S_{3}}{4}\bigg{]}+e^{ 4}\Bigg{[}\frac{S_{2}}{8}-\frac{r^{2}S_{3}}{4}-\frac{r^{2}\big{(}r^{2}-16\varphi ^{2}\big{)}S_{4}}{16}-\frac{r^{4}\big{(}r^{2}+2\varphi^{2}\big{)}S_{5}}{8}+ \frac{r^{8}S_{6}}{32}\bigg{]}\Bigg{\}} \tag{1}\]0  
Herein, the phase of the laser pulse can be expressed as \(\varphi=\varphi_{P}+\varphi_{0}+\varphi_{R}-\varphi_{G}\), where \(\varphi_{P}=\eta=\omega t-kz(\omega=\frac{\varsigma}{\lambda},k=\frac{2\pi}{\lambda})\) represents the phase of the plane wave, and \(\varphi_{0}\) represents a constant phase. \(\varphi_{R}=k\rho^{2}/2z(1+z_{0}{}^{2}/z^{2})\), which is related to the curvature of the wavefront. The Gouy phase \(\varphi_{G}=tan^{-1}(z/z_{0})\), indicating that during the process where z increases from \(-\infty\) to \(+\infty\), the Gaussian pulse undergoes a total phase change of \(\pi\).