## 2 Materials and Methods  
First, a few normalization parameters are introduced. The spatiotemporal coordinates are normalized by the laser wave number \(k_{0}^{-1}=\lambda_{0}/2\pi\), with the temporal scale defined via the laser angular frequency \(\omega_{0}^{-1}=\lambda_{0}/2\pi c\) (where \(c\) is the speed of light, \(\lambda=1\)\(\mu m\)). This study examines the propagation dynamics of Laguerre-Gaussian (LG) laser modes in linear dielectric media (\(\varepsilon_{r}>1,\mu_{r}=1,\sigma_{in}=0\)), focusing on incidence along the z-axis. In a tightly focused Gaussian laser field, the electric and magnetic fields following [PERSON]'s equations can be expressed as follows:  
\[\mathrm{E}=\
abla\times\mathrm{A} \tag{1}\]  
\[\mathrm{B}=\sqrt{\epsilon}\bigg{[}\bigg{(}\frac{i}{k}\bigg{)}\
abla(\
abla \cdot A)+ikA\bigg{]} \tag{2}\]  
where the vector potential A is the relevant solution of the Helmholtz equation, describing the distribution in the space in which it is located. \[\
abla^{2}\mathrm{A}+\mathrm{k}_{0}^{2}\mathrm{A}=0 \tag{3}\]For linearly polarized light in a laser field, the electric field oscillates in a fixed plane and can be resolved into two orthogonal components along the x-axis and y-axis. These components have a phase difference of either 0 or \(\pi\), allowing the total electromagnetic field to be expressed as the vector sum \(\mathrm{E}=\mathrm{E}_{\mathrm{xp}}+\mathrm{E}_{\mathrm{yp}}\), \(\mathrm{B}=\mathrm{B}_{\mathrm{xp}}+\mathrm{B}_{\mathrm{yp}}\). The x-polarized component of the laser field was analytically obtained from Equations (1) to (3) in the work of [PERSON] et al. [21]. For the y-polarized component, equivalent field expressions were derived through a similar theoretical framework by [PERSON] [22] and [PERSON] [23]. Figure 1 shows the interaction between a tightly focused laser pulse and electrons. Since the motion of electrons in a high-intensity laser field is strongly nonlinear; an exact analytical representation of the electromagnetic field expression for a Gaussian beam is required to obtain the exact trajectory of the electrons. We can extend the diffraction angle to the fifth order to accurately describe the electromagnetic field. \[E_{x}=E\{S_{0}+\varepsilon^{2}[\ \xi^{2}-\frac{\rho^{4}S_{3}}{4}]+\varepsilon^{4}[ \frac{S_{2}}{8}-\frac{\rho^{2}S_{3}}{4}-\frac{\rho^{2}(\rho^{2}-16\ \xi^{2})S_{4}}{16}]\} \tag{4}\]  
\[E_{y}=E\xi\zeta\{\varepsilon^{2}S_{2}+\varepsilon^{4}[\rho^{2}S_{4}-\frac{ \rho^{4}C_{4}}{4}]\} \tag{5}\]  
\[E_{z}=E\xi\{\epsilon C_{1}+\varepsilon^{3}[-\frac{C_{2}}{2}+\rho^{2}C_{3}- \frac{\rho^{4}C_{4}}{4}]+\varepsilon^{5}[-\frac{3C_{3}}{8}-\frac{3\rho^{4}C_{ 4}}{8}+\frac{17\rho^{4}C_{5}}{16}-\frac{3\rho^{6}C_{6}}{8}+\frac{\rho^{8}C_{7}} {32}]\} \tag{6}\]  
\[B_{x}=0 \tag{7}\]  
\[B_{y}=E\{S_{0}+\varepsilon^{2}[\frac{\rho_{2}^{2S}}{2}-\frac{\rho_{3}^{4S}}{4 }]+\varepsilon^{4}[-\frac{S_{2}}{8}+\frac{\rho_{3}^{2S}}{4}+\frac{5\rho_{4}^{ 4S}}{16}-\frac{\rho_{5}^{6S}}{4}+\frac{\rho_{6}^{8S}}{32}]\} \tag{8}\]  
\[B_{z}=E\zeta\{\varepsilon C_{1}+\varepsilon^{3}[\frac{C_{2}}{2}+\frac{\rho^{ 2}C_{3}}{2}-\frac{\rho^{4}C_{4}}{4}]+\varepsilon^{5}[\frac{3C_{3}}{8}+\frac{ 3\rho^{2}C_{4}}{8}+\frac{3\rho^{4}C_{5}}{16}-\frac{\rho^{6}C_{6}}{8}+\frac{ \rho^{8}C_{7}}{32}]\} \tag{9}\]  
Let \(\xi=x/\omega_{0}\) and \(\zeta=y/\omega_{0}\) denote the normalized coordinates, where \(\omega=\omega_{0}(1+(\mathrm{z}/z_{\mathrm{R}})^{2})^{1/2}\) is the beam waist radius along the z-axis. The radial distance \(\mathrm{r}=\sqrt{(x^{2}+y^{2})}\) is normalized as \(\rho=\mathrm{r}/\omega 0\), and the angle of diffraction is characterized by \(\varepsilon=\omega_{0}/z_{r}\), with \(z_{r}=\mathrm{k}\ \omega_{0}^{2}\) (\(k^{-1}=\lambda/2\pi\)) being the Rayleigh length.