## 2 Materials and Methods  
To commence, several normalized fundamental parameters are introduced. The space-time coordinate associated with the laser wave number, as defined in the subsequent formulae, is given by \(k_{0}^{-1}=\lambda_{0}/2\pi\), while the laser frequency is expressed as \(\omega_{0}^{-1}=\lambda_{0}/2\pi c\), where \(\lambda_{0}\) represents the wavelength of 1 \(\upmu\)m, and c denotes the speed of light. This article investigates the Laguerre-Gaussian (LG) laser pulse propagation in an isotropic, homogeneous, non-magnetic, and non-conductive medium, focusing specifically on the cases where the spread parameter \(\sigma_{in}=0\) along the z-axis of the incidence angle. Within a tightly focused Gaussian laser field, the electric and magnetic fields that adhere to Maxwell's equations can be represented as detailed in [23; 24]:  
\[E=\
abla\times A \tag{1}\]  
\[B=\sqrt{\epsilon}\big{[}\bigg{(}\frac{i}{k}\bigg{)}\
abla(\
abla\cdot A)+ikA \big{]} \tag{2}\]  
Herein, A represents the solution pertaining to the subsequent Helmholtz equation:  
\[\
abla^{2}A+k_{0}^{2}A=0 \tag{3}\]  
In a laser field, circularly polarized light can be disintegrated into a pair of linearly polarized components, which possess phase dissimilarity of \(\pi/2\) along the x- and y-axes. Consequently, the electromagnetic field can be similarly expressed as the sum of its x- and y-components, namely \(E=E_{xp}+E_{yp}\) and \(B=B_{xp}+B_{yp}\). [PERSON] [25] and colleagues derived the x-axis polarization of the laser field from Equations (1)-(3). [PERSON] [26] and [PERSON] [23] derived the electromagnetic field expression for the y-axis-polarized laser field by employing a symmetric approach. Figure 1 illustrates the interaction between a tightly focused laser pulse and an electron. The highly nonlinear nature of the electron motion within an intense laser field necessitates the acquisition of a precise two-point electron trajectory, which requires a high-precision representation of the electromagnetic field of the Gaussian beam. Building on the aforementioned analysis, the electric field equation can be accurately described by a fifth-order expansion of the electromagnetic field, which is accurate to the diffraction angle:  
\[ \begin{split} E_{x}=E_{L}\bigg{\{}S_{0}&+\varepsilon ^{2}\Big{[}\xi^{2}S_{2}-\frac{\rho^{4}S_{5}}{4}\Big{]}\\ &+\varepsilon^{4}\Big{[}\frac{S_{2}}{8}-\frac{\rho^{2}S_{3}}{4}- \frac{\rho^{2}\big{(}\rho^{2}-16\delta^{2}\big{)}S_{4}}{16}-\frac{\rho^{4} \big{(}\rho^{2}+2\delta^{2}\big{)}S_{5}}{8}+\frac{\rho^{8}S_{6}}{32}\Big{]}\\ &+\varepsilon^{2}C_{2}+\varepsilon^{4}\left[\rho^{2}C_{4}-\frac{ \rho^{4}C_{5}}{4}\right]\bigg{\}}\end{split}  \tag{4}\]  
\[ \begin{split} E_{y}=E_{L}\bigg{\{}C_{0}&+\varepsilon ^{2}\Big{[}\beta^{2}C_{2}-\frac{\rho^{4}C_{5}}{4}\Big{]}\\ &+\varepsilon^{4}\Big{[}\frac{C_{2}}{8}-\frac{\rho^{2}C_{3}}{4}- \frac{\rho^{2}\big{(}\rho^{2}-16\delta^{2}\big{)}C_{4}}{16}-\frac{\rho^{4} \big{(}\rho^{2}+2\beta^{2}\big{)}C_{5}}{8}\\ &+\frac{\rho^{8}C_{6}}{32}\Big{]}+\varepsilon^{2}S_{2}+\varepsilon ^{4}\Big{[}\rho^{2}S_{4}-\frac{\rho^{4}S_{5}}{4}\Big{]}\bigg{\}}\end{split}  \tag{5}\]  
\[ \begin{split} E_{z}=E_{L}\xi\bigg{\{}\varepsilon C_{1}& +\varepsilon^{2}\Big{[}-\frac{C_{2}}{2}+\rho^{2}C_{3}-\frac{\rho^{4}C_{4}}{4} \Big{]}\\ &+\varepsilon^{5}\Big{[}-\frac{3C_{3}}{8}-\frac{3\rho^{2}C_{4}}{8 }+\frac{17\rho^{4}C_{5}}{16}-\frac{3\rho^{6}C_{5}}{16}-\frac{3\rho^{6}C_{6}}{8 }+\frac{\rho^{8}C_{7}}{32}\Big{]}\bigg{\}}\\ &-K_{L}\beta\bigg{\{}\varepsilon S_{1}+\varepsilon^{3}\Big{[}- \frac{S_{2}}{2}+\rho^{2}S_{3}-\frac{\rho^{4}S_{4}}{4}\Big{]}\\ &+\varepsilon^{5}\Big{[}-\frac{35_{3}}{8}-\frac{3\rho^{2}S_{4}}{8 }+\frac{17\rho^{4}S_{5}}{16}-\frac{3\rho^{6}S_{5}}{16}-\frac{3\rho^{6}S_{6}}{8 }+\frac{\rho^{8}S_{7}}{32}\Big{]}\bigg{\}}\end{split}  \tag{6}\]  
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