## 2 Materials and Methods
\[a_{0}=\frac{eA_{0}}{mc^{2}} \tag{7}\]  
\[S_{n}=\left(\frac{b_{0}}{b}\right)^{n}sin(\varphi+n\varphi_{G}),\ \ n=0,1,2,\ldots \tag{8}\]  
\[C_{n}=\left(\frac{b_{0}}{b}\right)^{n}cos(\varphi+n\varphi_{G}) \tag{9}\]  
The phase structure of the laser pulse is constituted by the superposition of four components: \(\varphi=\varphi_{P}+\varphi_{0}+\varphi_{R}-\varphi_{G}\). Herein, the plane wave propagation phase term \(\varphi_{P}=\omega t-kz\big{(}\omega=\frac{c}{\lambda},k=\frac{2\pi}{\lambda} \big{)}\) is jointly determined by the optical angular frequency \(\omega=c/\lambda\) (where c is the speed of light) and the wave vector \(k=2\pi/\lambda\). The fixed phase term \(\varphi_{0}\) is a constant value, independent of space and time. The curvature phase \(\varphi_{R}=k\rho^{2}/2z\big{(}1+{z_{0}}^{2}/{z^{2}}\big{)}\) characterizes the wavefront curvature effect, and its mathematical form indicates that this phase component has a quadratic relationship with the propagation distance z and the transverse coordinate \(\rho\). The Gouy phase term \(\varphi_{G}=tan^{-1}(z/z_{0})\) embodies the specific geometric phase accumulation during the beam propagation. When the beam propagates along the z-axis from negative infinity to positive infinity, this phase component will generate a cumulative phase shift of \(\pi\) radians, forming the core phase feature that distinguishes Gaussian beams from plane waves. The motion of the electron can be depicted by employing the Lorentz equation and the electron energy function. The formulae are presented as follows:  
\[\left\{ \begin{array}{c}\frac{d}{dt}\mathbf{p}=-e\big{(}\mathbf{E}+\frac{u}{c}\times \mathbf{B}\big{)}\\ \frac{dK}{dt}=-e\big{(}\mathbf{v}\cdot\mathbf{E}\big{)}\end{array} \right. \tag{10}\]  
Among them, \(K=\gamma m_{e}c^{2}\) denotes the electron energy, with the Lorentz factor \(\gamma=\big{(}1-v^{2}/c^{2}\big{)}^{-\frac{1}{2}}\). \(c\) represents the speed of light; \(v\) represents the velocity of the electron; and \(m_{e}\) represents the rest mass of the electron. \(\mathbf{p}=K\rho/c^{2}\) indicates the momentum of the electron. The velocity of an electron can be represented as \(u=\frac{v}{c}=\hat{x}u_{x}+\hat{y}u_{y}+\hat{z}u_{z}\). Employing the 4-5 th-order Runge-Kutta-Fehlberg method (RKF45) and conducting numerical simulation via MATLAB-R2020a, parameters such as the velocity and acceleration of the high-energy electron during the interaction process are obtainable. The dynamic electromagnetic field excited by high-speed charged particles can be analytically modeled via the Lienard-Wiechert potentials. In the polar coordinate system, the angular distribution characteristics of the radiation field can be depicted by the unit solid angle parameter, where the spatial orientation vector is defined as \(\hat{n}=\hat{x}sin\theta cos\varphi+\hat{y}sin\theta sin\varphi+\hat{z}cos\varphi\), where \(\theta\) represents the polar angle, and \(\phi\) characterizes the azimuthal angle. The radiation power per unit solid angle can be computed in accordance with the following formula [21]:  
\[\frac{dP}{d\Omega}=\left\{\frac{|\hat{n}\times[(\hat{n}-\mathbf{u})\times d_{t}| ]^{2}}{\big{(}1-\mathbf{u}\cdot\hat{n}\big{)}^{6}}\right\}_{\tau} \tag{11}\]  
Among them, the radiation power is normalized by \(\frac{c^{2}\omega^{2}}{4\pi c}\). In the coordinate system setting, the displacement of the electron along the x-axis is denoted as \(x_{d}\). In the delay calculation requirement, the key parameter \(\tau=t+(R-x_{d}\cdot\hat{n})/c\) is introduced, where the observation distance \(R\) is required to satisfy the far-field approximation condition \((R\gg\lambda)\), that is, a collinear distance needs to be maintained between the measurement point and the laser-electron interaction point.