## 2 Materials and Methods
Regarding the motion of the electron, it can be described by the Lorentz equation and the electron energy function. We can describe the motion of the electron using the Lorentz equation and the electron energy function. The formulas are presented as follows:  
\[\left\{ \begin{array}{c}\frac{d}{dt}\mathbf{p}=-e\big{(}\mathbf{E}+\frac{\pi}{c}\times \mathbf{B}\big{)}\\ \frac{dK}{dt}=-e\big{(}\mathbf{v}\cdot\mathbf{E}\big{)}\end{array} \right. \tag{10}\]  
Among them, \(K=\gamma m_{e}c^{2}\) represents the energy of the electron, where the definition is given by the Lorentz factor \(\gamma=\left(1-v^{2}/c^{2}\right)^{-\frac{1}{2}}\). In this expression, \(m_{e}\) represents the rest mass of the electron, c represents the velocity of light, and \(v\) represents the velocity of the electron. The momentum of the electron is expressed as \(\mathbf{p}=K\varphi/c^{2}\). In the physical model of Figure 1, the incident laser emitted along the \(+\)z-axis will undergo cross-collision with the electrons. The initial coordinates of the electrons are set as \((x_{0},y_{0},z_{0})\), and by setting \(y_{0}\) and \(z_{0}\) as 0, the coordinates can be simplified to \((x_{0},0,0)\). The velocity is represented as \(u=\frac{v}{c}=\hat{x}u_{x}+\hat{y}u_{y}+\hat{z}u_{z}\). Utilizing the 4-5 th order Runge-Kutta-Fehlberg method (RKF45) and performing numerical simulation computations in MATLAB R2020a, the specific variations in the displacement, velocity, and acceleration of the electrons before and after the collision can be acquired. The time-varying electromagnetic field produced by the moving electron can be deduced using the Lienard-Wiechert potentials. The use of polar coordinates makes it easier to describe the radiation situation in terms of unit angles. Here, the direction vector is defined as \(\hat{n}=\hat{x}sin\theta cos\varphi+\hat{y}sin\theta sin\phi+\hat{z}cos\varphi\), where \(\phi\) represents the azimuth angle and \(\theta\) represents the polar angle. The following formula is used to calculate the radiation power per unit solid angle:  
\[\frac{dP}{d\Omega}=\left\{\frac{|\hat{\mathbf{n}}\times[(\hat{\mathbf{n}}-\mathbf{u}) \times d_{t}\mathbf{u}]|^{2}}{(1-\mathbf{u}\cdot\hat{\mathbf{n}})^{6}}\right\}_{\tau} \tag{11}\]  
Herein, \(\frac{e^{2}\omega^{2}}{4\pi c}\) is the normalized radiation power, \(x_{d}\) is the displacement distance of the electron on the x-axis, the \(\hat{n}\) observation vector represents the observation direction, and the subscript delay time \(\tau=t+(R-x_{d}\cdot\hat{n})/c\) is a condition that must be employed in the equation calculation. Among them, R is the straight-line distance from the observation point to the interaction point between the intense laser and the electron, and this distance should be sufficiently large. The radiated energy is normalized to \(e^{2}/4\pi c^{2}\) per unit angular frequency and per unit azimuth angle, which can be represented by the following equation:  
\[\frac{d^{2}I}{d\omega d\Omega}=\left|\int_{-\infty}^{+\infty}\frac{\hat{\mathbf{n}} \times[(\hat{\mathbf{n}}-\mathbf{u})\times d_{t}\mathbf{u}]}{(1-\mathbf{u}\cdot\hat{\mathbf{n}})^{ 2}}e^{is(t-\hat{\mathbf{n}}\cdot\hat{x}_{d})}dt\right|^{2} \tag{12}\]  
Among them, \(s=\omega_{s}/\omega\) denotes the order of the radiated harmonic, where \(\omega_{s}\) is the scattering angular frequency. Equations (11) and (12) are used to calculate the spatiotemporal distribution and spectrum of the nonlinear cross Thomson scattering (NCTS).