## 2 Materials and Methods  
The spatial and temporal coordinates in all the physical quantities are normalized by \(k_{0}^{-1}=\lambda_{0}/2\pi\) and \(\omega_{0}^{-1}=\lambda_{0}/2\pi c\), where \(k_{0}^{-1}\) represents the wave number, \(\omega_{0}^{-1}\) represents the laser pulse frequency, \(\lambda_{0}=1\) um is the laser pulse wavelength, and \(c=2.998\times 10^{8}\) m/s is the speed of light. In this study, the observer is used as the reference frame, with \(t\) defined as the time in the observer's frame. For ease of analysis, we introduce the electron's reference frame and define \(t_{e}=t-R^{\prime}+\mathbf{n}\cdot\mathbf{j}\) as the time in the electron's reference frame for the derivation, where \(R^{\prime}\) is the distance between the observer and the interaction center, and \(\mathbf{j}=(x_{te},y_{te},z_{te})\) is the electron's displacement at time \(t_{e}\). The radiation direction is \(\mathbf{n}=(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)\), where \(\theta\) is the polar angle, and \(\phi\) is the azimuthal angle. The initial position of the electron, \((x_{0},y_{0},z_{0})\), is set as \((0,0,0)\), where the electron moves along the negative \(z\)-axis from the origin and undergoes a coaxial collision with the laser pulse propagating along the positive \(z\)-axis. The interaction process between chirped pulses and electrons is illustrated in Figure 1. In a tightly focused Gaussian laser pulse field, the motion of electrons can be described by the electric field \(\mathbf{E}\) and magnetic field \(\mathbf{B}\), which satisfy [PERSON]'s equations [21]:  
\[\mathbf{E}=\
abla\times\mathbf{A} \tag{1}\]  
\[\mathbf{B}=\sqrt{\epsilon}\left[(i/k)\
abla(\
abla\cdot\mathbf{A})+ik\mathbf{A}\right] \tag{2}\]  
where \(\epsilon\) is the permittivity, the solution \(\mathbf{A}\) to the Helmholtz equation is as follows:  
\[\
abla^{2}\mathbf{A}+k_{0}^{2}\mathbf{A}=0 \tag{3}\]  
In the Cartesian coordinate system, circularly polarized laser pulse is synthesized using a pair of linearly polarized lasers with a phase difference of \(\pi/2\), whose polarization planes are perpendicular to the \(y\)-axis and \(x\)-axis, respectively. Thus, the electromagnetic field of the circularly polarized laser pulse can be decomposed as \(\mathbf{E}=\mathbf{E}_{\mathbf{x}\mathbf{p}}+\mathbf{E}_{\mathbf{y}\mathbf{p}}\) and \(\mathbf{B}=\mathbf{B}_{\mathbf{x}\mathbf{p}}+\mathbf{B}_{\mathbf{y}\mathbf{p}}\). In this model, the Gaussian laser field has a spot size of \(2w_{0}\), a pulse width of \(L\), and an intensity of \(l\). Therefore, we can define the following:  
\[E_{0}=a_{0}\frac{b_{0}}{b}\mathrm{exp}\left(-\frac{\eta^{2}}{L^{2}}\right) \mathrm{exp}\left(-\frac{\rho^{2}}{B^{2}}\right) \tag{4}\]  
Figure 1: The laser pulse and electron coaxial collision model diagram. Here, \(a_{0}=\sqrt{I\lambda_{0}^{2}/1.384}\times 10^{-9}\) represents the peak amplitude, and \(b=b_{0}\sqrt{1+z^{2}/z_{0}^{2}}\) is the beam waist radius, where the Rayleigh range is \(z_{0}=k_{0}b_{0}^{2}/2\). The longitudinal distance is \(\eta=z-t\), and the transverse distance is \(\rho=\sqrt{x^{2}+y^{2}}\). Under tightly focused laser conditions [22], the expressions for the electromagnetic fields must be expanded to higher orders that remain accurate with the diffraction angle \(\varepsilon=b_{0}/z_{0}\). [PERSON] et al. [2] derived the components \(\mathbf{A}\), \(\mathbf{B}\), and \(\mathbf{E}\) in the \(x\)-axis linearly polarized laser field from Equations (1)-(3). [PERSON] [23] and [PERSON] [22] proposed the electromagnetic field expressions for \(y\)-axis linearly polarized laser fields using symmetric formulations. Through vector synthesis, the fifth-order expansion of the electric field components \(\mathbf{E  
laser acceleration use low-order Gaussian or Bessel beams, our goal is to achieve electron radiation at GeV or even TeV energy levels, which requires extremely high-intensity laser fields. In [PERSON] et al.