## 3 Results
### Temporal Spectrum  
The time-domain radiation spectrum was further examined by sweeping the initial phase from 0 to \(2\pi\) (steps of \(\pi/18\)) at an initial electron energy of 20.511 MeV. As shown in Figure 5a-h, we can see that the time spectra at different initial phases are not the same, showing a symmetric bimodal distribution at initial phases 0 and \(\pi\), and a single-peak structure at the rest of the phases, while at the same time we observe that the time spectra repeat the same variation at every cycle \(\pi\). In reality, during the microscopic transients, the periodic variation of the time spectrum is synchronized with the laser electric field period with a period of \(2\pi\), which is due to the half-period inversion symmetry. The indistinguishability of positive and negative half-period radiation leads to macroscopic observations with a time-spectrum period equal to the laser electric field half-period (\(\pi\)). Figure 4: Dependence of radiation emission on polar angle across different initial energies (initial phase = 0). (**a**) initial energy \(gm_{0}=20\); (**b**) initial energy \(gm_{0}=40\); (**c**) initial energy \(gm_{0}=60\); (**d**) initial energy \(gm_{0}=80\); (**e**) initial energy \(gm_{0}=100\). The time spectrum under tightly focused line polarization is highly sensitive to the initial phase and shows a periodic modulation of the peak high harmonic radiation with a period of \(\pi\). This is due to the unidirectional electric field of the line-polarized light, resulting in a periodic energy modulation of the electron motion directly coupled to the phase. The electron motion undergoes a complete acceleration-deceleration process every half laser cycle (\(\pi\)), corresponding to an increase and then a decrease in the peak energy, which is strictly related to the initial phase. This physical mechanism has a good application for phase control to achieve an active modulation of harmonic intensity. On the contrary, the initial phase under tightly focused circular polarization has a minimal effect on the time spectrum and shows significant similarity across spectra. As the initial phase changes, the pulse width under linearly polarized light varies, fluctuating in the range of 0.0415-0.0477 as. Figure 6 illustrates the dependence of pulse width on initial phase. Through the images as well as the related data, we can see that the pulse width fluctuates significantly with the change of the initial phase, showing phase sensitivity and nonmonotonicity, and, at the same time, periodicity, with a period of \(2\pi\). In addition, we can see that the pulse width reaches a minimum near \(\phi_{0}=\pi/2\); a subminimum near \(\phi_{0}=2\pi/3\); and an extreme maximum near \(\phi_{0}=\pi\). The core physical mechanism is due to the nonlinear coupling effect of the field component. At the same time, the longitudinal field distorts the electron trajectory, and introduces the additional acceleration component, the electron is subjected to longitudinal acceleration, and the motion trajectory is complicated, which widens the radiation pulse. When \(\phi_{0}=0\) (\(E_{x}\) peak, \(B_{y}=0\)), a single electric field accelerates and the acceleration lasts for a longer period, creating a moderate pulse width. Thus, in the range 0-\(\pi/2\), electric field acceleration dominates, and the pulse width decreases rapidly. When \(\phi_{0}=\pi/2\) (\(E_{x}=0\), \(B_{y}\) peak), the magnetic field deflection dominates, and the Lorentz force induces transient transverse deflection, forming ultrafast transverse oscillations, at which time the pulse is the narrowest;at the same time, in the range of \(\pi/2\)-\(\pi\), the electric field reverses and couples with the longitudinal field at the same time, resulting in a sharp increase in the pulse width.