## 3 Results
### Temporal Spectrum
When \(\phi_{0}=\pi\) (\(E_{x}\) negative peak, \(B_{y}=0\)), the electrons undergo a two-stage process of deceleration followed by acceleration, and the prolongation of the interaction time leads to the widest pulse width; at the same time, in the range of \(\pi\)-\(3\pi/2\), the reversed electric field carries out acceleration leading to a decrease in the pulse width. When \(\phi_{0}=3\pi/2\), the reverse magnetic field is injected, and again, the reverse Lorentz force induces a transverse deflection, but due to the different direction of the initial velocity being asymmetric concerning that at \(\phi_{0}=\pi/2\), there is a subminimal value of the pulse width; at the same time, in the range of \(3\pi/2\)-\(2\pi\), the electric field starts to recover, while the longitudinal field modulates, causing the pulse width to increase again. Thus, the pulse width has a duration of \(2\pi\). As shown in Figure 7a, when the initial phase is 0, we observe a double-peaked structure at the crest of the linearly polarized laser pulse, and subsequently, by enhancing the laser pulse intensity, the double-peak structure remains prominently observable at the peak intensity of the linearly polarized laser pulse, as evidenced in Figure 7b,c. In the online polarized light laser field, when the electron oscillatory motion and the laser electric field phase reach a specific matching condition, its radiation electromagnetic wave in space and time will occur in the phase length interference, resulting in the concentration of radiant energy, pulse width compression, this phenomenon is the coherent  
Figure 6: The dependence of crest pulse width on initial phase (for \(gm_{0}=20\)). Figure 7: Time spectra at different initial energies of electrons: (**a**) initial energy \(gm_{0}=20\); (**b**) initial energy \(gm_{0}=40\); (**c**) initial energy \(gm_{0}=100\). enhancement effect. As illustrated in Figure 2, the motion of the electron remains in phase with the laser's electric field, resulting in a driven oscillatory motion under the influence of the field, the electron's initial phase is zero, then the electron will get the maximum acceleration every half-cycle, and the waveform of the radiation field is synchronized and superimposed. At the same time, the phase difference of electromagnetic waves radiated in adjacent half-cycles is \(2\pi\), which is coherently superimposed in the observation direction. When \(N\) cycles are coherently superimposed, the radiation intensity is enhanced (\(\propto N^{2}\)). In addition, the time-domain interference produces pulse-width compression, resulting in the main peak width \(\Delta\tau\approx\mathrm{T}/(2\gamma_{0}{}^{2})\) (T for the laser field cycle). It is worth noting that due to the combined effects of time-reversal symmetry, parity conservation, and nonlinear polarization selection, even-order harmonics are suppressed. Only odd-order harmonics satisfy the phase-matching conditions and participate in interference, forming a train of pulses. This mechanism has promising applications in generating isolated attosecond pulses. In contrast, when the electron trajectory is misaligned with the polarization direction or phase of the laser field, radiative energy loss occurs, accompanied by pulse broadening. Experimental measurements reveal a clear dependence of pulse duration on initial electron energy: at 20 \(\times\) 0.511 MeV, the pulse width is approximately 0.0443 as; at 40 \(\times\) 0.511 MeV, it decreases to 0.0114 as; and at 60 \(\times\) 0.511 MeV, further compression to 0.0051 as is observed. As illustrated in Figure 7, a comparative analysis of laser pulse profiles for initial energies of 40 \(\times\) 0.511 MeV and 60 \(\times\) 0.511 MeV against the 20 \(\times\) 0.511 MeV reference demonstrates that higher initial electron energies enhance the peak intensity of the laser pulse while simultaneously reducing its temporal width.