## 3 Numerical Simulations
### Modulation before the Radiator  
The initial phase space of the electron beam is shown in Figure 2a. Modulated by a few-cycle laser in the wiggler, energy modulation is imprinted on the longitudinal phase space of the electron beam, as shown in Figure 2b. The laser-induced energy modulation amplitude is A = \(\Delta\gamma\)/\(\Delta\sigma=12\). After passing through a chicane with an optimized strength of R56 = 0.75 mm [29; 35], energy modulation is transformed into density modulation, as shown in Figure 2c. Figure 2d presents the current profile after the chicane with a peak current of about 4800 A.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Section** & **Parameter** & **Value** & **Unit** \\ \hline \multirow{4}{*}{Initial beam} & Average energy & 2.5 & GeV \\  & Peak current & 800 & A \\  & Energy spread & 0.25 & MeV \\  & Average beam radius (RMS) & 30 & \(\upmu\)m \\ \hline \multirow{4}{*}{Modulative laser} & Wavelength & 2400 & nm \\  & Power & 40 & GW \\  & Radius & 0.4 & mm \\  & Pulse duration (FWHM) & 8 & fs \\ \hline \multirow{4}{*}{Wiggler} & K & 39.27 & - \\  & Period & 16 & cm \\  & Period number & 1 & - \\ \hline \multirow{4}{*}{Radiator} & K & 2.75 & - \\  & Period & 3 & cm \\  & Period number & 443 & - \\ \hline \multirow{4}{*}{Afterburner} & K & 2.75 & - \\  & Period & 3 & cm \\  & Period number & 24 & - \\ \hline \multirow{4}{*}{Kicker} & Length of the quadrupole & 10 & cm \\  & Focal length & 10 & m \\ \cline{1-1}  & Transverse offset in y axis & 0.3 & mm \\ \cline{1-1}  & Angle & 15 & \(\upmu\)rad \\ \hline \hline \end{tabular}
\end{table}
Table 1: Main parameters used in the proposed technique.  
Figure 2: _Cont._  
The electron beam with an enhanced current spike is sent into a radiator resonant at 3 nm. The simulation results for the evolutions of the FEL peak power and the pulse duration along the radiator are summarized in Figure 3. The peak power keeps increasing in the 21 m long radiator, while the pulse duration fluctuates from 500 as to 1400 as, due to the slippage effect. To obtain profound insight into the influence of the slippage effect, the evolution of the FEL pulse along the radiator is plotted in Figure 3. In the radiator, the FEL pulse slips ahead by about 1.7 fs for every 5 m with respect to the current peak. This slippage causes a decrease in the gain of the main pulse with a decline in the local current. With a sufficiently large bunching factor in the current peak, new pulses are continually generated, resulting in a large fluctuation of the pulse duration in the radiator.  
Figure 3: FEL performances in the radiator of ESASE: (**a**) Evolutions of the peak power and the pulse duration along the radiator; (**b**) evolution of normalized power along the radiator; (**c**) evolutions of the local bunching factor at the current peak and the maximum bunching factor along the radiator; (**d**) bunching factor profiles at 16 m and 20 m of the radiator, \(t=0\) represents the current peak.  
Figure 2: Modulation processes of ESASE: (**a**) Initial longitudinal phase space distribution; (**b**) phase space at the entrance of the chicane; (**c**) phase space at the entrance of the radiator; (**d**) current profile at the entrance of the radiator.