## 4 Conclusions  
In conclusion, in this study, we demonstrated that some properties of the ESASE process can be improved by using attosecond afterburners. The simulation results show that this method can produce FEL pulses with a pulse duration of 389 as and a peak power of 1 GW, while the pulse duration in the conventional ESASE process is 720 as. In addition, simulations were also performed to verify the superior stabilities of attosecond afterburners against SASE fluctuation as compared with the conventional ESASE process. The results indicate that the fluctuations of the output pulse duration and peak power from the attosecond afterburner is about half of that in the conventional ESASE process. The optimizing methods and the results demonstrated here are also applicable to other soft X-ray FELs with similar parameters. In practical operation, a reverse-tapered radiator [49] can be applied upstream of the afterburner, producing attosecond pulses with minimal background. The radiator may be at a harmonic enabling two color pulses, that could be recombined on a sample, proper selection of the recombination beam paths could allow time coincidence, or even allow some delay scan capabilities. In this paper, we took ESASE as an example to show the performance of the proposed technique. It should be highlighted that the proposed technique is also suitable for other SASE-based ultrafast FEL generation schemes, such as emittance spoiler [17], nonlinear bunch compression [24], and fresh-slice [25] based methods. A proof-of-principle experiment for the attosecond afterburner is under preparation at the Shanghai soft X-ray FEL facility (SXFEL) [50]. Studies with the realistic parameters of the SXFEL and considering various three-dimensional effects, such as LCS in undulators, are ongoing. Conceptualization, S.Z. and C.F. ; methodology, L.T., C.F., and Z.W. ; software, X.W., H.S., W.F., and Y.L. ; validation, C.F. and Z.Z. ; formal analysis, L.T. and X.W. ; investigation, L.T. and C.F. ; resources, C.F. and Z.W. ; data curation, X.W. ; writing--original draft preparation, L.T. ; writing--review and editing, L.T., Z.W., Z.Q., S.Z., and C.F; visualization, L.T. and S.Z. ; supervision, Z.W. and Z.Z. ; project administration, C.F. and Z.Z. ; funding acquisition, Z.W., C.F., and Z.Z. All authors have read and agreed to the published version of the manuscript. Figure 10: FEL performance in the afterburner with the LSC considered: **(a)** Development of the peak power and the pulse duration in the afterburner; **(b)** FEL profile at 1.47 m in the afterburner. **Funding:** This work was supported by the National Natural Science Foundation of China (12122514 and 11975300) and the Shanghai Science and Technology Committee Rising-Star Program (20 QA1410100). **Institutional Review Board Statement:** Not applicable. **Informed Consent Statement:** Not applicable. **Data Availability Statement:** Not applicable. **Acknowledgments:** We would like to thank [PERSON] and [PERSON] for useful discussions and help in the simulations. **Conflicts of Interest:** The authors declare no conflict of interest. * [PERSON] et al. (2017) [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON] Generating single-spike hard X-ray pulses with nonlinear bunch compression in free-electron lasers. _Phys. Rev. Lett._**2017**, _119_, 154801. [CrossRef]
* [PERSON] et al. (2018) [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON] [PERSON]; [PERSON]; [PERSON] High-power femtosecond soft x rays from fresh-slice multistage free-electron lasers. _Phys. Rev. Lett._**2018**, _120_, 264801. [CrossRef]
* [PERSON] et al. (2022) [PERSON]; [PERSON]; [PERSON] Generating Isolated Attosecond X-ray Pulses by Wavefront Control in a Seeded Free-Electron Laser. _Ultrafast Sci._**2022**. [CrossRef]
* [PERSON] et al. (2019) [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON]; et al. Tunable isolated attosecond X-ray pulses with gigawatt peak power from a free-electron laser. _Nat.