# Chirped-Pulse Amplification in an Echo-Enabled Harmonic-Generation Free-Electron Laser
###### Abstract  
The field of ultrafast science has experienced significant growth over the last decade, largely attributed to advancements in optical and laser technologies such as chirped-pulse amplification and high-harmonic generation. The distinctive characteristics of intense ultrafast free-electron lasers (FELs) have introduced novel prospects for investigating molecular dynamics, as well as providing an opportunity to gain deeper insights into nonlinear processes in materials. Therefore, high-power ultrafast FELs can be widely used for both fundamental research and practical applications. This study presents a novel approach for producing high-power femtosecond FEL pulses, utilizing chirped-pulse amplification in echo-enabled harmonic generation. Chirped seed pulses are employed to induce frequency-chirped energy modulation in the electron beam. The generated FEL pulse, which inherits the chirped frequency, can be compressed through the gratings in the off-plane mount geometry to provide ultraintense ultrafast pulses. The numerical modeling results indicate that peak power exceeding 20 GW and a pulse duration in the order of several femtoseconds can be achieved.  
free-electron laser; ultrafast pulses; high-power; chirped-pulse amplification +
Footnote †: journal: _Applied_  
CPA-SASE has been studied both theoretically and experimentally [20; 21; 22]. However, it is difficult to guarantee the phase relationship starting from the shot noise, leading to a limited compression ratio. By virtue of the coherent interplay between crystal optics and FEL, a novel \"self-seeding\" CPA scheme has been proposed recently, which utilizes Bragg crystals as the pulse stretcher and compressor instead of gratings [23]. This approach can potentially deliver femtosecond hard X-ray pulses with high peak power. Represented by high-gain harmonic generation (HGHG) [24] and echo-enabled harmonic generation (EEHG) [25; 26; 27; 28], the seeded FELs can inherit the proprieties of the external seed lasers, exhibiting notable advantages such as full coherence, precise phase control, and so on. Therefore, employing external seeded FEL schemes appears to be a more-viable approach towards integrating the CPA technique. A preliminary CPA experiment based on direct seeding at 800 nm has been performed at BNL, which showed a noticeable shortening of the FEL pulse after compression [29]. Theoretical studies on CPA-HGHG have demonstrated the potential to generate femtosecond UV pulses (\(\sim\)260 nm) with peak powers of \(\sim\)100 MW [30]. In [31], the researchers demonstrated the possibility of carrying out CPA-HGHG at FERMI@Elettra. However, the harmonic up-conversion coefficient of HGHG imposes a constraint on its potential for broadening applicable wavelengths towards shorter regimes [32].  
The aforementioned limitations can potentially be overcome by employing the EEHG technique, which involves the implementation of two modulator-chicane modules to induce substantial bunching at high harmonic numbers. Successful attainment of coherent emission at the 75 th harmonic of the seed laser has been demonstrated at SLAC using the EEHG technique [33]. Furthermore, coherent signals up to the 101 st harmonic have been observed at FERMI@Elettra [34].  
This paper presents a proposal that explores the generation of high-power and ultrafast FEL pulses through the combination of the CPA technique with EEHG. The simulation results demonstrate that this approach has the potential to produce coherent 13.5 nm FEL pulses, with a peak power exceeding 20 GW and a pulse duration in the order of several femtoseconds. The off-plane mount (OPM) geometry [35] was employed to implement the compressor in grazing incidence. The detailed design and transmission efficiency simulation of the compressor are also included.