Fully coherent soft X-ray pulse generation based on ERL

Energy recovery linacs (ERLs) possess the bright prospect of fully coherent X-ray generation. Recently, we designed a 600 MeV energy recovery linac capable of producing high power fully coherent radiation pulses at 13.5 nm with a relatively low-intensity 256.5 nm seed laser profiting from the employment of angular-dispersion-induced microbunching (ADM) technology. We also designed a matched multiplexed system that can deflect each radiator by 8 mrad with a carefully choreographed multi-bend achromat (MBA) scheme. As a result of downstream MBA’s dispersion compensation, bunching factors will be enhanced both at the fundamental wavelength and high harmonics. The bunching factor of the 19th harmonic increased from 10% to 26%, and that of the 57th harmonic became 8.4%, which is sufficient to generate fully coherent radiation in the soft X-ray range.


Introduction
Energy recovery linacs (ERLs) can recycle the kinetic energy of a used beam for accelerating a newly injected beam, thus reducing power consumption, utilizing the high injector brightness and dumping at injection energy [1][2][3].Compared with storage rings, ERLs can produce fresh electron beams with lower emittance and energy spread [4], which will optimize the laser beams' flux and spectral brightness while minimizing the source's transverse size and the pulse's duration as a result.These advantages are because every beam only passes the whole circle once and inherits the initial beam quality from the injector.Moreover, ERLs' high energy recovery efficiency makes GHz repetition rates more accessible than normal linacs.The above properties endow ERLs with the capacity to generate radiation with short pulse durations, high spatial coherence, and high average power, making ERLs attractive choices as X-ray sources.ERLs can be utilized in industrial and scientific applications such as photo lithography [5], free electron lasers [6], and inverse Compton scattering [7].
The fundamental principles of ERLs have been successfully demonstrated globally, and several research institutes are considering X-ray ERLs, such as LBNL [8], ANL [9], KEK [10], and Cornell [11].However, it is essential to emphasize that the achievable average current in ERLs is typically lower than that in storage rings.Also, maintaining high-quality electron beams in the arc remains challenging [12], limiting the peak current and the ability to produce shortwavelength, high-power free electron lasers (FELs).
Recent advancements in electron beam manipulation techniques, such as angular-dispersioninduced microbunching (ADM), have shown promise in reducing the required laser power to imprint highly coherent microbunching directly onto the electron beam [13].This technique has been combined with an 600 MeV ERL [14] to obtain fully coherent EUV pulses with high repetition rates and sub-meV spectral resolution, enabling new applications in fields such as materials science, chemistry, and biology.To extend the radiation wavelength to the X-ray band in this scheme, the modulation depth related to the seed laser intensity and the electron beam current needs to be amplified.However, the average power of the seed laser implemented in the complex has reached 26 W with 1.3 GHz repetition rate, which is two or three orders of magnitudes larger than currently achievable value from a commercial laser system [15] and nearly five times of that obtained from frequency-quadrupling of an all-fiberized ytterbium-doped fiber (YDF) master oscillator power amplifier (MOPA) [16], unpractical to be further augmented, thus insufficient to obtain adequate high-harmonic bunching factors for X-ray generation.
On the other hand, a bending system design is necessary to enhance the capacity of multiple beamlines operation in ERLs.In 2010, Y. Li et al. [17] designed a bending system to separate the linearly polarized light generated by the upstream planar undulator system from the circularly polarized radiation of the downstream radiator.However, only one deflection was demonstrated in their research.We have developed a bending system that can bend the same electron beam four times without destroying its microbunching and generate five fully coherent radiations [18].We found that after the first bending section, the high-harmonic bunching factor will increase greatly, making it possible to generate high-power, fully coherent soft X-ray radiation.
In this paper, we considered the secondary utilization of the bending system and conducted the simulation of soft X-ray generation to assess the potential of this scheme.The simulation results reveal the capacity to produce fully coherent 4.5 nm radiation with a peak power of 0.22 MW, indicating the practicality of this approach.

Layout
The schematic layout of the emitting system and the bending section design is shown in Fig. 1.This picture omits the last three repeating radiators and bending sections.The electron beam passing through the arc section of the ERL is transferred to the ADM section in the vertical plane.In this process, the electron beam first travels through a magnet dipole to couple the vertical divergences with the longitudinal locations.Then it will interact with the seed laser in the short modulator and obtain an enlarged energy spread.After the energy interaction, the following dogleg will convert the energy modulation to density modulation utilizing the dispersion and the momentum compaction.Next, the spatially modulated beam will be transmitted to the first radiator, generating the first coherent radiation.After that, the beam bunch will pass through the bending section in the horizontal plane and deflect without destroying the fine structure.Subsequently, the beam will be sent to the next radiator and undergo another radiation process.The bending and radiation processes will repeat several times in the downstream section.The bending section is composed of four identical FODO cells with phase advance of π/2.For the bending section, except for R 56 and T 566 terms, the first and second transfer matrices are optimized to unit ones [18].These optimizations can guarantee microbunching preservation to the fullest extent.

Simulation results
To demonstrate the efficacy of the proposed methodology, three-dimensional simulations were performed utilizing GENESIS [19] and ELEGANT [20], with typical ERL beam parameters [14] presented in Table 1.The beam longitudinal phase space and the calculated bunching factor after the ADM are shown in Fig. 2. In the ADM section, the bend angle is 1 mrad, and the distance between the dogleg is 0.7 m.From Fig. 2 (a), the electrons gather tightly at every wavelength of the seed laser.Here, the seed laser wavelength is 256.5 nm, R 56 is about 0.4 mm, and the ADM parameters are set to form an electron beam with 10% bunching factor at 19 th harmonic to exhibit the effect on the electron beam of subsequent operations.The electron beam will then be transported into the 3.75-meter long radiator to generate the 19 th harmonic radiation at the wavelength of 13.5 nm.The radiator is an in-vacuum undulator with a 1.2 T peak magnetic field at 3.5 mm gap.The K parameter is 1.75.The electron beam longitudinal distribution and the bunching factor after the first radiator are illustrated in Figs. 3 (a) and (b).Due to the pre-bunch, the fully coherent spontaneous radiation will soon appear at the front of the radiator.In addition, because of the slippage effect [21], the electrons at the front of the beam will be continuously overtaken by the radiation behind them, resulting in mutual interaction and energy modulation.The undulator dispersion will then convert energy modulation to density modulation.Fine microbunching within a radiation wavelength will be produced, benefiting from the electron beam's unique small slice energy spread in ERL.The fine structure between the 256.5 nm wavelength of the seed laser can be clearly observed in Fig. 3 (a), and the partially enlarged drawing reveals the sinusoidal shape of the phase space.The fine microbunching greatly enhances the radiation intensity, which in turn reinforces the bunching, creating positive feedback that leads to an exponential gain in radiation power.The first radiation has a peak power of approximately 0.93 MW and a pulse duration of about 40 fs (FWHM).At this point, the spectrum bandwidth is about 0.057%, 1.12 times the Fourier transform limit.
After the first radiation, the electron beam will travel through the bending system.Figures 3 (c) and (d) demonstrate the longitudinal phase space and the bunching factor of the electron beam after the bending system.The fine structure becomes distinctly erect compared with that in Fig. 3 (a), and the bunching increases greatly as a result.The 19 th bunching factor is nearly double that of the beam before the bending section.This phenomenon is because the dispersion of the bending system is designed to be opposite that of the radiator so that they can counteract 14th International Particle Accelerator Conference Journal of Physics: Conference Series 2687 (2024) 032024 each other.The bunching factor of the 57 th harmonic is about 8.4%, close to the value of the 19 th harmonic of the initial electron beam after the ADM part, which means the generation of radiation in the soft X-ray range is possible.
To illustrate the potential of extending the wavelength range of this scheme, the generations of the 19 th , 38 th and 57 th harmonic radiation are simulated in GENESIS.The parameters of the radiators are listed in Table 2.The electron beam in the ERL is smaller than in the storage ring, allowing a smaller gap and higher field for the radiator [22]..75nm and 4.5 nm wavelength radiation, respectively.It is worth noting that the x-axis represents the photon energy relative to the respective central photon energy value in Fig. 4 (b).The simulation results reveal that the 13.5 nm radiation has a peak power of approximately 2.9 MW, a pulse duration of 8.08 µm (FWHM), and a spectral bandwidth of 0.084% (FWHM), 1.14 times the Fourier transform limit.The 6.75 nm radiation, on the other hand, has a peak power of approximately 0.5 MW, a pulse duration of 7.69 µm (FWHM), and a spectral bandwidth of about 0.088% (FWHM), 2.27 times the Fourier transform limit.Finally, the 4.5 nm radiation has a peak power of 0.22 MW, a pulse duration of 7.99 µm (FWHM), and a spectral bandwidth of about 0.03% (FWHM), 1.23 times the Fourier transform limit.Overall, the average power of the 4.5 nm radiation is estimated to decrease to approximately one thirteenth of that of the 13.5 nm radiation, which is still acceptable for practical applications.Moreover, the spectral resolution of 4.5 nm pulse barely increases, which indicates that the radiation is nearly fully coherent.The other differences are relatively small and will not significantly impact the scheme's performance.Based on the simulation results of the radiation, the wavelength coverage has been effectively extended.

Conclusion
In this paper, we utilize the recently proposed MBA section to simultaneously realize multibeamline operation and wavelength extension.Firstly, we choose the ADM scheme to accomplish the pre-bunch of the initial electron beam.Next, the beam will be transported to a radiator to produce the first fully coherent radiation.Following the interaction, the bending section will kick the beam, and the microbunching will be adjusted to qualify for the generation of soft X-rays.After completing the above simulation, we use GENESIS to simulate the generation of radiation at wavelengths of 13.5 nm, 6.75 nm and 4.5 nm individually.The simulation results indicate that the proposed scheme can achieve higher harmonic up-conversion efficiency and implement at least 57 th harmonic fully coherent radiation.This scheme has the potential to extend the wavelength coverage of the ERL-based coherent light source to soft X-ray using a relatively low-intensity seed laser.

Figure 1 .
Figure 1.The schematic layout of the emitting system and the bending section design (left) and the optics of the MBA section (right).

Figure 2 .Figure 3 .
Figure 2. (a) Longitudinal distribution (γ is the Lorentz factor, z is the longitudinal position, λ s = 256.5nm,color map means the probability density of electrons in the vector z); (b) bunching factors b n as a function of the harmonic number n of the electron bunch after ADM.
Figures 4 (a) and (b) illustrate the pulse powers and spectra of 13.5 nm, 6.75 nm and 4.5 nm wavelength radiation, respectively.It is worth noting that the x-axis represents the photon energy relative to the respective central photon energy value in Fig.4 (b).The simulation results reveal that the 13.5 nm radiation has a peak power of approximately 2.9 MW, a pulse duration of 8.08 µm (FWHM), and a spectral bandwidth of 0.084% (FWHM), 1.14 times the Fourier transform limit.The 6.75 nm radiation, on the other hand, has a peak power of approximately 0.5 MW, a pulse duration of 7.69 µm (FWHM), and a spectral bandwidth of about 0.088% (FWHM), 2.27 times the Fourier transform limit.Finally, the 4.5 nm radiation has a peak power of 0.22 MW, a pulse duration of 7.99 µm (FWHM), and a spectral bandwidth of about 0.03% (FWHM), 1.23 times the Fourier transform limit.

Figure 4 .
Figure 4.The radiation pulses and the spectra.