###### Abstract  
Sub-Angstrom working regime refers to a working state of free-electron lasers which allows the generation of hard X-rays at a photon wavelength of 1 angstrom and below, that is, a photon energy of 12.5 keV and above. It is demonstrated that the accelerators of the European X-ray Free-Electron Laser can provide highly energetic electron beams of up to 17.5 GeV. Along with long variable-gap undulators, the facility offers superior conditions for exploring self-amplified spontaneous emission (SASE) in the sub-angstrom regime. However, the overall FEL performance relies quantitatively on achievable electron beam qualities through a kilometers-long accelerator beamline. Low-emittance electron beam production and the associated start-to-end beam physics thus becomes a prerequisite to dig in the potentials of SASE performance towards higher photon energies. In this article, we present the obtained results on electron beam qualities produced with different accelerating gradients of 40 MV/m-56 MV/m at the cathode, as well as the final beam qualities in front of the undulators via start-to-end simulations considering realistic conditions. SASE studies in the sub-angstrom regime, using optimized electron beams, are carried out at varied energy levels according to the present state of the facility, that is, a pulsed mode operating with a 10 Hz-repetition 0.65 ms-long bunch train energized to 14 GeV and 17.5 GeV. Millijoule-level SASE intensity is obtained at a photon energy of 25 keV at 14 GeV electron beam energy using a gain length of about 7 m. At 17.5 GeV, half-millijoule lasing is achieved at 40 keV. Lasing at up to 50 keV is demonstrated with pulse energies in the range of a few hundreds and tens of microjoules with existing undulators and currently achievable electron beam qualities. Sub-Angstrom; free-electron lasers; SASE; low-emittance; start-to-end simulations Article  
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Footnote †: Supported: Supported: Supported: Supported: Supported: 11 November 2021  
Footnote †: significantly from boosting the particle energy. The undulator K is proportional to the product of \(\lambda_{w}\) and B [12], where B stands for the undulator field. For a given undulator period, a larger undulator gap results in a weaker magnetic field characterized by a smaller K value, leading to a shorter photon wavelength. Practically, a reachable undulator K value depends on the control capability (range and accuracy) of the undulator gap. Reducing the undulator period certainly shifts the emission regime towards shorter photon wavelengths. The choice of short-period undulators can be either in-vacuum undulators or superconducting undulators [13]. The latter option offers a combination of a short period and a strong magnetic field. Furthermore, superconducting undulators enable period doubling [14], that is, dynamic change of the undulator period, which enhances the tunability and range of accessible photon energies. The standard undulators with longer periods standing out-of-vacuum can still be utilized along with advanced lasing concepts such as harmonic lasing [15; 16] and reverse tapering of the main undulator plus an afterburner for background-free harmonics generation [17; 18]. The interests in operating the European XFEL in the sub-angstrom regime have been initially expressed by our on-site users. This includes, for example, the X-ray microscopy at a MHz rate of various dynamic samples, the MHz X-ray stereography or triscopy via diamond splitters, and so on. Specific properties of the required radiation can be defined by the users once the machine capability is first demonstrated in terms of the SASE wavelength and intensity. At the European XFEL, the superconducting linac can accelerate the electron beam up to 17.5 GeV. The baseline hard X-ray variable-gap undulators have a period of 4 cm. The total undulator length is about 175 m for SA1 undulators [see Figure 1a]. Such a configuration allows the production of photon beams with high energy at 25 keV and above.