###### Abstract
However, the FEL performance depends on the achievable qualities of electron beams in the undulators. The transverse coherence depends on the ratio of geometric emittance to photon wavelength [19]. The FEL gain length increases with the increase of the transverse emittance and uncorrelated energy spread of the electron bunch, and decreases with an increase in peak current [12]. The obtained gain length should be sufficiently small, such that the SASE process at a resonant wavelength can grow within a given undulator length until saturation. Typically, an undulator length of about 20 times the obtained gain length allows to stably deliver high-power photon beams to the users at a suitable wavelength. In this article, transverse emittance budgets of the electron bunches are first optimized in the photo-injector via transverse laser pulse shaping at a series of accelerating gradients at the cathode in the simulations. The obtained bunches are transported through the present XFEL beamline [20] via start-to-end simulations with collective effects. This also  
Figure 1: Sketch of the European XFEL: (**a**) machine layout; (**b**) injector section layout. [GUN: electron gun; II: injector section; L1, L2 and L3: the first, second and third (main) acceleration section; BC0, BC1 and BC2: the first, second and third bunch compressors; SA1 and SA2: two hard X-ray undulator beamlines; SA3: a soft X-ray undulator beamline; A\({}_{ij=125}\) denotes 25 RF stations; AH1 stands for a 3.9 GHz module; TDS: transverse deflecting structure]. includes compression of the electron bunches in the multi-stage compression system [21] to a few kiloamperes' peak current by the front of the undulator beamline. Using the optimized electron bunches, SASE performance in the sub-angstrom working regime of the facility, here covering a photon energy range from 9.3 to 45 keV is explored based on existing undulators at the European XFEL. This considers a typical electron energy of 14 GeV as in routine user runs as well as a full beam energy of 17.5 GeV which was demonstrated in previous operation. It should be noted that, while some of the advanced lasing concepts mentioned above have been under development and/or are being planned at the European XFEL, they are not considered in this study. Additionally, in light of global developments towards continuous-wave (CW) operation [22; 23; 24; 25; 26; 27; 28; 29] of FEL facilities at LCLS-II, SHINE, MariX, and so forth, a potential dual-mode operation of the European XFEL is under consideration. This refers to a present pulsed operation mode with a duty cycle of about 1%, that is, a burst of up to 2700 bunches within a 0.6 ms-long 10 Hz-repetition bunch train, and a CW mode at 100% duty cycle. Compared to the pulsed mode, both the cathode accelerating gradient and the total beam energy are reduced in the CW mode [30; 31; 32]. Therefore, the SASE performance of a potential CW mode, operating at reduced cathode gradients of 40 MV/m and 50 MV/m and a reduced final beam energy of about 7 GeV, is studied in parallel to the pulsed mode of the present machine. This aims to justify the electron bunch qualities under distinct working-mode scenarios. It should also be pointed out that only the existing undulators at the European XFEL with a period of 4 cm are taken into account for computations. Detailed design studies of a complementary CW mode with requirements and considerations of modifying the present facility will be reported in [33]. The paper is organized as follows: In Section 2, the transverse shape of the cathode drive laser pulse is optimized based on particle tracking simulations, where three-dimensional space-charge fields are computed iteratively. An optimum transverse distribution with a truncated-Gaussian profile is selected for linearizing space-charge fields in close vicinity of the cathode, while keeping the longitudinal characteristics (length and shape) of the cathode drive laser pulse the same as the ones which are currently used at the facility.