## 3 Budgeting Emittance at Multiple Accelerating Gradients at the Cathode  
In this section, the performance of the EuXFEL injector [34,38] [as shown in Figure 1b] with the currently existing cathode laser pulse shaping capabilities, intrinsic emittance of the cathode, and position of the beamline components is studied for varying peak accelerating gradients at the cathode position. The gun solenoid strength, operation phase of the gun and rms size of the cathode laser spot are considered for the optimization. An optimal transverse shape of the cathode drive laser is applied according to the findings as presented in Section 2. The rms length of the longitudinal Gaussian laser pulse is kept at 3 ps, which is the same as that currently used in routine operation. The electron bunch is generated at the photocathode positioned at 0 m, then accelerated to about 6.5 MeV/c after the gun and boosted to 130 MeV/c downstream cathode at about 22 m, at which position the transverse projected emittance is calculated. The intrinsic energy of the emitted electrons is set to a nominal value of 0.55 eV [44] for cesium-telluride cathodes. A series of numerical simulations are carried out in ASTRA [45], driven by a simple script written in Matlab. The simulations were carried out on the MAXWELL cluster at DESY. Up to 2 million macro-particles are used for simulating 100 pC electron bunches. Interested readers are also referred to Refs. [46, 47, 48] for detailed descriptions about the concept of emittance growth sources and compensation schemes in photocathode RF injectors. Figure 4 shows the optimized overall transverse projected emittance for 100 pC electron bunches at the injector exit using peak accelerating gradients of 56 MV/m, 50 MV/m and 40 MV/m at the cathode. The 56 MV/m case is close to an optimal state of the present operation with the pulsed injector at the European XFEL. The lower gradients, 50 MV/m and 40 MV/m, would have relevance to the case with a CW injector, in which the peak accelerating gradient is reduced to 40 MV/m with a potential improvement to 50 MV/m in superconducting RF guns [49,50]. The beam energy at the gun exit is reduced from about 5.96 MeV in the 56 MV/m case to 5.28 MeV and 4.21 MeV for the 50 MV/m and 40 MV/m cases, respectively. The optimal solenoid field strengths for each optimized case are shown on the right axis of Figure 4. It can be seen, that at a reduced cathode gradient of 50 MV/m, the optimal transverse projected emittance (blue bars) is still comparable to the 56 MV/m case. The comparable emittance is achieved by increasing the laser spot size to relax the volumetric space-charge density. While this comes at the cost of higher intrinsic emittance (Figure 4, green bars), the stronger space-charge effects at lower cathode fields are mitigated to a similar extent. However, the central slice emittance (orange bars) at 50 MV/m is increased, implying the slice formation of the bunch is still affected at a lower cathode gradient and could not be fully compensated. By further decreasing the cathode gradient to 40 MV/m, one can see that both the transverse projected emittance and the central slice emittance are degraded. It should be noted that an increase of the cathode laser pulse length may be able to compensate for the overall emittance growth at 40 MV/m. However, since this study aims to evaluate the injector performance under the conditions more close to practical operation of the facility, further lengthening and/or shaping of the temporal cathode drive laser distribution is not considered. As a reference, in previous facility operation at 100 pC, the projected transverse emittance is typically about 0.33 um at a nominal gun gradient of about 56 MV/m, which is measured by the quadrupole scan method. This is about 30% higher than the optimized emittance one can achieve in the simulations as presented in this paper. One discrepancy source may attribute to the fact that the transverse pulse shape of the cathode drive laser is not yet experimentally optimized.