# Advanced Scheme to Generate MHz, Fully Coherent FEL Pulses at nm Wavelength
## 3 Resonator Considerations
### Cavity Design Considerations
For a wavelength of 15 nm with a mirror 15 m downstream of the undulator under a glancing angle of 1 degree, from Equation (6) the spot size is approximately 0.3 cm\({}^{2}\), assuming a 100 \(\upmu\)m beam size. With a reflectivity of 99% (\(R=0.99\)) and 1 mJ of pulse energy per second for a single pulse, the power density is around 1 mW/0.3 cm\({}^{2}\), or up to 17 W/cm\({}^{2}\) for a pulse train of 5000 pulses per second. Under these assumptions and taking into account the reflectivity, the absorbed power of FLASH2 on the mirror is up to 170 mW/cm\({}^{2}\) for 15 nm. For a modulator with the mirror at normal incidence at the same distance of 15 m, the same electron beam size and a wavelength of 50 nm, the spot is from Equation (5) approximately 2.4 by 2.4 mm. Assuming again Molybdenum mirrors with 95% absorption, the pulse energy should not exceed 2 mJ in order to avoid an absorbed power density higher than 170 mW/cm\({}^{2}\). At 45 degrees with 60% absorption, the pulse energy would be approximately 5 mJ. Assuming a typical pulse duration of 100 fs, the peak power is therefore 20 MW (or 50 MW for the 45 degree mirror case), which is consistent with the values mentioned earlier for FLASH. For a CW-FEL, the numbers are more critical because of the larger number of bunches per second. Finally, we would like to comment on the geometry of the optical feedback system. There is a number of components needed in order to maintain a stable operation and diagnose the radiation field properties. The intensity of the seed laser, which in this case is the intensity inside the resonator, needs to be regulated and therefore measured for a large wavelength range without significant distortion of the radiation field. Furthermore, with the system starting from noise, the noise needs to be suppressed, which is best done with a grating. Finally, the radiation needs to be refocused in the middle of the modulator. Therefore, the actual resonator will have a more complicated geometry than depicted earlier. A ring resonator could include all needed elements, but other geometries should be considered and compared depending on the wavelength requirements and space constraints of a specific facility. The technical design and specifications are, however, beyond the scope of this paper. Simulation Results and Implementation Considerations for Oscillator-Based Seeding Starting from Shot Noise  
In this section, we focus on an HGHG-based oscillator scheme as shown in Figure 1 and more specifically, in the case of an oscillator-FEL starting from shot noise. As shown in Figure 5b, when the process in the cavity starts from shot noise, there are two separate operation regimes to be considered. For a transition from positive net gain (\"build-up\") to zero net gain (\"steady-state\"), the gain has to be reduced. Here, we discuss different methods that could be applied in order to achieve control over the power gain in the resonator. In all cases we use the same set of simulation parameters, which is summarized in Table 1, and the modulator is resonant with 50 nm wavelength. For the sake of simplicity, here we restrict ourselves to the case of a relative energy modulation \(A=\Delta E/\sigma_{E}=7\), meaning that the amplitude of the energy modulation \(\Delta E\) after the modulator is seven times larger than the initial energy spread \(\sigma_{E}\) in the steady-state regime. As seen in Equation (3), for given lattice, electron beam parameters and constant waist size, the energy modulation is stabilized if the input peak power in the modulator \(P_{in}\) is stable too. All simulations here are done with Genesis 1.3 for the FEL process [39], while the radiation field in the cavity is treated with ocelot [50], which accounts for the slippage, reflectivity, focusing, and monochromatization.