## 3 Results  
Figure 3 shows the scheme of the coherent source, constituted by a short (about 7-10 m long) undulator module with period \(\lambda_{w}\) = 1.2 cm, embedded within a ring cavity. For an electron beam repetition rate of 1 MHz, the round trip of the cavity must be 300 m. The sequence of the accelerated beam packets entering the undulator is synchronized with the radiation reflected and recirculated by hard X-ray mirrors. Three Diamond mirrors and one beam splitter, made by Zincblend perfect single crystals (lattice parameters: \(a=b=c=3.566\) A, \(\alpha=\beta=\gamma=90\)) operating in the range of about 3.0-3.5 keV, namely 3.6-4 A(with an incidence angle \(\theta<30\) from normal), have been considered. At least two of the mirrors are assumed to be bent and focusing. Figure 4 presents the transfer function T of the total optical line as a function of the energy of the photons for a central energy of 3.015 keV and a quasi-orthogonal reflection (incidence angle \(\Theta_{0}=3\)), compared with the natural SASE spectral line. The mirror transfer function is narrower than the natural FEL spectral line and the spectral filtering of the mirrors is the dominant effect in the reduction of the spectral width. The angular filtering is instead absolutely inefficient, since the angular width of the transfer function is about 2 mrad, to be compared to the much smaller SASE divergence of 25 \(\upmu\)rad. FEL simulations were performed for a 50 pC electron beam. Figure 5 shows the intracavity radiation energy as a function of the number of round trips. The saturation is reached in about 50 cycles, with an intracavity energy level of about 20 \(\upmu\)J. Due to the short undulator and wavelength, the slippage is very low: the simulated time window (12 \(\upmu\)m) is thus adjusted to the electron beam and reported in the inner boxes of Figure 5.  
Figure 3: Scheme of the X-ray FEL Oscillator: \(\lambda_{w}=1.2\) cm undulator module within a ring cavity made of three Diamond mirrors (\(M_{2}\), \(M_{3}\) and \(M_{4}\)) and one beam splitter (\(M_{1}\), outcoupling the radiation). The X-ray radiation is emitted from right to left. \(\Theta_{0}\) is the incidence angle on the cavity mirrors with respect to the normal. The distance \(M_{1,4}\)–\(M_{3,2}\) between two mirrors is 75 m, giving a cavity round trip length of 300 m, equal to the distance between two successive electron bunches. Figure 4: Transfer function T of an optical line made by three diamond mirrors and one beam splitter (in blue) vs. the photon energy for an incidence angle \(\Theta_{0}\) of 3 with respect to the normal. SASE spectrum (in red) in arbitrary units. The reflectivities are simulated with the XOP code [55]. Figure 5: FEL Oscillator. **Left window**: intracavity energy growth as a function of the number of round trips. Inner boxes: temporal and spectral distributions at saturation. **Right windows**: auto (\(\Gamma\)) and mutual (\(\Gamma_{1,2}\)) coherence degrees vs. \(s=c\tau\). presented. The right windows of Figure 5 show the auto (\(\Gamma\)) and mutual (\(\Gamma_{1,2}\)) coherence degrees of the XFEL-Oscillator, pointing out its enhanced coherence and stability with respect to the SASE's reported in Figure 1.  
The longitudinal distributions along \(s=c\tau\) of the power (a) and the spectral amplitude structure (b) are shown in Figure 6 as a function of the round trip number in the X-ray cavity. The output radiation is much more stable and quasi monochromatic if compared to the analogous patterns of the SASE radiation already shown in Figure 2. The studied X-FEL Oscillator results in an almost fully coherent pulse, characterized by a very small spectral bandwidth. Table 2 summarizes the main properties of SASE (third and fourth columns) and XFELO (fifth column) radiation. Despite the lower number of photons produced by the XFELO with respect to the SASE case, its pulses' stability, directionality and brilliance are higher.