# Three-Dimensional, Time-Dependent Analysis of High- and Low-\(Q\) Free-Electron Laser Oscillators
## 5 High-Gain/Low-Q Oscillators
### An X-ray RAFEL
The crystals are arranged in a non-coplanar (3-D) scattering geometry. There are two backscattering units comprising three crystals (\(C_{1}\), \(C_{2}\), and \(C_{3}\)) on one side of the undulator and three crystals (\(C_{4}\), \(C_{5}\), and \(C_{6}\)) on the other side. Collimating and focusing elements are shown as CRL1,2, which could be grazing incidence mirrors, but are represented in Figure 21 by another possible alternative--compound refractive lense [242; 243]. In each backscattering unit, three successive Bragg reflections take place from three individual crystals to reverse the direction of the beam from the undulator. Assuming that all the crystals and Bragg reflections are the same, the Bragg angles can be chosen within the range \(30^{\circ}<\theta<90^{\circ}\); however, Bragg angles close to \(\theta=45^{\circ}\) should be avoided to ensure high reflectivity for both linear polarization components, as the reflection plane orientations for each crystal change. The cavity allows for tuning the photon energy in a large spectral range by synchronously changing all Bragg angles. In addition, to ensure constant time of flight, the distance \(L\) (which brackets the undulator), and the distance between crystals, as characterized by \(H\), have to be changed with \(\theta\). The lateral size \(G\) is kept constant as the resonator is tuned. Because the \(C_{1}C_{6}\) and \(C_{3}C_{4}\) lines are fixed, intracavity radiation can be out-coupled simultaneously for several users at different places in the cavity, although we only consider out-coupling through \(C_{6}\) at the present time. Out-coupling through crystals \(C_{3}\) and \(C_{6}\) are most favourable, since the direction of the out-coupled beams do not change with photon energy, but out-coupling for more users through crystals \(C_{1}\) and \(C_{4}\) is also possible. Such multi-user capability is in stark contrast with present SASE beamlines, which support one user at a time. We consider that the electron beam propagates from left to right through the undulator and the out-coupling is accomplished through a pinhole in the first downstream mirror (\(C_{6}\)). Consider the LCLS-II beamline [239] with the HXR undulator corresponding to an electron energy of 4.0 GeV, a bunch charge in the range of 10-30 pC with an rms bunch duration (length) at the undulator of 2-173 fs (0.6-52 \(\upmu\)m) and a repetition rate of 1 MHz. The peak current at the undulator is 1000 A, with a normalized emittance of 0.2-0.7 mm-mrad, and an rms energy spread of about 125-1500 keV. The HXR undulator [239] is a plane-polarized, hybrid permanent magnet undulator with a variable gap, a period of 2.6 cm, and a peak field of 10 kG. Each HXR undulator has 130 periods. In the simulation, each undulator is modeled using Equation (17) and we consider that the first and last period  
Figure 21: Schematic view of a 6-mirror ring resonator for an X-ray FEL oscillator using plane Bragg mirrors \(C_{1}\dots C_{6}\) and compound refractive lenses, CRL1 and CRL2 for focusing. After [236]. describe an entry/exit taper. There is a total of 32 segments that can be installed. The break sections between the undulators are 1.0 m in length and contain steering, focusing and diagnostic elements, although we only consider the focusing quadrupoles in the simulation, which we position in the center of the breaks. The quadrupoles are assumed to be 7.4 cm in length with a field gradient of 1.71 kG/cm. A fundamental resonance at 3.05 keV (=4.07 A) implies an undulator field of 5.61 kG. We assume that the electron beam has a normalized emittance of 0.45 mm-mrad and a relative energy spread of \(1.25\times 10^{-4}\), corresponding to the nominal design specification for LCLS-II. This yields a Pierce parameter of \(\rho=5.4\times 10^{-4}\). In order to match this beam into the undulator/FODO line, the initial beam size in the \(x(y)\)-direction is 37.87 (31.99) with Twiss \(\alpha_{x}=1.205\) (\(\alpha_{y}=-0.8656\)).