# Three-Dimensional, Time-Dependent Analysis of High- and Low-\(Q\) Free-Electron Laser Oscillators
## 5 High-Gain/Low-Q Oscillators
### An X-ray RAFEL
Note that this yields Twiss \(\beta_{x}=24.9\) m and \(\beta_{y}=17.80\) m.  
The resonator dimensions were fixed by means of estimates of the gain using the Ming Xie parameterization [214], and MINERVA simulations indicated that about 40-60 m of undulator would be required to operate as a RAFEL. As such, the distance, L, between the two mirrors framing the undulator was chosen to be 130 m, which is also the distance separating the two mirrors on the back side of the resonator (elements \(C_{3}\) and \(C_{4}\)). In studying the cavity tuning via time-dependent simulations, these two distances are allowed to vary while holding fixed the configurations of the backscattering units. The compound refractive lenses, which are modeled as thin lenses by OPC, are placed symmetrically around the undulator and are designed to place the optical focus at the center of the undulator in vacuo. In this study, the focal length is approximately 94.5 m.  
In order to out-couple the X-rays through a transmissive mirror at the wavelength of interest, the diamond crystal would need to be impractically thin (about 5 um); hence, we consider out-coupling through a hole in the first downstream mirror (\(C_{6}\)). We consider all the mirrors to be 100 um thick. Due to the high computational requirements of time-dependent simulations, we begin with an optimization of the RAFEL with respect to the hole radius and the undulator length using steady-state (i.e., time-independent) simulations. The choice of hole radius is important, because if the hole is too small then the bulk of the power remains within the resonator while if the hole is too large then the losses become too great and the RAFEL cannot lose. The results for the optimization of the hole radius indicate that the optimum hole radius is 135 um, which allows for 90 percent out-coupling, where we fixed the undulator line to consist of 11 HXR undulator segments. This is shown in Figure 22, where we plot the output power as a function of pass number for the optimum hole radius and the variation in the saturated power with the hole radius (inset). Figure 22: Output power versus number of passes through the resonator for a whole radius of 135 μm. The inset shows the output power as a function of hole radius. Other simulation parameters are described in the text. A local optimization on the undulator length for a hole radius of 135 \(\upmu\)m is shown in Figure 23, where we plot the peak recirculating power (left axis) and the average output power (right axis). The error bars in the figure indicate the level of pass-to-pass fluctuations in the power which is generally smaller than the level of shot-to-shot fluctuations in SASE. Note that while this represents steady-state simulations, the average power is calculated under the assumption of an electron bunch with a flat-top temporal profile having a duration of 24 fs which yields a duty factor of \(2.4\times 10^{-8}\). Each point in the figure refers to a given number of HXR undulators ranging from 9-13 segments. It is evident from the figure that the optimum length is 47.18 m, corresponding to 11 segments. Time-dependent simulations were performed under the assumption of electron bunches with a flat-top temporal profile having a full width duration of 24 fs and a peak current of 1000 A. This corresponds to a bunch charge of 24 pC. The detuning curve defining what cavity lengths are synchronized with the repetition rate of the electrons is shown in Figure 24. For simplicity, we take the synchronous cavity length \(L_{0}\) as \(L_{0}=c/f_{\text{rep}}\) (_cf._ Equation (46)), where a single optical pulse is assumed to be inside the cavity and the effect of gain on the group velocity is ignored. Here \(L_{0}=299.7924580\) m. The range of cavity lengths with overlap between the optical pulse and an electron pulse with a length of about \(7.2\,\upmu\)m is given by \(L_{0}-7.2\,\upmu\)m\(<L_{\text{cav}}<L_{0}+7.2\,\upmu\)m, where \(L_{\text{cav}}\) is taken here to be the total roundtrip length of the cavity.