# 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 impetus driving research into X-ray free-electron laser oscillators (XFELOs) and RAFELs is the character of SASE emission. Since there are no seed lasers available at X-ray wavelengths, fourth generation FEL light sources [198; 217; 218; 219; 220; 221] at these wavelengths are based on SASE over a single pass through a long undulator line. While pulse energies of the order of 2 mJ have been achieved at Angstrom to sub-Angstrom wavelengths, with the potential to reach multi-TW peak powers [222; 223; 224; 225; 226; 227], SASE exhibits shot-to-shot fluctuations in the output spectra and power of about 10-20 percent. For many applications, these fluctuations are undesirable, and efforts are underway to find alternatives. The utility of an XFELO has been under study for a decade [162; 228; 229; 230; 231; 232; 233], making use of resonators based upon Bragg scattering from atomic layers within diamond crystals [234; 235; 236; 237; 238]. The development of these crystals is a major breakthrough in the path toward an XFELO. Estimates indicate that using a superconducting RF linac producing 8 GeV electrons at a 1 MHz repetition rate is capable of producing \(10^{10}\) photons per pulse at a 0.86 A wavelength, with a FWHM bandwidth of about \(2.1\times 10^{-7}\). This design is consistent with the LCLS-II High Energy Upgrade [239]. As a consequence, an XFELO on a facility, such as the LCLS-II and LCLS-II-HE, is expected to result in a decrease in SASE fluctuations in the power and spectrum, and to narrow the spectral linewidth. As with the majority of FELOs to date, the aforementioned XFELOs use low gain/high-\(Q\) resonators with transmissive out-coupling through thin diamond crystals [229]. Potential difficulties with low-gain/high-\(Q\) resonators derive from sensitivities to electron beam properties, mirror loading and alignments. In addition, transmissive out-coupling with high intra-cavity power can result in mirror damage. While experiments show that diamond crystals can sustain relatively high thermal and radiation loads [238], transmissive out-coupling cannot be easily achieved at the photon energies of interest here. Because of this, X-ray RAFELs [213; 240; 241] using a variety of out-coupling schemes are receiving a great deal of attention, and we consider a RAFEL design using a pinhole in one of the mirrors [210; 213; 240] here. Optics propagation codes such as OPC must treat reflections from the diamond crystal Bragg mirrors, where the mirror losses and angles of reflection depend on the crystal orientation/geometry and the X-ray photon energy. X-ray Bragg mirrors typically have a very narrow reflection bandwidth and a narrow angle of acceptance [205]. For the X-ray RAFEL, and for computational efficiency, a temporal Fourier transform is applied at the beginning of the optical path, when the optical field is passed from MINERVA to OPC and the propagation is performed in the wavelength domain, i.e., each wavelength is independently propagated through the resonator using the modified Fresnel propagator (Equation (35)). The inverse Fourier transform is calculated at the end of the optical path, before the field is handed back to MINERVA. As the optical field inside the cavity is typically not collimated, a spatial Fourier transform in the transverse coordinates is calculated for each of the wavelengths when a Bragg mirror is encountered. Each combination of transverse and longitudinal wavenumber corresponds to a certain photon energy and angle of incidence on the Bragg mirror, and these parameters are used to calculate the complex reflection and transmission coefficients of the Bragg mirror [205]. After applying the appropriate parameter to the optical field, depending on whether it is reflected or transmitted, the inverse spatial Fourier transform is calculated and the field is propagated to the next optical element along the path until the end is reached. Here, we consider a six-crystal, tunable, compact cavity [236], as illustrated in Figure 21, which shows both a top and a side view.