# Three-Dimensional, Time-Dependent Analysis of High- and Low-\(Q\) Free-Electron Laser Oscillators
## 1 Introduction
Furthermore, electrostatic accelerators are capable of long to continuous wave electron pulses, that can result in single longitudinal mode oscillation, with a relative linewidth of about \(10^{-8}\) over a time of 5 \(\upmu\)s, which was set by the voltage droop present in the accelerator [6] at the time of measurement. On the other hand, the short duration of, and spread in, electron energy for the electron pulses produced by RF linacs typically leads to a relative linewidth of the order of \(10^{-4}\) to \(10^{-2}\)[13; 29; 40; 41; 42; 43; 44; 45; 46; 47; 48; 49; 50; 51; 52; 53], unless additional spectral filtering is used. For example, using the Vernier effect of a double cavity resonator to reduce the number of intra-cavity longitudinal modes and subsequent external filtering, an RF-linac-based FEL oscillator was able to provide a single longitudinal mode [54]. FEL oscillators based on storage rings provide a lower relative linewidth within the range \(10^{-5}\) to \(5\times 10^{-4}\)[34; 55; 56; 57; 58; 59; 60; 61; 62; 63], due to the higher electron beam quality of storage rings. Finally, the average and peak power are, to a large extent, determined by the accelerator technology. Superconducting energy recovery linacs are the primary drivers for high average power FEL oscillators [64; 65; 66; 67; 68; 69; 70]; however, other types of FEL oscillators have used various techniques to increase the output pulse energy including, but not limited to, optical klystrons [34; 36; 48; 57; 58; 60; 63; 71; 72; 73; 74; 75; 76], cavity dumping [77; 78; 79; 80; 81], pulse stacking in an external cavity [82; 83; 84], electron beam energy ramping [85; 86] or dynamic cavity desynchronization [87; 88; 89; 90; 91; 92]. Tapering of the undulator [93; 94; 95] has also been used to improve the  
Figure 1: Schematic representation of a typical layout for an FEL oscillator. The electron beam consisting of electron pulses is guided through the undulator and around the mirrors. The optical pulse that is amplified within the undulator is recirculating within the resonator and extracted through one of the mirrors. Reproduced with permission from [1]. Copyright Springer, 2018.  
peak power in the pulse [96; 97; 98; 99; 100], however, tapering of the undulator in FEL oscillators leads to complicated dynamics and a strong dependence on system parameters [101; 102; 103; 104]. The wavelength tuning of FEL oscillators is typically obtained by selecting a set of electron beam energies and varying the undulator gap. Operation at other wavelength ranges is possible through harmonic generation [105; 106; 107] and such harmonics were quickly observed [40]. Lasing on the third harmonic was first demonstrated using the Standford MARK-III FEL oscillator [108], quickly followed by others [43; 48; 58; 61; 97; 109; 110; 121]. The simultaneous generation of multiple wavelengths may be beneficial for certain applications. The fixed relation between the fundamental and its harmonics can be overcome using different undulators in a single optical cavity [122; 123; 124; 125], two undulators driven by a single electron beam at a single beam energy, but each having its own optical resonator [126] or having different beam energies [127; 128] to allow operation at two independently chosen wavelengths. FEL oscillators share many characteristics with amplifier systems, however, there are also many differences. For example, slippage between the optical and electron pulse, in combination with cavity desynchronization, will effect the FEL oscillator dynamics. Here, a cavity is considered synchronized when it is tuned to a length where the roundtrip time for the optical pulse matches an integer multiple of the time separating two subsequent electron pulses. Maxima in the gain and extraction efficiency are found for different cavity desynchronizations.