# Three-Dimensional, Time-Dependent Analysis of High- and Low-\(Q\) Free-Electron Laser Oscillators
## 1 Introduction
In particular, dynamic cavity desynchronization [87] can be used to first set a cavity length corresponding to maximum gain for a quick initial growth of the optical pulse energy and then switched to a cavity length for maximum extraction efficiency that is obtained at small cavity desynchronization. The latter is typically also associated with the generation of short optical pulses [129; 130; 131; 132; 133; 134]. For short pulse FEL oscillators, i.e., where the root-mean-square (rms) width of the electron bunch is smaller than the total slippage distance, the oscillator is operating in the superradiant regime [132; 134; 135; 136; 137; 138; 139; 140; 141; 142]. An analytic theory for this regime predicts a scaling of the optical pulse energy \(E_{L}\) with the bunch charge \(q\) as \(E_{L}\propto q^{3/2}\) and a scaling of the temporal width \(\tau_{L}\) of the optical pulse as \(\tau_{L}\propto q^{-1/2}\)[138], which has been experimentally confirmed [139], although higher extraction efficiencies have also been observed for a perfectly synchronized cavity length, where the analytic theory breaks down [92; 132]. Furthermore, short pulse FEL oscillators operating in the large-slippage regime also show a stable oscillation in the macropulse power, known as limit-cycle oscillations, for certain cavity lengths detuned from perfect synchronization, which has been observed both experimentally [137; 138; 143; 144; 145; 146] and in numerical simulations [136; 137; 147; 148; 149]. Techniques well known from laser physics have also been applied to FEL oscillators, either experimentally or in a conceptual study, such as injection seeding [150; 151; 152; 153; 154; 155] and mode locking [151; 152; 156; 133; 155]. On the other hand, the unique nature of the gain process in FELs allows for the micro-structuring of the gain medium [157; 158; 159] to enhance the gain, which is not readily available for ordinary lasers. One important application of this technique is to transfer coherence from a low-frequency optical beam to a higher frequency optical beam. This process is known as high-gain harmonic generation (HGHG) [160]. This technique has been successfully used to generate fully coherent light at soft X-ray frequencies [161]. Alternative methods currently investigated, in particular to reach even shorter wavelengths, and rely on Bragg reflections from atomic layers using appropriated crystals [162]. The drive to higher gain per unit length and to shorter wavelengths has pushed accelerator development; in particular, research into photo-cathode based electron sources [163; 164] that typically produce higher quality electron beams has contributed to today's state-of-the-art accelerator technology. The availability of electron beams with improved beam quality has driven FEL development over the years, especially the capability to generate shorter wavelengths. Due to a lack of suitable mirrors, initial focus was on FELs relying on self-amplified spontaneous emission (SASE), with a proof-of-principle experiment at 530 nm [165], followed by demonstrations at shorter wavelengths [166]. To overcome poor temporal coherence, HGHG is used to provide completely coherent output pulses down to 4-nm wavelength [167], while for even smaller wavelengths, FEL oscillators using the above-mentioned Bragg mirrors are considered [162]. Along with increasingly sophisticated FEL oscillator designs and experiments, analytical theories [168, 169, 170, 138, 171, 101] have been developed for an initial assessment of oscillator performance. An initial theoretical description of the FEL gain was quantum mechanical in nature [172, 173], however, it was quickly demonstrated that the gain could be described using classical theory [174]. Furthermore, the more complete simulation codes have also become adept at treating increasingly complicated designs and include both steady-state and time-dependent simulations in one- and three-dimensions. In the remainder of this paper, we will focus on the simulation of a few different FEL oscillators.