# Three-Dimensional, Time-Dependent Analysis of High- and Low-\(Q\) Free-Electron Laser Oscillators
## 5 High-Gain/Low-Q Oscillators  
An alternate approach to oscillator design is to use a high-gain undulator where the radiation grows exponentially on a single pass through the undulator [2]. Because the gain is high, the permissible resonator loss can be relatively high; hence, a large fraction of the power can be coupled out of the resonator. This type of oscillator has been referred  
Figure 8: Tuning curve for the IR upgrade experiment. Simulation parameters as described in the text.  
Figure 9: Temporal profiles of the power in the optical pulse after 100 passes at the undulator entrance (green) and exit (blue), as well as the current in the electron bunch (right axis, red) for the IR upgrade experiment. The cavity length is \(L_{\text{cav}}=32.041946079\) m and other parameters, as described in the text.  
to as a regenerative amplifier FEL (RAFEL), and the concept has been experimentally demonstrated at the Los Alamos National Laboratory [210]. Hence, a RAFEL may also be thought of as a low-\(Q\) oscillator and has advantages, both for (1) high power designs, since the mirror loading can be kept below mirror deformation or damage thresholds, and (2) for VUV and X-ray oscillators.  
RAFELs differ from low-gain oscillators in a number of ways. One difference is that, as shown in [PERSON]'s theorem [211], a low-gain oscillator exhibits no gain directly on the resonance. In contrast, the growth rate in the exponential gain regime has a peak on-resonance, and this is reflected in the wavelengths excited in a RAFEL. A second difference is the overall efficiency. The saturation efficiency, \(\eta\), of a low-gain oscillator is \(\eta=1/2.4N_{u}\), where \(N_{u}\) is the number of periods in the undulator. Since the radiation exponentiates in each pass through the undulator in the RAFEL, the efficiency is given by that found in the high-gain Compton regime, where \(\eta=\rho\) where \(\rho\) is the Pierce parameter. A third difference is in the linewidth, which scales inversely with \(N_{u}\) in a low-gain oscillator, but which is given by the linewidth of the exponential interaction in the high-gain Compton regime. A fourth difference is in the longitudinal and transverse mode structure, which is determined largely by the resonator properties in a low-gain oscillator. In a RAFEL, by contrast, the exponential gain leads to saturation in a very small number of passes through the resonator, and the mode structure is largely governed by the interaction in the undulator. A fifth difference is in the effect of slippage. Slippage in a low-gain oscillator scales with \(N_{u}\). However, the high-gain in a RAFEL results in a reduction in the group velocity, such that slippage scales with \(N_{u}/3\). However, one point of similarity that the RAFEL shares with low-gain oscillators, is the presence of limit-cycle oscillations.  
In this section, we discuss simulations of an infrared RAFEL with the intention of illustrating many of the general properties of a RAFEL and how it compares both to low-gain/high-\(Q\) oscillators and SASE FELs [212] and simulations of an X-ray RAFEL concept, making use of hole out-coupling [213]. We note that the infrared RAFEL simulations were performed with MEDUSA/OPC, while the X-ray RAFEL simulations were performed with MINERVA/OPC.