# Three-Dimensional, Time-Dependent Analysis of High- and Low-\(Q\) Free-Electron Laser Oscillators
## 5 High-Gain/Low-Q Oscillators
### An Infrared RAFEL
The on-axis undulator field is \(B_{u}=6.658\) kG (\(K_{\text{rms}}=1.055\)), while the other parameters are given in Figure 10:4  
Figure 10:5 Cavity detuning curve of the IR RAFEL for an on-axis undulator field of \(B_{u}=6.658\) kG (\(K_{\text{rms}}=1.055\)), while the other parameters are given in Figure 10:6  
The fluctuations seen in simulation can be rapid and irregular. There are two possible explanations for this. One is that due to the high gain and high out-coupling, small changes in the mode structure from pass to pass can result in relatively large changes in the gain and, hence, the pulse energy. These \"small\" changes can include variations in the transverse mode structure (both in terms of the modal decomposition and spot size), and the temporal pulse shape. The second explanation, related to the first, is that since we have employed hole out-coupling, these relatively small changes in the transverse mode structure at the mirror can give rise to large differences in the out-coupling of the optical mode. It is not surprising, therefore, that the magnitude of the fluctuations varies depending on the cavity detuning. In Figure 15, we show the variation in the rms magnitude of the fluctuations in the out-coupled pulse energy as a function of the cavity detuning. It is clear from Figure 15 that the fluctuation level is relatively constant, at about the 0.03 mJ level over most of the detuning range, but with rapid declines at the end of the detuning range. Furthermore, the oscillation period is of the order of a few passes through the resonator. Fluctuations/oscillations have been observed in low-gain oscillators and are referred to as limit-cycle oscillations. The observation of limit-cycle behavior in the low-gain FELIX FEL oscillator corresponds to an oscillation period of [143]  
Figure 10:7  
where \(\tau_{\text{slip}}=l_{\text{slip}}/c\) is the slippage time. For the case of FELIX, \(L_{\text{cav}}=6\) m, \(\lambda=40\) um, and \(N_{u}=38\), and the cavity detuning ranges over about 160 um. As a result, \(\tau_{\text{slip}}=5.1\) ps and \(\tau_{r}=\)(=2\(L_{\text{cav}}/c\)) = 40 ns is the nominal roundtrip time; hence, this implies that the limit cycle oscillation occurs over a period of about 3 us or 75 passes for a cavity detuning of \(-100\) um. In contrast, if we apply the slippage time for the high gain RAFEL under consideration  
Figure 10:8  
As such, we expect the oscillation period to occur on the scale of a small number of passes for the indicated cavity detuning range. This is indeed what is observed in Figure 10:9 For example, the oscillations occur approximately every 2-4 passes for \(\Delta L_{\text{cav}}/\lambda=-8\), which is consistent with Equation (50). However, there is not a great deal of variation with detuning possible when the oscillations occur on such a fast time scale, and we must take Equations (49) and (50) as approximate measures of the oscillation period. Still, the observed oscillation period is well described by the formula for the oscillation period for limit-cycle oscillations found in low-gain oscillators when the appropriate slippage is  
Table 1.0 Standard deviation of the optical pulse energy as a function of cavity detuning \(\Delta L_{\text{cav}}/\lambda\) for a undulator field strength of \(B_{u}=6.658\) kG (\(K_{\text{rms}}=1.055\)), while the other parameters are given in Table 1.1  
taken into account. For a high-gain RAFEL, the much lower slippage results in very short oscillation periods. The limit-cycle-like oscillations in the RAFEL are correlated with the fluctuations/oscillations in the transverse mode structure. The transverse mode structure in a low-gain oscillator is largely (but not completely) determined by the mode structure in the cold cavity, since the optical guiding of the radiation in the undulator is weak. This is not the case in a RAFEL where the mode is guided through the undulator.