# Three-Dimensional, Time-Dependent Analysis of High- and Low-\(Q\) Free-Electron Laser Oscillators
## 5 High-Gain/Low-Q Oscillators
### An Infrared RAFEL
Converting this to a tuning range in the undulator field and going from the rms width to a FWHM tuning range, we obtain \((\Delta B_{u}/B_{u})_{\rm FWHM}=0.022\), which compares well with the simulation results that give \((\Delta B_{u}/B_{u})_{\rm FWHM}=0.019\). Hence, the RAFEL behaves more like a SASE FEL than a typical low-gain oscillator in regards to the spectral linewidth. Another way in which the RAFEL differs from a low-gain oscillator is in the cavity detuning. In a low-gain FEL oscillator, the zero-detuning length is obtained by assuming that the group velocity \(v_{g}\) equals the speed of light in vacuo \(c\) throughout the resonator. However, \(v_{g}\) is reduced in a RAFEL by the interaction in the undulator, and results in smaller synchronous cavity length. This makes the cavity detuning dependent on the gain of the FEL. The change in group velocity also affects slippage. In a low-gain oscillator, the group velocity reduction is small and the slippage is one wavelength per undulator period; hence, the slippage distance is \(l_{\rm slip}=N_{u}\lambda\). However, the slippage per undulator period is reduced in a high-gain RAFEL, or in any FEL where there is exponential growth because they have medium decreases, in both the phase and group velocities. The reduced phase velocity results in the optical guiding of the radiation, while the reduced group velocity results in less slippage. It has been shown that \(l_{\rm slip}=N_{u}\lambda/3\) at the resonant wavelength [216]. For the example under consideration, this yields \(l_{\rm slip}=72\) um, which is much less than the slippage length of \(220\) um if the RAFEL behaved as a low-gain oscillator. In order to estimate the effect of this on the detuning length, we note that the zero-detuning length is found by equating the roundtrip time of the radiation through the cavity with the spacing between electron bunches (\(1/f_{\rm rep}\)), cf. Equation (45). As a result, in the  
Figure 10:0 Output pulse energy for the IR RAFEL (blue circles) and an equivalent SASE FEL (red triangles) as a function of the undulator field strength \(B_{u}\). The cavity detuning for the IR RAFEL is \(\Delta L_{\rm cav}=-8\lambda\), while the other simulation parameters are given in Figure 10:1 The error bars indicate energy fluctuations due to limit-cycle-like oscillations for the RAFEL and pulse-to-pulse rms energy fluctuations in case of SASE. high gain regime where \(v_{g}=c/(1+1/3\gamma_{\parallel}^{2})\), the difference in synchronous cavity length \(\Delta L_{0}\) for a high-gain RAFEL and a low-gain FEL oscillator is given by Equation (45) as  
Figure 10:2  
This is comparable to what is found in the simulation. The detuning curve found in the simulation is shown in Figure 13, where we plot the output pulse energy versus cavity detuning for an undulator field of 6.658 kG (\(K_{\text{rms}}=1.055\)). Here, we define the cavity detuning relative to the nominal zero detuning length (Equation (46) with \(M=1\)), so that \(\Delta L_{\text{cav}}=L_{\text{cav}}-L_{0}\). As shown in Figure 13, we find a full width detuning range of about 50-110 um and an FWHM detuning range of about 40 um, which are in reasonable agreement with the estimate based on the one-dimensional analysis of slippage. The temporal evolution of the pulse energy is shown for an undulator field of 6.658 kG (\(K_{\text{rms}}=1.055\)) in Figure 14, where we plot the pulse energy versus pass number through the undulator for the choice of several cavity detunings that samples the complete detuning curve. It is clear that significant fluctuations are found over a large range of detunings and that both the magnitude and period of the fluctuations decrease as the magnitude of the detuning increases, although the magnitude of the fluctuations decreases as well near the zero-detuning length. Figure 10:3 Temporal evolution of the pulse energy for cavity detunings of \(\Delta L_{\text{cav}}=0\), \(-18\lambda\), \(-33\lambda\), \(-43\lambda\) and \(-50\lambda\).