# Three-Dimensional, Time-Dependent Analysis of High- and Low-\(Q\) Free-Electron Laser Oscillators
## 5 High-Gain/Low-Q Oscillators
### An Infrared RAFEL
We take an average because there are fluctuations in the gain on a pass-to-pass basis (i.e., limit-cycle like oscillations), which will be discussed in more detail below. The average gain is shown as a function of the on-axis undulator field under the assumption of a cavity length of 8.65238 in Figure 11. This represents a cavity detuning with respect to the zero-detuning length of \(\Delta L_{\text{cav}}=-8\lambda\). It is clear from Figure 11 that the gain is relatively constant over the range of about 6.65-6.85 kG (\(K_{\text{rms}}=1.054\)-1.085) and falls off rapidly as the field diverges outside this range, which is consistent with the behavior of the gain length shown in Figure 10. The cutoff for a gain of about 3300 percent occurs for field levels of about 6.518 kG (\(K_{\text{rms}}=1.03\)) at the low end and 6.878 kG (\(K_{\text{rms}}=1.09\)) at the high end, and we do not expect the RAFEL to function outside of this range of undulator fields. In order to demonstrate how the saturation efficiency of a RAFEL differs from that of a low-gain oscillator, it is instructive to compare the RAFEL with an equivalent SASE FEL. The performance of the RAFEL is shown in Figure 12, where we plot the average pulse energy (blue circles) in the steady-state as a function of the on-axis undulator field for the same cavity detuning (\(\Delta L_{\text{cav}}=-8\lambda\)), as used for Figure 11. The error bars characterize the limit-cycle oscillations. The RAFEL reaches its peak pulse energy for an undulator field of 6.678 kG (\(K_{\text{rms}}=1.058\)) and falls to zero outside the range predicted in Figure 10. We also plot the equivalent SASE saturated pulse energy (red triangles). In order to deal with the statistical fluctuations inherent in the SASE output, a large number of runs were made with different shot noise distributions, and found the average (red triangles) and standard deviations (error bars). Note that the SASE results represent the pulse energies over whatever length of undulator is required to reach saturation. Observe that (1) the RAFEL configuration saturates with a higher pulse energy than the SASE configuration, (2) the fluctuations in the RAFEL in the steady-state regime are comparable in magnitude to the statistical fluctuations found in SASE, however, the fluctuations of the RAFEL are deterministic in nature, and (3) the FWHM of the tuning range in \(K_{\text{rms}}\) is comparable for both the RAFEL and SASE configurations. The RAFEL saturates with about a 0.28 mJ pulse energy, which corresponds to an extraction efficiency of about 0.64 percent. This compares well with the empirical formula [214] that predicts a saturation efficiency of about 0.76 percent. In contrast, the saturation efficiency of a low-gain oscillator is predicted to be \(\eta=1/(2.4N_{u})=0.43\%\). Figure 11: Average gain produced by the IR RAFEL over the first 10 passes versus the undulator field \(B_{u}\) for a cavity length of \(L_{\text{cav}}=8.62538144\) m (\(\Delta L_{\text{cav}}=-8\lambda\). Other parameters as in Table 1.  
The spectral linewidth of the RAFEL also differs from that of a low-gain oscillator. The full width of the spectrum for a typical low-gain oscillator is given by \(\Delta\omega/\omega=1/N_{u}=0.01\) for the example under consideration. This can be translated into a tuning range over the undulator field, as follows  
\[\left|\frac{\Delta B_{u}}{B_{u}}\right|=\frac{1+K_{\rm rms}^{2}}{K_{\rm rms}^{ 2}}\left|\frac{\Delta\omega}{\omega}\right|. \tag{47}\]  
This implies a full width tuning range of \(\Delta B_{u}=0.063\) kG (\(\Delta K_{\rm rms}=0.001\)), which is narrower than what we find in the simulation. The relative SASE linewidth is given by \((\Delta\omega/\omega)_{\rm rms}=\rho\)[215] where \(\rho\) denotes the Pierce parameter. Here, \(\rho=0.0097\) and \((\Delta\omega/\omega)_{\rm rms}=0.0097\).