# Three-Dimensional, Time-Dependent Analysis of High- and Low-\(Q\) Free-Electron Laser Oscillators
## 5 High-Gain/Low-Q Oscillators
### An Infrared RAFEL  
The electron beam, undulator, and resonator parameters are summarized in Table 1. Observe that the temporal profile of the electron bunch is parabolic with a full width of 1.2 ps, and that the undulator is a two-plane-focusing (i.e., parabolic pole face) design. Consequently, Equation (17) is used to model the undulator with the first and last periods tapered up and down to model the injection and ejection of the beam. Since there is exponential growth and, hence, the optical guiding of the radiation, we use a matched beam in the undulator. The resonator is concentric with the power coupled out through a 5.0 mm hole in the downstream mirror, which provides for a typical average out-coupling of about 97 percent. Given the repetition rate, \(f_{\rm rep}\), the nominal zero-detuning cavity length, \(L_{0}\), is 6.85239904 m when \(M=4\) and assuming \(v_{g}=c\) (see Equation (46)). The temporal window is an important numerical consideration, and must be chosen to be large enough to accommodate the maximum cavity detuning length that is consistent with pass-to-pass amplification so that the optical pulse remains within the time window for all the usable choices of cavity length. In practice, for this example, we choose a temporal window of 4.0 ps and include 182 temporal slices, which corresponds to the inclusion of one temporal slice every three wavelengths. OPC uses the modified Fresnel propagator (Equation (35)) to handle the divergence and convergence of the optical beam inside the resonator, using a grid with a fixed number of grid points. We first consider the single-pass gain because this will affect the performance of the RAFEL. Since the undulator is long enough to achieve exponential growth, we show the gain length, \(L_{\text{G}}\), found in simulation versus the undulator field strength in Figure 10. The optimal (i.e., minimal) gain length of 0.176 m occurs for an on-axis undulator field strength of about 6.7 kG, which corresponds to an rms undulator strength parameter of \(K_{\text{rms}}=1.06\). This is in good agreement with the prediction based on the parameterization of the interaction by [PERSON] [214]. It should be noted that the well-known resonance condition \(\lambda=\lambda_{u}(1+K_{\text{rms}}^{2})/2\gamma^{2}\) predicts an undulator field of about 6.81 kG (\(K_{\text{rms}}=1.08\)) at a 2.2 \(\upmu\)m wavelength. This shift in the resonance is due to three-dimensional effects and is in disagreement with the shift in the resonance associated with low-gain oscillators. The gain length has implications over the permissible range of \(K_{\text{rms}}\) for which the RAFEL will operate (i.e., over which there is pass-to-pass amplification). Since the RAFEL will saturate when the gain balances the loss, and the loss for the resonator is about 97 percent, this implies that the RAFEL \begin{table}
\begin{tabular}{l l l l} \hline \hline Electron Beam & & & \\ \hline Energy & 55 & MeV & \\ Charge & 800 & pC & \\ Bunch duration & 1.2 & ps & parabolic \\ Repetition rate & 87.5 & MHz & \\ Normalized emittance & 15 & mm-mrad & \\ Energy spread & 0.25\% & & \\ Matched beam radius & 392 & \(\upmu\)m & \\ \hline Undulator & & & two-plane focusing \\ \hline Period & 2.4 & cm & \\ Magnitude & 6.5–7.0 & kG & \\ \(K_{\text{rms}}\) & 1.03–1.11 & & \\ Length & \(100\lambda_{u}\) & & 98 uniform \\ \hline Resonator & & & Concentric \\ \hline Wavelength & 2.2 & \(\upmu\)m & \\ Length & 6.852 & m & \\ Radii of curvature & 3.5 & m & \\ Rayleigh range & 0.5 & m & \\ Hole radius & 5.0 & mm & \\ Out-coupling & 97\% & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Electron beam, undulator and resonator parameters for the IR RAFEL. Figure 10: Gain length \(L_{\text{G}}\) versus on-axis undulator field amplitude \(B_{u}\) for the IR RAFEL. Other parameters as in Table 1.  
will operate as long as the single-pass gain exceeds about 3200-3300 percent. In order to identify this range more closely, we perform multi-pass simulations and take the average gain over the first 10 passes.