# Three-Dimensional, Time-Dependent Analysis of High- and Low-\(Q\) Free-Electron Laser Oscillators
## 5 High-Gain/Low-Q Oscillators
### An Infrared RAFEL
This near-Gaussian mode then propagates to the downstream mirror, during which it expands by about a factor of three, as shown in Figure 18b, where the FWHM is about 3.9 mm in width. The FWHM of the modal superposition at the undulator exit is about 1.2 mm. Ignoring the hole in the out-coupling mirror, the waist (0.59 mm) of the fundamental cold cavity mode is designed to be about \(\sqrt{2}\) times the matched electron beam radius in the undulator. The FWHM of the fundamental cold cavity mode at the undulator exit and downstream mirror are 1.84 and 4.82 mm, respectively. We thus observe that the optical mode in the RAFEL expands faster from the undulator exit to the downstream mirror than the fundamental cold cavity mode (factor 3.25 and 2.62 respectively). That the RAFEL mode size is still smaller at the downstream mirror is the result of a balance between the faster expansion and smaller spot size of the RAFEL optical mode at the undulator exit compared to the cold cavity mode. The smaller spot size at the undulator exit is due to gain guiding as described above. Both the smaller spot size at the undulator exit and faster expansion of the RAFEL optical beam again indicate that at the undulator exit, the optical field consists of fundamental and higher order cold-cavity modes. Note, a fundamental Gaussian beam having a waist at the undulator exit with the same size as the RAFEL optical beam would have a Rayleigh range of 1.49 m. Finally, the cross section of the mode incident on the upstream mirror is shown in Figure 18c. After reflection from the upstream mirror and propagation through the resonator to the undulator entrance, this field results in a modal pattern similar to that shown in Table 1.5  
Table 1.6 Cross-section of the optical field at the undulator entrance on pass 60 for a cavity detuning \(\Delta L_{\mathrm{cav}}=-18\lambda\) and an undulator field strength of \(B_{u}=6.658\) kG (\(K_{\mathrm{rms}}=1.055\)). The other parameters are given in Table 1.7  
Now consider the formation of temporal coherence. Since the RAFEL starts from shot noise on the beam, the initial growth of the mode starts from spiky noise and, just as in a SASE FEL, develops coherence as it propagates through the undulator. However, the undulator is not long enough to reach saturation on a single pass, so that the evolution to temporal coherence is expected to develop over multiple passes. In Figure 19, we plot the temporal pulse shapes found on the first pass (a) at z = 0.5 m (blue) and 1.0 m (red), and (b) at z = 2.0 m (blue) and 2.4 m (red), the latter being at the undulator exit, for an on-axis undulator field amplitude \(B_{u}=6.658\) kG (\(K_{\text{rms}}=1.055\)). Figure 19 shows the full simulation time window, and it should be noted that the electron beam is centered in the time window with a full (parabolic) width of 1.2 ps. The pulse shows many spikes at 0.5 m, but that it has coalesced into about 5 spikes after 1.0 m. The development of temporal coherence continues, until at the undulator exit at 2.4 m, only two spikes remain. The multi-pass development of temporal coherence after the first pass depends strongly on the cavity detuning. The temporal pulse shapes on the 60 th pass at the undulator exit for \(B_{u}=6.658\) kG (\(K_{\text{rms}}=1.055\)) and for cavity detunings of \(\Delta L_{\text{cav}}=0\), \(-8\lambda\), \(-18\lambda\) and \(-43\lambda\) are shown in Table 1.8 As demonstrated previously, the synchronized cavity length for a RAFEL is shorter than the synchronized cavity length of a low-gain FEL oscillator. Therefore, as we have used the speed of light in vacuo, instead of the actual group velocity to define the cavity detuning, we note that \(\Delta L_{\text{cav}}=0\) actually corresponds to a cavity length that is larger than the synchronized length for the high-gain RAFEL. Consequently, the returning optical pulse will lag behind the center of the electron bunch.