# Three-Dimensional, Time-Dependent Analysis of High- and Low-\(Q\) Free-Electron Laser Oscillators
## 4 Low-Gain/High-\(Q\) Oscillators
### The 10-kW Upgrade Experiment at JLab  
This experiment represented an upgrade to the original IR-Demo experiment [65] and numerous elements represent upgrades. For example, the accelerating modules were upgraded to achieve higher energy, bunch charge, and average current. Effort was also made to reduce/mitigate the beam breakup instability in view of the higher average current. The original undulator was replaced to achieve high gain and resonance at a shorter wavelength. In order to handle the higher power, the resonator was lengthened to reduce the mirror loading and cryogenic, edge-cooling was used for the mirrors. As a result, the kinetic energy was increased to 115 MeV, while the energy spread of 0.3 percent remained approximately the same. The charge of the electron bunch was almost doubled to 115 pC with a pulse length of 390 fs. The normalized emittance of 9 mm-mrad in the wiggle plane and 7 mm-mrad in the plane orthogonal to the wiggle plane was similar as in the IR demo experiment, and the maximum repetition rate of 74.85 MHz for the electron bunches remained unchanged. Note, that although the IR-Demo experiment was capable of running at 74.85 MHz, the results reported in Section 4.1 corresponded to operation at 17.85 MHz. To operate at somewhat shorter wavelengths, the planar undulator, which is modeled in the simulation using Equation (19) with one period up- and down-taper, had a longer period of 5.5 cm, a total of 30 periods, and a peak on-axis magnetic field of 3.75 kG. The electron beam was focused into the undulator with the focus at the center of the undulator. The concentric resonator was also updated [209] and had a length of about 32 m with a cold-cavity Rayleigh length of 0.75 m. The total loss in the resonator was 21 percent with about 18 percent out-coupled per pass from the downstream mirror. For these settings, the wavelength was 1.6 \(\upmu\)m.  
In simulating this experiment, the number of particles in MINERVA was 5832 per slice, while the separation between slices was 5.4 fs. The number of optical modes was dynamically adjusted each roundtrip to accommodate the evolution of the optical field inside the resonator. OPC uses, again, the modified Fresnel propagator (Equation (35)) to handle the divergence and convergence of the optical beam inside the resonator. The length of the optical cavity must be selected so that the returning optical pulse is in synchronism with the electron bunches. The roundtrip time for the optical pulses in the cavity is \(\tau_{r}=2L_{\text{cav}}/c\) and the separation between electron bunches is \(\tau_{\text{sep}}=1/f_{\text{rep}}\), where \(L_{\text{cav}}\) is the cavity length and \(f_{\text{rep}}\) is the electron bunch repetition rate. Perfect synchronism (referred to as zero-detuning) is obtained when \(\tau_{r}=M\tau_{\text{sep}}\), which leads to Equation (46). Here, \(M\) is the number of optical pulses in the cavity. In this case, there were 16 optical pulses in the cavity and the zero-detuning length is \(L_{0}\) = 32.041946079 m. The cavity detuning curve obtained from simulations is shown in Figure 8 as a function of the difference between the cavity length \(L_{\text{cav}}\) and the zero-detuning length. With a maximum pulse energy of 0.194 mJ and a repetition rate of 74.85 MHz, we find that the maximum output power of 14.52 kW occurs for a positive detuning of 2 \(\upmu\)m and is close to the measured value of 14.3 \(\pm\) 0.72 kW [69]. As a result, the predicted extraction efficiency is about 1.4 percent, which is close to the theoretical value of \(\eta\) = 1.7 percent. We remark that the previous simulation of this experiment with MEDUSA/OPC [183] yielded an average output power of 12.3 kW, and the present formulation is in better agreement with the experiment than in the earlier simulation. As in the previous simulation [183], the roughly triangular shape of the detuning curve is also in agreement with the experimental observation.