# Three-Dimensional, Time-Dependent Analysis of High- and Low-\(Q\) Free-Electron Laser Oscillators
## 4 Low-Gain/High-\(Q\) Oscillators
### The IR-Demo Experiment at JLab  
The IR Demo experiment [50, 64, 65] was based on a superconducting energy recovery linac at JLab that produced 0.4 (\(\pm\)0.1) ps rms electron bunches, with energies of about 38 MeV bunch charges of 60 pC (60 A peak current) at a repetition rate of 18.7 MHz corresponding to an average current of 1.2 mA. The transverse emittance of the bunches was \(7.5(\pm\)1.5) mm-mrad (rms) and the rms energy spread was about 0.25 percent. The experiment uses a 42-period flat-pole-face planar undulator with a period of 2.7 cm and an on-axis amplitude of 5.56 kG, yielding an undulator strength parameter of \(K_{\text{rms}}=0.99\). Therefore, the simulation model uses Equation (17) to model the undulator with one period entry and exit tapers. The resonant wavelength was 4.8 \(\upmu\)m and a concentric resonator was used with a optical cavity length of about 8 m [208]. The radii of curvature of the mirrors were 4.045 m with a cold cavity Rayleigh range of 40 cm. Transmissive out-coupling through the downstream mirror was used and had a transmittance of approximately 10 percent. The outer radius of the mirrors was 2.54 cm. OPC uses the modified Fresnel propagator (Equation (35)) to handle the divergence and convergence of the optical beam inside the resonator using a fixed number of grid points. Simulations of this experiment were conducted with MEDUSA/OPC. MEDUSA [184] and MINERVA both employ Gaussian representations for the optical field and integrate the three-dimensional Lorentz force equations for the electron trajectories without performing an average over the wiggle-motion. However, MEDUSA is the more primitive code and MINERVA contains many additions and incorporates superior algorithms not present in MEDUSA. Nevertheless, in cases where the two codes have been compared, their predictions for FEL performance are in agreement to within about 10 percent. The evolution of the output energy found in simulation versus pass is shown in Figure 6 for a cavity length near the peak of the detuning curve of 8.01049 m. Observe that saturation is found after about 70 passes at a pulse energy of 0.17 mJ, corresponding to an average power of about 318 W, at a repetition rate of 18.7 MHz. This represents an average efficiency of about 0.70 percent, which is close to the theoretically predicted efficiency of about 1.0 percent for an undulator with 40 uniform periods. The average power found in simulation near the peak in the detuning curve is consistent with the observation of 311 W during CW operation of the IR Demo [50, 64, 65]. This is also shown in Figure 7, which shows the detuning curves found in simulation (blue) and during pulsed operation of the IR Demo (green). The average power during CW operation is indicated by the dashed line. Observe that the zero-detuning point in the simulation is shifted by about 5 \(\upmu\)m from that observed in the experiment. This may be due to an uncertainty in the cavity length measurement by up to 10 \(\upmu\)m. The full width of the detuning curve is found to be within about 30-35 \(\upmu\)m.  
Higher bunch charges were used in the pulsed runs than in the CW runs in the IR Demo. Lower bunch charges were used in the CW runs in order to reduce distortion due to mirror heating. Indeed, mirror distortion made it difficult to obtain a detuning curve  
Figure 6: Simulated pulse energy as a function of roundtrip number \(n\) at a near optimal cavity detuning of \(L_{\text{cav}}=8.01049\) m for the IR Demo experiment. Remaining parameters as described in the text. Figure 7: Tuning curve for the IR Demo experiment including measurements from CW and pulsed runs (green) and from simulation (blue). Other parameters as described in the text. during CW operation. Mirror heating was not a serious problem in the pulsed runs.