# Three-Dimensional, Time-Dependent Analysis of High- and Low-\(Q\) Free-Electron Laser Oscillators
## 5 High-Gain/Low-Q Oscillators
### An X-ray RAFEL
The evolution of the output energy at the fundamental and the 3 rd harmonic, and the spectral linewidth of the fundamental, vs. pass is shown in Figure 25 for a detuning of 5 \(\upmu\)m, which is close to the peak in the detuning curve (Figure 24). While it is not evident in the figure, the rms fluctuation in the energy from pass to pass is of the order of  
Figure 23: Recirculating output power versus length of the undulator \(L_{u}\) for a hole radius of 135 \(\upmu\)m. Other simulation parameters as described in the text. Figure 24: Output energy as a function of the cavity detuning \(\Delta L_{\text{cav}}=L_{\text{cav}}-L_{0}\) for a hole radius of 135 \(\upmu\)m. The simulation parameters are as described in the text. about 3 percent, which is lower than the shot-to-shot fluctuations expected from SASE. At least as important as the output power is that the linewidth contracts substantially during the exponential growth phase and remains constant through saturation. Starting with a linewidth of about \(3.7\times 10^{-4}\) after the first pass corresponding to SASE, the linewidth contracts to about \(6.0\times 10^{-5}\) at saturation. The SASE linewidth after the first pass through the undulator is slightly smaller than the predicted linewidth based on 1-D theory [240], which is approximately \(4.5\times 10^{-4}\). Hence, the RAFEL is expected to have both high average power and a stable narrow linewidth, which is to a large extent determined by the spectral filtering of the Bragg mirrors. The 3 rd harmonic grows parasitically from high powers/pulse energies at the fundamental in a single pass through the undulator [184] and has been shown to reach output intensities of 0.1 percent that of the fundamental in a variety of FEL configurations, and this is what we find in the RAFEL simulations. As shown in Figure 25, the 3 rd harmonic intensity remains small until the fundamental pulse energy reaches about 1 \(\upmu\)J after which it grows rapidly and saturates after about 12 passes. This is close to the point at which the fundamental saturates as well. The saturated pulse energies at the 3 rd harmonic reach about 0.067 \(\upmu\)J. Given a repetition rate of 1 MHz, this corresponds to a long-term average power of 67 mW. The reduction in the linewidth after saturation shown in Figure 25 indicates that a substantial level of longitudinal coherence has been achieved in the saturated regime. The RAFEL starts from shot noise on the beam during the first pass through the undulator, and longitudinal coherence develops over the subsequent passes. Hence, we expect that the temporal profile of the optical field will exhibit the typical spiky structure associated with SASE at the undulator exit after the first pass and this is indeed depicted in Figure 26a which shows the temporal profile at the undulator exit after the first pass. The number of spikes expected, \(N_{\text{spikes}}\), is given approximately by \(N_{\text{spikes}}=l_{b}/(2\pi l_{c})\), where \(l_{b}\) is the rms bunch length and \(l_{c}\) is the coherence length. For the present case, \(l_{b}=7.2\) um and \(l_{c}=60\) nm; hence, we expect that \(N_{\text{spikes}}=19\). We observe about 14 spikes in Figure 26a which is in reasonable agreement with the expectation. Note that the time axis encompasses the time window used in the simulation. As indicated in Figure 25, the linewidth after the first pass is of the order of \(4.3\times 10^{-4}\) which is relatively broad and corresponds to the interaction due to SASE. This is reflected in the output spectrum from the undulator after the first pass, which is shown in Figure 26b. Figure 25: Evolution of the fundamental pulse energy (blue, left) and the 3 rd harmonic (green, left), as well as the relative linewidth (red, right) when the cavity detuning is \(\Delta L_{\text{cav}}=5\) μm and for an electron beam energy spread \(\Delta E_{b}/E_{b}=1.25\times 10^{-4}\). Other parameters as described in the text.