# Three-Dimensional, Time-Dependent Analysis of High- and Low-\(Q\) Free-Electron Laser Oscillators
## 3 Some Considerations for Optical Resonators Used in Free-Electron Lasers
This zero-detuning cavity length, \(L_{0}\), is given by  
\[\frac{M}{f_{\text{rep}}}=\frac{2L_{0}-L_{u}}{c}+\frac{L_{u}}{v_{g}}, \tag{44}\]  
where \(M\) is the number of simultaneous electron bunches in the optical cavity, \(f_{\text{rep}}\) is the repetition rate of the electron bunches, \(L_{u}\) is the undulator length and \(v_{g}\) is the group velocity of the light within the undulator. Solving for \(L_{0}\) gives  
\[L_{0}=\frac{cM}{2f_{\text{rep}}}-\frac{L_{u}}{2}\frac{c}{v_{g}}\Big{(}1-\frac {v_{g}}{c}\Big{)}, \tag{45}\]  
which reduces to the well-known expression  
\[L_{0}=\frac{cM}{2f_{\text{rep}}}, \tag{46}\]  
for low gain oscillators where \(v_{g}\approx c\). As will be discussed in the Section 5 on high-gain/low-\(Q\) oscillators, exponential gain in the undulator results in a reduction in the group velocity, which has the effect of shortening the zero-detuning cavity length compared to that for low-gain/high-\(Q\) oscillators (Equation (46)). In general, the dynamics in an oscillator are set by the interplay between the instantaneous gain and loss, where in a free-electron laser, the gain is provided by the electrons streaming through the undulator and loss is due to out-coupling of the radiation and other loss mechanisms that may be present within the resonator. The total loss of the resonator determines the quality factor \(Q\) of the resonator [200]. To obtain laser oscillation, the roundtrip small-signal gain has to be higher than the loss per roundtrip. The so-called threshold gain is the roundtrip small-signal gain needed to just balance the loss per roundtrip. When the light builds up inside the resonator, the gain is reduced and a stationary state is obtained when the saturated gain equals the loss per roundtrip. Note that this is the same condition as for threshold, i.e., the saturated gain is equal to the threshold gain. If we denote the output power on the \(n\)th pass as \(P_{n}\), then the power after the (\(n\)+1)th pass is \(P_{n+1}=(1-L)(1+G)P_{n}\). Equilibrium is characterized by \(P_{n+1}=P_{n}\); at which point, \(G=L/(1-L)\). In FELs, where the bunch charge/peak current is relatively small and the gain cannot reach the exponential regime in a single pass through the undulator, the resonator \(Q\) must be high enough that the oscillator can lase. In such cases, most of the energy extracted from the electrons remains circulating in the resonator and a small fraction of the power is out-coupled. This implies that the loss, which is often dominated by the fraction of radiation that is coupled out of the resonator, but also includes losses at the mirrors or within the resonator, can impose difficulties if mirror absorption is large enough to result in excessive heating or degradation of the reflectivity. In such cases, the mirrors must be cooled or otherwise protected. For example, storage ring FELs have a low small-signal gain and thus require high-quality mirrors that are susceptible to the higher-harmonics generated. Short-wavelength X-ray FEL oscillators also have a low small-signal gain and the available X-ray mirror technology is pushed to its limits to produce mirrors with sufficiently low loss. High average power operation may require the cryogenic cooling of the mirrors to prevent mirror distortion due to thermal loading. The efficiency of a low-gain/high-\(Q\) oscillator is predicted to be \(\eta=1/2.4N_{u}\)[1], where \(N_{u}\) denotes the number of uniform periods in the undulator; hence, high efficiency requires relatively short undulators. Because of this, oscillator design is a balance between having a sufficiently long undulator for the gain to exceed the losses, set by the quality \(Q\) of the resonator, but not so long that the efficiency is negatively impacted.