# 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  
Most optical resonators used in laser oscillators consist of two spherical mirrors separated by a distance \(L_{\text{cav}}\), as schematically shown in Figure 1. When the radius of curvature \(R_{\text{c}}\) of the mirrors is equal, the resonator is called concentric when \(L_{\text{cav}}=2R_{\text{c}}\) and confocal when \(L_{\text{cav}}=R_{\text{c}}\). Other configurations with an unequal radii of curvature of different distances between the mirrors are known as generalized resonators. Most of the low-gain FEL oscillators employ stable concentric resonators to ensure that the optical pulse train does not \"walk\" out of the cavity. Consider a generalized resonator with a cavity length \(L_{\text{cav}}\), as shown in Figure 5, which shows the radii of curvature, \(R_{c,i}(i=1,2)\), for the two mirrors, as well as the distances \(z_{i}\) from the mode waist to each of the mirrors. The transverse optical modes of the cold resonator, i.e., without a gain medium (electron beam) present, can be described as either Gauss-Hermite _cf._ Equations (1)-(3) or Gauss-Laguerre modes. Defining  
\[g_{i}=1-\frac{L_{\text{cav}}}{R_{c,i}} \tag{40}\]  
then the Rayleigh range \(z_{R}\) for the Gaussian eigenmodes in the cavity is given by  
\[z_{R}=L_{\text{cav}}\sqrt{\frac{g_{1}g_{2}(1-g_{1}g_{2})}{g_{1}+g_{2}-2g_{1}g_{ 2}}} \tag{41}\]  
Since the Rayleigh range must be positive, we must have \(0\leq g_{1}g_{2}\leq 1\), which is the condition for resonator stability [200]. This means that Gaussian beams can only be eigenmodes of stable resonators. The stability condition implies that  
\[L_{\text{cav}}<R_{1}+R_{2}. \tag{42}\]  
Figure 5: Schematic representation of a generalized two-mirror resonator, defining some of its parameters. We also note that the waist of the Gaussian mode is at a distance  
\[z_{1}=L_{\text{cav}}\frac{g_{2}(1-g_{1})}{g_{1}+g_{2}-2g_{1}g_{2}} \tag{43}\]  
from mirror \(1\) and at a distance \(z_{2}=L_{\text{cav}}-z_{1}\) from mirror \(2\). Note that the sign convention used for the ray-optic ABCD formalism [200] applies here as well, meaning that \(R_{c}>0\) for a concave mirror and \(R_{c}<0\) for a convex mirror and that the distances \(z_{i}\) have a sign. The Gauss-Hermite transverse modes as described by Equations (1)-(3) are eigenmodes of an empty generalized resonator with a particular Rayleigh range and location of the waist defined by the resonator. Whenever the resonator contains internal apertures clipping the optical field, partial waveguiding, or if the light is extracted via a hole, a single transverse Gaussian mode can no longer be an eigenmode of the resonator. The Fox-Li method [206] can be used to find the eigenmodes of such resonators [207]. The resonant frequencies, or longitudinal modes, of the resonator are those frequencies for which the total roundtrip phase is an integer multiple of \(2\pi\). As a result, the resonant frequencies differ for the different transverse optical modes [200]. The frequency spacing between two subsequent resonances of the same transverse mode is known as the free-spectral range \(\Delta v_{\text{FSR}}\). FEL resonators typically have a \(\Delta v_{\text{FSR}}\) that is much smaller than the gain bandwidth, due to the large mirror separation used. Consequently, most FEL oscillators are operating on multiple longitudinal modes. As shown in Figure 1, the resonator may contain more than one optical pulse if the roundtrip time in the resonator is larger than the time between subsequent electron bunches. The resonator design must ensure synchronism between the optical pulses and the electron bunches and optimal performance is typically found near the synchronous or zero-detuning cavity length, where the incoming electron bunch coincides with the returning optical pulse train.