# Three-Dimensional, Time-Dependent Analysis of High- and Low-\(Q\) Free-Electron Laser Oscillators
## 5 High-Gain/Low-Q Oscillators
### An Infrared RAFEL
As a result, the mode structure that forms as the RAFEL saturates differs substantially from the cold cavity modes, and our choice of a Rayleigh range of 0.5 m serves mainly to determine the radii of curvature of the mirrors. Since the radiation is guided in the undulator, the spot size at the undulator exit may be smaller than it would be in the cold cavity, which means that the Rayleigh range of the radiation as it exits the undulator is smaller than it would be in the cold cavity. This implies, in turn, that the optical mode will expand more rapidly as it propagates to the downstream mirror. Alternatively, decomposing the smaller spot size at the undulator exit in cold cavity modes, necessarily leads to higher order transverse modes in the optical field. After propagating to the outcoupler, the superposition of these modes determine the fraction of the optical field coupled out through the hole, and, similarly, after propagation to the undulator entrance, the superposition sets the field profile at undulator entrance. Small variations in the exponential growth rate, e.g., due to changing coupling of the electrons to the optical field at the undulator entrance, lead to relatively larger effects on the optical guiding of the radiation. This in turn changes the spot size at the undulator exit and, hence, the energy coupled out of the resonator and the spot size at the undulator entrance. The oscillation in mode size has also been observed in simulation of low-gain oscillators [183] where the magnitude of the oscillation depends on the amount of optical guiding, which may vary between the FEL gain codes used [204]. Furthermore, it is found that mirror aberrations, e.g., spherical aberration, also effect the variation in optical mode size observed from pass to pass [204]. This is illustrated in Figure 16, where we plot the pass-to-pass variation in the width of the optical mode on the downstream and upstream mirrors for \(B_{u}=6.585\) kG (\(K_{\text{rms}}=1.043\)) and \(\Delta L_{\text{cav}}=-8\lambda\). It is clear that both the spot size and the fluctuations of the spot size on the upstream mirror are greater than those on the downstream mirror, due to the optical properties of the resonator. At saturation, the location of the smallest optical beam size moves over the axis of the undulator and this changes the optical magnification. Consequently, the size of the optical field at both mirrors as well as at the entrance of the undulator changes from pass to pass, as can be observed in Table 1.2  
Table 1.3 Variation in the width (rms diameter) of the optical beam incident on the upstream mirror (red) and downstream mirror (blue) as a function of the number of passes for a cavity detuning of \(\Delta L_{\text{cav}}=-8\lambda\) and an undulator field strength of \(B_{u}=6.658\) kG (\(K_{\text{rms}}=1.055\)). The other parameters are given in Table 1.4  
The transverse mode structure is not only a result of optical guiding; it is also affected by the hole out-coupling. Consider the case of \(B_{u}=6.658\) kG (\(K_{\mathrm{rms}}=1.055\)) and \(\Delta L_{\mathrm{cav}}=-18\lambda\). The cross-section of the field delivered to the undulator entrance on pass 60 is shown in Figure 17, where we plot the normalized power in the \(x\)-direction (i.e., the wiggle plane). Observe that the bulk of the power is at the edge of the optical field, but there is a spike at the center that seeds the subsequent pass through the undulator. Despite the multiple peaks in the cross section at the undulator entrance, the strength of the interaction in the undulator yields a near-Gaussian mode peaked on-axis at the undulator exit, as shown in Figure 18a. What has happened is that the interaction with the electron beam, which has a diameter of about 0.784 mm, essentially amplifies and guides the central peak shown in Figure 17, while the power in the wings falls outside the electron beam and is not amplified.