# Three-Dimensional, Time-Dependent Analysis of High- and Low-\(Q\) Free-Electron Laser Oscillators
## 1 Introduction  
Free-electron laser (FEL) oscillators (FELO) have been part of the overall research activity since the beginnings of the field. An FEL oscillator consists of an undulator placed within an optical resonator, also known as an optical cavity, that typically consists of two spherical mirrors separated by a distance \(L_{\text{cav}}\). Note that the undulator is not necessarily centered within the optical resonator. Furthermore, an electron beam is directed into the cavity through the undulator, and out of the cavity, as is schematically shown in Figure 1, which is taken from Reference [1]. The optical pulse generated by the electrons streaming through the undulator is extracted from the cavity either by a partially reflective (i.e., transmissive) mirror or through a hole in the mirror. Observe that multiple optical pulses may be contained within the resonator depending upon the repetition rate of the electron bunches and the roundtrip time for the optical pulses to circulate through the resonator. We also note that, when the growth of the optical field exponentiates in a single pass through the undulator, i.e., in case of a high-gain FEL [2], a large fraction of the light can be extracted from the resonator. The first FEL oscillator driven by a relativistic electron beam was realized in 1977 [3], shortly after the coherent amplification of radiation of the same wavelength was experimentally demonstrated [4]. Coherent undulator radiation was first demonstrated using the so-called Ubitron, which was configured both as amplifier and oscillator, and which is nowadays considered to be the first (non-relativistic) FEL [5]. In the early days of FEL development, the technology available to generate high-quality, high-brightness electron beams, that are necessary for efficient FEL operation, was relatively primitive by today'sstandards. At that time, the limited available gain, which in part may be due to space restrictions, favoured oscillator configurations to bring the laser into saturation and provide fully coherent output, as mirrors were readily available all the way from the visible to well into the microwave spectral range. Although we limit ourselves here to infrared wavelengths or shorter, exciting FEL research also takes place at longer wavelengths. The unique properties of FEL oscillators, in particular to fill the spectral gap for coherent light sources in the visible to microwave spectral range, resulted quickly in the establishment of several facilities for applied research. The characteristics of user facilities using FEL oscillators depends, in part, on the accelerator technology used, which includes electrostatic acceleration [6; 7; 8; 9], energy recovery linear accelerators (linacs) [10; 11; 12; 13; 14; 15], non-recovery linacs [16; 17; 18; 19; 20], microtrons [31; 32] and synchrotrons [33; 34; 35; 36; 37]. Research performed at these user facilities includes, amongst others, biological and material science [38]. A more complete overview of the various FEL oscillators, together with the main system parameters, can be found in Ref. [39]. As mentioned, the optical properties of FEL oscillators are in part determined by the electron accelerator used. For example, electrostatic accelerators can only accelerate electrons up to a kinetic energy of a few MeV and, consequently, the spectral range is limited to microwave and THz frequencies [39]. The much higher electron energies of storage rings allow the FEL oscillators to operate in the visible down to the deep UV wavelengths [39], while RF linacs can, in principle, accelerate electrons to still higher energies, from a few MeV up to tens of GeV, allowing FELs to cover a large fraction of the electromagnetic spectrum, from microwaves all the way down to hard X-rays [39].