| Free-electron Lasers(FELs) have some unique features such as wavelength tenability and design-ability, and potentialities for high efficiency and good beam quality etc. Therefore FELs may be found wide-range applications in many areas such as science, defense, medicine and industry. The optical cavity is one of important segments in oscillator FEL system. Presently, there are some of the most difficult issues and key problems associated with the oscillator FEL design and experiment. These problems include optical damage, optical loss, coupling, detuning, sensitivity to vibration and misalignment, and extraction efficiency etc. The study on three-dimension resonator problems is still on the frontiers of FEL theory. In this dissertation, comprehensive three-dimensional simulations and theoretical studies of FEL oscillators resonator are carried out with special attention on the CAEP FIR FEL and JLAB IR FEL. The following results are achieved:The research and comparison of the two kinds of output couplers have been done. The 3D simulations with inserted scraper output coupler are carried out for the first time with the help of our 3D FEL oscillator code. The saturated power, the net gain and the coupling efficiency are evaluated as a function of the scraper radius and the distance between scraper and the center of mirror. It has been proved that the inserted scraper is effective and applicable for the FIR FEL. Using inserted scraper output coupler, the optical cavity Q can be tuned freely. These will be helpful to the design of the Far-Infrared FEL device.The transverse construction of optical modes in the wiggler is calculated and simulated to two kinds of output couplers considering the gain and optical guiding effects. These studies include the evolvement of optical field, distributing of modes in the entrance and exit of wiggler and on the mirrors. The numerical simulations confirm that the fundamental mode is still dominant whenthe output mirror shift from the center to the edge. Based on these facts, the estimate of optical loss and coupling efficiency is calculated by a brief method.The waveguide's influence on the gain is estimated approximately by calculating the transverse coupling factor between the optical beam and electron beam. Furthermore, the waveguide's influence is simulated by 3D code. Then due to the main problem of CAEP FIR FEL experiment is that the gain is too small, the simulation have been done by the methods of reducing the wave-guide gap and enhancing field intensity to increase the optical gain. Finally, non-coaxial problems such as the tilts of the mirror in two direction are studies, which will be helpful to the designs of the FEL device.The detuning curves of the CAEP FIR FEL and the Jefferson Lab IR Demo FEL are calculated with the help of our 3D code, which are compared with those from one-dimensional simulations and super-mode theory, and the experiments. The influence of the optical guiding and other three- dimensional effects on the detuning curve is studied. It is found that the length of the detuning curve from three-dimensional simulations is shorter than that from one-dimensional simulations and super-mode theory.The introduction and analyses have been done to the characteristic of a new style optical resonator used in the regenerative amplifier FEL (RAFEL). Furthermore, the simulations have been carried out to RAFEL device in the LOS ALAMOS Lab. These results include the wiggler gain, net gain, output coupling etc. , which are compared with the results from theory and the experiments. It shows both results accord well.
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