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The use of two-dimensional (2D) distributed feedback is considered as a method of providing spatially coherent radiation from an oversized annular electron beam. To realize the feedback mechanism, 2D Bragg structures formed from doubly-corrugated waveguide sections of coaxial geometry are suggested. The properties of two types of coaxial cavities formed using such structures are compared: a single-section 2D Bragg cavity and a two-mirror cavity. The eigenmodes of both cavities are found and their high selectivity over both azimuthal and longitudinal indices was demonstrated. Time-domain analyses of the excitation of the cavities by an annular electron beam were carried out. The influence of the cavity parameters on the oscillation regime is analyzed and discussed. It was shown that for a specific set of 2D Bragg cavity parameters it is possible to obtain a regime of steady-state oscillations when the transverse size of the beam exceeds the wavelength by a few orders of magnitude, while outside this parameter space multimode oscillation takes place. The design of a 2D Bragg free-electron maser oscillator based on a high-current accelerator at the University of Strathclyde is discussed. © 2002 American Institute of Physics.