The use of two-dimensional Bragg resonators of planar geometry, realizing two-dimensional (2D) distributed feedback, is considered as a method of producing spatially coherent radiation from a large sheet electron beam. The spectrum of eigenmodes is found for a 2D Bragg resonator when the sides of the resonator are open and also when they are closed. The higher selectivity of the open resonator in comparison with the closed one is shown. A time-domain analysis of the excitation of an open 2D Bragg resonator by a sheet electron beam demonstrates that a single-mode steady-state oscillation regime may be obtained for a sheet electron beam of width 100-1000 wavelengths. Nevertheless, for a free-electron maser (FEM) with a closed 2D Bragg resonator, a steady-state regime can also be realized if the beam width does not exceed 50-100 wavelengths. The parameters for a FEM with a 2D planar Bragg resonator driven by a sheet electron beam based on the U-2 accelerator (INP RAS, Novosibirsk) are estimated and the project is described.

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