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We have developed 585-GHz quasi-optical mixers and focal-plane arrays (FPAs) comprised of planar annular slot antennas (ASAs) with integrated niobium hot-electron bolometers for imaging applications. In order to optimize the single-element mixer design, the embedding impedance of the single ASA presented to the bolometer is analyzed using the induced electromotive force (EMF) method by including the antenna feed contribution. This approach has been further expanded to analyze the ASA self-impedance and mutual impedance in an array by utilizing the even-odd mode analysis. In addition, the far-field radiation patterns of the ASAs mounted to an extended hemispherical high-resistivity silicon lens have been calculated using the ray-tracing techniques. The details of circuit design and fabrication are presented in this study. Single mixer element measurement results have shown that a conversion gain of -11.9 dB and a double-sideband (DSB) receiver noise temperature of ~650 K have been achieved. Initial array imaging experiment results are presented and show excellent agreement with theory and simulation data. A spatial resolution of ~2.75 mm has been demonstrated at 585 GHz for a 1-D mixer FPA that is capable of diffraction-limited imaging. Y-factor measurements show DSB mixer noise temperatures of 1675 and 3517 K with mixer conversion gains of -14.73 and -17.74 dB, respectively, have been obtained for two adjacent elements in a mixer array, which is comparable to the results reported in the literature.