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A new design of two-dimensional cylindrical Luneberg lens is introduced based on TE10 mode propagation between parallel plates, with special focus on ease of manufacturing. The parallel plates are partially filled with low cost polymer material (Rexolite epsivr = 2.54) to match Luneberg's law. A planar linear tapered slot antenna (LTSA) is inserted into the air region between the parallel plates at the edge of the Luneberg lens as a feed antenna, with fine positioning to the focal point of the Luneberg lens to optimize the antenna system performance. A combined ray-optics/diffraction method is used to obtain the radiation pattern of the system and results are compared with predictions of a time domain numerical solver. Measurements done on a 10-cm Luneberg lens designed for operation at 30 GHz agree very well with predictions. For this prototype, 3-dB E- and if-plane beamwidths of 6.6deg and 54deg respectively were obtained, and the sidelobe level in the E-plane was -17.7-dB. Although the parallel plate configuration should lead to a narrow band design due to the dispersion characteristics of the TE10 mode, the measurement results demonstrate broadband characteristics with radiation efficiencies varying between 43% and 72% over the tested frequency band of 26.5-37 GHz. The designed cylindrical Luneberg lens can be used to launch multiple beams by implementing an arc array of planar LTSA elements at the periphery of the lens, and can be easily extended to higher mm-wave frequencies.