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This paper introduces a new explanation of the electromagnetic bandgap (EBG) material properties using the study of the EBG structures in the frequency domain and reciprocal space. Once the behavior of such a material is understood, the properties of the EBG are used in order to make an EBG antenna. The antenna is realized with dielectric EBG rods. Its directivity is increased compared to a simple patch antenna. Such a device allows us to obtain a high gain with a very thin structure.