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Substrate integrated waveguides (SIW) are built up of periodically arranged metallic via-holes or via-slots. The leakage loss of SIW structures increases with the distance between the via-holes or via-slots. An open periodic waveguide with a large via distance supports the propagation of leaky-wave modes and can thus be used for the design of a leaky-wave antenna. In this paper, this leakage loss is studied in detail and used to design a periodic leaky-wave antenna. The proposed concept represents an excellent choice for applications in the millimeter-wave band. Due to its versatility, the finite difference frequency domain method for periodic guided-wave or leaky-wave structures is used to analyze the characteristics of the proposed periodic leaky-wave antenna. Two modes (TE10 and TE20) are investigated and their different leaky-wave properties are analyzed. Based on the proposed leaky-mode analysis method, a novel periodic leaky-wave antenna at 28-34 GHz is designed and fabricated.