Optical nanoantennas support surface plasmon polariton (SPP) with a confinement of light breaking through the diffraction limit, and thereby achieving an enhancement and regulation of electromagnetic field on a deep-subwavelength scale. In this paper, a periodic array of optical nanoantennas on a metallic mirror is proposed, where the antennas are gold nanocubes forming a two-dimensional periodic array, and a single point emission source is located in the nanogap between the antenna of gold nanocube and the gold mirror. The nanogap between the antenna and mirror can support gap surface plasmon, which results in an enhanced spontaneous emission rate. Meanwhile, the periodic array of nanoantennas can support the surface lattice resonance (SLR), which can achieve directional far-field radiation perpendicular to the substrate or in a specified direction by properly designing the array period. To design the antenna that can simultaneously achieve an enhancement of spontaneous emission rate and a directional radiation of far field, the calculation of the radiation field of a single point source in a periodic structure is transformed into the calculation of the radiation fields of a set of pseudoperiodic point-source arrays by combining the array scanning method (ASM) and full-wave rigorous numerical method, thus giving the spontaneous emission rate of the emitter and the near-field distribution of the antenna. Concerning the calculation of the angular distribution of far-field radiation intensity, we start from the Maxwell’s equations and provide a rigorous formulation and proof of the reciprocity-theorem method. This proof is different from those reported in existing literature and has a more rigorous applicability for infinite-extent periodic structures or has a lower amount of computational work. Based on the reciprocity-theorem method, the antenna parameters are designed so that the enhancement factor of far-field radiation intensity reaches a maximum value of 2756 in the direction perpendicular to the substrate, and simultaneously, the enhancement factors of total spontaneous emission rate and far-field spontaneous emission rate of the point source reach 1097 and 55.50, respectively. The proposed antenna has a simple structure that is easy to design and fabricate, and the proposed design method is intuitive and easy to implement, which can be used to guide the design of high-speed, high-brightness and directional-radiation light sources.