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Green's function representations are presented to rapidly compute the fields resulting from a linear (1D) periodic array of dipole current sources on or near a planarly layered medium in 2D and 3D space. The representation is formulated as spectral integral, which accounts for the reflected continuous spectrum of fields, and a series that accounts for the discrete spectrum of guided modes. It is exponentially convergent for observation points on and near the array axis and surface, and for complex phase shifts between periodic unit cells. It can be defined on alternate Riemann sheets with respect to any of the diffraction modes characterizing the array. A complete dyadic Green's function is derived to fully account for the reflected fields for all source current orientations. This Green's function representation can greatly accelerate the simulation of printed 1D periodic structures in optics and microwave engineering.