A novel and systematic method is developed to evaluate periodic Green's functions on empty lattices through extractions of an imaginary wavenumber component of the lattice Green's function and its associated derivatives. We consider cases of volumetric periodicity where the dimensionality of the periodicity has the same dimensionality as the physical problem. This includes one-dimensional (1D) problem with 1D periodicity, two-dimensional (2D) problem with 2D periodicity, and three-dimensional (3D) problem with 3D periodicity, respectively. The remainder of the Green's function is put in spectral series with high-order power-law convergence rates, while the extracted imaginary wavenumber parts are put in spatial series with super-fast and close-to exponential convergence rate. The formulation is free of transcendental functions and thus simple in implementation. It is especially efficient for broadband evaluations of the Green's function as the spatial series are defined on fixed wavenumbers that take small CPU to compute, and the spectral series have simple and separable wavenumber dependences. Keeping only a few terms in both the spatial and spectral series, results of the lattice Green's function are in good agreement with benchmark solutions in 1D, 2D, and 3D, respectively, demonstrating the high accuracy and computational efficiency of the proposed method. The proposed method can be readily generalized to deal with Green's functions including arbitrary periodic scatterers.
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