A fast periodic interpolation method (FPIM) is presented for rapidly computing fields in a unit cell of an infinitely periodic array. For low and moderate frequencies (for unit cells smaller than or on the order of the wavelength) the FPIM has the computational cost of ( ) and it requires only (1) periodic Green's function (PGF) evaluations, for sources and observers. For high-or mixed-frequencies the computational cost scales aswhere is the domain size within the unit cell and is the wavelength. FPIM is based on splitting the field into the near-field from the sources around the unit cell and the far-field from the remaining sources. The near-field component can be evaluated rapidly using any available fast method. The far-field component is computed by tabulating the PGF at sparse source and observer grids, using this table to calculate the field at the observation grid, and interpolating from the observation grid to the actual observers. The FPIM is kernel independent and allows using any method for evaluating the PGF, including simple Floquet expansions. The computational times can be comparable to those of conventional (non-periodic) -body electromagnetic problems. The presented method can be used to accelerate integral equations for periodic unit cell problems with many applications in microwave engineering and optics. Diego, where he currently holds the position of Associate Professor. His research interests include computational electromagnetics, micromagnetics, the analysis of microwave phenomena on structured surfaces, the analysis of optical phenomena in photonic nanostructures, and the analysis of magnetization dynamics in magnetic nanostructures.