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This paper presents a high-order vector finite element analysis of periodic structures. In the modeling, the computational domain is confined to a unit cell by enforcing appropriate boundary conditions. The region inside the unit cell is discretized with high-order curvilinear tetrahedral elements to obtain the best geometric modeling. Since the FEM typically produces a large number of unknowns, high-order elements are used to reduce computation time and memory for a given accuracy. Furthermore, an asymptotic waveform evaluation (AWE)-based technique is used to perform fast frequency and angle sweeps. Numerical examples, including periodic absorbers and FSS structures, are presented to demonstrate the accuracy and versatility of the method.