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Periodic structures like gratings or frequency selective surfaces are widely met in antenna design and applications. Some problems require calculating reflectivity in a wide frequency band. Time domain methods like FDTD are beneficial in such problems since they provide the whole spectrum in a single simulation. A periodic structure is modeled as a Floquet waveguide. In FDTD the simulation region needs to be bound with absorbing boundary condition that imitates a semi-infinite waveguide. We tested several such conditions, which are widely used, and reveal that none of these performs well near cutoff frequencies. The problem is demonstrated on a simple illustrative diffraction problem. Some possible ways of improving ABC to overcome the difficulty are discussed.

absorbing boundary condition; Floquet waveguide; Klein-Gordon equation; mode basis method; perfectly matched layer