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An eigenvalue solution algorithm is formulated based on the finite-difference frequency-domain (FDFD) method for determining guided modes, including the surface plasmon modes, supported by periodic metallic structures. The Yee-mesh grids which have been popularly adopted in the finite-difference time-domain (FDTD) method are used in the FDFD method and standard eigenvalue matrix equations are obtained for easily searching for the guided eigenmodes. Both two-dimensional (2-D) and three-dimensional (3-D) structures are considered and the periodicity is along the propagation direction. The metals are assumed to be perfect ones or real ones without loss. For 2-D structures, an array of grooves drilled in a perfect conductor and a real-metal structure with a periodic arrangement of subwavelength slits in air are analyzed and the dispersion diagrams and mode-field profiles are obtained. For the latter structure, surface plasmon modes and dielectric slab modes are identified to be in agreement with published results based on a different numerical scheme. This subwavelength-slit structure is then extended to a 3-D one having an additional depth and it is demonstrated that the formulated algorithm can solve the same two kinds of modes for the more complicated 3-D problem. The modes guided along drilled periodic rectangle holes on a perfect conductor surface are also calculated.