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A general relation considering an interface condition between a sampled point and its nearby points in the finite-difference analysis of optical waveguides, is derived. Our formulation is a full-vectorial one, and the interface can be oblique to the global coordinate axes or even can have curvature. Using the derived formulation, optical waveguide structures with arbitrary step-index profiles can be dealt with. The validity of the proposed method is examined by comparing the numerical calculation with the analytical solution of a step-index optical fiber. The method is also applied to the analysis of fused fiber-optic couplers and compared with prior analysis using the surface integral equation method.