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An eigenvalue analysis numerical technique for curved closed waveguiding structures loaded with inhomogeneous and/or anisotropic materials is presented. For this purpose, a frequency-domain finite-difference method is developed for a general orthogonal curvilinear coordinate system. The main strength of the technique is the accurate modeling of curved transverse boundaries, as well as curved geometries along the propagation direction, under certain limitations for the latter. This feature avoids the necessity of a fine mesh, while it retains a high accuracy and it is free of any staircase effects. In general, the proposed method shares the finite-element technique capabilities in modeling complex boundaries, while it preserves the finite-difference convenience in handling inhomogeneous and anisotropic material loadings. The resulting eigenvalue-based problem is solved using the Arnoldi algorithm, which exploits the system matrix sparcity and the overall technique is robust, unconditionally stable with minimal computational and memory requirements. Numerical results are validated against analytical results and results from 3-D commercial electromagnetic simulators. Finally, novel results are also given.