Impedance of Evanescent Modes for Determining Interface States in One-Dimensional Chain

Abstract

Interface states and edge states in periodic structures have been extensively investigated in the context of topological dynamics over the past decades. In this study, we propose an impedance method based on surface impedance to analyze interface and edge states in one-dimensional (1D) periodic chains. The impedances are defined analytically from the Bloch eigen-modes of the periodic chains. At the interface between two periodic structures, interface states arise at the frequencies where the impedances of the two structures become the same. Likewise, edge states occur when the impedance of the structure matches the boundary impedance. This approach is universal for studying trivial and topological interface and edge states in 1D chain with different types of boundary conditions. We demonstrate this point with three representative examples: a chain comprising two periodic lattices, a chain anchored to ground springs at both ends, and a symmetric chain with interfacial defects. The analysis of topological interface states offers a vivid physical perspective, revealing that the topological interface states are either symmetric or antisymmetric modes. Furthermore, we show that the frequency of the symmetric topological state can be tuned via a single spring at the interface. This finding can be used to design tunable topological devices.