Eigenmode and eigenpropagation of the electromagnetic waves in Möbius and Klein networks

To explore the distribution of characteristic frequencies and the propagation properties of eigenmodes in topological networks at the zero-energy level, we design optical waveguide networks with two typical topologies: Möbius network and Klein network, inspired by the Möbius strip and Klein bottle, respectively. We investigate the degeneracy at characteristic frequencies and the propagation properties of the eigenmodes of these networks, both theoretically and experimentally. We discovered an intriguing eigenpropagation in the Möbius network and multiple degenerate eigenmodes in the Klein network, analyzing the propagation characteristics and distribution patterns of electromagnetic waves within them. In our experiments, we utilize coaxial cables as one-dimensional waveguides to construct transmission line networks for the two networks. We observe the distinct transmission paths of the Möbius network's eigenmode and the two degenerate eigenmodes of the Klein network. Our findings provide a theoretical foundation for new optical modal transmission devices and novel nanoarrays, with potential implications for theoretical and experimental research in other quantum systems and topological networks.