Bulk and Surface Correspondence through Geometric Phases in Classical Wave systems

This thesis is concerned with the relationship between the geometric phase of the bulk bands in a periodic classical wave system and the surface impedance of a truncated bulk. As the Brillouin zone of a periodic system can be regarded as having a torus topology, the integral of Berry curvature over the closed torus is known to be topological and quantized as integers. This topological classification allows one to identify some systems as topologically trivial and others are topologically non-trivial. A fundamental consequence of this classification is the prediction of the existence of surface states in the boundary separating gapped topologically different systems. For a non-absorptive and closed classical wave system, we can also use a Hamiltonian description and thus we should also be able to give a topological classification for classical wave systems. This thesis is devoted to calculating the geometric phase for classical wave periodic systems and giving a topological description for classical waves which include but are not limited to electromagnetic waves and acoustic waves. We derived a new relationship between the surface impedance of a one-dimensional Photonic Crystal (PC) and its bulk properties through the geometrical (Zak) phases of the bulk bands. This relation applies to any PCs with inversion symmetry and can also be extended to acoustic systems. This relationship shows that the bulk properties determine the surface impedance and the relationship can be used to predict the existene of interface modes. Based on this relation, we are able to experimentally measure the Zak phases of bulk bands in acoustic systems. We also designed for the first time an acoustic system which has a topological transition point and band inversion. We then extended the idea to 2D systems, and we found the relationship between surface impedance and bulk geometric phase for 2D photonic crystals. In order to achieve this goal, we have to find an analytic formulation of the surface impedance using layer-by-layer scattering theory. The derived impedance can explain the existence of various kinds of interface states. In particular, we find and explain why a small perturbation guarantees the existence of interface states in some 2D photonic crystals possessing Dirac-like cone dispersions at k=0. Through the new knowledge of bulk and interface correspondence, the existence of interface states can also be understood from the geometric phases of the bulk bands.