Surface Impedance and Bulk Band Geometric Phases in One-Dimensional Systems

Surface impedance is an important concept in classical wave systems such as photonic crystals (PCs). For example, the condition of an interface state formation in the interfacial region of two different one-dimensional PCs is simplyZ(SL) + Z(SR) = 0, where Z(SL)(Z(SR)) is the surface impedance of the semi-infinite PC on the left-hand (right-hand) side of the interface. Here, we also show a rigorous relation between the surface impedance of a one-dimensional PC and its bulk properties through the geometrical (Zak) phases of the bulk bands, which can be used to determine the existence or nonexistence of interface states at the interface of the two PCs in a particular band gap. Our results hold for any PCs with inversion symmetry, independent of the frequency of the gap and the symmetry point where the gap lies in the Brillouin zone. Our results provide new insights into the relationship between surface scattering properties, the bulk band properties, and the formation of interface states, which in turn can enable the design of systems with interface states in a rational manner.