Abstract
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 simply, where 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.
- Received 7 January 2014
DOI:https://doi.org/10.1103/PhysRevX.4.021017
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Published by the American Physical Society
Popular Summary
When a light wave hits a photonic crystal, it can be both reflected by the surface it hits and transmitted through the surface. Surface impedance, which measures the ratio of the electric to magnetic field at the surface of a material, governs this “surface” process and is used as the key design parameter for the engineering of artificial materials, from antennae and efficient absorbers to stealthy objects.
The propagation of a wave inside a photonic crystal is, on the other hand, determined by a so-called “band structure”—“bands” showing the multiple allowed wave frequencies for each wavelength. Is the bulk band structure, then, at all related to the surface impedance? And if so, how exactly are they related, and what kind of predictions can such a relationship offer us? In this paper, we rigorously show that the surface impedance for a one-dimensional photonic crystal is, in fact, uniquely determined by the geometric property of the bulk band structure—an understanding that is not only of fundamental importance but can also be used to engineer photonic states at the interface of two photonic crystals.
The one-dimensional photonic crystal we have studied consists of a stack of alternating layers of two optically different materials. This photonic crystal has spatial inversion symmetry even when the two types of layers have different thicknesses. We have found a rigorous relationship between the surface impedance of the photonic crystal and the geometric property (the so-called ZaK phase) of its bulk band structure.
This relationship not only establishes a new fundamental understanding of the electromagnetic surface impedance, but it also offers a deterministic recipe for designing propagating waves confined at the interface of two photonic crystals, based entirely on the knowledge of their bulk band structures instead of the surface impedance, which is more difficult to obtain.
