Microscopic theory of photonic one-way edge mode

Unusual wave propagation effects can occur in systems with broken time-reversal symmetry. In a two-dimensional electron gas, for example, time-reversal symmetry can be broken by a perpendicular static magnetic field. As a result, the system can exhibit a quantum Hall effect, with a unidirectional flow of electrons on the edge of the system.1–4 Analogous one-way modes of photons have recently been proposed using surface plasmon polaritons,5 photonic crystals,6–10 and the interface of two magneto-optical materials.11 In all cases, time-reversal symmetry is broken by the use of materials exhibiting magneto-optical effects. Most recently, such a oneway edge mode in a photonic crystal has been experimentally observed.12–14 In a magneto-optical photonic crystal system,15–17 the existence of a one-way edge mode is typically linked to a nonzero Chern’s number of bulk band structure.6–8 Such a link is important because it provides a general condition for designing the magneto-optical photonic crystal structure. However, since the Chern’s number is a universal number characterizing the topological properties of the band over the entire first Brillouin zone, the microscopic connection, between the properties of the bulk mode and the properties of the one-way edge mode, is not evident in the Chern’s number analysis. In this paper, we consider a model system consisting of a honeycomb lattice of magneto-optical resonators. We show that the existence and the detailed physical properties of the one-way edge modes in this system can be understood in terms of the properties of the bulk modes at the edge of the bulk photonic band gap. This understanding then provides insights into the design of many properties of one-way edge modes. The paper is organized as follows: in Sec. II we describe our physical model consisting of a honeycomb lattice of resonators, and we present numerical results of the band structure and the eigenmode field distribution at the band edges for this system. In Sec. III we construct a tight-binding model to analytically calculate the band structure of such a honeycomb lattice of resonators. We obtain excellent agreement between the tight-binding model and the numerical results. In Sec. IV, we present a microscopic picture of the emergence of the oneway edge modes in this system, combining the numerical and the tight-binding analyses outlined in the previous sections. In Sec. V, we apply the microscopic picture developed in Sec. IV to other kind of edges, and we design a one-way edge state with a small group velocity.