Topological photonics in metamaterials

Originally a pure mathematical concept, topology has been vigorously developed in various physical systems in recent years, and underlies many interesting phenomena such as the quantum Hall effect and quantum spin Hall effect. Its widespread influence in physics led the award of the 2016 Nobel Prize in Physics to this field. Topological photonics further expands the research field of topology to classical wave systems and holds promise for novel devices and applications, e.g., topological quantum computation and topological lasers. Here, we review recent developments in topological photonics but focus mainly on their realizations based on metamaterials. Through artificially designed resonant units, metamaterials provide vast degrees of freedom for realizing various topological states, e.g., the Weyl point, nodal line, Dirac point, topological insulator, and even the Yang monopole and Weyl surface in higher-dimensional synthetic spaces, wherein each specific topological nontrivial state endows novel metamaterial responses that originate from the feature of some high-energy physics.

Keywords

constitutive relation metamaterial semimetal topological photonics

1 Introduction

In geometry, topology^[1] concerns the global features of a shape, independent of the detail—a famous example being that a coffee mug and a torus are topologically equivalent because they can be smoothly transformed into each other without experiencing dramatic changes, e.g., opening holes, tearing, gluing. The Euler characteristic $χ$ is introduced to describe such a global invariant, which is defined by the integral of the Gaussian curvature $\vec{K}$ over a closed surface, $χ = \frac{1}{2 π} \int_{S} \vec{K} \cdot d \vec{A} \in Z$ , which is always an integer. Hence, it cannot vary continuously and is topologically stable. The surfaces of a sphere ( $χ = 2$ ) and torus ( $χ = 0$ ) are distinguished topologically by their Euler characteristics $χ$ .