These works ushered in topological photonics research, which extended topological physics from quantum to classical systems. Shortly afterward, the idea was experimentally implemented by Wang et al. The photonic bands would have nontrivial topological invariants in such an electromagnetic system. In 2008, Haldane and Raghu made the crucial generalization of topological physics into photonics by proposing that the presence of “nonreciprocal” (Faraday-effect) media in photonic crystals can introduce a direct analog of the chiral edge states of electrons in the quantum Hall effect. įor a long time, research on topological physics had focused mainly on condensed-matter systems. More detailed information can be found in previous reviews. Most early topological systems are found in the Hall effect family, e.g., quantum spin Hall effect, quantum anomalous Hall effect. A topologically protected surface state is guaranteed at the interface between two topologically distinct systems with Δ C = 1, which possesses features of gaplessness, unidirectional propagation, and immunity to structural defects. A most remarkable feature for such a topological nontrivial system is the celebrated bulk boundary correspondence, which indicates that the multiplicities of edge modes on the boundary are characterized by differences in the topological invariants of the bulk energy bands. Just like the Euler characteristic χ in mathematics, such a flat quantized resistance is related to the integer-valued integral of Berry curvature F → over the filled portion of the bands in a crystal, the so-called Chern number, or TKNN number, C = 1 2 π ∫ B Z F → The measured values of Hall conductance are integer or fractional (in the fractional quantum Hall effect ) multiples of e 2 / h to nearly one part in a billion. In a two-dimensional electron system subject to low temperature and a strong magnetic field, the Hall resistance R x y exhibits plateaus that take on quantized values R x y = h / v e 2, with v being an integer number, e the elementary charge, and h the Planck’s constant. The earliest discovered topology-governed physical phenomenon was the celebrated quantum Hall effect. The surfaces of a sphere ( χ = 2) and torus ( χ = 0) are distinguished topologically by their Euler characteristics χ. Hence, it cannot vary continuously and is topologically stable. d A → ∈ Z, which is always an integer.The Euler characteristic χ is introduced to describe such a global invariant, which is defined by the integral of the Gaussian curvature K → over a closed surface, χ = 1 2 π ∫ S K → In geometry, topology 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. 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. Here, we review recent developments in topological photonics but focus mainly on their realizations based on metamaterials. 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. Its widespread influence in physics led the award of the 2016 Nobel Prize in Physics to this field. 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.
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