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Topological Insulators in Photonics

Topological insulators (TIs) are a class of materials which possesses properties of both conductors and insulators. TIs conduct close to the edge or boundary of a material and act as insulators in the interior or bulk of a material. These states were orginally identified in condensced matter physics systems [1], but more recently they have been observed in optical waveguide arrays [2].

Topologically Protected Modes

One property that distinguishes topological insulators is the presence of topological invariants such as the Chern number. These integers are independent of the edge boundaries and possible defects they may possess. A typical set of dispersion bands are shown in Fig. 1. Localized edge modes on the left and right edges of a semi-infinite domain correspond to a set of bands that span the gapbetween two bulk (non-localized modes) bands. The bulk bands possess a nontrivial Chern number that indicates the states are indeed topologically protected. If we deform the waveguide arrays, then the non-topological bands lose their gapless edge modes and the corresponding Chern numbers are zero.

Fig. 1: Dispersion bands found by solving for edge modes along a semi-infinite domain. Dirichlet zero BCs are used on the left and right zig-zag edges. The Chern number for the bulk bands is included.

The dynamics of the edge modes corresponding to the bands found in Fig. 1 are shown in Fig. 2. When a topologically protected mode propagates into a defect barrier it and does not backscatter. This mode exhibits nearly 100% transmission of its intensity. If, on the other hand, the mode is not topologically protected, then the mode will reflect backward off the barrier.

Fig. 2: (Top row) Evolution of a topologically protected mode into a defect barrier. The output intensity is nearly equal to the input intensity. (Bottom row) Progression of edge mode that is not topologically protected into same defect barrier. Considerable reflection and loss of intenisty is observed.

[1] Thouless, Kohmoto, Nightingale, and den Nijs, Quantized Hall Conductance in a Two-Dimensional Periodic Potential, PRL (1982).

[2] Rechtsman, Zeuner, Plotnik, Lumer, Podolsky, Dreisow, Nolte, Segev, and Szameit, Photonic Floquet topological insulators, Nature (2013).

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