Visualization of Higher-Order Topological Insulating Phases in Two-Dimensional Dielectric Photonic Crystals

The studies of topological phases of matter have been developed from condensed matter physics to photonic systems, resulting in fascinating designs of robust photonic devices. Recently, higher-order topological insulators have been investigated as a novel topological phase of matter beyond the conventional bulk-boundary correspondence. Previous studies of higher-order topological insulators have been mainly focused on the topological multipole systems with negative coupling between lattice sites. Here we experimentally demonstrate that second-order topological insulating phases without negative coupling can be realized in two-dimensional dielectric photonic crystals. We visualize both one-dimensional topological edge states and zero-dimensional topological corner states by using the near-field scanning technique. Our findings open new research frontiers for photonic topological phases and provide a new mechanism for light manipulating in a hierarchical way.

Figure 1

Crystal structures and bulk band structures. (a) The lattice structure with $d_1$ and $d_2$ representing the intracell distance and intercell distance, respectively, of the nearest atoms. (b) Different configurations of photonic crystals. The dashed blue (red) circles represent the shrunken (expanded) configuration, and gray circles represent the normal square lattice without tetramerization. (c) Three-dimensional structure of a unit cell with the upper plate removed. (d) Band structures for $\mathrm{\Delta }=-0.11a$ (left), $\mathrm{\Delta }=0$ (middle), and $\mathrm{\Delta }=0.11a$ (right), which represent a shrunken, normal, and expanded lattice, respectively. For shrunken and expanded lattices, the band structures have the same dispersion but different parities of the first and second bands at the $X$ point (denoted by $+$ and $-$ symbols). (e) The electric field distributions in a unit cell of the first band (lower panels) and the second band (upper panels) at the $X$ points in the first Brillouin zone which show the parities.

Figure 2

Phase diagram and topological edge states. (a) The OI phase, metallic phase, and SOTI phase with different $\mathrm{\Delta }$. (b) The projected band structure of two photonic crystals within different topological phases. A band gap with a 1D edge state is presented. (c),(d) The projected band structure of two photonic crystals within the same trivial and nontrivial phases, respectively. There is no edge state in the gap. (e) A simulated edge state with a frequency at 6.00 GHz [represented by a red star in (b)].

Figure 3

Experimental visualization of corner states in a metastructure. (a) Photograph of a metastructure with the upper metallic plate removed. The boundary of two photonic crystals is denoted by the blue dashed line. The enlarged corner structure is presented. (b) Eigenmodes calculation of the metastructure with the same parameters in Fig. 1. Corner states, edge states, and bulk states are represented by blue, yellow, and gray dots, respectively. (c) A simulated electric field distribution of one of the four corner states. (d) Experimental visualization of corner states at 6.26 GHz. The scanning area is a rectangle instead of a square. The excitation source is depicted by the green dot.

Figure 4

Hierarchical structure of topological insulating phases. (a) Normalized local field intensity measured at bulk, edge, and corner points as depicted in the inset. The bulk, edge, and corner eigenstates from the numerical calculation are represented by gray, yellow, and blue dots, respectively. We provide the visualization of (b) a bulk state at 5.32 GHz, (c) an edge state at 6.00 GHz, (d) a corner state at 6.26 GHz, and (e) a bulk state at 6.52 GHz with the excitation source placed at the upper-right corner of the boundary (represented by the white dashed line).