Fig. 1. Proposed C2v symmetry triangular PC with anisotropy. (a) Schematic illustration of the metal pillars. (b) Periodically arranged triangular array of the elliptical metal pillars (b1, b2, and b3 are the basis vectors of three mirror symmetrical directions). The right panel presents the complex first Brillouin zone. (c) Dispersion relation of the ΓK and ΓK′ paths.

Fig. 2. Arrangement of the PC structures and the band structures of the different configurations with (a) φ=0°, (b) φ=30°, and (c) φ=−30°. Upper panels represent the arrangement of the PC with various crystal orientation vectors b1, b2, and b3 induced by anisotropy. The orange, blue, and dark gray ellipses represent the b1, b2, and b3 basis vectors, respectively. Lower panels show the band diagram of the different unit cells. Orange shaded regions represent the band gap.

Fig. 3. Anisotropy-induced zigzag domain walls. (a)–(c) Simulated electric field component Ez at 14.47 GHz corresponding to e1, e2, and e3 interface, respectively. (d)–(e) Dispersion diagrams of the anisotropy-induced zigzag edge states along the b1 and b2(b3) basis vectors. (f) Left panel shows the simulated transmission of the zigzag domain walls, and right panel shows the density of states (DOS) of the band dispersion. (g) Group velocity of the edge states e1 and e2. Black dotted line represents the zero-horizontal line of zero value. (h) Enlarged view of the flat dispersion curve of (e). (i) Intrinsic field intensity distribution at the frequencies of 14.462 GHz, 14.482 GHz, and 15.502 GHz. Electric field strength distribution of e2 edge mode is along the x direction.

Fig. 4. Robustness and evolution of topological states with anisotropy. (a) Simulated electric field distributions of disorder defect (left panel) and cavity defect (right panel). (b) Evolution of the edge modes with varying anisotropy strength ρ, where both sides of the domain wall have the same ρ. (c) Evolution of the edge modes with varying anisotropy strength ρ where one side of the domain wall has a fixed ρ=4.5. Lower panels of (b) and (c) show the calculated band dispersion diagrams with ρ=1.5, 3, and 5.5. Bulk states, edge states, and band gap are marked as gray shaded regions, green shaded regions, and white shaded regions, respectively.

Fig. 5. Higher-order topology in corner structure. (a) Schematic image of the 60° zigzag domain walls corner structure. The blue lines show the interface of the domain walls. (b) Eigen modes of the parallelograms supercell corresponding to (a). Inset shows the schematic of the supercell structure. (c) Numerically simulated electric field distribution Ez of the two corner modes at 13.33 GHz and 14.29 GHz. (d) Left panel shows the schematic diagram of experimental set-up and the fabricated corner configuration sample. The green dot represents the source, and four yellow dots indicate different detection positions (one, two, three, and four dots for edge, corner I, corner II, and bulk antenna-probes, respectively). Right panel shows the measured transmission spectra for four different types of antenna-probes.

Fig. 6. Schematic diagram of dispersion relation with different ellipticities. (a)–(d) Change in projection band corresponding to the ΓK and ΓK′ paths with increasing dimensionless shape deviation factor ρ from 1 to 4.5.

Fig. 7. Anisotropic phase diagrams. (a)–(c) Phase diagrams of lowest two energy bands at the high-symmetry points M, M′, and M′′ as a function of φ, respectively.

Fig. 8. Topological edge states between two PCs with opposite signs of the Dirac mass. (a)–(c) Projected band of edge states corresponding to φ=±0°, φ=±15°, and φ=±30°, respectively.