Fig. 1. (a) Schematic of the “bearded” interface composed of unit cells with negative and positive valley Chern numbers in the top and bottom components, respectively, and (b) the corresponding projected band diagram, group index diagram when r₁ − r₂ = 0.1857a, and |H_z| field distribution at k_x = 0.4 (2π/a).

Fig. 2. Evolution of (a), (b) chiral and (c), (d) nontrivial state frequency and the corresponding bandwidth of the bearded LN valley photonic crystal as a function of r₁ − r₂. (e) Transmission contrast and (f), (g) the corresponding |H_z| field distributions when the source is set to show two opposite chiralities. (h) Band diagram of LN valley photonic crystal when r₁ − r₂ = 0.2857a, and (i), (j) |H_z| field distributions excited by a plane wave source at ω = 0.435 (2π/a) and ω = 0.42 (2π/a).

Fig. 3. (a) Structure of the proposed broadband valley LN photonic crystal waveguide with a sharp bend. The |H_z| field distributions of (b) the waveguide with the sharp bend and (c) the straight waveguide at ω = 0.42 (2π/a). (d) Structure of trivial LN photonic crystal waveguides with sharp bend. The |H_z| field distributions of (e) the waveguide with the sharp bend and (f) the straight waveguide at ω = 0.42 (2π/a). (g), (h) Transmission spectra for the topological and ordinary waveguides.

Fig. 4. Schematics of (a), (b) the bearded interface and (c)–(f) the zigzag interface composed of circle holes and their corresponding projected band diagrams and group index diagrams. In (a) and (e), the unit cells possess positive and negative valley Chern numbers in the top and bottom parts, while in (c) the unit cells possess negative and positive valley Chern numbers in the top and bottom parts.