Fig. 1: Schematic and band structure of the valley photonic crystal.

a Valley photonic crystal composed of first-order dendritic unit cell arranged periodically along a triangular lattice. Solid black lines label the rhombic lattice, where the black dashed lines divide the rhombic lattice into regions I and II. Light orange shaded regions represent the geometric perturbation region. b Schematics and the band structures of the first-order, second-order and third-order fractal dendritic unit cell. Yellow-shaded regions represent the topological bandgaps. Geometric parameters of the first-order dendritic unit cell are a = 30 mm, w = 2.1 mm, θ = -30°, l = 12.15 mm. The lengths of the three branches are the same indicated by l, and the angle between each branch is 120°. Geometric parameters of the second-order dendritic unit cell are α = 60°, b₁ = 6 mm, b₂ = 7.1 mm, l = b₁ + \(\frac{\sqrt{3}}{2}\)b₂. Geometric parameters of the third-order dendritic unit cell are b₁ = 6 mm, b₂ = 0.75b₁, b₃ = 0.5b₁, l = b₁ + \(\frac{\sqrt{3}}{2}\)(b₂ + \(\frac{\sqrt{3}}{2}\)b₃). c Electric field distribution of the valley states at the K point for the different fractal dendritic unit cells. White arrows represent the Poynting vectors. d Variation of the relative bandwidth γ and ΔE at the K point versus l and w for the first-order dendritic unit cell.