Fig. 1. Schematic diagrams of two-dimensional photonic crystal structure with C_3V symmetry of arc-cut triangular units arranged in triangle and its band structures. (a) Photonic crystal structure with θ=0°; (b) photonic crystal structure with θ=30°; (c) photonic crystal structure with θ=-30°; (d) photonic crystal band structure with θ=0°; (e) photonic crystal band structure with θ=30°; (f) photonic crystal band structure with θ=-30°

Fig. 2. Schematic diagrams of two types of interfaces of photonic crystal structure arranged in triangle. (a) zigzag interface; (b) armchair interface

Fig. 3. Supercell structure model and band analysis of zigzag interface of the first valley topological edge state. (a) Schematic of supercell structure model of zigzag interface of the first valley topological edge state (the forward propagation modes are ψ1+ and ψ2+ for interfacesI1, and the backward propagation modes are ψ1- and ψ2- for interfacesI2); (b) calculation results of energy band at the first valley topological edge state of supercell with zigzag interface (solid line represents body state, and dashed line and dotted line representI1 andI2 valley edge states,respectively); (c) electric field distributions and energy flow directions (indicated by arrow) of four valley edge states at the same frequency (four insets are enlarged views of I1 and I2); (d) the first valley topological edge state of I1; (e) the first valley topological edge state of I2

Fig. 4. Supercell structure model and band analysis of zigzag interface of the second valley topological edge state. (a) Schematic diagram of supercell structure model of zigzag interface of the second valley topological edge state; (b) calculation results of energy band at the second edge state of supercell of zigzag interface; (c) electric field distributions and energy flow directions (indicated by arrows) of four valley topology interface states at the same frequency (four insets are enlarged views of I1 and I2); (d) the second valley topological edge state of I1; (e) the second valley topological edge state of I2

Fig. 5. Supercell structure model and band analysis of armchair interface of the first valley topological edge state. (a) Schematic of supercell model of armchair interface (forward propagation modes are ψ3+ and ψ4+ for interface I3, and backward propagation modes are ψ3- and ψ4- for interface I4); (b) calculation results of energy band at the first edge state of supercell with armchair interface (solid lines represent body state, and dashed line and dotted line represent I3 and I4 valley edge states, respectively)

Fig. 6. Electric field distributions and energy flow directions indicated by cone arrows of four valley edge states at the same frequency(four insets are enlarged views ofI3 and I4)

Fig. 7. Supercell structure model and band analysis of armchair interface of the second valley topological edge state. (a) Schematic of supercell structure model of armchair interface in the second valley topology edge state; (b) calculation results of energy band at the second valley topological edge state of supercells with armchair interface (solid lines represent body state, and dashed line, dotted line, and dash-dotted line represent valley interface states of I3 and I4 interfaces)

Fig. 8. Electric field distributions and energy flow directions corresponding to four valley topological interface states at the same frequency. (a) I3 interface at point 1; (b) I4 interface at point 1; (c) I3 interface at point 2; (d) I4 interface at point 2

Fig. 9. Electric field distributions and energy flow directions corresponding to eight valley topological interface states at the same frequency. (a) I3 interface at point 3; (b) I4 interface at point 3; (c) I3 interface at point 4; (d) I4 interface at point 4; (e) I3 interface at point 5; (f) I4 interface at point 5; (g) I3 interface at point 6; (h) I4 interface at point 6

Fig. 10. Electric field distributions when rightward propagating plane wave is incident at left edge of interface. (a) First valley topological edge state can be excited at interface I1; (b) first valley topological edge state cannot be excited at interface I2

Fig. 11. Electric field distribution when rightward propagating plane wave is incident on whole left side of interface I1

Fig. 12. Electric field distributions when upward propagating plane wave is incident at bottom edge of interface. (a) Valley topological edge state can be excited at interface I3; (b) valley topological edge state can be excited at interface I4

Fig. 13. Robustness of valley topological edge state of zigzag interface I1. (a) Structure of interface I1 with six scatterers removed, and inset is enlarged view of removed scatterers; (b) calculated electric field distribution of structure in Fig. 13(a), and inset is enlarged view of electric field distribution near removed scatterers; (c) structure of interface I1 with four scatterers of index 1; (d) calculated electric field distribution of structure in Fig. 13(c)

Fig. 14. Robustness of valley topological edge state of zigzag interface I1. (a) Structure of interface I1 with six deformed scatterers, and inset is enlarged view of deformed scatterers; (b) calculated electric field distribution of structure in Fig. 14(a), and inset is enlarged view of electric field distribution near deformed scatterers; (c) schematic of Z-shaped waveguide structure composed of interface I1, and broken line represents I1 interface; (d) calculated electric field distribution of Z-shaped waveguide structure

Fig. 15. Robustness of valley topological edge state of armchair interfaces I3 and I4. (a) 30° waveguide without defects in I and with defects in Ⅱ, and two sides of broken line are different types of photonic crystals; (b) calculated electric field distributions of interfaceI3 in I and Ⅱ of Fig. 15(a); (c) calculated electric field distributions of interface I4 in I and Ⅱ of Fig. 15(a)

Fig. 16. Electric field distributions when rightward propagating plane wave is incident at left edge of interface. (a) Second valley topological edge state can be excited at interface I1; (b) second valley topological edge state can be excited at interface I2

Fig. 17. Robustness of the second valley topological edge state of zigzag interfaces I1 and I2. (a) Z-shaped waveguide formed by interface I1; (b) Z-shaped waveguide formed by interface I2

Fig. 18. Transport characteristics of the second valley topological edge state of armchair interface I3. (a) Plane wave excitation with normalized frequency of 0.93; (b) plane wave excitation with normalized frequency of 0.95; (c) plane wave excitation with normalized frequency of 0.97

Fig. 19. Transport characteristics of the second valley topological edge state of armchair interface I4. (a) Plane wave excitation with normalized frequency of 0.93; (b) plane wave excitation with normalized frequency of 0.95; (c) plane wave excitation with normalized frequency of 0.97

Fig. 20. Transport robustness of the second valley topology edge state of armchair interface I3. (a) Plane wave with normalized frequency of 0.93 is excited and transmitted in 30° waveguide; (b) plane wave with normalized frequency of 0.95 is excited and transmitted in 30° waveguide; (c) plane wave with normalized frequency of 0.97 is excited and transmitted in 30° waveguide; (d) plane wave with normalized frequency of 0.93 is excited and transmitted in 30° waveguide with defects; (e) plane wave with normalized frequency of 0.95 is excited and transmitted in 30° waveguide with defects; (f) plane wave with normalized frequency of 0.97 is excited and transmitted in 30° waveguide with defects

Fig. 21. Transport robustness of the second valley topology edge state of the armchair interface I4. (a) Plane wave with normalized frequency of 0.93 is excited and transmitted in 30° waveguide; (b) plane wave with normalized frequency of 0.95 is excited and transmitted in 30° waveguide; (c) plane wave with normalized frequency of 0.97 is excited and transmitted in 30° waveguide