Beijing Key Laboratory of Nanophotonics and Ultrafine Optoelectronic Systems, School of Physics, Beijing Institute of Technology, Beijing 100081, China

Chern number is one of the most important criteria by which the existence of a topological photonic state among various photonic crystals can be judged; however, few reports have presented a universal numerical calculation method to directly calculate the Chern numbers of different topological photonic crystals and have denoted the influence of different structural parameters. Herein, we demonstrate a direct and universal method based on the finite element method to calculate the Chern number of the typical topological photonic crystals by dividing the Brillouin zone into small zones, establishing new properties to obtain the discrete Chern number, and simultaneously drawing the Berry curvature of the first Brillouin zone. We also explore the manner in which the topological properties are influenced by the different structure types, air duty ratios, and rotating operations of the unit cells; meanwhile, we obtain large Chern numbers from −2 to 4. Furthermore, we can tune the topological phase change via different rotation operations of triangular dielectric pillars. This study provides a highly efficient and simple method for calculating the Chern numbers and plays a major role in the prediction of novel topological photonic states.

Fig.1 Discretization of the Brillouin zone of the square lattices. Each node ${{\displaystyle \mathit{k}}}_{l}$ represents a Bloch wave vector; ${{\displaystyle \delta \mathit{k}}}_{\mathrm{1}}$ and ${{\displaystyle \delta \mathit{k}}}_{\mathrm{2}}$ are the finite differences along each reciprocal lattice vector

Fig.2 Deformation and discretization of the Brillouin zone. (a) Deformation of the Brillouin zone; (b) discretization of the deformed Brillouin zone, where each node ${{\displaystyle \mathit{k}}}_{l}$ represents the Bloch wave vector, ${{\displaystyle \delta {\displaystyle \mathit{k}}}}_{\mathrm{1}}$ and ${{\displaystyle \delta {\displaystyle \mathit{k}}}}_{\mathrm{2}}$ denote the finite differences along each reciprocal lattice vector

Fig.3 A gyromagnetic cylinder model denoting the dispersion relation of the TM mode, band Chern numbers, and Berry curvature. (a) Geometry of a unit cell. The blue part represents the gyromagnetic cylinder with r = 0.13${{\displaystyle a}}_{\mathrm{0}}$, e = 13, and κ = 0.4. The gray part symbolizes pure air; (b) first four bands of the TM modes. The Chern numbers are shown at the bottom, whereas the gap Chern numbers are marked at the bandgaps; (c)–(f) Calculated Berry curvature within the first Brillouin zone of the first four bands of the TM modes: (c) for band 1; (d) for band 2; (e) for band 3; and (f) for band 4. The square red dashed line represents the first Brillouin zone, and the triangular red dashed line represents the irreducible Brillouin zone

Fig.4 A gyromagnetic square column model denoting the dispersion relation of the TM mode, band Chern numbers, and Berry curvature. (a) Geometry of a unit cell. The blue part represents the gyromagnetic square column when l = 0.26${{\displaystyle a}}_{\mathrm{0}}$, e = 13, and k = 0.4. The gray part symbolizes pure air; (b) first four bands of the TM modes. The Chern numbers are shown at the bottom, whereas the gap Chern numbers are marked at the bandgaps; (c)–(f) Calculated Berry curvature within the first Brillouin zone of the first four bands of the TM modes: (c) for band 1; (d) for band 2; (e) for band 3; and (f) for band 4. The red dashed line represents the first Brillouin zone, and the triangular red dashed line represents the irreducible Brillouin zone

Fig.5 Gyromagnetic cylinder model denoting the dispersion relation of the TM mode, band Chern numbers, and Berry curvature. (a) Hexagon cell of a single gyromagnetic cylinder. The blue part represents the gyromagnetic cylinder when r = 0.13${{\displaystyle a}}_{\mathrm{0}}$, e = 13, and k = 0.4. The gray part symbolizes pure air; (b) first four bands of the TM modes. The Chern numbers are shown at the bottom, whereas the gap Chern numbers are marked at the bandgaps; (c)–(f) calculated Berry curvature within the first Brillouin zone of the first four bands of the TM modes: (c) for band 1; (d) for band 2; (e) for band 3; and (f) for band 4. The red dashed line represents the first Brillouin zone, and the triangular red dashed line represents the irreducible Brillouin zone

Fig.6 Dielectric triangular column model denoting the dispersion relation of the TM mode, band Chern numbers, and Berry curvature. (a) Hexagon cell of a single air triangular column. The gray part represents the air triangular column with a side length of $l={{\displaystyle \mathrm{0.808}a}}_{\mathrm{0}}$. The blue part symbolizes a dielectric slab when e = 8.9 and m = 1; (b) dispersion relation of the first four bands of the TM modes. The Chern number of each band is zero; (c)–(f) calculated Berry curvature within the first Brillouin zone of the first four bands of the TM modes: (c) for band 1; (d) for band 2; (e) for band 3; and (f) for band 4. The red dashed line represents the first Brillouin zone

Fig.7 Calculation results of a hexagon cell with three geometric shapes but with the same air duty ratio of 0.192. (a) Hexagon cell with a gyromagnetic cylinder, gyromagnetic square column, and gyromagnetic triangle column at its center; (b) first four band structures of the TM mode and the Chern number of the three structures presented in (a); (c) calculated band structures of the TM mode and Chern number from the fifth to the eighth band of a hexagon cell with a gyromagnetic cylinder, gyromagnetic square column, and gyromagnetic triangle column at its center

Fig.8 Gyromagnetic cylinders with different radii in the center of a hexagon cell, maintaining the hexagon unchanged. (a) Band structures of the TM mode and the corresponding Chern numbers of a gyromagnetic cylinder with a radius of 0.13${{\displaystyle a}}_{\mathrm{0}}$; (b) band structures of the TM mode and corresponding Chern numbers of a gyromagnetic cylinder with a radius of 0.23${{\displaystyle a}}_{\mathrm{0}}$; (c) band structures of the TM mode and the corresponding Chern numbers of a gyromagnetic cylinder with a radius of 0.33${{\displaystyle a}}_{\mathrm{0}}$

Fig.9 Band structures of the TE mode and Berry curvature of the hexagon cell after the anticlockwise rotation operation. (a) Band structure of the TE mode of the hexagon cell in which the inside triangle is in the state of origin location; (b) band structure of the TE mode of the hexagon cell in which the inside triangle is rotated 15° anticlockwise; (c) band structure of the TE mode of the hexagon cell in which the inside triangle is rotated 30° anticlockwise; (d) calculated Berry curvature of the second band corresponding to the structures without rotation; (e)–(f) calculated Berry curvature of the second band corresponding to the structures that are rotated by 15° and 30°, respectively: (e) for the hexagon cell in which the inside triangle is rotated 15° anticlockwise and (f) for the hexagon cell in which the inside triangle is rotated 30° anticlockwise

Fig.10 Band structures of the TE mode and Berry curvature of the hexagon cell after various rotation operations. (a) Band structure of the TE mode of the hexagon cell in which the inside triangle is rotated 15° clockwise; (b) band structure of the TE mode of the hexagon cell in which the inside triangle is rotated 15° anticlockwise; (c) band structure of the TE mode of the hexagon cell in which the inside triangle is rotated 45° anticlockwise; (d)–(f) calculated Berry curvature corresponding to the structures rotated by −15°, 15°, and 45°, respectively: (d) for the hexagon cell in which the inside triangle is rotated 15° clockwise; (e) for the hexagon cell in which the inside triangle is rotated 15° anticlockwise; and (f) for the hexagon cell in which the inside triangle is rotated 45° anticlockwise

Fig.11 Edge state in the photonic-crystal-based topological insulator. (a) First type of structure whose unit cell comprises four hexagon cells with a black line. The unit cell with the black line is periodic along the x direction. The triangles inside the hexagon cells that were located above the red line were rotated by 15° clockwise. The triangles inside the hexagon cells that were located below the red line were rotated by 15° anticlockwise. (b) Band diagram of the TE mode of the periodic 2D structure located in (a) shows the edge state, which is denoted using a red line. The blue lines represent the bulk state, and the green rectangle represents the bandgap. (c) Energy-density distribution of the edge state propagating unidirectionally toward the left, corresponding to the structure shown in (a). The six yellow points at the top-right corner represent the point source comprising six vertical magnetic currents. The arrow indicates the direction in which the phase increased. The phases of the currents shown in (c) increased counterclockwise. (d) Energy-density distribution of the edge state that propagated unidirectionally toward the right, corresponding to the structure shown in (a). The source is identical with (c) apart from the phases of the currents shown in (d) that increased clockwise. (e) Second type of structure whose unit cell comprised four hexagon cells with a black line. The unit cell with the black line is periodic along the x direction. The triangles inside the hexagon cells that were located above the red line were rotated by 45° anticlockwise. The triangles inside the hexagon cells that were located below the red line were rotated by 15° anticlockwise. (f) Band diagram of the TE mode of the periodic 2D structure located in (b) shows no edge state. The blue lines represent the bulk state, and the green rectangle represents the bandgap. (g) Energy-density distribution for structures shown in (e). The source is the same as (c), where the phase of the currents increased counterclockwise. No edge state existed that propagated unidirectionally. (h) Energy-density distribution for the structure shown in (e). The source is the same as (d), where the phase of currents increased clockwise. No unidirectionally propagating edge state existed

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