Universal numerical calculation method for the Berry curvature and Chern numbers of typical topological photonic crystals

doi:10.1007/s12200-019-0963-9

Front. Optoelectron.

2020, Vol. 13

Issue (1) : 73-88 https://doi.org/10.1007/s12200-019-0963-9

RESEARCH ARTICLE

Universal numerical calculation method for the Berry curvature and Chern numbers of typical topological photonic crystals

Chenyang WANG, Hongyu ZHANG, Hongyi YUAN, Jinrui ZHONG, Cuicui LU(

)

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

Download: PDF(4609 KB) HTML
Export: BibTeX | EndNote | Reference Manager | ProCite | RefWorks

Abstract

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.

Keywords Chern number topological photonic crystal finite element method symmetry

Corresponding Author(s): Cuicui LU

Just Accepted Date: 17 December 2019 Online First Date: 20 January 2020 Issue Date: 03 April 2020

Cite this article:

Chenyang WANG,Hongyu ZHANG,Hongyi YUAN, et al. Universal numerical calculation method for the Berry curvature and Chern numbers of typical topological photonic crystals[J]. Front. Optoelectron., 2020, 13(1): 73-88.

URL:

https://academic.hep.com.cn/foe/EN/10.1007/s12200-019-0963-9
https://academic.hep.com.cn/foe/EN/Y2020/V13/I1/73

Fig.1 Discretization of the Brillouin zone of the square lattices. Each node

k l

represents a Bloch wave vector;

δ k 1

and

δ k 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

k l

represents the Bloch wave vector,

δ k 1

and

δ k 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

a 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

a 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

a 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 = 0.808 a 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

a 0

; (b) band structures of the TM mode and corresponding Chern numbers of a gyromagnetic cylinder with a radius of 0.23

a 0

; (c) band structures of the TM mode and the corresponding Chern numbers of a gyromagnetic cylinder with a radius of 0.33

a 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

1	S Raghu, F D M Haldane. Analogs of quantum-Hall-effect edge states in photonic crystals. Physical Review A, 2008, 78(3): 033834 https://doi.org/10.1103/PhysRevA.78.033834
2	Y Wu, C Li, X Hu, Y Ao, Y Zhao, Q Gong. Applications of topological photonics in integrated photonic devices. Advanced Optical Materials, 2017, 5(18): 1700357 https://doi.org/10.1002/adom.201700357
3	N Den, M Quantized. Hall conductance in a two dimensional periodic potential. Physica A, 1984, 124(1): 199–210
4	L Lu, J D Joannopoulos, M Soljačić. Topological photonics. Nature Photonics, 2014, 8(11): 821–829 https://doi.org/10.1038/nphoton.2014.248
5	X D Chen, F L Zhao, M Chen, J W Dong. Valley-contrasting physics in all-dielectric photonic crystals: Orbital angular momentum and topological propagation. Physical Review B, 2017, 96(2): 020202 https://doi.org/10.1103/PhysRevB.96.020202
6	Z Gao, Z J Yang, F Gao, H R Xue, Y H Yang, J W Dong, B L Zhang. Valley surface-wave photonic crystal and its bulk/edge transport. Physical Review B, 2017, 96(20): 201402 https://doi.org/10.1103/PhysRevB.96.201402
7	Y Kang, X Ni, X Cheng, A B Khanikaev, A Z Genack. Pseudo-spin-valley coupled edge states in a photonic topological insulator. Nature Communications, 2018, 9(1): 3029 https://doi.org/10.1038/s41467-018-05408-w pmid: 30072759
8	M V Berry. Quantal phase factors accompanying adiabatic changes. Proceedings of the Royal Society of London, Series A, 1802, 1984(392): 45–57
9	A Tomita, R Y Chiao. Observation of Berry’s topological phase by use of an optical fiber. Physical Review Letters, 1986, 57(8): 937–940 https://doi.org/10.1103/PhysRevLett.57.937 pmid: 10034204
10	H X Wang, G Y Guo, J H Jiang. Band topology in classical waves: Wilson-loop approach to topological numbers and fragile topology. New Journal of Physics, 2019, 21(9): 093029 https://doi.org/10.1088/1367-2630/ab3f71
11	Y Hatsugai. Chern number and edge states in the integer quantum Hall effect. Physical Review Letters, 1993, 71(22): 3697–3700 https://doi.org/10.1103/PhysRevLett.71.3697 pmid: 10055049
12	Z Wang, Y D Chong, J D Joannopoulos, M Soljacić. Reflection-free one-way edge modes in a gyromagnetic photonic crystal. Physical Review Letters, 2008, 100(1): 013905 https://doi.org/10.1103/PhysRevLett.100.013905 pmid: 18232767
13	S A Skirlo, L Lu, M Soljačić. Multimode one-way waveguides of large Chern numbers. Physical Review Letters, 2014, 113(11): 113904 https://doi.org/10.1103/PhysRevLett.113.113904 pmid: 25259982
14	B Yang, H F Zhang, T Wu, R Dong, X Yan, X Zhang. Topological states in amorphous magnetic photonic lattices. Physical Review B, 2019, 99(4): 045307 https://doi.org/10.1103/PhysRevB.99.045307
15	T Ochiai, M Onoda. Photonic analog of graphene model and its extension: Dirac cone, symmetry, and edge states. Physical Review B, 2009, 80(15): 155103 https://doi.org/10.1103/PhysRevB.80.155103
16	T Fukui, Y Hatsugai, H Suzuki. Chern numbers in discretized Brillouin Zone: efficient method of computing (spin) hall conductances. Journal of the Physical Society of Japan, 2005, 74(6): 1674–1677 https://doi.org/10.1143/JPSJ.74.1674
17	Z Ringel, Y E Kraus. Determining topological order from a local ground-state correlation function. Physical Review B, 2011, 83(24): 245115 https://doi.org/10.1103/PhysRevB.83.245115
18	R Yu, X L Qi, A Bernevig, Z Fang, X Dai. Equivalent expression of Z(2) topological invariant for band insulators using the non-Abelian Berry connection. Physical Review B, 2011, 84(7): 075119 https://doi.org/10.1103/PhysRevB.84.075119
19	T Ma, G Shvets. All-Si valley-Hall photonic topological insulator. New Journal of Physics, 2016, 18(2): 025012 https://doi.org/10.1088/1367-2630/18/2/025012
20	T Ma, G Shvets. Scattering-free edge states between heterogeneous photonic topological insulators. Physical Review B, 2017, 95(16): 165102 https://doi.org/10.1103/PhysRevB.95.165102
21	L Ye, Y T Yang, Z H Hang, C Y Qiu, Z Y Liu. Observation of valley-selective microwave transport in photonic crystals. Applied Physics Letters, 2017, 111(25): 251107 https://doi.org/10.1063/1.5009597
22	F Gao, H R Xue, Z J Yang, K Lai, Y Yu, X Lin, Y Chong, G Shvets, B Zhang. Topologically protected refraction of robust kink states in valley photonic crystals. Nature Physics, 2018, 14(2): 140–144 https://doi.org/10.1038/nphys4304
23	D Xiao, W Yao, Q Niu. Valley-contrasting physics in graphene: magnetic moment and topological transport. Physical Review Letters, 2007, 99(23): 236809 https://doi.org/10.1103/PhysRevLett.99.236809 pmid: 18233399
24	M I Shalaev, W Walasik, A Tsukernik, Y Xu, N M Litchinitser. Robust topologically protected transport in photonic crystals at telecommunication wavelengths. Nature Nanotechnology, 2019, 14(1): 31–34 https://doi.org/10.1038/s41565-018-0297-6 pmid: 30420760
25	X T He, E T Liang, J J Yuan, H Y Qiu, X D Chen, F L Zhao, J W Dong. A silicon-on-insulator slab for topological valley transport. Nature Communications, 2019, 10(1): 872 https://doi.org/10.1038/s41467-019-08881-z pmid: 30787288
26	J D Joannopoulos, S G Johnson, J N Winn, R D Meade. Photonic Crystals Molding the Flow of Light. 2nd ed. America: Princeton University Press, 2008, 1–283
27	B Yang, T Wu, X Zhang. Engineering topological edge states in two dimensional magnetic photonic crystal. Applied Physics Letters, 2017, 110(2): 021109 https://doi.org/10.1063/1.4973990
28	H C Chan, G Y Guo. Tuning topological phase transitions in hexagonal photonic lattices made of triangular rods. Physical Review B, 2018, 97(4): 045422 https://doi.org/10.1103/PhysRevB.97.045422

[1]	Hongfei WANG, Samit Kumar GUPTA, Biye XIE, Minghui LU. Topological photonic crystals: a review[J]. Front. Optoelectron., 2020, 13(1): 50-72.
[2]	Kambiz ABEDI, Habib VAHIDI. Design optimization of microwave properties for polymer electro-optic modulator using full vectorial finite element method[J]. Front Optoelec, 2013, 6(3): 290-296.
[3]	Kambiz ABEDI, Habib VAHIDI. Structure and microwave properties analysis of substrate removed GaAs/AlGaAs electro-optic modulator structure by finite element method[J]. Front Optoelec, 2013, 6(1): 108-113.

Viewed

Full text

Abstract

Cited

Shared

Discussed