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

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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.

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References

  1. Raghu S, Haldane F D M. Analogs of quantum-Hall-effect edge states in photonic crystals. Physical Review A, 2008, 78(3): 033834

    Article  Google Scholar 

  2. Wu Y, Li C, Hu X, Ao Y, Zhao Y, Gong Q. Applications of topological photonics in integrated photonic devices. Advanced Optical Materials, 2017, 5(18): 1700357

    Article  Google Scholar 

  3. Den N, Quantized M. Hall conductance in a two dimensional periodic potential. Physica A, 1984, 124(1): 199–210

    Google Scholar 

  4. Lu L, Joannopoulos J D, Soljačić M. Topological photonics. Nature Photonics, 2014, 8(11): 821–829

    Article  Google Scholar 

  5. Chen X D, Zhao F L, Chen M, Dong J W. Valley-contrasting physics in all-dielectric photonic crystals: Orbital angular momentum and topological propagation. Physical Review B, 2017, 96(2): 020202

    Article  Google Scholar 

  6. Gao Z, Yang Z J, Gao F, Xue H R, Yang Y H, Dong J W, Zhang B L. Valley surface-wave photonic crystal and its bulk/edge transport. Physical Review B, 2017, 96(20): 201402

    Article  Google Scholar 

  7. Kang Y, Ni X, Cheng X, Khanikaev A B, Genack A Z. Pseudo-spin-valley coupled edge states in a photonic topological insulator. Nature Communications, 2018, 9(1): 3029

    Article  Google Scholar 

  8. Berry M V. Quantal phase factors accompanying adiabatic changes. Proceedings of the Royal Society of London, Series A, 1802, 1984 (392): 45–57

    MathSciNet  MATH  Google Scholar 

  9. Tomita A, Chiao R Y. Observation of Berry’s topological phase by use of an optical fiber. Physical Review Letters, 1986, 57(8): 937–940

    Article  Google Scholar 

  10. Wang H X, Guo G Y, Jiang J H. Band topology in classical waves: Wilson-loop approach to topological numbers and fragile topology. New Journal of Physics, 2019, 21(9): 093029

    Article  Google Scholar 

  11. Hatsugai Y. Chern number and edge states in the integer quantum Hall effect. Physical Review Letters, 1993, 71(22): 3697–3700

    Article  MathSciNet  Google Scholar 

  12. Wang Z, Chong Y D, Joannopoulos J D, Soljacić M. Reflection-free one-way edge modes in a gyromagnetic photonic crystal. Physical Review Letters, 2008, 100(1): 013905

    Article  Google Scholar 

  13. Skirlo S A, Lu L, Soljačić M. Multimode one-way waveguides of large Chern numbers. Physical Review Letters, 2014, 113(11): 113904

    Article  Google Scholar 

  14. Yang B, Zhang H F, Wu T, Dong R, Yan X, Zhang X. Topological states in amorphous magnetic photonic lattices. Physical Review B, 2019, 99(4): 045307

    Article  Google Scholar 

  15. Ochiai T, Onoda M. Photonic analog of graphene model and its extension: Dirac cone, symmetry, and edge states. Physical Review B, 2009, 80(15): 155103

    Article  Google Scholar 

  16. Fukui T, Hatsugai Y, Suzuki H. 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

    Article  Google Scholar 

  17. Ringel Z, Kraus Y E. Determining topological order from a local ground-state correlation function. Physical Review B, 2011, 83(24): 245115

    Article  Google Scholar 

  18. Yu R, Qi X L, Bernevig A, Fang Z, Dai X. Equivalent expression of Z(2) topological invariant for band insulators using the non-Abelian Berry connection. Physical Review B, 2011, 84(7): 075119

    Article  Google Scholar 

  19. Ma T, Shvets G. All-Si valley-Hall photonic topological insulator. New Journal of Physics, 2016, 18(2): 025012

    Article  Google Scholar 

  20. Ma T, Shvets G. Scattering-free edge states between heterogeneous photonic topological insulators. Physical Review B, 2017, 95(16): 165102

    Article  Google Scholar 

  21. Ye L, Yang Y T, Hang Z H, Qiu C Y, Liu Z Y. Observation of valley-selective microwave transport in photonic crystals. Applied Physics Letters, 2017, 111(25): 251107

    Article  Google Scholar 

  22. Gao F, Xue H R, Yang Z J, Lai K, Yu Y, Lin X, Chong Y, Shvets G, Zhang B. Topologically protected refraction of robust kink states in valley photonic crystals. Nature Physics, 2018, 14(2): 140–144

    Article  Google Scholar 

  23. Xiao D, Yao W, Niu Q. Valley-contrasting physics in graphene: magnetic moment and topological transport. Physical Review Letters, 2007, 99(23): 236809

    Article  Google Scholar 

  24. Shalaev M I, Walasik W, Tsukernik A, Xu Y, Litchinitser N M. Robust topologically protected transport in photonic crystals at telecommunication wavelengths. Nature Nanotechnology, 2019, 14 (1): 31–34

    Article  Google Scholar 

  25. He X T, Liang E T, Yuan J J, Qiu H Y, Chen X D, Zhao F L, Dong J W. A silicon-on-insulator slab for topological valley transport. Nature Communications, 2019, 10(1): 872

    Article  Google Scholar 

  26. Joannopoulos J D, Johnson S G, Winn J N, Meade R D. Photonic Crystals Molding the Flow of Light. 2nd ed. America: Princeton University Press, 2008, 1–283

    MATH  Google Scholar 

  27. Yang B, Wu T, Zhang X. Engineering topological edge states in two dimensional magnetic photonic crystal. Applied Physics Letters, 2017, 110(2): 021109

    Article  Google Scholar 

  28. Chan H C, Guo G Y. Tuning topological phase transitions in hexagonal photonic lattices made of triangular rods. Physical Review B, 2018, 97(4): 045422

    Article  Google Scholar 

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Acknowledgements

This paper was supported by the National Natural Science Foundation of China (Grant Nos. 11604378, 91850117, and 11654003), Beijing Institute of Technology Research Fund Program for Young Scholars, and Double First Class University Plan. We would like to thank Prof. Xiangdong Zhang, Dr. Lu He, Dr. Yujing Wang, and Dr. Changyin Ji from the Beijing Institute of Technology for the useful discussion.

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Authors and Affiliations

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

    Chenyang Wang, Hongyu Zhang, Hongyi Yuan, Jinrui Zhong & Cuicui Lu

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  1. Chenyang Wang
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  2. Hongyu Zhang
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  3. Hongyi Yuan
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  4. Jinrui Zhong
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  5. Cuicui Lu
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Correspondence to Cuicui Lu.

Additional information

Mr. Chenyang Wang is an undergraduate student majoring in Applied Physics at the Beijing Institute of Technology, China. His research interests include nanophotonics and topological photonic crystals. He successfully applied for one national invention patent as the co-inventor. Furthermore, he received the excellent poster award during the 2019 CPS Fall Meeting as well as a National Scholarship.

Dr. Hongyu Zhang received her bachelor’s degree from the Hebei Normal University in 2018. She is currently a graduate student at the Beijing Institute of Technology and studies the interaction between light and matter in nanostructures, including the metal nanoparticles and photonic crystal cavities. She successfully applied for two national invention patents as the second inventor and received a National Scholarship during her first year in graduate school. Currently, she is focused on researching topological photonics and hybrid cavity systems.

Mr. Hongyi Yuan is currently an undergraduate in the Physics Department at the Beijing Institute of Technology. His research is focused on the design, calculation and simulation of nanophotonic devices. He is currently focused on general designing using algorithms. He successfully applied for two national invention patents as the co-inventor.

Mr. Jinrui Zhong is an undergraduate student at the Beijing Institute of Technology. He has been awarded a scholarship every semester and received an A–in the astrophysics course during the summer school program at the University of California, Berkeley. His major research interests include the theory of topological fields, Weyl semimetal, chiral anomaly, and axion electric dynamics in condensed matter physics. As a big fan of Richard Feynman, he is independently pursuing Ph.D. while conducting quantum transport experiments.

Prof. Cuicui Lu received her Ph.D. degree from the Peking University in 2015 and is currently an associate research scientist at the Beijing Institute of Technology. Her research interests include nanophotonic devices based on algorithm, topological photonics, photonic crystal cavity and applications, and plasmonics. She has published 30 papers in several journals, including Light: Science & Applications, Optica, Nano Letters, Laser & Photonics Reviews, and Advanced Optical Materials. She has led six scientific research projects, including three projects of National Natural Science Foundations of China, two projects of the Beijing Institute of Technology, and one independent innovation project of the Qian Xuesen Space Technology Laboratory. She received the SPIE Optics and Photonics Education Scholarship and World Quantitative and Science Scholarship in 2014. Email: cuicuilu@bit.edu.cn

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Wang, C., Zhang, H., Yuan, H. et al. Universal numerical calculation method for the Berry curvature and Chern numbers of typical topological photonic crystals. Front. Optoelectron. 13, 73–88 (2020). https://doi.org/10.1007/s12200-019-0963-9

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  • DOI: https://doi.org/10.1007/s12200-019-0963-9

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