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Intensity tuning of the edge states in the imperfect topological waveguides based on the photonic crystals with the \(C_{3}\) point group symmetry

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Abstract

We explore the topological behavior of a two-dimensional honeycomb photonic crystal (2D HPC) based on the presence of double Dirac-cone connected the orbitals \(p\) and \(d\), due to the \(C_{6}\) point group symmetry. Removing the four-fold degeneracy between the bands at the Dirac point can be achieved by introducing three small dielectric rods near the bigger ones to realize the perturbed PCs with the \(C_{3}\) point group symmetry with different topological features. By proposing the unique structure involving two PCs with different topological effects, one may study the one-way light distribution along the local boundary in spite of the defects, cavities, and disorders. Moreover, we investigate the variation of the transmitted intensity values under different defect conditions and realize that the size, location and material type affect the transmitted light. In the other words, tunable intensity of the edge states can be achieved through adjusting the defects such that by decreasing the radius of rods, the intensity of the edge states can be decreased or by increasing their distance from the unit cell center, the intensity will be decreased too. Finally, topological rhombic resonator enables unidirectional filtering of guided mode. The fact that different light manipulation scenarios can be realized provides a unique aspect for topological photonic insulators.

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source along the interface of two types of HPCs; b–e helical edge state distributions at the interface, Z shape, against the cavity, Ag rod and Al blocks, respectively. Yellow star indicates the point-like source at the interface; f the intensity of the edge states versus frequency for different cases: without defect, cavity, Al and Ag defects

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References

  • Bandres, M.A., Rechtsman, M.C., Segev, M.: Topological photonic quasicrystals: fractal topological spectrum and protected transport. Phys. Rev. X 6, 011016 (2016)

    Google Scholar 

  • Bansil, A., Lin, H., Das, T.: Colloquium: topological band theory. Rev. Mod. Phys. 88(2), 1–16 (2016)

    Google Scholar 

  • Barik, S., Miyake, H., DeGottardi, W., Waks, E., Hafezi, M.: Two-dimensionally confined topological edge states in photonic crystals. New J. Phys. 18, 113013 (2016)

    ADS  Google Scholar 

  • Barik, S., et al.: A topological quantum optics interface. Science 359, 666–668 (2018)

    ADS  MathSciNet  MATH  Google Scholar 

  • Chen, W.J., Jiang, S.J., Chen, X.D., Zhu, B., Zhou, L., Dong, J.W., Chan, C.T.: Experimental realization of photonic topological insulator in a uniaxial metacrystal waveguide. Nat. Commun. 5, 1–7 (2014)

    ADS  Google Scholar 

  • Estep, N.A., Sounas, D.L., Soric, J., Alù, A.: Nonreciprocity and magnetic-free isolation based on optomechanical interactions. Nat. Phys. 10, 923–927 (2014)

    Google Scholar 

  • Fan, H., Xia, B., Tong, L., Zheng, S., Yu, D.: Elastic higher-order topological insulator with topologically protected corner states. Phys. Rev. Lett. 122, 204301 (2019)

    ADS  Google Scholar 

  • Fang, K., Yu, Z., Fan, S.: Realizing effective magnetic field for photons by controlling the phase of dynamic modulation. Nat. Photonics 6, 782–787 (2012)

    ADS  Google Scholar 

  • Hafezi, M., Demler, E.A., Lukin, M.D., Taylor, J.M.: Robust optical delay lines with topological protection. Nat. Phys. 7, 907–912 (2011)

    Google Scholar 

  • Hafezi, M., Mittal, S., Fan, J., Migdall, A., Taylor, J.M.: Imaging topological edge states in silicon photonics. Nat. Photonics 7, 1001–1005 (2013)

    ADS  Google Scholar 

  • Hajivandi, J., Kurt, H.: Topological photonic states and directional emission of the light exiting from the photonic topological structure composed of two dimensional Honeycomb photonic crystals with different point group symmetries, submitted in JOURNAL OF OPTICS, OPTI-D-20-00101 (2020)

  • Hajivandi, J., Kurt, H.: Robust transport of the edge modes along the photonic topological interfaces of different configurations. Phys. B Condens. Matter. https://doi.org/10.1016/j.physb.2020.412550(2020)

    Article  Google Scholar 

  • Hajivandi, J., Kurt, H.: Topological phase transition of the centered rectangular photonic lattice, arXiv:2005.11916 [physics.app-ph] (2020)

  • Hajivandi, J., Kaya, E., Edwards G., Kurt, H.: Simulating topological robustness of Fano resonance in rotated Honeycomb photonic crystals, submitted in PNFA, PNFA-D-20-00069 (2020)

  • Haldane, F.D., Raghu, S.: Possible realization of directional optical waveguides in photonic crystals with broken Time-Reversal symmetry. Phys. Rev. Lett. 100, 013904 (2008)

    ADS  Google Scholar 

  • Hasan, M.Z., Kane, C.L.: Topological insulators. Rev. Mod. Phys. 82(4), 3045 (2010)

    ADS  Google Scholar 

  • He, C., Sun, X.C., Liu, X.P., Lu, M.H., Chen, Y., Feng, L., Chen, Y.F.: Photonic topological insulator with broken time-reversal symmetry. Proc. Natl. Acad. Sci. USA 113, 4924–4928 (2016)

    ADS  Google Scholar 

  • Huo, S.Y., Chen, J.J., Feng, L.Y., Huang, H.B.: Pseudospins and topological edge states for fundamental antisymmetric Lamb modes in snowflake like phononic crystal slabs. J. Acoust. Soc. Am. 146, 729–735 (2019)

    ADS  Google Scholar 

  • Lumerical Inc. https://www.lumerical.com/tcad-pucts/fdtd/

  • Johnson, P.B., Christy, R.W.: Optical constants of the noble metals. Phys. Rev. B 6, 4370–4379 (1972)

    ADS  Google Scholar 

  • Johnson, S.G., Joannopoulos, J.D.: Block-iterative frequency-domain methods for Maxwell’s equations in a planewave, basis. Opt. Express 8, 173–190 (2001)

    ADS  Google Scholar 

  • Kane, C.L., Mele, E.J.: Quantum spin Hall effect in graphene. Phys. Rev. Lett. 95(22), 226801 (2005)

    ADS  Google Scholar 

  • Kapit, E., Hafezi, M., Simon, S.H.: Induced self-stabilization in fractional quantum Hall states of light. Phys. Rev. X 4(3), 031039 (2014)

    Google Scholar 

  • Khanikaev, A.B., Mousavi, S.H., Tse, W.K., Kargarian, M., MacDonald, A.H., Shvets, G.: Photonic topological insulators. Nat. Mater. 12, 233–239 (2013)

    ADS  Google Scholar 

  • Leykam, D., Chong, Y.D.: Edge solitons in nonlinear-photonic topological insulators. Phys. Rev. Lett. 117, 143901 (2016)

    ADS  Google Scholar 

  • Leykam, D., Rechtsman, M.C., Chong, Y.D.: Anomalous topological phases and unpaired Dirac cones in photonic floquet topological insulators. Phys. Rev. Lett. 117, 013902 (2016)

    ADS  Google Scholar 

  • Liu, C.X., Zhang, S.C., Qi, X.L.: The quantum anomalous Hall effect. Annu. Rev. Condens. Matter Phys. 7, 301–321 (2016)

    ADS  Google Scholar 

  • Lu, L., Joannopoulos, J.D., Soljačić, M.: Topological photonics. Nat. Photon 8, 821–829 (2014)

    ADS  Google Scholar 

  • Ma, T., Shvets, G.: All-Si valley-Hall photonic topological insulator. New J. Phys. 18, 025012 (2016)

    ADS  Google Scholar 

  • Noh, J., Benalcazar, W.A., Huang, S., Collins, M.J., Chen, K., Hughes, T.L., Rechtsman, M.C.: Topological protection of photonic mid-gap defect modes. Nat. Photonics 12, 408–415 (2018)

    ADS  Google Scholar 

  • Palik, E.D.: Handbook of optical constants of solids. Academic Press, Orlando (1985)

    Google Scholar 

  • Qi, X.L., Zhang, S.C.: Topological insulators and superconductors. Rev. Mod. Phys. 83(4), 1057 (2011)

    ADS  Google Scholar 

  • Raghu, S., Haldane, F.D.M.: Analogs of quantum Hall effect edge states in photonic crystals. Phys. Rev. A 78, 033834 (2018)

    ADS  Google Scholar 

  • Rechtsman, M.C., Plotnik, Y., Zeuner, J.M., Song, D., Chen, Z., Szameit, A., Segev, M.: Topological creation and destruction of edge states in photonic graphene. Phys. Rev. Lett. 111, 103901 (2013)

    ADS  Google Scholar 

  • Rechtsman, M.C., Zeuner, J.M., Plotnik, Y., Lumer, Y., Podolsky, D., Dreisow, F., Nolte, S., Segev, M., Szameit, A.: Photonic floquet topological insulators. Nature (London) 496, 196–200 (2013)

    ADS  Google Scholar 

  • Sakoda, K.: Dirac cone in two and three-dimensional metamaterials. Opt. Express 20, 3898–3917 (2012)

    ADS  Google Scholar 

  • Song, Z., Liu, H.J., Huang, N., Wang, Z.: Prediction of two-dimensional organic topological insulator in metal-DCB lattices. Appl. Opt 57, 29 (2018)

    Google Scholar 

  • Sounas, D.L., Caloz, C., Alu, A.: Magnetic-free non-reciprocity based on staggered commutation. Nat. Commun. 4, 2407 (2013)

    ADS  Google Scholar 

  • Sun, X.C., He, C., Liu, X.P., Lu, M.H., Zhu, S.N., Chen, Y.F.: Photonics meets topology. Prog. Quantum Electron 55, 52–73 (2017)

    ADS  Google Scholar 

  • Umucalilar, R.O., Carusotto, I.: Artificial gauge field for photons in coupled cavity arrays. Phys. Rev. A 84, 043804 (2011)

    ADS  Google Scholar 

  • Wang, Z., Chong, Y.D., Joannopoulos, J.D., Soljacic, M.: Reflection-free one-way edge modes in a gyromagnetic photonic crystal. Phys. Rev. Lett. 100, 013905 (2008)

    ADS  Google Scholar 

  • Wang, Z., Chong, Y., Joannopoulos, J.D., Soljacic, M.: Observation of unidirectional backscattering-immune topological electromagnetic states. Nature (London) 461, 772–775 (2009)

    ADS  Google Scholar 

  • Wang, H.X., Xu, L., Chen, H.Y., Jiang, J.H.: Three-dimensional photonic Dirac points stabilized by point group symmetry. Phys. Rev. B 93, 235155 (2016)

    ADS  Google Scholar 

  • Wang, H.X., Chen, Y., Hang, Z.H., Kee, H.Y., Jiang, J.H.: Type-II. Dirac photons. NPJ Quantum Mater 2, 54 (2017)

    ADS  Google Scholar 

  • Wu, L.H., Hu, X.: Scheme for achieving a topological photonic crystal by using dielectric material. Phys. Rev. Lett. 114, 223901 (2015)

    ADS  Google Scholar 

  • Xia, B., Fan, H., Liu, T.: Topologically protected edge states of phoxonic crystals. Int. J. Mech. Sci. 155, 197–205 (2019)

    Google Scholar 

  • Xie, B.Y., Wang, H.F., Zhu, X.Y., Lu, M.H., Wang, Z.D., Chen, Y.F.: Feature issue introduction: topological photonics and materials. Opt. Express 26, 19 (2018)

    ADS  Google Scholar 

  • Xu, L., Wang, H.X., Xu, Y.D., Chen, H.Y., Jiang, J.H.: Accidental degeneracy in photonic bands and topological phase transitions in two-dimensional core-shell dielectric photonic crystals. Opt. Express 24, 18059 (2016)

    ADS  Google Scholar 

  • Yang, Y., Xu, Y.F., Xu, T., Wang, H.X., Jiang, J.H., Hu, X., Hang, Z.H.: Visualization of unidirectional optical waveguide using topological photonic crystals made of dielectric material, arXiv:1610.07780

  • Zhu, X., Wang, H.X., Xu, C., Lai, Y., Jiang, J.H., John, S.: Topological transitions in continuously deformed photonic crystals. J. Phys. Rev. B 97, 085148 (2018)

    ADS  Google Scholar 

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Acknowledgements

H. Kurt acknowledges partial support of the Turkish Academy of Sciences.

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Hajivandi, J., Pakarzadeh, H. & Kurt, H. Intensity tuning of the edge states in the imperfect topological waveguides based on the photonic crystals with the \(C_{3}\) point group symmetry. Opt Quant Electron 53, 102 (2021). https://doi.org/10.1007/s11082-021-02745-x

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