volume | PIER Journals

Abstract

We introduce the concept of gap Zak or Chern topological invariants for photonic crystals of various dimensionalities. Specifically, we consider a case where Tamm states are formed at an interface of two semi-infinite Bragg mirrors and derive the formulism for gap Zak phases of two constituent Bragg mirrors. We demonstrate that gap topological numbers are instrumental in studies of interface states both in conventional and photonic crystals.

1. Xiao, D., M.-C. Chang, and Q. Niu, "Berry phase effects on electronic properties," Rev. Mod. Phys., Vol. 82, 1959-Jul. 2007, 2010.
doi:10.1103/RevModPhys.82.1959 Google Scholar

2. Berry, M. V., "Quantal phase factors accompanying adiabatic changes," Proc. R. Soc. Lond., Vol. 392, No. 1802, 45-57, 1996. Google Scholar

3. Qiang, W., X. Meng, L. Hui, S. Zhu, and C. T. Chan, "Measurement of the zak phase of photonic bands through the interface states of metasurface/photonic crystal," Physical Review B, Vol. 93, No. 4, 041415.1-041415.5, 2016. Google Scholar

4. Gao, W. S., M. Xiao, C. T. Chan, and W. Y. Tam, "Determination of zak phase by reflection phase in 1D photonic crystals," Optics Letters, Vol. 40, No. 22, 5259, 2015.
doi:10.1364/OL.40.005259 Google Scholar

5. Ozawa, T., H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. C. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, and I. Carusotto, "Topological photonics," Rev. Mod. Phys., Vol. 91, 015006, Mar. 2019.
doi:10.1103/RevModPhys.91.015006 Google Scholar

6. Wang, F. and Y. Ran, "Nearly flat band with chern number c = 2 on the dice lattice," Phys. Rev. B, Vol. 84, 241103, Dec. 2011.
doi:10.1103/PhysRevB.84.241103 Google Scholar

7. Hatsugai, Y., T. Fukui, and H. Aoki, "Topological analysis of the quantum hall effect in graphene: Dirac-fermi transition across van hove singularities and edge versus bulk quantum numbers," Phys. Rev. B, Vol. 74, 205414, Nov. 2006. Google Scholar

8. Kavokin, A., Microcavities. Number No. 16 in Series on Semiconductor Science and Technology, Oxford University Press, 2007, OCLC: ocn153553936.

9. Afinogenov, B. I., V. O. Bessonov, A. A. Nikulin, and A. A. Fedyanin, "Observation of hybrid state of tamm and surface plasmon-polaritons in one-dimensional photonic crystals," Applied Physics Letters, Vol. 103, No. 6, 1800, 2013.
doi:10.1063/1.4817999 Google Scholar

10. Sasin, M. E., R. P. Seisyan, M. A. Kalitteevski, S. Brand, R. A. Abram, J. M. Chamberlain, A. Y. Egorov, A. P. Vasil'Ev, V. S. Mikhrin, and A. V. Kavokin, "Tamm plasmon polaritons: Slow and spatially compact light," Applied Physics Letters, Vol. 92, No. 25, 824, 2008.
doi:10.1063/1.2952486 Google Scholar

11. Kavokin, A. V., I. A. Shelykh, and G. Malpuech, "Lossless interface modes at the boundary between two periodic dielectric structures," Physical Review B, Vol. 72, No. 23, 233102, Dec. 2005.
doi:10.1103/PhysRevB.72.233102 Google Scholar

12. Su, Y., C. Y. Lin, R. C. Hong, W. X. Yang, and R. K. Lee, "Lasing on surface states in vertical-cavity surface-emission lasers," Optics Letters, Vol. 39, No. 19, 2014.
doi:10.1364/OL.39.005582 Google Scholar

13. Symonds, C., G. Lheureux, J. P. Hugonin, J. J. Greffet, and J. Bellessa, "Confined tamm plasmon lasers," Nano Letters, Vol. 13, No. 7, 3179, 2013.
doi:10.1021/nl401210b Google Scholar

14. Symonds, C., A. Lematre, E. Homeyer, J. C. Plenet, and J. Bellessa, "Emission of Tamm plasmon/exciton polaritons," Applied Physics Letters, Vol. 95, No. 15, 151114-151114-3, 2009.
doi:10.1063/1.3251073 Google Scholar

15. Kavokin, A., I. Shelykh, and G. Malpuech, "Optical Tamm states for the fabrication of polariton lasers," Applied Physics Letters, Vol. 87, No. 26, 193, 2005.
doi:10.1063/1.2136414 Google Scholar

16. Henriques, J. C. G., T. G. Rappoport, Y. V. Bludov, M. I. Vasilevskiy, and N. M. R. Peres, "Topological photonic Tamm states and the Su-Schrieffer-Heeger model," Phys. Rev. A, Vol. 101, 043811, Apr. 2020.
doi:10.1103/PhysRevA.101.043811 Google Scholar

17. Xiao, M., Z. Q. Zhang, and C. T. Chan, "Surface impedance and bulk band geometric phases in one-dimensional systems," Phys. Rev. X, Vol. 4, 021017, Apr. 2014. Google Scholar

18. Zak, J., "Berry's phase for energy bands in solids," Physical Review Letters, Vol. 62, No. 23, 2747-2750, Jun. 1989.
doi:10.1103/PhysRevLett.62.2747 Google Scholar

19. Ryder, L. H., "The optical berry phase and the gauss-bonnet theorem," European Journal of Physics, Vol. 12, No. 1, 15, 1991.
doi:10.1088/0143-0807/12/1/003 Google Scholar

20. Holstein, B. R., "The adiabatic theorem and Berry's phase," American Journal of Physics, 57, 1989. Google Scholar

21. Wang, H.-X., G.-Y. Guo, and J.-H. Jiang, "Band topology in classical waves: Wilsonloop approach to topological numbers and fragile topology," New Journal of Physics, Vol. 21, No. 9, 093029, Sep. 2019.
doi:10.1088/1367-2630/ab3f71 Google Scholar

22. Gubarev, F. V. and V. I. Zakharov, "The Berry phase and monopoles in non-Abelian gauge theories," International Journal of Modern Physics A, Vol. 17, No. 2, 157-174, 2002.
doi:10.1142/S0217751X02005840 Google Scholar

23. Kondo, K.-I., "Wilson loop and magnetic monopole through a non-Abelian stokes theorem," Physical Review D, Vol. 77, No. 8, 284-299, 2008.
doi:10.1103/PhysRevD.77.085029 Google Scholar

Dr. Junhui Cao

Prof. Alexey V. Kavokin

Dr. Anton V. Nalitov