Vol. 173
Latest Volume
All Volumes
PIER 179 [2024] PIER 178 [2023] PIER 177 [2023] PIER 176 [2023] PIER 175 [2022] PIER 174 [2022] PIER 173 [2022] PIER 172 [2021] PIER 171 [2021] PIER 170 [2021] PIER 169 [2020] PIER 168 [2020] PIER 167 [2020] PIER 166 [2019] PIER 165 [2019] PIER 164 [2019] PIER 163 [2018] PIER 162 [2018] PIER 161 [2018] PIER 160 [2017] PIER 159 [2017] PIER 158 [2017] PIER 157 [2016] PIER 156 [2016] PIER 155 [2016] PIER 154 [2015] PIER 153 [2015] PIER 152 [2015] PIER 151 [2015] PIER 150 [2015] PIER 149 [2014] PIER 148 [2014] PIER 147 [2014] PIER 146 [2014] PIER 145 [2014] PIER 144 [2014] PIER 143 [2013] PIER 142 [2013] PIER 141 [2013] PIER 140 [2013] PIER 139 [2013] PIER 138 [2013] PIER 137 [2013] PIER 136 [2013] PIER 135 [2013] PIER 134 [2013] PIER 133 [2013] PIER 132 [2012] PIER 131 [2012] PIER 130 [2012] PIER 129 [2012] PIER 128 [2012] PIER 127 [2012] PIER 126 [2012] PIER 125 [2012] PIER 124 [2012] PIER 123 [2012] PIER 122 [2012] PIER 121 [2011] PIER 120 [2011] PIER 119 [2011] PIER 118 [2011] PIER 117 [2011] PIER 116 [2011] PIER 115 [2011] PIER 114 [2011] PIER 113 [2011] PIER 112 [2011] PIER 111 [2011] PIER 110 [2010] PIER 109 [2010] PIER 108 [2010] PIER 107 [2010] PIER 106 [2010] PIER 105 [2010] PIER 104 [2010] PIER 103 [2010] PIER 102 [2010] PIER 101 [2010] PIER 100 [2010] PIER 99 [2009] PIER 98 [2009] PIER 97 [2009] PIER 96 [2009] PIER 95 [2009] PIER 94 [2009] PIER 93 [2009] PIER 92 [2009] PIER 91 [2009] PIER 90 [2009] PIER 89 [2009] PIER 88 [2008] PIER 87 [2008] PIER 86 [2008] PIER 85 [2008] PIER 84 [2008] PIER 83 [2008] PIER 82 [2008] PIER 81 [2008] PIER 80 [2008] PIER 79 [2008] PIER 78 [2008] PIER 77 [2007] PIER 76 [2007] PIER 75 [2007] PIER 74 [2007] PIER 73 [2007] PIER 72 [2007] PIER 71 [2007] PIER 70 [2007] PIER 69 [2007] PIER 68 [2007] PIER 67 [2007] PIER 66 [2006] PIER 65 [2006] PIER 64 [2006] PIER 63 [2006] PIER 62 [2006] PIER 61 [2006] PIER 60 [2006] PIER 59 [2006] PIER 58 [2006] PIER 57 [2006] PIER 56 [2006] PIER 55 [2005] PIER 54 [2005] PIER 53 [2005] PIER 52 [2005] PIER 51 [2005] PIER 50 [2005] PIER 49 [2004] PIER 48 [2004] PIER 47 [2004] PIER 46 [2004] PIER 45 [2004] PIER 44 [2004] PIER 43 [2003] PIER 42 [2003] PIER 41 [2003] PIER 40 [2003] PIER 39 [2003] PIER 38 [2002] PIER 37 [2002] PIER 36 [2002] PIER 35 [2002] PIER 34 [2001] PIER 33 [2001] PIER 32 [2001] PIER 31 [2001] PIER 30 [2001] PIER 29 [2000] PIER 28 [2000] PIER 27 [2000] PIER 26 [2000] PIER 25 [2000] PIER 24 [1999] PIER 23 [1999] PIER 22 [1999] PIER 21 [1999] PIER 20 [1998] PIER 19 [1998] PIER 18 [1998] PIER 17 [1997] PIER 16 [1997] PIER 15 [1997] PIER 14 [1996] PIER 13 [1996] PIER 12 [1996] PIER 11 [1995] PIER 10 [1995] PIER 09 [1994] PIER 08 [1994] PIER 07 [1993] PIER 06 [1992] PIER 05 [1991] PIER 04 [1991] PIER 03 [1990] PIER 02 [1990] PIER 01 [1989]
2022-04-22
Tamm States and Gap Topological Numbers in Photonic Crystals (Invited Paper)
By
Progress In Electromagnetics Research, Vol. 173, 141-149, 2022
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.
Citation
Junhui Cao, Alexey V. Kavokin, and Anton V. Nalitov, "Tamm States and Gap Topological Numbers in Photonic Crystals (Invited Paper)," Progress In Electromagnetics Research, Vol. 173, 141-149, 2022.
doi:10.2528/PIER22011601
References

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

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

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.

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

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

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

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.

8. Kavokin, A., Microcavities. Number No. 16 in Series on Semiconductor Science and Technology, Oxford University Press, Oxford, New York, 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

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

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

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

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

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

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

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

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.

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

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

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

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

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

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