Abstract

The Zak phase and topological plasmonic Tamm states in plasmonic crystals based on periodic metal-insulator-metal waveguides are systematically investigated. We reveal that robust topological interfacial states against structural defects exist when the Zak phase between two adjoining plasmonic lattices are different in a common band gap. A kind of efficient admittance-based transfer matrix method is proposed to calculate and optimize the configuration with inverse symmetry. The topologically protected states are favorable for the spatial confinement and enhancement of electromagnetic fields, which open a new avenue for topological photonic applications.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2018 (5)

G. Harari, M. A. Bandres, Y. Lumer, M. C. Rechtsman, Y. D. Chong, M. Khajavikhan, D. N. Christodoulides, and M. Segev, “Topological insulator laser: Theory,” Science 359, aar4003 (2018).
[Crossref]

M. A. Bandres, S. Wittek, G. Harari, M. Parto, J. Ren, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Topological insulator laser: Experiments,” Science 359, aar4005 (2018).
[Crossref]

S. R. Pocock, X. Xiao, P. A. Huidobro, and V. Giannini, “Topological plasmonic chain with retardation and radiative effects,” ACS Photonics 5, 2271–2279 (2018).
[Crossref]

H. Zhao, P. Miao, M. H. Teimourpour, S. Malzard, R. El-Ganainy, H. Schomerus, and L. Feng, “Topological hybrid silicon microlasers,” Nat. Commun. 9, 981 (2018).
[Crossref] [PubMed]

B. I. Afinogenov, A. A. Popkova, V. O. Bessonov, B. Lukyanchuk, and A. A. Fedyanin, “Phase matching with Tamm plasmons for enhanced second- and third-harmonic generation,” Phys. Rev. B 97, 115438 (2018).
[Crossref]

2017 (6)

W. Gao, M. Xiao, B. Chen, E. Y. B. Pun, C. T. Chan, and W. Y. Tam, “Controlling interface states in 1d photonic crystals by tuning bulk geometric phases,” Opt. Lett. 42, 1500–1503 (2017).
[Crossref] [PubMed]

X. Wu, Y. Meng, J. Tian, Y. Huang, H. Xiang, D. Han, and W. Wen, “Direct observation of valley-polarized topological edge states in designer surface plasmon crystals,” Nat. Commun. 8, 1304 (2017).
[Crossref] [PubMed]

D. Jin, T. Christensen, M. Soljačić, N. X. Fang, L. Lu, and X. Zhang, “Infrared topological plasmons in graphene,” Phys. Rev. Lett. 118, 245301 (2017).
[Crossref] [PubMed]

D. Pan, R. Yu, H. Xu, and F. J. García de Abajo, “Topologically protected dirac plasmons in a graphene superlattice,” Nat. Commun. 8, 1243 (2017).
[Crossref] [PubMed]

D. E. Gómez, Y. Hwang, J. Lin, T. J. Davis, and A. Roberts, “Plasmonic edge states: An electrostatic eigenmode description,” ACS Photonics 4, 1607–1614 (2017).
[Crossref]

B. Bahari, A. Ndao, F. Vallini, A. El Amili, Y. Fainman, and B. Kanté, “Nonreciprocal lasing in topological cavities of arbitrary geometries,” Science 358, 636–640 (2017).
[Crossref] [PubMed]

2016 (6)

2015 (5)

S. D. Choudhury, R. Badugu, and J. R. Lakowicz, “Directing fluorescence with plasmonic and photonic structures,” Accounts Chem. Res. 48, 2171–2180 (2015).
[Crossref]

C. W. Ling, M. Xiao, C. T. Chan, S. F. Yu, and K. H. Fung, “Topological edge plasmon modes between diatomic chains of plasmonic nanoparticles,” Opt. Express 23, 2021–2031 (2015).
[Crossref] [PubMed]

Q. Cheng, Y. Pan, Q. Wang, T. Li, and S. Zhu, “Topologically protected interface mode in plasmonic waveguide arrays,” Laser & Photonics Rev. 9, 392–398 (2015).
[Crossref]

L. Ge, L. Wang, M. Xiao, W. Wen, C. T. Chan, and D. Han, “Topological edge modes in multilayer graphene systems,” Opt. Express 23, 21585–21595 (2015).
[Crossref] [PubMed]

L.-H. Wu and X. Hu, “Scheme for achieving a topological photonic crystal by using dielectric material,” Phys. Rev. Lett. 114, 223901 (2015).
[Crossref] [PubMed]

2014 (3)

A. Poddubny, A. Miroshnichenko, A. Slobozhanyuk, and Y. Kivshar, “Topological majorana states in zigzag chains of plasmonic nanoparticles,” ACS Photonics 1, 101–105 (2014).
[Crossref]

Y. Xiang, P. Wang, W. Cai, C.-F. Ying, X. Zhang, and J. Xu, “Plasmonic tamm states: dual enhancement of light inside the plasmonic waveguide,” J. Opt. Soc. Am. B 31, 2769–2772 (2014).
[Crossref]

M. Xiao, Z. Q. Zhang, and C. T. Chan, “Surface impedance and bulk band geometric phases in one-dimensional systems,” Phys. Rev. X 4, 021017 (2014).

2013 (3)

Y. Xiang, X. Zhang, W. Cai, L. Wang, C. Ying, and J. Xu, “Optical bistability based on Bragg grating resonators in metal-insulator-metal plasmonic waveguides,” AIP Adv. 3, 012106 (2013).
[Crossref]

M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. M. Taylor, “Imaging topological edge states in silicon photonics,” Nat. Photon. 7, 1001–1005 (2013).
[Crossref]

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic floquet topological insulators,” Nature 496, 196–200 (2013).
[Crossref] [PubMed]

2011 (2)

L. Fu, “Topological crystalline insulators,” Phys. Rev. Lett. 106, 106802 (2011).
[Crossref] [PubMed]

M. Hafezi, E. A. Demler, M. D. Lukin, and J. M. Taylor, “Robust optical delay lines with topological protection,” Nat. Phys. 7, 907–912 (2011).
[Crossref]

2009 (1)

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljačić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature 461, 772–775 (2009).
[Crossref] [PubMed]

2008 (1)

T. Goto, A. V. Dorofeenko, A. M. Merzlikin, A. V. Baryshev, A. P. Vinogradov, M. Inoue, A. A. Lisyansky, and A. B. Granovsky, “Optical tamm states in one-dimensional magnetophotonic structures,” Phys. Rev. Lett. 101, 113902 (2008).
[Crossref] [PubMed]

2007 (1)

M. Kaliteevski, I. Iorsh, S. Brand, R. A. Abram, J. M. Chamberlain, A. V. Kavokin, and I. A. Shelykh, “Tamm plasmon-polaritons: Possible electromagnetic states at the interface of a metal and a dielectric bragg mirror,” Phys. Rev. B 76, 165415 (2007).
[Crossref]

2006 (1)

B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, “Quantum spin hall effect and topological phase transition in HgTe quantum wells,” Science 314, 1757–1761 (2006).
[Crossref] [PubMed]

1989 (1)

J. Zak, “Berry’s phase for energy bands in solids,” Phys. Rev. Lett. 62, 2747–2750 (1989).
[Crossref] [PubMed]

1984 (1)

M. V. Berry, “Quantal phase factors accompanying adiabatic changes,” Proc. Royal Soc. Lond. 392, 45–57 (1984).
[Crossref]

1959 (1)

W. Kohn, “Analytic properties of Bloch waves and Wannier functions,” Phys. Rev. 115, 809–821 (1959).
[Crossref]

1932 (1)

I. Tamm, “On the possible bound states of electrons on a crystal surface,” Phys. Z. Sov. Union 1, 733 (1932).

Abram, R. A.

M. Kaliteevski, I. Iorsh, S. Brand, R. A. Abram, J. M. Chamberlain, A. V. Kavokin, and I. A. Shelykh, “Tamm plasmon-polaritons: Possible electromagnetic states at the interface of a metal and a dielectric bragg mirror,” Phys. Rev. B 76, 165415 (2007).
[Crossref]

Afinogenov, B. I.

B. I. Afinogenov, A. A. Popkova, V. O. Bessonov, B. Lukyanchuk, and A. A. Fedyanin, “Phase matching with Tamm plasmons for enhanced second- and third-harmonic generation,” Phys. Rev. B 97, 115438 (2018).
[Crossref]

Amili, A. El

B. Bahari, A. Ndao, F. Vallini, A. El Amili, Y. Fainman, and B. Kanté, “Nonreciprocal lasing in topological cavities of arbitrary geometries,” Science 358, 636–640 (2017).
[Crossref] [PubMed]

Badugu, R.

S. D. Choudhury, R. Badugu, and J. R. Lakowicz, “Directing fluorescence with plasmonic and photonic structures,” Accounts Chem. Res. 48, 2171–2180 (2015).
[Crossref]

Bahari, B.

B. Bahari, A. Ndao, F. Vallini, A. El Amili, Y. Fainman, and B. Kanté, “Nonreciprocal lasing in topological cavities of arbitrary geometries,” Science 358, 636–640 (2017).
[Crossref] [PubMed]

Bandres, M. A.

G. Harari, M. A. Bandres, Y. Lumer, M. C. Rechtsman, Y. D. Chong, M. Khajavikhan, D. N. Christodoulides, and M. Segev, “Topological insulator laser: Theory,” Science 359, aar4003 (2018).
[Crossref]

M. A. Bandres, S. Wittek, G. Harari, M. Parto, J. Ren, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Topological insulator laser: Experiments,” Science 359, aar4005 (2018).
[Crossref]

Baryshev, A. V.

T. Goto, A. V. Dorofeenko, A. M. Merzlikin, A. V. Baryshev, A. P. Vinogradov, M. Inoue, A. A. Lisyansky, and A. B. Granovsky, “Optical tamm states in one-dimensional magnetophotonic structures,” Phys. Rev. Lett. 101, 113902 (2008).
[Crossref] [PubMed]

Bernevig, B. A.

B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, “Quantum spin hall effect and topological phase transition in HgTe quantum wells,” Science 314, 1757–1761 (2006).
[Crossref] [PubMed]

Berry, M. V.

M. V. Berry, “Quantal phase factors accompanying adiabatic changes,” Proc. Royal Soc. Lond. 392, 45–57 (1984).
[Crossref]

Bessonov, V. O.

B. I. Afinogenov, A. A. Popkova, V. O. Bessonov, B. Lukyanchuk, and A. A. Fedyanin, “Phase matching with Tamm plasmons for enhanced second- and third-harmonic generation,” Phys. Rev. B 97, 115438 (2018).
[Crossref]

Brand, S.

M. Kaliteevski, I. Iorsh, S. Brand, R. A. Abram, J. M. Chamberlain, A. V. Kavokin, and I. A. Shelykh, “Tamm plasmon-polaritons: Possible electromagnetic states at the interface of a metal and a dielectric bragg mirror,” Phys. Rev. B 76, 165415 (2007).
[Crossref]

Cai, W.

L. Niu, Y. Xiang, W. Luo, W. Cai, J. Qi, X. Zhang, and J. Xu, “Nanofocusing of the free-space optical energy with plasmonic tamm states,” Sci. Rep. 6, 39125 (2016).
[Crossref] [PubMed]

Y. Xiang, P. Wang, W. Cai, C.-F. Ying, X. Zhang, and J. Xu, “Plasmonic tamm states: dual enhancement of light inside the plasmonic waveguide,” J. Opt. Soc. Am. B 31, 2769–2772 (2014).
[Crossref]

Y. Xiang, X. Zhang, W. Cai, L. Wang, C. Ying, and J. Xu, “Optical bistability based on Bragg grating resonators in metal-insulator-metal plasmonic waveguides,” AIP Adv. 3, 012106 (2013).
[Crossref]

Chamberlain, J. M.

M. Kaliteevski, I. Iorsh, S. Brand, R. A. Abram, J. M. Chamberlain, A. V. Kavokin, and I. A. Shelykh, “Tamm plasmon-polaritons: Possible electromagnetic states at the interface of a metal and a dielectric bragg mirror,” Phys. Rev. B 76, 165415 (2007).
[Crossref]

Chan, C. T.

Chen, B.

Chen, H.

L. Xu, H. Wang, Y. Xu, H. Chen, and J. Jiang, “Accidental degeneracy in photonic bands and topological phase transitions in two-dimensional core-shell dielectric photonic crystals,” Opt. Express 24, 18059–18071 (2016).
[Crossref] [PubMed]

F. Gao, Z. Gao, X. Shi, Z. Yang, X. Lin, H. Xu, J. D. Joannopoulos, M. Soljačić, H. Chen, L. Lu, Y. Chong, and B. Zhang, “Probing topological protection using a designer surface plasmon structure,” Nat. Commun. 7, 11619 (2016).
[Crossref] [PubMed]

Chen, X.

Cheng, Q.

Q. Cheng, Y. Pan, Q. Wang, T. Li, and S. Zhu, “Topologically protected interface mode in plasmonic waveguide arrays,” Laser & Photonics Rev. 9, 392–398 (2015).
[Crossref]

Choi, K. H.

Chong, Y.

F. Gao, Z. Gao, X. Shi, Z. Yang, X. Lin, H. Xu, J. D. Joannopoulos, M. Soljačić, H. Chen, L. Lu, Y. Chong, and B. Zhang, “Probing topological protection using a designer surface plasmon structure,” Nat. Commun. 7, 11619 (2016).
[Crossref] [PubMed]

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljačić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature 461, 772–775 (2009).
[Crossref] [PubMed]

Chong, Y. D.

G. Harari, M. A. Bandres, Y. Lumer, M. C. Rechtsman, Y. D. Chong, M. Khajavikhan, D. N. Christodoulides, and M. Segev, “Topological insulator laser: Theory,” Science 359, aar4003 (2018).
[Crossref]

Choudhury, S. D.

S. D. Choudhury, R. Badugu, and J. R. Lakowicz, “Directing fluorescence with plasmonic and photonic structures,” Accounts Chem. Res. 48, 2171–2180 (2015).
[Crossref]

Christensen, T.

D. Jin, T. Christensen, M. Soljačić, N. X. Fang, L. Lu, and X. Zhang, “Infrared topological plasmons in graphene,” Phys. Rev. Lett. 118, 245301 (2017).
[Crossref] [PubMed]

Christodoulides, D. N.

G. Harari, M. A. Bandres, Y. Lumer, M. C. Rechtsman, Y. D. Chong, M. Khajavikhan, D. N. Christodoulides, and M. Segev, “Topological insulator laser: Theory,” Science 359, aar4003 (2018).
[Crossref]

M. A. Bandres, S. Wittek, G. Harari, M. Parto, J. Ren, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Topological insulator laser: Experiments,” Science 359, aar4005 (2018).
[Crossref]

Davis, T. J.

D. E. Gómez, Y. Hwang, J. Lin, T. J. Davis, and A. Roberts, “Plasmonic edge states: An electrostatic eigenmode description,” ACS Photonics 4, 1607–1614 (2017).
[Crossref]

Demler, E. A.

M. Hafezi, E. A. Demler, M. D. Lukin, and J. M. Taylor, “Robust optical delay lines with topological protection,” Nat. Phys. 7, 907–912 (2011).
[Crossref]

Deng, H.

Dorofeenko, A. V.

T. Goto, A. V. Dorofeenko, A. M. Merzlikin, A. V. Baryshev, A. P. Vinogradov, M. Inoue, A. A. Lisyansky, and A. B. Granovsky, “Optical tamm states in one-dimensional magnetophotonic structures,” Phys. Rev. Lett. 101, 113902 (2008).
[Crossref] [PubMed]

Dreisow, F.

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic floquet topological insulators,” Nature 496, 196–200 (2013).
[Crossref] [PubMed]

El-Ganainy, R.

H. Zhao, P. Miao, M. H. Teimourpour, S. Malzard, R. El-Ganainy, H. Schomerus, and L. Feng, “Topological hybrid silicon microlasers,” Nat. Commun. 9, 981 (2018).
[Crossref] [PubMed]

Fainman, Y.

B. Bahari, A. Ndao, F. Vallini, A. El Amili, Y. Fainman, and B. Kanté, “Nonreciprocal lasing in topological cavities of arbitrary geometries,” Science 358, 636–640 (2017).
[Crossref] [PubMed]

Fan, J.

M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. M. Taylor, “Imaging topological edge states in silicon photonics,” Nat. Photon. 7, 1001–1005 (2013).
[Crossref]

Fang, N. X.

D. Jin, T. Christensen, M. Soljačić, N. X. Fang, L. Lu, and X. Zhang, “Infrared topological plasmons in graphene,” Phys. Rev. Lett. 118, 245301 (2017).
[Crossref] [PubMed]

Fedyanin, A. A.

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Qi, J.

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M. Kaliteevski, I. Iorsh, S. Brand, R. A. Abram, J. M. Chamberlain, A. V. Kavokin, and I. A. Shelykh, “Tamm plasmon-polaritons: Possible electromagnetic states at the interface of a metal and a dielectric bragg mirror,” Phys. Rev. B 76, 165415 (2007).
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A. Poddubny, A. Miroshnichenko, A. Slobozhanyuk, and Y. Kivshar, “Topological majorana states in zigzag chains of plasmonic nanoparticles,” ACS Photonics 1, 101–105 (2014).
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D. Jin, T. Christensen, M. Soljačić, N. X. Fang, L. Lu, and X. Zhang, “Infrared topological plasmons in graphene,” Phys. Rev. Lett. 118, 245301 (2017).
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M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic floquet topological insulators,” Nature 496, 196–200 (2013).
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H. Zhao, P. Miao, M. H. Teimourpour, S. Malzard, R. El-Ganainy, H. Schomerus, and L. Feng, “Topological hybrid silicon microlasers,” Nat. Commun. 9, 981 (2018).
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X. Wu, Y. Meng, J. Tian, Y. Huang, H. Xiang, D. Han, and W. Wen, “Direct observation of valley-polarized topological edge states in designer surface plasmon crystals,” Nat. Commun. 8, 1304 (2017).
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B. Bahari, A. Ndao, F. Vallini, A. El Amili, Y. Fainman, and B. Kanté, “Nonreciprocal lasing in topological cavities of arbitrary geometries,” Science 358, 636–640 (2017).
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L. Ge, L. Wang, M. Xiao, W. Wen, C. T. Chan, and D. Han, “Topological edge modes in multilayer graphene systems,” Opt. Express 23, 21585–21595 (2015).
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X. Wu, Y. Meng, J. Tian, Y. Huang, H. Xiang, D. Han, and W. Wen, “Direct observation of valley-polarized topological edge states in designer surface plasmon crystals,” Nat. Commun. 8, 1304 (2017).
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M. A. Bandres, S. Wittek, G. Harari, M. Parto, J. Ren, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Topological insulator laser: Experiments,” Science 359, aar4005 (2018).
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L.-H. Wu and X. Hu, “Scheme for achieving a topological photonic crystal by using dielectric material,” Phys. Rev. Lett. 114, 223901 (2015).
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X. Wu, Y. Meng, J. Tian, Y. Huang, H. Xiang, D. Han, and W. Wen, “Direct observation of valley-polarized topological edge states in designer surface plasmon crystals,” Nat. Commun. 8, 1304 (2017).
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F. Gao, Z. Gao, X. Shi, Z. Yang, X. Lin, H. Xu, J. D. Joannopoulos, M. Soljačić, H. Chen, L. Lu, Y. Chong, and B. Zhang, “Probing topological protection using a designer surface plasmon structure,” Nat. Commun. 7, 11619 (2016).
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Y. Xiang, P. Wang, W. Cai, C.-F. Ying, X. Zhang, and J. Xu, “Plasmonic tamm states: dual enhancement of light inside the plasmonic waveguide,” J. Opt. Soc. Am. B 31, 2769–2772 (2014).
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Y. Xiang, X. Zhang, W. Cai, L. Wang, C. Ying, and J. Xu, “Optical bistability based on Bragg grating resonators in metal-insulator-metal plasmonic waveguides,” AIP Adv. 3, 012106 (2013).
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L. Niu, Y. Xiang, W. Luo, W. Cai, J. Qi, X. Zhang, and J. Xu, “Nanofocusing of the free-space optical energy with plasmonic tamm states,” Sci. Rep. 6, 39125 (2016).
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Y. Xiang, P. Wang, W. Cai, C.-F. Ying, X. Zhang, and J. Xu, “Plasmonic tamm states: dual enhancement of light inside the plasmonic waveguide,” J. Opt. Soc. Am. B 31, 2769–2772 (2014).
[Crossref]

Y. Xiang, X. Zhang, W. Cai, L. Wang, C. Ying, and J. Xu, “Optical bistability based on Bragg grating resonators in metal-insulator-metal plasmonic waveguides,” AIP Adv. 3, 012106 (2013).
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F. Gao, Z. Gao, X. Shi, Z. Yang, X. Lin, H. Xu, J. D. Joannopoulos, M. Soljačić, H. Chen, L. Lu, Y. Chong, and B. Zhang, “Probing topological protection using a designer surface plasmon structure,” Nat. Commun. 7, 11619 (2016).
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D. Pan, R. Yu, H. Xu, and F. J. García de Abajo, “Topologically protected dirac plasmons in a graphene superlattice,” Nat. Commun. 8, 1243 (2017).
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Figures (6)

Fig. 1
Fig. 1 (a) Sketch of two adjoining one dimensional plasmonic crystals (PCs) composed of symmetric ABA layers. The insets show the side views of PC1 and PC2, of which the unit cells Λ1 = 2dA1 + dB1 and Λ2 = 2dA2 + dB2 are highlighted by blue and red dashed frames, respectively. Here the width w = 40 nm, and the refractive index nA = 1 and nB = 1.5, respectively. (b) The band structure of the PCs as a function of dA when δA + δB = 3π is fixed, where δA and δB are the phase delay in slab A and slab B for a unit cell, respectively. The blank zones indicate the stopband. TPI and TPII are two topological phase points, and the integer pairs in parentheses are δA and δB in the unit of π, respectively. Moreover, the dashed lines show the parameters adopt to construct edge states. (c, d) The dispersion relation (c) and mode distributions (d) of the edge modes, where the magnetic field distributions H are calculated at kx = 0.
Fig. 2
Fig. 2 (a, b) The band structure and Zak phase of plasmonic crystals with parameters (a) d A1 = 319.3 nm, and dB2 = 2dA1 = 638.6 nm for PC1, and (b) dA2 = 210 nm, and dB2 = 738.8 nm for PC2, respectively. The Zak phase of each individual band is labeled in green, and the numbers of the bands and gaps are listed with red and black labels. The magenta strip represents the gap with reflection phase φPC > 0, while the cyan strip represents the gap with φPC < 0. Moreover, the symmetries of the eight band edge states, K, L, M, N, O, P, Q, R are labeled with red (antisymmetric) and indigo (symmetric) circles, respectively. (c) The black and blue broken curves represent the reflection phase φPC1 and the negative value of the reflection phase −φPC2 of the plasmonic crystals consisting of 50 periods calculated by Eq. (6). The intersection point denotes the location of the interface states. (d) The Bloch magnetic field eigenfunctions of the band-edge states at z = 0. The zero amplitude at the origin represents the Bloch magnetic fields are antisymmetric, while the non-zero values indicate the the Bloch magnetic fields are symmetric in the plasmonic crystals.
Fig. 3
Fig. 3 (a-c) The reflection spectra of 4 unit cells of PC1 (a), 8 unit cells of PC2 (b) and a system composed of 4 unit cells of PC1 connecting with 8 unit cells of PC2 (c). The solid lines indicate the spectra calculated by the TMM, while the cross marks indicate the spectra simulated by FEM. (d,e) The electric and magnetic field amplitude distributions inside the MIM structure. The contour plots show the normalized field amplitude along the central axis. The magenta dash line shows the position of interface between PC1 and PC2.
Fig. 4
Fig. 4 (a) The reflection spectra of PC1/defect/PC2 composites, the thickness Δd are 0, 100, 300, and 450 nm, respectively. (b) The resonance energy of TPTSs as a function of the thickness of defect. The upper scale denotes the phase delay Δδ in the defect.
Fig. 5
Fig. 5 (a, b) The band structure and Zak phase of plasmonic crystals consist of HfO2/SiO2/HfO2 cores with parameters (a) dA1 = 227.1 nm, and dB2 = 2dA1 = 454.2 nm for PC1, and (b) dA2 = 151.4 nm, and dB2 = 302.8 nm for PC2, respectively. The Zak phase of each individual band is labeled in green, and the numbers of the bands and gaps are listed with red and black labels. The magenta strip represents the gap with reflection phase φPC > 0, while the cyan strip represents the gap with φPC < 0. Moreover, the symmetries of the eight band edge states, K, L, M, N, O, P, Q, R are labeled with red (antisymmetric) and indigo (symmetric) circles, respectively. (c) The black and blue broken curves represent the reflection phase φPC1 and the negative value of the reflection phase −φPC2 of the plasmonic crystals consisting of 50 periods. The intersection point denotes the location of the interface states. (d) The Bloch magnetic field eigenfunctions of the band-edge states at z = 0. The zero amplitude at the origin represent the Bloch magnetic fields are antisymmetric, while the non-zero values indicate the the Bloch magnetic fields are symmetric in the plasmonic crystals.
Fig. 6
Fig. 6 (a) The reflection spectra of 4 unit cells of PC1 (a), 8 unit cells of PC2 (b) and a system composed of 4 unit cells of PC1 connecting with 8 unit cells of PC2 (c). The solid lines indicate the spectra calculated by the TMM, while the cross marks indicate the spectra simulated by FEM. (d,e) The electric and magnetic field amplitude distributions inside the MIM structure. The contour plots show the normalized field amplitude along the central axis. The magenta dash line shows the position of interface between PC1 and PC2.

Equations (26)

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η = I / V = c H x d x / + E y d y = ω ϵ d ϵ 0 d x β w ,
cos δ = cos δ A cos δ B 1 2 ( η A / η B + η B / η A ) sin δ A sin δ B ,
α = δ A δ B = m 1 m 2 . m 1 , m 2 +
θ n Zak = π / Λ π / Λ [ i u n i t c e l l d z u n , q * ( z ) q u n , q ( z ) ] d q ,
sgn [ φ ( n ) ] = ( 1 ) n exp ( i m = 0 n 1 θ m Zak ) , n +
r PC = | r | exp ( i φ PC ) = η 0 / η η / η 0 2 i cot ( N δ ) + η 0 / η + η / η 0 ,
η = sin δ / { 1 / η B sin δ B ( cos 2 δ A 2 η B 2 / η A 2 sin 2 δ A 2 ) + 1 / η A sin δ A cos δ B } .
E y = E 0 + cosh ( k d l y ) exp ( i β l z ) + E 0 cosh ( k d l y ) exp ( i β l z ) ;
T A ( E A j + E A j ) = T B ( E B j + E B j ) ,
P α = [ exp ( i β α d α ) 0 0 exp ( i β α d α ) ] . ( α = A , B )
T A E N = T A P A E 1 l = T A P A T A 1 T B E 1 r = T A P A T A 1 T B P B E 2 l = T A P A T A 1 T B P B T B 1 T A E 2 r = T A P A T A 1 T B P B T B 1 T A P A T A 1 T A E N + 1 = M A M B M A T A E N + 1
M α T α P α T α 1 = [ cos ( β α d α ) i sin ( β α d α ) / η α i η α sin ( β α d α ) cos ( β α d α ) ]
M = M A M B M A = [ cos δ i sin δ / η i η sin δ cos δ ] ,
cos δ = cos δ A cos δ B 1 2 ( η A / η B + η B / η A ) sin δ A sin δ B
η = ± M 21 M 12 = sin δ / { 1 η B sin δ B ( cos 2 δ A 2 η B 2 η A 2 sin 2 δ A 2 ) + 1 / η A sin δ A cos δ B } .
M N = M N = [ cos ( N δ ) i sin ( N δ ) / η i η sin ( N δ ) cos ( N δ ) ] [ M ^ 11 M ^ 12 M ^ 21 M ^ 22 ] .
T 0 E 0 l = T A E 0 r = M N T A E N + 1 l = M N T N + 1 E N + 1 r .
( E 0 + E 0 ) = T 0 1 M N T N + 1 ( E N + 1 + 0 ) .
r PC = E 0 E 0 + = M ^ 11 η 0 + M ^ 12 η 0 η N + 1 M ^ 21 M ^ 22 η N + 1 M ^ 11 η 0 + M ^ 12 η 0 η N + 1 + M ^ 21 + M ^ 22 η N + 1 .
r PC = i sin ( N δ ) ( η 0 / η η / η 0 ) 2 cos ( N δ ) i sin ( N δ ) ( η 0 / η + η / η 0 ) = η 0 / η η / η 0 2 i cot ( N δ ) + η 0 / η + η / η 0
M ˜ 11 = cos ( N 1 δ 1 ) cos ( N 2 δ 2 ) η 2 / η 1 sin ( N 1 δ 1 ) sin ( N 2 δ 2 ) , M ˜ 12 = i / η 1 sin ( N 1 δ 1 ) cos ( N 2 δ 2 ) i / η 2 cos ( N 1 δ 1 ) sin ( N 2 δ 2 ) , M ˜ 21 = i η 1 sin ( N 1 δ 1 ) cos ( N 2 δ 2 ) i η 2 cos ( N 1 δ 1 ) sin ( N 2 δ 2 ) , M ˜ 22 = cos ( N 1 δ 1 ) cos ( N 2 δ 2 ) η 1 / η 2 sin ( N 1 δ 1 ) sin ( N 2 δ 2 ) ;
r PC = M ˜ 11 η 0 + M ˜ 12 η 0 η N + 1 M ˜ 21 M ˜ 22 η N + 1 M ˜ 11 η 0 + M ˜ 12 η 0 η N + 1 + M ˜ 21 + M ˜ 22 η N + 1 .
M 11 E = exp ( i δ A ) [ cos δ B + i 2 ( η A η B + η B η A ) sin δ B ] , M 12 E = i 2 sin δ B ( η A η B η B η A ) .
E y ( z ) = E 1 r + exp [ i β B ( z d A ) ] + E 1 r exp [ i β B ( z d A ) ]
d x w H x ( z ) = η B E 1 r + exp [ i β B ( z d A ) ] η B E 1 r exp [ i β B ( z d A ) ] ;
u ( z ) = w d x exp ( i K z ) × { η A exp ( i β A z ) η A Γ exp ( i β A z ) , x [ n Λ d A , n Λ + d A ] η B E 1 r + exp [ i β B ( z d A ) ] η B E 1 r exp [ i β B ( z d A ) ] . x [ n Λ + d A , ( n + 1 ) Λ d A ]

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