Abstract

One of the fascinating topological phenomena is the edge state in one-dimensional system. In this work, the topological photonics in the dimer chains composed by the split ring resonators are revealed based on the Su-Schrieffer-Heeger model. The topologically protected photonic edge state is observed directly with the in situ measurements of the local density of states in the topological nontrivial chain. Moreover, we experimentally demonstrate that the edge state localized at both ends is robust against a varied of perturbations, such as losses and disorder. Our results not only provide a versatile platform to study the topological physics in photonics but also may have potential applications in the robust power transfer.

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

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  29. Z. Guo, H. Jiang, Y. Li, H. Chen, and G. S. Agarwal, “Enhancement of electromagnetically induced transparency in metamaterials using long range coupling mediated by a hyperbolic material,” Opt. Express 26(2), 627–641 (2018).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]

2018 (2)

2017 (4)

C. Liu, W. Gao, B. Yang, and S. Zhang, “Disorder-induced topological state transition in photonic metamaterials,” Phys. Rev. Lett. 119(18), 183901 (2017).
[Crossref] [PubMed]

Z. Guo, H. Jiang, Y. Long, K. Yu, J. Ren, C. Xue, and H. Chen, “Photonic spin Hall efect in waveguides composed of two types of single-negative metamaterials,” Sci. Rep. 7(1), 7742 (2017).
[Crossref] [PubMed]

S. Weimann, M. Kremer, Y. Plotnik, Y. Lumer, S. Nolte, K. G. Makris, M. Segev, M. C. Rechtsman, and A. Szameit, “Topologically protected bound states in photonic parity-time-symmetric crystals,” Nat. Mater. 16(4), 433–438 (2017).
[Crossref] [PubMed]

F. Baboux, E. Levy, A. Lemaître, C. Gómez, E. Galopin, L. L. Gratiet, I. Sagnes, A. Amo, J. Bloch, and E. Akkermans, “Measuring topological invariants from generalized edge states in polaritonic quasicrystals,” Phys. Rev. B 95(16), 161114 (2017).
[Crossref]

2016 (4)

L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological states in photonic systems,” Nat. Phys. 12(7), 626–629 (2016).
[Crossref]

A. Blanco-Redondo, I. Andonegui, M. J. Collins, G. Harari, Y. Lumer, M. C. Rechtsman, B. J. Eggleton, and M. Segev, “Topological optical waveguiding in silicon and the transition between topological and trivial defect states,” Phys. Rev. Lett. 116(16), 163901 (2016).
[Crossref] [PubMed]

E. J. Meier, F. A. An, and B. Gadway, “Observation of the topological soliton state in the Su-Schrieffer-Heeger model,” Nat. Commun. 7, 13986 (2016).
[Crossref] [PubMed]

Y. Hadad, A. B. Khanikaev, and A. Alu, “Self-induced topological transitions and edge states supported by nonlinear staggered potentials,” Phys. Rev. B 93(15), 155112 (2016).
[Crossref]

2015 (8)

S. Cheon, T. H. Kim, S. H. Lee, and H. W. Yeom, “Chiral solitons in a coupled double Peierls chain,” Science 350(6257), 182–185 (2015).
[Crossref] [PubMed]

M. Xiao, G. Ma, Z. Yang, P. Sheng, Z. Q. Zhang, and C. T. Chan, “Geometric phase and band inversion in periodic acoustic systems,” Nat. Phys. 11(3), 240–244 (2015).
[Crossref]

W. S. Gao, M. Xiao, C. T. Chan, and W. Y. Tam, “Determination of Zak phase by reflection phase in 1D photonic crystals,” Opt. Lett. 40(22), 5259–5262 (2015).
[Crossref] [PubMed]

L. H. Wu and X. Hu, “Scheme for achieving a topological photonic crystal by using dielectric material,” Phys. Rev. Lett. 114(22), 223901 (2015).
[Crossref] [PubMed]

W. Tan, Y. Sun, H. Chen, and S. Q. Shen, “Photonic simulation of topological excitations in metamaterials,” Sci. Rep. 4(1), 3842 (2015).
[Crossref] [PubMed]

A. P. Slobozhanyuk, A. N. Poddubny, A. E. Miroshnichenko, P. A. Belov, and Y. S. Kivshar, “Subwavelength topological edge States in optically resonant dielectric structures,” Phys. Rev. Lett. 114(12), 123901 (2015).
[Crossref] [PubMed]

C. Poli, M. Bellec, U. Kuhl, F. Mortessagne, and H. Schomerus, “Selective enhancement of topologically induced interface states in a dielectric resonator chain,” Nat. Commun. 6(1), 6710 (2015).
[Crossref] [PubMed]

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(3), 2021–2031 (2015).
[Crossref] [PubMed]

2014 (3)

M. Xiao, Z. Q. Zhang, and C. T. Chan, “Surface impedance and bulk band geometric phases in one-dimensional systems,” Phys. Rev. X 4(2), 021017 (2014).
[Crossref]

L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological photonics,” Nat. Photonics 8(11), 821–829 (2014).
[Crossref]

L. H. Li, Z. H. Xu, and S. Chen, “Topological phases of generalized Su-Schrieffer-Heeger models,” Phys. Rev. B 89(8), 085111 (2014).
[Crossref]

2013 (4)

M. Bellec, U. Kuhl, G. Montambaux, and F. Mortessagne, “Tight-binding couplings in microwave artificial graphene,” Phys. Rev. B 88(11), 115437 (2013).
[Crossref]

A. B. Khanikaev, S. H. Mousavi, W. K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12(3), 233–239 (2013).
[Crossref] [PubMed]

M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. M. Taylor, “Imaging topological edge states in silicon photonics,” Nat. Photonics 7(12), 1001–1005 (2013).
[Crossref]

G. Q. Liang and Y. D. Chong, “Optical Resonator Analog of a two-dimensional topological insulator,” Phys. Rev. Lett. 110(20), 203904 (2013).
[Crossref] [PubMed]

2012 (1)

E. Tatarschuk, N. Gneiding, F. Hesmer, A. Radkovskaya, and E. Shamonina, “Mapping inter-element coupling in metamaterials: Scaling down to infrared,” J. Appl. Phys. 111(9), 094904 (2012).
[Crossref]

2009 (3)

O. Sydoruk, E. Tatartschuk, E. Shamonina, and L. Solymar, “Analytical formulation for the resonant frequency of split rings,” J. Appl. Phys. 105(1), 014903 (2009).
[Crossref]

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljacić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature 461(7265), 772–775 (2009).
[Crossref] [PubMed]

N. Malkova, I. Hromada, X. Wang, G. Bryant, and Z. Chen, “Transition between Tamm-like and Shockley-like surface states in optically induced photonic superlattices,” Phys. Rev. A 80(4), 043806 (2009).
[Crossref]

2008 (2)

F. D. Haldane and S. Raghu, “Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry,” Phys. Rev. Lett. 100(1), 013904 (2008).
[Crossref] [PubMed]

S. Zhang, D. A. Genov, Y. Wang, M. Liu, and X. Zhang, “Plasmon-induced transparency in metamaterials,” Phys. Rev. Lett. 101(4), 047401 (2008).
[Crossref] [PubMed]

2007 (1)

A. Kurs, A. Karalis, R. Moffatt, J. D. Joannopoulos, P. Fisher, and M. Soljačić, “Wireless power transfer via strongly coupled magnetic resonances,” Science 317(5834), 83–86 (2007).
[Crossref] [PubMed]

2005 (1)

C. L. Kane and E. J. Mele, “Quantum spin Hall effect in graphene,” Phys. Rev. Lett. 95(22), 226801 (2005).
[Crossref] [PubMed]

1993 (1)

R. D. King-Smith and D. Vanderbilt, “Theory of polarization of crystalline solids,” Phys. Rev. B Condens. Matter 47(3), 1651–1654 (1993).
[Crossref] [PubMed]

1961 (1)

J. E. Allen and S. E. Segre, “The electric field in single-turn and multi-sector coils,” Nuovo Cim. 21(6), 980–987 (1961).
[Crossref]

Agarwal, G. S.

Akkermans, E.

F. Baboux, E. Levy, A. Lemaître, C. Gómez, E. Galopin, L. L. Gratiet, I. Sagnes, A. Amo, J. Bloch, and E. Akkermans, “Measuring topological invariants from generalized edge states in polaritonic quasicrystals,” Phys. Rev. B 95(16), 161114 (2017).
[Crossref]

Allen, J. E.

J. E. Allen and S. E. Segre, “The electric field in single-turn and multi-sector coils,” Nuovo Cim. 21(6), 980–987 (1961).
[Crossref]

Alu, A.

Y. Hadad, J. C. Soric, A. B. Khanikaev, and A. Alu, “Self-induced topological protection in nonlinear circuit arrays,” Nat. Electron. 1(3), 178–182 (2018).
[Crossref]

Y. Hadad, A. B. Khanikaev, and A. Alu, “Self-induced topological transitions and edge states supported by nonlinear staggered potentials,” Phys. Rev. B 93(15), 155112 (2016).
[Crossref]

Amo, A.

F. Baboux, E. Levy, A. Lemaître, C. Gómez, E. Galopin, L. L. Gratiet, I. Sagnes, A. Amo, J. Bloch, and E. Akkermans, “Measuring topological invariants from generalized edge states in polaritonic quasicrystals,” Phys. Rev. B 95(16), 161114 (2017).
[Crossref]

An, F. A.

E. J. Meier, F. A. An, and B. Gadway, “Observation of the topological soliton state in the Su-Schrieffer-Heeger model,” Nat. Commun. 7, 13986 (2016).
[Crossref] [PubMed]

Andonegui, I.

A. Blanco-Redondo, I. Andonegui, M. J. Collins, G. Harari, Y. Lumer, M. C. Rechtsman, B. J. Eggleton, and M. Segev, “Topological optical waveguiding in silicon and the transition between topological and trivial defect states,” Phys. Rev. Lett. 116(16), 163901 (2016).
[Crossref] [PubMed]

Baboux, F.

F. Baboux, E. Levy, A. Lemaître, C. Gómez, E. Galopin, L. L. Gratiet, I. Sagnes, A. Amo, J. Bloch, and E. Akkermans, “Measuring topological invariants from generalized edge states in polaritonic quasicrystals,” Phys. Rev. B 95(16), 161114 (2017).
[Crossref]

Bellec, M.

C. Poli, M. Bellec, U. Kuhl, F. Mortessagne, and H. Schomerus, “Selective enhancement of topologically induced interface states in a dielectric resonator chain,” Nat. Commun. 6(1), 6710 (2015).
[Crossref] [PubMed]

M. Bellec, U. Kuhl, G. Montambaux, and F. Mortessagne, “Tight-binding couplings in microwave artificial graphene,” Phys. Rev. B 88(11), 115437 (2013).
[Crossref]

Belov, P. A.

A. P. Slobozhanyuk, A. N. Poddubny, A. E. Miroshnichenko, P. A. Belov, and Y. S. Kivshar, “Subwavelength topological edge States in optically resonant dielectric structures,” Phys. Rev. Lett. 114(12), 123901 (2015).
[Crossref] [PubMed]

Blanco-Redondo, A.

A. Blanco-Redondo, I. Andonegui, M. J. Collins, G. Harari, Y. Lumer, M. C. Rechtsman, B. J. Eggleton, and M. Segev, “Topological optical waveguiding in silicon and the transition between topological and trivial defect states,” Phys. Rev. Lett. 116(16), 163901 (2016).
[Crossref] [PubMed]

Bloch, J.

F. Baboux, E. Levy, A. Lemaître, C. Gómez, E. Galopin, L. L. Gratiet, I. Sagnes, A. Amo, J. Bloch, and E. Akkermans, “Measuring topological invariants from generalized edge states in polaritonic quasicrystals,” Phys. Rev. B 95(16), 161114 (2017).
[Crossref]

Bryant, G.

N. Malkova, I. Hromada, X. Wang, G. Bryant, and Z. Chen, “Transition between Tamm-like and Shockley-like surface states in optically induced photonic superlattices,” Phys. Rev. A 80(4), 043806 (2009).
[Crossref]

Chan, C. T.

W. S. Gao, M. Xiao, C. T. Chan, and W. Y. Tam, “Determination of Zak phase by reflection phase in 1D photonic crystals,” Opt. Lett. 40(22), 5259–5262 (2015).
[Crossref] [PubMed]

M. Xiao, G. Ma, Z. Yang, P. Sheng, Z. Q. Zhang, and C. T. Chan, “Geometric phase and band inversion in periodic acoustic systems,” Nat. Phys. 11(3), 240–244 (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(3), 2021–2031 (2015).
[Crossref] [PubMed]

M. Xiao, Z. Q. Zhang, and C. T. Chan, “Surface impedance and bulk band geometric phases in one-dimensional systems,” Phys. Rev. X 4(2), 021017 (2014).
[Crossref]

Chen, H.

Z. Guo, H. Jiang, Y. Li, H. Chen, and G. S. Agarwal, “Enhancement of electromagnetically induced transparency in metamaterials using long range coupling mediated by a hyperbolic material,” Opt. Express 26(2), 627–641 (2018).
[Crossref] [PubMed]

Z. Guo, H. Jiang, Y. Long, K. Yu, J. Ren, C. Xue, and H. Chen, “Photonic spin Hall efect in waveguides composed of two types of single-negative metamaterials,” Sci. Rep. 7(1), 7742 (2017).
[Crossref] [PubMed]

W. Tan, Y. Sun, H. Chen, and S. Q. Shen, “Photonic simulation of topological excitations in metamaterials,” Sci. Rep. 4(1), 3842 (2015).
[Crossref] [PubMed]

Chen, S.

L. H. Li, Z. H. Xu, and S. Chen, “Topological phases of generalized Su-Schrieffer-Heeger models,” Phys. Rev. B 89(8), 085111 (2014).
[Crossref]

Chen, Z.

N. Malkova, I. Hromada, X. Wang, G. Bryant, and Z. Chen, “Transition between Tamm-like and Shockley-like surface states in optically induced photonic superlattices,” Phys. Rev. A 80(4), 043806 (2009).
[Crossref]

Cheon, S.

S. Cheon, T. H. Kim, S. H. Lee, and H. W. Yeom, “Chiral solitons in a coupled double Peierls chain,” Science 350(6257), 182–185 (2015).
[Crossref] [PubMed]

Chong, Y.

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljacić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature 461(7265), 772–775 (2009).
[Crossref] [PubMed]

Chong, Y. D.

G. Q. Liang and Y. D. Chong, “Optical Resonator Analog of a two-dimensional topological insulator,” Phys. Rev. Lett. 110(20), 203904 (2013).
[Crossref] [PubMed]

Collins, M. J.

A. Blanco-Redondo, I. Andonegui, M. J. Collins, G. Harari, Y. Lumer, M. C. Rechtsman, B. J. Eggleton, and M. Segev, “Topological optical waveguiding in silicon and the transition between topological and trivial defect states,” Phys. Rev. Lett. 116(16), 163901 (2016).
[Crossref] [PubMed]

Eggleton, B. J.

A. Blanco-Redondo, I. Andonegui, M. J. Collins, G. Harari, Y. Lumer, M. C. Rechtsman, B. J. Eggleton, and M. Segev, “Topological optical waveguiding in silicon and the transition between topological and trivial defect states,” Phys. Rev. Lett. 116(16), 163901 (2016).
[Crossref] [PubMed]

Fan, J.

M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. M. Taylor, “Imaging topological edge states in silicon photonics,” Nat. Photonics 7(12), 1001–1005 (2013).
[Crossref]

Fisher, P.

A. Kurs, A. Karalis, R. Moffatt, J. D. Joannopoulos, P. Fisher, and M. Soljačić, “Wireless power transfer via strongly coupled magnetic resonances,” Science 317(5834), 83–86 (2007).
[Crossref] [PubMed]

Fung, K. H.

Gadway, B.

E. J. Meier, F. A. An, and B. Gadway, “Observation of the topological soliton state in the Su-Schrieffer-Heeger model,” Nat. Commun. 7, 13986 (2016).
[Crossref] [PubMed]

Galopin, E.

F. Baboux, E. Levy, A. Lemaître, C. Gómez, E. Galopin, L. L. Gratiet, I. Sagnes, A. Amo, J. Bloch, and E. Akkermans, “Measuring topological invariants from generalized edge states in polaritonic quasicrystals,” Phys. Rev. B 95(16), 161114 (2017).
[Crossref]

Gao, W.

C. Liu, W. Gao, B. Yang, and S. Zhang, “Disorder-induced topological state transition in photonic metamaterials,” Phys. Rev. Lett. 119(18), 183901 (2017).
[Crossref] [PubMed]

Gao, W. S.

Genov, D. A.

S. Zhang, D. A. Genov, Y. Wang, M. Liu, and X. Zhang, “Plasmon-induced transparency in metamaterials,” Phys. Rev. Lett. 101(4), 047401 (2008).
[Crossref] [PubMed]

Gneiding, N.

E. Tatarschuk, N. Gneiding, F. Hesmer, A. Radkovskaya, and E. Shamonina, “Mapping inter-element coupling in metamaterials: Scaling down to infrared,” J. Appl. Phys. 111(9), 094904 (2012).
[Crossref]

Gómez, C.

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L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological states in photonic systems,” Nat. Phys. 12(7), 626–629 (2016).
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M. Xiao, G. Ma, Z. Yang, P. Sheng, Z. Q. Zhang, and C. T. Chan, “Geometric phase and band inversion in periodic acoustic systems,” Nat. Phys. 11(3), 240–244 (2015).
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A. B. Khanikaev, S. H. Mousavi, W. K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12(3), 233–239 (2013).
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S. Weimann, M. Kremer, Y. Plotnik, Y. Lumer, S. Nolte, K. G. Makris, M. Segev, M. C. Rechtsman, and A. Szameit, “Topologically protected bound states in photonic parity-time-symmetric crystals,” Nat. Mater. 16(4), 433–438 (2017).
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M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. M. Taylor, “Imaging topological edge states in silicon photonics,” Nat. Photonics 7(12), 1001–1005 (2013).
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M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. M. Taylor, “Imaging topological edge states in silicon photonics,” Nat. Photonics 7(12), 1001–1005 (2013).
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A. Kurs, A. Karalis, R. Moffatt, J. D. Joannopoulos, P. Fisher, and M. Soljačić, “Wireless power transfer via strongly coupled magnetic resonances,” Science 317(5834), 83–86 (2007).
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M. Bellec, U. Kuhl, G. Montambaux, and F. Mortessagne, “Tight-binding couplings in microwave artificial graphene,” Phys. Rev. B 88(11), 115437 (2013).
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C. Poli, M. Bellec, U. Kuhl, F. Mortessagne, and H. Schomerus, “Selective enhancement of topologically induced interface states in a dielectric resonator chain,” Nat. Commun. 6(1), 6710 (2015).
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A. B. Khanikaev, S. H. Mousavi, W. K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12(3), 233–239 (2013).
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S. Weimann, M. Kremer, Y. Plotnik, Y. Lumer, S. Nolte, K. G. Makris, M. Segev, M. C. Rechtsman, and A. Szameit, “Topologically protected bound states in photonic parity-time-symmetric crystals,” Nat. Mater. 16(4), 433–438 (2017).
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C. Poli, M. Bellec, U. Kuhl, F. Mortessagne, and H. Schomerus, “Selective enhancement of topologically induced interface states in a dielectric resonator chain,” Nat. Commun. 6(1), 6710 (2015).
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E. Tatarschuk, N. Gneiding, F. Hesmer, A. Radkovskaya, and E. Shamonina, “Mapping inter-element coupling in metamaterials: Scaling down to infrared,” J. Appl. Phys. 111(9), 094904 (2012).
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F. D. Haldane and S. Raghu, “Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry,” Phys. Rev. Lett. 100(1), 013904 (2008).
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Z. Guo, H. Jiang, Y. Long, K. Yu, J. Ren, C. Xue, and H. Chen, “Photonic spin Hall efect in waveguides composed of two types of single-negative metamaterials,” Sci. Rep. 7(1), 7742 (2017).
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F. Baboux, E. Levy, A. Lemaître, C. Gómez, E. Galopin, L. L. Gratiet, I. Sagnes, A. Amo, J. Bloch, and E. Akkermans, “Measuring topological invariants from generalized edge states in polaritonic quasicrystals,” Phys. Rev. B 95(16), 161114 (2017).
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C. Poli, M. Bellec, U. Kuhl, F. Mortessagne, and H. Schomerus, “Selective enhancement of topologically induced interface states in a dielectric resonator chain,” Nat. Commun. 6(1), 6710 (2015).
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S. Weimann, M. Kremer, Y. Plotnik, Y. Lumer, S. Nolte, K. G. Makris, M. Segev, M. C. Rechtsman, and A. Szameit, “Topologically protected bound states in photonic parity-time-symmetric crystals,” Nat. Mater. 16(4), 433–438 (2017).
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A. B. Khanikaev, S. H. Mousavi, W. K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12(3), 233–239 (2013).
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A. P. Slobozhanyuk, A. N. Poddubny, A. E. Miroshnichenko, P. A. Belov, and Y. S. Kivshar, “Subwavelength topological edge States in optically resonant dielectric structures,” Phys. Rev. Lett. 114(12), 123901 (2015).
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L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological states in photonic systems,” Nat. Phys. 12(7), 626–629 (2016).
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Y. Hadad, J. C. Soric, A. B. Khanikaev, and A. Alu, “Self-induced topological protection in nonlinear circuit arrays,” Nat. Electron. 1(3), 178–182 (2018).
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S. Weimann, M. Kremer, Y. Plotnik, Y. Lumer, S. Nolte, K. G. Makris, M. Segev, M. C. Rechtsman, and A. Szameit, “Topologically protected bound states in photonic parity-time-symmetric crystals,” Nat. Mater. 16(4), 433–438 (2017).
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Tan, W.

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M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. M. Taylor, “Imaging topological edge states in silicon photonics,” Nat. Photonics 7(12), 1001–1005 (2013).
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A. B. Khanikaev, S. H. Mousavi, W. K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12(3), 233–239 (2013).
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S. Weimann, M. Kremer, Y. Plotnik, Y. Lumer, S. Nolte, K. G. Makris, M. Segev, M. C. Rechtsman, and A. Szameit, “Topologically protected bound states in photonic parity-time-symmetric crystals,” Nat. Mater. 16(4), 433–438 (2017).
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Z. Guo, H. Jiang, Y. Long, K. Yu, J. Ren, C. Xue, and H. Chen, “Photonic spin Hall efect in waveguides composed of two types of single-negative metamaterials,” Sci. Rep. 7(1), 7742 (2017).
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Figures (8)
Fig. 1
Fig. 1 Experimental setup. The one dimensional dimer chain composed by equally spaced 32 identical SRRs (not shown all the SRRs) is arranged on a foam substrate, and sandwiched by two metallic plates in experiments (here the top metal plate is taken away in order to take a picture of the chain). The near-field probe made of a non-resonant loop is used to measure the density of states.

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Fig. 2
Fig. 2 (a) Sketch of a pair of arbitrarily rotated split rings. (b) Controlling coupling strengthen realized by tuning the relevant angle between two resonators. The strong and weak coupling strength used in our SSH chains are shown in the upper inset and down inset respectively.

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Fig. 3
Fig. 3 Two types of dimer chains differ in their topological properties. (a-b) Schematic of the unit cell, calculated Eigen frequencies (black dots), and measured DOS spectrum (blue profile) of the Type I and the Type II chains are given in (a) and (b), respectively. (c) Experimental (red triangles) and theoretical (grey dashed line) LDOS distribution of the edge state (f = 1.9GHz). (d) LDOS profiles of the ordinary state in pass band (f = 1.95GHz).

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Fig. 4
Fig. 4 Robust edge state against the losses perturbation. (a) The topological nontrivial chain with losses added into the central 20 SRRs (indicated by the grey background). (b) Measured DOS spectrum with loss. DOS of the edge state is much more robust than that of the bulk state. (c) Measured LDOS distribution of the edge state (f = 1.9GHz) with losses (red triangles), along with the theoretical calculations without losses (grey dashed line). (d) Measured LDOS distribution of the ordinary bulk state (f = 1.95GHz) with losses (red triangles), along with the theoretical calculations without losses (grey dashed line).

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Fig. 5
Fig. 5 Robust edge states against the disorder perturbation. (a) Schematic representation of the topological nontrivial chain with disorder (random coupling strengths by rotating the central 20 SRRs). Detail of rotation is shown in the inset of Fig. 4(b). (b, c) Measured LDOS distributions of the edge state and the bulk state at various disorder levels, including α = 1 ° (red circles), α = 3 ° (blue stars), and α = 5 ° (green triangles). As a comparison, the calculated LDOS distribution of the edge (f = 1.9GHz) and the bulk (f = 1.95GHz) states in the original chain are also presented (gray dashed line).

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Fig. 6
Fig. 6 Measured LDOS distributions of the edge states in the chain with both losses and disorder perturbations in the central 20 SRRs. As a comparison, the calculated LDOS distribution of the edge state (f = 1.9GHz) in the original chain are also presented (grey dashed line).

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Fig. 7
Fig. 7 (a) The schematic of a single SRR. (b) A pair of coupled SRRs in the planar configuration. The angle of rotation of two resonators are marked by φ 1 and φ 2 .

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Fig. 8
Fig. 8 The relationship between κ s l (l = 1, 2, 3, 4, and 5) and the rotation angle of the pair of SRRs.

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Equations (21)

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− ω n o r 2 ( a k b k ) = ( − 1 κ intra + κ inter e − i k d κ intra + κ inter e i k d − 1 ) ( a k b k ) ,
w = 1 2 π ∫ π / d − π / d θ k d k ,
M = μ 0 4 π ∬ I → m 1 ( r → 1 ) ⋅ I → m 2 ( r → 2 ) | r → 1 − r → 2 | d s 1 d s 2 ,
K − 1 = 1 4 π ε 0 ∬ ρ m 1 ( r → 1 ) ρ m 2 ( r → 2 ) | r → 1 − r → 2 | d τ 1 d τ 2 ,
ρ m ( θ ) = ( tan θ 2 ) / 2 ln ( g 4 r ) ,
θ < θ g
I m ( θ ) = ln ( cos θ g 2 cos θ 2 ) ln − 1 ( cos θ g 2 ) .
L = μ 0 r ( ln 8 r h + w − 1 2 ) ,
C = ε 0 [ ( w + g ) ( h + g ) g + 2 ( h + w ) π ln ( 4 r g ) ] .
κ H = 2 M L , κ E = − 2 C K .
l = ∑ n [ L 2 ( p ˙ n 2 + q ˙ n 2 ) − p n 2 + q n 2 2 C + M 2 p ˙ n q ˙ n − 1 − p n q n − 1 K 2 − p n q n K 1 + M 1 p ˙ n q ˙ n ] .
d d t ( ∂ l ∂ α ˙ n ) − ∂ l ∂ α n = 0 ,   ( α = p , q ) .
d d t ( L p ˙ n + M 2 q ˙ n − 1 + M 1 q ˙ n ) + 1 − i ω C p ˙ n + 1 − i ω K 2 q ˙ n − 1 + 1 − i ω K 1 q ˙ n = 0 d d t ( L q ˙ n + M 2 p ˙ n + 1 + M 1 p ˙ n ) + 1 − i ω C q ˙ n + 1 − i ω K 2 p ˙ n + 1 + 1 − i ω K 1 p ˙ n = 0 .
ω n o r 2 [ κ H 2 2 + κ H 1 2 + 2 cos ( k d ) κ H 2 κ H 1 − 1 ] ( a k b k ) = H ' ( a k b k ) .
− ω n o r 2 ( a k b k ) = ( − 1 κ 1 + κ 2 e − i k d κ 1 + κ 2 e i k d − 1 ) ( a k b k ) ,
ω n o r 2 ( a k b k ) = D k ( a k b k ) ,
| u k , ± 〉 = ( a k b k ) = ( κ 1 + κ 2 cos ( k d ) − i κ 2 sin ( k d ) ( κ 1 + κ 2 cos ( k d ) ) 2 + ( κ 2 sin ( k d ) ) 2 ∓ 1 ) ,
| u k , ± 〉 = ( e − i θ k ∓ 1 ) ,
| u k , ± 〉 = 1 2 ( e − i θ k ∓ 1 ) ,
w w i n d i n g = i π ∫ − π / d π / d ( a k * ∂ k a k + b k * ∂ k b k ) d k .
w = i π ∫ − π / d π / d ( a k * ∂ k a k + b k * ∂ k b k ) d k = 1 2 π ∫ − π / d π / d θ k d k ,

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