Abstract

We propose using a topological plasmonic crystal structure composed of an array of nearly parallel nanowires with unequal spacing for manipulating light. In the paraxial approximation, the Helmholtz equation that describes the propagation of light along the nanowires maps onto the Schrödinger equation of the Su-Schrieffer-Heeger (SSH) model. Using a full three-dimensional finite difference time domain solution of the Maxwell equations, we verify the existence of topological defect modes, with sub-wavelength localization, bound to domain walls of the plasmonic crystal. We show that by manipulating domain walls we can construct spatial mode filters that couple bulk modes to topological defect modes, and topological beam-splitters that couple two topological defect modes. Finally, we show that the structures are tolerant to fabrication errors with an inverse length-scale smaller than the topological band gap.

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

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References

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    [Crossref]
  28. W. P. Su, J. R. Schrieffer, and A. J. Heeger, “Solitons in polyacetylene,” Phys. Rev. Lett. 42, 1698–1701 (1979).
    [Crossref]
  29. F. Bleckmann, Z. Cherpakova, S. Linden, and A. Alberti, “Spectral imaging of topological edge states in plasmonic waveguide arrays,” Phys. Rev. B 96, 045417 (2017).
    [Crossref]
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    [Crossref] [PubMed]
  31. W. Huang, C. Xu, S. T. Chu, and S. K. Chaudhuri, “The finite-difference vector beam propagation method: analysis and assessment,” J. Lightwave Technol. 10, 295–305 (1992).
    [Crossref]
  32. W. P. Huang and C. L. Xu, “Simulation of three-dimensional optical waveguides by a full-vector beam propagation method,” IEEE J. Quantum Electron. 29, 2639–2649 (1993).
    [Crossref]
  33. M. V. Berry, “Quantal phase factors accompanying adiabatic changes,” Proc. R. Soc. Lond. A 392, 45–57 (1984).
    [Crossref]
  34. J. Zak, “Berry’s phase for energy bands in solids,” Phys. Rev. Lett. 62, 2747–2750 (1989).
    [Crossref] [PubMed]
  35. M. F. Atiyah and I. M. Singer, “The index of elliptic operators: I,” Annals Math. 87, 484–530 (1968).
    [Crossref]
  36. X. Yi, “Note on the electron-hole symmetry breaking of the energy band structure of polyacetylene,” Commun. Theoretical Phys. 14, 209 (1990).
    [Crossref]
  37. Z. J. Li, Z. An, K. L. Yao, and Z. G. Li, “Effects of the second and third neighbor hopping interactions on the electronic states of soliton in polyacetylene,” Zeitschrift für Physik B Condensed Matter 92, 195–197 (1993).
    [Crossref]
  38. L. Li, Z. Xu, and S. Chen, “Topological phases of generalized su-schrieffer-heeger models,” Phys. Rev. B 89, 085111 (2014).
    [Crossref]
  39. B. Narayan, “End modes in arrays of modulated su–schrieffer–heeger chains,” Pramana 87, 25 (2016).
    [Crossref]

2017 (4)

S. Longhi, “Non-hermitian bidirectional robust transport,” Phys. Rev. B 95, 014201 (2017).
[Crossref]

W. Zhang, X. Chen, and F. Ye, “Plasmonic topological insulators for topological nanophotonics,” Opt. Lett. 42, 4063–4066 (2017).
[Crossref] [PubMed]

F. Bleckmann, Z. Cherpakova, S. Linden, and A. Alberti, “Spectral imaging of topological edge states in plasmonic waveguide arrays,” Phys. Rev. B 96, 045417 (2017).
[Crossref]

S. Ke, B. Wang, H. Long, K. Wang, and P. Lu, “Topological edge modes in non-hermitian plasmonic waveguide arrays,” Opt. Express 25, 11132–11143 (2017).
[Crossref] [PubMed]

2016 (3)

L. Lu, C. Fang, L. Fu, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Symmetry-protected topological photonic crystal in three dimensions,” Nat. Phys. 12, 337–340 (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, 163901 (2016).
[Crossref] [PubMed]

B. Narayan, “End modes in arrays of modulated su–schrieffer–heeger chains,” Pramana 87, 25 (2016).
[Crossref]

2015 (2)

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]

I. S. Sinev, I. S. Mukhin, A. P. Slobozhanyuk, A. N. Poddubny, A. E. Miroshnichenko, A. K. Samusev, and Y. S. Kivshar, “Mapping plasmonic topological states at the nanoscale,” Nanoscale 7, 11904–11908 (2015).
[Crossref] [PubMed]

2014 (4)

A. R. Mellnik, J. S. Lee, A. Richardella, J. L. Grab, P. J. Mintun, M. H. Fischer, A. Vaezi, A. Manchon, E.-A. Kim, N. Samarth, and D. C. Ralph, “Spin-transfer torque generated by a topological insulator,” Nature 511, 449–451 (2014). Letter.
[Crossref] [PubMed]

S. A. Skirlo, L. Lu, and M. Soljačić, “Multimode one-way waveguides of large chern numbers,” Phys. Rev. Lett. 113, 113904 (2014).
[Crossref] [PubMed]

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]

L. Li, Z. Xu, and S. Chen, “Topological phases of generalized su-schrieffer-heeger models,” Phys. Rev. B 89, 085111 (2014).
[Crossref]

2013 (4)

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

L. Lu, L. Fu, J. D. Joannopoulos, and M. Soljacic, “Weyl points and line nodes in gyroid photonic crystals,” Nat. Photonics 7, 294–299 (2013).
[Crossref]

M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. M. Taylor, “Imaging topological edge states in silicon photonics,” Nat. Photonics 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]

2012 (1)

K. Nakayama, K. Eto, Y. Tanaka, T. Sato, S. Souma, T. Takahashi, K. Segawa, and Y. Ando, “Manipulation of topological states and the bulk band gap using natural heterostructures of a topological insulator,” Phys. Rev. Lett. 109, 236804 (2012).
[Crossref]

2011 (1)

F. Xiu, L. He, Y. Wang, L. Cheng, L.-T. Chang, M. Lang, G. Huang, X. Kou, Y. Zhou, X. Jiang, Z. Chen, J. Zou, A. Shailos, and K. L. Wang, “Manipulating surface states in topological insulator nanoribbons,” Nat. Nano. 6, 216–221 (2011).
[Crossref]

2010 (1)

M. Z. Hasan and C. L. Kane, “Colloquium : Topological insulators,” Rev. Mod. Phys. 82, 3045–3067 (2010).
[Crossref]

2009 (2)

A. Kitaev, “Periodic table for topological insulators and superconductors,” AIP Conference Proc. 1134, 22–30 (2009).
[Crossref]

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

2008 (3)

C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma, “Non-abelian anyons and topological quantum computation,” Rev. Mod. Phys. 80, 1083–1159 (2008).
[Crossref]

A. P. Schnyder, S. Ryu, A. Furusaki, and A. W. W. Ludwig, “Classification of topological insulators and superconductors in three spatial dimensions,” Phys. Rev. B 78, 195125 (2008).
[Crossref]

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

2007 (2)

L. Fu and C. L. Kane, “Topological insulators with inversion symmetry,” Phys. Rev. B 76, 045302 (2007).
[Crossref]

M. König, S. Wiedmann, C. Brüne, A. Roth, H. Buhmann, L. W. Molenkamp, X.-L. Qi, and S.-C. Zhang, “Quantum spin hall insulator state in hgte quantum wells,” Science 318, 766–770 (2007).
[Crossref] [PubMed]

2006 (1)

B. A. Bernevig and S.-C. Zhang, “Quantum spin hall effect,” Phys. Rev. Lett. 96, 106802 (2006).
[Crossref] [PubMed]

2005 (1)

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

1993 (2)

W. P. Huang and C. L. Xu, “Simulation of three-dimensional optical waveguides by a full-vector beam propagation method,” IEEE J. Quantum Electron. 29, 2639–2649 (1993).
[Crossref]

Z. J. Li, Z. An, K. L. Yao, and Z. G. Li, “Effects of the second and third neighbor hopping interactions on the electronic states of soliton in polyacetylene,” Zeitschrift für Physik B Condensed Matter 92, 195–197 (1993).
[Crossref]

1992 (1)

W. Huang, C. Xu, S. T. Chu, and S. K. Chaudhuri, “The finite-difference vector beam propagation method: analysis and assessment,” J. Lightwave Technol. 10, 295–305 (1992).
[Crossref]

1990 (1)

X. Yi, “Note on the electron-hole symmetry breaking of the energy band structure of polyacetylene,” Commun. Theoretical Phys. 14, 209 (1990).
[Crossref]

1989 (1)

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

1988 (1)

A. J. Heeger, S. Kivelson, J. R. Schrieffer, and W. P. Su, “Solitons in conducting polymers,” Rev. Mod. Phys. 60, 781–850 (1988).
[Crossref]

1984 (1)

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

1979 (1)

W. P. Su, J. R. Schrieffer, and A. J. Heeger, “Solitons in polyacetylene,” Phys. Rev. Lett. 42, 1698–1701 (1979).
[Crossref]

1968 (1)

M. F. Atiyah and I. M. Singer, “The index of elliptic operators: I,” Annals Math. 87, 484–530 (1968).
[Crossref]

Alberti, A.

F. Bleckmann, Z. Cherpakova, S. Linden, and A. Alberti, “Spectral imaging of topological edge states in plasmonic waveguide arrays,” Phys. Rev. B 96, 045417 (2017).
[Crossref]

An, Z.

Z. J. Li, Z. An, K. L. Yao, and Z. G. Li, “Effects of the second and third neighbor hopping interactions on the electronic states of soliton in polyacetylene,” Zeitschrift für Physik B Condensed Matter 92, 195–197 (1993).
[Crossref]

Ando, Y.

K. Nakayama, K. Eto, Y. Tanaka, T. Sato, S. Souma, T. Takahashi, K. Segawa, and Y. Ando, “Manipulation of topological states and the bulk band gap using natural heterostructures of a topological insulator,” Phys. Rev. Lett. 109, 236804 (2012).
[Crossref]

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, 163901 (2016).
[Crossref] [PubMed]

Atiyah, M. F.

M. F. Atiyah and I. M. Singer, “The index of elliptic operators: I,” Annals Math. 87, 484–530 (1968).
[Crossref]

Bernevig, B. A.

B. A. Bernevig and S.-C. Zhang, “Quantum spin hall effect,” Phys. Rev. Lett. 96, 106802 (2006).
[Crossref] [PubMed]

Berry, M. V.

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

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, 163901 (2016).
[Crossref] [PubMed]

Bleckmann, F.

F. Bleckmann, Z. Cherpakova, S. Linden, and A. Alberti, “Spectral imaging of topological edge states in plasmonic waveguide arrays,” Phys. Rev. B 96, 045417 (2017).
[Crossref]

Brüne, C.

M. König, S. Wiedmann, C. Brüne, A. Roth, H. Buhmann, L. W. Molenkamp, X.-L. Qi, and S.-C. Zhang, “Quantum spin hall insulator state in hgte quantum wells,” Science 318, 766–770 (2007).
[Crossref] [PubMed]

Buhmann, H.

M. König, S. Wiedmann, C. Brüne, A. Roth, H. Buhmann, L. W. Molenkamp, X.-L. Qi, and S.-C. Zhang, “Quantum spin hall insulator state in hgte quantum wells,” Science 318, 766–770 (2007).
[Crossref] [PubMed]

Chang, L.-T.

F. Xiu, L. He, Y. Wang, L. Cheng, L.-T. Chang, M. Lang, G. Huang, X. Kou, Y. Zhou, X. Jiang, Z. Chen, J. Zou, A. Shailos, and K. L. Wang, “Manipulating surface states in topological insulator nanoribbons,” Nat. Nano. 6, 216–221 (2011).
[Crossref]

Chaudhuri, S. K.

W. Huang, C. Xu, S. T. Chu, and S. K. Chaudhuri, “The finite-difference vector beam propagation method: analysis and assessment,” J. Lightwave Technol. 10, 295–305 (1992).
[Crossref]

Chen, S.

L. Li, Z. Xu, and S. Chen, “Topological phases of generalized su-schrieffer-heeger models,” Phys. Rev. B 89, 085111 (2014).
[Crossref]

Chen, X.

Chen, Z.

F. Xiu, L. He, Y. Wang, L. Cheng, L.-T. Chang, M. Lang, G. Huang, X. Kou, Y. Zhou, X. Jiang, Z. Chen, J. Zou, A. Shailos, and K. L. Wang, “Manipulating surface states in topological insulator nanoribbons,” Nat. Nano. 6, 216–221 (2011).
[Crossref]

Cheng, L.

F. Xiu, L. He, Y. Wang, L. Cheng, L.-T. Chang, M. Lang, G. Huang, X. Kou, Y. Zhou, X. Jiang, Z. Chen, J. Zou, A. Shailos, and K. L. Wang, “Manipulating surface states in topological insulator nanoribbons,” Nat. Nano. 6, 216–221 (2011).
[Crossref]

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]

Cherpakova, Z.

F. Bleckmann, Z. Cherpakova, S. Linden, and A. Alberti, “Spectral imaging of topological edge states in plasmonic waveguide arrays,” Phys. Rev. B 96, 045417 (2017).
[Crossref]

Chong, Y.

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

Chu, S. T.

W. Huang, C. Xu, S. T. Chu, and S. K. Chaudhuri, “The finite-difference vector beam propagation method: analysis and assessment,” J. Lightwave Technol. 10, 295–305 (1992).
[Crossref]

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, 163901 (2016).
[Crossref] [PubMed]

Das Sarma, S.

C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma, “Non-abelian anyons and topological quantum computation,” Rev. Mod. Phys. 80, 1083–1159 (2008).
[Crossref]

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]

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, 163901 (2016).
[Crossref] [PubMed]

Eto, K.

K. Nakayama, K. Eto, Y. Tanaka, T. Sato, S. Souma, T. Takahashi, K. Segawa, and Y. Ando, “Manipulation of topological states and the bulk band gap using natural heterostructures of a topological insulator,” Phys. Rev. Lett. 109, 236804 (2012).
[Crossref]

Fan, J.

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

Fang, C.

L. Lu, C. Fang, L. Fu, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Symmetry-protected topological photonic crystal in three dimensions,” Nat. Phys. 12, 337–340 (2016).
[Crossref]

Fischer, M. H.

A. R. Mellnik, J. S. Lee, A. Richardella, J. L. Grab, P. J. Mintun, M. H. Fischer, A. Vaezi, A. Manchon, E.-A. Kim, N. Samarth, and D. C. Ralph, “Spin-transfer torque generated by a topological insulator,” Nature 511, 449–451 (2014). Letter.
[Crossref] [PubMed]

Freedman, M.

C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma, “Non-abelian anyons and topological quantum computation,” Rev. Mod. Phys. 80, 1083–1159 (2008).
[Crossref]

Fu, L.

L. Lu, C. Fang, L. Fu, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Symmetry-protected topological photonic crystal in three dimensions,” Nat. Phys. 12, 337–340 (2016).
[Crossref]

L. Lu, L. Fu, J. D. Joannopoulos, and M. Soljacic, “Weyl points and line nodes in gyroid photonic crystals,” Nat. Photonics 7, 294–299 (2013).
[Crossref]

L. Fu and C. L. Kane, “Topological insulators with inversion symmetry,” Phys. Rev. B 76, 045302 (2007).
[Crossref]

Furusaki, A.

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A. R. Mellnik, J. S. Lee, A. Richardella, J. L. Grab, P. J. Mintun, M. H. Fischer, A. Vaezi, A. Manchon, E.-A. Kim, N. Samarth, and D. C. Ralph, “Spin-transfer torque generated by a topological insulator,” Nature 511, 449–451 (2014). Letter.
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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, 1001–1005 (2013).
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A. R. Mellnik, J. S. Lee, A. Richardella, J. L. Grab, P. J. Mintun, M. H. Fischer, A. Vaezi, A. Manchon, E.-A. Kim, N. Samarth, and D. C. Ralph, “Spin-transfer torque generated by a topological insulator,” Nature 511, 449–451 (2014). Letter.
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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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Poddubny, A. N.

I. S. Sinev, I. S. Mukhin, A. P. Slobozhanyuk, A. N. Poddubny, A. E. Miroshnichenko, A. K. Samusev, and Y. S. Kivshar, “Mapping plasmonic topological states at the nanoscale,” Nanoscale 7, 11904–11908 (2015).
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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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A. R. Mellnik, J. S. Lee, A. Richardella, J. L. Grab, P. J. Mintun, M. H. Fischer, A. Vaezi, A. Manchon, E.-A. Kim, N. Samarth, and D. C. Ralph, “Spin-transfer torque generated by a topological insulator,” Nature 511, 449–451 (2014). Letter.
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A. R. Mellnik, J. S. Lee, A. Richardella, J. L. Grab, P. J. Mintun, M. H. Fischer, A. Vaezi, A. Manchon, E.-A. Kim, N. Samarth, and D. C. Ralph, “Spin-transfer torque generated by a topological insulator,” Nature 511, 449–451 (2014). Letter.
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M. König, S. Wiedmann, C. Brüne, A. Roth, H. Buhmann, L. W. Molenkamp, X.-L. Qi, and S.-C. Zhang, “Quantum spin hall insulator state in hgte quantum wells,” Science 318, 766–770 (2007).
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A. P. Schnyder, S. Ryu, A. Furusaki, and A. W. W. Ludwig, “Classification of topological insulators and superconductors in three spatial dimensions,” Phys. Rev. B 78, 195125 (2008).
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A. R. Mellnik, J. S. Lee, A. Richardella, J. L. Grab, P. J. Mintun, M. H. Fischer, A. Vaezi, A. Manchon, E.-A. Kim, N. Samarth, and D. C. Ralph, “Spin-transfer torque generated by a topological insulator,” Nature 511, 449–451 (2014). Letter.
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I. S. Sinev, I. S. Mukhin, A. P. Slobozhanyuk, A. N. Poddubny, A. E. Miroshnichenko, A. K. Samusev, and Y. S. Kivshar, “Mapping plasmonic topological states at the nanoscale,” Nanoscale 7, 11904–11908 (2015).
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K. Nakayama, K. Eto, Y. Tanaka, T. Sato, S. Souma, T. Takahashi, K. Segawa, and Y. Ando, “Manipulation of topological states and the bulk band gap using natural heterostructures of a topological insulator,” Phys. Rev. Lett. 109, 236804 (2012).
[Crossref]

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A. P. Schnyder, S. Ryu, A. Furusaki, and A. W. W. Ludwig, “Classification of topological insulators and superconductors in three spatial dimensions,” Phys. Rev. B 78, 195125 (2008).
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A. J. Heeger, S. Kivelson, J. R. Schrieffer, and W. P. Su, “Solitons in conducting polymers,” Rev. Mod. Phys. 60, 781–850 (1988).
[Crossref]

W. P. Su, J. R. Schrieffer, and A. J. Heeger, “Solitons in polyacetylene,” Phys. Rev. Lett. 42, 1698–1701 (1979).
[Crossref]

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K. Nakayama, K. Eto, Y. Tanaka, T. Sato, S. Souma, T. Takahashi, K. Segawa, and Y. Ando, “Manipulation of topological states and the bulk band gap using natural heterostructures of a topological insulator,” Phys. Rev. Lett. 109, 236804 (2012).
[Crossref]

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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, 163901 (2016).
[Crossref] [PubMed]

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]

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F. Xiu, L. He, Y. Wang, L. Cheng, L.-T. Chang, M. Lang, G. Huang, X. Kou, Y. Zhou, X. Jiang, Z. Chen, J. Zou, A. Shailos, and K. L. Wang, “Manipulating surface states in topological insulator nanoribbons,” Nat. Nano. 6, 216–221 (2011).
[Crossref]

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A. B. Khanikaev, S. Hossein Mousavi, W.-K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12, 233–239 (2013).
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C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma, “Non-abelian anyons and topological quantum computation,” Rev. Mod. Phys. 80, 1083–1159 (2008).
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S. A. Skirlo, L. Lu, and M. Soljačić, “Multimode one-way waveguides of large chern numbers,” Phys. Rev. Lett. 113, 113904 (2014).
[Crossref] [PubMed]

Slobozhanyuk, A.

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]

Slobozhanyuk, A. P.

I. S. Sinev, I. S. Mukhin, A. P. Slobozhanyuk, A. N. Poddubny, A. E. Miroshnichenko, A. K. Samusev, and Y. S. Kivshar, “Mapping plasmonic topological states at the nanoscale,” Nanoscale 7, 11904–11908 (2015).
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Soljacic, M.

L. Lu, C. Fang, L. Fu, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Symmetry-protected topological photonic crystal in three dimensions,” Nat. Phys. 12, 337–340 (2016).
[Crossref]

S. A. Skirlo, L. Lu, and M. Soljačić, “Multimode one-way waveguides of large chern numbers,” Phys. Rev. Lett. 113, 113904 (2014).
[Crossref] [PubMed]

L. Lu, L. Fu, J. D. Joannopoulos, and M. Soljacic, “Weyl points and line nodes in gyroid photonic crystals,” Nat. Photonics 7, 294–299 (2013).
[Crossref]

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

Souma, S.

K. Nakayama, K. Eto, Y. Tanaka, T. Sato, S. Souma, T. Takahashi, K. Segawa, and Y. Ando, “Manipulation of topological states and the bulk band gap using natural heterostructures of a topological insulator,” Phys. Rev. Lett. 109, 236804 (2012).
[Crossref]

Stern, A.

C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma, “Non-abelian anyons and topological quantum computation,” Rev. Mod. Phys. 80, 1083–1159 (2008).
[Crossref]

Su, W. P.

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Opt. Express (1)

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Figures (10)

Fig. 1
Fig. 1 (a) Single-double bond pattern in a polyacetylene chain with a kink defect. The red disks show the carbon atoms. (b) Analogous nanowire array with staggered spacing and a kink defect. (c) Staggered nanowire spacing leads to a band gap for the bulk modes (red), the topological defect mode appears in the middle of the band gap (blue). (d,f) The positions of nanowires in the array without (d) and with (f) a domain wall. (e,g) Spreading of light in the nanowire array without (e) and with (g) a domain wall. The topological defect mode bound to the domain wall is observed guiding light in (g). For this figure we used = −45.83 + 2.49 × 10−9i (the real part of the permittivity is identical to that of silver and the imaginary part was chosen to be very small to highlight the guidance by the topological defect mode.) Geometry parameters used: diameter 200 nm, center-to center spacing 300 and 500 nm, wavelength λ = 1 μm.
Fig. 2
Fig. 2 (a,d) Nanowire array geometry for a spatial mode filter (a) and a beam splitter (d). Blue lines indicate the positions of the nanowires and red dashed lines the position of the kinks. (b,e) Spectral flow of bulk modes (blue) and the topological defect modes (red) as a function of z for the spatial mode filter (b) and the beam splitter (e). (c,f) Light propagation through the spatial mode filter (c) and the beam splitter (f), red dashed lines indicate the position of the kinks (see text). Parameters used are same as Fig. 1.
Fig. 3
Fig. 3 (a,b) Topological defect mode propagation in a nanowire array with a perturbation. The red nanowire is shifted away from its original position indicated by the black dashed line. The smoothness parameter is w = 25 μm in (a) and w = 6.25 μm in (b). (see text for details) (c) Fraction of flux retained in the topological defect mode η in a structure with a shifted nanowire perturbation. We vary the smoothness parameter w and fix the displacement parameter δx = 150 nm. (d) η in a structure with a nanowire diameter perturbation, the nanowire diameter parameter δd is varied and w = 25 μm is fixed. Materials and nanowire geometry parameters are same as Fig. 1.
Fig. 4
Fig. 4 (a) The splitting of the symmetric and antisymmetric plasmon modes (for λ = 1 μm in vacuum) in a system of two parallel silver nanowires with 100 nm radius, as a function of the nanowire center-to-center separation s. The black dashed line indicates the single nanowire eigenvalue β1. (b) Extracted tight-binding parameter t as a function of spacing between two nanowires. The tight-binding model starts to break down when the spacing between the two nanowire goes below 0.5 μm (indicated by the dashed red line).
Fig. 5
Fig. 5 Mode rejection by the mode filtering nanowire array. On the input (left) side of the nanowire array, a bulk mode that is orthogonal to the two select modes, is injected into the nanowire array. On the output (right) side there is essentially no light in the vicinity of the two topological defect modes.
Fig. 6
Fig. 6 Light propagation along structures with a perturbation on the position of one of the nanowires. (a) Light flux (top panel) and the spectral flow (bottom panel) in a structure with no perturbation. (b)–(f) Light flux (top panel), meander M(z) (middle panel), and spectral flow (bottom panel) in structures with one of the nanowires displaced from its original position (dashed line in top panel) to a new position (red line in top panel). The nanowire is displaced by δx = 150 nm over a width of w = 25.00 μm (b), 18.75 μm (c), 12.50 μm (d), 10.00 μm (e), 6.25 μm (f).
Fig. 7
Fig. 7 Same as Fig. 6, except we perturb the diameter D(z) of the red nanowire. We fix the perturbation width at w = 25μm, and shrink the nanowire diameter by δD = 25.0 nm (a), δD = 50.0 nm (b), δD = 62.5 nm (c), δD = 75.0 nm (d), δD = 87.5 nm (e), δD = 100 nm (f).
Fig. 8
Fig. 8 The Figure of Merit (ΔβLd/(2π)) as a function of the free space wavelength (λ0) for a silver nanowire array with nanowire diameter 400 nm, 800 nm and 1200 nm. The major and minor surface-to-surface spacing is 350 nm and 50 nm respectively. The dashed lines show the topological defect with minor spacing around the defect while the solid lines show the topological defect with major spacing, as indicated in the insets.
Fig. 9
Fig. 9 The electric field z component (Ez) of the topological defect mode around two kinds of defects, (a) a topological defect with major spacing, and (b) a defect with minor spacing. The Ez field is calculated based on a 400 nm diameter silver nanowire array with a staggered surface-to-surface spacing 50 nm and 350 nm. The dashed circles show the position of the nanowires.
Fig. 10
Fig. 10 The schematic nanowire array geometry at the beginning of an array with minor-spacing defect (a), and major-spacing defect (b). The beginning of the nanowire array is staggered to match the phase different between the nearby nanowires in topological defect mode. The length difference between the nanowire is determined by the exaction method and plasmonic mode wavelength in the nanowire array.

Equations (8)

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[ c 2 β 0 2 + V ] ( ψ x ψ y ) = i c z ( ψ x ψ y ) ,
H SSH = i t i , i + 1 ( c i c i + 1 + c i + 1 c i )
H SSH = k ( t 1 + t 2 e i k R ) a k b k + h . c .
P z ( x , y = 0 , z ) = S ( x , y = 0 , z ) z ^ d x d y S ( x , y , z ) z ^ ,
η = 1 2 d x d y ( E α + H + E + H α ) z ^ ,
( 2 + ω 2 c 2 ) { E B } = { ( 1 E ) 1 × ( × B ) }
V ( x , y , z ) = ω 2 ( 1 ) + c ( x ( x ) x ( y ) y ( x ) y ( y ) ) ,
H SSH = i t i , i + 1 ( c i c i + 1 + c i + 1 c i ) + v i c i c i .

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