Abstract

We study the topological edge plasmon modes between two “diatomic” chains of identical plasmonic nanoparticles. Zak phase for longitudinal plasmon modes in each chain is calculated analytically by solutions of macroscopic Maxwell’s equations for particles in quasi-static dipole approximation. This approximation provides a direct analogy with the Su-Schrieffer-Heeger model such that the eigenvalue is mapped to the frequency dependent inverse-polarizability of the nanoparticles. The edge state frequency is found to be the same as the single-particle resonance frequency, which is insensitive to the separation distances within a unit cell. Finally, full electrodynamic simulations with realistic parameters suggest that the edge plasmon mode can be realized through near-field optical spectroscopy.

© 2015 Optical Society of America

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References

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  4. X. L. Qi, “Symmetry meets topology,” Science,  338(6114), 1550–1551 (2012).
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  6. L. Lu, J. D. Joannopoulos, and M. Soljacic, “Topological photonics,” Nat. Photon. 8, 821–829 (2014).
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  7. M. Xiao, Z. Q. Zhang, and C. T. Chan, “Surface impedance and bulk band geometric hases in one-dimensional systems,” Phys. Rev. X 4, 021017 (2014).
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  13. A. Poddubny, A. Miroshnichenko, A. Slobozhanyuk, and Y. Kivshar, “Topological Majorana states in zigzag chains of plasmonic nanoparticles,” ACS Photonics 1(2), 101–105 (2014).
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  23. K. H. Fung and C. T. Chan, “Analytical study of the plasmonic modes of a metal nanoparticle circular array,” Phys. Rev. B 77(20), 205423 (2008).
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    [Crossref]
  26. M. J. Puska, R. M. Nieminen, and M. Manninen, “Electronic polarizability of small metal spheres,” Phys. Rev. B 31(6), 3486 (1985).
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  27. C. W. Ling, M. J. Zheng, and K. W. Yu, “Slowing light in diatomic nanoshelled chains,” Opt. Commun. 283(9), 1945–1949 (2010).
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  31. M. B. Walker and J. Zak, “Geometrical phase for phonons,” Europhys. Lett. 26(7), 481 (1994).
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    [Crossref] [PubMed]

2014 (5)

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

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

V. Yannopapas, “Dirac points, topological edge modes and nonreciprocal transmission in One-dimensional metamaterial-based coupled-cavity arrays,” Int. J. Mod. Phys. B 28, 1441006 (2014).
[Crossref]

A. Poddubny, A. Miroshnichenko, A. Slobozhanyuk, and Y. Kivshar, “Topological Majorana states in zigzag chains of plasmonic nanoparticles,” ACS Photonics 1(2), 101–105 (2014).
[Crossref]

A. V. Poshakinskiy, A. N. Poddubny, L. Pilozzi, and E. L. Ivchenko, “Radiative topological states in resonant photonic crystals,” Phys. Rev. Lett. 112, 107403 (2014).
[Crossref] [PubMed]

2013 (3)

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

G. Weick, C. Woolacott, W. L. Barnes, O. Hess, and E. Mariani, “Dirac-like Plasmons in honeycomb lattices of metallic nanoparticles,” Phys. Rev. Lett. 110(10), 106801 (2013).
[Crossref] [PubMed]

S. Longhi, “Zak phase of photons in optical waveguide lattices,” Optics Letters 38(9), 3716 (2013).
[Crossref] [PubMed]

2012 (2)

X. L. Qi, “Symmetry meets topology,” Science,  338(6114), 1550–1551 (2012).
[Crossref] [PubMed]

V. Yannopapas, “Topological photonic bands in two-dimensional networks of metamaterial elements,” New J. Phys. 14, 113017 (2012).
[Crossref]

2011 (4)

B. Willingham and S. Link, “Energy transport in metal nanoparticle chains via sub-radiant plasmon modes,” Optics Express 19(7), 6450–6461 (2011)
[Crossref] [PubMed]

V. Yannopapas, “Gapless surface states in a lattice of coupled cavities: A photonic analog of topological crystalline insulators,” Phys. Rev. B 84, 195126 (2011).
[Crossref]

K. T. Chen and P. A. Lee, “Static electric field in one-dimensional insulators without boundaries,” Phys. Rev. B 84(11), 113111 (2011).
[Crossref]

P. Delplace, D. Ullmo, and G. Montambaux, “Zak phase and the existence of edge states in graphene,” Phys. Rev. B 84(19), 195452 (2011).
[Crossref]

2010 (2)

C. W. Ling, M. J. Zheng, and K. W. Yu, “Slowing light in diatomic nanoshelled chains,” Opt. Commun. 283(9), 1945–1949 (2010).
[Crossref]

D. Xiao, M. Chang, and Q. Niu, “Berry phase effects on electronic properties,” Rev. Mod. Phys. 82(3), 1959–2007 (2010).
[Crossref]

2008 (2)

Z. Wang, Y. D. Chong, J. D. Joannopoulos, and M. Soljacic, “Reflection-free one-way edge modes in a gyromagnetic photonic crystal,” Phy. Rev. Lett. 100(1), 013905 (2008).
[Crossref]

K. H. Fung and C. T. Chan, “Analytical study of the plasmonic modes of a metal nanoparticle circular array,” Phys. Rev. B 77(20), 205423 (2008).
[Crossref]

2007 (1)

K. H. Fung and C. T. Chan, “Plasmonic modes in periodic metal nanoparticle chains: a direct dynamic eigenmode analysis,” Optics Letters 32(8), 973–975 (2007).
[Crossref] [PubMed]

2006 (1)

A. Alu and N. Engheta, “Theory of linear chains of metamaterial/plasmonic particles as subdiffraction optical nanotransmission lines,” Phys. Rev. B 74(20), 205436 (2006).
[Crossref]

2004 (2)

W. H. Weber and G. W. Ford, “Propagation of optical excitations by dipolar interactions in metal nanoparticle chains,” Phys. Rev. B 70(12), 125429 (2004).
[Crossref]

R. Quidant, C. Girard, J. C. Weeber, and A. Dereux, “Tailoring the transmittance of integrated optical waveguides with short metallic nanoparticle chains,” Phys. Rev. B 69(8), 085407 (2004).
[Crossref]

2003 (2)

J. E. Avron, D. Osadchy, and R. Seiler, “A topological look at the quantum hall effect,” Phys. Today 56(8), 38–42 (2003).
[Crossref]

E. Prodan, C. Radloff, N. J. Halas, and P. Nordlander, “A hybridization model for the plasmon response of complex nanostructures,” Science 302, 419 (2003).
[Crossref] [PubMed]

2002 (1)

S. Ryu and Y. Hatsugai, “Topological origin of zero-energy edge states in particle-Hole Symmetric Systems,” Phys. Rev. Lett. 89(7), 077002 (2002).
[Crossref] [PubMed]

2000 (2)

M. L. Brongersma, J. W. Hartman, and H. A. Atwater, “Electromagnetic energy transfer and switching in nanoparticle chain arrays below the diffraction limit,” Phys. Rev. B 62, 16 (2000).
[Crossref]

R. Resta, “Manifestations of Berry’s phase in molecules and condensed matter,” J. P.: Condens. Matter 12(9), R107–R142 (2000).

1994 (1)

M. B. Walker and J. Zak, “Geometrical phase for phonons,” Europhys. Lett. 26(7), 481 (1994).
[Crossref]

1989 (1)

J. Zak, “Berrys phase for energy bands in solids,” Phys. Rev. Lett. 62(23), 2747–2750 (1989).
[Crossref] [PubMed]

1985 (2)

M. J. Puska, R. M. Nieminen, and M. Manninen, “Electronic polarizability of small metal spheres,” Phys. Rev. B 31(6), 3486 (1985).
[Crossref]

M. Kohmoto, “Topological invariant and the quantization of the hall conductance,” Ann. Phys. (New York) 160(2), 343–354 (1985).
[Crossref]

Alu, A.

A. Alu and N. Engheta, “Theory of linear chains of metamaterial/plasmonic particles as subdiffraction optical nanotransmission lines,” Phys. Rev. B 74(20), 205436 (2006).
[Crossref]

Atwater, H. A.

M. L. Brongersma, J. W. Hartman, and H. A. Atwater, “Electromagnetic energy transfer and switching in nanoparticle chain arrays below the diffraction limit,” Phys. Rev. B 62, 16 (2000).
[Crossref]

Avron, J. E.

J. E. Avron, D. Osadchy, and R. Seiler, “A topological look at the quantum hall effect,” Phys. Today 56(8), 38–42 (2003).
[Crossref]

Barnes, W. L.

G. Weick, C. Woolacott, W. L. Barnes, O. Hess, and E. Mariani, “Dirac-like Plasmons in honeycomb lattices of metallic nanoparticles,” Phys. Rev. Lett. 110(10), 106801 (2013).
[Crossref] [PubMed]

Brongersma, M. L.

M. L. Brongersma, J. W. Hartman, and H. A. Atwater, “Electromagnetic energy transfer and switching in nanoparticle chain arrays below the diffraction limit,” Phys. Rev. B 62, 16 (2000).
[Crossref]

Chan, C. T.

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

K. H. Fung and C. T. Chan, “Analytical study of the plasmonic modes of a metal nanoparticle circular array,” Phys. Rev. B 77(20), 205423 (2008).
[Crossref]

K. H. Fung and C. T. Chan, “Plasmonic modes in periodic metal nanoparticle chains: a direct dynamic eigenmode analysis,” Optics Letters 32(8), 973–975 (2007).
[Crossref] [PubMed]

Chang, M.

D. Xiao, M. Chang, and Q. Niu, “Berry phase effects on electronic properties,” Rev. Mod. Phys. 82(3), 1959–2007 (2010).
[Crossref]

Chen, K. T.

K. T. Chen and P. A. Lee, “Static electric field in one-dimensional insulators without boundaries,” Phys. Rev. B 84(11), 113111 (2011).
[Crossref]

Chong, Y. D.

Z. Wang, Y. D. Chong, J. D. Joannopoulos, and M. Soljacic, “Reflection-free one-way edge modes in a gyromagnetic photonic crystal,” Phy. Rev. Lett. 100(1), 013905 (2008).
[Crossref]

Davison, S. G.

S. G. Davison and M. Steslicka, Basics Theory of Surface States (Oxford, 1992), Chap. 3.

Delplace, P.

P. Delplace, D. Ullmo, and G. Montambaux, “Zak phase and the existence of edge states in graphene,” Phys. Rev. B 84(19), 195452 (2011).
[Crossref]

Dereux, A.

R. Quidant, C. Girard, J. C. Weeber, and A. Dereux, “Tailoring the transmittance of integrated optical waveguides with short metallic nanoparticle chains,” Phys. Rev. B 69(8), 085407 (2004).
[Crossref]

Engheta, N.

A. Alu and N. Engheta, “Theory of linear chains of metamaterial/plasmonic particles as subdiffraction optical nanotransmission lines,” Phys. Rev. B 74(20), 205436 (2006).
[Crossref]

Ford, G. W.

W. H. Weber and G. W. Ford, “Propagation of optical excitations by dipolar interactions in metal nanoparticle chains,” Phys. Rev. B 70(12), 125429 (2004).
[Crossref]

Fung, K. H.

K. H. Fung and C. T. Chan, “Analytical study of the plasmonic modes of a metal nanoparticle circular array,” Phys. Rev. B 77(20), 205423 (2008).
[Crossref]

K. H. Fung and C. T. Chan, “Plasmonic modes in periodic metal nanoparticle chains: a direct dynamic eigenmode analysis,” Optics Letters 32(8), 973–975 (2007).
[Crossref] [PubMed]

Girard, C.

R. Quidant, C. Girard, J. C. Weeber, and A. Dereux, “Tailoring the transmittance of integrated optical waveguides with short metallic nanoparticle chains,” Phys. Rev. B 69(8), 085407 (2004).
[Crossref]

Halas, N. J.

E. Prodan, C. Radloff, N. J. Halas, and P. Nordlander, “A hybridization model for the plasmon response of complex nanostructures,” Science 302, 419 (2003).
[Crossref] [PubMed]

Hartman, J. W.

M. L. Brongersma, J. W. Hartman, and H. A. Atwater, “Electromagnetic energy transfer and switching in nanoparticle chain arrays below the diffraction limit,” Phys. Rev. B 62, 16 (2000).
[Crossref]

Hatsugai, Y.

S. Ryu and Y. Hatsugai, “Topological origin of zero-energy edge states in particle-Hole Symmetric Systems,” Phys. Rev. Lett. 89(7), 077002 (2002).
[Crossref] [PubMed]

Hess, O.

G. Weick, C. Woolacott, W. L. Barnes, O. Hess, and E. Mariani, “Dirac-like Plasmons in honeycomb lattices of metallic nanoparticles,” Phys. Rev. Lett. 110(10), 106801 (2013).
[Crossref] [PubMed]

Ivchenko, E. L.

A. V. Poshakinskiy, A. N. Poddubny, L. Pilozzi, and E. L. Ivchenko, “Radiative topological states in resonant photonic crystals,” Phys. Rev. Lett. 112, 107403 (2014).
[Crossref] [PubMed]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (John Wiley and Sons, 1998) 3 Edition.

Joannopoulos, J. D.

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

Z. Wang, Y. D. Chong, J. D. Joannopoulos, and M. Soljacic, “Reflection-free one-way edge modes in a gyromagnetic photonic crystal,” Phy. Rev. Lett. 100(1), 013905 (2008).
[Crossref]

Kargarian, M.

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

Khanikaev, A. B.

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

Kivshar, Y.

A. Poddubny, A. Miroshnichenko, A. Slobozhanyuk, and Y. Kivshar, “Topological Majorana states in zigzag chains of plasmonic nanoparticles,” ACS Photonics 1(2), 101–105 (2014).
[Crossref]

Kohmoto, M.

M. Kohmoto, “Topological invariant and the quantization of the hall conductance,” Ann. Phys. (New York) 160(2), 343–354 (1985).
[Crossref]

Lee, P. A.

K. T. Chen and P. A. Lee, “Static electric field in one-dimensional insulators without boundaries,” Phys. Rev. B 84(11), 113111 (2011).
[Crossref]

Ling, C. W.

C. W. Ling, M. J. Zheng, and K. W. Yu, “Slowing light in diatomic nanoshelled chains,” Opt. Commun. 283(9), 1945–1949 (2010).
[Crossref]

Link, S.

B. Willingham and S. Link, “Energy transport in metal nanoparticle chains via sub-radiant plasmon modes,” Optics Express 19(7), 6450–6461 (2011)
[Crossref] [PubMed]

Longhi, S.

S. Longhi, “Zak phase of photons in optical waveguide lattices,” Optics Letters 38(9), 3716 (2013).
[Crossref] [PubMed]

Lu, L.

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

MacDonald, A. H.

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

Manninen, M.

M. J. Puska, R. M. Nieminen, and M. Manninen, “Electronic polarizability of small metal spheres,” Phys. Rev. B 31(6), 3486 (1985).
[Crossref]

Mariani, E.

G. Weick, C. Woolacott, W. L. Barnes, O. Hess, and E. Mariani, “Dirac-like Plasmons in honeycomb lattices of metallic nanoparticles,” Phys. Rev. Lett. 110(10), 106801 (2013).
[Crossref] [PubMed]

Miroshnichenko, A.

A. Poddubny, A. Miroshnichenko, A. Slobozhanyuk, and Y. Kivshar, “Topological Majorana states in zigzag chains of plasmonic nanoparticles,” ACS Photonics 1(2), 101–105 (2014).
[Crossref]

Montambaux, G.

P. Delplace, D. Ullmo, and G. Montambaux, “Zak phase and the existence of edge states in graphene,” Phys. Rev. B 84(19), 195452 (2011).
[Crossref]

Mousavi, S. H.

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

Nieminen, R. M.

M. J. Puska, R. M. Nieminen, and M. Manninen, “Electronic polarizability of small metal spheres,” Phys. Rev. B 31(6), 3486 (1985).
[Crossref]

Niu, Q.

D. Xiao, M. Chang, and Q. Niu, “Berry phase effects on electronic properties,” Rev. Mod. Phys. 82(3), 1959–2007 (2010).
[Crossref]

Nordlander, P.

E. Prodan, C. Radloff, N. J. Halas, and P. Nordlander, “A hybridization model for the plasmon response of complex nanostructures,” Science 302, 419 (2003).
[Crossref] [PubMed]

Osadchy, D.

J. E. Avron, D. Osadchy, and R. Seiler, “A topological look at the quantum hall effect,” Phys. Today 56(8), 38–42 (2003).
[Crossref]

Pilozzi, L.

A. V. Poshakinskiy, A. N. Poddubny, L. Pilozzi, and E. L. Ivchenko, “Radiative topological states in resonant photonic crystals,” Phys. Rev. Lett. 112, 107403 (2014).
[Crossref] [PubMed]

Poddubny, A.

A. Poddubny, A. Miroshnichenko, A. Slobozhanyuk, and Y. Kivshar, “Topological Majorana states in zigzag chains of plasmonic nanoparticles,” ACS Photonics 1(2), 101–105 (2014).
[Crossref]

Poddubny, A. N.

A. V. Poshakinskiy, A. N. Poddubny, L. Pilozzi, and E. L. Ivchenko, “Radiative topological states in resonant photonic crystals,” Phys. Rev. Lett. 112, 107403 (2014).
[Crossref] [PubMed]

Poshakinskiy, A. V.

A. V. Poshakinskiy, A. N. Poddubny, L. Pilozzi, and E. L. Ivchenko, “Radiative topological states in resonant photonic crystals,” Phys. Rev. Lett. 112, 107403 (2014).
[Crossref] [PubMed]

Prodan, E.

E. Prodan, C. Radloff, N. J. Halas, and P. Nordlander, “A hybridization model for the plasmon response of complex nanostructures,” Science 302, 419 (2003).
[Crossref] [PubMed]

Puska, M. J.

M. J. Puska, R. M. Nieminen, and M. Manninen, “Electronic polarizability of small metal spheres,” Phys. Rev. B 31(6), 3486 (1985).
[Crossref]

Qi, X. L.

X. L. Qi, “Symmetry meets topology,” Science,  338(6114), 1550–1551 (2012).
[Crossref] [PubMed]

Quidant, R.

R. Quidant, C. Girard, J. C. Weeber, and A. Dereux, “Tailoring the transmittance of integrated optical waveguides with short metallic nanoparticle chains,” Phys. Rev. B 69(8), 085407 (2004).
[Crossref]

Radloff, C.

E. Prodan, C. Radloff, N. J. Halas, and P. Nordlander, “A hybridization model for the plasmon response of complex nanostructures,” Science 302, 419 (2003).
[Crossref] [PubMed]

Resta, R.

R. Resta, “Manifestations of Berry’s phase in molecules and condensed matter,” J. P.: Condens. Matter 12(9), R107–R142 (2000).

Ryu, S.

S. Ryu and Y. Hatsugai, “Topological origin of zero-energy edge states in particle-Hole Symmetric Systems,” Phys. Rev. Lett. 89(7), 077002 (2002).
[Crossref] [PubMed]

Seiler, R.

J. E. Avron, D. Osadchy, and R. Seiler, “A topological look at the quantum hall effect,” Phys. Today 56(8), 38–42 (2003).
[Crossref]

Shvets, G.

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

Slobozhanyuk, A.

A. Poddubny, A. Miroshnichenko, A. Slobozhanyuk, and Y. Kivshar, “Topological Majorana states in zigzag chains of plasmonic nanoparticles,” ACS Photonics 1(2), 101–105 (2014).
[Crossref]

Soljacic, M.

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

Z. Wang, Y. D. Chong, J. D. Joannopoulos, and M. Soljacic, “Reflection-free one-way edge modes in a gyromagnetic photonic crystal,” Phy. Rev. Lett. 100(1), 013905 (2008).
[Crossref]

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R. Quidant, C. Girard, J. C. Weeber, and A. Dereux, “Tailoring the transmittance of integrated optical waveguides with short metallic nanoparticle chains,” Phys. Rev. B 69(8), 085407 (2004).
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G. Weick, C. Woolacott, W. L. Barnes, O. Hess, and E. Mariani, “Dirac-like Plasmons in honeycomb lattices of metallic nanoparticles,” Phys. Rev. Lett. 110(10), 106801 (2013).
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G. Weick, C. Woolacott, W. L. Barnes, O. Hess, and E. Mariani, “Dirac-like Plasmons in honeycomb lattices of metallic nanoparticles,” Phys. Rev. Lett. 110(10), 106801 (2013).
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C. W. Ling, M. J. Zheng, and K. W. Yu, “Slowing light in diatomic nanoshelled chains,” Opt. Commun. 283(9), 1945–1949 (2010).
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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(2), 101–105 (2014).
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M. B. Walker and J. Zak, “Geometrical phase for phonons,” Europhys. Lett. 26(7), 481 (1994).
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Int. J. Mod. Phys. B (1)

V. Yannopapas, “Dirac points, topological edge modes and nonreciprocal transmission in One-dimensional metamaterial-based coupled-cavity arrays,” Int. J. Mod. Phys. B 28, 1441006 (2014).
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Opt. Commun. (1)

C. W. Ling, M. J. Zheng, and K. W. Yu, “Slowing light in diatomic nanoshelled chains,” Opt. Commun. 283(9), 1945–1949 (2010).
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M. Xiao, Z. Q. Zhang, and C. T. Chan, “Surface impedance and bulk band geometric hases in one-dimensional systems,” Phys. Rev. X 4, 021017 (2014).

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S. G. Davison and M. Steslicka, Basics Theory of Surface States (Oxford, 1992), Chap. 3.

Lumerical Solutions, Inc., http://www.lumerical.com/tcad-products/fdtd/

J. D. Jackson, Classical Electrodynamics (John Wiley and Sons, 1998) 3 Edition.

Supplementary Material (4)

» Media 1: MP4 (3383 KB)     
» Media 2: MP4 (3513 KB)     
» Media 3: MP4 (3605 KB)     
» Media 4: MP4 (2834 KB)     

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Figures (6)

Fig. 1
Fig. 1 Band dispersion of the longitudinal dipolar modes in a “diatomic” chain of plasmonic nanoparticles. (a) Schematic of the chain aligned in the z-direction. There are two spherical metal nanoparticles in a unit cell, and are denoted by sphere A and B. The length of unit cell, the separation between spheres (within a unit cell), and the radii of the spheres are denoted by d, t, and a respectively. The chain is embedded in air. We use d0d/2 as the reference length. This figure is drawn in scale such that a = 0.33d/2, and t = 0.8d/2. (b) shows the longitudinal mode dispersion relation of the diatomic chain. There are two non-degenerated longitudinal bands (solid lines) as there are two atoms in a unit cell. The dispersion of a monatomic chain is also plotted (dashed line) for comparison. In that case, t = d/2, which means the spheres are equally separated.
Fig. 2
Fig. 2 Mapping between plasmon frequency ω and the eigenvalue α−1 for Drude model of different damping coefficient 1/τ (see Eq. 5). The mapping is nearly linear in the range of interested shown in Fig. 1 (b) (i.e., 0.64ωp > ω > 0.50ωp). The real part decreases when one increases the damping coefficient.
Fig. 3
Fig. 3 Representation of (dt)3 + t3exp(−ikd) in complex plane. The complex number evolves as kd changes from −π to π for (a) t < d/2 and (b) t > d/2. In (b), winding number is non-zero, which leads to non-zero Zak phase.
Fig. 4
Fig. 4 Edge states between diatomic chains. (a) shows the geometry of a connected chain system. Separation t = tL for the left chain and t = tR for the right chain. The interface is in between n = −1 and n = 0. (b) shows the plasmon frequency obtained by solving eigenvalue problem of a connected chain with 123 spheres. tL, tR, and a are set to 1.2d/2, 0.8d/2, and 0.33d/2. (a) is drawn in scale with these parameters. (c) and (e) show the eigenstate of the connected chain just on top and below the band gap. (d) shows the edge state at the interface.
Fig. 5
Fig. 5 Dependence of bands and edge state on tL. (a) tR is set equal to dtL. For example when tL = 1.1d/2, tR = 0.9d/2. (b) tR is fixed and equals 0.9d/2. The frequency of edge state does not vary with tL in both cases.
Fig. 6
Fig. 6 Photon emission rates in plasmonic nanoparticle chains calculated by finite-difference time-domain simulations. A dipole source is placed inside a particular sphere with a small hole of negligible size, acting as an emitter. The emission rate is defined by Eq. (16), which represents approximately the local density of states (LDOS). (a) Emission rate in single sphere, diatomic chain, and connected chain. The last one reveals the existence of an edge state. Videos of corresponding time-domain fields are attached, in which longitudinal component of electric fields are shown. Media 1: connected chain; Media 2: diatomic chain; Media 3: monatomic chain; Media 4: single sphere. (b) Emission rate for connected case with different tL. d/2 = 75 nm and tR = dtL.

Tables (1)

Tables Icon

Table 1 Zak phase γ of diatomic chains. The upper and the lower bands share the same value of Zak phase.

Equations (17)

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α 1 p n = m n G n m p m ,
α 1 p n = { 2 4 π ε 0 [ p n 1 ( d t ) 3 + p n + 1 t 3 ] , for n is even 2 4 π ε 0 [ p n 1 t 3 + p n + 1 ( d t ) 3 ] , for n is odd ,
p n ( k ) = { p A ( k ) e i k n 2 d , for n is even p B ( k ) e i k ( n 1 2 d ) , for n is odd ,
( 0 a 12 ( k ) a 21 ( k ) 0 ) ( p A p B ) = α 1 ( p A p B ) ,
α 1 ( ω ) = 1 3 ω 2 ω p 2 i ( 1 τ ) ( ω ω p 2 ) V ε 0 .
α 1 ( ω ) = ± a 12 ( k ) a 21 ( k ) .
Re ( ω ) = ω p 2 3 1 4 τ 2 V ε 0 ω p 2 a 12 ( k ) a 21 ( k ) , Im ( ω ) = 1 2 τ ,
( p A ( k ) p B ( k ) ) = 1 2 ( ± e i ϕ ( k ) 1 ) ,
γ = i π d π d ( p A * p A k + p B * p B k ) d k = ϕ ( π d ) ϕ ( π d ) 2
a 12 , 21 R = 2 4 π ε 0 ( 1 t R 3 1 ( d t R ) 3 e ± μ R d ) a 12 , 21 L = 2 4 π ε 0 ( 1 t L 3 1 ( d t L ) 3 e μ L d )
( α 1 ) 2 = a 12 R a 21 R = a 12 L a 21 L .
( p A R p B R ) = ( 0 1 ) and ( p A L p B L ) = ( 0 1 ) .
p n = { C p n R , for n 0 ( right region ) D p n L , for n < 0 ( left region ) ,
{ 4 π ε 0 α 1 p 1 = 2 ( d t 0 ) 3 p 0 + 2 t L 3 p 2 4 π ε 0 α 1 p 0 = 2 ( d t 0 ) 3 p 1 + 2 t R 3 p 1 ,
p n = { 0 , for n is even ( 1 ) n + 1 2 e n + 1 2 μ L d , for n is odd and n < 0 ( 1 ) n + 1 2 t R 3 ( d ( t L + t R ) / 2 ) 3 × e n 1 2 μ R d , for n is odd and n > 0
( p A R p B R ) = ( a 12 R / α 1 1 ) and ( p A L p B L ) = ( a 12 L / α 1 1 ) .
emission rate ( ω ) = power flowing out ( ω ) source power ( ω ) ,

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