Abstract

We introduce a new class of plasmonic crystals possessing graphene-like internal symmetries and Dirac-type spectrum in k-space. We study dynamics of surface plasmon polaritons supported in the plasmonic crystals by employing the formalism of Dirac dynamics for relativistic quantum particles. Through an analogy with graphene, we introduce a concept of pseudo-spin and chirality to indicate built-in symmetry of the plasmonic crystals near Dirac point. The surface plasmon polaritons with different pseudo-spin states are shown to split in the crystals into two beams, analogous to spin Hall effect.

© 2010 OSA

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  1. A. K. Geim, “Graphene: status and prospects,” Science 324(5934), 1530–1534 (2009).
    [CrossRef] [PubMed]
  2. A. K. Geim and K. S. Novoselov, “The rise of graphene,” Nat. Mater. 6(3), 183–191 (2007).
    [CrossRef] [PubMed]
  3. S. Raghu and F. D. M. Haldane, “Analogs of quantum-Hall-effect edge states in photonic crystals,” Phys. Rev. A 78(3), 033834 (2008).
    [CrossRef]
  4. T. Ochiai and M. Onoda, “Photonic analog of graphene model and its extension: Dirac cone, symmetry, and edge states,” Phys. Rev. B 80(15), 155103 (2009).
    [CrossRef]
  5. X. Zhang, “Observing Zitterbewegung for photons near the Dirac point of a two-dimensional photonic crystal,” Phys. Rev. Lett. 100(11), 113903 (2008).
    [CrossRef] [PubMed]
  6. O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, “Conical diffraction and gap solitons in honeycomb photonic lattices,” Phys. Rev. Lett. 98(10), 103901 (2007).
    [CrossRef] [PubMed]
  7. O. Bahat-Treidel, O. Peleg, M. Grobman, N. Shapira, M. Segev, and T. Pereg-Barnea, “Klein tunneling in deformed honeycomb lattices,” Phys. Rev. Lett. 104(6), 063901 (2010).
    [CrossRef] [PubMed]
  8. S. H. Nam, A. J. Taylor, and A. Efimov, “Diabolical point and conical-like diffraction in periodic plasmonic nanostructures,” Opt. Express 18(10), 10120–10126 (2010).
    [CrossRef] [PubMed]
  9. S. Hyun Nam, E. Ulin-Avila, G. Bartal, and X. Zhang, “Deep subwavelength surface modes in metal-dielectric metamaterials,” Opt. Lett. 35(11), 1847–1849 (2010).
    [CrossRef] [PubMed]
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    [CrossRef]
  11. A. H. Castro Neto, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81(1), 109–162 (2009).
    [CrossRef]
  12. M. I. Katsnelson, K. S. Novoselov, and A. K. Geim, “Chiral tunnelling and the Klein paradox in graphene,” Nat. Phys. 2(9), 620–625 (2006).
    [CrossRef]
  13. T. Ando, T. Nakanishi, and R. Saito, “Berry's phase and absence of back scattering in carbon nanotubes,” J. Phys. Soc. Jpn. 67(8), 2857–2862 (1998).
    [CrossRef]
  14. Z. H. Ni, T. Yu, Y. H. Lu, Y. Y. Wang, Y. P. Feng, and Z. X. Shen, “Uniaxial strain on graphene: Raman spectroscopy study and band-gap opening,” ACS Nano 2(11), 2301–2305 (2008).
    [CrossRef]
  15. F. Guinea, M. I. Katsnelson, and A. K. Geim, “Energy gaps and a zero-field quantum Hall effect in graphene by strain engineering,” Nat. Phys. 6(1), 30–33 (2010).
    [CrossRef]
  16. A. K. Geim and A. H. MacDonald, “Graphene: Exploring carbon flatland,” Phys. Today 60(8), 35–41 (2007).
    [CrossRef]
  17. J. K. Furdyna, “Split light,” Physics 3, 56 (2010).
    [CrossRef]
  18. M. Onoda, S. Murakami, and N. Nagaosa, “Hall effect of light,” Phys. Rev. Lett. 93(8), 083901 (2004).
    [CrossRef] [PubMed]
  19. O. Hosten and P. Kwiat, “Observation of the spin hall effect of light via weak measurements,” Science 319(5864), 787–790 (2008).
    [CrossRef] [PubMed]
  20. B. Thaller, Advanced visual quantum mechanics, (Springer, 2005).

2010 (5)

O. Bahat-Treidel, O. Peleg, M. Grobman, N. Shapira, M. Segev, and T. Pereg-Barnea, “Klein tunneling in deformed honeycomb lattices,” Phys. Rev. Lett. 104(6), 063901 (2010).
[CrossRef] [PubMed]

S. H. Nam, A. J. Taylor, and A. Efimov, “Diabolical point and conical-like diffraction in periodic plasmonic nanostructures,” Opt. Express 18(10), 10120–10126 (2010).
[CrossRef] [PubMed]

S. Hyun Nam, E. Ulin-Avila, G. Bartal, and X. Zhang, “Deep subwavelength surface modes in metal-dielectric metamaterials,” Opt. Lett. 35(11), 1847–1849 (2010).
[CrossRef] [PubMed]

F. Guinea, M. I. Katsnelson, and A. K. Geim, “Energy gaps and a zero-field quantum Hall effect in graphene by strain engineering,” Nat. Phys. 6(1), 30–33 (2010).
[CrossRef]

J. K. Furdyna, “Split light,” Physics 3, 56 (2010).
[CrossRef]

2009 (3)

A. H. Castro Neto, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81(1), 109–162 (2009).
[CrossRef]

A. K. Geim, “Graphene: status and prospects,” Science 324(5934), 1530–1534 (2009).
[CrossRef] [PubMed]

T. Ochiai and M. Onoda, “Photonic analog of graphene model and its extension: Dirac cone, symmetry, and edge states,” Phys. Rev. B 80(15), 155103 (2009).
[CrossRef]

2008 (4)

X. Zhang, “Observing Zitterbewegung for photons near the Dirac point of a two-dimensional photonic crystal,” Phys. Rev. Lett. 100(11), 113903 (2008).
[CrossRef] [PubMed]

S. Raghu and F. D. M. Haldane, “Analogs of quantum-Hall-effect edge states in photonic crystals,” Phys. Rev. A 78(3), 033834 (2008).
[CrossRef]

Z. H. Ni, T. Yu, Y. H. Lu, Y. Y. Wang, Y. P. Feng, and Z. X. Shen, “Uniaxial strain on graphene: Raman spectroscopy study and band-gap opening,” ACS Nano 2(11), 2301–2305 (2008).
[CrossRef]

O. Hosten and P. Kwiat, “Observation of the spin hall effect of light via weak measurements,” Science 319(5864), 787–790 (2008).
[CrossRef] [PubMed]

2007 (3)

A. K. Geim and A. H. MacDonald, “Graphene: Exploring carbon flatland,” Phys. Today 60(8), 35–41 (2007).
[CrossRef]

O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, “Conical diffraction and gap solitons in honeycomb photonic lattices,” Phys. Rev. Lett. 98(10), 103901 (2007).
[CrossRef] [PubMed]

A. K. Geim and K. S. Novoselov, “The rise of graphene,” Nat. Mater. 6(3), 183–191 (2007).
[CrossRef] [PubMed]

2006 (1)

M. I. Katsnelson, K. S. Novoselov, and A. K. Geim, “Chiral tunnelling and the Klein paradox in graphene,” Nat. Phys. 2(9), 620–625 (2006).
[CrossRef]

2004 (1)

M. Onoda, S. Murakami, and N. Nagaosa, “Hall effect of light,” Phys. Rev. Lett. 93(8), 083901 (2004).
[CrossRef] [PubMed]

2002 (1)

1998 (1)

T. Ando, T. Nakanishi, and R. Saito, “Berry's phase and absence of back scattering in carbon nanotubes,” J. Phys. Soc. Jpn. 67(8), 2857–2862 (1998).
[CrossRef]

Ando, T.

T. Ando, T. Nakanishi, and R. Saito, “Berry's phase and absence of back scattering in carbon nanotubes,” J. Phys. Soc. Jpn. 67(8), 2857–2862 (1998).
[CrossRef]

Bahat-Treidel, O.

O. Bahat-Treidel, O. Peleg, M. Grobman, N. Shapira, M. Segev, and T. Pereg-Barnea, “Klein tunneling in deformed honeycomb lattices,” Phys. Rev. Lett. 104(6), 063901 (2010).
[CrossRef] [PubMed]

Bartal, G.

S. Hyun Nam, E. Ulin-Avila, G. Bartal, and X. Zhang, “Deep subwavelength surface modes in metal-dielectric metamaterials,” Opt. Lett. 35(11), 1847–1849 (2010).
[CrossRef] [PubMed]

O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, “Conical diffraction and gap solitons in honeycomb photonic lattices,” Phys. Rev. Lett. 98(10), 103901 (2007).
[CrossRef] [PubMed]

Castro Neto, A. H.

A. H. Castro Neto, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81(1), 109–162 (2009).
[CrossRef]

Christodoulides, D. N.

O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, “Conical diffraction and gap solitons in honeycomb photonic lattices,” Phys. Rev. Lett. 98(10), 103901 (2007).
[CrossRef] [PubMed]

Efimov, A.

Feng, Y. P.

Z. H. Ni, T. Yu, Y. H. Lu, Y. Y. Wang, Y. P. Feng, and Z. X. Shen, “Uniaxial strain on graphene: Raman spectroscopy study and band-gap opening,” ACS Nano 2(11), 2301–2305 (2008).
[CrossRef]

Freedman, B.

O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, “Conical diffraction and gap solitons in honeycomb photonic lattices,” Phys. Rev. Lett. 98(10), 103901 (2007).
[CrossRef] [PubMed]

Furdyna, J. K.

J. K. Furdyna, “Split light,” Physics 3, 56 (2010).
[CrossRef]

Geim, A. K.

F. Guinea, M. I. Katsnelson, and A. K. Geim, “Energy gaps and a zero-field quantum Hall effect in graphene by strain engineering,” Nat. Phys. 6(1), 30–33 (2010).
[CrossRef]

A. K. Geim, “Graphene: status and prospects,” Science 324(5934), 1530–1534 (2009).
[CrossRef] [PubMed]

A. H. Castro Neto, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81(1), 109–162 (2009).
[CrossRef]

A. K. Geim and K. S. Novoselov, “The rise of graphene,” Nat. Mater. 6(3), 183–191 (2007).
[CrossRef] [PubMed]

A. K. Geim and A. H. MacDonald, “Graphene: Exploring carbon flatland,” Phys. Today 60(8), 35–41 (2007).
[CrossRef]

M. I. Katsnelson, K. S. Novoselov, and A. K. Geim, “Chiral tunnelling and the Klein paradox in graphene,” Nat. Phys. 2(9), 620–625 (2006).
[CrossRef]

Grobman, M.

O. Bahat-Treidel, O. Peleg, M. Grobman, N. Shapira, M. Segev, and T. Pereg-Barnea, “Klein tunneling in deformed honeycomb lattices,” Phys. Rev. Lett. 104(6), 063901 (2010).
[CrossRef] [PubMed]

Guinea, F.

F. Guinea, M. I. Katsnelson, and A. K. Geim, “Energy gaps and a zero-field quantum Hall effect in graphene by strain engineering,” Nat. Phys. 6(1), 30–33 (2010).
[CrossRef]

Haldane, F. D. M.

S. Raghu and F. D. M. Haldane, “Analogs of quantum-Hall-effect edge states in photonic crystals,” Phys. Rev. A 78(3), 033834 (2008).
[CrossRef]

Hosten, O.

O. Hosten and P. Kwiat, “Observation of the spin hall effect of light via weak measurements,” Science 319(5864), 787–790 (2008).
[CrossRef] [PubMed]

Hyun Nam, S.

Katsnelson, M. I.

F. Guinea, M. I. Katsnelson, and A. K. Geim, “Energy gaps and a zero-field quantum Hall effect in graphene by strain engineering,” Nat. Phys. 6(1), 30–33 (2010).
[CrossRef]

M. I. Katsnelson, K. S. Novoselov, and A. K. Geim, “Chiral tunnelling and the Klein paradox in graphene,” Nat. Phys. 2(9), 620–625 (2006).
[CrossRef]

Kivshar, Y. S.

Kwiat, P.

O. Hosten and P. Kwiat, “Observation of the spin hall effect of light via weak measurements,” Science 319(5864), 787–790 (2008).
[CrossRef] [PubMed]

Lu, Y. H.

Z. H. Ni, T. Yu, Y. H. Lu, Y. Y. Wang, Y. P. Feng, and Z. X. Shen, “Uniaxial strain on graphene: Raman spectroscopy study and band-gap opening,” ACS Nano 2(11), 2301–2305 (2008).
[CrossRef]

MacDonald, A. H.

A. K. Geim and A. H. MacDonald, “Graphene: Exploring carbon flatland,” Phys. Today 60(8), 35–41 (2007).
[CrossRef]

Manela, O.

O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, “Conical diffraction and gap solitons in honeycomb photonic lattices,” Phys. Rev. Lett. 98(10), 103901 (2007).
[CrossRef] [PubMed]

Murakami, S.

M. Onoda, S. Murakami, and N. Nagaosa, “Hall effect of light,” Phys. Rev. Lett. 93(8), 083901 (2004).
[CrossRef] [PubMed]

Nagaosa, N.

M. Onoda, S. Murakami, and N. Nagaosa, “Hall effect of light,” Phys. Rev. Lett. 93(8), 083901 (2004).
[CrossRef] [PubMed]

Nakanishi, T.

T. Ando, T. Nakanishi, and R. Saito, “Berry's phase and absence of back scattering in carbon nanotubes,” J. Phys. Soc. Jpn. 67(8), 2857–2862 (1998).
[CrossRef]

Nam, S. H.

Ni, Z. H.

Z. H. Ni, T. Yu, Y. H. Lu, Y. Y. Wang, Y. P. Feng, and Z. X. Shen, “Uniaxial strain on graphene: Raman spectroscopy study and band-gap opening,” ACS Nano 2(11), 2301–2305 (2008).
[CrossRef]

Novoselov, K. S.

A. H. Castro Neto, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81(1), 109–162 (2009).
[CrossRef]

A. K. Geim and K. S. Novoselov, “The rise of graphene,” Nat. Mater. 6(3), 183–191 (2007).
[CrossRef] [PubMed]

M. I. Katsnelson, K. S. Novoselov, and A. K. Geim, “Chiral tunnelling and the Klein paradox in graphene,” Nat. Phys. 2(9), 620–625 (2006).
[CrossRef]

Ochiai, T.

T. Ochiai and M. Onoda, “Photonic analog of graphene model and its extension: Dirac cone, symmetry, and edge states,” Phys. Rev. B 80(15), 155103 (2009).
[CrossRef]

Onoda, M.

T. Ochiai and M. Onoda, “Photonic analog of graphene model and its extension: Dirac cone, symmetry, and edge states,” Phys. Rev. B 80(15), 155103 (2009).
[CrossRef]

M. Onoda, S. Murakami, and N. Nagaosa, “Hall effect of light,” Phys. Rev. Lett. 93(8), 083901 (2004).
[CrossRef] [PubMed]

Peleg, O.

O. Bahat-Treidel, O. Peleg, M. Grobman, N. Shapira, M. Segev, and T. Pereg-Barnea, “Klein tunneling in deformed honeycomb lattices,” Phys. Rev. Lett. 104(6), 063901 (2010).
[CrossRef] [PubMed]

O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, “Conical diffraction and gap solitons in honeycomb photonic lattices,” Phys. Rev. Lett. 98(10), 103901 (2007).
[CrossRef] [PubMed]

Pereg-Barnea, T.

O. Bahat-Treidel, O. Peleg, M. Grobman, N. Shapira, M. Segev, and T. Pereg-Barnea, “Klein tunneling in deformed honeycomb lattices,” Phys. Rev. Lett. 104(6), 063901 (2010).
[CrossRef] [PubMed]

Peres, N. M. R.

A. H. Castro Neto, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81(1), 109–162 (2009).
[CrossRef]

Raghu, S.

S. Raghu and F. D. M. Haldane, “Analogs of quantum-Hall-effect edge states in photonic crystals,” Phys. Rev. A 78(3), 033834 (2008).
[CrossRef]

Saito, R.

T. Ando, T. Nakanishi, and R. Saito, “Berry's phase and absence of back scattering in carbon nanotubes,” J. Phys. Soc. Jpn. 67(8), 2857–2862 (1998).
[CrossRef]

Segev, M.

O. Bahat-Treidel, O. Peleg, M. Grobman, N. Shapira, M. Segev, and T. Pereg-Barnea, “Klein tunneling in deformed honeycomb lattices,” Phys. Rev. Lett. 104(6), 063901 (2010).
[CrossRef] [PubMed]

O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, “Conical diffraction and gap solitons in honeycomb photonic lattices,” Phys. Rev. Lett. 98(10), 103901 (2007).
[CrossRef] [PubMed]

Shapira, N.

O. Bahat-Treidel, O. Peleg, M. Grobman, N. Shapira, M. Segev, and T. Pereg-Barnea, “Klein tunneling in deformed honeycomb lattices,” Phys. Rev. Lett. 104(6), 063901 (2010).
[CrossRef] [PubMed]

Shen, Z. X.

Z. H. Ni, T. Yu, Y. H. Lu, Y. Y. Wang, Y. P. Feng, and Z. X. Shen, “Uniaxial strain on graphene: Raman spectroscopy study and band-gap opening,” ACS Nano 2(11), 2301–2305 (2008).
[CrossRef]

Sukhorukov, A. A.

Taylor, A. J.

Ulin-Avila, E.

Wang, Y. Y.

Z. H. Ni, T. Yu, Y. H. Lu, Y. Y. Wang, Y. P. Feng, and Z. X. Shen, “Uniaxial strain on graphene: Raman spectroscopy study and band-gap opening,” ACS Nano 2(11), 2301–2305 (2008).
[CrossRef]

Yu, T.

Z. H. Ni, T. Yu, Y. H. Lu, Y. Y. Wang, Y. P. Feng, and Z. X. Shen, “Uniaxial strain on graphene: Raman spectroscopy study and band-gap opening,” ACS Nano 2(11), 2301–2305 (2008).
[CrossRef]

Zhang, X.

S. Hyun Nam, E. Ulin-Avila, G. Bartal, and X. Zhang, “Deep subwavelength surface modes in metal-dielectric metamaterials,” Opt. Lett. 35(11), 1847–1849 (2010).
[CrossRef] [PubMed]

X. Zhang, “Observing Zitterbewegung for photons near the Dirac point of a two-dimensional photonic crystal,” Phys. Rev. Lett. 100(11), 113903 (2008).
[CrossRef] [PubMed]

ACS Nano (1)

Z. H. Ni, T. Yu, Y. H. Lu, Y. Y. Wang, Y. P. Feng, and Z. X. Shen, “Uniaxial strain on graphene: Raman spectroscopy study and band-gap opening,” ACS Nano 2(11), 2301–2305 (2008).
[CrossRef]

J. Phys. Soc. Jpn. (1)

T. Ando, T. Nakanishi, and R. Saito, “Berry's phase and absence of back scattering in carbon nanotubes,” J. Phys. Soc. Jpn. 67(8), 2857–2862 (1998).
[CrossRef]

Nat. Mater. (1)

A. K. Geim and K. S. Novoselov, “The rise of graphene,” Nat. Mater. 6(3), 183–191 (2007).
[CrossRef] [PubMed]

Nat. Phys. (2)

F. Guinea, M. I. Katsnelson, and A. K. Geim, “Energy gaps and a zero-field quantum Hall effect in graphene by strain engineering,” Nat. Phys. 6(1), 30–33 (2010).
[CrossRef]

M. I. Katsnelson, K. S. Novoselov, and A. K. Geim, “Chiral tunnelling and the Klein paradox in graphene,” Nat. Phys. 2(9), 620–625 (2006).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

Phys. Rev. A (1)

S. Raghu and F. D. M. Haldane, “Analogs of quantum-Hall-effect edge states in photonic crystals,” Phys. Rev. A 78(3), 033834 (2008).
[CrossRef]

Phys. Rev. B (1)

T. Ochiai and M. Onoda, “Photonic analog of graphene model and its extension: Dirac cone, symmetry, and edge states,” Phys. Rev. B 80(15), 155103 (2009).
[CrossRef]

Phys. Rev. Lett. (4)

X. Zhang, “Observing Zitterbewegung for photons near the Dirac point of a two-dimensional photonic crystal,” Phys. Rev. Lett. 100(11), 113903 (2008).
[CrossRef] [PubMed]

O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, “Conical diffraction and gap solitons in honeycomb photonic lattices,” Phys. Rev. Lett. 98(10), 103901 (2007).
[CrossRef] [PubMed]

O. Bahat-Treidel, O. Peleg, M. Grobman, N. Shapira, M. Segev, and T. Pereg-Barnea, “Klein tunneling in deformed honeycomb lattices,” Phys. Rev. Lett. 104(6), 063901 (2010).
[CrossRef] [PubMed]

M. Onoda, S. Murakami, and N. Nagaosa, “Hall effect of light,” Phys. Rev. Lett. 93(8), 083901 (2004).
[CrossRef] [PubMed]

Phys. Today (1)

A. K. Geim and A. H. MacDonald, “Graphene: Exploring carbon flatland,” Phys. Today 60(8), 35–41 (2007).
[CrossRef]

Physics (1)

J. K. Furdyna, “Split light,” Physics 3, 56 (2010).
[CrossRef]

Rev. Mod. Phys. (1)

A. H. Castro Neto, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81(1), 109–162 (2009).
[CrossRef]

Science (2)

O. Hosten and P. Kwiat, “Observation of the spin hall effect of light via weak measurements,” Science 319(5864), 787–790 (2008).
[CrossRef] [PubMed]

A. K. Geim, “Graphene: status and prospects,” Science 324(5934), 1530–1534 (2009).
[CrossRef] [PubMed]

Other (1)

B. Thaller, Advanced visual quantum mechanics, (Springer, 2005).

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Figures (4)

Fig. 1
Fig. 1

(a) Metal-dielectric plasmonic crystal structure with sublattice A and B, inset: graphene’s honeycomb lattices with sublattices A and B. (b) Dirac-type dispersion for balanced couplings ( η = 1 ). σ + and σ denote ‘up’ and ‘down’ pseudo-spin, respectively, and their projections onto the momentum direction are indicated by solid arrows. γis the eigenvalue of the chirality. (c) In the graphene-like plasmonic crystal, an input Gaussian beam splits into two different directions depending on their pseudo-spin state, analogous to spin Hall effect.

Fig. 2
Fig. 2

SPP dynamics when η = 0.99 . (a) Probability density of the initial state of Dirac spinor | ϕ ^ ( k ) | 2 , of the projection to positive K components | ϕ ^ + ( k ) | 2 , and negative K components | ϕ ^ ( k ) | 2 from left to right in momemtum space. (b) | ψ ( x ) | 2 , | ϕ p o s ( x ) | 2 , | ϕ n e g ( x ) | 2 from left to right in position space. (c) Left: Evolution of Gaussian input wave packet | ψ ( x , z ) | 2 , upper right: dispersion curves, lower right: diffraction of a normally incident Gaussian beam in the binary plasmonic lattice by numerical simulation from Fig. 2(c) in Ref. [8].

Fig. 3
Fig. 3

SPP dynamics when with η = 0.85 . Configuration of the figures is the same as in Fig. 2.

Fig. 4
Fig. 4

SPP dynamics when η = 0.5 . Configuration of the figures is the same as in Fig. 2 and 3.

Equations (21)

Equations on this page are rendered with MathJax. Learn more.

H 0 | ψ = K | ψ ,
H 0 ( η , κ ) = ( 0 Δ * Δ 0 ) , | ψ = ( A B ) ,
H 0 ' = ( K 0 0 K ) ,
u K = 1 2 ( | Δ | Δ 1 ) , u K = 1 2 ( | Δ | Δ 1 ) ,
H 0 ( 0 i κ i κ 0 ) = σ 2 κ , ​ ​       when     η = 1 , | κ | << 1       ,
C σ 2 κ | κ | , γ = ± 1 ,
u p o s ( κ ; x , z ) = 1 2 π u K ( κ ) e i κ x + i K ( κ ) z ,
u n e g ( κ ; x , z ) = 1 2 π u K ( κ ) e i κ x i K ( κ ) z .
ψ ( x , z = 0 ) = ϕ ( x ) = A e ( x / w 0 ) 2 ( 1 1 ) ,
ϕ ^ ( κ ) = 1 2 π e i κ x ϕ ( x ) d x = A w 0 2 2 e ( w 0 κ / 2 ) 2 ( 1 1 ) ,
ϕ ^ ( κ ) = ϕ ^ + ( κ ) u K ( κ ) + ϕ ^ ( κ ) u K ( κ ) ,
ϕ ^ + ( κ ) = u K ( κ ) , ϕ ^ ( κ ) = A w 0 2 e ( w 0 κ / 2 ) 2 ( 1 + | Δ | / Δ * ) ,
ϕ ^ ( κ ) = u K ( κ ) , ϕ ^ ( κ ) = A w 0 2 e ( w 0 κ / 2 ) 2 ( 1 | Δ | / Δ * ) .
ψ ^ ( κ , z ) = ϕ ^ + ( κ ) u K ( κ ) e i K z + ϕ ^ ( κ ) u K ( κ ) e i K z ,
ψ ( x , z ) = 1 2 π e i κ x ψ ^ ( κ , z ) d κ = ϕ ^ + ( κ ) 1 2 π u K ( κ ) e i κ x + i λ z d κ + ϕ ^ ( κ ) 1 2 π u K ( κ ) e i κ x i λ z d κ = ϕ p o s ( x ) + ϕ n e g ( x ) .
E = m 2 c 4 + p 2 c 2 ,
H D = ( m c 2 c p c p m c 2 ) = σ 1 c p + σ 3 m c 2 .
K 2 = η 2 + 1 + 2 η cos κ ( η + 1 ) 2 η κ 2     ( κ << 1 ) ,
H p = ( η + 1 η κ η κ ( η + 1 ) ) ( δ υ D κ υ D κ δ ) = σ 1 υ D κ + σ 3 δ .
H 0 ( 0 ( 1 + η ) i η κ η κ 2 / 2 ( 1 + η ) + i η κ η κ 2 / 2 0 ) , ​ ​       when   | κ | << 1
H p = T U + H 0 U T + ,

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