Abstract

The giant optical nonlocality near the Dirac point in lossless metal-dielectric multilayer metamaterials is revealed and investigated through the analysis of the band structure of the multilayer stack in the three-dimensional ω-k space, according to the transfer-matrix method with the optical nonlocal effect. The position of the Dirac point is analytically located in the ω-k space. It is revealed that the emergence of the Dirac point is due to the degeneracy of the symmetric and the asymmetric eigenmodes of the coupled surface plasmon polaritons. The optical nonlocality induced epsilon-near-zero frequency shift for the multilayer stack compared to the effective medium is studied. Furthermore, the giant optical nonlocality around the Dirac point is explored with the iso-frequency contour analysis, while the beam splitting phenomenon at the Dirac point due to the optical nonlocal effect is also demonstrated.

© 2013 OSA

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  1. P. R. Wallace, “The band theory of graphite,” Phys. Rev.71, 622–634 (1947).
    [CrossRef]
  2. K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, “Two-dimensional gas of massless Dirac fermions in graphene,” Nature438, 197–200 (2005).
    [CrossRef] [PubMed]
  3. Y. Zhang, Y.-W. Tan, H. L. Stormer, and P. Kim, “Experimental observation of the quantum Hall effect and Berry’s phase in graphene,” Nature438, 201–204 (2005).
    [CrossRef] [PubMed]
  4. A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys.81, 109–162 (2009).
    [CrossRef]
  5. M. Plihal and A. A. Maradudin, “Photonic band structure of two-dimensional systems: The triangular lattice,” Phys. Rev. B44, 8565–8571 (1991).
    [CrossRef]
  6. R. A. Sepkhanov, Ya. B. Bazaliy, and C. W. J. Beenakker, “Extremal transmission at the Dirac point of a photonic band structure,” Phys. Rev. A75, 063813 (2007).
    [CrossRef]
  7. T. Ochiai and M. Onoda, “Photonic analog of graphene model and its extension: Dirac cone, symmetry, and edge states,” Phys. Rev. B80, 155103 (2009).
    [CrossRef]
  8. M. Diema, T. Koschnya, and C. M. Soukoulis, “Transmission in the vicinity of the Dirac point in hexagonal photonic crystals,” Physica B405, 2990–2995 (2010).
    [CrossRef]
  9. V. Yannopapas and A. Vanakaras, “Dirac point in the photon dispersion relation of a negative/zero/positive-index plasmonic metamaterial,” Phys. Rev. B84, 045128 (2011).
    [CrossRef]
  10. D. A. Abanin, S. V. Morozov, L. A. Ponomarenko, R. V. Gorbachev, A. S. Mayorov, M. I. Katsnelson, K. Watanabe, T. Taniguchi, K. S. Novoselov, L. S. Levitov, and A. K. Geim, “Giant nonlocality near the Dirac point in graphene,” Science332, 328–330 (2011).
    [CrossRef] [PubMed]
  11. 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, 103901 (2007).
    [CrossRef] [PubMed]
  12. X. Chen, L.-G. Wang, and C.-F. Li, “Transmission gap, Bragg-like reflection, and Goos-Hänchen shifts near the Dirac point inside a negative-zero-positive index metamaterial slab,” Phys. Rev. A80, 043839 (2009).
    [CrossRef]
  13. X. Zhang, “Observing Zitterbewegung for photons near the Dirac point of a two-dimensional photonic crystal,” Phys. Rev. Lett.100, 113903 (2008).
    [CrossRef] [PubMed]
  14. L.-G. Wang, Z.-G. Wang, and S.-Y. Zhu, “Zitterbewegung of optical pulses near the Dirac point inside a negative-zero-positive index metamaterial,” EPL86, 47008 (2009).
    [CrossRef]
  15. K. Sakoda, “Dirac cone in two- and three-dimensional metamaterials,” Opt. Express20, 3898–3917 (2012).
    [CrossRef] [PubMed]
  16. K. Sakoda, “Proof of the universality of mode symmetries in creating photonic Dirac cones,” Opt. Express20, 25181–25194 (2012).
    [CrossRef] [PubMed]
  17. L.-G. Wang, Z.-G. Wang, J.-X. Zhang, and S.-Y. Zhu, “Realization of Dirac point with double cones in optics,” Opt. Lett.34, 1510–1512 (2009).
    [CrossRef] [PubMed]
  18. X. Huang, Y. Lai, Z. H. Hang, H. Zheng, and C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nat. Mater.10, 582–586 (2011).
    [CrossRef] [PubMed]
  19. Y. Bliokh, V. Freilikher, and F. Nori, “Charge transport in graphene and light propagation in periodic dielectric structures with metamaterials: a comparative study,” arXiv:1302.5533v2 [physics.optics].
  20. A. Alù, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: Tailoring the radiation phase pattern,” Phys. Rev. B75, 155410 (2007).
    [CrossRef]
  21. L. Sun, S. Feng, and X. Yang, “Loss enhanced transmission and collimation in anisotropic epsilon-near-zero metamaterials,” Appl. Phys. Lett.101, 241101 (2012).
    [CrossRef]
  22. L. Sun, J. Gao, and X. Yang, “Broadband epsilon-near-zero metamaterials with steplike metal-dielectric multilayer structures,” Phys. Rev. B87, 165134 (2013).
    [CrossRef]
  23. M. J. Roberts, S. Feng, M. Moran, and L. Johnson, “Effective permittivity near zero in nanolaminates of silver and amorphous polycarbonate,” J. Nanophotonics4, 043511 (2010).
    [CrossRef]
  24. L. V. Alekseyev, E. E. Narimanov, T. Tumkur, H. Li, Yu. A. Barnakov, and M. A. Noginov, “Uniaxial epsilon-near-zero metamaterial for angular filtering and polarization control,” Appl. Phys. Lett.97, 131107 (2010).
    [CrossRef]
  25. G. Subramania, A. J. Fischer, and T. S. Luk, “Optical properties of metal-dielectric based epsilon near zero metamaterials,” Appl. Phys. Lett.101, 241107 (2012).
    [CrossRef]
  26. E. J. R. Vesseur, T. Coenen, H. Caglayan, N. Engheta, and A. Polman, “Experimental verification of n= 0 structures for visible light,” Phys. Rev. Lett.110, 013902 (2013).
    [CrossRef]
  27. J. Elser, V. A. Podolskiy, I. Salakhutdinov, and I. Avrutsky, “Nonlocal effects in effective-medium response of nanolayered metamaterials,” Appl. Phys. Lett.90, 191109 (2007).
    [CrossRef]
  28. A. A. Orlov, P. M. Voroshilov, P. A. Belov, and Y. S. Kivshar, “Engineered optical nonlocality in nanostructured metamaterials,” Phys. Rev. B84, 045424 (2011).
    [CrossRef]
  29. S. S. Kruk, D. A. Powell, A. Minovich, D. N. Neshev, and Y. S. Kivshar, “Spatial dispersion of multilayer fishnet metamaterials,” Opt. Express20, 15100–15105 (2012).
    [CrossRef] [PubMed]
  30. A. V. Chebykin, A. A. Orlov, C. R. Simovski, Yu. S. Kivshar, and P. A. Belov, “Nonlocal effective parameters of multilayered metal-dielectric metamaterials,” Phy. Rev. B86, 115420 (2012).
    [CrossRef]
  31. G. A. Wurtz, R. Pollard, W. Hendren, G. P. Wiederrecht, D. J. Gosztola, V. A. Podolskiy, and A. V. Zayats, “Designed ultrafast optical nonlinearity in a plasmonic nanorod metamaterial enhanced by nonlocality,” Nat. Nanotechnol.6, 107–111 (2011).
    [CrossRef] [PubMed]
  32. R. J. Pollard, A. Murphy, W. R. Hendren, P. R. Evans, R. Atkinson, G. A. Wurtz, A. V. Zayats, and V. A. Podolskiy, “Optical nonlocalities and additional waves in epsilon-near-zero metamaterials,” Phys. Rev. Lett.102, 127405 (2009).
    [CrossRef] [PubMed]
  33. G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “Nonlocal transformation optics,” Phys. Rev. Lett.108, 063902 (2012).
    [CrossRef] [PubMed]

2013

L. Sun, J. Gao, and X. Yang, “Broadband epsilon-near-zero metamaterials with steplike metal-dielectric multilayer structures,” Phys. Rev. B87, 165134 (2013).
[CrossRef]

E. J. R. Vesseur, T. Coenen, H. Caglayan, N. Engheta, and A. Polman, “Experimental verification of n= 0 structures for visible light,” Phys. Rev. Lett.110, 013902 (2013).
[CrossRef]

2012

G. Subramania, A. J. Fischer, and T. S. Luk, “Optical properties of metal-dielectric based epsilon near zero metamaterials,” Appl. Phys. Lett.101, 241107 (2012).
[CrossRef]

A. V. Chebykin, A. A. Orlov, C. R. Simovski, Yu. S. Kivshar, and P. A. Belov, “Nonlocal effective parameters of multilayered metal-dielectric metamaterials,” Phy. Rev. B86, 115420 (2012).
[CrossRef]

G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “Nonlocal transformation optics,” Phys. Rev. Lett.108, 063902 (2012).
[CrossRef] [PubMed]

K. Sakoda, “Dirac cone in two- and three-dimensional metamaterials,” Opt. Express20, 3898–3917 (2012).
[CrossRef] [PubMed]

S. S. Kruk, D. A. Powell, A. Minovich, D. N. Neshev, and Y. S. Kivshar, “Spatial dispersion of multilayer fishnet metamaterials,” Opt. Express20, 15100–15105 (2012).
[CrossRef] [PubMed]

K. Sakoda, “Proof of the universality of mode symmetries in creating photonic Dirac cones,” Opt. Express20, 25181–25194 (2012).
[CrossRef] [PubMed]

L. Sun, S. Feng, and X. Yang, “Loss enhanced transmission and collimation in anisotropic epsilon-near-zero metamaterials,” Appl. Phys. Lett.101, 241101 (2012).
[CrossRef]

2011

X. Huang, Y. Lai, Z. H. Hang, H. Zheng, and C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nat. Mater.10, 582–586 (2011).
[CrossRef] [PubMed]

V. Yannopapas and A. Vanakaras, “Dirac point in the photon dispersion relation of a negative/zero/positive-index plasmonic metamaterial,” Phys. Rev. B84, 045128 (2011).
[CrossRef]

D. A. Abanin, S. V. Morozov, L. A. Ponomarenko, R. V. Gorbachev, A. S. Mayorov, M. I. Katsnelson, K. Watanabe, T. Taniguchi, K. S. Novoselov, L. S. Levitov, and A. K. Geim, “Giant nonlocality near the Dirac point in graphene,” Science332, 328–330 (2011).
[CrossRef] [PubMed]

G. A. Wurtz, R. Pollard, W. Hendren, G. P. Wiederrecht, D. J. Gosztola, V. A. Podolskiy, and A. V. Zayats, “Designed ultrafast optical nonlinearity in a plasmonic nanorod metamaterial enhanced by nonlocality,” Nat. Nanotechnol.6, 107–111 (2011).
[CrossRef] [PubMed]

A. A. Orlov, P. M. Voroshilov, P. A. Belov, and Y. S. Kivshar, “Engineered optical nonlocality in nanostructured metamaterials,” Phys. Rev. B84, 045424 (2011).
[CrossRef]

2010

M. J. Roberts, S. Feng, M. Moran, and L. Johnson, “Effective permittivity near zero in nanolaminates of silver and amorphous polycarbonate,” J. Nanophotonics4, 043511 (2010).
[CrossRef]

L. V. Alekseyev, E. E. Narimanov, T. Tumkur, H. Li, Yu. A. Barnakov, and M. A. Noginov, “Uniaxial epsilon-near-zero metamaterial for angular filtering and polarization control,” Appl. Phys. Lett.97, 131107 (2010).
[CrossRef]

M. Diema, T. Koschnya, and C. M. Soukoulis, “Transmission in the vicinity of the Dirac point in hexagonal photonic crystals,” Physica B405, 2990–2995 (2010).
[CrossRef]

2009

T. Ochiai and M. Onoda, “Photonic analog of graphene model and its extension: Dirac cone, symmetry, and edge states,” Phys. Rev. B80, 155103 (2009).
[CrossRef]

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

X. Chen, L.-G. Wang, and C.-F. Li, “Transmission gap, Bragg-like reflection, and Goos-Hänchen shifts near the Dirac point inside a negative-zero-positive index metamaterial slab,” Phys. Rev. A80, 043839 (2009).
[CrossRef]

L.-G. Wang, Z.-G. Wang, and S.-Y. Zhu, “Zitterbewegung of optical pulses near the Dirac point inside a negative-zero-positive index metamaterial,” EPL86, 47008 (2009).
[CrossRef]

R. J. Pollard, A. Murphy, W. R. Hendren, P. R. Evans, R. Atkinson, G. A. Wurtz, A. V. Zayats, and V. A. Podolskiy, “Optical nonlocalities and additional waves in epsilon-near-zero metamaterials,” Phys. Rev. Lett.102, 127405 (2009).
[CrossRef] [PubMed]

L.-G. Wang, Z.-G. Wang, J.-X. Zhang, and S.-Y. Zhu, “Realization of Dirac point with double cones in optics,” Opt. Lett.34, 1510–1512 (2009).
[CrossRef] [PubMed]

2008

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

2007

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, 103901 (2007).
[CrossRef] [PubMed]

A. Alù, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: Tailoring the radiation phase pattern,” Phys. Rev. B75, 155410 (2007).
[CrossRef]

R. A. Sepkhanov, Ya. B. Bazaliy, and C. W. J. Beenakker, “Extremal transmission at the Dirac point of a photonic band structure,” Phys. Rev. A75, 063813 (2007).
[CrossRef]

J. Elser, V. A. Podolskiy, I. Salakhutdinov, and I. Avrutsky, “Nonlocal effects in effective-medium response of nanolayered metamaterials,” Appl. Phys. Lett.90, 191109 (2007).
[CrossRef]

2005

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, “Two-dimensional gas of massless Dirac fermions in graphene,” Nature438, 197–200 (2005).
[CrossRef] [PubMed]

Y. Zhang, Y.-W. Tan, H. L. Stormer, and P. Kim, “Experimental observation of the quantum Hall effect and Berry’s phase in graphene,” Nature438, 201–204 (2005).
[CrossRef] [PubMed]

1991

M. Plihal and A. A. Maradudin, “Photonic band structure of two-dimensional systems: The triangular lattice,” Phys. Rev. B44, 8565–8571 (1991).
[CrossRef]

1947

P. R. Wallace, “The band theory of graphite,” Phys. Rev.71, 622–634 (1947).
[CrossRef]

Abanin, D. A.

D. A. Abanin, S. V. Morozov, L. A. Ponomarenko, R. V. Gorbachev, A. S. Mayorov, M. I. Katsnelson, K. Watanabe, T. Taniguchi, K. S. Novoselov, L. S. Levitov, and A. K. Geim, “Giant nonlocality near the Dirac point in graphene,” Science332, 328–330 (2011).
[CrossRef] [PubMed]

Alekseyev, L. V.

L. V. Alekseyev, E. E. Narimanov, T. Tumkur, H. Li, Yu. A. Barnakov, and M. A. Noginov, “Uniaxial epsilon-near-zero metamaterial for angular filtering and polarization control,” Appl. Phys. Lett.97, 131107 (2010).
[CrossRef]

Alù, A.

G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “Nonlocal transformation optics,” Phys. Rev. Lett.108, 063902 (2012).
[CrossRef] [PubMed]

A. Alù, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: Tailoring the radiation phase pattern,” Phys. Rev. B75, 155410 (2007).
[CrossRef]

Atkinson, R.

R. J. Pollard, A. Murphy, W. R. Hendren, P. R. Evans, R. Atkinson, G. A. Wurtz, A. V. Zayats, and V. A. Podolskiy, “Optical nonlocalities and additional waves in epsilon-near-zero metamaterials,” Phys. Rev. Lett.102, 127405 (2009).
[CrossRef] [PubMed]

Avrutsky, I.

J. Elser, V. A. Podolskiy, I. Salakhutdinov, and I. Avrutsky, “Nonlocal effects in effective-medium response of nanolayered metamaterials,” Appl. Phys. Lett.90, 191109 (2007).
[CrossRef]

Barnakov, Yu. A.

L. V. Alekseyev, E. E. Narimanov, T. Tumkur, H. Li, Yu. A. Barnakov, and M. A. Noginov, “Uniaxial epsilon-near-zero metamaterial for angular filtering and polarization control,” Appl. Phys. Lett.97, 131107 (2010).
[CrossRef]

Bartal, G.

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, 103901 (2007).
[CrossRef] [PubMed]

Bazaliy, Ya. B.

R. A. Sepkhanov, Ya. B. Bazaliy, and C. W. J. Beenakker, “Extremal transmission at the Dirac point of a photonic band structure,” Phys. Rev. A75, 063813 (2007).
[CrossRef]

Beenakker, C. W. J.

R. A. Sepkhanov, Ya. B. Bazaliy, and C. W. J. Beenakker, “Extremal transmission at the Dirac point of a photonic band structure,” Phys. Rev. A75, 063813 (2007).
[CrossRef]

Belov, P. A.

A. V. Chebykin, A. A. Orlov, C. R. Simovski, Yu. S. Kivshar, and P. A. Belov, “Nonlocal effective parameters of multilayered metal-dielectric metamaterials,” Phy. Rev. B86, 115420 (2012).
[CrossRef]

A. A. Orlov, P. M. Voroshilov, P. A. Belov, and Y. S. Kivshar, “Engineered optical nonlocality in nanostructured metamaterials,” Phys. Rev. B84, 045424 (2011).
[CrossRef]

Bliokh, Y.

Y. Bliokh, V. Freilikher, and F. Nori, “Charge transport in graphene and light propagation in periodic dielectric structures with metamaterials: a comparative study,” arXiv:1302.5533v2 [physics.optics].

Caglayan, H.

E. J. R. Vesseur, T. Coenen, H. Caglayan, N. Engheta, and A. Polman, “Experimental verification of n= 0 structures for visible light,” Phys. Rev. Lett.110, 013902 (2013).
[CrossRef]

Castaldi, G.

G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “Nonlocal transformation optics,” Phys. Rev. Lett.108, 063902 (2012).
[CrossRef] [PubMed]

Castro Neto, A. H.

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

Chan, C. T.

X. Huang, Y. Lai, Z. H. Hang, H. Zheng, and C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nat. Mater.10, 582–586 (2011).
[CrossRef] [PubMed]

Chebykin, A. V.

A. V. Chebykin, A. A. Orlov, C. R. Simovski, Yu. S. Kivshar, and P. A. Belov, “Nonlocal effective parameters of multilayered metal-dielectric metamaterials,” Phy. Rev. B86, 115420 (2012).
[CrossRef]

Chen, X.

X. Chen, L.-G. Wang, and C.-F. Li, “Transmission gap, Bragg-like reflection, and Goos-Hänchen shifts near the Dirac point inside a negative-zero-positive index metamaterial slab,” Phys. Rev. A80, 043839 (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, 103901 (2007).
[CrossRef] [PubMed]

Coenen, T.

E. J. R. Vesseur, T. Coenen, H. Caglayan, N. Engheta, and A. Polman, “Experimental verification of n= 0 structures for visible light,” Phys. Rev. Lett.110, 013902 (2013).
[CrossRef]

Diema, M.

M. Diema, T. Koschnya, and C. M. Soukoulis, “Transmission in the vicinity of the Dirac point in hexagonal photonic crystals,” Physica B405, 2990–2995 (2010).
[CrossRef]

Dubonos, S. V.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, “Two-dimensional gas of massless Dirac fermions in graphene,” Nature438, 197–200 (2005).
[CrossRef] [PubMed]

Elser, J.

J. Elser, V. A. Podolskiy, I. Salakhutdinov, and I. Avrutsky, “Nonlocal effects in effective-medium response of nanolayered metamaterials,” Appl. Phys. Lett.90, 191109 (2007).
[CrossRef]

Engheta, N.

E. J. R. Vesseur, T. Coenen, H. Caglayan, N. Engheta, and A. Polman, “Experimental verification of n= 0 structures for visible light,” Phys. Rev. Lett.110, 013902 (2013).
[CrossRef]

G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “Nonlocal transformation optics,” Phys. Rev. Lett.108, 063902 (2012).
[CrossRef] [PubMed]

A. Alù, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: Tailoring the radiation phase pattern,” Phys. Rev. B75, 155410 (2007).
[CrossRef]

Evans, P. R.

R. J. Pollard, A. Murphy, W. R. Hendren, P. R. Evans, R. Atkinson, G. A. Wurtz, A. V. Zayats, and V. A. Podolskiy, “Optical nonlocalities and additional waves in epsilon-near-zero metamaterials,” Phys. Rev. Lett.102, 127405 (2009).
[CrossRef] [PubMed]

Feng, S.

L. Sun, S. Feng, and X. Yang, “Loss enhanced transmission and collimation in anisotropic epsilon-near-zero metamaterials,” Appl. Phys. Lett.101, 241101 (2012).
[CrossRef]

M. J. Roberts, S. Feng, M. Moran, and L. Johnson, “Effective permittivity near zero in nanolaminates of silver and amorphous polycarbonate,” J. Nanophotonics4, 043511 (2010).
[CrossRef]

Firsov, A. A.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, “Two-dimensional gas of massless Dirac fermions in graphene,” Nature438, 197–200 (2005).
[CrossRef] [PubMed]

Fischer, A. J.

G. Subramania, A. J. Fischer, and T. S. Luk, “Optical properties of metal-dielectric based epsilon near zero metamaterials,” Appl. Phys. Lett.101, 241107 (2012).
[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, 103901 (2007).
[CrossRef] [PubMed]

Freilikher, V.

Y. Bliokh, V. Freilikher, and F. Nori, “Charge transport in graphene and light propagation in periodic dielectric structures with metamaterials: a comparative study,” arXiv:1302.5533v2 [physics.optics].

Galdi, V.

G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “Nonlocal transformation optics,” Phys. Rev. Lett.108, 063902 (2012).
[CrossRef] [PubMed]

Gao, J.

L. Sun, J. Gao, and X. Yang, “Broadband epsilon-near-zero metamaterials with steplike metal-dielectric multilayer structures,” Phys. Rev. B87, 165134 (2013).
[CrossRef]

Geim, A. K.

D. A. Abanin, S. V. Morozov, L. A. Ponomarenko, R. V. Gorbachev, A. S. Mayorov, M. I. Katsnelson, K. Watanabe, T. Taniguchi, K. S. Novoselov, L. S. Levitov, and A. K. Geim, “Giant nonlocality near the Dirac point in graphene,” Science332, 328–330 (2011).
[CrossRef] [PubMed]

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

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, “Two-dimensional gas of massless Dirac fermions in graphene,” Nature438, 197–200 (2005).
[CrossRef] [PubMed]

Gorbachev, R. V.

D. A. Abanin, S. V. Morozov, L. A. Ponomarenko, R. V. Gorbachev, A. S. Mayorov, M. I. Katsnelson, K. Watanabe, T. Taniguchi, K. S. Novoselov, L. S. Levitov, and A. K. Geim, “Giant nonlocality near the Dirac point in graphene,” Science332, 328–330 (2011).
[CrossRef] [PubMed]

Gosztola, D. J.

G. A. Wurtz, R. Pollard, W. Hendren, G. P. Wiederrecht, D. J. Gosztola, V. A. Podolskiy, and A. V. Zayats, “Designed ultrafast optical nonlinearity in a plasmonic nanorod metamaterial enhanced by nonlocality,” Nat. Nanotechnol.6, 107–111 (2011).
[CrossRef] [PubMed]

Grigorieva, I. V.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, “Two-dimensional gas of massless Dirac fermions in graphene,” Nature438, 197–200 (2005).
[CrossRef] [PubMed]

Guinea, F.

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

Hang, Z. H.

X. Huang, Y. Lai, Z. H. Hang, H. Zheng, and C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nat. Mater.10, 582–586 (2011).
[CrossRef] [PubMed]

Hendren, W.

G. A. Wurtz, R. Pollard, W. Hendren, G. P. Wiederrecht, D. J. Gosztola, V. A. Podolskiy, and A. V. Zayats, “Designed ultrafast optical nonlinearity in a plasmonic nanorod metamaterial enhanced by nonlocality,” Nat. Nanotechnol.6, 107–111 (2011).
[CrossRef] [PubMed]

Hendren, W. R.

R. J. Pollard, A. Murphy, W. R. Hendren, P. R. Evans, R. Atkinson, G. A. Wurtz, A. V. Zayats, and V. A. Podolskiy, “Optical nonlocalities and additional waves in epsilon-near-zero metamaterials,” Phys. Rev. Lett.102, 127405 (2009).
[CrossRef] [PubMed]

Huang, X.

X. Huang, Y. Lai, Z. H. Hang, H. Zheng, and C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nat. Mater.10, 582–586 (2011).
[CrossRef] [PubMed]

Jiang, D.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, “Two-dimensional gas of massless Dirac fermions in graphene,” Nature438, 197–200 (2005).
[CrossRef] [PubMed]

Johnson, L.

M. J. Roberts, S. Feng, M. Moran, and L. Johnson, “Effective permittivity near zero in nanolaminates of silver and amorphous polycarbonate,” J. Nanophotonics4, 043511 (2010).
[CrossRef]

Katsnelson, M. I.

D. A. Abanin, S. V. Morozov, L. A. Ponomarenko, R. V. Gorbachev, A. S. Mayorov, M. I. Katsnelson, K. Watanabe, T. Taniguchi, K. S. Novoselov, L. S. Levitov, and A. K. Geim, “Giant nonlocality near the Dirac point in graphene,” Science332, 328–330 (2011).
[CrossRef] [PubMed]

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, “Two-dimensional gas of massless Dirac fermions in graphene,” Nature438, 197–200 (2005).
[CrossRef] [PubMed]

Kim, P.

Y. Zhang, Y.-W. Tan, H. L. Stormer, and P. Kim, “Experimental observation of the quantum Hall effect and Berry’s phase in graphene,” Nature438, 201–204 (2005).
[CrossRef] [PubMed]

Kivshar, Y. S.

S. S. Kruk, D. A. Powell, A. Minovich, D. N. Neshev, and Y. S. Kivshar, “Spatial dispersion of multilayer fishnet metamaterials,” Opt. Express20, 15100–15105 (2012).
[CrossRef] [PubMed]

A. A. Orlov, P. M. Voroshilov, P. A. Belov, and Y. S. Kivshar, “Engineered optical nonlocality in nanostructured metamaterials,” Phys. Rev. B84, 045424 (2011).
[CrossRef]

Kivshar, Yu. S.

A. V. Chebykin, A. A. Orlov, C. R. Simovski, Yu. S. Kivshar, and P. A. Belov, “Nonlocal effective parameters of multilayered metal-dielectric metamaterials,” Phy. Rev. B86, 115420 (2012).
[CrossRef]

Koschnya, T.

M. Diema, T. Koschnya, and C. M. Soukoulis, “Transmission in the vicinity of the Dirac point in hexagonal photonic crystals,” Physica B405, 2990–2995 (2010).
[CrossRef]

Kruk, S. S.

Lai, Y.

X. Huang, Y. Lai, Z. H. Hang, H. Zheng, and C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nat. Mater.10, 582–586 (2011).
[CrossRef] [PubMed]

Levitov, L. S.

D. A. Abanin, S. V. Morozov, L. A. Ponomarenko, R. V. Gorbachev, A. S. Mayorov, M. I. Katsnelson, K. Watanabe, T. Taniguchi, K. S. Novoselov, L. S. Levitov, and A. K. Geim, “Giant nonlocality near the Dirac point in graphene,” Science332, 328–330 (2011).
[CrossRef] [PubMed]

Li, C.-F.

X. Chen, L.-G. Wang, and C.-F. Li, “Transmission gap, Bragg-like reflection, and Goos-Hänchen shifts near the Dirac point inside a negative-zero-positive index metamaterial slab,” Phys. Rev. A80, 043839 (2009).
[CrossRef]

Li, H.

L. V. Alekseyev, E. E. Narimanov, T. Tumkur, H. Li, Yu. A. Barnakov, and M. A. Noginov, “Uniaxial epsilon-near-zero metamaterial for angular filtering and polarization control,” Appl. Phys. Lett.97, 131107 (2010).
[CrossRef]

Luk, T. S.

G. Subramania, A. J. Fischer, and T. S. Luk, “Optical properties of metal-dielectric based epsilon near zero metamaterials,” Appl. Phys. Lett.101, 241107 (2012).
[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, 103901 (2007).
[CrossRef] [PubMed]

Maradudin, A. A.

M. Plihal and A. A. Maradudin, “Photonic band structure of two-dimensional systems: The triangular lattice,” Phys. Rev. B44, 8565–8571 (1991).
[CrossRef]

Mayorov, A. S.

D. A. Abanin, S. V. Morozov, L. A. Ponomarenko, R. V. Gorbachev, A. S. Mayorov, M. I. Katsnelson, K. Watanabe, T. Taniguchi, K. S. Novoselov, L. S. Levitov, and A. K. Geim, “Giant nonlocality near the Dirac point in graphene,” Science332, 328–330 (2011).
[CrossRef] [PubMed]

Minovich, A.

Moran, M.

M. J. Roberts, S. Feng, M. Moran, and L. Johnson, “Effective permittivity near zero in nanolaminates of silver and amorphous polycarbonate,” J. Nanophotonics4, 043511 (2010).
[CrossRef]

Morozov, S. V.

D. A. Abanin, S. V. Morozov, L. A. Ponomarenko, R. V. Gorbachev, A. S. Mayorov, M. I. Katsnelson, K. Watanabe, T. Taniguchi, K. S. Novoselov, L. S. Levitov, and A. K. Geim, “Giant nonlocality near the Dirac point in graphene,” Science332, 328–330 (2011).
[CrossRef] [PubMed]

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, “Two-dimensional gas of massless Dirac fermions in graphene,” Nature438, 197–200 (2005).
[CrossRef] [PubMed]

Murphy, A.

R. J. Pollard, A. Murphy, W. R. Hendren, P. R. Evans, R. Atkinson, G. A. Wurtz, A. V. Zayats, and V. A. Podolskiy, “Optical nonlocalities and additional waves in epsilon-near-zero metamaterials,” Phys. Rev. Lett.102, 127405 (2009).
[CrossRef] [PubMed]

Narimanov, E. E.

L. V. Alekseyev, E. E. Narimanov, T. Tumkur, H. Li, Yu. A. Barnakov, and M. A. Noginov, “Uniaxial epsilon-near-zero metamaterial for angular filtering and polarization control,” Appl. Phys. Lett.97, 131107 (2010).
[CrossRef]

Neshev, D. N.

Noginov, M. A.

L. V. Alekseyev, E. E. Narimanov, T. Tumkur, H. Li, Yu. A. Barnakov, and M. A. Noginov, “Uniaxial epsilon-near-zero metamaterial for angular filtering and polarization control,” Appl. Phys. Lett.97, 131107 (2010).
[CrossRef]

Nori, F.

Y. Bliokh, V. Freilikher, and F. Nori, “Charge transport in graphene and light propagation in periodic dielectric structures with metamaterials: a comparative study,” arXiv:1302.5533v2 [physics.optics].

Novoselov, K. S.

D. A. Abanin, S. V. Morozov, L. A. Ponomarenko, R. V. Gorbachev, A. S. Mayorov, M. I. Katsnelson, K. Watanabe, T. Taniguchi, K. S. Novoselov, L. S. Levitov, and A. K. Geim, “Giant nonlocality near the Dirac point in graphene,” Science332, 328–330 (2011).
[CrossRef] [PubMed]

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

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, “Two-dimensional gas of massless Dirac fermions in graphene,” Nature438, 197–200 (2005).
[CrossRef] [PubMed]

Ochiai, T.

T. Ochiai and M. Onoda, “Photonic analog of graphene model and its extension: Dirac cone, symmetry, and edge states,” Phys. Rev. B80, 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. B80, 155103 (2009).
[CrossRef]

Orlov, A. A.

A. V. Chebykin, A. A. Orlov, C. R. Simovski, Yu. S. Kivshar, and P. A. Belov, “Nonlocal effective parameters of multilayered metal-dielectric metamaterials,” Phy. Rev. B86, 115420 (2012).
[CrossRef]

A. A. Orlov, P. M. Voroshilov, P. A. Belov, and Y. S. Kivshar, “Engineered optical nonlocality in nanostructured metamaterials,” Phys. Rev. B84, 045424 (2011).
[CrossRef]

Peleg, 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, 103901 (2007).
[CrossRef] [PubMed]

Peres, N. M. R.

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

Plihal, M.

M. Plihal and A. A. Maradudin, “Photonic band structure of two-dimensional systems: The triangular lattice,” Phys. Rev. B44, 8565–8571 (1991).
[CrossRef]

Podolskiy, V. A.

G. A. Wurtz, R. Pollard, W. Hendren, G. P. Wiederrecht, D. J. Gosztola, V. A. Podolskiy, and A. V. Zayats, “Designed ultrafast optical nonlinearity in a plasmonic nanorod metamaterial enhanced by nonlocality,” Nat. Nanotechnol.6, 107–111 (2011).
[CrossRef] [PubMed]

R. J. Pollard, A. Murphy, W. R. Hendren, P. R. Evans, R. Atkinson, G. A. Wurtz, A. V. Zayats, and V. A. Podolskiy, “Optical nonlocalities and additional waves in epsilon-near-zero metamaterials,” Phys. Rev. Lett.102, 127405 (2009).
[CrossRef] [PubMed]

J. Elser, V. A. Podolskiy, I. Salakhutdinov, and I. Avrutsky, “Nonlocal effects in effective-medium response of nanolayered metamaterials,” Appl. Phys. Lett.90, 191109 (2007).
[CrossRef]

Pollard, R.

G. A. Wurtz, R. Pollard, W. Hendren, G. P. Wiederrecht, D. J. Gosztola, V. A. Podolskiy, and A. V. Zayats, “Designed ultrafast optical nonlinearity in a plasmonic nanorod metamaterial enhanced by nonlocality,” Nat. Nanotechnol.6, 107–111 (2011).
[CrossRef] [PubMed]

Pollard, R. J.

R. J. Pollard, A. Murphy, W. R. Hendren, P. R. Evans, R. Atkinson, G. A. Wurtz, A. V. Zayats, and V. A. Podolskiy, “Optical nonlocalities and additional waves in epsilon-near-zero metamaterials,” Phys. Rev. Lett.102, 127405 (2009).
[CrossRef] [PubMed]

Polman, A.

E. J. R. Vesseur, T. Coenen, H. Caglayan, N. Engheta, and A. Polman, “Experimental verification of n= 0 structures for visible light,” Phys. Rev. Lett.110, 013902 (2013).
[CrossRef]

Ponomarenko, L. A.

D. A. Abanin, S. V. Morozov, L. A. Ponomarenko, R. V. Gorbachev, A. S. Mayorov, M. I. Katsnelson, K. Watanabe, T. Taniguchi, K. S. Novoselov, L. S. Levitov, and A. K. Geim, “Giant nonlocality near the Dirac point in graphene,” Science332, 328–330 (2011).
[CrossRef] [PubMed]

Powell, D. A.

Roberts, M. J.

M. J. Roberts, S. Feng, M. Moran, and L. Johnson, “Effective permittivity near zero in nanolaminates of silver and amorphous polycarbonate,” J. Nanophotonics4, 043511 (2010).
[CrossRef]

Sakoda, K.

Salakhutdinov, I.

J. Elser, V. A. Podolskiy, I. Salakhutdinov, and I. Avrutsky, “Nonlocal effects in effective-medium response of nanolayered metamaterials,” Appl. Phys. Lett.90, 191109 (2007).
[CrossRef]

Salandrino, A.

A. Alù, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: Tailoring the radiation phase pattern,” Phys. Rev. B75, 155410 (2007).
[CrossRef]

Segev, M.

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, 103901 (2007).
[CrossRef] [PubMed]

Sepkhanov, R. A.

R. A. Sepkhanov, Ya. B. Bazaliy, and C. W. J. Beenakker, “Extremal transmission at the Dirac point of a photonic band structure,” Phys. Rev. A75, 063813 (2007).
[CrossRef]

Silveirinha, M. G.

A. Alù, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: Tailoring the radiation phase pattern,” Phys. Rev. B75, 155410 (2007).
[CrossRef]

Simovski, C. R.

A. V. Chebykin, A. A. Orlov, C. R. Simovski, Yu. S. Kivshar, and P. A. Belov, “Nonlocal effective parameters of multilayered metal-dielectric metamaterials,” Phy. Rev. B86, 115420 (2012).
[CrossRef]

Soukoulis, C. M.

M. Diema, T. Koschnya, and C. M. Soukoulis, “Transmission in the vicinity of the Dirac point in hexagonal photonic crystals,” Physica B405, 2990–2995 (2010).
[CrossRef]

Stormer, H. L.

Y. Zhang, Y.-W. Tan, H. L. Stormer, and P. Kim, “Experimental observation of the quantum Hall effect and Berry’s phase in graphene,” Nature438, 201–204 (2005).
[CrossRef] [PubMed]

Subramania, G.

G. Subramania, A. J. Fischer, and T. S. Luk, “Optical properties of metal-dielectric based epsilon near zero metamaterials,” Appl. Phys. Lett.101, 241107 (2012).
[CrossRef]

Sun, L.

L. Sun, J. Gao, and X. Yang, “Broadband epsilon-near-zero metamaterials with steplike metal-dielectric multilayer structures,” Phys. Rev. B87, 165134 (2013).
[CrossRef]

L. Sun, S. Feng, and X. Yang, “Loss enhanced transmission and collimation in anisotropic epsilon-near-zero metamaterials,” Appl. Phys. Lett.101, 241101 (2012).
[CrossRef]

Tan, Y.-W.

Y. Zhang, Y.-W. Tan, H. L. Stormer, and P. Kim, “Experimental observation of the quantum Hall effect and Berry’s phase in graphene,” Nature438, 201–204 (2005).
[CrossRef] [PubMed]

Taniguchi, T.

D. A. Abanin, S. V. Morozov, L. A. Ponomarenko, R. V. Gorbachev, A. S. Mayorov, M. I. Katsnelson, K. Watanabe, T. Taniguchi, K. S. Novoselov, L. S. Levitov, and A. K. Geim, “Giant nonlocality near the Dirac point in graphene,” Science332, 328–330 (2011).
[CrossRef] [PubMed]

Tumkur, T.

L. V. Alekseyev, E. E. Narimanov, T. Tumkur, H. Li, Yu. A. Barnakov, and M. A. Noginov, “Uniaxial epsilon-near-zero metamaterial for angular filtering and polarization control,” Appl. Phys. Lett.97, 131107 (2010).
[CrossRef]

Vanakaras, A.

V. Yannopapas and A. Vanakaras, “Dirac point in the photon dispersion relation of a negative/zero/positive-index plasmonic metamaterial,” Phys. Rev. B84, 045128 (2011).
[CrossRef]

Vesseur, E. J. R.

E. J. R. Vesseur, T. Coenen, H. Caglayan, N. Engheta, and A. Polman, “Experimental verification of n= 0 structures for visible light,” Phys. Rev. Lett.110, 013902 (2013).
[CrossRef]

Voroshilov, P. M.

A. A. Orlov, P. M. Voroshilov, P. A. Belov, and Y. S. Kivshar, “Engineered optical nonlocality in nanostructured metamaterials,” Phys. Rev. B84, 045424 (2011).
[CrossRef]

Wallace, P. R.

P. R. Wallace, “The band theory of graphite,” Phys. Rev.71, 622–634 (1947).
[CrossRef]

Wang, L.-G.

X. Chen, L.-G. Wang, and C.-F. Li, “Transmission gap, Bragg-like reflection, and Goos-Hänchen shifts near the Dirac point inside a negative-zero-positive index metamaterial slab,” Phys. Rev. A80, 043839 (2009).
[CrossRef]

L.-G. Wang, Z.-G. Wang, J.-X. Zhang, and S.-Y. Zhu, “Realization of Dirac point with double cones in optics,” Opt. Lett.34, 1510–1512 (2009).
[CrossRef] [PubMed]

L.-G. Wang, Z.-G. Wang, and S.-Y. Zhu, “Zitterbewegung of optical pulses near the Dirac point inside a negative-zero-positive index metamaterial,” EPL86, 47008 (2009).
[CrossRef]

Wang, Z.-G.

L.-G. Wang, Z.-G. Wang, and S.-Y. Zhu, “Zitterbewegung of optical pulses near the Dirac point inside a negative-zero-positive index metamaterial,” EPL86, 47008 (2009).
[CrossRef]

L.-G. Wang, Z.-G. Wang, J.-X. Zhang, and S.-Y. Zhu, “Realization of Dirac point with double cones in optics,” Opt. Lett.34, 1510–1512 (2009).
[CrossRef] [PubMed]

Watanabe, K.

D. A. Abanin, S. V. Morozov, L. A. Ponomarenko, R. V. Gorbachev, A. S. Mayorov, M. I. Katsnelson, K. Watanabe, T. Taniguchi, K. S. Novoselov, L. S. Levitov, and A. K. Geim, “Giant nonlocality near the Dirac point in graphene,” Science332, 328–330 (2011).
[CrossRef] [PubMed]

Wiederrecht, G. P.

G. A. Wurtz, R. Pollard, W. Hendren, G. P. Wiederrecht, D. J. Gosztola, V. A. Podolskiy, and A. V. Zayats, “Designed ultrafast optical nonlinearity in a plasmonic nanorod metamaterial enhanced by nonlocality,” Nat. Nanotechnol.6, 107–111 (2011).
[CrossRef] [PubMed]

Wurtz, G. A.

G. A. Wurtz, R. Pollard, W. Hendren, G. P. Wiederrecht, D. J. Gosztola, V. A. Podolskiy, and A. V. Zayats, “Designed ultrafast optical nonlinearity in a plasmonic nanorod metamaterial enhanced by nonlocality,” Nat. Nanotechnol.6, 107–111 (2011).
[CrossRef] [PubMed]

R. J. Pollard, A. Murphy, W. R. Hendren, P. R. Evans, R. Atkinson, G. A. Wurtz, A. V. Zayats, and V. A. Podolskiy, “Optical nonlocalities and additional waves in epsilon-near-zero metamaterials,” Phys. Rev. Lett.102, 127405 (2009).
[CrossRef] [PubMed]

Yang, X.

L. Sun, J. Gao, and X. Yang, “Broadband epsilon-near-zero metamaterials with steplike metal-dielectric multilayer structures,” Phys. Rev. B87, 165134 (2013).
[CrossRef]

L. Sun, S. Feng, and X. Yang, “Loss enhanced transmission and collimation in anisotropic epsilon-near-zero metamaterials,” Appl. Phys. Lett.101, 241101 (2012).
[CrossRef]

Yannopapas, V.

V. Yannopapas and A. Vanakaras, “Dirac point in the photon dispersion relation of a negative/zero/positive-index plasmonic metamaterial,” Phys. Rev. B84, 045128 (2011).
[CrossRef]

Zayats, A. V.

G. A. Wurtz, R. Pollard, W. Hendren, G. P. Wiederrecht, D. J. Gosztola, V. A. Podolskiy, and A. V. Zayats, “Designed ultrafast optical nonlinearity in a plasmonic nanorod metamaterial enhanced by nonlocality,” Nat. Nanotechnol.6, 107–111 (2011).
[CrossRef] [PubMed]

R. J. Pollard, A. Murphy, W. R. Hendren, P. R. Evans, R. Atkinson, G. A. Wurtz, A. V. Zayats, and V. A. Podolskiy, “Optical nonlocalities and additional waves in epsilon-near-zero metamaterials,” Phys. Rev. Lett.102, 127405 (2009).
[CrossRef] [PubMed]

Zhang, J.-X.

Zhang, X.

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

Zhang, Y.

Y. Zhang, Y.-W. Tan, H. L. Stormer, and P. Kim, “Experimental observation of the quantum Hall effect and Berry’s phase in graphene,” Nature438, 201–204 (2005).
[CrossRef] [PubMed]

Zheng, H.

X. Huang, Y. Lai, Z. H. Hang, H. Zheng, and C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nat. Mater.10, 582–586 (2011).
[CrossRef] [PubMed]

Zhu, S.-Y.

L.-G. Wang, Z.-G. Wang, and S.-Y. Zhu, “Zitterbewegung of optical pulses near the Dirac point inside a negative-zero-positive index metamaterial,” EPL86, 47008 (2009).
[CrossRef]

L.-G. Wang, Z.-G. Wang, J.-X. Zhang, and S.-Y. Zhu, “Realization of Dirac point with double cones in optics,” Opt. Lett.34, 1510–1512 (2009).
[CrossRef] [PubMed]

Appl. Phys. Lett.

L. Sun, S. Feng, and X. Yang, “Loss enhanced transmission and collimation in anisotropic epsilon-near-zero metamaterials,” Appl. Phys. Lett.101, 241101 (2012).
[CrossRef]

L. V. Alekseyev, E. E. Narimanov, T. Tumkur, H. Li, Yu. A. Barnakov, and M. A. Noginov, “Uniaxial epsilon-near-zero metamaterial for angular filtering and polarization control,” Appl. Phys. Lett.97, 131107 (2010).
[CrossRef]

G. Subramania, A. J. Fischer, and T. S. Luk, “Optical properties of metal-dielectric based epsilon near zero metamaterials,” Appl. Phys. Lett.101, 241107 (2012).
[CrossRef]

J. Elser, V. A. Podolskiy, I. Salakhutdinov, and I. Avrutsky, “Nonlocal effects in effective-medium response of nanolayered metamaterials,” Appl. Phys. Lett.90, 191109 (2007).
[CrossRef]

EPL

L.-G. Wang, Z.-G. Wang, and S.-Y. Zhu, “Zitterbewegung of optical pulses near the Dirac point inside a negative-zero-positive index metamaterial,” EPL86, 47008 (2009).
[CrossRef]

J. Nanophotonics

M. J. Roberts, S. Feng, M. Moran, and L. Johnson, “Effective permittivity near zero in nanolaminates of silver and amorphous polycarbonate,” J. Nanophotonics4, 043511 (2010).
[CrossRef]

Nat. Mater.

X. Huang, Y. Lai, Z. H. Hang, H. Zheng, and C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nat. Mater.10, 582–586 (2011).
[CrossRef] [PubMed]

Nat. Nanotechnol.

G. A. Wurtz, R. Pollard, W. Hendren, G. P. Wiederrecht, D. J. Gosztola, V. A. Podolskiy, and A. V. Zayats, “Designed ultrafast optical nonlinearity in a plasmonic nanorod metamaterial enhanced by nonlocality,” Nat. Nanotechnol.6, 107–111 (2011).
[CrossRef] [PubMed]

Nature

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, “Two-dimensional gas of massless Dirac fermions in graphene,” Nature438, 197–200 (2005).
[CrossRef] [PubMed]

Y. Zhang, Y.-W. Tan, H. L. Stormer, and P. Kim, “Experimental observation of the quantum Hall effect and Berry’s phase in graphene,” Nature438, 201–204 (2005).
[CrossRef] [PubMed]

Opt. Express

Opt. Lett.

Phy. Rev. B

A. V. Chebykin, A. A. Orlov, C. R. Simovski, Yu. S. Kivshar, and P. A. Belov, “Nonlocal effective parameters of multilayered metal-dielectric metamaterials,” Phy. Rev. B86, 115420 (2012).
[CrossRef]

Phys. Rev.

P. R. Wallace, “The band theory of graphite,” Phys. Rev.71, 622–634 (1947).
[CrossRef]

Phys. Rev. A

R. A. Sepkhanov, Ya. B. Bazaliy, and C. W. J. Beenakker, “Extremal transmission at the Dirac point of a photonic band structure,” Phys. Rev. A75, 063813 (2007).
[CrossRef]

X. Chen, L.-G. Wang, and C.-F. Li, “Transmission gap, Bragg-like reflection, and Goos-Hänchen shifts near the Dirac point inside a negative-zero-positive index metamaterial slab,” Phys. Rev. A80, 043839 (2009).
[CrossRef]

Phys. Rev. B

L. Sun, J. Gao, and X. Yang, “Broadband epsilon-near-zero metamaterials with steplike metal-dielectric multilayer structures,” Phys. Rev. B87, 165134 (2013).
[CrossRef]

A. Alù, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: Tailoring the radiation phase pattern,” Phys. Rev. B75, 155410 (2007).
[CrossRef]

T. Ochiai and M. Onoda, “Photonic analog of graphene model and its extension: Dirac cone, symmetry, and edge states,” Phys. Rev. B80, 155103 (2009).
[CrossRef]

V. Yannopapas and A. Vanakaras, “Dirac point in the photon dispersion relation of a negative/zero/positive-index plasmonic metamaterial,” Phys. Rev. B84, 045128 (2011).
[CrossRef]

M. Plihal and A. A. Maradudin, “Photonic band structure of two-dimensional systems: The triangular lattice,” Phys. Rev. B44, 8565–8571 (1991).
[CrossRef]

A. A. Orlov, P. M. Voroshilov, P. A. Belov, and Y. S. Kivshar, “Engineered optical nonlocality in nanostructured metamaterials,” Phys. Rev. B84, 045424 (2011).
[CrossRef]

Phys. Rev. Lett.

E. J. R. Vesseur, T. Coenen, H. Caglayan, N. Engheta, and A. Polman, “Experimental verification of n= 0 structures for visible light,” Phys. Rev. Lett.110, 013902 (2013).
[CrossRef]

R. J. Pollard, A. Murphy, W. R. Hendren, P. R. Evans, R. Atkinson, G. A. Wurtz, A. V. Zayats, and V. A. Podolskiy, “Optical nonlocalities and additional waves in epsilon-near-zero metamaterials,” Phys. Rev. Lett.102, 127405 (2009).
[CrossRef] [PubMed]

G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “Nonlocal transformation optics,” Phys. Rev. Lett.108, 063902 (2012).
[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, 103901 (2007).
[CrossRef] [PubMed]

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

Physica B

M. Diema, T. Koschnya, and C. M. Soukoulis, “Transmission in the vicinity of the Dirac point in hexagonal photonic crystals,” Physica B405, 2990–2995 (2010).
[CrossRef]

Rev. Mod. Phys.

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

Science

D. A. Abanin, S. V. Morozov, L. A. Ponomarenko, R. V. Gorbachev, A. S. Mayorov, M. I. Katsnelson, K. Watanabe, T. Taniguchi, K. S. Novoselov, L. S. Levitov, and A. K. Geim, “Giant nonlocality near the Dirac point in graphene,” Science332, 328–330 (2011).
[CrossRef] [PubMed]

Other

Y. Bliokh, V. Freilikher, and F. Nori, “Charge transport in graphene and light propagation in periodic dielectric structures with metamaterials: a comparative study,” arXiv:1302.5533v2 [physics.optics].

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Figures (4)

Fig. 1
Fig. 1

(a) The metal-dielectric multilayer stack consists of alternating layers of gold (Au) and alumina (Al2O3), with d1 = 15nm and d2 = 35nm and the permittivity ε1 and ε2, respectively. (b) The band structure of the multilayer stack in the first BZ calculated from Eq. (3) in three dimensional ω-k space. (c) The band 1 and band 2 in the band structure of the multilayer stack is connected by two Dirac points at the positions determined by Eq. (5).

Fig. 2
Fig. 2

(a) The dispersion relation kx/kpω/ωp based on Eq. (3), under the condition of k y / k p = ε 2 d 1 d 2 / ( ( d 2 d 1 ) ( ε d 1 + ε 2 d 2 ) ) in red curves. The band 1 and band 2 are connected at the Dirac point, located at the position of kx/kp = 0 according to Eq. (5). The black lines indicate the linear dispersion in the neighborhood of the Dirac point consistent with Eq. (6). (b) The dispersion relation ky/kpω/ωp near the Dirac point based on Eq. (3), with respect to kx/kp = 0 in red curves. The dispersion relation obtained from the EMT and the SPP dispersion relation are plotted in dot-dashed black curve and dashed blue curves, respectively. All the dispersion curves intersect at the Dirac point. The dispersion curves obtained from the multilayer stack converge to the SPP dispersion when the wave vector ky/kp increases, due to the SPR. The linear dispersion relation in the neighborhood of the Dirac point is plotted in black lines based on Eq. (10). (c) The positions of the ENZ determined from the EMT and the multilayer stack including the optical nonlocality are marked for comparison.

Fig. 3
Fig. 3

The variations of the IFCs at five different frequencies around the Dirac point (a) 636.577 THz (the nonlocal ENZ frequency), (b) 641 THz (slightly below the Dirac point), (c) 647.027 THz (at the Dirac point), (d) 651 THz (slightly above the Dirac point), and (e) 671 THz (far above the Dirac point). The IFCs from Eq. (3) with the optical nonlocality are plotted in red curves, while the EMT calculated IFCs are plotted in blue curves. The IFC of air is plotted in green circle for reference. The IFCs from Eq. (3) consist of two branches, and two eigenmodes degenerate at the Dirac point. On the contrary, only a single branch exists in the EMT calculated IFCs, which reduces into a straight line at the frequency of the Dirac point.

Fig. 4
Fig. 4

The eigenmodes and the propagating patterns of the electromagnetic wave in the multilayer stack at three different frequencies of (a) 641 THz, (b) 647.027 THz, and (c) 651 THz near the Dirac point, corresponding to the IFCs in Fig. 3. The symmetry eigen-mode and the asymmetry eigenmode in one Al2O3-Au-Al2O3 unit cell of the multilayer stack are represented by the amplitude and the absolute value of the magnetic field Hz. The two eigenmodes degenerate as one symmetric mode at the frequency of the Dirac point, and invert as the frequency across the frequency of the Dirac point. The propagating patterns are plotted for both the multilayer stack and the corresponding effective medium for a normal incident Gaussian beam of TM mode (Ex, Hz, and ky), represented by the distribution of the absolute value of the magnetic field abs(Hz). Similar diverging patterns can be observed when the frequency is above and below the Dirac point, due to the sharp curvature of the IFC. The giant optical nonlocality at the Dirac point leads to a beam splitting in the multilayer stack that is dramatically different from the propagating property in the effective medium.

Equations (26)

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ε x ( 0 ) = ε 1 ε 2 ( d 1 + d 2 ) ε 1 d 2 + ε 2 d 1 = ε 1 ε 2 ( 1 f 1 ) ε 1 + f 1 ε 2 ,
ε y ( 0 ) = ε z ( 0 ) = ε 1 d 1 + ε 2 d 2 d 1 + d 2 = f 1 ε 1 + ( 1 f 1 ) ε 2 .
cos ( k x ( d 1 + d 2 ) ) = cos ( k x ( 1 ) d 1 ) cos ( k x ( 2 ) d 2 ) 1 2 ( ε 1 k x ( 2 ) ε 2 k x ( 1 ) + ε 2 k x ( 1 ) ε 1 k x ( 2 ) ) sin ( k x ( 1 ) d 1 ) sin ( k x ( 2 ) d 2 ) ,
cos ( k x ( d 1 + d 2 ) ) = cos ( k ε 1 d 1 + ε 2 d 2 ε 1 + ε 2 ) .
{ k x / k p = 0 k y / k p = ε 2 d 1 d 2 ( d 2 d 1 ) ( ε d 1 + ε 2 d 2 ) = f 1 ( 1 f 1 ) ε 2 ( 1 2 f 1 ) ( f 1 ε + ( 1 f 1 ) ε 2 ) , ω / ω p = d 1 ε d 1 + ε 2 d 2 = f 1 f 1 ε + ( 1 f 1 ) ε 2
ω / ω p = ± k p ( d 1 + d 2 ) 2 Δ ( k x / k p ) + d 1 ε d 1 + ε d d 2 ,
k x 2 ε y ( 0 ) + k y 2 ε x ( 0 ) = ( ω c ) 2 ,
k y / k p = ε x ( 0 ) ω / ω p ,
{ k y / k p ω / ω p = d 1 / ( ε d 1 + ε 2 d 2 ) ,
ω ω p d 1 ε d 1 + ε 2 d 2 = C 2 ± C 2 2 4 C 1 C 3 2 C 1 ( k y k p ε 2 d 1 d 2 ( d 2 d 1 ) ( ε d 1 + ε 2 d 2 ) ) .
k x 2 ε y eff + k y 2 ε x eff = ( ω c ) 2 ,
δ x = A x 1 + A x 2 + A x 3 + A x 4 + A x 5 + A x 6 B x 0 ( B x 1 + B x 2 + B x 3 + B x 4 + B x 5 ) ,
δ y = A y 1 + A y 2 + A y 3 + A y 4 + A y 5 B y 1 + B y 2 + B y 3 + B y 4 + B y 5 ,
lim ( ε y eff ) = ε 2 2 d 1 d 2 2 k p 2 12 ( ε d 1 + ε 2 d 2 ) = 0.108129
Δ = 1 4 ε 2 2 d 1 3 d 2 4 ( Δ 1 + Δ 2 + Δ 3 + Δ 4 ) ,
{ Δ 1 = ε ε 2 d 1 d 2 ( ε d 1 2 + ε 2 d 2 2 ) ( d 1 + d 2 ) 3 Δ 2 = ( ε d 1 2 + ε 2 d 2 2 ) ( ε 2 d 1 3 + ε 2 2 d 2 3 ) ( d 1 + d 2 ) 2 Δ 3 = 2 ε 2 d 1 2 d 2 2 k p 2 ( ε d 1 2 ε 2 d 2 2 ) ( d 2 d 2 ) Δ 4 = ( ε d 1 + ε 2 d 2 ) ( ε d 1 2 + ε 2 d 2 2 ) 2 ( d 1 + d 2 ) 2 cosh ( 2 ε 2 d 1 d 2 k p ( d 2 d 1 ) ( ε d 1 + ε 2 d 2 ) ) .
C 1 = 1 4 ε 2 2 d 1 3 d 2 4 ( C 1 ( 1 ) + C 1 ( 2 ) + C 1 ( 3 ) + C 1 ( 4 ) + C 1 ( 5 ) ) ,
C 2 = d 1 2 d 2 2 2 ε 2 d 1 3 d 2 3 d 2 d 1 ε 2 d 2 ( C 2 ( 1 ) + C 2 ( 2 ) + C 2 ( 3 ) + C 2 ( 4 ) ) ,
C 3 = ( d 1 2 d 2 2 ) 2 4 ε 2 d 1 3 d 2 3 ( C 3 ( 1 ) + C 3 ( 2 ) + C 3 ( 3 ) ) ,
{ C 1 ( 1 ) = ε 2 3 d 2 5 ( ( d 1 + d 2 ) 2 2 d 1 2 d 2 ( d 1 d 2 ) k p 2 ) C 1 ( 2 ) = ε ε 2 2 d 1 d 2 3 ( ( d 1 + d 2 ) 2 ( 2 d 1 + d 2 ) + 4 d 1 3 d 2 ( d 1 d 2 ) k p 2 ) C 1 ( 3 ) = ε 2 ε 2 d 1 3 d 2 ( ( d 1 + d 2 ) 2 ( d 1 + 2 d 2 ) 2 d 1 3 d 2 ( d 1 d 2 ) k p 2 ) C 1 ( 4 ) = ε 3 d 1 5 ( d 1 + d 2 ) 2 C 1 ( 5 ) = ( d 1 + d 2 ) 2 ( ε d 1 + ε 2 d 2 ) ( ε d 1 2 + ε 2 d 2 2 ) 2 cosh ( 2 ε 2 d 1 d 2 k p ( d 2 d 1 ) ( ε d 1 + ε 2 d 2 ) ) ,
{ C 2 ( 1 ) = ε 2 2 d 2 3 ( d 1 + d 2 + 2 d 1 2 d 2 k p 2 ) C 2 ( 2 ) = ε ε 2 d 1 d 2 ( ( d 1 + d 2 ) 2 2 d 1 3 d 2 k p 2 ) C 2 ( 3 ) = ε 2 d 1 3 ( d 1 + d 2 ) C 2 ( 4 ) = ( d 1 + d 2 ) ( ε d 1 + ε 2 d 2 ) ( ε d 1 2 + ε 2 d 2 2 ) cosh ( 2 ε 2 d 1 d 2 k p ( d 2 d 1 ) ( ε d 1 + ε 2 d 2 ) ) ,
{ C 3 ( 1 ) = ε 2 d 2 ( d 1 + d 2 + 2 d 1 2 d 2 k p 2 ) C 3 ( 2 ) = ε d 1 ( d 1 + d 2 ) C 3 ( 3 ) = ( d 1 d 2 ) ( ε d 1 + ε 2 d 2 ) cosh ( 2 ε 2 d 1 d 2 k p ( d 2 d 1 ) ( ε d 1 + ε 2 d 2 ) ) .
{ A x 1 = ε 1 ε 2 ( k y 2 ε 1 k 2 ) d 1 5 A x 2 = ( ( 2 ε 1 2 + ε 1 ε 2 + 2 ε 2 2 ) k y 2 ε 1 ( ε 1 2 + 2 ε 1 ε 2 + 2 ε 2 2 ) k 2 ) d 1 4 d 2 A x 3 = 2 ( ( ε 1 2 + 3 ε 1 ε 2 + ε 2 2 ) k y 2 ε 1 ε 2 ( 3 ε 1 + 2 ε 2 ) k 2 ) d 1 3 d 2 2 A x 4 = 2 ( ( ε 1 2 + 3 ε 1 ε 2 + ε 2 2 ) k y 2 ε 1 ε 2 ( 2 ε 1 + 3 ε 2 ) k 2 ) d 1 2 d 2 3 A x 5 = ( ( 2 ε 1 2 + ε 1 ε 2 + 2 ε 2 2 ) k y 2 ε 1 ( 2 ε 1 2 + 2 ε 1 ε 2 + ε 2 2 ) k 2 ) d 1 d 2 4 A x 6 = ε 1 ε 2 ( k y 2 ε 2 k 2 ) d 2 5 ,
{ B x 0 = ε 1 d 2 + ε 2 d 1 B x 1 = ε 1 2 k 2 d 1 4 B x 2 = 2 ε 1 ( ε 1 + ε 2 ) k 2 d 1 3 d 2 B x 3 = 6 ε 1 ( ε 2 k 2 d 2 2 2 ) d 1 2 B x 4 = 2 ( ε 1 + ε 2 ) ( ε 2 k 2 d 2 2 6 ) d 1 d 2 B x 5 = ε 2 ( ε 2 k 2 d 2 2 12 ) d 2 2 ,
{ A y 1 = ε 1 ( ε 1 k 2 k x 2 ) d 1 4 A y 2 = ( 2 ε 1 ( ε 1 + ε 2 ) k 2 ( 3 ε 1 + ε 2 ) k x 2 ) d 1 3 d 2 A y 3 = 3 ( 2 ε 1 ε 2 k 2 ( ε 1 + ε 2 ) k x 2 ) d 1 2 d 2 2 A y 4 = ( 2 ε 2 ( ε 1 + ε 2 ) k 2 ( ε 1 + 3 ε 2 ) k x 2 ) d 1 d 2 3 A y 5 = ε 2 ( ε 2 k 2 k x 2 ) d 2 4 ,
{ B y 1 = ε 1 2 k 2 d 1 4 B y 2 = 2 ε 1 ( ε 1 + ε 2 ) k 2 d 1 3 d 2 B y 3 = 6 ε 1 ( ε 2 k 2 d 2 2 2 ) d 1 2 B y 4 = 2 ( ε 1 + ε 2 ) ( ε 2 k 2 d 2 2 6 ) d 1 d 2 B y 5 = ε 2 ( ε 2 k 2 d 2 2 12 ) d 2 2 ,

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