Abstract

Accidental degeneracy in a photonic crystal consisting of a square array of elliptical dielectric cylinders leads to both a semi-Dirac point at the center of the Brillouin zone and an electromagnetic topological transition (ETT). A perturbation method is deduced to affirm the peculiar linear-parabolic dispersion near the semi-Dirac point. An effective medium theory is developed to explain the simultaneous semi-Dirac point and ETT and to show that the photonic crystal is either a zero-refractive-index material or an epsilon-near-zero material at the semi-Dirac point. Drastic changes in the wave manipulation properties at the semi-Dirac point, resulting from ETT, are described.

© 2014 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  34. B. Edwards, A. Alù, M. E. Young, M. Silveirinha, N. Engheta, “Experimental verification of epsilon-near-zero metamaterial coupling and energy squeezing using a microwave waveguide,” Phys. Rev. Lett. 100(3), 033903 (2008).
    [CrossRef] [PubMed]
  35. N. Engheta, “Materials Science. Pursuing Near-Zero Response,” Science 340(6130), 286–287 (2013).
    [CrossRef] [PubMed]
  36. B. A. Foreman, “Theory of the effective Hamiltonian for degenerate bands in an electric field,” J. Phys. Condens. Matter 12(34), R435–R461 (2000).
    [CrossRef]
  37. Y. Wu, J. Li, Z.-Q. Zhang, C. T. Chan, “Effective medium theory for magnetodielectric composites: Beyond the long-wavelength limit,” Phys. Rev. B 74(8), 085111 (2006).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  41. Q. Cheng, W. X. Jiang, T. J. Cui, “Spatial Power Combination for Omnidirectional Radiation via Anisotropic Metamaterials,” Phys. Rev. Lett. 108(21), 213903 (2012).
    [CrossRef] [PubMed]

2013 (7)

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, A. Szameit, “Photonic Floquet topological insulators,” Nature 496(7444), 196–200 (2013).
[CrossRef] [PubMed]

Y. P. Bliokh, V. Freilikher, F. Nori, “Ballistic charge transport in graphene and light propagation in periodic dielectric structures with metamaterials: A comparative study,” Phys. Rev. B 87(24), 245134 (2013).
[CrossRef]

D. Torrent, D. Mayou, J. Sánchez-Dehesa, “Elastic analog of graphene: Dirac cones and edge states for flexural waves in thin plates,” Phys. Rev. B 87(11), 115143 (2013).
[CrossRef]

Y. Li, Y. Wu, X. Chen, J. Mei, “Selection rule for Dirac-like points in two-dimensional dielectric photonic crystals,” Opt. Express 21(6), 7699–7711 (2013).
[CrossRef] [PubMed]

Y. Wu, J. Li, “Total reflection and cloaking by zero index metamaterials loaded with rectangular dielectric defects,” Appl. Phys. Lett. 102(18), 183105 (2013).
[CrossRef]

A. A. Basharin, C. Mavidis, M. Kafesaki, E. N. Economou, C. M. Soukoulis, “Epsilon near zero based phenomena in metamaterials,” Phys. Rev. B 87(15), 155130 (2013).
[CrossRef]

N. Engheta, “Materials Science. Pursuing Near-Zero Response,” Science 340(6130), 286–287 (2013).
[CrossRef] [PubMed]

2012 (10)

H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, V. M. Menon, “Topological Transitions in Metamaterials,” Science 336(6078), 205–209 (2012).
[CrossRef] [PubMed]

D. Torrent, J. Sánchez-Dehesa, “Acoustic Analogue of Graphene: Observation of Dirac Cones in Acoustic Surface Waves,” Phys. Rev. Lett. 108(17), 174301 (2012).
[CrossRef] [PubMed]

K. Sakoda, “Dirac cone in two- and three-dimensional metamaterials,” Opt. Express 20(4), 3898–3917 (2012).
[CrossRef] [PubMed]

K. Sakoda, “Proof of the universality of mode symmetries in creating photonic Dirac cones,” Opt. Express 20(22), 25181–25194 (2012).
[CrossRef] [PubMed]

J. Mei, Y. Wu, C. T. Chan, Z.-Q. Zhang, “First-principles study of Dirac and Dirac-like cones in phononic and photonic crystals,” Phys. Rev. B 86(3), 035141 (2012).
[CrossRef]

J. Bravo-Abad, J. D. Joannopoulos, M. Soljačić, “Enabling single-mode behavior over large areas with photonic Dirac cones,” Proc. Natl. Acad. Sci. U.S.A. 109(25), 9761–9765 (2012).
[CrossRef] [PubMed]

A. B. Khanikaev, S. H. Mousavi, W.-K. Tse, M. Kargarian, A. H. MacDonald, G. Shvets, “Photonic topological insulators,” Nat. Mater. 12(3), 233–239 (2012).
[CrossRef] [PubMed]

H. F. Ma, J. H. Shi, B. G. Cai, T. J. Cui, “Total transmission and super reflection realized by anisotropic zero-index materials,” New J. Phys. 14(12), 123010 (2012).
[CrossRef]

J. Luo, P. Xu, H. Chen, B. Hou, L. Gao, Y. Lai, “Realizing almost perfect bending waveguides with anisotropic epsilon-near-zero metamaterials,” Appl. Phys. Lett. 100(22), 221903 (2012).
[CrossRef]

Q. Cheng, W. X. Jiang, T. J. Cui, “Spatial Power Combination for Omnidirectional Radiation via Anisotropic Metamaterials,” Phys. Rev. Lett. 108(21), 213903 (2012).
[CrossRef] [PubMed]

2011 (4)

Y. Lai, Y. Wu, P. Sheng, Z.-Q. Zhang, “Hybrid elastic solids,” Nat. Mater. 10(8), 620–624 (2011).
[CrossRef] [PubMed]

X. Huang, Y. Lai, Z. H. Hang, H. Zheng, C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nat. Mater. 10(8), 582–586 (2011).
[CrossRef] [PubMed]

V. Yannopapas, “Photonic analog of a spin-polarized system with Rashba spin-orbit coupling,” Phys. Rev. B 83(11), 113101 (2011).
[CrossRef]

M. O. Goerbig, “Electronic properties of graphene in a strong magnetic field,” Rev. Mod. Phys. 83(4), 1193–1243 (2011).
[CrossRef]

2010 (3)

J. Hao, W. Yan, M. Qiu, “Super-reflection and cloaking based on zero index metamaterial,” Appl. Phys. Lett. 96(10), 101109 (2010).
[CrossRef]

V. C. Nguyen, L. Chen, K. Halterman, “Total Transmission and Total Reflection by Zero Index Metamaterials with Defects,” Phys. Rev. Lett. 105(23), 233908 (2010).
[CrossRef] [PubMed]

S. R. Zandbergen, M. J. A. de Dood, “Experimental observation of strong edge effects on the pseudodiffusive transport of light in photonic graphene,” Phys. Rev. Lett. 104(4), 043903 (2010).
[CrossRef] [PubMed]

2009 (5)

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

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

V. Pardo, W. E. Pickett, “Half-Metallic Semi-Dirac-Point Generated by Quantum Confinement in TiO2/VO2 Nanostructures,” Phys. Rev. Lett. 102(16), 166803 (2009).
[CrossRef] [PubMed]

S. Banerjee, R. R. P. Singh, V. Pardo, W. E. Pickett, “Tight-Binding Modeling and Low-Energy Behavior of the Semi-Dirac Point,” Phys. Rev. Lett. 103(1), 016402 (2009).
[CrossRef] [PubMed]

G. Montambaux, F. Piéchon, J. N. Fuchs, M. O. Goerbig, “Merging of Dirac points in a two-dimensional crystal,” Phys. Rev. B 80(15), 153412 (2009).
[CrossRef]

2008 (5)

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

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

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]

X. Zhang, Z. Liu, “Extremal Transmission and Beating Effect of Acoustic Waves in Two-Dimensional Sonic Crystals,” Phys. Rev. Lett. 101(26), 264303 (2008).
[CrossRef] [PubMed]

B. Edwards, A. Alù, M. E. Young, M. Silveirinha, N. Engheta, “Experimental verification of epsilon-near-zero metamaterial coupling and energy squeezing using a microwave waveguide,” Phys. Rev. Lett. 100(3), 033903 (2008).
[CrossRef] [PubMed]

2007 (3)

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

R. A. Sepkhanov, Y. B. Bazaliy, C. W. J. Beenakker, “Extremal transmission at the Dirac point of a photonic band structure,” Phys. Rev. A 75(6), 063813 (2007).
[CrossRef]

A. Alu, M. G. Silveirinha, A. Salandrino, N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: Tailoring the radiation phase pattern,” Phys. Rev. B 75(15), 155410 (2007).
[CrossRef]

2006 (2)

M. Silveirinha, N. Engheta, “Tunneling of electromagnetic energy through subwavelength channels and bends using epsilon-near-zero materials,” Phys. Rev. Lett. 97(15), 157403 (2006).
[CrossRef] [PubMed]

Y. Wu, J. Li, Z.-Q. Zhang, C. T. Chan, “Effective medium theory for magnetodielectric composites: Beyond the long-wavelength limit,” Phys. Rev. B 74(8), 085111 (2006).
[CrossRef]

2000 (1)

B. A. Foreman, “Theory of the effective Hamiltonian for degenerate bands in an electric field,” J. Phys. Condens. Matter 12(34), R435–R461 (2000).
[CrossRef]

1947 (1)

P. R. Wallace, “The band theory of Graphite,” Phys. Rev. 71(9), 622–634 (1947).
[CrossRef]

Alu, A.

A. Alu, M. G. Silveirinha, A. Salandrino, N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: Tailoring the radiation phase pattern,” Phys. Rev. B 75(15), 155410 (2007).
[CrossRef]

Alù, A.

B. Edwards, A. Alù, M. E. Young, M. Silveirinha, N. Engheta, “Experimental verification of epsilon-near-zero metamaterial coupling and energy squeezing using a microwave waveguide,” Phys. Rev. Lett. 100(3), 033903 (2008).
[CrossRef] [PubMed]

Banerjee, S.

S. Banerjee, R. R. P. Singh, V. Pardo, W. E. Pickett, “Tight-Binding Modeling and Low-Energy Behavior of the Semi-Dirac Point,” Phys. Rev. Lett. 103(1), 016402 (2009).
[CrossRef] [PubMed]

Basharin, A. A.

A. A. Basharin, C. Mavidis, M. Kafesaki, E. N. Economou, C. M. Soukoulis, “Epsilon near zero based phenomena in metamaterials,” Phys. Rev. B 87(15), 155130 (2013).
[CrossRef]

Bazaliy, Y. B.

R. A. Sepkhanov, Y. B. Bazaliy, C. W. J. Beenakker, “Extremal transmission at the Dirac point of a photonic band structure,” Phys. Rev. A 75(6), 063813 (2007).
[CrossRef]

Beenakker, C. W. J.

R. A. Sepkhanov, Y. B. Bazaliy, C. W. J. Beenakker, “Extremal transmission at the Dirac point of a photonic band structure,” Phys. Rev. A 75(6), 063813 (2007).
[CrossRef]

Bliokh, Y. P.

Y. P. Bliokh, V. Freilikher, F. Nori, “Ballistic charge transport in graphene and light propagation in periodic dielectric structures with metamaterials: A comparative study,” Phys. Rev. B 87(24), 245134 (2013).
[CrossRef]

Bravo-Abad, J.

J. Bravo-Abad, J. D. Joannopoulos, M. Soljačić, “Enabling single-mode behavior over large areas with photonic Dirac cones,” Proc. Natl. Acad. Sci. U.S.A. 109(25), 9761–9765 (2012).
[CrossRef] [PubMed]

Cai, B. G.

H. F. Ma, J. H. Shi, B. G. Cai, T. J. Cui, “Total transmission and super reflection realized by anisotropic zero-index materials,” New J. Phys. 14(12), 123010 (2012).
[CrossRef]

Castro Neto, A. H.

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

Chan, C. T.

J. Mei, Y. Wu, C. T. Chan, Z.-Q. Zhang, “First-principles study of Dirac and Dirac-like cones in phononic and photonic crystals,” Phys. Rev. B 86(3), 035141 (2012).
[CrossRef]

X. Huang, Y. Lai, Z. H. Hang, H. Zheng, C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nat. Mater. 10(8), 582–586 (2011).
[CrossRef] [PubMed]

Y. Wu, J. Li, Z.-Q. Zhang, C. T. Chan, “Effective medium theory for magnetodielectric composites: Beyond the long-wavelength limit,” Phys. Rev. B 74(8), 085111 (2006).
[CrossRef]

Chen, H.

J. Luo, P. Xu, H. Chen, B. Hou, L. Gao, Y. Lai, “Realizing almost perfect bending waveguides with anisotropic epsilon-near-zero metamaterials,” Appl. Phys. Lett. 100(22), 221903 (2012).
[CrossRef]

Chen, L.

V. C. Nguyen, L. Chen, K. Halterman, “Total Transmission and Total Reflection by Zero Index Metamaterials with Defects,” Phys. Rev. Lett. 105(23), 233908 (2010).
[CrossRef] [PubMed]

Chen, X.

Cheng, Q.

Q. Cheng, W. X. Jiang, T. J. Cui, “Spatial Power Combination for Omnidirectional Radiation via Anisotropic Metamaterials,” Phys. Rev. Lett. 108(21), 213903 (2012).
[CrossRef] [PubMed]

Cui, T. J.

Q. Cheng, W. X. Jiang, T. J. Cui, “Spatial Power Combination for Omnidirectional Radiation via Anisotropic Metamaterials,” Phys. Rev. Lett. 108(21), 213903 (2012).
[CrossRef] [PubMed]

H. F. Ma, J. H. Shi, B. G. Cai, T. J. Cui, “Total transmission and super reflection realized by anisotropic zero-index materials,” New J. Phys. 14(12), 123010 (2012).
[CrossRef]

de Dood, M. J. A.

S. R. Zandbergen, M. J. A. de Dood, “Experimental observation of strong edge effects on the pseudodiffusive transport of light in photonic graphene,” Phys. Rev. Lett. 104(4), 043903 (2010).
[CrossRef] [PubMed]

Dreisow, F.

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, A. Szameit, “Photonic Floquet topological insulators,” Nature 496(7444), 196–200 (2013).
[CrossRef] [PubMed]

Economou, E. N.

A. A. Basharin, C. Mavidis, M. Kafesaki, E. N. Economou, C. M. Soukoulis, “Epsilon near zero based phenomena in metamaterials,” Phys. Rev. B 87(15), 155130 (2013).
[CrossRef]

Edwards, B.

B. Edwards, A. Alù, M. E. Young, M. Silveirinha, N. Engheta, “Experimental verification of epsilon-near-zero metamaterial coupling and energy squeezing using a microwave waveguide,” Phys. Rev. Lett. 100(3), 033903 (2008).
[CrossRef] [PubMed]

Engheta, N.

N. Engheta, “Materials Science. Pursuing Near-Zero Response,” Science 340(6130), 286–287 (2013).
[CrossRef] [PubMed]

B. Edwards, A. Alù, M. E. Young, M. Silveirinha, N. Engheta, “Experimental verification of epsilon-near-zero metamaterial coupling and energy squeezing using a microwave waveguide,” Phys. Rev. Lett. 100(3), 033903 (2008).
[CrossRef] [PubMed]

A. Alu, M. G. Silveirinha, A. Salandrino, N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: Tailoring the radiation phase pattern,” Phys. Rev. B 75(15), 155410 (2007).
[CrossRef]

M. Silveirinha, N. Engheta, “Tunneling of electromagnetic energy through subwavelength channels and bends using epsilon-near-zero materials,” Phys. Rev. Lett. 97(15), 157403 (2006).
[CrossRef] [PubMed]

Foreman, B. A.

B. A. Foreman, “Theory of the effective Hamiltonian for degenerate bands in an electric field,” J. Phys. Condens. Matter 12(34), R435–R461 (2000).
[CrossRef]

Freilikher, V.

Y. P. Bliokh, V. Freilikher, F. Nori, “Ballistic charge transport in graphene and light propagation in periodic dielectric structures with metamaterials: A comparative study,” Phys. Rev. B 87(24), 245134 (2013).
[CrossRef]

Fuchs, J. N.

G. Montambaux, F. Piéchon, J. N. Fuchs, M. O. Goerbig, “Merging of Dirac points in a two-dimensional crystal,” Phys. Rev. B 80(15), 153412 (2009).
[CrossRef]

Gao, L.

J. Luo, P. Xu, H. Chen, B. Hou, L. Gao, Y. Lai, “Realizing almost perfect bending waveguides with anisotropic epsilon-near-zero metamaterials,” Appl. Phys. Lett. 100(22), 221903 (2012).
[CrossRef]

Geim, A. K.

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

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

Goerbig, M. O.

M. O. Goerbig, “Electronic properties of graphene in a strong magnetic field,” Rev. Mod. Phys. 83(4), 1193–1243 (2011).
[CrossRef]

G. Montambaux, F. Piéchon, J. N. Fuchs, M. O. Goerbig, “Merging of Dirac points in a two-dimensional crystal,” Phys. Rev. B 80(15), 153412 (2009).
[CrossRef]

Guinea, F.

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

Haldane, F. D. M.

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

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

Halterman, K.

V. C. Nguyen, L. Chen, K. Halterman, “Total Transmission and Total Reflection by Zero Index Metamaterials with Defects,” Phys. Rev. Lett. 105(23), 233908 (2010).
[CrossRef] [PubMed]

Hang, Z. H.

X. Huang, Y. Lai, Z. H. Hang, H. Zheng, C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nat. Mater. 10(8), 582–586 (2011).
[CrossRef] [PubMed]

Hao, J.

J. Hao, W. Yan, M. Qiu, “Super-reflection and cloaking based on zero index metamaterial,” Appl. Phys. Lett. 96(10), 101109 (2010).
[CrossRef]

Hou, B.

J. Luo, P. Xu, H. Chen, B. Hou, L. Gao, Y. Lai, “Realizing almost perfect bending waveguides with anisotropic epsilon-near-zero metamaterials,” Appl. Phys. Lett. 100(22), 221903 (2012).
[CrossRef]

Huang, X.

X. Huang, Y. Lai, Z. H. Hang, H. Zheng, C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nat. Mater. 10(8), 582–586 (2011).
[CrossRef] [PubMed]

Jacob, Z.

H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, V. M. Menon, “Topological Transitions in Metamaterials,” Science 336(6078), 205–209 (2012).
[CrossRef] [PubMed]

Jiang, W. X.

Q. Cheng, W. X. Jiang, T. J. Cui, “Spatial Power Combination for Omnidirectional Radiation via Anisotropic Metamaterials,” Phys. Rev. Lett. 108(21), 213903 (2012).
[CrossRef] [PubMed]

Joannopoulos, J. D.

J. Bravo-Abad, J. D. Joannopoulos, M. Soljačić, “Enabling single-mode behavior over large areas with photonic Dirac cones,” Proc. Natl. Acad. Sci. U.S.A. 109(25), 9761–9765 (2012).
[CrossRef] [PubMed]

Kafesaki, M.

A. A. Basharin, C. Mavidis, M. Kafesaki, E. N. Economou, C. M. Soukoulis, “Epsilon near zero based phenomena in metamaterials,” Phys. Rev. B 87(15), 155130 (2013).
[CrossRef]

Kargarian, M.

A. B. Khanikaev, S. H. Mousavi, W.-K. Tse, M. Kargarian, A. H. MacDonald, G. Shvets, “Photonic topological insulators,” Nat. Mater. 12(3), 233–239 (2012).
[CrossRef] [PubMed]

Khanikaev, A. B.

A. B. Khanikaev, S. H. Mousavi, W.-K. Tse, M. Kargarian, A. H. MacDonald, G. Shvets, “Photonic topological insulators,” Nat. Mater. 12(3), 233–239 (2012).
[CrossRef] [PubMed]

Kretzschmar, I.

H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, V. M. Menon, “Topological Transitions in Metamaterials,” Science 336(6078), 205–209 (2012).
[CrossRef] [PubMed]

Krishnamoorthy, H. N. S.

H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, V. M. Menon, “Topological Transitions in Metamaterials,” Science 336(6078), 205–209 (2012).
[CrossRef] [PubMed]

Lai, Y.

J. Luo, P. Xu, H. Chen, B. Hou, L. Gao, Y. Lai, “Realizing almost perfect bending waveguides with anisotropic epsilon-near-zero metamaterials,” Appl. Phys. Lett. 100(22), 221903 (2012).
[CrossRef]

Y. Lai, Y. Wu, P. Sheng, Z.-Q. Zhang, “Hybrid elastic solids,” Nat. Mater. 10(8), 620–624 (2011).
[CrossRef] [PubMed]

X. Huang, Y. Lai, Z. H. Hang, H. Zheng, C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nat. Mater. 10(8), 582–586 (2011).
[CrossRef] [PubMed]

Li, J.

Y. Wu, J. Li, “Total reflection and cloaking by zero index metamaterials loaded with rectangular dielectric defects,” Appl. Phys. Lett. 102(18), 183105 (2013).
[CrossRef]

Y. Wu, J. Li, Z.-Q. Zhang, C. T. Chan, “Effective medium theory for magnetodielectric composites: Beyond the long-wavelength limit,” Phys. Rev. B 74(8), 085111 (2006).
[CrossRef]

Li, Y.

Liu, Z.

X. Zhang, Z. Liu, “Extremal Transmission and Beating Effect of Acoustic Waves in Two-Dimensional Sonic Crystals,” Phys. Rev. Lett. 101(26), 264303 (2008).
[CrossRef] [PubMed]

Lumer, Y.

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, A. Szameit, “Photonic Floquet topological insulators,” Nature 496(7444), 196–200 (2013).
[CrossRef] [PubMed]

Luo, J.

J. Luo, P. Xu, H. Chen, B. Hou, L. Gao, Y. Lai, “Realizing almost perfect bending waveguides with anisotropic epsilon-near-zero metamaterials,” Appl. Phys. Lett. 100(22), 221903 (2012).
[CrossRef]

Ma, H. F.

H. F. Ma, J. H. Shi, B. G. Cai, T. J. Cui, “Total transmission and super reflection realized by anisotropic zero-index materials,” New J. Phys. 14(12), 123010 (2012).
[CrossRef]

MacDonald, A. H.

A. B. Khanikaev, S. H. Mousavi, W.-K. Tse, M. Kargarian, A. H. MacDonald, G. Shvets, “Photonic topological insulators,” Nat. Mater. 12(3), 233–239 (2012).
[CrossRef] [PubMed]

Mavidis, C.

A. A. Basharin, C. Mavidis, M. Kafesaki, E. N. Economou, C. M. Soukoulis, “Epsilon near zero based phenomena in metamaterials,” Phys. Rev. B 87(15), 155130 (2013).
[CrossRef]

Mayou, D.

D. Torrent, D. Mayou, J. Sánchez-Dehesa, “Elastic analog of graphene: Dirac cones and edge states for flexural waves in thin plates,” Phys. Rev. B 87(11), 115143 (2013).
[CrossRef]

Mei, J.

Y. Li, Y. Wu, X. Chen, J. Mei, “Selection rule for Dirac-like points in two-dimensional dielectric photonic crystals,” Opt. Express 21(6), 7699–7711 (2013).
[CrossRef] [PubMed]

J. Mei, Y. Wu, C. T. Chan, Z.-Q. Zhang, “First-principles study of Dirac and Dirac-like cones in phononic and photonic crystals,” Phys. Rev. B 86(3), 035141 (2012).
[CrossRef]

Menon, V. M.

H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, V. M. Menon, “Topological Transitions in Metamaterials,” Science 336(6078), 205–209 (2012).
[CrossRef] [PubMed]

Montambaux, G.

G. Montambaux, F. Piéchon, J. N. Fuchs, M. O. Goerbig, “Merging of Dirac points in a two-dimensional crystal,” Phys. Rev. B 80(15), 153412 (2009).
[CrossRef]

Mousavi, S. H.

A. B. Khanikaev, S. H. Mousavi, W.-K. Tse, M. Kargarian, A. H. MacDonald, G. Shvets, “Photonic topological insulators,” Nat. Mater. 12(3), 233–239 (2012).
[CrossRef] [PubMed]

Narimanov, E.

H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, V. M. Menon, “Topological Transitions in Metamaterials,” Science 336(6078), 205–209 (2012).
[CrossRef] [PubMed]

Nguyen, V. C.

V. C. Nguyen, L. Chen, K. Halterman, “Total Transmission and Total Reflection by Zero Index Metamaterials with Defects,” Phys. Rev. Lett. 105(23), 233908 (2010).
[CrossRef] [PubMed]

Nolte, S.

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, A. Szameit, “Photonic Floquet topological insulators,” Nature 496(7444), 196–200 (2013).
[CrossRef] [PubMed]

Nori, F.

Y. P. Bliokh, V. Freilikher, F. Nori, “Ballistic charge transport in graphene and light propagation in periodic dielectric structures with metamaterials: A comparative study,” Phys. Rev. B 87(24), 245134 (2013).
[CrossRef]

Novoselov, K. S.

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

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

Ochiai, T.

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

Pardo, V.

S. Banerjee, R. R. P. Singh, V. Pardo, W. E. Pickett, “Tight-Binding Modeling and Low-Energy Behavior of the Semi-Dirac Point,” Phys. Rev. Lett. 103(1), 016402 (2009).
[CrossRef] [PubMed]

V. Pardo, W. E. Pickett, “Half-Metallic Semi-Dirac-Point Generated by Quantum Confinement in TiO2/VO2 Nanostructures,” Phys. Rev. Lett. 102(16), 166803 (2009).
[CrossRef] [PubMed]

Peres, N. M. R.

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

Pickett, W. E.

V. Pardo, W. E. Pickett, “Half-Metallic Semi-Dirac-Point Generated by Quantum Confinement in TiO2/VO2 Nanostructures,” Phys. Rev. Lett. 102(16), 166803 (2009).
[CrossRef] [PubMed]

S. Banerjee, R. R. P. Singh, V. Pardo, W. E. Pickett, “Tight-Binding Modeling and Low-Energy Behavior of the Semi-Dirac Point,” Phys. Rev. Lett. 103(1), 016402 (2009).
[CrossRef] [PubMed]

Piéchon, F.

G. Montambaux, F. Piéchon, J. N. Fuchs, M. O. Goerbig, “Merging of Dirac points in a two-dimensional crystal,” Phys. Rev. B 80(15), 153412 (2009).
[CrossRef]

Plotnik, Y.

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, A. Szameit, “Photonic Floquet topological insulators,” Nature 496(7444), 196–200 (2013).
[CrossRef] [PubMed]

Podolsky, D.

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, A. Szameit, “Photonic Floquet topological insulators,” Nature 496(7444), 196–200 (2013).
[CrossRef] [PubMed]

Qiu, M.

J. Hao, W. Yan, M. Qiu, “Super-reflection and cloaking based on zero index metamaterial,” Appl. Phys. Lett. 96(10), 101109 (2010).
[CrossRef]

Raghu, S.

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

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

Rechtsman, M. C.

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, A. Szameit, “Photonic Floquet topological insulators,” Nature 496(7444), 196–200 (2013).
[CrossRef] [PubMed]

Sakoda, K.

Salandrino, A.

A. Alu, M. G. Silveirinha, A. Salandrino, N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: Tailoring the radiation phase pattern,” Phys. Rev. B 75(15), 155410 (2007).
[CrossRef]

Sánchez-Dehesa, J.

D. Torrent, D. Mayou, J. Sánchez-Dehesa, “Elastic analog of graphene: Dirac cones and edge states for flexural waves in thin plates,” Phys. Rev. B 87(11), 115143 (2013).
[CrossRef]

D. Torrent, J. Sánchez-Dehesa, “Acoustic Analogue of Graphene: Observation of Dirac Cones in Acoustic Surface Waves,” Phys. Rev. Lett. 108(17), 174301 (2012).
[CrossRef] [PubMed]

Segev, M.

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, A. Szameit, “Photonic Floquet topological insulators,” Nature 496(7444), 196–200 (2013).
[CrossRef] [PubMed]

Sepkhanov, R. A.

R. A. Sepkhanov, Y. B. Bazaliy, C. W. J. Beenakker, “Extremal transmission at the Dirac point of a photonic band structure,” Phys. Rev. A 75(6), 063813 (2007).
[CrossRef]

Sheng, P.

Y. Lai, Y. Wu, P. Sheng, Z.-Q. Zhang, “Hybrid elastic solids,” Nat. Mater. 10(8), 620–624 (2011).
[CrossRef] [PubMed]

Shi, J. H.

H. F. Ma, J. H. Shi, B. G. Cai, T. J. Cui, “Total transmission and super reflection realized by anisotropic zero-index materials,” New J. Phys. 14(12), 123010 (2012).
[CrossRef]

Shvets, G.

A. B. Khanikaev, S. H. Mousavi, W.-K. Tse, M. Kargarian, A. H. MacDonald, G. Shvets, “Photonic topological insulators,” Nat. Mater. 12(3), 233–239 (2012).
[CrossRef] [PubMed]

Silveirinha, M.

B. Edwards, A. Alù, M. E. Young, M. Silveirinha, N. Engheta, “Experimental verification of epsilon-near-zero metamaterial coupling and energy squeezing using a microwave waveguide,” Phys. Rev. Lett. 100(3), 033903 (2008).
[CrossRef] [PubMed]

M. Silveirinha, N. Engheta, “Tunneling of electromagnetic energy through subwavelength channels and bends using epsilon-near-zero materials,” Phys. Rev. Lett. 97(15), 157403 (2006).
[CrossRef] [PubMed]

Silveirinha, M. G.

A. Alu, M. G. Silveirinha, A. Salandrino, N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: Tailoring the radiation phase pattern,” Phys. Rev. B 75(15), 155410 (2007).
[CrossRef]

Singh, R. R. P.

S. Banerjee, R. R. P. Singh, V. Pardo, W. E. Pickett, “Tight-Binding Modeling and Low-Energy Behavior of the Semi-Dirac Point,” Phys. Rev. Lett. 103(1), 016402 (2009).
[CrossRef] [PubMed]

Soljacic, M.

J. Bravo-Abad, J. D. Joannopoulos, M. Soljačić, “Enabling single-mode behavior over large areas with photonic Dirac cones,” Proc. Natl. Acad. Sci. U.S.A. 109(25), 9761–9765 (2012).
[CrossRef] [PubMed]

Soukoulis, C. M.

A. A. Basharin, C. Mavidis, M. Kafesaki, E. N. Economou, C. M. Soukoulis, “Epsilon near zero based phenomena in metamaterials,” Phys. Rev. B 87(15), 155130 (2013).
[CrossRef]

Szameit, A.

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, A. Szameit, “Photonic Floquet topological insulators,” Nature 496(7444), 196–200 (2013).
[CrossRef] [PubMed]

Torrent, D.

D. Torrent, D. Mayou, J. Sánchez-Dehesa, “Elastic analog of graphene: Dirac cones and edge states for flexural waves in thin plates,” Phys. Rev. B 87(11), 115143 (2013).
[CrossRef]

D. Torrent, J. Sánchez-Dehesa, “Acoustic Analogue of Graphene: Observation of Dirac Cones in Acoustic Surface Waves,” Phys. Rev. Lett. 108(17), 174301 (2012).
[CrossRef] [PubMed]

Tse, W.-K.

A. B. Khanikaev, S. H. Mousavi, W.-K. Tse, M. Kargarian, A. H. MacDonald, G. Shvets, “Photonic topological insulators,” Nat. Mater. 12(3), 233–239 (2012).
[CrossRef] [PubMed]

Wallace, P. R.

P. R. Wallace, “The band theory of Graphite,” Phys. Rev. 71(9), 622–634 (1947).
[CrossRef]

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Y. Li, Y. Wu, X. Chen, J. Mei, “Selection rule for Dirac-like points in two-dimensional dielectric photonic crystals,” Opt. Express 21(6), 7699–7711 (2013).
[CrossRef] [PubMed]

Y. Wu, J. Li, “Total reflection and cloaking by zero index metamaterials loaded with rectangular dielectric defects,” Appl. Phys. Lett. 102(18), 183105 (2013).
[CrossRef]

J. Mei, Y. Wu, C. T. Chan, Z.-Q. Zhang, “First-principles study of Dirac and Dirac-like cones in phononic and photonic crystals,” Phys. Rev. B 86(3), 035141 (2012).
[CrossRef]

Y. Lai, Y. Wu, P. Sheng, Z.-Q. Zhang, “Hybrid elastic solids,” Nat. Mater. 10(8), 620–624 (2011).
[CrossRef] [PubMed]

Y. Wu, J. Li, Z.-Q. Zhang, C. T. Chan, “Effective medium theory for magnetodielectric composites: Beyond the long-wavelength limit,” Phys. Rev. B 74(8), 085111 (2006).
[CrossRef]

Xu, P.

J. Luo, P. Xu, H. Chen, B. Hou, L. Gao, Y. Lai, “Realizing almost perfect bending waveguides with anisotropic epsilon-near-zero metamaterials,” Appl. Phys. Lett. 100(22), 221903 (2012).
[CrossRef]

Yan, W.

J. Hao, W. Yan, M. Qiu, “Super-reflection and cloaking based on zero index metamaterial,” Appl. Phys. Lett. 96(10), 101109 (2010).
[CrossRef]

Yannopapas, V.

V. Yannopapas, “Photonic analog of a spin-polarized system with Rashba spin-orbit coupling,” Phys. Rev. B 83(11), 113101 (2011).
[CrossRef]

Young, M. E.

B. Edwards, A. Alù, M. E. Young, M. Silveirinha, N. Engheta, “Experimental verification of epsilon-near-zero metamaterial coupling and energy squeezing using a microwave waveguide,” Phys. Rev. Lett. 100(3), 033903 (2008).
[CrossRef] [PubMed]

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S. R. Zandbergen, M. J. A. de Dood, “Experimental observation of strong edge effects on the pseudodiffusive transport of light in photonic graphene,” Phys. Rev. Lett. 104(4), 043903 (2010).
[CrossRef] [PubMed]

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M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, A. Szameit, “Photonic Floquet topological insulators,” Nature 496(7444), 196–200 (2013).
[CrossRef] [PubMed]

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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]

X. Zhang, Z. Liu, “Extremal Transmission and Beating Effect of Acoustic Waves in Two-Dimensional Sonic Crystals,” Phys. Rev. Lett. 101(26), 264303 (2008).
[CrossRef] [PubMed]

Zhang, Z.-Q.

J. Mei, Y. Wu, C. T. Chan, Z.-Q. Zhang, “First-principles study of Dirac and Dirac-like cones in phononic and photonic crystals,” Phys. Rev. B 86(3), 035141 (2012).
[CrossRef]

Y. Lai, Y. Wu, P. Sheng, Z.-Q. Zhang, “Hybrid elastic solids,” Nat. Mater. 10(8), 620–624 (2011).
[CrossRef] [PubMed]

Y. Wu, J. Li, Z.-Q. Zhang, C. T. Chan, “Effective medium theory for magnetodielectric composites: Beyond the long-wavelength limit,” Phys. Rev. B 74(8), 085111 (2006).
[CrossRef]

Zheng, H.

X. Huang, Y. Lai, Z. H. Hang, H. Zheng, C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nat. Mater. 10(8), 582–586 (2011).
[CrossRef] [PubMed]

Appl. Phys. Lett. (3)

J. Hao, W. Yan, M. Qiu, “Super-reflection and cloaking based on zero index metamaterial,” Appl. Phys. Lett. 96(10), 101109 (2010).
[CrossRef]

Y. Wu, J. Li, “Total reflection and cloaking by zero index metamaterials loaded with rectangular dielectric defects,” Appl. Phys. Lett. 102(18), 183105 (2013).
[CrossRef]

J. Luo, P. Xu, H. Chen, B. Hou, L. Gao, Y. Lai, “Realizing almost perfect bending waveguides with anisotropic epsilon-near-zero metamaterials,” Appl. Phys. Lett. 100(22), 221903 (2012).
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J. Phys. Condens. Matter (1)

B. A. Foreman, “Theory of the effective Hamiltonian for degenerate bands in an electric field,” J. Phys. Condens. Matter 12(34), R435–R461 (2000).
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Nat. Mater. (4)

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

X. Huang, Y. Lai, Z. H. Hang, H. Zheng, C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nat. Mater. 10(8), 582–586 (2011).
[CrossRef] [PubMed]

A. B. Khanikaev, S. H. Mousavi, W.-K. Tse, M. Kargarian, A. H. MacDonald, G. Shvets, “Photonic topological insulators,” Nat. Mater. 12(3), 233–239 (2012).
[CrossRef] [PubMed]

Y. Lai, Y. Wu, P. Sheng, Z.-Q. Zhang, “Hybrid elastic solids,” Nat. Mater. 10(8), 620–624 (2011).
[CrossRef] [PubMed]

Nature (1)

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, A. Szameit, “Photonic Floquet topological insulators,” Nature 496(7444), 196–200 (2013).
[CrossRef] [PubMed]

New J. Phys. (1)

H. F. Ma, J. H. Shi, B. G. Cai, T. J. Cui, “Total transmission and super reflection realized by anisotropic zero-index materials,” New J. Phys. 14(12), 123010 (2012).
[CrossRef]

Opt. Express (3)

Phys. Rev. (1)

P. R. Wallace, “The band theory of Graphite,” Phys. Rev. 71(9), 622–634 (1947).
[CrossRef]

Phys. Rev. A (2)

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

R. A. Sepkhanov, Y. B. Bazaliy, C. W. J. Beenakker, “Extremal transmission at the Dirac point of a photonic band structure,” Phys. Rev. A 75(6), 063813 (2007).
[CrossRef]

Phys. Rev. B (9)

J. Mei, Y. Wu, C. T. Chan, Z.-Q. Zhang, “First-principles study of Dirac and Dirac-like cones in phononic and photonic crystals,” Phys. Rev. B 86(3), 035141 (2012).
[CrossRef]

Y. P. Bliokh, V. Freilikher, F. Nori, “Ballistic charge transport in graphene and light propagation in periodic dielectric structures with metamaterials: A comparative study,” Phys. Rev. B 87(24), 245134 (2013).
[CrossRef]

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

V. Yannopapas, “Photonic analog of a spin-polarized system with Rashba spin-orbit coupling,” Phys. Rev. B 83(11), 113101 (2011).
[CrossRef]

D. Torrent, D. Mayou, J. Sánchez-Dehesa, “Elastic analog of graphene: Dirac cones and edge states for flexural waves in thin plates,” Phys. Rev. B 87(11), 115143 (2013).
[CrossRef]

G. Montambaux, F. Piéchon, J. N. Fuchs, M. O. Goerbig, “Merging of Dirac points in a two-dimensional crystal,” Phys. Rev. B 80(15), 153412 (2009).
[CrossRef]

Y. Wu, J. Li, Z.-Q. Zhang, C. T. Chan, “Effective medium theory for magnetodielectric composites: Beyond the long-wavelength limit,” Phys. Rev. B 74(8), 085111 (2006).
[CrossRef]

A. A. Basharin, C. Mavidis, M. Kafesaki, E. N. Economou, C. M. Soukoulis, “Epsilon near zero based phenomena in metamaterials,” Phys. Rev. B 87(15), 155130 (2013).
[CrossRef]

A. Alu, M. G. Silveirinha, A. Salandrino, N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: Tailoring the radiation phase pattern,” Phys. Rev. B 75(15), 155410 (2007).
[CrossRef]

Phys. Rev. Lett. (11)

B. Edwards, A. Alù, M. E. Young, M. Silveirinha, N. Engheta, “Experimental verification of epsilon-near-zero metamaterial coupling and energy squeezing using a microwave waveguide,” Phys. Rev. Lett. 100(3), 033903 (2008).
[CrossRef] [PubMed]

M. Silveirinha, N. Engheta, “Tunneling of electromagnetic energy through subwavelength channels and bends using epsilon-near-zero materials,” Phys. Rev. Lett. 97(15), 157403 (2006).
[CrossRef] [PubMed]

V. C. Nguyen, L. Chen, K. Halterman, “Total Transmission and Total Reflection by Zero Index Metamaterials with Defects,” Phys. Rev. Lett. 105(23), 233908 (2010).
[CrossRef] [PubMed]

V. Pardo, W. E. Pickett, “Half-Metallic Semi-Dirac-Point Generated by Quantum Confinement in TiO2/VO2 Nanostructures,” Phys. Rev. Lett. 102(16), 166803 (2009).
[CrossRef] [PubMed]

S. Banerjee, R. R. P. Singh, V. Pardo, W. E. Pickett, “Tight-Binding Modeling and Low-Energy Behavior of the Semi-Dirac Point,” Phys. Rev. Lett. 103(1), 016402 (2009).
[CrossRef] [PubMed]

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Proc. Natl. Acad. Sci. U.S.A. (1)

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Figures (5)

Fig. 1
Fig. 1

(a) The band structure of the 2D PhC composed of a square array of elliptical dielectric cylinders. The inset shows the unit cell of the PhC. A doubly-degenerate state in the center of the Brillouin zone is found near the dimensionless frequency, 0.540, marked as “A”. In the vicinity of this point, the dispersion relation is linear along the ΓX direction and quadratic along the ΓY direction, which is shown more clearly in Fig. 3(d). Near point “A”, there is another state in the center of the Brillouin zone, marked as “B”. The states at points “A” and “B” are used in the perturbation theory. The branches highlighted by black and blue dots are used to compute the effective medium parameters, which are shown in Fig. 4(a). (b) and (c) Enlarged views of the band structure for smaller and larger elliptical cylinders. The doubly-degenerate state shown in (a) splits into two single states, marked as A1 and A2, where A1 corresponds to a dipolar state and A2 corresponds to a monopolar state.

Fig. 2
Fig. 2

(a) and (b) The three-dimensional band structure of the PhC. The upper surface is a semi-Dirac cone. Near its bottom, it is linear in Δk along all directions except for the ΓY direction, which is quadratic. It touches the lower surface at the Brillouin zone center near the dimensionless frequency, 0.54. The lower surface is flat in one direction and bends down along the other directions. (c) and (d) The iso-frequency surfaces of the lower and higher branches, where hyperbolic and elliptical surfaces are found, respectively.

Fig. 3
Fig. 3

(a) The electric field pattern of the eigenstate marked as “B” in Fig. 1(a). Dark red and dark blue indicate the maximum positive and negative values, respectively. This is a dipolar state with a magnetic field parallel to the x-axis, indicated by the arrows. (b) and (c) The electric field patterns of the doubly-degenerate states marked as “A” in Fig. 1(a). A monopolar and a dipolar state with the magnetic field (arrows) perpendicular to the x-axis are evident. (d) An enlarged view of the band structure near the doubly-degenerate point. The dots are calculated by COMSOL. Linear dispersion is seen along the ΓX direction, while a quadratic dispersion relation is manifest along the ΓY direction. Red solid lines and green solid curves are obtained from the perturbation theory. The blue dashed curves represent the results of quadratic fitting. (e) The same as (d) but along the ΓM direction. A linear dispersion relation is seen again.

Fig. 4
Fig. 4

(a) Effective medium parameters evaluated with a boundary effective medium theory using the eigenstates highlighted by solid dots in Fig. 1(a). The blue triangles and black squares represent the effective permittivity ε eff calculated by using the eigenstates along the ΓY and ΓX directions, respectively. They almost overlap, indicating that ε eff is a scalar and does not depend on the direction. The red circles represent μ y eff , which crosses zero simultaneously with ε eff at the semi-Dirac point. The green triangles represent μ x eff , which crosses zero at dimensionless frequency 0.487. Note that both the blue and green triangles are missing from the frequency regime at 0.487 to 0.540, which corresponds to a band gap along the ΓY direction. No eigenstates are thus available to evaluate the related effective medium parameters. (b)-(d) The electric field for a plane wave impinging on a PhC slab in a waveguide whose walls have perfect magnetic conductor boundary conditions at the semi-Dirac frequency 0.540. Dark red and dark blue indicate the maximum positive and negative values, respectively. (b) The real part of the electric field when the incident wave is along the ΓY direction. The transmitted field is very weak. The imaginary part is orders of magnitude smaller than the real part, which is why it is not shown here. (c) and (d) The real and imaginary parts of the electric field when the incident wave is along the ΓX direction. Both suggest that there is no phase change in the sample, which is a typical property of a ZIM.

Fig. 5
Fig. 5

A point source is placed inside the center of a square sample of 16-by-16 rods. (a) and (c) show the electric field patterns when the source frequency is below (0.520) and slightly above (0.544) the semi-Dirac point, respectively. Beam splitting and directional beam shaping are observed. (b) The radial flux as a function of the angle for the case simulated in (a). (d) The same as (c) but the sample is replaced by its effective medium. A similar pattern to that shown in (c) is found. Dark red and dark blue indicate the maximum positive and negative values, respectively.

Equations (12)

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k x 2 μ y + k y 2 μ x = ω 2 ε.
det| ω k 2 ω B 2 c 2 + P 11 P 12 P 13 P 21 ω k 2 ω A 2 c 2 + P 22 P 23 P 31 P 32 ω k 2 ω A 2 c 2 + P 33 |=0,
Δ ω ˜ k =(±0.0459cosβ)Δk+O(Δ k 2 ),
D ¯ z = ε eff E ¯ z , and ( B ¯ x B ¯ y )=( μ x eff 0 0 μ y eff )( H ¯ x H ¯ y ),
E ¯ z =( 0 a E z ( x=0 )dy + 0 a E z ( x=a )dy )/2a,
H ¯ y =( 0 a H y ( x=0 )dy + 0 a H y ( x=a )dy )/2a,
D ¯ z =( 0 a H y ( x=a )dy 0 a H y ( x=0 )dy )/( iω a 2 ),
B ¯ y =( 0 a E z ( x=a )dy 0 a E z ( x=0 )dy )/( iω a 2 ).
×( 1 μ( r ) × E z z ^ )= ω 2 c 0 2 ε( r ) E z z ^ ,
Ψ n k ( r )= j A nj ( k ) e i( k k 0 ) r Ψ j k 0 ( r ),
j [ ω j0 2 ω n k 2 c 0 2 δ lj P lj ( k ) ] A nj ( k ) =0
P lj ( k )=( k k 0 ) p lj ( k k 0 ) 2 q lj ,

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