Abstract

Trapping light within cavities or waveguides in photonic crystals is an effective technology in modern integrated optics. Traditionally, cavities rely on total internal reflection or a photonic bandgap to achieve field confinement. Recent investigations have examined new localized modes that occur at a Dirac frequency that is beyond any complete photonic bandgap. We design Al2O3 dielectric cylinders placed on a triangular lattice in air, and change the central rod size to form a photonic crystal microcavity. It is predicted that waves can be localized at the Dirac frequency in this device without photonic bandgaps or total internal reflections. We perform a theoretical analysis of this new wave localization and verify it experimentally. This work paves the way for exploring localized defect modes at the Dirac point in the visible and infrared bands, with potential applicability to new optical devices.

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

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References

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2017 (1)

2015 (2)

K. Xie, W. Zhang, A. D. Boardman, H. Jiang, Z. Hu, Y. Liu, M. Xie, Q. Mao, L. Hu, Q. Li, T. Yang, F. Wen, and E. Wang, “Fiber guiding at the Dirac frequency beyond photonic bandgaps,” Light Sci. Appl. 4(6), e304 (2015).
[Crossref]

J. Böhm, M. Bellec, F. Mortessagne, U. Kuhl, S. Barkhofen, S. Gehler, H. J. Stöckmann, I. Foulger, S. Gnutzmann, and G. Tanner, “Microwave Experiments Simulating Quantum Search and Directed Transport in Artificial Graphene,” Phys. Rev. Lett. 114(11), 110501 (2015).
[Crossref] [PubMed]

2014 (2)

M. Bellec, U. Kuhl, G. Montambaux, and F. Mortessagne, “Manipulation of edge states in microwave artificial grapheme,” New J. Phys. 16(11), 113023 (2014).
[Crossref]

K. Xie, H. Jiang, A. D. Boardman, Y. Liu, Z. Wu, M. Xie, P. Jiang, Q. Xu, M. Yu, and L. E. Davis, “Trapped photons at a Dirac point: a new horizon for photonic crystals,” Laser Photonics Rev. 8(4), 583–589 (2014).
[Crossref]

2013 (1)

M. Bellec, U. Kuhl, G. Montambaux, and F. Mortessagne, “Tight-binding couplings in microwave artificial graphene,” Phys. Rev. B 88(11), 115437 (2013).
[Crossref]

2012 (2)

J. Mei, Y. Wu, C. T. Chan, and 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]

S. Bittner, B. Dietz, M. Miski-Oglu, and A. Richter, “Extremal transmission through a microwave photonic crystal and the observation of edge states in a rectangular Dirac billiard,” Phys. Rev. B 85(6), 064301 (2012).
[Crossref]

2010 (4)

S. R. Zandbergen and 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]

S. Bittner, B. Dietz, M. Miski-Oglu, P. O. Iriarte, A. Richter, and F. Schäfer, “Observation of a Dirac point in microwave experiments with a photonic crystal modeling graphene,” Phys. Rev. B 82(1), 014301 (2010).
[Crossref]

U. Kuhl, S. Barkhofen, T. Tudorovskiy, H. J. Stöckmann, T. Hossain, L. de Forges de Parny, and F. Mortessagne, “Dirac point and edge states in a microwave realization of tight-binding graphene-like structures,” Phys. Rev. B 82(9), 094308 (2010).
[Crossref]

M. Diem, T. Koschny, and C. M. Soukoulis, “Transmission in the vicinity of the Dirac point in hexagonal photonic crystals,” Physica B 405(14), 2990–2995 (2010).
[Crossref]

2009 (4)

H. M. Guo and M. Franz, “Topological insulator on the kagome lattice,” Phys. Rev. B 80(11), 113102 (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(1), 109–162 (2009).
[Crossref]

S. Longhi, “Quantum - optical analogies using photonic structures,” Laser Photonics Rev. 3(3), 243–261 (2009).
[Crossref]

C. Wiesmann, K. Bergenek, N. Linder, and U. T. Schwarz, “Photonic crystal LEDs–designing light extraction,” Laser Photonics Rev. 3(3), 262–286 (2009).
[Crossref]

2008 (3)

O. Frazão, J. L. Santos, F. M. Araújo, and L. A. Ferreira, “Optical sensing with photonic crystal fibers,” Laser Photonics Rev. 2(6), 449–459 (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]

B. Wunsch, F. Guinea, and F. Sols, “Dirac-point engineering and topological phase transitions in honeycomb optical lattices,” New J. Phys. 10(10), 103027 (2008).
[Crossref]

2007 (1)

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

2006 (1)

2005 (2)

Y. Akahane, T. Asano, B.-S. Song, and S. Noda, “Fine-tuned high-Q photonic-crystal nanocavity,” Opt. Express 13(4), 1202–1214 (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,” Nature 438(7065), 201–204 (2005).
[Crossref] [PubMed]

2004 (3)

2003 (1)

1997 (1)

J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, “Photonic crystals: putting a new twist on light,” Nature 386(6621), 143–149 (1997).
[Crossref]

1994 (1)

1991 (1)

M. Plihal and A. A. Maradudin, “Photonic band structure of two-dimensional systems: The triangular lattice,” Phys. Rev. B Condens. Matter 44(16), 8565–8571 (1991).
[Crossref] [PubMed]

Akahane, Y.

Araújo, F. M.

O. Frazão, J. L. Santos, F. M. Araújo, and L. A. Ferreira, “Optical sensing with photonic crystal fibers,” Laser Photonics Rev. 2(6), 449–459 (2008).
[Crossref]

Arjavalingam, G.

Asano, T.

Bang, O.

Barkhofen, S.

J. Böhm, M. Bellec, F. Mortessagne, U. Kuhl, S. Barkhofen, S. Gehler, H. J. Stöckmann, I. Foulger, S. Gnutzmann, and G. Tanner, “Microwave Experiments Simulating Quantum Search and Directed Transport in Artificial Graphene,” Phys. Rev. Lett. 114(11), 110501 (2015).
[Crossref] [PubMed]

U. Kuhl, S. Barkhofen, T. Tudorovskiy, H. J. Stöckmann, T. Hossain, L. de Forges de Parny, and F. Mortessagne, “Dirac point and edge states in a microwave realization of tight-binding graphene-like structures,” Phys. Rev. B 82(9), 094308 (2010).
[Crossref]

Bazaliy, Y. B.

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

Bellec, M.

J. Böhm, M. Bellec, F. Mortessagne, U. Kuhl, S. Barkhofen, S. Gehler, H. J. Stöckmann, I. Foulger, S. Gnutzmann, and G. Tanner, “Microwave Experiments Simulating Quantum Search and Directed Transport in Artificial Graphene,” Phys. Rev. Lett. 114(11), 110501 (2015).
[Crossref] [PubMed]

M. Bellec, U. Kuhl, G. Montambaux, and F. Mortessagne, “Manipulation of edge states in microwave artificial grapheme,” New J. Phys. 16(11), 113023 (2014).
[Crossref]

M. Bellec, U. Kuhl, G. Montambaux, and F. Mortessagne, “Tight-binding couplings in microwave artificial graphene,” Phys. Rev. B 88(11), 115437 (2013).
[Crossref]

Bergenek, K.

C. Wiesmann, K. Bergenek, N. Linder, and U. T. Schwarz, “Photonic crystal LEDs–designing light extraction,” Laser Photonics Rev. 3(3), 262–286 (2009).
[Crossref]

Bittner, S.

S. Bittner, B. Dietz, M. Miski-Oglu, and A. Richter, “Extremal transmission through a microwave photonic crystal and the observation of edge states in a rectangular Dirac billiard,” Phys. Rev. B 85(6), 064301 (2012).
[Crossref]

S. Bittner, B. Dietz, M. Miski-Oglu, P. O. Iriarte, A. Richter, and F. Schäfer, “Observation of a Dirac point in microwave experiments with a photonic crystal modeling graphene,” Phys. Rev. B 82(1), 014301 (2010).
[Crossref]

Boardman, A. D.

K. Xie, A. D. Boardman, Q. Li, Z. Shi, H. Jiang, H. Xia, Z. Hu, J. Zhang, W. Zhang, Q. Mao, L. Hu, T. Yang, F. Wen, and E. Wang, “Spatial algebraic solitons at the Dirac point in optically induced nonlinear photonic lattices,” Opt. Express 25(24), 30349–30364 (2017).
[Crossref] [PubMed]

K. Xie, W. Zhang, A. D. Boardman, H. Jiang, Z. Hu, Y. Liu, M. Xie, Q. Mao, L. Hu, Q. Li, T. Yang, F. Wen, and E. Wang, “Fiber guiding at the Dirac frequency beyond photonic bandgaps,” Light Sci. Appl. 4(6), e304 (2015).
[Crossref]

K. Xie, H. Jiang, A. D. Boardman, Y. Liu, Z. Wu, M. Xie, P. Jiang, Q. Xu, M. Yu, and L. E. Davis, “Trapped photons at a Dirac point: a new horizon for photonic crystals,” Laser Photonics Rev. 8(4), 583–589 (2014).
[Crossref]

Böhm, J.

J. Böhm, M. Bellec, F. Mortessagne, U. Kuhl, S. Barkhofen, S. Gehler, H. J. Stöckmann, I. Foulger, S. Gnutzmann, and G. Tanner, “Microwave Experiments Simulating Quantum Search and Directed Transport in Artificial Graphene,” Phys. Rev. Lett. 114(11), 110501 (2015).
[Crossref] [PubMed]

Broeng, J.

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(1), 109–162 (2009).
[Crossref]

Chan, C. T.

J. Mei, Y. Wu, C. T. Chan, and 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]

Davis, L. E.

K. Xie, H. Jiang, A. D. Boardman, Y. Liu, Z. Wu, M. Xie, P. Jiang, Q. Xu, M. Yu, and L. E. Davis, “Trapped photons at a Dirac point: a new horizon for photonic crystals,” Laser Photonics Rev. 8(4), 583–589 (2014).
[Crossref]

de Dood, M. J. A.

S. R. Zandbergen and 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]

de Forges de Parny, L.

U. Kuhl, S. Barkhofen, T. Tudorovskiy, H. J. Stöckmann, T. Hossain, L. de Forges de Parny, and F. Mortessagne, “Dirac point and edge states in a microwave realization of tight-binding graphene-like structures,” Phys. Rev. B 82(9), 094308 (2010).
[Crossref]

Diem, M.

M. Diem, T. Koschny, and C. M. Soukoulis, “Transmission in the vicinity of the Dirac point in hexagonal photonic crystals,” Physica B 405(14), 2990–2995 (2010).
[Crossref]

Dietz, B.

S. Bittner, B. Dietz, M. Miski-Oglu, and A. Richter, “Extremal transmission through a microwave photonic crystal and the observation of edge states in a rectangular Dirac billiard,” Phys. Rev. B 85(6), 064301 (2012).
[Crossref]

S. Bittner, B. Dietz, M. Miski-Oglu, P. O. Iriarte, A. Richter, and F. Schäfer, “Observation of a Dirac point in microwave experiments with a photonic crystal modeling graphene,” Phys. Rev. B 82(1), 014301 (2010).
[Crossref]

Dufva, M.

Fan, S.

V. Lousse, W. Suh, O. Kilic, S. Kim, O. Solgaard, and S. Fan, “Angular and polarization properties of a photonic crystal slab mirror,” Opt. Express 12(8), 1575–1582 (2004).
[Crossref] [PubMed]

J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, “Photonic crystals: putting a new twist on light,” Nature 386(6621), 143–149 (1997).
[Crossref]

Ferreira, L. A.

O. Frazão, J. L. Santos, F. M. Araújo, and L. A. Ferreira, “Optical sensing with photonic crystal fibers,” Laser Photonics Rev. 2(6), 449–459 (2008).
[Crossref]

Foulger, I.

J. Böhm, M. Bellec, F. Mortessagne, U. Kuhl, S. Barkhofen, S. Gehler, H. J. Stöckmann, I. Foulger, S. Gnutzmann, and G. Tanner, “Microwave Experiments Simulating Quantum Search and Directed Transport in Artificial Graphene,” Phys. Rev. Lett. 114(11), 110501 (2015).
[Crossref] [PubMed]

Franz, M.

H. M. Guo and M. Franz, “Topological insulator on the kagome lattice,” Phys. Rev. B 80(11), 113102 (2009).
[Crossref]

Frazão, O.

O. Frazão, J. L. Santos, F. M. Araújo, and L. A. Ferreira, “Optical sensing with photonic crystal fibers,” Laser Photonics Rev. 2(6), 449–459 (2008).
[Crossref]

Gehler, S.

J. Böhm, M. Bellec, F. Mortessagne, U. Kuhl, S. Barkhofen, S. Gehler, H. J. Stöckmann, I. Foulger, S. Gnutzmann, and G. Tanner, “Microwave Experiments Simulating Quantum Search and Directed Transport in Artificial Graphene,” Phys. Rev. Lett. 114(11), 110501 (2015).
[Crossref] [PubMed]

Geim, A. K.

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(1), 109–162 (2009).
[Crossref]

Gnutzmann, S.

J. Böhm, M. Bellec, F. Mortessagne, U. Kuhl, S. Barkhofen, S. Gehler, H. J. Stöckmann, I. Foulger, S. Gnutzmann, and G. Tanner, “Microwave Experiments Simulating Quantum Search and Directed Transport in Artificial Graphene,” Phys. Rev. Lett. 114(11), 110501 (2015).
[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(1), 109–162 (2009).
[Crossref]

B. Wunsch, F. Guinea, and F. Sols, “Dirac-point engineering and topological phase transitions in honeycomb optical lattices,” New J. Phys. 10(10), 103027 (2008).
[Crossref]

Guo, H. M.

H. M. Guo and M. Franz, “Topological insulator on the kagome lattice,” Phys. Rev. B 80(11), 113102 (2009).
[Crossref]

Høiby, P. E.

Hossain, T.

U. Kuhl, S. Barkhofen, T. Tudorovskiy, H. J. Stöckmann, T. Hossain, L. de Forges de Parny, and F. Mortessagne, “Dirac point and edge states in a microwave realization of tight-binding graphene-like structures,” Phys. Rev. B 82(9), 094308 (2010).
[Crossref]

Hu, L.

K. Xie, A. D. Boardman, Q. Li, Z. Shi, H. Jiang, H. Xia, Z. Hu, J. Zhang, W. Zhang, Q. Mao, L. Hu, T. Yang, F. Wen, and E. Wang, “Spatial algebraic solitons at the Dirac point in optically induced nonlinear photonic lattices,” Opt. Express 25(24), 30349–30364 (2017).
[Crossref] [PubMed]

K. Xie, W. Zhang, A. D. Boardman, H. Jiang, Z. Hu, Y. Liu, M. Xie, Q. Mao, L. Hu, Q. Li, T. Yang, F. Wen, and E. Wang, “Fiber guiding at the Dirac frequency beyond photonic bandgaps,” Light Sci. Appl. 4(6), e304 (2015).
[Crossref]

Hu, Z.

K. Xie, A. D. Boardman, Q. Li, Z. Shi, H. Jiang, H. Xia, Z. Hu, J. Zhang, W. Zhang, Q. Mao, L. Hu, T. Yang, F. Wen, and E. Wang, “Spatial algebraic solitons at the Dirac point in optically induced nonlinear photonic lattices,” Opt. Express 25(24), 30349–30364 (2017).
[Crossref] [PubMed]

K. Xie, W. Zhang, A. D. Boardman, H. Jiang, Z. Hu, Y. Liu, M. Xie, Q. Mao, L. Hu, Q. Li, T. Yang, F. Wen, and E. Wang, “Fiber guiding at the Dirac frequency beyond photonic bandgaps,” Light Sci. Appl. 4(6), e304 (2015).
[Crossref]

Iliew, R.

Iriarte, P. O.

S. Bittner, B. Dietz, M. Miski-Oglu, P. O. Iriarte, A. Richter, and F. Schäfer, “Observation of a Dirac point in microwave experiments with a photonic crystal modeling graphene,” Phys. Rev. B 82(1), 014301 (2010).
[Crossref]

Jakobsen, C.

Jensen, J. B.

Jiang, H.

K. Xie, A. D. Boardman, Q. Li, Z. Shi, H. Jiang, H. Xia, Z. Hu, J. Zhang, W. Zhang, Q. Mao, L. Hu, T. Yang, F. Wen, and E. Wang, “Spatial algebraic solitons at the Dirac point in optically induced nonlinear photonic lattices,” Opt. Express 25(24), 30349–30364 (2017).
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K. Xie, W. Zhang, A. D. Boardman, H. Jiang, Z. Hu, Y. Liu, M. Xie, Q. Mao, L. Hu, Q. Li, T. Yang, F. Wen, and E. Wang, “Fiber guiding at the Dirac frequency beyond photonic bandgaps,” Light Sci. Appl. 4(6), e304 (2015).
[Crossref]

K. Xie, H. Jiang, A. D. Boardman, Y. Liu, Z. Wu, M. Xie, P. Jiang, Q. Xu, M. Yu, and L. E. Davis, “Trapped photons at a Dirac point: a new horizon for photonic crystals,” Laser Photonics Rev. 8(4), 583–589 (2014).
[Crossref]

Jiang, P.

K. Xie, H. Jiang, A. D. Boardman, Y. Liu, Z. Wu, M. Xie, P. Jiang, Q. Xu, M. Yu, and L. E. Davis, “Trapped photons at a Dirac point: a new horizon for photonic crystals,” Laser Photonics Rev. 8(4), 583–589 (2014).
[Crossref]

Joannopoulos, J. D.

J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, “Photonic crystals: putting a new twist on light,” Nature 386(6621), 143–149 (1997).
[Crossref]

Kilic, O.

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,” Nature 438(7065), 201–204 (2005).
[Crossref] [PubMed]

Kim, S.

Koschny, T.

M. Diem, T. Koschny, and C. M. Soukoulis, “Transmission in the vicinity of the Dirac point in hexagonal photonic crystals,” Physica B 405(14), 2990–2995 (2010).
[Crossref]

Kuhl, U.

J. Böhm, M. Bellec, F. Mortessagne, U. Kuhl, S. Barkhofen, S. Gehler, H. J. Stöckmann, I. Foulger, S. Gnutzmann, and G. Tanner, “Microwave Experiments Simulating Quantum Search and Directed Transport in Artificial Graphene,” Phys. Rev. Lett. 114(11), 110501 (2015).
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M. Bellec, U. Kuhl, G. Montambaux, and F. Mortessagne, “Manipulation of edge states in microwave artificial grapheme,” New J. Phys. 16(11), 113023 (2014).
[Crossref]

M. Bellec, U. Kuhl, G. Montambaux, and F. Mortessagne, “Tight-binding couplings in microwave artificial graphene,” Phys. Rev. B 88(11), 115437 (2013).
[Crossref]

U. Kuhl, S. Barkhofen, T. Tudorovskiy, H. J. Stöckmann, T. Hossain, L. de Forges de Parny, and F. Mortessagne, “Dirac point and edge states in a microwave realization of tight-binding graphene-like structures,” Phys. Rev. B 82(9), 094308 (2010).
[Crossref]

Kuramochi, E.

Lederer, F.

Li, Q.

K. Xie, A. D. Boardman, Q. Li, Z. Shi, H. Jiang, H. Xia, Z. Hu, J. Zhang, W. Zhang, Q. Mao, L. Hu, T. Yang, F. Wen, and E. Wang, “Spatial algebraic solitons at the Dirac point in optically induced nonlinear photonic lattices,” Opt. Express 25(24), 30349–30364 (2017).
[Crossref] [PubMed]

K. Xie, W. Zhang, A. D. Boardman, H. Jiang, Z. Hu, Y. Liu, M. Xie, Q. Mao, L. Hu, Q. Li, T. Yang, F. Wen, and E. Wang, “Fiber guiding at the Dirac frequency beyond photonic bandgaps,” Light Sci. Appl. 4(6), e304 (2015).
[Crossref]

Liem, A.

Limpert, J.

Lin, S. Y.

Linder, N.

C. Wiesmann, K. Bergenek, N. Linder, and U. T. Schwarz, “Photonic crystal LEDs–designing light extraction,” Laser Photonics Rev. 3(3), 262–286 (2009).
[Crossref]

Liu, Y.

K. Xie, W. Zhang, A. D. Boardman, H. Jiang, Z. Hu, Y. Liu, M. Xie, Q. Mao, L. Hu, Q. Li, T. Yang, F. Wen, and E. Wang, “Fiber guiding at the Dirac frequency beyond photonic bandgaps,” Light Sci. Appl. 4(6), e304 (2015).
[Crossref]

K. Xie, H. Jiang, A. D. Boardman, Y. Liu, Z. Wu, M. Xie, P. Jiang, Q. Xu, M. Yu, and L. E. Davis, “Trapped photons at a Dirac point: a new horizon for photonic crystals,” Laser Photonics Rev. 8(4), 583–589 (2014).
[Crossref]

Longhi, S.

S. Longhi, “Quantum - optical analogies using photonic structures,” Laser Photonics Rev. 3(3), 243–261 (2009).
[Crossref]

Lousse, V.

Mao, Q.

K. Xie, A. D. Boardman, Q. Li, Z. Shi, H. Jiang, H. Xia, Z. Hu, J. Zhang, W. Zhang, Q. Mao, L. Hu, T. Yang, F. Wen, and E. Wang, “Spatial algebraic solitons at the Dirac point in optically induced nonlinear photonic lattices,” Opt. Express 25(24), 30349–30364 (2017).
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K. Xie, W. Zhang, A. D. Boardman, H. Jiang, Z. Hu, Y. Liu, M. Xie, Q. Mao, L. Hu, Q. Li, T. Yang, F. Wen, and E. Wang, “Fiber guiding at the Dirac frequency beyond photonic bandgaps,” Light Sci. Appl. 4(6), e304 (2015).
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Maradudin, A. A.

M. Plihal and A. A. Maradudin, “Photonic band structure of two-dimensional systems: The triangular lattice,” Phys. Rev. B Condens. Matter 44(16), 8565–8571 (1991).
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J. Mei, Y. Wu, C. T. Chan, and Z. Q. Zhang, “First-principles study of Dirac and Dirac-like cones in phononic and photonic crystals,” Phys. Rev. B 86(3), 035141 (2012).
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Miski-Oglu, M.

S. Bittner, B. Dietz, M. Miski-Oglu, and A. Richter, “Extremal transmission through a microwave photonic crystal and the observation of edge states in a rectangular Dirac billiard,” Phys. Rev. B 85(6), 064301 (2012).
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S. Bittner, B. Dietz, M. Miski-Oglu, P. O. Iriarte, A. Richter, and F. Schäfer, “Observation of a Dirac point in microwave experiments with a photonic crystal modeling graphene,” Phys. Rev. B 82(1), 014301 (2010).
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Mitsugi, S.

Montambaux, G.

M. Bellec, U. Kuhl, G. Montambaux, and F. Mortessagne, “Manipulation of edge states in microwave artificial grapheme,” New J. Phys. 16(11), 113023 (2014).
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M. Bellec, U. Kuhl, G. Montambaux, and F. Mortessagne, “Tight-binding couplings in microwave artificial graphene,” Phys. Rev. B 88(11), 115437 (2013).
[Crossref]

Mortessagne, F.

J. Böhm, M. Bellec, F. Mortessagne, U. Kuhl, S. Barkhofen, S. Gehler, H. J. Stöckmann, I. Foulger, S. Gnutzmann, and G. Tanner, “Microwave Experiments Simulating Quantum Search and Directed Transport in Artificial Graphene,” Phys. Rev. Lett. 114(11), 110501 (2015).
[Crossref] [PubMed]

M. Bellec, U. Kuhl, G. Montambaux, and F. Mortessagne, “Manipulation of edge states in microwave artificial grapheme,” New J. Phys. 16(11), 113023 (2014).
[Crossref]

M. Bellec, U. Kuhl, G. Montambaux, and F. Mortessagne, “Tight-binding couplings in microwave artificial graphene,” Phys. Rev. B 88(11), 115437 (2013).
[Crossref]

U. Kuhl, S. Barkhofen, T. Tudorovskiy, H. J. Stöckmann, T. Hossain, L. de Forges de Parny, and F. Mortessagne, “Dirac point and edge states in a microwave realization of tight-binding graphene-like structures,” Phys. Rev. B 82(9), 094308 (2010).
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Nolte, S.

Notomi, M.

Novoselov, K. S.

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Peres, N. M. R.

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Plihal, M.

M. Plihal and A. A. Maradudin, “Photonic band structure of two-dimensional systems: The triangular lattice,” Phys. Rev. B Condens. Matter 44(16), 8565–8571 (1991).
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Richter, A.

S. Bittner, B. Dietz, M. Miski-Oglu, and A. Richter, “Extremal transmission through a microwave photonic crystal and the observation of edge states in a rectangular Dirac billiard,” Phys. Rev. B 85(6), 064301 (2012).
[Crossref]

S. Bittner, B. Dietz, M. Miski-Oglu, P. O. Iriarte, A. Richter, and F. Schäfer, “Observation of a Dirac point in microwave experiments with a photonic crystal modeling graphene,” Phys. Rev. B 82(1), 014301 (2010).
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Ryu, H.

Santos, J. L.

O. Frazão, J. L. Santos, F. M. Araújo, and L. A. Ferreira, “Optical sensing with photonic crystal fibers,” Laser Photonics Rev. 2(6), 449–459 (2008).
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Schäfer, F.

S. Bittner, B. Dietz, M. Miski-Oglu, P. O. Iriarte, A. Richter, and F. Schäfer, “Observation of a Dirac point in microwave experiments with a photonic crystal modeling graphene,” Phys. Rev. B 82(1), 014301 (2010).
[Crossref]

Schreiber, T.

Schwarz, U. T.

C. Wiesmann, K. Bergenek, N. Linder, and U. T. Schwarz, “Photonic crystal LEDs–designing light extraction,” Laser Photonics Rev. 3(3), 262–286 (2009).
[Crossref]

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

Solgaard, O.

Sols, F.

B. Wunsch, F. Guinea, and F. Sols, “Dirac-point engineering and topological phase transitions in honeycomb optical lattices,” New J. Phys. 10(10), 103027 (2008).
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Soukoulis, C. M.

M. Diem, T. Koschny, and C. M. Soukoulis, “Transmission in the vicinity of the Dirac point in hexagonal photonic crystals,” Physica B 405(14), 2990–2995 (2010).
[Crossref]

Stöckmann, H. J.

J. Böhm, M. Bellec, F. Mortessagne, U. Kuhl, S. Barkhofen, S. Gehler, H. J. Stöckmann, I. Foulger, S. Gnutzmann, and G. Tanner, “Microwave Experiments Simulating Quantum Search and Directed Transport in Artificial Graphene,” Phys. Rev. Lett. 114(11), 110501 (2015).
[Crossref] [PubMed]

U. Kuhl, S. Barkhofen, T. Tudorovskiy, H. J. Stöckmann, T. Hossain, L. de Forges de Parny, and F. Mortessagne, “Dirac point and edge states in a microwave realization of tight-binding graphene-like structures,” Phys. Rev. B 82(9), 094308 (2010).
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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,” Nature 438(7065), 201–204 (2005).
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Suh, W.

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,” Nature 438(7065), 201–204 (2005).
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Tanner, G.

J. Böhm, M. Bellec, F. Mortessagne, U. Kuhl, S. Barkhofen, S. Gehler, H. J. Stöckmann, I. Foulger, S. Gnutzmann, and G. Tanner, “Microwave Experiments Simulating Quantum Search and Directed Transport in Artificial Graphene,” Phys. Rev. Lett. 114(11), 110501 (2015).
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Tudorovskiy, T.

U. Kuhl, S. Barkhofen, T. Tudorovskiy, H. J. Stöckmann, T. Hossain, L. de Forges de Parny, and F. Mortessagne, “Dirac point and edge states in a microwave realization of tight-binding graphene-like structures,” Phys. Rev. B 82(9), 094308 (2010).
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Tünnermann, T.

Vienne, G.

Villeneuve, P. R.

J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, “Photonic crystals: putting a new twist on light,” Nature 386(6621), 143–149 (1997).
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Wang, E.

K. Xie, A. D. Boardman, Q. Li, Z. Shi, H. Jiang, H. Xia, Z. Hu, J. Zhang, W. Zhang, Q. Mao, L. Hu, T. Yang, F. Wen, and E. Wang, “Spatial algebraic solitons at the Dirac point in optically induced nonlinear photonic lattices,” Opt. Express 25(24), 30349–30364 (2017).
[Crossref] [PubMed]

K. Xie, W. Zhang, A. D. Boardman, H. Jiang, Z. Hu, Y. Liu, M. Xie, Q. Mao, L. Hu, Q. Li, T. Yang, F. Wen, and E. Wang, “Fiber guiding at the Dirac frequency beyond photonic bandgaps,” Light Sci. Appl. 4(6), e304 (2015).
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Wen, F.

K. Xie, A. D. Boardman, Q. Li, Z. Shi, H. Jiang, H. Xia, Z. Hu, J. Zhang, W. Zhang, Q. Mao, L. Hu, T. Yang, F. Wen, and E. Wang, “Spatial algebraic solitons at the Dirac point in optically induced nonlinear photonic lattices,” Opt. Express 25(24), 30349–30364 (2017).
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K. Xie, W. Zhang, A. D. Boardman, H. Jiang, Z. Hu, Y. Liu, M. Xie, Q. Mao, L. Hu, Q. Li, T. Yang, F. Wen, and E. Wang, “Fiber guiding at the Dirac frequency beyond photonic bandgaps,” Light Sci. Appl. 4(6), e304 (2015).
[Crossref]

Wiesmann, C.

C. Wiesmann, K. Bergenek, N. Linder, and U. T. Schwarz, “Photonic crystal LEDs–designing light extraction,” Laser Photonics Rev. 3(3), 262–286 (2009).
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Wu, Y.

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

K. Xie, H. Jiang, A. D. Boardman, Y. Liu, Z. Wu, M. Xie, P. Jiang, Q. Xu, M. Yu, and L. E. Davis, “Trapped photons at a Dirac point: a new horizon for photonic crystals,” Laser Photonics Rev. 8(4), 583–589 (2014).
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Wunsch, B.

B. Wunsch, F. Guinea, and F. Sols, “Dirac-point engineering and topological phase transitions in honeycomb optical lattices,” New J. Phys. 10(10), 103027 (2008).
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Xia, H.

Xie, K.

K. Xie, A. D. Boardman, Q. Li, Z. Shi, H. Jiang, H. Xia, Z. Hu, J. Zhang, W. Zhang, Q. Mao, L. Hu, T. Yang, F. Wen, and E. Wang, “Spatial algebraic solitons at the Dirac point in optically induced nonlinear photonic lattices,” Opt. Express 25(24), 30349–30364 (2017).
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K. Xie, W. Zhang, A. D. Boardman, H. Jiang, Z. Hu, Y. Liu, M. Xie, Q. Mao, L. Hu, Q. Li, T. Yang, F. Wen, and E. Wang, “Fiber guiding at the Dirac frequency beyond photonic bandgaps,” Light Sci. Appl. 4(6), e304 (2015).
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K. Xie, H. Jiang, A. D. Boardman, Y. Liu, Z. Wu, M. Xie, P. Jiang, Q. Xu, M. Yu, and L. E. Davis, “Trapped photons at a Dirac point: a new horizon for photonic crystals,” Laser Photonics Rev. 8(4), 583–589 (2014).
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Xie, M.

K. Xie, W. Zhang, A. D. Boardman, H. Jiang, Z. Hu, Y. Liu, M. Xie, Q. Mao, L. Hu, Q. Li, T. Yang, F. Wen, and E. Wang, “Fiber guiding at the Dirac frequency beyond photonic bandgaps,” Light Sci. Appl. 4(6), e304 (2015).
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K. Xie, H. Jiang, A. D. Boardman, Y. Liu, Z. Wu, M. Xie, P. Jiang, Q. Xu, M. Yu, and L. E. Davis, “Trapped photons at a Dirac point: a new horizon for photonic crystals,” Laser Photonics Rev. 8(4), 583–589 (2014).
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K. Xie, H. Jiang, A. D. Boardman, Y. Liu, Z. Wu, M. Xie, P. Jiang, Q. Xu, M. Yu, and L. E. Davis, “Trapped photons at a Dirac point: a new horizon for photonic crystals,” Laser Photonics Rev. 8(4), 583–589 (2014).
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K. Xie, A. D. Boardman, Q. Li, Z. Shi, H. Jiang, H. Xia, Z. Hu, J. Zhang, W. Zhang, Q. Mao, L. Hu, T. Yang, F. Wen, and E. Wang, “Spatial algebraic solitons at the Dirac point in optically induced nonlinear photonic lattices,” Opt. Express 25(24), 30349–30364 (2017).
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K. Xie, W. Zhang, A. D. Boardman, H. Jiang, Z. Hu, Y. Liu, M. Xie, Q. Mao, L. Hu, Q. Li, T. Yang, F. Wen, and E. Wang, “Fiber guiding at the Dirac frequency beyond photonic bandgaps,” Light Sci. Appl. 4(6), e304 (2015).
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K. Xie, H. Jiang, A. D. Boardman, Y. Liu, Z. Wu, M. Xie, P. Jiang, Q. Xu, M. Yu, and L. E. Davis, “Trapped photons at a Dirac point: a new horizon for photonic crystals,” Laser Photonics Rev. 8(4), 583–589 (2014).
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S. R. Zandbergen and 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).
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K. Xie, A. D. Boardman, Q. Li, Z. Shi, H. Jiang, H. Xia, Z. Hu, J. Zhang, W. Zhang, Q. Mao, L. Hu, T. Yang, F. Wen, and E. Wang, “Spatial algebraic solitons at the Dirac point in optically induced nonlinear photonic lattices,” Opt. Express 25(24), 30349–30364 (2017).
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K. Xie, W. Zhang, A. D. Boardman, H. Jiang, Z. Hu, Y. Liu, M. Xie, Q. Mao, L. Hu, Q. Li, T. Yang, F. Wen, and E. Wang, “Fiber guiding at the Dirac frequency beyond photonic bandgaps,” Light Sci. Appl. 4(6), e304 (2015).
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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).
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Y. Zhang, Y. W. Tan, H. L. Stormer, and P. Kim, “Experimental observation of the quantum Hall effect and Berry’s phase in graphene,” Nature 438(7065), 201–204 (2005).
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Zhang, Z. Q.

J. Mei, Y. Wu, C. T. Chan, and Z. Q. Zhang, “First-principles study of Dirac and Dirac-like cones in phononic and photonic crystals,” Phys. Rev. B 86(3), 035141 (2012).
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J. Opt. Soc. Am. B (1)

Laser Photonics Rev. (4)

O. Frazão, J. L. Santos, F. M. Araújo, and L. A. Ferreira, “Optical sensing with photonic crystal fibers,” Laser Photonics Rev. 2(6), 449–459 (2008).
[Crossref]

C. Wiesmann, K. Bergenek, N. Linder, and U. T. Schwarz, “Photonic crystal LEDs–designing light extraction,” Laser Photonics Rev. 3(3), 262–286 (2009).
[Crossref]

S. Longhi, “Quantum - optical analogies using photonic structures,” Laser Photonics Rev. 3(3), 243–261 (2009).
[Crossref]

K. Xie, H. Jiang, A. D. Boardman, Y. Liu, Z. Wu, M. Xie, P. Jiang, Q. Xu, M. Yu, and L. E. Davis, “Trapped photons at a Dirac point: a new horizon for photonic crystals,” Laser Photonics Rev. 8(4), 583–589 (2014).
[Crossref]

Light Sci. Appl. (1)

K. Xie, W. Zhang, A. D. Boardman, H. Jiang, Z. Hu, Y. Liu, M. Xie, Q. Mao, L. Hu, Q. Li, T. Yang, F. Wen, and E. Wang, “Fiber guiding at the Dirac frequency beyond photonic bandgaps,” Light Sci. Appl. 4(6), e304 (2015).
[Crossref]

Nature (2)

Y. Zhang, Y. W. Tan, H. L. Stormer, and P. Kim, “Experimental observation of the quantum Hall effect and Berry’s phase in graphene,” Nature 438(7065), 201–204 (2005).
[Crossref] [PubMed]

J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, “Photonic crystals: putting a new twist on light,” Nature 386(6621), 143–149 (1997).
[Crossref]

New J. Phys. (2)

B. Wunsch, F. Guinea, and F. Sols, “Dirac-point engineering and topological phase transitions in honeycomb optical lattices,” New J. Phys. 10(10), 103027 (2008).
[Crossref]

M. Bellec, U. Kuhl, G. Montambaux, and F. Mortessagne, “Manipulation of edge states in microwave artificial grapheme,” New J. Phys. 16(11), 113023 (2014).
[Crossref]

Opt. Express (7)

J. Limpert, T. Schreiber, S. Nolte, H. Zellmer, T. Tünnermann, R. Iliew, F. Lederer, J. Broeng, G. Vienne, A. Petersson, and C. Jakobsen, “High-power air-clad large-mode-area photonic crystal fiber laser,” Opt. Express 11(7), 818–823 (2003).
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J. Limpert, A. Liem, M. Reich, T. Schreiber, S. Nolte, H. Zellmer, A. Tünnermann, J. Broeng, A. Petersson, and C. Jakobsen, “Low-nonlinearity single-transverse-mode ytterbium-doped photonic crystal fiber amplifier,” Opt. Express 12(7), 1313–1319 (2004).
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M. Notomi, A. Shinya, S. Mitsugi, E. Kuramochi, and H. Ryu, “Waveguides, resonators and their coupled elements in photonic crystal slabs,” Opt. Express 12(8), 1551–1561 (2004).
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V. Lousse, W. Suh, O. Kilic, S. Kim, O. Solgaard, and S. Fan, “Angular and polarization properties of a photonic crystal slab mirror,” Opt. Express 12(8), 1575–1582 (2004).
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Y. Akahane, T. Asano, B.-S. Song, and S. Noda, “Fine-tuned high-Q photonic-crystal nanocavity,” Opt. Express 13(4), 1202–1214 (2005).
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L. Rindorf, J. B. Jensen, M. Dufva, L. H. Pedersen, P. E. Høiby, and O. Bang, “Photonic crystal fiber long-period gratings for biochemical sensing,” Opt. Express 14(18), 8224–8231 (2006).
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K. Xie, A. D. Boardman, Q. Li, Z. Shi, H. Jiang, H. Xia, Z. Hu, J. Zhang, W. Zhang, Q. Mao, L. Hu, T. Yang, F. Wen, and E. Wang, “Spatial algebraic solitons at the Dirac point in optically induced nonlinear photonic lattices,” Opt. Express 25(24), 30349–30364 (2017).
[Crossref] [PubMed]

Phys. Rev. A (1)

R. A. Sepkhanov, Y. B. Bazaliy, and 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 (6)

H. M. Guo and M. Franz, “Topological insulator on the kagome lattice,” Phys. Rev. B 80(11), 113102 (2009).
[Crossref]

S. Bittner, B. Dietz, M. Miski-Oglu, P. O. Iriarte, A. Richter, and F. Schäfer, “Observation of a Dirac point in microwave experiments with a photonic crystal modeling graphene,” Phys. Rev. B 82(1), 014301 (2010).
[Crossref]

S. Bittner, B. Dietz, M. Miski-Oglu, and A. Richter, “Extremal transmission through a microwave photonic crystal and the observation of edge states in a rectangular Dirac billiard,” Phys. Rev. B 85(6), 064301 (2012).
[Crossref]

M. Bellec, U. Kuhl, G. Montambaux, and F. Mortessagne, “Tight-binding couplings in microwave artificial graphene,” Phys. Rev. B 88(11), 115437 (2013).
[Crossref]

U. Kuhl, S. Barkhofen, T. Tudorovskiy, H. J. Stöckmann, T. Hossain, L. de Forges de Parny, and F. Mortessagne, “Dirac point and edge states in a microwave realization of tight-binding graphene-like structures,” Phys. Rev. B 82(9), 094308 (2010).
[Crossref]

J. Mei, Y. Wu, C. T. Chan, and 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]

Phys. Rev. B Condens. Matter (1)

M. Plihal and A. A. Maradudin, “Photonic band structure of two-dimensional systems: The triangular lattice,” Phys. Rev. B Condens. Matter 44(16), 8565–8571 (1991).
[Crossref] [PubMed]

Phys. Rev. Lett. (3)

S. R. Zandbergen and 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).
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Physica B (1)

M. Diem, T. Koschny, and C. M. Soukoulis, “Transmission in the vicinity of the Dirac point in hexagonal photonic crystals,” Physica B 405(14), 2990–2995 (2010).
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Rev. Mod. Phys. (1)

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(1), 109–162 (2009).
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K. Busch, S. Lölkes, R. B. Wehrspohn, and H. Föll, Photonic Crystals: Advances in Design, Fabrication, and Characterization (John Wiley & Sons, 2006).

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Figures (7)

Fig. 1
Fig. 1 (a) Calculated photonic band structure of a triangular lattice of dielectric rods in air. The rods are constituted of Al2O3 material with a dielectric constant εd = 9.8 and a radius-to-pitch ratio r/a = 0.25. Blue lines: TE polarization. Red dashed lines: TM polarization. Black dot-dashed line at fda/c = 0.5244 marks the Dirac point in the TE band structure. Lower-left inset: top view of the 2D triangular lattice PC. Lower-middle inset: first Brillouin zone of the reciprocal lattice, revealing the high-symmetry points Γ, M, K at the corners of the irreducible Brillouin zone of the structure. (b) The variation of the Dirac cone edge with the radius-to-pitch ratio r/a. Blue diamond lines: Dirac cone edges. Black circle line: Dirac point.
Fig. 2
Fig. 2 Dependence of the frequency of the localized modes on the defect rod radius R. PC structure parameters are: εd = 9.8, r/a = 0.30. Magenta solid curve: localized defect modes calculated by PWE method. Red diamonds: data points of the localized modes simulated by FDTD method. Green line: the Dirac point with the normalized frequency fd = 0.4763c/a. Lower-left inset: profile of the localized modes at Dirac frequency superimposed on geometric structures of the crystals. Upper-right inset: conventional band structure in dimensionless frequency units. The Dirac cone is formed between the second and third photonic band for TE polarized waves.
Fig. 3
Fig. 3 Numerical investigations (by FDTD) of the Dirac mode at the Dirac frequency fd = 12.24 GHz for the structure: lattice constant a = 11.67 mm, rods diameters d = 7 mm, defect diameter D = 6 mm, material relative permittivity εd = 9.8 for both normal rods and the defect rod. The numerical computations are set with a step size ∆x = a/50, ∆y = ∆x·sin(60°), time step ct = 0.6484∆x. Cavity area are S = 50 cm × 50 cm, and the boundary conditions are PML boundary. (a) Time domain evolution of the magnetic field amplitude hz at the cavity center. (b) Pattern of frequency component of the magnetic field |Hz| at fd. (c) Dependence of product of the |Hz| and |r3/2| on the length in the x-axis. (d) Phase of the mode profile along the x axis. (e) Total quality factor Qtotal variation with resonant frequency. The resonant frequency is dependent on the diameter D of the defect rod. The vertical green line marks the position of the Dirac frequency. The value of Qtotal reaches its maximum at this frequency.
Fig. 4
Fig. 4 Experimental setup for measurement of the transmission spectra of the PC cavity. Parameters of the cavity are: a = 11.7 mm, εd = 9.8 ± 0.7, d = 6.85 ± 0.1 mm, and D = 6 mm.
Fig. 5
Fig. 5 (a) Transmitted power vs. frequency for the PC along ΓM direction with 6 or 13 layers in the propagation direction. (b) The band structure in the ΓM direction. The grey regions indicate two partial bandgaps within the effective frequency of the horn antennas. The lower-left inset shows the transmission direction. (c) Transmitted power vs. frequency for the PC along the MK direction with 10 or 20 layers in the propagation direction. The dashed lines mark the frequency range of the Dirac cone. (d) The band structure in the MK direction. The light grey regions indicate two partial bandgaps within the effective frequency of horn antennas. The dark grey region is the frequency range occupied by the Dirac cone. The lower-left inset shows the transmission direction.
Fig. 6
Fig. 6 (a) Transmitted power vs. frequency for the PC cavity with a defect. Red line: ΓM direction with 13 layers in the propagation direction. Blue line: MK direction with 19 layers in the transmission direction. The vertical green line marks the location of the resonant peak in both the ΓM and MK cases. (b) Numerically calculated (by FDTD) transmissivity of the photonic cavity with a defect (Red line: 13 layers on ΓM direction, blue line: 19 layers on MK direction). The vertical green line marks the location of the resonant peak in both the ΓM and MK cases.
Fig. 7
Fig. 7 (a) The partial bandgap edges and the Dirac point vary with the relative permittivity εd. Red rectangle lines: first partial gap edges in the ΓM direction as well as the Dirac cone edge. Blue diamond lines: second partial gap edges in the ΓM direction. Black circle line: Dirac point. (b) Red lines: Folded band structure of a 5 × 5 lattice with uneven rod sizes randomly varied between d = 6.85 ± 0.1 mm. Blue lines: The standard photonic band structure of an ideal, uniform lattice for rod diameters d = 7 mm. The Dirac points (black dot-dashed line) for the two cases almost coincide with each other.

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