Abstract

We present a multiple-scattering method in conjunction with supercell calculations to study the electromagnetic surface states in three-dimensional photonic crystal slab. Using our technique, we obtain the first prediction of Dirac-cone photonic surface state in some three-dimensional photonic crystal slabs. Such a state can be used to investigate some extremal transmission phenomena of electromagnetic waves near the Dirac point on the surface of the crystal, which is similar to the case of electron on the surface of topological insulators.

© 2011 OSA

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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2010 (4)

J. E. Moore, “The birth of topological insulators,” Nature 464(7286), 194–198 (2010).
[CrossRef] [PubMed]

X.-L. Qi and S.-C. Zhang, “The quantum spin Hall effect and topological insulators,” Phys. Today 63(1), 33 (2010).
[CrossRef]

H. Lin, R. S. Markiewicz, L. A. Wray, L. Fu, M. Z. Hasan, and A. Bansil, “Single-Dirac-cone topological surface states in the TlBiSe(2) class of topological semiconductors,” Phys. Rev. Lett. 105(3), 036404 (2010).
[CrossRef] [PubMed]

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]

2009 (8)

K. Ishizaki and S. Noda, “Manipulation of photons at the surface of three-dimensional photonic crystals,” Nature 460(7253), 367–370 (2009).
[CrossRef] [PubMed]

D. Hsieh, Y. Xia, L. Wray, D. Qian, A. Pal, J. H. Dil, J. Osterwalder, F. Meier, G. Bihlmayer, C. L. Kane, Y. S. Hor, R. J. Cava, and M. Z. Hasan, “Observation of unconventional quantum spin textures in topological insulators,” Science 323(5916), 919–922 (2009).
[CrossRef] [PubMed]

Y. Xia, D. Qian, D. Hsieh, L. Wray, A. Pal, H. Lin, A. Bansil, D. Grauer, Y. S. Hor, R. J. Cava, and M. Z. Hasan, “Observation of a large-gap topological-insulator class with a single Dirac cone on the surface,” Nat. Phys. 5(6), 398–402 (2009).
[CrossRef]

Y. L. Chen, J. G. Analytis, J. H. Chu, Z. K. Liu, S.-K. Mo, X. L. Qi, H. J. Zhang, D. H. Lu, X. Dai, Z. Fang, S. C. Zhang, I. R. Fisher, Z. Hussain, and Z. X. Shen, “Experimental realization of a three-dimensional topological insulator, Bi2Te3,” Science 325(5937), 178–181 (2009).
[CrossRef] [PubMed]

D. Hsieh, Y. Xia, D. Qian, L. Wray, J. H. Dil, F. Meier, J. Osterwalder, L. Patthey, J. G. Checkelsky, N. P. Ong, A. V. Fedorov, H. Lin, A. Bansil, D. Grauer, Y. S. Hor, R. J. Cava, and M. Z. Hasan, “A tunable topological insulator in the spin helical Dirac transport regime,” Nature 460(7259), 1101–1105 (2009).
[CrossRef] [PubMed]

D. Hsieh, Y. Xia, D. Qian, L. Wray, F. Meier, J. H. Dil, J. Osterwalder, L. Patthey, A. V. Fedorov, H. Lin, A. Bansil, D. Grauer, Y. S. Hor, R. J. Cava, and M. Z. Hasan, “Observation of time-reversal-protected single-dirac-cone topological-insulator states in Bi2Te3 and Sb2Te3,” Phys. Rev. Lett. 103(14), 146401 (2009).
[CrossRef] [PubMed]

H. J. Zhang, C. X. Liu, X. L. Qi, X. Dai, Z. Fang, and S.-C. Zhang, “Topological insulators in Bi2Se3, Bi2Te3 and Sb2Te3 with a single Dirac cone on the surface,” Nat. Phys. 5(6), 438–442 (2009).
[CrossRef]

T. Zhang, P. Cheng, X. Chen, J.-F. Jia, X. Ma, K. He, L. Wang, H. Zhang, X. Dai, Z. Fang, X. Xie, and Q.-K. Xue, “Experimental demonstration of topological surface states protected by time-reversal symmetry,” Phys. Rev. Lett. 103(26), 266803 (2009).
[CrossRef] [PubMed]

2008 (5)

D. Hsieh, D. Qian, L. Wray, Y. Xia, Y. S. Hor, R. J. Cava, and M. Z. Hasan, “A topological Dirac insulator in a quantum spin Hall phase,” Nature 452(7190), 970–974 (2008).
[CrossRef] [PubMed]

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

X. D. 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. D. Zhang and Z. Y. Liu, “Extremal transmission and beating effect of acoustic waves in two-dimensional sonic crystals,” Phys. Rev. Lett. 101(26), 264303 (2008).
[CrossRef] [PubMed]

2007 (1)

R. A. Sepkhanov, Ya. 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)

B. A. Bernevig and S. C. Zhang, “Quantum spin Hall effect,” Phys. Rev. Lett. 96(10), 106802 (2006).
[CrossRef] [PubMed]

2005 (2)

C. L. Kane and E. J. Mele, “Z2 topological order and the quantum spin Hall effect,” Phys. Rev. Lett. 95(14), 146802 (2005).
[CrossRef] [PubMed]

S. Enoch, E. Popov, and N. Bonod, “Analysis of the physical origin of surface modes on finite-size photonic crystals,” Phys. Rev. B 72(15), 155101 (2005).
[CrossRef]

2001 (2)

X. Zhang, L.-M. Li, Z.-Q. Zhang, and C. Chan, “Surface states in two-dimensional metallodielectric photonic crystals studied by a multiple-scattering method,” Phys. Rev. B 63(12), 125114 (2001).
[CrossRef]

W. Y. Zhang, C. T. Chan, and P. Sheng, “Multiple scattering theory and its application to photonic band gap systems consisting of coated spheres,” Opt. Express 8(3), 203–208 (2001).
[CrossRef] [PubMed]

2000 (1)

W. Y. Zhang, X. Y. Lei, Z. L. Wang, D. G. Zheng, W. Y. Tam, C. T. Chan, and P. Sheng, “Robust photonic band gap from tunable scatterers,” Phys. Rev. Lett. 84(13), 2853–2856 (2000).
[CrossRef] [PubMed]

1999 (3)

A. Moroz, “Three-dimensional complete photonic-band-gap structures in the visible,” Phys. Rev. Lett. 83(25), 5274–5277 (1999).
[CrossRef]

F. Ramos-Mendieta and P. Halevi, “Surface electromagnetic waves in two-dimensional photonic crystals: Effect of the position of the surface plane,” Phys. Rev. B 59(23), 15112–15120 (1999).
[CrossRef]

W. M. Robertson and M. S. May, “Surface electromagnetic wave excitation on one-dimensional photonic band-gap arrays,” Appl. Phys. Lett. 74(13), 1800–1803 (1999).
[CrossRef]

1998 (1)

N. Stefanou, V. Yannopapas, and A. Modinos, “Heterostructures of photonic crystals: frequency bands and transmission coefficients,” Comput. Phys. Commun. 113(1), 49–77 (1998).
[CrossRef]

1996 (2)

P. Etchegoin and R. T. Phillips, “Photon focusing, internal diffraction, and surface states in periodic dielectric structures,” Phys. Rev. B Condens. Matter 53(19), 12674–12683 (1996).
[CrossRef] [PubMed]

J. Elson and P. Tran, “Coupled-mode calculation with the R-matrix propagator for the dispersion of surface waves on a truncated photonic crystal,” Phys. Rev. B 54(3), 1711–1715 (1996).
[CrossRef]

1993 (1)

1991 (1)

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Electromagnetic Bloch waves at the surface of a photonic crystal,” Phys. Rev. B 44(19), R10961 (1991).
[CrossRef]

1987 (3)

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58(20), 2059–2062 (1987).
[CrossRef] [PubMed]

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58(23), 2486–2489 (1987).
[CrossRef] [PubMed]

A. Modinos, “Scattering of electromagnetic waves by a plane of spheres-formalism,” Physica A 141(2-3), 575–588 (1987).
[CrossRef]

1961 (1)

F. S. Ham and B. Segall, “Energy bands in periodic lattices: Green’s function method,” Phys. Rev. 124(6), 1786–1796 (1961).
[CrossRef]

Analytis, J. G.

Y. L. Chen, J. G. Analytis, J. H. Chu, Z. K. Liu, S.-K. Mo, X. L. Qi, H. J. Zhang, D. H. Lu, X. Dai, Z. Fang, S. C. Zhang, I. R. Fisher, Z. Hussain, and Z. X. Shen, “Experimental realization of a three-dimensional topological insulator, Bi2Te3,” Science 325(5937), 178–181 (2009).
[CrossRef] [PubMed]

Arjavalingam, G.

Bansil, A.

H. Lin, R. S. Markiewicz, L. A. Wray, L. Fu, M. Z. Hasan, and A. Bansil, “Single-Dirac-cone topological surface states in the TlBiSe(2) class of topological semiconductors,” Phys. Rev. Lett. 105(3), 036404 (2010).
[CrossRef] [PubMed]

Y. Xia, D. Qian, D. Hsieh, L. Wray, A. Pal, H. Lin, A. Bansil, D. Grauer, Y. S. Hor, R. J. Cava, and M. Z. Hasan, “Observation of a large-gap topological-insulator class with a single Dirac cone on the surface,” Nat. Phys. 5(6), 398–402 (2009).
[CrossRef]

D. Hsieh, Y. Xia, D. Qian, L. Wray, J. H. Dil, F. Meier, J. Osterwalder, L. Patthey, J. G. Checkelsky, N. P. Ong, A. V. Fedorov, H. Lin, A. Bansil, D. Grauer, Y. S. Hor, R. J. Cava, and M. Z. Hasan, “A tunable topological insulator in the spin helical Dirac transport regime,” Nature 460(7259), 1101–1105 (2009).
[CrossRef] [PubMed]

D. Hsieh, Y. Xia, D. Qian, L. Wray, F. Meier, J. H. Dil, J. Osterwalder, L. Patthey, A. V. Fedorov, H. Lin, A. Bansil, D. Grauer, Y. S. Hor, R. J. Cava, and M. Z. Hasan, “Observation of time-reversal-protected single-dirac-cone topological-insulator states in Bi2Te3 and Sb2Te3,” Phys. Rev. Lett. 103(14), 146401 (2009).
[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. A 75(6), 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. A 75(6), 063813 (2007).
[CrossRef]

Bernevig, B. A.

B. A. Bernevig and S. C. Zhang, “Quantum spin Hall effect,” Phys. Rev. Lett. 96(10), 106802 (2006).
[CrossRef] [PubMed]

Bihlmayer, G.

D. Hsieh, Y. Xia, L. Wray, D. Qian, A. Pal, J. H. Dil, J. Osterwalder, F. Meier, G. Bihlmayer, C. L. Kane, Y. S. Hor, R. J. Cava, and M. Z. Hasan, “Observation of unconventional quantum spin textures in topological insulators,” Science 323(5916), 919–922 (2009).
[CrossRef] [PubMed]

Bonod, N.

S. Enoch, E. Popov, and N. Bonod, “Analysis of the physical origin of surface modes on finite-size photonic crystals,” Phys. Rev. B 72(15), 155101 (2005).
[CrossRef]

Brommer, K. D.

W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Observation of surface photons on periodic dielectric arrays,” Opt. Lett. 18(7), 528–530 (1993).
[CrossRef] [PubMed]

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Electromagnetic Bloch waves at the surface of a photonic crystal,” Phys. Rev. B 44(19), R10961 (1991).
[CrossRef]

Cava, R. J.

D. Hsieh, Y. Xia, L. Wray, D. Qian, A. Pal, J. H. Dil, J. Osterwalder, F. Meier, G. Bihlmayer, C. L. Kane, Y. S. Hor, R. J. Cava, and M. Z. Hasan, “Observation of unconventional quantum spin textures in topological insulators,” Science 323(5916), 919–922 (2009).
[CrossRef] [PubMed]

D. Hsieh, Y. Xia, D. Qian, L. Wray, F. Meier, J. H. Dil, J. Osterwalder, L. Patthey, A. V. Fedorov, H. Lin, A. Bansil, D. Grauer, Y. S. Hor, R. J. Cava, and M. Z. Hasan, “Observation of time-reversal-protected single-dirac-cone topological-insulator states in Bi2Te3 and Sb2Te3,” Phys. Rev. Lett. 103(14), 146401 (2009).
[CrossRef] [PubMed]

D. Hsieh, Y. Xia, D. Qian, L. Wray, J. H. Dil, F. Meier, J. Osterwalder, L. Patthey, J. G. Checkelsky, N. P. Ong, A. V. Fedorov, H. Lin, A. Bansil, D. Grauer, Y. S. Hor, R. J. Cava, and M. Z. Hasan, “A tunable topological insulator in the spin helical Dirac transport regime,” Nature 460(7259), 1101–1105 (2009).
[CrossRef] [PubMed]

Y. Xia, D. Qian, D. Hsieh, L. Wray, A. Pal, H. Lin, A. Bansil, D. Grauer, Y. S. Hor, R. J. Cava, and M. Z. Hasan, “Observation of a large-gap topological-insulator class with a single Dirac cone on the surface,” Nat. Phys. 5(6), 398–402 (2009).
[CrossRef]

D. Hsieh, D. Qian, L. Wray, Y. Xia, Y. S. Hor, R. J. Cava, and M. Z. Hasan, “A topological Dirac insulator in a quantum spin Hall phase,” Nature 452(7190), 970–974 (2008).
[CrossRef] [PubMed]

Chan, C.

X. Zhang, L.-M. Li, Z.-Q. Zhang, and C. Chan, “Surface states in two-dimensional metallodielectric photonic crystals studied by a multiple-scattering method,” Phys. Rev. B 63(12), 125114 (2001).
[CrossRef]

Chan, C. T.

W. Y. Zhang, C. T. Chan, and P. Sheng, “Multiple scattering theory and its application to photonic band gap systems consisting of coated spheres,” Opt. Express 8(3), 203–208 (2001).
[CrossRef] [PubMed]

W. Y. Zhang, X. Y. Lei, Z. L. Wang, D. G. Zheng, W. Y. Tam, C. T. Chan, and P. Sheng, “Robust photonic band gap from tunable scatterers,” Phys. Rev. Lett. 84(13), 2853–2856 (2000).
[CrossRef] [PubMed]

Checkelsky, J. G.

D. Hsieh, Y. Xia, D. Qian, L. Wray, J. H. Dil, F. Meier, J. Osterwalder, L. Patthey, J. G. Checkelsky, N. P. Ong, A. V. Fedorov, H. Lin, A. Bansil, D. Grauer, Y. S. Hor, R. J. Cava, and M. Z. Hasan, “A tunable topological insulator in the spin helical Dirac transport regime,” Nature 460(7259), 1101–1105 (2009).
[CrossRef] [PubMed]

Chen, X.

T. Zhang, P. Cheng, X. Chen, J.-F. Jia, X. Ma, K. He, L. Wang, H. Zhang, X. Dai, Z. Fang, X. Xie, and Q.-K. Xue, “Experimental demonstration of topological surface states protected by time-reversal symmetry,” Phys. Rev. Lett. 103(26), 266803 (2009).
[CrossRef] [PubMed]

Chen, Y. L.

Y. L. Chen, J. G. Analytis, J. H. Chu, Z. K. Liu, S.-K. Mo, X. L. Qi, H. J. Zhang, D. H. Lu, X. Dai, Z. Fang, S. C. Zhang, I. R. Fisher, Z. Hussain, and Z. X. Shen, “Experimental realization of a three-dimensional topological insulator, Bi2Te3,” Science 325(5937), 178–181 (2009).
[CrossRef] [PubMed]

Cheng, P.

T. Zhang, P. Cheng, X. Chen, J.-F. Jia, X. Ma, K. He, L. Wang, H. Zhang, X. Dai, Z. Fang, X. Xie, and Q.-K. Xue, “Experimental demonstration of topological surface states protected by time-reversal symmetry,” Phys. Rev. Lett. 103(26), 266803 (2009).
[CrossRef] [PubMed]

Chu, J. H.

Y. L. Chen, J. G. Analytis, J. H. Chu, Z. K. Liu, S.-K. Mo, X. L. Qi, H. J. Zhang, D. H. Lu, X. Dai, Z. Fang, S. C. Zhang, I. R. Fisher, Z. Hussain, and Z. X. Shen, “Experimental realization of a three-dimensional topological insulator, Bi2Te3,” Science 325(5937), 178–181 (2009).
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D. Hsieh, Y. Xia, L. Wray, D. Qian, A. Pal, J. H. Dil, J. Osterwalder, F. Meier, G. Bihlmayer, C. L. Kane, Y. S. Hor, R. J. Cava, and M. Z. Hasan, “Observation of unconventional quantum spin textures in topological insulators,” Science 323(5916), 919–922 (2009).
[CrossRef] [PubMed]

Y. Xia, D. Qian, D. Hsieh, L. Wray, A. Pal, H. Lin, A. Bansil, D. Grauer, Y. S. Hor, R. J. Cava, and M. Z. Hasan, “Observation of a large-gap topological-insulator class with a single Dirac cone on the surface,” Nat. Phys. 5(6), 398–402 (2009).
[CrossRef]

Patthey, L.

D. Hsieh, Y. Xia, D. Qian, L. Wray, J. H. Dil, F. Meier, J. Osterwalder, L. Patthey, J. G. Checkelsky, N. P. Ong, A. V. Fedorov, H. Lin, A. Bansil, D. Grauer, Y. S. Hor, R. J. Cava, and M. Z. Hasan, “A tunable topological insulator in the spin helical Dirac transport regime,” Nature 460(7259), 1101–1105 (2009).
[CrossRef] [PubMed]

D. Hsieh, Y. Xia, D. Qian, L. Wray, F. Meier, J. H. Dil, J. Osterwalder, L. Patthey, A. V. Fedorov, H. Lin, A. Bansil, D. Grauer, Y. S. Hor, R. J. Cava, and M. Z. Hasan, “Observation of time-reversal-protected single-dirac-cone topological-insulator states in Bi2Te3 and Sb2Te3,” Phys. Rev. Lett. 103(14), 146401 (2009).
[CrossRef] [PubMed]

Phillips, R. T.

P. Etchegoin and R. T. Phillips, “Photon focusing, internal diffraction, and surface states in periodic dielectric structures,” Phys. Rev. B Condens. Matter 53(19), 12674–12683 (1996).
[CrossRef] [PubMed]

Popov, E.

S. Enoch, E. Popov, and N. Bonod, “Analysis of the physical origin of surface modes on finite-size photonic crystals,” Phys. Rev. B 72(15), 155101 (2005).
[CrossRef]

Qi, X. L.

Y. L. Chen, J. G. Analytis, J. H. Chu, Z. K. Liu, S.-K. Mo, X. L. Qi, H. J. Zhang, D. H. Lu, X. Dai, Z. Fang, S. C. Zhang, I. R. Fisher, Z. Hussain, and Z. X. Shen, “Experimental realization of a three-dimensional topological insulator, Bi2Te3,” Science 325(5937), 178–181 (2009).
[CrossRef] [PubMed]

H. J. Zhang, C. X. Liu, X. L. Qi, X. Dai, Z. Fang, and S.-C. Zhang, “Topological insulators in Bi2Se3, Bi2Te3 and Sb2Te3 with a single Dirac cone on the surface,” Nat. Phys. 5(6), 438–442 (2009).
[CrossRef]

Qi, X.-L.

X.-L. Qi and S.-C. Zhang, “The quantum spin Hall effect and topological insulators,” Phys. Today 63(1), 33 (2010).
[CrossRef]

Qian, D.

Y. Xia, D. Qian, D. Hsieh, L. Wray, A. Pal, H. Lin, A. Bansil, D. Grauer, Y. S. Hor, R. J. Cava, and M. Z. Hasan, “Observation of a large-gap topological-insulator class with a single Dirac cone on the surface,” Nat. Phys. 5(6), 398–402 (2009).
[CrossRef]

D. Hsieh, Y. Xia, L. Wray, D. Qian, A. Pal, J. H. Dil, J. Osterwalder, F. Meier, G. Bihlmayer, C. L. Kane, Y. S. Hor, R. J. Cava, and M. Z. Hasan, “Observation of unconventional quantum spin textures in topological insulators,” Science 323(5916), 919–922 (2009).
[CrossRef] [PubMed]

D. Hsieh, Y. Xia, D. Qian, L. Wray, F. Meier, J. H. Dil, J. Osterwalder, L. Patthey, A. V. Fedorov, H. Lin, A. Bansil, D. Grauer, Y. S. Hor, R. J. Cava, and M. Z. Hasan, “Observation of time-reversal-protected single-dirac-cone topological-insulator states in Bi2Te3 and Sb2Te3,” Phys. Rev. Lett. 103(14), 146401 (2009).
[CrossRef] [PubMed]

D. Hsieh, Y. Xia, D. Qian, L. Wray, J. H. Dil, F. Meier, J. Osterwalder, L. Patthey, J. G. Checkelsky, N. P. Ong, A. V. Fedorov, H. Lin, A. Bansil, D. Grauer, Y. S. Hor, R. J. Cava, and M. Z. Hasan, “A tunable topological insulator in the spin helical Dirac transport regime,” Nature 460(7259), 1101–1105 (2009).
[CrossRef] [PubMed]

D. Hsieh, D. Qian, L. Wray, Y. Xia, Y. S. Hor, R. J. Cava, and M. Z. Hasan, “A topological Dirac insulator in a quantum spin Hall phase,” Nature 452(7190), 970–974 (2008).
[CrossRef] [PubMed]

Raghu, S.

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

Ramos-Mendieta, F.

F. Ramos-Mendieta and P. Halevi, “Surface electromagnetic waves in two-dimensional photonic crystals: Effect of the position of the surface plane,” Phys. Rev. B 59(23), 15112–15120 (1999).
[CrossRef]

Rappe, A. M.

W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Observation of surface photons on periodic dielectric arrays,” Opt. Lett. 18(7), 528–530 (1993).
[CrossRef] [PubMed]

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Electromagnetic Bloch waves at the surface of a photonic crystal,” Phys. Rev. B 44(19), R10961 (1991).
[CrossRef]

Robertson, W. M.

W. M. Robertson and M. S. May, “Surface electromagnetic wave excitation on one-dimensional photonic band-gap arrays,” Appl. Phys. Lett. 74(13), 1800–1803 (1999).
[CrossRef]

W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Observation of surface photons on periodic dielectric arrays,” Opt. Lett. 18(7), 528–530 (1993).
[CrossRef] [PubMed]

Segall, B.

F. S. Ham and B. Segall, “Energy bands in periodic lattices: Green’s function method,” Phys. Rev. 124(6), 1786–1796 (1961).
[CrossRef]

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. A 75(6), 063813 (2007).
[CrossRef]

Shen, Z. X.

Y. L. Chen, J. G. Analytis, J. H. Chu, Z. K. Liu, S.-K. Mo, X. L. Qi, H. J. Zhang, D. H. Lu, X. Dai, Z. Fang, S. C. Zhang, I. R. Fisher, Z. Hussain, and Z. X. Shen, “Experimental realization of a three-dimensional topological insulator, Bi2Te3,” Science 325(5937), 178–181 (2009).
[CrossRef] [PubMed]

Sheng, P.

W. Y. Zhang, C. T. Chan, and P. Sheng, “Multiple scattering theory and its application to photonic band gap systems consisting of coated spheres,” Opt. Express 8(3), 203–208 (2001).
[CrossRef] [PubMed]

W. Y. Zhang, X. Y. Lei, Z. L. Wang, D. G. Zheng, W. Y. Tam, C. T. Chan, and P. Sheng, “Robust photonic band gap from tunable scatterers,” Phys. Rev. Lett. 84(13), 2853–2856 (2000).
[CrossRef] [PubMed]

Stefanou, N.

N. Stefanou, V. Yannopapas, and A. Modinos, “Heterostructures of photonic crystals: frequency bands and transmission coefficients,” Comput. Phys. Commun. 113(1), 49–77 (1998).
[CrossRef]

Tam, W. Y.

W. Y. Zhang, X. Y. Lei, Z. L. Wang, D. G. Zheng, W. Y. Tam, C. T. Chan, and P. Sheng, “Robust photonic band gap from tunable scatterers,” Phys. Rev. Lett. 84(13), 2853–2856 (2000).
[CrossRef] [PubMed]

Tran, P.

J. Elson and P. Tran, “Coupled-mode calculation with the R-matrix propagator for the dispersion of surface waves on a truncated photonic crystal,” Phys. Rev. B 54(3), 1711–1715 (1996).
[CrossRef]

Wang, L.

T. Zhang, P. Cheng, X. Chen, J.-F. Jia, X. Ma, K. He, L. Wang, H. Zhang, X. Dai, Z. Fang, X. Xie, and Q.-K. Xue, “Experimental demonstration of topological surface states protected by time-reversal symmetry,” Phys. Rev. Lett. 103(26), 266803 (2009).
[CrossRef] [PubMed]

Wang, Z. L.

W. Y. Zhang, X. Y. Lei, Z. L. Wang, D. G. Zheng, W. Y. Tam, C. T. Chan, and P. Sheng, “Robust photonic band gap from tunable scatterers,” Phys. Rev. Lett. 84(13), 2853–2856 (2000).
[CrossRef] [PubMed]

Wray, L.

D. Hsieh, Y. Xia, D. Qian, L. Wray, J. H. Dil, F. Meier, J. Osterwalder, L. Patthey, J. G. Checkelsky, N. P. Ong, A. V. Fedorov, H. Lin, A. Bansil, D. Grauer, Y. S. Hor, R. J. Cava, and M. Z. Hasan, “A tunable topological insulator in the spin helical Dirac transport regime,” Nature 460(7259), 1101–1105 (2009).
[CrossRef] [PubMed]

D. Hsieh, Y. Xia, D. Qian, L. Wray, F. Meier, J. H. Dil, J. Osterwalder, L. Patthey, A. V. Fedorov, H. Lin, A. Bansil, D. Grauer, Y. S. Hor, R. J. Cava, and M. Z. Hasan, “Observation of time-reversal-protected single-dirac-cone topological-insulator states in Bi2Te3 and Sb2Te3,” Phys. Rev. Lett. 103(14), 146401 (2009).
[CrossRef] [PubMed]

D. Hsieh, Y. Xia, L. Wray, D. Qian, A. Pal, J. H. Dil, J. Osterwalder, F. Meier, G. Bihlmayer, C. L. Kane, Y. S. Hor, R. J. Cava, and M. Z. Hasan, “Observation of unconventional quantum spin textures in topological insulators,” Science 323(5916), 919–922 (2009).
[CrossRef] [PubMed]

Y. Xia, D. Qian, D. Hsieh, L. Wray, A. Pal, H. Lin, A. Bansil, D. Grauer, Y. S. Hor, R. J. Cava, and M. Z. Hasan, “Observation of a large-gap topological-insulator class with a single Dirac cone on the surface,” Nat. Phys. 5(6), 398–402 (2009).
[CrossRef]

D. Hsieh, D. Qian, L. Wray, Y. Xia, Y. S. Hor, R. J. Cava, and M. Z. Hasan, “A topological Dirac insulator in a quantum spin Hall phase,” Nature 452(7190), 970–974 (2008).
[CrossRef] [PubMed]

Wray, L. A.

H. Lin, R. S. Markiewicz, L. A. Wray, L. Fu, M. Z. Hasan, and A. Bansil, “Single-Dirac-cone topological surface states in the TlBiSe(2) class of topological semiconductors,” Phys. Rev. Lett. 105(3), 036404 (2010).
[CrossRef] [PubMed]

Xia, Y.

D. Hsieh, Y. Xia, L. Wray, D. Qian, A. Pal, J. H. Dil, J. Osterwalder, F. Meier, G. Bihlmayer, C. L. Kane, Y. S. Hor, R. J. Cava, and M. Z. Hasan, “Observation of unconventional quantum spin textures in topological insulators,” Science 323(5916), 919–922 (2009).
[CrossRef] [PubMed]

Y. Xia, D. Qian, D. Hsieh, L. Wray, A. Pal, H. Lin, A. Bansil, D. Grauer, Y. S. Hor, R. J. Cava, and M. Z. Hasan, “Observation of a large-gap topological-insulator class with a single Dirac cone on the surface,” Nat. Phys. 5(6), 398–402 (2009).
[CrossRef]

D. Hsieh, Y. Xia, D. Qian, L. Wray, F. Meier, J. H. Dil, J. Osterwalder, L. Patthey, A. V. Fedorov, H. Lin, A. Bansil, D. Grauer, Y. S. Hor, R. J. Cava, and M. Z. Hasan, “Observation of time-reversal-protected single-dirac-cone topological-insulator states in Bi2Te3 and Sb2Te3,” Phys. Rev. Lett. 103(14), 146401 (2009).
[CrossRef] [PubMed]

D. Hsieh, Y. Xia, D. Qian, L. Wray, J. H. Dil, F. Meier, J. Osterwalder, L. Patthey, J. G. Checkelsky, N. P. Ong, A. V. Fedorov, H. Lin, A. Bansil, D. Grauer, Y. S. Hor, R. J. Cava, and M. Z. Hasan, “A tunable topological insulator in the spin helical Dirac transport regime,” Nature 460(7259), 1101–1105 (2009).
[CrossRef] [PubMed]

D. Hsieh, D. Qian, L. Wray, Y. Xia, Y. S. Hor, R. J. Cava, and M. Z. Hasan, “A topological Dirac insulator in a quantum spin Hall phase,” Nature 452(7190), 970–974 (2008).
[CrossRef] [PubMed]

Xie, X.

T. Zhang, P. Cheng, X. Chen, J.-F. Jia, X. Ma, K. He, L. Wang, H. Zhang, X. Dai, Z. Fang, X. Xie, and Q.-K. Xue, “Experimental demonstration of topological surface states protected by time-reversal symmetry,” Phys. Rev. Lett. 103(26), 266803 (2009).
[CrossRef] [PubMed]

Xue, Q.-K.

T. Zhang, P. Cheng, X. Chen, J.-F. Jia, X. Ma, K. He, L. Wang, H. Zhang, X. Dai, Z. Fang, X. Xie, and Q.-K. Xue, “Experimental demonstration of topological surface states protected by time-reversal symmetry,” Phys. Rev. Lett. 103(26), 266803 (2009).
[CrossRef] [PubMed]

Yablonovitch, E.

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58(20), 2059–2062 (1987).
[CrossRef] [PubMed]

Yannopapas, V.

N. Stefanou, V. Yannopapas, and A. Modinos, “Heterostructures of photonic crystals: frequency bands and transmission coefficients,” Comput. Phys. Commun. 113(1), 49–77 (1998).
[CrossRef]

Zandbergen, S. R.

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]

Zhang, H.

T. Zhang, P. Cheng, X. Chen, J.-F. Jia, X. Ma, K. He, L. Wang, H. Zhang, X. Dai, Z. Fang, X. Xie, and Q.-K. Xue, “Experimental demonstration of topological surface states protected by time-reversal symmetry,” Phys. Rev. Lett. 103(26), 266803 (2009).
[CrossRef] [PubMed]

Zhang, H. J.

H. J. Zhang, C. X. Liu, X. L. Qi, X. Dai, Z. Fang, and S.-C. Zhang, “Topological insulators in Bi2Se3, Bi2Te3 and Sb2Te3 with a single Dirac cone on the surface,” Nat. Phys. 5(6), 438–442 (2009).
[CrossRef]

Y. L. Chen, J. G. Analytis, J. H. Chu, Z. K. Liu, S.-K. Mo, X. L. Qi, H. J. Zhang, D. H. Lu, X. Dai, Z. Fang, S. C. Zhang, I. R. Fisher, Z. Hussain, and Z. X. Shen, “Experimental realization of a three-dimensional topological insulator, Bi2Te3,” Science 325(5937), 178–181 (2009).
[CrossRef] [PubMed]

Zhang, S. C.

Y. L. Chen, J. G. Analytis, J. H. Chu, Z. K. Liu, S.-K. Mo, X. L. Qi, H. J. Zhang, D. H. Lu, X. Dai, Z. Fang, S. C. Zhang, I. R. Fisher, Z. Hussain, and Z. X. Shen, “Experimental realization of a three-dimensional topological insulator, Bi2Te3,” Science 325(5937), 178–181 (2009).
[CrossRef] [PubMed]

B. A. Bernevig and S. C. Zhang, “Quantum spin Hall effect,” Phys. Rev. Lett. 96(10), 106802 (2006).
[CrossRef] [PubMed]

Zhang, S.-C.

X.-L. Qi and S.-C. Zhang, “The quantum spin Hall effect and topological insulators,” Phys. Today 63(1), 33 (2010).
[CrossRef]

H. J. Zhang, C. X. Liu, X. L. Qi, X. Dai, Z. Fang, and S.-C. Zhang, “Topological insulators in Bi2Se3, Bi2Te3 and Sb2Te3 with a single Dirac cone on the surface,” Nat. Phys. 5(6), 438–442 (2009).
[CrossRef]

Zhang, T.

T. Zhang, P. Cheng, X. Chen, J.-F. Jia, X. Ma, K. He, L. Wang, H. Zhang, X. Dai, Z. Fang, X. Xie, and Q.-K. Xue, “Experimental demonstration of topological surface states protected by time-reversal symmetry,” Phys. Rev. Lett. 103(26), 266803 (2009).
[CrossRef] [PubMed]

Zhang, W. Y.

W. Y. Zhang, C. T. Chan, and P. Sheng, “Multiple scattering theory and its application to photonic band gap systems consisting of coated spheres,” Opt. Express 8(3), 203–208 (2001).
[CrossRef] [PubMed]

W. Y. Zhang, X. Y. Lei, Z. L. Wang, D. G. Zheng, W. Y. Tam, C. T. Chan, and P. Sheng, “Robust photonic band gap from tunable scatterers,” Phys. Rev. Lett. 84(13), 2853–2856 (2000).
[CrossRef] [PubMed]

Zhang, X.

X. Zhang, L.-M. Li, Z.-Q. Zhang, and C. Chan, “Surface states in two-dimensional metallodielectric photonic crystals studied by a multiple-scattering method,” Phys. Rev. B 63(12), 125114 (2001).
[CrossRef]

Zhang, X. D.

X. D. 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. D. Zhang and Z. Y. 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.

X. Zhang, L.-M. Li, Z.-Q. Zhang, and C. Chan, “Surface states in two-dimensional metallodielectric photonic crystals studied by a multiple-scattering method,” Phys. Rev. B 63(12), 125114 (2001).
[CrossRef]

Zheng, D. G.

W. Y. Zhang, X. Y. Lei, Z. L. Wang, D. G. Zheng, W. Y. Tam, C. T. Chan, and P. Sheng, “Robust photonic band gap from tunable scatterers,” Phys. Rev. Lett. 84(13), 2853–2856 (2000).
[CrossRef] [PubMed]

Appl. Phys. Lett. (1)

W. M. Robertson and M. S. May, “Surface electromagnetic wave excitation on one-dimensional photonic band-gap arrays,” Appl. Phys. Lett. 74(13), 1800–1803 (1999).
[CrossRef]

Comput. Phys. Commun. (1)

N. Stefanou, V. Yannopapas, and A. Modinos, “Heterostructures of photonic crystals: frequency bands and transmission coefficients,” Comput. Phys. Commun. 113(1), 49–77 (1998).
[CrossRef]

Nat. Phys. (2)

H. J. Zhang, C. X. Liu, X. L. Qi, X. Dai, Z. Fang, and S.-C. Zhang, “Topological insulators in Bi2Se3, Bi2Te3 and Sb2Te3 with a single Dirac cone on the surface,” Nat. Phys. 5(6), 438–442 (2009).
[CrossRef]

Y. Xia, D. Qian, D. Hsieh, L. Wray, A. Pal, H. Lin, A. Bansil, D. Grauer, Y. S. Hor, R. J. Cava, and M. Z. Hasan, “Observation of a large-gap topological-insulator class with a single Dirac cone on the surface,” Nat. Phys. 5(6), 398–402 (2009).
[CrossRef]

Nature (4)

D. Hsieh, Y. Xia, D. Qian, L. Wray, J. H. Dil, F. Meier, J. Osterwalder, L. Patthey, J. G. Checkelsky, N. P. Ong, A. V. Fedorov, H. Lin, A. Bansil, D. Grauer, Y. S. Hor, R. J. Cava, and M. Z. Hasan, “A tunable topological insulator in the spin helical Dirac transport regime,” Nature 460(7259), 1101–1105 (2009).
[CrossRef] [PubMed]

K. Ishizaki and S. Noda, “Manipulation of photons at the surface of three-dimensional photonic crystals,” Nature 460(7253), 367–370 (2009).
[CrossRef] [PubMed]

J. E. Moore, “The birth of topological insulators,” Nature 464(7286), 194–198 (2010).
[CrossRef] [PubMed]

D. Hsieh, D. Qian, L. Wray, Y. Xia, Y. S. Hor, R. J. Cava, and M. Z. Hasan, “A topological Dirac insulator in a quantum spin Hall phase,” Nature 452(7190), 970–974 (2008).
[CrossRef] [PubMed]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. (1)

F. S. Ham and B. Segall, “Energy bands in periodic lattices: Green’s function method,” Phys. Rev. 124(6), 1786–1796 (1961).
[CrossRef]

Phys. Rev. A (2)

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

R. A. Sepkhanov, Ya. 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 (5)

J. Elson and P. Tran, “Coupled-mode calculation with the R-matrix propagator for the dispersion of surface waves on a truncated photonic crystal,” Phys. Rev. B 54(3), 1711–1715 (1996).
[CrossRef]

F. Ramos-Mendieta and P. Halevi, “Surface electromagnetic waves in two-dimensional photonic crystals: Effect of the position of the surface plane,” Phys. Rev. B 59(23), 15112–15120 (1999).
[CrossRef]

X. Zhang, L.-M. Li, Z.-Q. Zhang, and C. Chan, “Surface states in two-dimensional metallodielectric photonic crystals studied by a multiple-scattering method,” Phys. Rev. B 63(12), 125114 (2001).
[CrossRef]

S. Enoch, E. Popov, and N. Bonod, “Analysis of the physical origin of surface modes on finite-size photonic crystals,” Phys. Rev. B 72(15), 155101 (2005).
[CrossRef]

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Electromagnetic Bloch waves at the surface of a photonic crystal,” Phys. Rev. B 44(19), R10961 (1991).
[CrossRef]

Phys. Rev. B Condens. Matter (1)

P. Etchegoin and R. T. Phillips, “Photon focusing, internal diffraction, and surface states in periodic dielectric structures,” Phys. Rev. B Condens. Matter 53(19), 12674–12683 (1996).
[CrossRef] [PubMed]

Phys. Rev. Lett. (13)

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58(20), 2059–2062 (1987).
[CrossRef] [PubMed]

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58(23), 2486–2489 (1987).
[CrossRef] [PubMed]

C. L. Kane and E. J. Mele, “Z2 topological order and the quantum spin Hall effect,” Phys. Rev. Lett. 95(14), 146802 (2005).
[CrossRef] [PubMed]

B. A. Bernevig and S. C. Zhang, “Quantum spin Hall effect,” Phys. Rev. Lett. 96(10), 106802 (2006).
[CrossRef] [PubMed]

X. D. 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. D. Zhang and Z. Y. Liu, “Extremal transmission and beating effect of acoustic waves in two-dimensional sonic crystals,” Phys. Rev. Lett. 101(26), 264303 (2008).
[CrossRef] [PubMed]

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]

A. Moroz, “Three-dimensional complete photonic-band-gap structures in the visible,” Phys. Rev. Lett. 83(25), 5274–5277 (1999).
[CrossRef]

W. Y. Zhang, X. Y. Lei, Z. L. Wang, D. G. Zheng, W. Y. Tam, C. T. Chan, and P. Sheng, “Robust photonic band gap from tunable scatterers,” Phys. Rev. Lett. 84(13), 2853–2856 (2000).
[CrossRef] [PubMed]

D. Hsieh, Y. Xia, D. Qian, L. Wray, F. Meier, J. H. Dil, J. Osterwalder, L. Patthey, A. V. Fedorov, H. Lin, A. Bansil, D. Grauer, Y. S. Hor, R. J. Cava, and M. Z. Hasan, “Observation of time-reversal-protected single-dirac-cone topological-insulator states in Bi2Te3 and Sb2Te3,” Phys. Rev. Lett. 103(14), 146401 (2009).
[CrossRef] [PubMed]

T. Zhang, P. Cheng, X. Chen, J.-F. Jia, X. Ma, K. He, L. Wang, H. Zhang, X. Dai, Z. Fang, X. Xie, and Q.-K. Xue, “Experimental demonstration of topological surface states protected by time-reversal symmetry,” Phys. Rev. Lett. 103(26), 266803 (2009).
[CrossRef] [PubMed]

H. Lin, R. S. Markiewicz, L. A. Wray, L. Fu, M. Z. Hasan, and A. Bansil, “Single-Dirac-cone topological surface states in the TlBiSe(2) class of topological semiconductors,” Phys. Rev. Lett. 105(3), 036404 (2010).
[CrossRef] [PubMed]

F. D. M. Haldane and 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]

Phys. Today (1)

X.-L. Qi and S.-C. Zhang, “The quantum spin Hall effect and topological insulators,” Phys. Today 63(1), 33 (2010).
[CrossRef]

Physica A (1)

A. Modinos, “Scattering of electromagnetic waves by a plane of spheres-formalism,” Physica A 141(2-3), 575–588 (1987).
[CrossRef]

Science (2)

Y. L. Chen, J. G. Analytis, J. H. Chu, Z. K. Liu, S.-K. Mo, X. L. Qi, H. J. Zhang, D. H. Lu, X. Dai, Z. Fang, S. C. Zhang, I. R. Fisher, Z. Hussain, and Z. X. Shen, “Experimental realization of a three-dimensional topological insulator, Bi2Te3,” Science 325(5937), 178–181 (2009).
[CrossRef] [PubMed]

D. Hsieh, Y. Xia, L. Wray, D. Qian, A. Pal, J. H. Dil, J. Osterwalder, F. Meier, G. Bihlmayer, C. L. Kane, Y. S. Hor, R. J. Cava, and M. Z. Hasan, “Observation of unconventional quantum spin textures in topological insulators,” Science 323(5916), 919–922 (2009).
[CrossRef] [PubMed]

Other (5)

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, “Photonic Crystal-Molding the Flow of Light,” Princeton University Press, Princeton, NJ, (1995).

C. M. Soukoulis, “Photonic Band Gap Materials,” Kluwer, Academic, Dordrecht (1996).

K. Sakoda, “Optical properties of photonic crystals,” Springer (2001).

J. B. Pendry, “Low energy electron diffraction” (Academic, London, 1974).

K. Kambe, “Theory of low-energy electron diffraction,” 23a, 1280–1294 (1968).

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Figures (5)

Fig. 1
Fig. 1

(a) Schematic structure of the 3D photonic crystal slab and supercell with 7 monolayers. Here A, B and C denote the different monolayers in the supercell. The pink lines mark the nearest neighbors of the spheres. (b) Schematic structure of the diamond, and the light gray plane is the (111) plane of the diamond. The Cartesian-coordinates inside the figure (red marks) correspond to those in (a). (c) The first Brillouin zone (BZ) of the diamond structure and the corresponding surface 2D BZ. W, K and L represent the symmetric points in the 3D BZ; Γ ¯ , K ¯ and M ¯ are the symmetric points in the surface 2D BZ.

Fig. 2
Fig. 2

(a) Calculated photonic band structures of a 3D diamond structure consisting of metallo-dielectric spheres in air, the filling ratio is f = 0.34 , the metal core is modeled by ε = 200 , and coating layer is 2% in radius with a dielectric constant ε = 12.96 . (b) The corresponding in-plane band structures for the 3D PC slab with type I surfaces and ten-monolayer thickness. Red dot lines represent the surface modes, green solid lines are the lightlines. (c) The intensity distribution of the magnetic-field for a surface state in the supercell of the yz plane for the Γ ¯ M ¯ direction and a frequency of f = 0.466 ω a / 2 π c (blue=low, red=high).

Fig. 3
Fig. 3

in-plane band structures for the 3D PC slab with type II surfaces and ten-layer thickness in diamond structure consisting of metallo-dielectric spheres in air. The coating layer is 10% in radius. The other parameters are identical with those in Fig. 2. Red dot lines represent the surface modes and green solid lines are the lightlines.

Fig. 4
Fig. 4

The in-plane band structures for the 3D PC slab with type II surfaces and ten-layer thickness in diamond structure consisting of metallo-dielectric spheres in air under the modification of the surface layers. (a) corresponds to the case without being modified; (b), (c), (d), (e), (f), (g) and (h) correspond to the cases with the thickness of the dielectric coating 10%, 20%, 30%, 40%, 50%, 60% and 65% of the sphere radius, respectively. The other parameters are identical with those in Fig. 2. Red dot lines represent the surface modes (all in the gap region) and green solid lines are the lightlines.

Fig. 5
Fig. 5

Distributions of the electric field intensity on the surface of 3D finite PC sample with 16 layers under the incidence of a fundamental Gaussian beam with numerical aperture NA=0.3. The length (y direction), width (x direction) of the sample are taken as 10a, 11a respectively. (a), (c) and (e) correspond to the field patterns in yz plane, and (b), (d) and (f) to those in xy plane. (a) and (b) at 0.43 ω a / 2 π c for the structure as shown in Fig. 2; (c) and (d) at 0.54 ω a / 2 π c for the structure as shown in Fig. 2; (e) and (f) at 0.481 ω a / 2 π c (Dirac point) for the structure as shown in Fig. 4 (d).

Equations (23)

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E s c ( r ) = j = 1 N l = 1 m = l l [ i q b j l m + E × R n exp ( i k R n ) h l + ( q r n j ) X l m ( r ^ n j ) + b j l m + H R n exp ( i k R n ) h l + ( q r n j ) X l m ( r ^ n j ) ]
E j s c ' ( r ) = l = 1 m = l l ( i q b j l m E × j l ( q r n j ) X l m ( r ^ n j ) + b j l m H j l ( q r n j ) X l m ( r ^ n j ) )
l ( l + 1 ) b j l m E = q j l ( q r ) r E j s c Y l m * ( Ω ^ ) d Ω ^ ,
l ( l + 1 ) b j l m H = q j l ( q r ) r H j s c Y l m * ( Ω ^ ) d Ω ^ ( ε ε 0 μ μ 0 ) 1 ,
h l ( 1 ) ( q | r R j | ) Y l m ( Ω ^ ( r R j ) ) = l = 0 m = l l g l m , l m ( R i R j ) j l ( q | r R i | ) Y l m ( Ω ^ ( r R i ) )
g l m , l m ( R i R j ) = l = 0 m = l l 4 π ( 1 ) ( l l l ) / 2 ( 1 ) m + m h l ( 1 ) ( q | R i R j | ) Y l m ( Ω ^ ( R i R j ) ) Y l m ( Ω ^ ) Y l m ( Ω ^ ) Y l m ( Ω ^ ) d Ω
b j l m P = j = 1 N P = E , H l m Ω j l m , j l m P P b j l m + P .
Ω j l m ; j l m E E = Ω j l m ; j l m H H = ( ψ l ψ l ) 1 [ 2 α l m α l m Z j l m 1 ; j l m 1 + m m Z j l m ; j l m + 2 α l m α l m Z j l m + 1 ; j l m + 1 ] ,
Ω j l m ; j l m H E = Ω j l m ; j l m E H = ( 2 l + 1 ) ( ψ l ψ l ) 1 [ 2 α l m γ l m Z j l m 1 ; j l 1 m 1 + m ζ l m Z j l m ; j l 1 m + 2 α l m γ l m Z j l m + 1 ; j l 1 m + 1 ] ,
ψ l = l ( l + 1 ) ,
α l m = 1 2 ( l m ) ( l + m + 1 ) ,
γ l m = 1 2 [ ( l + m ) ( l + m 1 ) ] 1 / 2 / [ ( 2 l 1 ) ( 2 l + 1 ) ] 1 / 2 ,
ζ l m = [ ( l + m ) ( l m ) ] 1 / 2 / [ ( 2 l 1 ) ( 2 l + 1 ) ] 1 / 2 ,
Z j l m , j l m = R n g l m , l m ( δ j δ j R n ) e i k R n ,
g l m , l m ( r ) = l = 0 m = l l 4 π ( 1 ) ( l l l ) / 2 ( 1 ) m + m h l ( 1 ) ( q r ) Y l m ( Ω ^ ( r ) ) Y l m ( Ω ^ ) Y l m ( Ω ^ ) Y l m ( Ω ^ ) d Ω .
b j l m + P = T j l m P ( j = 1 N P = E , H l m Ω j l m , j l m P P b j l m + P + a j l m 0 P ) ,
det | δ P P δ j j δ l l δ m m l m P Ω j l m , j l m P P T j l m , j l m P | = 0.
D l m = R n e i k R n h l ( q | δ R n | ) Y l m ( Ω ^ ( δ R n ) ) .
D l m = D l m ( 1 ) + D l m ( 2 ) + D l m ( 3 ) .
D l m ( 1 ) = 1 i q ( 1 ) l m 1 A q l i m + 1 2 l [ ( 2 l + 1 ) ( l | m | ) ! ( l + | m | ) ! ] 1 2 K n e i ( k + K n ) δ x y i m φ k + K n n = 0 l | m | 1 n ! ( Γ K ) 2 n 1 Δ K n s = n min ( 2 n , l | m | ) ( n 2 n s ) δ z 2 n s ( | k + K n | ) l s ( l | m | s 2 ) ! ( l + | m | s 2 ) ! ,
D l m ( 2 ) = i q 1 2 π R n e i k R j Y l m ( Ω ^ ( δ R n ) ) | δ R n | l q l 1 η ξ l 1 2 e 1 2 ( | δ R n | 2 ξ q 2 ξ ) d ξ ,
D l m ( 3 ) = [ 1 4 π i q π j = 0 ( 1 2 η ) 1 2 ( q 2 η 2 ) j j ! ( 2 j 1 ) ] δ l , 0 δ m , 0 δ δ , 0 ,
Δ K n = e π i Γ K 2 2 η ξ 1 2 n e ξ + Γ K 2 δ z 2 4 ξ d ξ .

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