Abstract

Two-dimensional (2D) coupled resonant optical waveguide (CROW), exhibiting topological edge states, provides an efficient platform for designing integrated topological photonic devices. In this paper, we propose an experimentally feasible design of 2D honeycomb CROW photonic structure. The characteristic optical system possesses two-fold and three-fold Dirac points at different positions in the Brillouin zone. The effective gauge fields implemented by the intrinsic pseudo-spin-orbit interaction open up topologically nontrivial bandgaps through the Dirac points. Spatial lattice geometries allow destructive wave interference, leading to a dispersionless, near-flat energy band in the vicinity of the three-fold Dirac point in the telecommunication frequency regime. This nontrivial structure with a near-flat band yields topologically protected edge states. These characteristics underpin the fundamental importance as well as the potential applications in various optical devices. Based on the honeycomb CROW lattice, we design the shape-independent topological cavity and the beam splitter, which demonstrate the relevance for a wide range of photonic applications.

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

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2018 (4)

C. E. Whittaker, E. Cancellieri, P. M. Walker, D. R. Gulevich, H. Schomerus, D. Vaitiekus, B. Royall, D. M. Whittaker, E. Clarke, I. V. Iorsh, I. A. Shelykh, M. S. Skolnick, and D. N. Krizhanovskii, “Exciton Polaritons in a Two-Dimensional Lieb Lattice with Spin-Orbit Coupling,” Phys. Rev. Lett. 120(9), 097401 (2018).
[Crossref] [PubMed]

G. Harari, M. A. Bandres, Y. Lumer, M. C. Rechtsman, Y. D. Chong, M. Khajavikhan, D. N. Christodoulides, and M. Segev, “Topological insulator laser: Theory,” Science 359(6381), eaar4003 (2018).
[Crossref] [PubMed]

M. A. Bandres, S. Wittek, G. Harari, M. Parto, J. Ren, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Topological insulator laser: Experiments,” Science 359(6381), eaar4005 (2018).
[Crossref] [PubMed]

D. Leykam, S. Mittal, M. Hafezi, and Y. D. Chong, “Reconfigurable topological phases in next-nearest-neighbor coupled resonator lattices,” Phys. Rev. Lett. 121(2), 023901 (2018).
[Crossref] [PubMed]

2017 (2)

B. Bahari, A. Ndao, F. Vallini, A. El Amili, Y. Fainman, and B. Kanté, “Nonreciprocal lasing in topological cavities of arbitrary geometries,” Science 358(6363), 636–640 (2017).
[Crossref] [PubMed]

D. Leykam, K. Y. Bliokh, C. Huang, Y. D. Chong, and F. Nori, “Edge modes, degeneracies, and topological numbers in non-hermitian systems,” Phys. Rev. Lett. 118(4), 040401 (2017).
[Crossref] [PubMed]

2016 (6)

C. He, X.-C. Sun, X.-P. Liu, M.-H. Lu, Y. Chen, L. Feng, and Y.-F. Chen, “Photonic topological insulator with broken time-reversal symmetry,” Proc. Natl. Acad. Sci. U.S.A. 113(18), 4924–4928 (2016).
[Crossref] [PubMed]

Y. G. Peng, C. Z. Qin, D. G. Zhao, Y. X. Shen, X. Y. Xu, M. Bao, H. Jia, and X. F. Zhu, “Experimental demonstration of anomalous Floquet topological insulator for sound,” Nat. Commun. 7, 13368 (2016).
[Crossref] [PubMed]

C. He, X. Ni, H. Ge, X.-C. Sun, Y.-B. Chen, M.-H. Lu, X.-P. Liu, and Y.-F. Chen, “Acoustic topological insulator and robust one-way sound transport,” Nat. Phys. 12(12), 1124–1129 (2016).
[Crossref]

C. He, Z. Li, X. Ni, X.-C. Sun, S.-Y. Yu, M.-H. Lu, X.-P. Liu, and Y.-F. Chen, “Topological phononic states of underwater sound based on coupled ring resonators,” Appl. Phys. Lett. 108(3), 031904 (2016).
[Crossref]

F. Gao, Z. Gao, X. Shi, Z. Yang, X. Lin, H. Xu, J. D. Joannopoulos, M. Soljačić, H. Chen, L. Lu, Y. Chong, and B. Zhang, “Probing topological protection using a designer surface plasmon structure,” Nat. Commun. 7, 11619 (2016).
[Crossref] [PubMed]

F. Diebel, D. Leykam, S. Kroesen, C. Denz, and A. S. Desyatnikov, “Conical diffraction and composite Lieb bosons in photonic lattices,” Phys. Rev. Lett. 116(18), 183902 (2016).
[Crossref] [PubMed]

2015 (4)

S. Mukherjee, A. Spracklen, D. Choudhury, N. Goldman, P. Öhberg, E. Andersson, and R. R. Thomson, “Observation of a localized flat-band state in a photonic Lieb lattice,” Phys. Rev. Lett. 114(24), 245504 (2015).
[Crossref] [PubMed]

R. A. Vicencio, C. Cantillano, L. Morales-Inostroza, B. Real, C. Mejía-Cortés, S. Weimann, A. Szameit, and M. I. Molina, “Observation of localized States in Lieb photonic lattices,” Phys. Rev. Lett. 114(24), 245503 (2015).
[Crossref] [PubMed]

Q. Lin and S. Fan, “Resonator-free realization of effective magnetic field for photons,” New J. Phys. 17(7), 075008 (2015).
[Crossref]

X. Ni, C. He, X.-C. Sun, X.-P. Liu, M.-H. Lu, L. Feng, and Y.-F. Chen, “Topologically protected one-way edge mode in networks of acoustic resonators with circulating air flow,” New J. Phys. 17(5), 053016 (2015).
[Crossref]

2014 (1)

M. Pasek and Y. D. Chong, “Network models of photonic Floquet topological insulators,” Phys. Rev. B 89(7), 075113 (2014).
[Crossref]

2013 (5)

P. Moitra, Y. Yang, Z. Anderson, I. I. Kravchenko, D. P. Briggs, and J. Valentine, “Realization of an all-dielectric zero-index optical metamaterial,” Nat. Photonics 7(10), 791–795 (2013).
[Crossref]

A. B. Khanikaev, S. H. Mousavi, W. K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12(3), 233–239 (2013).
[Crossref] [PubMed]

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

G.-Q. Liang and Y. D. Chong, “Optical resonator analog of a two-dimensional topological insulator,” Phys. Rev. Lett. 110(20), 203904 (2013).
[Crossref] [PubMed]

M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. M. Taylor, “Imaging topological edge states in silicon photonics,” Nat. Photonics 7(12), 1001–1005 (2013).
[Crossref]

2012 (3)

K. Fang, Z. Yu, and S. Fan, “Realizing effective magnetic field for photons by controlling the phase of dynamic modulation,” Nat. Photonics 6(11), 782–787 (2012).
[Crossref]

Z. Lan, N. Goldman, and P. Öhberg, “Coexistence of spin-1/2 and spin-1 Dirac-Weyl fermions in the edge-centered honeycomb lattice,” Phys. Rev. B 85(15), 155451 (2012).
[Crossref]

L. Deák and T. Fülöp, “Reciprocity in quantum, electromagnetic and other wave scattering,” Ann. Phys. 327(4), 1050–1077 (2012).
[Crossref]

2011 (6)

T. L. Hughes, E. Prodan, and B. A. Bernevig, “Inversion-symmetric topological insulators,” Phys. Rev. B 83(24), 245132 (2011).
[Crossref]

K. Sun, Z. Gu, H. Katsura, and S. Das Sarma, “Nearly flatbands with nontrivial topology,” Phys. Rev. Lett. 106(23), 236803 (2011).
[Crossref] [PubMed]

M. Hafezi, E. A. Demler, M. D. Lukin, and J. M. Taylor, “Robust optical delay lines with topological protection,” Nat. Phys. 7(11), 907–912 (2011).
[Crossref]

J. Li, L. O’Faolain, I. H. Rey, and T. F. Krauss, “Four-wave mixing in photonic crystal waveguides: slow light enhancement and limitations,” Opt. Express 19(5), 4458–4463 (2011).
[Crossref] [PubMed]

M. Shinkawa, N. Ishikura, Y. Hama, K. Suzuki, and T. Baba, “Nonlinear enhancement in photonic crystal slow light waveguides fabricated using CMOS-compatible process,” Opt. Express 19(22), 22208–22218 (2011).
[Crossref] [PubMed]

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

2009 (4)

C. Monat, B. Corcoran, M. Ebnali-Heidari, C. Grillet, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, “Slow light enhancement of nonlinear effects in silicon engineered photonic crystal waveguides,” Opt. Express 17(4), 2944–2953 (2009).
[Crossref] [PubMed]

J. Hou, D. Gao, H. Wu, R. Hao, and Z. Zhou, “Flat band slow light in symmetric line defect photonic crystal waveguides,” IEEE Photonics Technol. Lett. 21(20), 1571–1573 (2009).
[Crossref]

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljacić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature 461(7265), 772–775 (2009).
[Crossref] [PubMed]

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

2008 (4)

Z. Wang, Y. D. Chong, J. D. Joannopoulos, and M. Soljacić, “Reflection-free one-way edge modes in a gyromagnetic photonic crystal,” Phys. Rev. Lett. 100(1), 013905 (2008).
[Crossref] [PubMed]

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

F. D. 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]

J. Li, T. P. White, L. O’Faolain, A. Gomez-Iglesias, and T. F. Krauss, “Systematic design of flat band slow light in photonic crystal waveguides,” Opt. Express 16(9), 6227–6232 (2008).
[Crossref] [PubMed]

2007 (2)

2005 (2)

C. L. Kane and E. J. Mele, “Quantum spin Hall effect in graphene,” Phys. Rev. Lett. 95(22), 226801 (2005).
[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]

1986 (1)

B. Sutherland, “Localization of electronic wave functions due to local topology,” Phys. Rev. B Condens. Matter 34(8), 5208–5211 (1986).
[Crossref] [PubMed]

Anderson, Z.

P. Moitra, Y. Yang, Z. Anderson, I. I. Kravchenko, D. P. Briggs, and J. Valentine, “Realization of an all-dielectric zero-index optical metamaterial,” Nat. Photonics 7(10), 791–795 (2013).
[Crossref]

Andersson, E.

S. Mukherjee, A. Spracklen, D. Choudhury, N. Goldman, P. Öhberg, E. Andersson, and R. R. Thomson, “Observation of a localized flat-band state in a photonic Lieb lattice,” Phys. Rev. Lett. 114(24), 245504 (2015).
[Crossref] [PubMed]

Baba, T.

Bahari, B.

B. Bahari, A. Ndao, F. Vallini, A. El Amili, Y. Fainman, and B. Kanté, “Nonreciprocal lasing in topological cavities of arbitrary geometries,” Science 358(6363), 636–640 (2017).
[Crossref] [PubMed]

Bandres, M. A.

G. Harari, M. A. Bandres, Y. Lumer, M. C. Rechtsman, Y. D. Chong, M. Khajavikhan, D. N. Christodoulides, and M. Segev, “Topological insulator laser: Theory,” Science 359(6381), eaar4003 (2018).
[Crossref] [PubMed]

M. A. Bandres, S. Wittek, G. Harari, M. Parto, J. Ren, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Topological insulator laser: Experiments,” Science 359(6381), eaar4005 (2018).
[Crossref] [PubMed]

Bao, M.

Y. G. Peng, C. Z. Qin, D. G. Zhao, Y. X. Shen, X. Y. Xu, M. Bao, H. Jia, and X. F. Zhu, “Experimental demonstration of anomalous Floquet topological insulator for sound,” Nat. Commun. 7, 13368 (2016).
[Crossref] [PubMed]

Bernevig, B. A.

T. L. Hughes, E. Prodan, and B. A. Bernevig, “Inversion-symmetric topological insulators,” Phys. Rev. B 83(24), 245132 (2011).
[Crossref]

Bliokh, K. Y.

D. Leykam, K. Y. Bliokh, C. Huang, Y. D. Chong, and F. Nori, “Edge modes, degeneracies, and topological numbers in non-hermitian systems,” Phys. Rev. Lett. 118(4), 040401 (2017).
[Crossref] [PubMed]

Briggs, D. P.

P. Moitra, Y. Yang, Z. Anderson, I. I. Kravchenko, D. P. Briggs, and J. Valentine, “Realization of an all-dielectric zero-index optical metamaterial,” Nat. Photonics 7(10), 791–795 (2013).
[Crossref]

Cancellieri, E.

C. E. Whittaker, E. Cancellieri, P. M. Walker, D. R. Gulevich, H. Schomerus, D. Vaitiekus, B. Royall, D. M. Whittaker, E. Clarke, I. V. Iorsh, I. A. Shelykh, M. S. Skolnick, and D. N. Krizhanovskii, “Exciton Polaritons in a Two-Dimensional Lieb Lattice with Spin-Orbit Coupling,” Phys. Rev. Lett. 120(9), 097401 (2018).
[Crossref] [PubMed]

Cantillano, C.

R. A. Vicencio, C. Cantillano, L. Morales-Inostroza, B. Real, C. Mejía-Cortés, S. Weimann, A. Szameit, and M. I. Molina, “Observation of localized States in Lieb photonic lattices,” Phys. Rev. Lett. 114(24), 245503 (2015).
[Crossref] [PubMed]

Chan, C. T.

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

Chen, H.

F. Gao, Z. Gao, X. Shi, Z. Yang, X. Lin, H. Xu, J. D. Joannopoulos, M. Soljačić, H. Chen, L. Lu, Y. Chong, and B. Zhang, “Probing topological protection using a designer surface plasmon structure,” Nat. Commun. 7, 11619 (2016).
[Crossref] [PubMed]

Chen, Y.

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Figures (7)

Fig. 1
Fig. 1 (a) a schematic diagram of honeycomb CROW structures. The topological edge state are shown as the red line. The insets represent the size parameters and the transmission/reflection spectra of ring resonators, respectively. (b) The detailed unit cell diagram of honeycomb CROW structures. The black dashed line represents a single repeating unit, and the arrows denote three nearest neighbor vectors a 1 = ( 0 , 1 ) , a 2 = ( 3 2 , 1 2 ) , a 3 = ( 3 2 , 1 2 ) .
Fig. 2
Fig. 2 The energy bands E ( k ) corresponding to the Hamiltonian (a) without the Spin-Orbit term, and (b) with the Spin-Orbit coupling term.
Fig. 3
Fig. 3 (a) Schematic diagram of the energy band. (b) Schematic diagram of the projected band. The gapless edge states are shown in the red and blue dots, which corresponds to the upper and lower borders, respectively. The calculation interval of k x is Γ K Γ ( Γ and K are high symmetric points in the first Brillouin zone), and k y is always zero in the calculation.
Fig. 4
Fig. 4 The field distributions at the four high symmetric points for the energy bands.
Fig. 5
Fig. 5 The field distributions of topologically protected edge state and the bulk localized mode. (a) The field distribution of topologically protected edge state, white dashed box represents the input port of CROW structure. (b)Robust edge state propagation with the site ring defect.
Fig. 6
Fig. 6 The field distributions of the collective localization modes. (a) The flat band is activated by inputting the mode light into the lower right port of the structure. (b) The flat band is activated by using the dipole light source (at the end of the white arrows).
Fig. 7
Fig. 7 (a) The field distribution of edge state in an arbitrarily shaped topological cavity. The cavity mode is activated by using the dipole light source (at the end of the white arrows). (b) The field distribution of edge state in a topological beam splitter.

Tables (1)

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Table 1 Parity-eigenvalue patterns at the four high symmetric points for the four different energy bands. All the energy gaps are associated with a nontrivial Z2 index .

Equations (7)

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H 0 = < i j > t i j c i c j
H k = ( 0 e i k · a 2 / 2 0 e i k · a 2 / 2 0 e i k · a 2 / 2 0 e i k · a 1 / 2 0 e i k · a 3 / 2 0 e i k · a 1 / 2 0 e i k · a 1 / 2 0 e i k · a 2 / 2 0 e i k · a 1 / 2 0 e i k · a 3 / 2 0 e i k · a 3 / 2 0 e i k · a 3 / 2 0 )
ε Γ ( k ) ( ϕ 1 ϕ 3 ϕ 5 ) = 2 ( 0 cos k ( a 2 a 1 ) / 2 cos k ( a 2 a 3 ) / 2 cos k ( a 2 a 1 ) / 2 0 cos k ( a 3 a 1 ) / 2 cos k ( a 2 a 3 ) / 2 cos k ( a 3 a 1 ) / 2 0 ) ( ϕ 1 ϕ 3 ϕ 5 ) = H Γ ( k ) ( ϕ 1 ϕ 3 ϕ 5 )
ε K ( k ) ( ϕ 2 ϕ 4 ) = 2 ( 0 v = 1 3 exp ( i k a v ) v = 1 3 exp ( i k a v ) 0 ) ( ϕ 2 ϕ 4 ) = H K ( k ) ( ϕ 2 ϕ 4 )
ε K ( k ) = ( 1 ± 4 A k 3 ) , ε K ( k ) = 2 , ε Γ ( k ) = ± | e x p ( i k a 1 ) + e x p ( i k a 2 ) + e x p ( i k a 3 ) |
h p K = v K ( p x σ 1 + p y σ 2 ) 3 2 α λ s o σ 3 h p Γ = v Γ ( p x S 1 + p y S 2 ) 2 3 λ s o S 3
i = 0 3 m = 1 N ξ 2 m ( Γ i ) = ( 1 ) v N

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