Abstract

We investigate the topological bound modes in a binary optical waveguide array with anti-parity-time (PT) symmetry. The anti-PT-symmetric arrays are realized by incorporating additional waveguides to the bare arrays, such that the effective coupling coefficients are imaginary. The systems experience two kinds of phase transition, including global topological order transition and quantum phase transition. As a result, the system supports two kinds of robust bound modes, which are protected by the global topological order and the quantum phase, respectively. The study provides a promising approach to realizing robust light transport by utilizing mediating components.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2019 (2)

X. T. He, E. T. Liang, J. J. Yuan, H. Y. Qiu, X. D. Chen, F. L. Zhao, and J. W. Dong, “A silicon-on-insulator slab for topological valley transport,” Nat. Commun. 10(1), 872 (2019).
[Crossref] [PubMed]

D. Zhao, D. Zhong, Y. Hu, S. Ken, and W. Liu, “Imaginary modulation inducing giant spatial Goos-Hänchen shifts in one-dimensional defective photonic lattices,” Opt. Quantum Electron. 51(113), 1–11 (2019).

2018 (24)

S Ke, Q Liu, D Zhao, and W Liu, “Spectral discrete diffraction with non-Hermitian coupling,” J. Opt. Soc. Am. B 35(10), 2387–2393 (2018).

F. K. Kunst, E. Edvardsson, J. C. Budich, and E. J. Bergholtz, “Biorthogonal bulk-boundary correspondence in non-Hermitian systems,” Phys. Rev. Lett. 121(2), 026808 (2018).
[Crossref] [PubMed]

H. Zhao, P. Miao, M. H. Teimourpour, S. Malzard, R. El-Ganainy, H. Schomerus, and L. Feng, “Topological hybrid silicon microlasers,” Nat. Commun. 9(1), 981 (2018).
[Crossref] [PubMed]

Q. Liu, S. Ke, and W. Liu, “Mode conversion and absorption in an optical waveguide under cascaded complex modulations,” Opt. Quantum Electron. 50(9), 356 (2018).
[Crossref]

B. Xu, T. Li, and S. Zhu, “Simulation of massless Dirac dynamics in plasmonic waveguide arrays,” Opt. Express 26(10), 13416–13424 (2018).
[Crossref] [PubMed]

J. Tan, Y. Zhou, M. He, Y. Chen, Q. Ke, J. Liang, X. Zhu, M. Li, and P. Lu, “Determination of the ionization time using attosecond photoelectron Interferometry,” Phys. Rev. Lett. 121(25), 253203 (2018).
[Crossref] [PubMed]

M. Pan, H. Zhao, P. Miao, S. Longhi, and L. Feng, “Photonic zero mode in a non-Hermitian photonic lattice,” Nat. Commun. 9(1), 1308 (2018).
[Crossref] [PubMed]

S. Ke, D. Zhao, Q. Liu, S. Wu, B. Wang, and P. Lu, “Optical imaginary directional couplers,” J. Lightwave Technol. 36(12), 2510–2516 (2018).
[Crossref]

Y. Choi, C. Hahn, J. W. Yoon, and S. H. Song, “Observation of an anti-PT-symmetric exceptional point and energy-difference conserving dynamics in electrical circuit resonators,” Nat. Commun. 9(1), 2182 (2018).
[Crossref] [PubMed]

M. Parto, S. Wittek, H. Hodaei, G. Harari, M. A. Bandres, J. Ren, M. C. Rechtsman, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Edge-mode lasing in 1D topological active arrays,” Phys. Rev. Lett. 120(11), 113901 (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).

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).

S. R. Pocock, X. Xiao, P. A. Huidobro, and V. Giannini, “Topological plasmonic chain with retardation and radiative effects,” ACS Photonics 5(6), 2271–2279 (2018).
[Crossref]

H. Deng, Y. Chen, N. C. Panoiu, B. A. Malomed, and F. Ye, “Surface modes in plasmonic Bragg fibers with negative average permittivity,” Opt. Express 26(3), 2559–2568 (2018).
[Crossref] [PubMed]

C. Shang, X. Chen, W. Luo, and F. Ye, “Quantum anomalous Hall-quantum spin Hall effect in optical superlattices,” Opt. Lett. 43(2), 275–278 (2018).
[Crossref] [PubMed]

C. Liu, M. V. Gurudev Dutt, and D. Pekker, “Robust manipulation of light using topologically protected plasmonic modes,” Opt. Express 26(3), 2857–2872 (2018).
[Crossref] [PubMed]

F. Wang, S. Ke, C. Qin, B. Wang, H. Long, K. Wang, and P. Lu, “Topological interface modes in graphene multilayer arrays,” Opt. Laser Technol. 103, 272–278 (2018).
[Crossref]

R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, and D. N. Christodoulides, “Non-Hermitian physics and PT symmetry,” Nat. Phys. 14(1), 11–19 (2018).
[Crossref]

S. Yao and Z. Wang, “Edge states and topological invariants of non-Hermitian systems,” Phys. Rev. Lett. 121(8), 086803 (2018).
[Crossref] [PubMed]

S. Lieu, “Topological phases in the non-Hermitian Su-Schrieffer-Heeger model,” Phys. Rev. B 97(4), 045106 (2018).
[Crossref]

C. Yuce, “Edge states at the interface of non-Hermitian systems,” Phys. Rev. A (Coll. Park) 97(4), 042118 (2018).
[Crossref]

K. Takata and M. Notomi, “Photonic topological insulating phase induced solely by gain and loss,” Phys. Rev. Lett. 121(21), 213902 (2018).
[Crossref] [PubMed]

S. Ke, J. Liu, Q. Liu, D. Zhao, and W. Liu, “Strong absorption near exceptional points in plasmonic waveguide arrays,” Opt. Quantum Electron. 50(8), 318 (2018).
[Crossref]

S. Ke, J. Liu, Q. Liu, D. Zhao, and W. Liu, “Strong absorption near exceptional points in plasmonic waveguide arrays,” Opt. Quantum Electron. 50(8), 318 (2018).
[Crossref]

2017 (13)

L. Jin, “Topological phases and edge states in a non-Hermitian trimerized optical lattice,” Phys. Rev. A (Coll. Park) 96(3), 032103 (2017).
[Crossref]

B. Wu, J. Wang, M. Xiao, J. Xu, and Y. Chen, “Strong hybridization of edge and bulk states in dimerized PT-symmetric coupled waveguide chain,” Opt. Express 25(2), 1040–1049 (2017).
[Crossref] [PubMed]

S. Ke, B. Wang, H. Long, K. Wang, and P. Lu, “Topological edge modes in non-Hermitian plasmonic waveguide arrays,” Opt. Express 25(10), 11132–11143 (2017).
[Crossref] [PubMed]

S. Weimann, M. Kremer, Y. Plotnik, Y. Lumer, S. Nolte, K. G. Makris, M. Segev, M. C. Rechtsman, and A. Szameit, “Topologically protected bound states in photonic parity-time-symmetric crystals,” Nat. Mater. 16(4), 433–438 (2017).
[Crossref] [PubMed]

P. Qiu, R. Liang, W. Qiu, H. Chen, J. Ren, Z. Lin, J. X. Wang, Q. Kan, and J. Q. Pan, “Topologically protected edge states in graphene plasmonic crystals,” Opt. Express 25(19), 22587–22594 (2017).
[Crossref] [PubMed]

P. St-Jean, V. Goblot, E. Galopin, A. Lemaître, T. Ozawa, L. Le Gratiet, I. Sagnes, J. Bloch, and A. Amo, “Lasing in topological edge states of a one-dimensional lattice,” Nat. Photonics 11(10), 651–656 (2017).
[Crossref]

F. Bleckmann, Z. Cherpakova, S. Linden, and A. Alberti, “Spectral imaging of topological edge states in plasmonic waveguide arrays,” Phys. Rev. B 96(4), 045417 (2017).
[Crossref]

Z. Wang, B. Wang, H. Long, K. Wang, and P. Lu, “Surface plasmonic lattice solitons in semi-infinite graphene sheet arrays,” J. Lightwave Technol. 35(14), 2960–2965 (2017).
[Crossref]

D. Zhao, Z. Wang, H. Long, K. Wang, B. Wang, and P. Lu, “Optical bistability in defective photonic multilayers doped by graphene,” Opt. Quantum Electron. 49(4), 163 (2017).
[Crossref]

L. Ge, “Symmetry-protected zero-mode laser with a tunable spatial profile,” Phys. Rev. A (Coll. Park) 95(2), 023812 (2017).
[Crossref]

S. Ke, B. Wang, H. Long, K. Wang, and P. Lu, “Topological mode switching in a graphene doublet with exceptional points,” Opt. Quantum Electron. 49(6), 224 (2017).
[Crossref]

F. Yang, Y. Liu, and L. You, “Anti-PT symmetry in dissipatively coupled optical systems,” Phys. Rev. A (Coll. Park) 96(5), 053845 (2017).
[Crossref]

F. Wang, C. Qin, B. Wang, H. Long, K. Wang, and P. Lu, “Rabi oscillations of plasmonic supermodes in graphene multilayer arrays,” IEEE J. Sel. Top. Quantum Electron. 23(1), 125 (2017).
[Crossref]

2016 (9)

C. Qin, B. Wang, H. Long, K. Wang, and P. Lu, “Nonreciprocal phase shift and mode modulation in dynamic graphene waveguides,” J. Lightwave Technol. 34(16), 3877–3883 (2016).

S. Longhi, “Non-Hermitian tight-binding network engineering,” Phys. Rev. A (Coll. Park) 93(2), 022102 (2016).
[Crossref]

R. Keil, C. Poli, M. Heinrich, J. Arkinstall, G. Weihs, H. Schomerus, and A. Szameit, “Universal sign control of coupling in tight-binding lattices,” Phys. Rev. Lett. 116(21), 213901 (2016).
[Crossref] [PubMed]

S. Ke, B. Wang, C. Qin, H. Long, K. Wang, and P. Lu, “Exceptional points and asymmetric mode switching in plasmonic waveguides,” J. Lightwave Technol. 34(22), 5258–5262 (2016).
[Crossref]

J. Doppler, A. A. Mailybaev, J. Böhm, U. Kuhl, A. Girschik, F. Libisch, T. J. Milburn, P. Rabl, N. Moiseyev, and S. Rotter, “Dynamically encircling an exceptional point for asymmetric mode switching,” Nature 537(7618), 76–79 (2016).
[Crossref] [PubMed]

P. Peng, W. Cao, C. Shen, W. Qu, J. Wen, L. Jiang, and Y. Xiao, “Anti-parity–time symmetry with flying atoms,” Nat. Phys. 12(12), 1139–1145 (2016).
[Crossref]

A. Blanco-Redondo, I. Andonegui, M. J. Collins, G. Harari, Y. Lumer, M. C. Rechtsman, B. J. Eggleton, and M. Segev, “Topological optical waveguiding in silicon and the transition between topological and trivial defect states,” Phys. Rev. Lett. 116(16), 163901 (2016).
[Crossref] [PubMed]

H. Deng, X. Chen, N. C. Panoiu, and F. Ye, “Topological surface plasmons in superlattices with changing sign of the average permittivity,” Opt. Lett. 41(18), 4281–4284 (2016).
[Crossref] [PubMed]

X. Lin, R. Li, F. Gao, E. Li, X. Zhang, B. Zhang, and H. Chen, “Loss induced amplification of graphene plasmons,” Opt. Lett. 41(4), 681–684 (2016).
[Crossref] [PubMed]

2015 (6)

J. Xu and Y. Chen, “General coupled mode theory in non-Hermitian waveguides,” Opt. Express 23(17), 22619–22627 (2015).
[Crossref] [PubMed]

Q. Cheng, Y. Pan, Q. Wang, T. Li, and S. Zhu, “Topologically protected interface mode in plasmonic waveguide arrays,” Laser Photonics Rev. 9(4), 392–398 (2015).
[Crossref]

L. Ge, L. Wang, M. Xiao, W. Wen, C. T. Chan, and D. Han, “Topological edge modes in multilayer graphene systems,” Opt. Express 23(17), 21585–21595 (2015).
[Crossref] [PubMed]

H. Zhao, S. Longhi, and L. Feng, “Robust light state by quantum phase transition in Non-Hermitian optical materials,” Sci. Rep. 5(1), 17022 (2015).
[Crossref] [PubMed]

S. Malzard, C. Poli, and H. Schomerus, “Topologically protected defect states in open photonic systems with non-Hermitian charge-conjugation and Parity-Time symmetry,” Phys. Rev. Lett. 115(20), 200402 (2015).
[Crossref] [PubMed]

C. Poli, M. Bellec, U. Kuhl, F. Mortessagne, and H. Schomerus, “Selective enhancement of topologically induced interface states in a dielectric resonator chain,” Nat. Commun. 6(1), 6710 (2015).
[Crossref] [PubMed]

2014 (2)

M. Golshani, S. Weimann, Kh. Jafari, M. K. Nezhad, A. Langari, A. R. Bahrampour, T. Eichelkraut, S. M. Mahdavi, and A. Szameit, “Impact of loss on the wave dynamics in photonic waveguide lattices,” Phys. Rev. Lett. 113(12), 123903 (2014).
[Crossref] [PubMed]

L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological photonics,” Nat. Photonics 8(11), 821–829 (2014).
[Crossref]

2013 (5)

Y. Li, X. Guo, C. Xu, J. Yang, X. Jiang, and M. Wang, “Coupled mode theory under the parity-time symmetry frame,” J. Lightwave Technol. 31(15), 2477–2481 (2013).
[Crossref]

H. Schomerus, “Topologically protected midgap states in complex photonic lattices,” Opt. Lett. 38(11), 1912–1914 (2013).
[Crossref] [PubMed]

D. I. Pikulin and Y. V. Nazarov, “Two types of topological transitions in finite Majorana wires,” Phys. Rev. B Condens. Matter Mater. Phys. 87(23), 235421 (2013).
[Crossref]

S. Liang and G. Huang, “Topological invariance and global Berry phase in non-Hermitian systems,” Phys. Rev. A 87(1), 012118 (2013).
[Crossref]

L. Ge and H. E. Türeci, “Antisymmetric PT-photonic structures with balanced positive- and negative-index materials,” Phys. Rev. A 88(5), 053810 (2013).
[Crossref]

2012 (1)

2011 (1)

J. Wiersig, “Structure of whispering-gallery modes in optical microdisks perturbed by nanoparticles,” Phys. Rev. A 84(6), 063828 (2011).
[Crossref]

2010 (1)

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity–time symmetry in optics,” Nat. Phys. 6(3), 192–195 (2010).
[Crossref]

2009 (2)

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103(9), 093902 (2009).
[Crossref] [PubMed]

N. Malkova, I. Hromada, X. Wang, G. Bryant, and Z. Chen, “Observation of optical Shockley-like surface states in photonic superlattices,” Opt. Lett. 34(11), 1633–1635 (2009).
[Crossref] [PubMed]

2008 (1)

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100(10), 103904 (2008).
[Crossref] [PubMed]

1981 (1)

Aimez, V.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103(9), 093902 (2009).
[Crossref] [PubMed]

Alberti, A.

F. Bleckmann, Z. Cherpakova, S. Linden, and A. Alberti, “Spectral imaging of topological edge states in plasmonic waveguide arrays,” Phys. Rev. B 96(4), 045417 (2017).
[Crossref]

Amo, A.

P. St-Jean, V. Goblot, E. Galopin, A. Lemaître, T. Ozawa, L. Le Gratiet, I. Sagnes, J. Bloch, and A. Amo, “Lasing in topological edge states of a one-dimensional lattice,” Nat. Photonics 11(10), 651–656 (2017).
[Crossref]

Andonegui, I.

A. Blanco-Redondo, I. Andonegui, M. J. Collins, G. Harari, Y. Lumer, M. C. Rechtsman, B. J. Eggleton, and M. Segev, “Topological optical waveguiding in silicon and the transition between topological and trivial defect states,” Phys. Rev. Lett. 116(16), 163901 (2016).
[Crossref] [PubMed]

Arkinstall, J.

R. Keil, C. Poli, M. Heinrich, J. Arkinstall, G. Weihs, H. Schomerus, and A. Szameit, “Universal sign control of coupling in tight-binding lattices,” Phys. Rev. Lett. 116(21), 213901 (2016).
[Crossref] [PubMed]

Bahrampour, A. R.

M. Golshani, S. Weimann, Kh. Jafari, M. K. Nezhad, A. Langari, A. R. Bahrampour, T. Eichelkraut, S. M. Mahdavi, and A. Szameit, “Impact of loss on the wave dynamics in photonic waveguide lattices,” Phys. Rev. Lett. 113(12), 123903 (2014).
[Crossref] [PubMed]

Bandres, M. A.

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).

M. Parto, S. Wittek, H. Hodaei, G. Harari, M. A. Bandres, J. Ren, M. C. Rechtsman, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Edge-mode lasing in 1D topological active arrays,” Phys. Rev. Lett. 120(11), 113901 (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).

Bellec, M.

C. Poli, M. Bellec, U. Kuhl, F. Mortessagne, and H. Schomerus, “Selective enhancement of topologically induced interface states in a dielectric resonator chain,” Nat. Commun. 6(1), 6710 (2015).
[Crossref] [PubMed]

Bergholtz, E. J.

F. K. Kunst, E. Edvardsson, J. C. Budich, and E. J. Bergholtz, “Biorthogonal bulk-boundary correspondence in non-Hermitian systems,” Phys. Rev. Lett. 121(2), 026808 (2018).
[Crossref] [PubMed]

Blanco-Redondo, A.

A. Blanco-Redondo, I. Andonegui, M. J. Collins, G. Harari, Y. Lumer, M. C. Rechtsman, B. J. Eggleton, and M. Segev, “Topological optical waveguiding in silicon and the transition between topological and trivial defect states,” Phys. Rev. Lett. 116(16), 163901 (2016).
[Crossref] [PubMed]

Bleckmann, F.

F. Bleckmann, Z. Cherpakova, S. Linden, and A. Alberti, “Spectral imaging of topological edge states in plasmonic waveguide arrays,” Phys. Rev. B 96(4), 045417 (2017).
[Crossref]

Bloch, J.

P. St-Jean, V. Goblot, E. Galopin, A. Lemaître, T. Ozawa, L. Le Gratiet, I. Sagnes, J. Bloch, and A. Amo, “Lasing in topological edge states of a one-dimensional lattice,” Nat. Photonics 11(10), 651–656 (2017).
[Crossref]

Böhm, J.

J. Doppler, A. A. Mailybaev, J. Böhm, U. Kuhl, A. Girschik, F. Libisch, T. J. Milburn, P. Rabl, N. Moiseyev, and S. Rotter, “Dynamically encircling an exceptional point for asymmetric mode switching,” Nature 537(7618), 76–79 (2016).
[Crossref] [PubMed]

Boscolo, S.

Bryant, G.

Budich, J. C.

F. K. Kunst, E. Edvardsson, J. C. Budich, and E. J. Bergholtz, “Biorthogonal bulk-boundary correspondence in non-Hermitian systems,” Phys. Rev. Lett. 121(2), 026808 (2018).
[Crossref] [PubMed]

Cao, W.

P. Peng, W. Cao, C. Shen, W. Qu, J. Wen, L. Jiang, and Y. Xiao, “Anti-parity–time symmetry with flying atoms,” Nat. Phys. 12(12), 1139–1145 (2016).
[Crossref]

Capobianco, A. D.

Chan, C. T.

Chen, H.

Chen, X.

Chen, X. D.

X. T. He, E. T. Liang, J. J. Yuan, H. Y. Qiu, X. D. Chen, F. L. Zhao, and J. W. Dong, “A silicon-on-insulator slab for topological valley transport,” Nat. Commun. 10(1), 872 (2019).
[Crossref] [PubMed]

Chen, Y.

Chen, Z.

Cheng, Q.

Q. Cheng, Y. Pan, Q. Wang, T. Li, and S. Zhu, “Topologically protected interface mode in plasmonic waveguide arrays,” Laser Photonics Rev. 9(4), 392–398 (2015).
[Crossref]

Cherpakova, Z.

F. Bleckmann, Z. Cherpakova, S. Linden, and A. Alberti, “Spectral imaging of topological edge states in plasmonic waveguide arrays,” Phys. Rev. B 96(4), 045417 (2017).
[Crossref]

Choi, Y.

Y. Choi, C. Hahn, J. W. Yoon, and S. H. Song, “Observation of an anti-PT-symmetric exceptional point and energy-difference conserving dynamics in electrical circuit resonators,” Nat. Commun. 9(1), 2182 (2018).
[Crossref] [PubMed]

Chong, Y. D.

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).

Christodoulides, D. N.

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).

M. Parto, S. Wittek, H. Hodaei, G. Harari, M. A. Bandres, J. Ren, M. C. Rechtsman, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Edge-mode lasing in 1D topological active arrays,” Phys. Rev. Lett. 120(11), 113901 (2018).
[Crossref] [PubMed]

R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, and D. N. Christodoulides, “Non-Hermitian physics and PT symmetry,” Nat. Phys. 14(1), 11–19 (2018).
[Crossref]

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).

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity–time symmetry in optics,” Nat. Phys. 6(3), 192–195 (2010).
[Crossref]

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103(9), 093902 (2009).
[Crossref] [PubMed]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100(10), 103904 (2008).
[Crossref] [PubMed]

Collins, M. J.

A. Blanco-Redondo, I. Andonegui, M. J. Collins, G. Harari, Y. Lumer, M. C. Rechtsman, B. J. Eggleton, and M. Segev, “Topological optical waveguiding in silicon and the transition between topological and trivial defect states,” Phys. Rev. Lett. 116(16), 163901 (2016).
[Crossref] [PubMed]

De Angelis, C.

Deng, H.

Dong, J. W.

X. T. He, E. T. Liang, J. J. Yuan, H. Y. Qiu, X. D. Chen, F. L. Zhao, and J. W. Dong, “A silicon-on-insulator slab for topological valley transport,” Nat. Commun. 10(1), 872 (2019).
[Crossref] [PubMed]

Doppler, J.

J. Doppler, A. A. Mailybaev, J. Böhm, U. Kuhl, A. Girschik, F. Libisch, T. J. Milburn, P. Rabl, N. Moiseyev, and S. Rotter, “Dynamically encircling an exceptional point for asymmetric mode switching,” Nature 537(7618), 76–79 (2016).
[Crossref] [PubMed]

Duchesne, D.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103(9), 093902 (2009).
[Crossref] [PubMed]

Edvardsson, E.

F. K. Kunst, E. Edvardsson, J. C. Budich, and E. J. Bergholtz, “Biorthogonal bulk-boundary correspondence in non-Hermitian systems,” Phys. Rev. Lett. 121(2), 026808 (2018).
[Crossref] [PubMed]

Eggleton, B. J.

A. Blanco-Redondo, I. Andonegui, M. J. Collins, G. Harari, Y. Lumer, M. C. Rechtsman, B. J. Eggleton, and M. Segev, “Topological optical waveguiding in silicon and the transition between topological and trivial defect states,” Phys. Rev. Lett. 116(16), 163901 (2016).
[Crossref] [PubMed]

Eichelkraut, T.

M. Golshani, S. Weimann, Kh. Jafari, M. K. Nezhad, A. Langari, A. R. Bahrampour, T. Eichelkraut, S. M. Mahdavi, and A. Szameit, “Impact of loss on the wave dynamics in photonic waveguide lattices,” Phys. Rev. Lett. 113(12), 123903 (2014).
[Crossref] [PubMed]

El-Ganainy, R.

H. Zhao, P. Miao, M. H. Teimourpour, S. Malzard, R. El-Ganainy, H. Schomerus, and L. Feng, “Topological hybrid silicon microlasers,” Nat. Commun. 9(1), 981 (2018).
[Crossref] [PubMed]

R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, and D. N. Christodoulides, “Non-Hermitian physics and PT symmetry,” Nat. Phys. 14(1), 11–19 (2018).
[Crossref]

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity–time symmetry in optics,” Nat. Phys. 6(3), 192–195 (2010).
[Crossref]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100(10), 103904 (2008).
[Crossref] [PubMed]

Feng, L.

H. Zhao, P. Miao, M. H. Teimourpour, S. Malzard, R. El-Ganainy, H. Schomerus, and L. Feng, “Topological hybrid silicon microlasers,” Nat. Commun. 9(1), 981 (2018).
[Crossref] [PubMed]

M. Pan, H. Zhao, P. Miao, S. Longhi, and L. Feng, “Photonic zero mode in a non-Hermitian photonic lattice,” Nat. Commun. 9(1), 1308 (2018).
[Crossref] [PubMed]

H. Zhao, S. Longhi, and L. Feng, “Robust light state by quantum phase transition in Non-Hermitian optical materials,” Sci. Rep. 5(1), 17022 (2015).
[Crossref] [PubMed]

Galopin, E.

P. St-Jean, V. Goblot, E. Galopin, A. Lemaître, T. Ozawa, L. Le Gratiet, I. Sagnes, J. Bloch, and A. Amo, “Lasing in topological edge states of a one-dimensional lattice,” Nat. Photonics 11(10), 651–656 (2017).
[Crossref]

Gao, F.

Gaylord, T. K.

Ge, L.

L. Ge, “Symmetry-protected zero-mode laser with a tunable spatial profile,” Phys. Rev. A (Coll. Park) 95(2), 023812 (2017).
[Crossref]

L. Ge, L. Wang, M. Xiao, W. Wen, C. T. Chan, and D. Han, “Topological edge modes in multilayer graphene systems,” Opt. Express 23(17), 21585–21595 (2015).
[Crossref] [PubMed]

L. Ge and H. E. Türeci, “Antisymmetric PT-photonic structures with balanced positive- and negative-index materials,” Phys. Rev. A 88(5), 053810 (2013).
[Crossref]

Giannini, V.

S. R. Pocock, X. Xiao, P. A. Huidobro, and V. Giannini, “Topological plasmonic chain with retardation and radiative effects,” ACS Photonics 5(6), 2271–2279 (2018).
[Crossref]

Girschik, A.

J. Doppler, A. A. Mailybaev, J. Böhm, U. Kuhl, A. Girschik, F. Libisch, T. J. Milburn, P. Rabl, N. Moiseyev, and S. Rotter, “Dynamically encircling an exceptional point for asymmetric mode switching,” Nature 537(7618), 76–79 (2016).
[Crossref] [PubMed]

Goblot, V.

P. St-Jean, V. Goblot, E. Galopin, A. Lemaître, T. Ozawa, L. Le Gratiet, I. Sagnes, J. Bloch, and A. Amo, “Lasing in topological edge states of a one-dimensional lattice,” Nat. Photonics 11(10), 651–656 (2017).
[Crossref]

Golshani, M.

M. Golshani, S. Weimann, Kh. Jafari, M. K. Nezhad, A. Langari, A. R. Bahrampour, T. Eichelkraut, S. M. Mahdavi, and A. Szameit, “Impact of loss on the wave dynamics in photonic waveguide lattices,” Phys. Rev. Lett. 113(12), 123903 (2014).
[Crossref] [PubMed]

Guo, A.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103(9), 093902 (2009).
[Crossref] [PubMed]

Guo, X.

Gurudev Dutt, M. V.

Hahn, C.

Y. Choi, C. Hahn, J. W. Yoon, and S. H. Song, “Observation of an anti-PT-symmetric exceptional point and energy-difference conserving dynamics in electrical circuit resonators,” Nat. Commun. 9(1), 2182 (2018).
[Crossref] [PubMed]

Han, D.

Harari, G.

M. Parto, S. Wittek, H. Hodaei, G. Harari, M. A. Bandres, J. Ren, M. C. Rechtsman, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Edge-mode lasing in 1D topological active arrays,” Phys. Rev. Lett. 120(11), 113901 (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).

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).

A. Blanco-Redondo, I. Andonegui, M. J. Collins, G. Harari, Y. Lumer, M. C. Rechtsman, B. J. Eggleton, and M. Segev, “Topological optical waveguiding in silicon and the transition between topological and trivial defect states,” Phys. Rev. Lett. 116(16), 163901 (2016).
[Crossref] [PubMed]

He, M.

J. Tan, Y. Zhou, M. He, Y. Chen, Q. Ke, J. Liang, X. Zhu, M. Li, and P. Lu, “Determination of the ionization time using attosecond photoelectron Interferometry,” Phys. Rev. Lett. 121(25), 253203 (2018).
[Crossref] [PubMed]

He, X. T.

X. T. He, E. T. Liang, J. J. Yuan, H. Y. Qiu, X. D. Chen, F. L. Zhao, and J. W. Dong, “A silicon-on-insulator slab for topological valley transport,” Nat. Commun. 10(1), 872 (2019).
[Crossref] [PubMed]

Heinrich, M.

R. Keil, C. Poli, M. Heinrich, J. Arkinstall, G. Weihs, H. Schomerus, and A. Szameit, “Universal sign control of coupling in tight-binding lattices,” Phys. Rev. Lett. 116(21), 213901 (2016).
[Crossref] [PubMed]

Hodaei, H.

M. Parto, S. Wittek, H. Hodaei, G. Harari, M. A. Bandres, J. Ren, M. C. Rechtsman, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Edge-mode lasing in 1D topological active arrays,” Phys. Rev. Lett. 120(11), 113901 (2018).
[Crossref] [PubMed]

Hromada, I.

Hu, Y.

D. Zhao, D. Zhong, Y. Hu, S. Ken, and W. Liu, “Imaginary modulation inducing giant spatial Goos-Hänchen shifts in one-dimensional defective photonic lattices,” Opt. Quantum Electron. 51(113), 1–11 (2019).

Huang, G.

S. Liang and G. Huang, “Topological invariance and global Berry phase in non-Hermitian systems,” Phys. Rev. A 87(1), 012118 (2013).
[Crossref]

Huidobro, P. A.

S. R. Pocock, X. Xiao, P. A. Huidobro, and V. Giannini, “Topological plasmonic chain with retardation and radiative effects,” ACS Photonics 5(6), 2271–2279 (2018).
[Crossref]

Jafari, Kh.

M. Golshani, S. Weimann, Kh. Jafari, M. K. Nezhad, A. Langari, A. R. Bahrampour, T. Eichelkraut, S. M. Mahdavi, and A. Szameit, “Impact of loss on the wave dynamics in photonic waveguide lattices,” Phys. Rev. Lett. 113(12), 123903 (2014).
[Crossref] [PubMed]

Jiang, L.

P. Peng, W. Cao, C. Shen, W. Qu, J. Wen, L. Jiang, and Y. Xiao, “Anti-parity–time symmetry with flying atoms,” Nat. Phys. 12(12), 1139–1145 (2016).
[Crossref]

Jiang, X.

Jin, L.

L. Jin, “Topological phases and edge states in a non-Hermitian trimerized optical lattice,” Phys. Rev. A (Coll. Park) 96(3), 032103 (2017).
[Crossref]

Joannopoulos, J. D.

L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological photonics,” Nat. Photonics 8(11), 821–829 (2014).
[Crossref]

Kan, Q.

Ke, Q.

J. Tan, Y. Zhou, M. He, Y. Chen, Q. Ke, J. Liang, X. Zhu, M. Li, and P. Lu, “Determination of the ionization time using attosecond photoelectron Interferometry,” Phys. Rev. Lett. 121(25), 253203 (2018).
[Crossref] [PubMed]

Ke, S

Ke, S.

F. Wang, S. Ke, C. Qin, B. Wang, H. Long, K. Wang, and P. Lu, “Topological interface modes in graphene multilayer arrays,” Opt. Laser Technol. 103, 272–278 (2018).
[Crossref]

Q. Liu, S. Ke, and W. Liu, “Mode conversion and absorption in an optical waveguide under cascaded complex modulations,” Opt. Quantum Electron. 50(9), 356 (2018).
[Crossref]

S. Ke, D. Zhao, Q. Liu, S. Wu, B. Wang, and P. Lu, “Optical imaginary directional couplers,” J. Lightwave Technol. 36(12), 2510–2516 (2018).
[Crossref]

S. Ke, J. Liu, Q. Liu, D. Zhao, and W. Liu, “Strong absorption near exceptional points in plasmonic waveguide arrays,” Opt. Quantum Electron. 50(8), 318 (2018).
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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).

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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).

M. Parto, S. Wittek, H. Hodaei, G. Harari, M. A. Bandres, J. Ren, M. C. Rechtsman, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Edge-mode lasing in 1D topological active arrays,” Phys. Rev. Lett. 120(11), 113901 (2018).
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A. Blanco-Redondo, I. Andonegui, M. J. Collins, G. Harari, Y. Lumer, M. C. Rechtsman, B. J. Eggleton, and M. Segev, “Topological optical waveguiding in silicon and the transition between topological and trivial defect states,” Phys. Rev. Lett. 116(16), 163901 (2016).
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Ren, J.

M. Parto, S. Wittek, H. Hodaei, G. Harari, M. A. Bandres, J. Ren, M. C. Rechtsman, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Edge-mode lasing in 1D topological active arrays,” Phys. Rev. Lett. 120(11), 113901 (2018).
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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).

P. Qiu, R. Liang, W. Qiu, H. Chen, J. Ren, Z. Lin, J. X. Wang, Q. Kan, and J. Q. Pan, “Topologically protected edge states in graphene plasmonic crystals,” Opt. Express 25(19), 22587–22594 (2017).
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Rotter, S.

R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, and D. N. Christodoulides, “Non-Hermitian physics and PT symmetry,” Nat. Phys. 14(1), 11–19 (2018).
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J. Doppler, A. A. Mailybaev, J. Böhm, U. Kuhl, A. Girschik, F. Libisch, T. J. Milburn, P. Rabl, N. Moiseyev, and S. Rotter, “Dynamically encircling an exceptional point for asymmetric mode switching,” Nature 537(7618), 76–79 (2016).
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C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity–time symmetry in optics,” Nat. Phys. 6(3), 192–195 (2010).
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P. St-Jean, V. Goblot, E. Galopin, A. Lemaître, T. Ozawa, L. Le Gratiet, I. Sagnes, J. Bloch, and A. Amo, “Lasing in topological edge states of a one-dimensional lattice,” Nat. Photonics 11(10), 651–656 (2017).
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Salamo, G. J.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103(9), 093902 (2009).
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Schomerus, H.

H. Zhao, P. Miao, M. H. Teimourpour, S. Malzard, R. El-Ganainy, H. Schomerus, and L. Feng, “Topological hybrid silicon microlasers,” Nat. Commun. 9(1), 981 (2018).
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R. Keil, C. Poli, M. Heinrich, J. Arkinstall, G. Weihs, H. Schomerus, and A. Szameit, “Universal sign control of coupling in tight-binding lattices,” Phys. Rev. Lett. 116(21), 213901 (2016).
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S. Malzard, C. Poli, and H. Schomerus, “Topologically protected defect states in open photonic systems with non-Hermitian charge-conjugation and Parity-Time symmetry,” Phys. Rev. Lett. 115(20), 200402 (2015).
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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).

M. Parto, S. Wittek, H. Hodaei, G. Harari, M. A. Bandres, J. Ren, M. C. Rechtsman, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Edge-mode lasing in 1D topological active arrays,” Phys. Rev. Lett. 120(11), 113901 (2018).
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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).

S. Weimann, M. Kremer, Y. Plotnik, Y. Lumer, S. Nolte, K. G. Makris, M. Segev, M. C. Rechtsman, and A. Szameit, “Topologically protected bound states in photonic parity-time-symmetric crystals,” Nat. Mater. 16(4), 433–438 (2017).
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A. Blanco-Redondo, I. Andonegui, M. J. Collins, G. Harari, Y. Lumer, M. C. Rechtsman, B. J. Eggleton, and M. Segev, “Topological optical waveguiding in silicon and the transition between topological and trivial defect states,” Phys. Rev. Lett. 116(16), 163901 (2016).
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C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity–time symmetry in optics,” Nat. Phys. 6(3), 192–195 (2010).
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Shang, C.

Shen, C.

P. Peng, W. Cao, C. Shen, W. Qu, J. Wen, L. Jiang, and Y. Xiao, “Anti-parity–time symmetry with flying atoms,” Nat. Phys. 12(12), 1139–1145 (2016).
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A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103(9), 093902 (2009).
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L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological photonics,” Nat. Photonics 8(11), 821–829 (2014).
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Y. Choi, C. Hahn, J. W. Yoon, and S. H. Song, “Observation of an anti-PT-symmetric exceptional point and energy-difference conserving dynamics in electrical circuit resonators,” Nat. Commun. 9(1), 2182 (2018).
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P. St-Jean, V. Goblot, E. Galopin, A. Lemaître, T. Ozawa, L. Le Gratiet, I. Sagnes, J. Bloch, and A. Amo, “Lasing in topological edge states of a one-dimensional lattice,” Nat. Photonics 11(10), 651–656 (2017).
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Szameit, A.

S. Weimann, M. Kremer, Y. Plotnik, Y. Lumer, S. Nolte, K. G. Makris, M. Segev, M. C. Rechtsman, and A. Szameit, “Topologically protected bound states in photonic parity-time-symmetric crystals,” Nat. Mater. 16(4), 433–438 (2017).
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R. Keil, C. Poli, M. Heinrich, J. Arkinstall, G. Weihs, H. Schomerus, and A. Szameit, “Universal sign control of coupling in tight-binding lattices,” Phys. Rev. Lett. 116(21), 213901 (2016).
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M. Golshani, S. Weimann, Kh. Jafari, M. K. Nezhad, A. Langari, A. R. Bahrampour, T. Eichelkraut, S. M. Mahdavi, and A. Szameit, “Impact of loss on the wave dynamics in photonic waveguide lattices,” Phys. Rev. Lett. 113(12), 123903 (2014).
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J. Tan, Y. Zhou, M. He, Y. Chen, Q. Ke, J. Liang, X. Zhu, M. Li, and P. Lu, “Determination of the ionization time using attosecond photoelectron Interferometry,” Phys. Rev. Lett. 121(25), 253203 (2018).
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Teimourpour, M. H.

H. Zhao, P. Miao, M. H. Teimourpour, S. Malzard, R. El-Ganainy, H. Schomerus, and L. Feng, “Topological hybrid silicon microlasers,” Nat. Commun. 9(1), 981 (2018).
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A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103(9), 093902 (2009).
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Wang, B.

S. Ke, D. Zhao, Q. Liu, S. Wu, B. Wang, and P. Lu, “Optical imaginary directional couplers,” J. Lightwave Technol. 36(12), 2510–2516 (2018).
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F. Wang, S. Ke, C. Qin, B. Wang, H. Long, K. Wang, and P. Lu, “Topological interface modes in graphene multilayer arrays,” Opt. Laser Technol. 103, 272–278 (2018).
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S. Ke, B. Wang, H. Long, K. Wang, and P. Lu, “Topological mode switching in a graphene doublet with exceptional points,” Opt. Quantum Electron. 49(6), 224 (2017).
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Z. Wang, B. Wang, H. Long, K. Wang, and P. Lu, “Surface plasmonic lattice solitons in semi-infinite graphene sheet arrays,” J. Lightwave Technol. 35(14), 2960–2965 (2017).
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F. Wang, C. Qin, B. Wang, H. Long, K. Wang, and P. Lu, “Rabi oscillations of plasmonic supermodes in graphene multilayer arrays,” IEEE J. Sel. Top. Quantum Electron. 23(1), 125 (2017).
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D. Zhao, Z. Wang, H. Long, K. Wang, B. Wang, and P. Lu, “Optical bistability in defective photonic multilayers doped by graphene,” Opt. Quantum Electron. 49(4), 163 (2017).
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S. Ke, B. Wang, H. Long, K. Wang, and P. Lu, “Topological edge modes in non-Hermitian plasmonic waveguide arrays,” Opt. Express 25(10), 11132–11143 (2017).
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S. Ke, B. Wang, C. Qin, H. Long, K. Wang, and P. Lu, “Exceptional points and asymmetric mode switching in plasmonic waveguides,” J. Lightwave Technol. 34(22), 5258–5262 (2016).
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C. Qin, B. Wang, H. Long, K. Wang, and P. Lu, “Nonreciprocal phase shift and mode modulation in dynamic graphene waveguides,” J. Lightwave Technol. 34(16), 3877–3883 (2016).

Wang, F.

F. Wang, S. Ke, C. Qin, B. Wang, H. Long, K. Wang, and P. Lu, “Topological interface modes in graphene multilayer arrays,” Opt. Laser Technol. 103, 272–278 (2018).
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F. Wang, C. Qin, B. Wang, H. Long, K. Wang, and P. Lu, “Rabi oscillations of plasmonic supermodes in graphene multilayer arrays,” IEEE J. Sel. Top. Quantum Electron. 23(1), 125 (2017).
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Wang, J.

Wang, J. X.

Wang, K.

F. Wang, S. Ke, C. Qin, B. Wang, H. Long, K. Wang, and P. Lu, “Topological interface modes in graphene multilayer arrays,” Opt. Laser Technol. 103, 272–278 (2018).
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Z. Wang, B. Wang, H. Long, K. Wang, and P. Lu, “Surface plasmonic lattice solitons in semi-infinite graphene sheet arrays,” J. Lightwave Technol. 35(14), 2960–2965 (2017).
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S. Ke, B. Wang, H. Long, K. Wang, and P. Lu, “Topological mode switching in a graphene doublet with exceptional points,” Opt. Quantum Electron. 49(6), 224 (2017).
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F. Wang, C. Qin, B. Wang, H. Long, K. Wang, and P. Lu, “Rabi oscillations of plasmonic supermodes in graphene multilayer arrays,” IEEE J. Sel. Top. Quantum Electron. 23(1), 125 (2017).
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D. Zhao, Z. Wang, H. Long, K. Wang, B. Wang, and P. Lu, “Optical bistability in defective photonic multilayers doped by graphene,” Opt. Quantum Electron. 49(4), 163 (2017).
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S. Ke, B. Wang, H. Long, K. Wang, and P. Lu, “Topological edge modes in non-Hermitian plasmonic waveguide arrays,” Opt. Express 25(10), 11132–11143 (2017).
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S. Ke, B. Wang, C. Qin, H. Long, K. Wang, and P. Lu, “Exceptional points and asymmetric mode switching in plasmonic waveguides,” J. Lightwave Technol. 34(22), 5258–5262 (2016).
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C. Qin, B. Wang, H. Long, K. Wang, and P. Lu, “Nonreciprocal phase shift and mode modulation in dynamic graphene waveguides,” J. Lightwave Technol. 34(16), 3877–3883 (2016).

Wang, L.

Wang, M.

Wang, Q.

Q. Cheng, Y. Pan, Q. Wang, T. Li, and S. Zhu, “Topologically protected interface mode in plasmonic waveguide arrays,” Laser Photonics Rev. 9(4), 392–398 (2015).
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S. Yao and Z. Wang, “Edge states and topological invariants of non-Hermitian systems,” Phys. Rev. Lett. 121(8), 086803 (2018).
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Z. Wang, B. Wang, H. Long, K. Wang, and P. Lu, “Surface plasmonic lattice solitons in semi-infinite graphene sheet arrays,” J. Lightwave Technol. 35(14), 2960–2965 (2017).
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D. Zhao, Z. Wang, H. Long, K. Wang, B. Wang, and P. Lu, “Optical bistability in defective photonic multilayers doped by graphene,” Opt. Quantum Electron. 49(4), 163 (2017).
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R. Keil, C. Poli, M. Heinrich, J. Arkinstall, G. Weihs, H. Schomerus, and A. Szameit, “Universal sign control of coupling in tight-binding lattices,” Phys. Rev. Lett. 116(21), 213901 (2016).
[Crossref] [PubMed]

Weimann, S.

S. Weimann, M. Kremer, Y. Plotnik, Y. Lumer, S. Nolte, K. G. Makris, M. Segev, M. C. Rechtsman, and A. Szameit, “Topologically protected bound states in photonic parity-time-symmetric crystals,” Nat. Mater. 16(4), 433–438 (2017).
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M. Golshani, S. Weimann, Kh. Jafari, M. K. Nezhad, A. Langari, A. R. Bahrampour, T. Eichelkraut, S. M. Mahdavi, and A. Szameit, “Impact of loss on the wave dynamics in photonic waveguide lattices,” Phys. Rev. Lett. 113(12), 123903 (2014).
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Wen, J.

P. Peng, W. Cao, C. Shen, W. Qu, J. Wen, L. Jiang, and Y. Xiao, “Anti-parity–time symmetry with flying atoms,” Nat. Phys. 12(12), 1139–1145 (2016).
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Wiersig, J.

J. Wiersig, “Structure of whispering-gallery modes in optical microdisks perturbed by nanoparticles,” Phys. Rev. A 84(6), 063828 (2011).
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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).

M. Parto, S. Wittek, H. Hodaei, G. Harari, M. A. Bandres, J. Ren, M. C. Rechtsman, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Edge-mode lasing in 1D topological active arrays,” Phys. Rev. Lett. 120(11), 113901 (2018).
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Wu, B.

Wu, S.

Xiao, M.

Xiao, X.

S. R. Pocock, X. Xiao, P. A. Huidobro, and V. Giannini, “Topological plasmonic chain with retardation and radiative effects,” ACS Photonics 5(6), 2271–2279 (2018).
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Xiao, Y.

P. Peng, W. Cao, C. Shen, W. Qu, J. Wen, L. Jiang, and Y. Xiao, “Anti-parity–time symmetry with flying atoms,” Nat. Phys. 12(12), 1139–1145 (2016).
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Xu, B.

Xu, C.

Xu, J.

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F. Yang, Y. Liu, and L. You, “Anti-PT symmetry in dissipatively coupled optical systems,” Phys. Rev. A (Coll. Park) 96(5), 053845 (2017).
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Yang, J.

Yao, S.

S. Yao and Z. Wang, “Edge states and topological invariants of non-Hermitian systems,” Phys. Rev. Lett. 121(8), 086803 (2018).
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Ye, F.

Yoon, J. W.

Y. Choi, C. Hahn, J. W. Yoon, and S. H. Song, “Observation of an anti-PT-symmetric exceptional point and energy-difference conserving dynamics in electrical circuit resonators,” Nat. Commun. 9(1), 2182 (2018).
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You, L.

F. Yang, Y. Liu, and L. You, “Anti-PT symmetry in dissipatively coupled optical systems,” Phys. Rev. A (Coll. Park) 96(5), 053845 (2017).
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Yuan, J. J.

X. T. He, E. T. Liang, J. J. Yuan, H. Y. Qiu, X. D. Chen, F. L. Zhao, and J. W. Dong, “A silicon-on-insulator slab for topological valley transport,” Nat. Commun. 10(1), 872 (2019).
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C. Yuce, “Edge states at the interface of non-Hermitian systems,” Phys. Rev. A (Coll. Park) 97(4), 042118 (2018).
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Zhang, B.

Zhang, X.

Zhao, D

Zhao, D.

D. Zhao, D. Zhong, Y. Hu, S. Ken, and W. Liu, “Imaginary modulation inducing giant spatial Goos-Hänchen shifts in one-dimensional defective photonic lattices,” Opt. Quantum Electron. 51(113), 1–11 (2019).

S. Ke, J. Liu, Q. Liu, D. Zhao, and W. Liu, “Strong absorption near exceptional points in plasmonic waveguide arrays,” Opt. Quantum Electron. 50(8), 318 (2018).
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S. Ke, D. Zhao, Q. Liu, S. Wu, B. Wang, and P. Lu, “Optical imaginary directional couplers,” J. Lightwave Technol. 36(12), 2510–2516 (2018).
[Crossref]

S. Ke, J. Liu, Q. Liu, D. Zhao, and W. Liu, “Strong absorption near exceptional points in plasmonic waveguide arrays,” Opt. Quantum Electron. 50(8), 318 (2018).
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D. Zhao, Z. Wang, H. Long, K. Wang, B. Wang, and P. Lu, “Optical bistability in defective photonic multilayers doped by graphene,” Opt. Quantum Electron. 49(4), 163 (2017).
[Crossref]

Zhao, F. L.

X. T. He, E. T. Liang, J. J. Yuan, H. Y. Qiu, X. D. Chen, F. L. Zhao, and J. W. Dong, “A silicon-on-insulator slab for topological valley transport,” Nat. Commun. 10(1), 872 (2019).
[Crossref] [PubMed]

Zhao, H.

M. Pan, H. Zhao, P. Miao, S. Longhi, and L. Feng, “Photonic zero mode in a non-Hermitian photonic lattice,” Nat. Commun. 9(1), 1308 (2018).
[Crossref] [PubMed]

H. Zhao, P. Miao, M. H. Teimourpour, S. Malzard, R. El-Ganainy, H. Schomerus, and L. Feng, “Topological hybrid silicon microlasers,” Nat. Commun. 9(1), 981 (2018).
[Crossref] [PubMed]

H. Zhao, S. Longhi, and L. Feng, “Robust light state by quantum phase transition in Non-Hermitian optical materials,” Sci. Rep. 5(1), 17022 (2015).
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Zhong, D.

D. Zhao, D. Zhong, Y. Hu, S. Ken, and W. Liu, “Imaginary modulation inducing giant spatial Goos-Hänchen shifts in one-dimensional defective photonic lattices,” Opt. Quantum Electron. 51(113), 1–11 (2019).

Zhou, Y.

J. Tan, Y. Zhou, M. He, Y. Chen, Q. Ke, J. Liang, X. Zhu, M. Li, and P. Lu, “Determination of the ionization time using attosecond photoelectron Interferometry,” Phys. Rev. Lett. 121(25), 253203 (2018).
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Zhu, S.

B. Xu, T. Li, and S. Zhu, “Simulation of massless Dirac dynamics in plasmonic waveguide arrays,” Opt. Express 26(10), 13416–13424 (2018).
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Q. Cheng, Y. Pan, Q. Wang, T. Li, and S. Zhu, “Topologically protected interface mode in plasmonic waveguide arrays,” Laser Photonics Rev. 9(4), 392–398 (2015).
[Crossref]

Zhu, X.

J. Tan, Y. Zhou, M. He, Y. Chen, Q. Ke, J. Liang, X. Zhu, M. Li, and P. Lu, “Determination of the ionization time using attosecond photoelectron Interferometry,” Phys. Rev. Lett. 121(25), 253203 (2018).
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ACS Photonics (1)

S. R. Pocock, X. Xiao, P. A. Huidobro, and V. Giannini, “Topological plasmonic chain with retardation and radiative effects,” ACS Photonics 5(6), 2271–2279 (2018).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (1)

F. Wang, C. Qin, B. Wang, H. Long, K. Wang, and P. Lu, “Rabi oscillations of plasmonic supermodes in graphene multilayer arrays,” IEEE J. Sel. Top. Quantum Electron. 23(1), 125 (2017).
[Crossref]

J. Lightwave Technol. (5)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. B (1)

Laser Photonics Rev. (1)

Q. Cheng, Y. Pan, Q. Wang, T. Li, and S. Zhu, “Topologically protected interface mode in plasmonic waveguide arrays,” Laser Photonics Rev. 9(4), 392–398 (2015).
[Crossref]

Nat. Commun. (5)

X. T. He, E. T. Liang, J. J. Yuan, H. Y. Qiu, X. D. Chen, F. L. Zhao, and J. W. Dong, “A silicon-on-insulator slab for topological valley transport,” Nat. Commun. 10(1), 872 (2019).
[Crossref] [PubMed]

H. Zhao, P. Miao, M. H. Teimourpour, S. Malzard, R. El-Ganainy, H. Schomerus, and L. Feng, “Topological hybrid silicon microlasers,” Nat. Commun. 9(1), 981 (2018).
[Crossref] [PubMed]

C. Poli, M. Bellec, U. Kuhl, F. Mortessagne, and H. Schomerus, “Selective enhancement of topologically induced interface states in a dielectric resonator chain,” Nat. Commun. 6(1), 6710 (2015).
[Crossref] [PubMed]

Y. Choi, C. Hahn, J. W. Yoon, and S. H. Song, “Observation of an anti-PT-symmetric exceptional point and energy-difference conserving dynamics in electrical circuit resonators,” Nat. Commun. 9(1), 2182 (2018).
[Crossref] [PubMed]

M. Pan, H. Zhao, P. Miao, S. Longhi, and L. Feng, “Photonic zero mode in a non-Hermitian photonic lattice,” Nat. Commun. 9(1), 1308 (2018).
[Crossref] [PubMed]

Nat. Mater. (1)

S. Weimann, M. Kremer, Y. Plotnik, Y. Lumer, S. Nolte, K. G. Makris, M. Segev, M. C. Rechtsman, and A. Szameit, “Topologically protected bound states in photonic parity-time-symmetric crystals,” Nat. Mater. 16(4), 433–438 (2017).
[Crossref] [PubMed]

Nat. Photonics (2)

L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological photonics,” Nat. Photonics 8(11), 821–829 (2014).
[Crossref]

P. St-Jean, V. Goblot, E. Galopin, A. Lemaître, T. Ozawa, L. Le Gratiet, I. Sagnes, J. Bloch, and A. Amo, “Lasing in topological edge states of a one-dimensional lattice,” Nat. Photonics 11(10), 651–656 (2017).
[Crossref]

Nat. Phys. (3)

R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, and D. N. Christodoulides, “Non-Hermitian physics and PT symmetry,” Nat. Phys. 14(1), 11–19 (2018).
[Crossref]

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity–time symmetry in optics,” Nat. Phys. 6(3), 192–195 (2010).
[Crossref]

P. Peng, W. Cao, C. Shen, W. Qu, J. Wen, L. Jiang, and Y. Xiao, “Anti-parity–time symmetry with flying atoms,” Nat. Phys. 12(12), 1139–1145 (2016).
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Nature (1)

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Figures (8)

Fig. 1
Fig. 1 Schematic of waveguide arrays with anti-PT symmetry. (a) The proposed sawtooth array to realize the effective imaginary coupling. The blue and red sites represent bare arrays and the gray and green are for assistant sites. (b) The equivalent array with imaginary coupling. (c) The geometry of proposed planar waveguide array with anti-PT symmetry.
Fig. 2
Fig. 2 The band structures of anti-PT-symmetric waveguide arrays for various coupling strength. (a), (c), and (e) The real part of band structure. (b), (d), and (f) The imaginary part of band structure. In all figures, the intra-layer coupling and the detuning of real part of propagation constant are fixed at c1 = 4 × 10−3 μm−1 and Δ = 4 × 10−3 μm−1, respectively. The inter-layer coupling varies with (a) and (b) c2 = 1 × 10−3 μm−1, (c) and (d) c2 = 4 × 10−3 μm−1, (e) and (f) c2 = 7 × 10−3 μm−1.
Fig. 3
Fig. 3 (a) and (b) are the mode profiles of Bloch modes for φ = 0 in upper and lower bands, respectively. (c) The field fidelity as a function of coupling strength J0. (d) The amount of loss added in assistant waveguides C and D for different coupling strength. In the calculation, J1 = J2 = J0, c1 = c2 = c0 = 4 × 10−3 μm−1.
Fig. 4
Fig. 4 Topological bound modes in finite waveguide arrays. (a) The geometry of finite lattice. (b) The propagation constants for all supermodes. (c) and (d) are the mode profiles for the topological edge modes. The effective coupling strength is c1 = 1 × 10−3 μm−1 and c2 = 4 × 10−3 μm−1. The detuning of real propagation constant is Δ = 4 × 10−3.
Fig. 5
Fig. 5 Propagation of topological edge modes. In (a) and (b), the input fields are eigenmodes. (c) and (d) are for a single waveguide excitation from the right and left termination.
Fig. 6
Fig. 6 The Berry phase as a function of detuning Δ. (a) Real part and (b) imaginary part of the Berry phase. The red and blue curves represent upper and lower bands, respectively. The effective coupling strength is c1 = 4 × 10−3 μm−1 and c2 = 1 × 10−3 μm−1.
Fig. 7
Fig. 7 Bound mode induced by quantum phase transition. (a) The proposed structures. (b) The spectrum of eigenmodes. The green dot denotes the robust bound modes. (c) The mode profiles of bound mode. The lines present the simulated results and the dots are the theoretical results calculated by the coupled mode theory. (d) Propagation of bound mode.
Fig. 8
Fig. 8 Robustness of the bound mode against perturbation. (a) The proposed structures by tuning the local coupling strength at the interface. (b) The realistic structure with assistant waveguides. (c) and (d) are the mode profiles of bound modes as the perturbation of coupling strength is δc = 1 × 10−3 μm−1 and 3 × 10−3 μm−1, respectively.

Equations (11)

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i d a n dz = δ 1 a n + J 1 c n + J 2 d n1 i d b n dz = δ 2 b n + J 1 c n + J 2 d n i d c n dz = δ 3 c n + J 1 ( a n + b n ) i d d n dz = δ 4 d n + J 2 ( a n+1 + b n )
c n = J 1 ( a n + b n ) δ 3 , d n = J 2 ( a n+1 + b n ) δ 4
i d a n dz =( δ 1 J 1 2 δ 3 J 2 2 δ 4 ) a n J 1 2 δ 3 b n J 2 2 δ 4 b n1 i d b n dz = J 1 2 δ 3 a n J 2 2 δ 4 a n+1 +( δ 2 J 1 2 δ 3 J 2 2 δ 4 ) b n
  δ 1 =i( c 1 + c 2 )Δ/2, δ 2 =i( c 1 + c 2 )+Δ/2, δ 3 =i J 1 2 / c 1 , δ 4 =i J 2 2 / c 2 ,
i d a n dz = Δ 2 a n +i c 1 b n +i c 2 b n1 , i d b n dz =i c 1 a n +i c 2 a n+1 + Δ 2 b n .
H=( Δ/2 i c 1 +i c 2 exp(iφ) i c 1 +i c 2 exp(iφ) Δ/2 ).
λ ± =± Δ 2 /4 c 1 2 c 2 2 2 c 1 c 2 cosφ .
| λ + =( cos( Δ k /2)exp(i ϕ k ) sin( Δ k /2) ),| λ =( sin( Δ k /2)exp(i ϕ k ) cos( Δ k /2) ) μ + |= ( cos( Δ k /2)exp(i ϕ k ) sin( Δ k /2) ) T , μ |= ( sin( Δ k /2)exp(i ϕ k ) cos( Δ k /2) ) T
φ B ± = d ϕ k /2 ± cos( γ k )d ϕ k /2
F= m=A,B | λ +, m * λ m ' | .
H=( Δ/2 c 1 + c 2 exp(iφ) c 1 c 2 exp(iφ) Δ/2 ).

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