Abstract

The coupling strength between two parity-time (PT) symmetric resonators determines whether the PT phase is broken or not. Here we investigate the scenario that two optical waveguides are spatially curved so that they switch periodically between unbroken and broken PT phases. We show that the existence of locally broken PT phase does not necessarily render a broken phase to waves propagating inside. Criteria are proposed to characterize the collective dynamics of wave near the Brillouin zone (BZ) edge, toward the cases of a totally broken phase, a partially broken phase, or a totally unbroken phase. We also discuss the characteristics of two special kinds of exceptional points (EPs) at the BZ edge, and show that their field patterns are displaced by half a period with each other. Full-wave numerical simulation proves our analysis. Potential applications especially these associated with EPs are discussed. This study helps us to understand how the locally PT-symmetric related eigenstate influences the globally collective dynamics of wave in spatially periodic configuration.

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

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References

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    [Crossref] [PubMed]
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    [Crossref] [PubMed]

2018 (3)

T. Liu, X. Zhu, F. Chen, S. Liang, and J. Zhu, “Unidirectional wave vector manipulation in two-dimensional space with an all passive acoustic parity-time-symmetric metamaterials crystal,” Phys. Rev. Lett. 120, 124502 (2018).
[Crossref] [PubMed]

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

T. Goldzak, A. A. Mailybaev, and N. Moiseyev, “Light stops at exceptional points,” Phys. Rev. Lett. 120, 013901 (2018).
[Crossref] [PubMed]

2017 (7)

W. Wang, L. Q. Wang, R. D. Xue, H. L. Chen, R. P. Guo, Y. Liu, and J. Chen, “Unidirectional excitation of radiative-loss-free surface plasmon polaritons in PT-symmetric systems,” Phys. Rev. Lett. 119, 077401 (2017).
[Crossref]

L. Feng, R. El-Ganainy, and L. Ge, “Non-Hermitian photonics based on parity-time symmetry,” Nat. Photo. 11, 752–762 (2017).
[Crossref]

H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. N. Christodoulides, and M. Khajavikhan, “Enhanced sensitivity at higher-order exceptional points,” Nature 548, 187–191 (2017).
[Crossref] [PubMed]

Y. Wu, B. Zhu, S. F. Hu, Z. Zhou, and H. H. Zhong, “Floquet control of the gain and loss in a PT-symmetric optical coupler,” Front. Phys. 12, 121102 (2017).
[Crossref]

M. Chitsazi, H. Li, F. M. Ellis, and T. Kottos, “Experimental realization of Floquet PT-symmetric systems,” Phys. Rev. Lett. 119, 093901 (2017).
[Crossref]

A. Pick, B. Zhen, O. D. Miller, C. W. Hsu, F. Hernandez, A. W. Rodriguez, M. Soljačić, and S. G. Johnson, “General theory of spontaneous emission near exceptional points,” Opt. Express 25, 12325–12348 (2017).
[Crossref] [PubMed]

L. Q. Wang, R. D. Xue, W. Wang, R. X. Wang, R. P. Guo, and J. Chen, “Coherent-trapped helical mode in parity-time symmetric metamaterials,” Opt. Express 25, 15231–15240 (2017).
[Crossref] [PubMed]

2016 (3)

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, 76–80 (2016).
[Crossref] [PubMed]

Z. Lin, A. Pick, M. Loncar, and A. W. Rodriguez, “Enhanced spontaneous emission at third-order Dirac exceptional points in inverse-designed photonic crystals,” Phys. Rev. Lett. 117, 107402 (2016).
[Crossref] [PubMed]

F. J. Shu, C. L. Zou, X. B. Zou, and L. Yang, “Chiral symmetry breaking in a microring optical cavity by engineered dissipation,” Phys. Rev. A 94, 013848 (2016).
[Crossref]

2015 (4)

H. Cao and J. Wiersig, “Dielectric microcavities: model systems for wave chaos and non-Hermitian physics,” Rev. Mod. Phys. 87, 61–111 (2015).
[Crossref]

B. Zhen, C. W. Hsu, Y. Igarashi, L. Lu, I. Kaminer, A. Pick, S. L. Chua, J. D. Joannopoulos, and M. Soljačić, “Spawning rings of exceptional points out of Dirac cones,” Nature 525, 354–358 (2015).
[Crossref] [PubMed]

J. Gear, F. Liu, S. T. Chu, S. Rotter, and J. Li, “Parity-time symmetry from stacking purely dielectric and magnetic slabs,” Phys. Rev. A 91, 033825 (2015).
[Crossref]

X. F. Zhu, “Defect states and exceptional point splitting in the band gaps of one-dimensional parity-time lattices,” Opt. Express 23, 22274–22284 (2015).
[Crossref] [PubMed]

2014 (8)

L. Ge and A. D. Stone, “Parity-time symmetry breaking beyond one dimension: the role of degeneracy,” Phys. Rev. X 4, 031011 (2014).

H. Hodaei, M. A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346, 975–978 (2014).
[Crossref] [PubMed]

X. F. Zhu, H. Ramezani, C. Shi, J. Zhu, and X. Zhang, “PT-symmetric acoustics,” Phys. Rev. X 4, 031042 (2014).

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photon. 8, 524–529 (2014).
[Crossref]

B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
[Crossref]

M. Lawrence, N. Xu, X. Zhang, L. Cong, J. Han, W. Zhang, and S. Zhang, “Manifestation of PT symmetry breaking in polarization space with terahertz metasurfaces,” Phys. Rev. Lett. 113, 093901 (2014).
[Crossref]

Y. N. Joglekar, R. Marathe, P. Durganandini, and R. K. Pathak, “PT spectroscopy of the Rabi problem,” Phys. Rev. A 90, 040101 (2014).
[Crossref]

X. F. Zhu, Y. G. Peng, and D. G. Zhao, “Anisotropic reflection oscillation in periodic multilayer structures of parity-time symmetry,” Opt. Express 22, 18401–18411 (2014).
[Crossref] [PubMed]

2013 (5)

H. X. Cui, X. W. Cao, M. Kang, T. F. Li, M. Yang, T. J. Guo, Q. H. Guo, and J. Chen, “Exceptional points in extraordinary optical transmission through dual subwavelength metallic gratings,” Opt. Express 21, 13368–13379 (2013).
[Crossref] [PubMed]

H. Cang, Y. Liu, Y. Wang, X. Yin, and X. Zhang, “Giant suppression of photobleaching for single molecule detection via the Purcell effect,” Nano Letters 13, 5949–5953 (2013).
[Crossref] [PubMed]

L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. B. Oliveira, V. R. Almeida, Y. F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12, 108–113 (2013).
[Crossref]

X. Luo, J. Huang, H. Zhong, X. Qin, Q. Xie, Y. S. Kivshar, and C. Lee, “Pseudo-parity-time symmetry in optical systems,” Phys. Rev. Lett. 110, 243902 (2013).
[Crossref] [PubMed]

G. D. Valle and S. Longhi, “Spectral and transport properties of time-periodic PT-symmetric tight-binding lattices,” Phys. Rev. A 87, 022119 (2013).
[Crossref]

2011 (1)

W. Wan, Y. Chong, L. Ge, H. Noh, A. D. Stone, and H. Cao, “Time-reversed lasing and interferometric control of absorption,” Science 331, 889–892 (2011).
[Crossref] [PubMed]

2010 (3)

S. Longhi, “PT-symmetric laser absorber,” Phys. Rev. A 82, 031801 (2010).
[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, 192–195 (2010).
[Crossref]

J. B. Khurgin, “Slow light in various media: a tutorial,” Adv. Opt. Photon. 2, 287–318 (2010).
[Crossref]

2009 (1)

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, 093902 (2009).
[Crossref]

2008 (2)

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

S. Klaiman, U. Gunther, and N. Moiseyev, “Visualization of branch points in PT-symmetric waveguides,” Phys. Rev. Lett. 101, 080402 (2008).
[Crossref]

2004 (1)

C. Dembowski, B. Dietz, H. D. Graf, H. L. Harney, A. Heine, W. D. Heiss, and A. Richter, “Encircling an exceptional point,” Phys. Rev. E 69, 056216 (2004).
[Crossref]

1998 (1)

C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80, 5243–5246 (1998).
[Crossref]

1946 (1)

E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 681 (1946).

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, 093902 (2009).
[Crossref]

Almeida, V. R.

L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. B. Oliveira, V. R. Almeida, Y. F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12, 108–113 (2013).
[Crossref]

Bender, C. M.

B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
[Crossref]

C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80, 5243–5246 (1998).
[Crossref]

Boettcher, S.

C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80, 5243–5246 (1998).
[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, 76–80 (2016).
[Crossref] [PubMed]

Cang, H.

H. Cang, Y. Liu, Y. Wang, X. Yin, and X. Zhang, “Giant suppression of photobleaching for single molecule detection via the Purcell effect,” Nano Letters 13, 5949–5953 (2013).
[Crossref] [PubMed]

Cao, H.

H. Cao and J. Wiersig, “Dielectric microcavities: model systems for wave chaos and non-Hermitian physics,” Rev. Mod. Phys. 87, 61–111 (2015).
[Crossref]

W. Wan, Y. Chong, L. Ge, H. Noh, A. D. Stone, and H. Cao, “Time-reversed lasing and interferometric control of absorption,” Science 331, 889–892 (2011).
[Crossref] [PubMed]

Cao, X. W.

Chang, L.

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photon. 8, 524–529 (2014).
[Crossref]

Chen, F.

T. Liu, X. Zhu, F. Chen, S. Liang, and J. Zhu, “Unidirectional wave vector manipulation in two-dimensional space with an all passive acoustic parity-time-symmetric metamaterials crystal,” Phys. Rev. Lett. 120, 124502 (2018).
[Crossref] [PubMed]

Chen, H. L.

W. Wang, L. Q. Wang, R. D. Xue, H. L. Chen, R. P. Guo, Y. Liu, and J. Chen, “Unidirectional excitation of radiative-loss-free surface plasmon polaritons in PT-symmetric systems,” Phys. Rev. Lett. 119, 077401 (2017).
[Crossref]

Chen, J.

Chen, Y. F.

L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. B. Oliveira, V. R. Almeida, Y. F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12, 108–113 (2013).
[Crossref]

Chitsazi, M.

M. Chitsazi, H. Li, F. M. Ellis, and T. Kottos, “Experimental realization of Floquet PT-symmetric systems,” Phys. Rev. Lett. 119, 093901 (2017).
[Crossref]

Chong, Y.

W. Wan, Y. Chong, L. Ge, H. Noh, A. D. Stone, and H. Cao, “Time-reversed lasing and interferometric control of absorption,” Science 331, 889–892 (2011).
[Crossref] [PubMed]

Christodoulides, D. N.

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

H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. N. Christodoulides, and M. Khajavikhan, “Enhanced sensitivity at higher-order exceptional points,” Nature 548, 187–191 (2017).
[Crossref] [PubMed]

H. Hodaei, M. A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346, 975–978 (2014).
[Crossref] [PubMed]

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, 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, 093902 (2009).
[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, 103904 (2008).
[Crossref]

Chu, S. T.

J. Gear, F. Liu, S. T. Chu, S. Rotter, and J. Li, “Parity-time symmetry from stacking purely dielectric and magnetic slabs,” Phys. Rev. A 91, 033825 (2015).
[Crossref]

Chua, S. L.

B. Zhen, C. W. Hsu, Y. Igarashi, L. Lu, I. Kaminer, A. Pick, S. L. Chua, J. D. Joannopoulos, and M. Soljačić, “Spawning rings of exceptional points out of Dirac cones,” Nature 525, 354–358 (2015).
[Crossref] [PubMed]

Cong, L.

M. Lawrence, N. Xu, X. Zhang, L. Cong, J. Han, W. Zhang, and S. Zhang, “Manifestation of PT symmetry breaking in polarization space with terahertz metasurfaces,” Phys. Rev. Lett. 113, 093901 (2014).
[Crossref]

Cui, H. X.

Dembowski, C.

C. Dembowski, B. Dietz, H. D. Graf, H. L. Harney, A. Heine, W. D. Heiss, and A. Richter, “Encircling an exceptional point,” Phys. Rev. E 69, 056216 (2004).
[Crossref]

Dietz, B.

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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, 76–80 (2016).
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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, 093902 (2009).
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Durganandini, P.

Y. N. Joglekar, R. Marathe, P. Durganandini, and R. K. Pathak, “PT spectroscopy of the Rabi problem,” Phys. Rev. A 90, 040101 (2014).
[Crossref]

El-Ganainy, R.

R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, and D. N. Christodoulides, “Non-Hermitian physics and P T symmetry,” Nat. Phys. 14, 11–19 (2018).
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L. Feng, R. El-Ganainy, and L. Ge, “Non-Hermitian photonics based on parity-time symmetry,” Nat. Photo. 11, 752–762 (2017).
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H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. N. Christodoulides, and M. Khajavikhan, “Enhanced sensitivity at higher-order exceptional points,” Nature 548, 187–191 (2017).
[Crossref] [PubMed]

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, 192–195 (2010).
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K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008).
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M. Chitsazi, H. Li, F. M. Ellis, and T. Kottos, “Experimental realization of Floquet PT-symmetric systems,” Phys. Rev. Lett. 119, 093901 (2017).
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B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
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L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. B. Oliveira, V. R. Almeida, Y. F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12, 108–113 (2013).
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L. Feng, R. El-Ganainy, and L. Ge, “Non-Hermitian photonics based on parity-time symmetry,” Nat. Photo. 11, 752–762 (2017).
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L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. B. Oliveira, V. R. Almeida, Y. F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12, 108–113 (2013).
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H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. N. Christodoulides, and M. Khajavikhan, “Enhanced sensitivity at higher-order exceptional points,” Nature 548, 187–191 (2017).
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L. Feng, R. El-Ganainy, and L. Ge, “Non-Hermitian photonics based on parity-time symmetry,” Nat. Photo. 11, 752–762 (2017).
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L. Ge and A. D. Stone, “Parity-time symmetry breaking beyond one dimension: the role of degeneracy,” Phys. Rev. X 4, 031011 (2014).

W. Wan, Y. Chong, L. Ge, H. Noh, A. D. Stone, and H. Cao, “Time-reversed lasing and interferometric control of absorption,” Science 331, 889–892 (2011).
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J. Gear, F. Liu, S. T. Chu, S. Rotter, and J. Li, “Parity-time symmetry from stacking purely dielectric and magnetic slabs,” Phys. Rev. A 91, 033825 (2015).
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B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
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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, 76–80 (2016).
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Goldzak, T.

T. Goldzak, A. A. Mailybaev, and N. Moiseyev, “Light stops at exceptional points,” Phys. Rev. Lett. 120, 013901 (2018).
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C. Dembowski, B. Dietz, H. D. Graf, H. L. Harney, A. Heine, W. D. Heiss, and A. Richter, “Encircling an exceptional point,” Phys. Rev. E 69, 056216 (2004).
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S. Klaiman, U. Gunther, and N. Moiseyev, “Visualization of branch points in PT-symmetric waveguides,” Phys. Rev. Lett. 101, 080402 (2008).
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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, 093902 (2009).
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Guo, R. P.

L. Q. Wang, R. D. Xue, W. Wang, R. X. Wang, R. P. Guo, and J. Chen, “Coherent-trapped helical mode in parity-time symmetric metamaterials,” Opt. Express 25, 15231–15240 (2017).
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W. Wang, L. Q. Wang, R. D. Xue, H. L. Chen, R. P. Guo, Y. Liu, and J. Chen, “Unidirectional excitation of radiative-loss-free surface plasmon polaritons in PT-symmetric systems,” Phys. Rev. Lett. 119, 077401 (2017).
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Guo, T. J.

Han, J.

M. Lawrence, N. Xu, X. Zhang, L. Cong, J. Han, W. Zhang, and S. Zhang, “Manifestation of PT symmetry breaking in polarization space with terahertz metasurfaces,” Phys. Rev. Lett. 113, 093901 (2014).
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C. Dembowski, B. Dietz, H. D. Graf, H. L. Harney, A. Heine, W. D. Heiss, and A. Richter, “Encircling an exceptional point,” Phys. Rev. E 69, 056216 (2004).
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Hassan, A. U.

H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. N. Christodoulides, and M. Khajavikhan, “Enhanced sensitivity at higher-order exceptional points,” Nature 548, 187–191 (2017).
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Heine, A.

C. Dembowski, B. Dietz, H. D. Graf, H. L. Harney, A. Heine, W. D. Heiss, and A. Richter, “Encircling an exceptional point,” Phys. Rev. E 69, 056216 (2004).
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Heinrich, M.

H. Hodaei, M. A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346, 975–978 (2014).
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Heiss, W. D.

C. Dembowski, B. Dietz, H. D. Graf, H. L. Harney, A. Heine, W. D. Heiss, and A. Richter, “Encircling an exceptional point,” Phys. Rev. E 69, 056216 (2004).
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Hernandez, F.

Hodaei, H.

H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. N. Christodoulides, and M. Khajavikhan, “Enhanced sensitivity at higher-order exceptional points,” Nature 548, 187–191 (2017).
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H. Hodaei, M. A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346, 975–978 (2014).
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Hsu, C. W.

A. Pick, B. Zhen, O. D. Miller, C. W. Hsu, F. Hernandez, A. W. Rodriguez, M. Soljačić, and S. G. Johnson, “General theory of spontaneous emission near exceptional points,” Opt. Express 25, 12325–12348 (2017).
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B. Zhen, C. W. Hsu, Y. Igarashi, L. Lu, I. Kaminer, A. Pick, S. L. Chua, J. D. Joannopoulos, and M. Soljačić, “Spawning rings of exceptional points out of Dirac cones,” Nature 525, 354–358 (2015).
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Hu, S. F.

Y. Wu, B. Zhu, S. F. Hu, Z. Zhou, and H. H. Zhong, “Floquet control of the gain and loss in a PT-symmetric optical coupler,” Front. Phys. 12, 121102 (2017).
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Hua, S.

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photon. 8, 524–529 (2014).
[Crossref]

Huang, J.

X. Luo, J. Huang, H. Zhong, X. Qin, Q. Xie, Y. S. Kivshar, and C. Lee, “Pseudo-parity-time symmetry in optical systems,” Phys. Rev. Lett. 110, 243902 (2013).
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Igarashi, Y.

B. Zhen, C. W. Hsu, Y. Igarashi, L. Lu, I. Kaminer, A. Pick, S. L. Chua, J. D. Joannopoulos, and M. Soljačić, “Spawning rings of exceptional points out of Dirac cones,” Nature 525, 354–358 (2015).
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Jiang, L.

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photon. 8, 524–529 (2014).
[Crossref]

Jiang, X.

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photon. 8, 524–529 (2014).
[Crossref]

Joannopoulos, J. D.

B. Zhen, C. W. Hsu, Y. Igarashi, L. Lu, I. Kaminer, A. Pick, S. L. Chua, J. D. Joannopoulos, and M. Soljačić, “Spawning rings of exceptional points out of Dirac cones,” Nature 525, 354–358 (2015).
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J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University, 2008).

Joglekar, Y. N.

Y. N. Joglekar, R. Marathe, P. Durganandini, and R. K. Pathak, “PT spectroscopy of the Rabi problem,” Phys. Rev. A 90, 040101 (2014).
[Crossref]

Johnson, S. G.

Kaminer, I.

B. Zhen, C. W. Hsu, Y. Igarashi, L. Lu, I. Kaminer, A. Pick, S. L. Chua, J. D. Joannopoulos, and M. Soljačić, “Spawning rings of exceptional points out of Dirac cones,” Nature 525, 354–358 (2015).
[Crossref] [PubMed]

Kang, M.

Khajavikhan, M.

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

H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. N. Christodoulides, and M. Khajavikhan, “Enhanced sensitivity at higher-order exceptional points,” Nature 548, 187–191 (2017).
[Crossref] [PubMed]

H. Hodaei, M. A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346, 975–978 (2014).
[Crossref] [PubMed]

Khurgin, J. B.

Kip, D.

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, 192–195 (2010).
[Crossref]

Kivshar, Y. S.

X. Luo, J. Huang, H. Zhong, X. Qin, Q. Xie, Y. S. Kivshar, and C. Lee, “Pseudo-parity-time symmetry in optical systems,” Phys. Rev. Lett. 110, 243902 (2013).
[Crossref] [PubMed]

Klaiman, S.

S. Klaiman, U. Gunther, and N. Moiseyev, “Visualization of branch points in PT-symmetric waveguides,” Phys. Rev. Lett. 101, 080402 (2008).
[Crossref]

Kottos, T.

M. Chitsazi, H. Li, F. M. Ellis, and T. Kottos, “Experimental realization of Floquet PT-symmetric systems,” Phys. Rev. Lett. 119, 093901 (2017).
[Crossref]

Kuhl, U.

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, 76–80 (2016).
[Crossref] [PubMed]

Lawrence, M.

M. Lawrence, N. Xu, X. Zhang, L. Cong, J. Han, W. Zhang, and S. Zhang, “Manifestation of PT symmetry breaking in polarization space with terahertz metasurfaces,” Phys. Rev. Lett. 113, 093901 (2014).
[Crossref]

Lee, C.

X. Luo, J. Huang, H. Zhong, X. Qin, Q. Xie, Y. S. Kivshar, and C. Lee, “Pseudo-parity-time symmetry in optical systems,” Phys. Rev. Lett. 110, 243902 (2013).
[Crossref] [PubMed]

Lei, F.

B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
[Crossref]

Li, G.

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photon. 8, 524–529 (2014).
[Crossref]

Li, H.

M. Chitsazi, H. Li, F. M. Ellis, and T. Kottos, “Experimental realization of Floquet PT-symmetric systems,” Phys. Rev. Lett. 119, 093901 (2017).
[Crossref]

Li, J.

J. Gear, F. Liu, S. T. Chu, S. Rotter, and J. Li, “Parity-time symmetry from stacking purely dielectric and magnetic slabs,” Phys. Rev. A 91, 033825 (2015).
[Crossref]

Li, T. F.

Liang, S.

T. Liu, X. Zhu, F. Chen, S. Liang, and J. Zhu, “Unidirectional wave vector manipulation in two-dimensional space with an all passive acoustic parity-time-symmetric metamaterials crystal,” Phys. Rev. Lett. 120, 124502 (2018).
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Libisch, F.

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, 76–80 (2016).
[Crossref] [PubMed]

Lin, Z.

Z. Lin, A. Pick, M. Loncar, and A. W. Rodriguez, “Enhanced spontaneous emission at third-order Dirac exceptional points in inverse-designed photonic crystals,” Phys. Rev. Lett. 117, 107402 (2016).
[Crossref] [PubMed]

Liu, F.

J. Gear, F. Liu, S. T. Chu, S. Rotter, and J. Li, “Parity-time symmetry from stacking purely dielectric and magnetic slabs,” Phys. Rev. A 91, 033825 (2015).
[Crossref]

Liu, T.

T. Liu, X. Zhu, F. Chen, S. Liang, and J. Zhu, “Unidirectional wave vector manipulation in two-dimensional space with an all passive acoustic parity-time-symmetric metamaterials crystal,” Phys. Rev. Lett. 120, 124502 (2018).
[Crossref] [PubMed]

Liu, Y.

W. Wang, L. Q. Wang, R. D. Xue, H. L. Chen, R. P. Guo, Y. Liu, and J. Chen, “Unidirectional excitation of radiative-loss-free surface plasmon polaritons in PT-symmetric systems,” Phys. Rev. Lett. 119, 077401 (2017).
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H. Cang, Y. Liu, Y. Wang, X. Yin, and X. Zhang, “Giant suppression of photobleaching for single molecule detection via the Purcell effect,” Nano Letters 13, 5949–5953 (2013).
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Loncar, M.

Z. Lin, A. Pick, M. Loncar, and A. W. Rodriguez, “Enhanced spontaneous emission at third-order Dirac exceptional points in inverse-designed photonic crystals,” Phys. Rev. Lett. 117, 107402 (2016).
[Crossref] [PubMed]

Long, G. L.

B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
[Crossref]

Longhi, S.

G. D. Valle and S. Longhi, “Spectral and transport properties of time-periodic PT-symmetric tight-binding lattices,” Phys. Rev. A 87, 022119 (2013).
[Crossref]

S. Longhi, “PT-symmetric laser absorber,” Phys. Rev. A 82, 031801 (2010).
[Crossref]

Lu, L.

B. Zhen, C. W. Hsu, Y. Igarashi, L. Lu, I. Kaminer, A. Pick, S. L. Chua, J. D. Joannopoulos, and M. Soljačić, “Spawning rings of exceptional points out of Dirac cones,” Nature 525, 354–358 (2015).
[Crossref] [PubMed]

Lu, M. H.

L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. B. Oliveira, V. R. Almeida, Y. F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12, 108–113 (2013).
[Crossref]

Luo, X.

X. Luo, J. Huang, H. Zhong, X. Qin, Q. Xie, Y. S. Kivshar, and C. Lee, “Pseudo-parity-time symmetry in optical systems,” Phys. Rev. Lett. 110, 243902 (2013).
[Crossref] [PubMed]

Mailybaev, A. A.

T. Goldzak, A. A. Mailybaev, and N. Moiseyev, “Light stops at exceptional points,” Phys. Rev. Lett. 120, 013901 (2018).
[Crossref] [PubMed]

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, 76–80 (2016).
[Crossref] [PubMed]

Makris, K. G.

R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, and D. N. Christodoulides, “Non-Hermitian physics and P T symmetry,” Nat. Phys. 14, 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, 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, 103904 (2008).
[Crossref]

Marathe, R.

Y. N. Joglekar, R. Marathe, P. Durganandini, and R. K. Pathak, “PT spectroscopy of the Rabi problem,” Phys. Rev. A 90, 040101 (2014).
[Crossref]

Meade, R. D.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University, 2008).

Milburn, T. 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, 76–80 (2016).
[Crossref] [PubMed]

Miller, O. D.

Miri, M. A.

H. Hodaei, M. A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346, 975–978 (2014).
[Crossref] [PubMed]

Moiseyev, N.

T. Goldzak, A. A. Mailybaev, and N. Moiseyev, “Light stops at exceptional points,” Phys. Rev. Lett. 120, 013901 (2018).
[Crossref] [PubMed]

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, 76–80 (2016).
[Crossref] [PubMed]

S. Klaiman, U. Gunther, and N. Moiseyev, “Visualization of branch points in PT-symmetric waveguides,” Phys. Rev. Lett. 101, 080402 (2008).
[Crossref]

Monifi, F.

B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
[Crossref]

Morandotti, R.

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, 093902 (2009).
[Crossref]

Musslimani, Z. H.

R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, and D. N. Christodoulides, “Non-Hermitian physics and P T symmetry,” Nat. Phys. 14, 11–19 (2018).
[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, 103904 (2008).
[Crossref]

Noh, H.

W. Wan, Y. Chong, L. Ge, H. Noh, A. D. Stone, and H. Cao, “Time-reversed lasing and interferometric control of absorption,” Science 331, 889–892 (2011).
[Crossref] [PubMed]

Nori, F.

B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
[Crossref]

Oliveira, J. E. B.

L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. B. Oliveira, V. R. Almeida, Y. F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12, 108–113 (2013).
[Crossref]

Ozdemir, S. K.

B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
[Crossref]

Pathak, R. K.

Y. N. Joglekar, R. Marathe, P. Durganandini, and R. K. Pathak, “PT spectroscopy of the Rabi problem,” Phys. Rev. A 90, 040101 (2014).
[Crossref]

Peng, B.

B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
[Crossref]

Peng, Y. G.

Pick, A.

A. Pick, B. Zhen, O. D. Miller, C. W. Hsu, F. Hernandez, A. W. Rodriguez, M. Soljačić, and S. G. Johnson, “General theory of spontaneous emission near exceptional points,” Opt. Express 25, 12325–12348 (2017).
[Crossref] [PubMed]

Z. Lin, A. Pick, M. Loncar, and A. W. Rodriguez, “Enhanced spontaneous emission at third-order Dirac exceptional points in inverse-designed photonic crystals,” Phys. Rev. Lett. 117, 107402 (2016).
[Crossref] [PubMed]

B. Zhen, C. W. Hsu, Y. Igarashi, L. Lu, I. Kaminer, A. Pick, S. L. Chua, J. D. Joannopoulos, and M. Soljačić, “Spawning rings of exceptional points out of Dirac cones,” Nature 525, 354–358 (2015).
[Crossref] [PubMed]

Purcell, E. M.

E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 681 (1946).

Qin, X.

X. Luo, J. Huang, H. Zhong, X. Qin, Q. Xie, Y. S. Kivshar, and C. Lee, “Pseudo-parity-time symmetry in optical systems,” Phys. Rev. Lett. 110, 243902 (2013).
[Crossref] [PubMed]

Rabl, P.

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, 76–80 (2016).
[Crossref] [PubMed]

Ramezani, H.

X. F. Zhu, H. Ramezani, C. Shi, J. Zhu, and X. Zhang, “PT-symmetric acoustics,” Phys. Rev. X 4, 031042 (2014).

Richter, A.

C. Dembowski, B. Dietz, H. D. Graf, H. L. Harney, A. Heine, W. D. Heiss, and A. Richter, “Encircling an exceptional point,” Phys. Rev. E 69, 056216 (2004).
[Crossref]

Rodriguez, A. W.

A. Pick, B. Zhen, O. D. Miller, C. W. Hsu, F. Hernandez, A. W. Rodriguez, M. Soljačić, and S. G. Johnson, “General theory of spontaneous emission near exceptional points,” Opt. Express 25, 12325–12348 (2017).
[Crossref] [PubMed]

Z. Lin, A. Pick, M. Loncar, and A. W. Rodriguez, “Enhanced spontaneous emission at third-order Dirac exceptional points in inverse-designed photonic crystals,” Phys. Rev. Lett. 117, 107402 (2016).
[Crossref] [PubMed]

Rotter, S.

R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, and D. N. Christodoulides, “Non-Hermitian physics and P T symmetry,” Nat. Phys. 14, 11–19 (2018).
[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, 76–80 (2016).
[Crossref] [PubMed]

J. Gear, F. Liu, S. T. Chu, S. Rotter, and J. Li, “Parity-time symmetry from stacking purely dielectric and magnetic slabs,” Phys. Rev. A 91, 033825 (2015).
[Crossref]

Rüter, C. E.

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, 192–195 (2010).
[Crossref]

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, 093902 (2009).
[Crossref]

Scherer, A.

L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. B. Oliveira, V. R. Almeida, Y. F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12, 108–113 (2013).
[Crossref]

Segev, M.

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, 192–195 (2010).
[Crossref]

Shi, C.

X. F. Zhu, H. Ramezani, C. Shi, J. Zhu, and X. Zhang, “PT-symmetric acoustics,” Phys. Rev. X 4, 031042 (2014).

Shu, F. J.

F. J. Shu, C. L. Zou, X. B. Zou, and L. Yang, “Chiral symmetry breaking in a microring optical cavity by engineered dissipation,” Phys. Rev. A 94, 013848 (2016).
[Crossref]

Siviloglou, G. 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, 093902 (2009).
[Crossref]

Soljacic, M.

A. Pick, B. Zhen, O. D. Miller, C. W. Hsu, F. Hernandez, A. W. Rodriguez, M. Soljačić, and S. G. Johnson, “General theory of spontaneous emission near exceptional points,” Opt. Express 25, 12325–12348 (2017).
[Crossref] [PubMed]

B. Zhen, C. W. Hsu, Y. Igarashi, L. Lu, I. Kaminer, A. Pick, S. L. Chua, J. D. Joannopoulos, and M. Soljačić, “Spawning rings of exceptional points out of Dirac cones,” Nature 525, 354–358 (2015).
[Crossref] [PubMed]

Stone, A. D.

L. Ge and A. D. Stone, “Parity-time symmetry breaking beyond one dimension: the role of degeneracy,” Phys. Rev. X 4, 031011 (2014).

W. Wan, Y. Chong, L. Ge, H. Noh, A. D. Stone, and H. Cao, “Time-reversed lasing and interferometric control of absorption,” Science 331, 889–892 (2011).
[Crossref] [PubMed]

Valle, G. D.

G. D. Valle and S. Longhi, “Spectral and transport properties of time-periodic PT-symmetric tight-binding lattices,” Phys. Rev. A 87, 022119 (2013).
[Crossref]

Volatier-Ravat, M.

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, 093902 (2009).
[Crossref]

Wan, W.

W. Wan, Y. Chong, L. Ge, H. Noh, A. D. Stone, and H. Cao, “Time-reversed lasing and interferometric control of absorption,” Science 331, 889–892 (2011).
[Crossref] [PubMed]

Wang, G.

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photon. 8, 524–529 (2014).
[Crossref]

Wang, L. Q.

L. Q. Wang, R. D. Xue, W. Wang, R. X. Wang, R. P. Guo, and J. Chen, “Coherent-trapped helical mode in parity-time symmetric metamaterials,” Opt. Express 25, 15231–15240 (2017).
[Crossref] [PubMed]

W. Wang, L. Q. Wang, R. D. Xue, H. L. Chen, R. P. Guo, Y. Liu, and J. Chen, “Unidirectional excitation of radiative-loss-free surface plasmon polaritons in PT-symmetric systems,” Phys. Rev. Lett. 119, 077401 (2017).
[Crossref]

Wang, R. X.

Wang, W.

L. Q. Wang, R. D. Xue, W. Wang, R. X. Wang, R. P. Guo, and J. Chen, “Coherent-trapped helical mode in parity-time symmetric metamaterials,” Opt. Express 25, 15231–15240 (2017).
[Crossref] [PubMed]

W. Wang, L. Q. Wang, R. D. Xue, H. L. Chen, R. P. Guo, Y. Liu, and J. Chen, “Unidirectional excitation of radiative-loss-free surface plasmon polaritons in PT-symmetric systems,” Phys. Rev. Lett. 119, 077401 (2017).
[Crossref]

Wang, Y.

H. Cang, Y. Liu, Y. Wang, X. Yin, and X. Zhang, “Giant suppression of photobleaching for single molecule detection via the Purcell effect,” Nano Letters 13, 5949–5953 (2013).
[Crossref] [PubMed]

Wen, J.

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photon. 8, 524–529 (2014).
[Crossref]

Wiersig, J.

H. Cao and J. Wiersig, “Dielectric microcavities: model systems for wave chaos and non-Hermitian physics,” Rev. Mod. Phys. 87, 61–111 (2015).
[Crossref]

Winn, J. N.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University, 2008).

Wittek, S.

H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. N. Christodoulides, and M. Khajavikhan, “Enhanced sensitivity at higher-order exceptional points,” Nature 548, 187–191 (2017).
[Crossref] [PubMed]

Wu, Y.

Y. Wu, B. Zhu, S. F. Hu, Z. Zhou, and H. H. Zhong, “Floquet control of the gain and loss in a PT-symmetric optical coupler,” Front. Phys. 12, 121102 (2017).
[Crossref]

Xiao, M.

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photon. 8, 524–529 (2014).
[Crossref]

Xie, Q.

X. Luo, J. Huang, H. Zhong, X. Qin, Q. Xie, Y. S. Kivshar, and C. Lee, “Pseudo-parity-time symmetry in optical systems,” Phys. Rev. Lett. 110, 243902 (2013).
[Crossref] [PubMed]

Xu, N.

M. Lawrence, N. Xu, X. Zhang, L. Cong, J. Han, W. Zhang, and S. Zhang, “Manifestation of PT symmetry breaking in polarization space with terahertz metasurfaces,” Phys. Rev. Lett. 113, 093901 (2014).
[Crossref]

Xu, Y. L.

L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. B. Oliveira, V. R. Almeida, Y. F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12, 108–113 (2013).
[Crossref]

Xue, R. D.

W. Wang, L. Q. Wang, R. D. Xue, H. L. Chen, R. P. Guo, Y. Liu, and J. Chen, “Unidirectional excitation of radiative-loss-free surface plasmon polaritons in PT-symmetric systems,” Phys. Rev. Lett. 119, 077401 (2017).
[Crossref]

L. Q. Wang, R. D. Xue, W. Wang, R. X. Wang, R. P. Guo, and J. Chen, “Coherent-trapped helical mode in parity-time symmetric metamaterials,” Opt. Express 25, 15231–15240 (2017).
[Crossref] [PubMed]

Yang, C.

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photon. 8, 524–529 (2014).
[Crossref]

Yang, L.

F. J. Shu, C. L. Zou, X. B. Zou, and L. Yang, “Chiral symmetry breaking in a microring optical cavity by engineered dissipation,” Phys. Rev. A 94, 013848 (2016).
[Crossref]

B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
[Crossref]

Yang, M.

Yin, X.

H. Cang, Y. Liu, Y. Wang, X. Yin, and X. Zhang, “Giant suppression of photobleaching for single molecule detection via the Purcell effect,” Nano Letters 13, 5949–5953 (2013).
[Crossref] [PubMed]

Zhang, S.

M. Lawrence, N. Xu, X. Zhang, L. Cong, J. Han, W. Zhang, and S. Zhang, “Manifestation of PT symmetry breaking in polarization space with terahertz metasurfaces,” Phys. Rev. Lett. 113, 093901 (2014).
[Crossref]

Zhang, W.

M. Lawrence, N. Xu, X. Zhang, L. Cong, J. Han, W. Zhang, and S. Zhang, “Manifestation of PT symmetry breaking in polarization space with terahertz metasurfaces,” Phys. Rev. Lett. 113, 093901 (2014).
[Crossref]

Zhang, X.

M. Lawrence, N. Xu, X. Zhang, L. Cong, J. Han, W. Zhang, and S. Zhang, “Manifestation of PT symmetry breaking in polarization space with terahertz metasurfaces,” Phys. Rev. Lett. 113, 093901 (2014).
[Crossref]

X. F. Zhu, H. Ramezani, C. Shi, J. Zhu, and X. Zhang, “PT-symmetric acoustics,” Phys. Rev. X 4, 031042 (2014).

H. Cang, Y. Liu, Y. Wang, X. Yin, and X. Zhang, “Giant suppression of photobleaching for single molecule detection via the Purcell effect,” Nano Letters 13, 5949–5953 (2013).
[Crossref] [PubMed]

Zhao, D. G.

Zhen, B.

A. Pick, B. Zhen, O. D. Miller, C. W. Hsu, F. Hernandez, A. W. Rodriguez, M. Soljačić, and S. G. Johnson, “General theory of spontaneous emission near exceptional points,” Opt. Express 25, 12325–12348 (2017).
[Crossref] [PubMed]

B. Zhen, C. W. Hsu, Y. Igarashi, L. Lu, I. Kaminer, A. Pick, S. L. Chua, J. D. Joannopoulos, and M. Soljačić, “Spawning rings of exceptional points out of Dirac cones,” Nature 525, 354–358 (2015).
[Crossref] [PubMed]

Zhong, H.

X. Luo, J. Huang, H. Zhong, X. Qin, Q. Xie, Y. S. Kivshar, and C. Lee, “Pseudo-parity-time symmetry in optical systems,” Phys. Rev. Lett. 110, 243902 (2013).
[Crossref] [PubMed]

Zhong, H. H.

Y. Wu, B. Zhu, S. F. Hu, Z. Zhou, and H. H. Zhong, “Floquet control of the gain and loss in a PT-symmetric optical coupler,” Front. Phys. 12, 121102 (2017).
[Crossref]

Zhou, Z.

Y. Wu, B. Zhu, S. F. Hu, Z. Zhou, and H. H. Zhong, “Floquet control of the gain and loss in a PT-symmetric optical coupler,” Front. Phys. 12, 121102 (2017).
[Crossref]

Zhu, B.

Y. Wu, B. Zhu, S. F. Hu, Z. Zhou, and H. H. Zhong, “Floquet control of the gain and loss in a PT-symmetric optical coupler,” Front. Phys. 12, 121102 (2017).
[Crossref]

Zhu, J.

T. Liu, X. Zhu, F. Chen, S. Liang, and J. Zhu, “Unidirectional wave vector manipulation in two-dimensional space with an all passive acoustic parity-time-symmetric metamaterials crystal,” Phys. Rev. Lett. 120, 124502 (2018).
[Crossref] [PubMed]

X. F. Zhu, H. Ramezani, C. Shi, J. Zhu, and X. Zhang, “PT-symmetric acoustics,” Phys. Rev. X 4, 031042 (2014).

Zhu, X.

T. Liu, X. Zhu, F. Chen, S. Liang, and J. Zhu, “Unidirectional wave vector manipulation in two-dimensional space with an all passive acoustic parity-time-symmetric metamaterials crystal,” Phys. Rev. Lett. 120, 124502 (2018).
[Crossref] [PubMed]

Zhu, X. F.

Zou, C. L.

F. J. Shu, C. L. Zou, X. B. Zou, and L. Yang, “Chiral symmetry breaking in a microring optical cavity by engineered dissipation,” Phys. Rev. A 94, 013848 (2016).
[Crossref]

Zou, X. B.

F. J. Shu, C. L. Zou, X. B. Zou, and L. Yang, “Chiral symmetry breaking in a microring optical cavity by engineered dissipation,” Phys. Rev. A 94, 013848 (2016).
[Crossref]

Adv. Opt. Photon. (1)

Front. Phys. (1)

Y. Wu, B. Zhu, S. F. Hu, Z. Zhou, and H. H. Zhong, “Floquet control of the gain and loss in a PT-symmetric optical coupler,” Front. Phys. 12, 121102 (2017).
[Crossref]

Nano Letters (1)

H. Cang, Y. Liu, Y. Wang, X. Yin, and X. Zhang, “Giant suppression of photobleaching for single molecule detection via the Purcell effect,” Nano Letters 13, 5949–5953 (2013).
[Crossref] [PubMed]

Nat. Mater. (1)

L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. B. Oliveira, V. R. Almeida, Y. F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12, 108–113 (2013).
[Crossref]

Nat. Photo. (1)

L. Feng, R. El-Ganainy, and L. Ge, “Non-Hermitian photonics based on parity-time symmetry,” Nat. Photo. 11, 752–762 (2017).
[Crossref]

Nat. Photon. (1)

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photon. 8, 524–529 (2014).
[Crossref]

Nat. Phys. (3)

B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
[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, 192–195 (2010).
[Crossref]

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

Nature (3)

B. Zhen, C. W. Hsu, Y. Igarashi, L. Lu, I. Kaminer, A. Pick, S. L. Chua, J. D. Joannopoulos, and M. Soljačić, “Spawning rings of exceptional points out of Dirac cones,” Nature 525, 354–358 (2015).
[Crossref] [PubMed]

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, 76–80 (2016).
[Crossref] [PubMed]

H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. N. Christodoulides, and M. Khajavikhan, “Enhanced sensitivity at higher-order exceptional points,” Nature 548, 187–191 (2017).
[Crossref] [PubMed]

Opt. Express (5)

Phys. Rev. (1)

E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 681 (1946).

Phys. Rev. A (5)

G. D. Valle and S. Longhi, “Spectral and transport properties of time-periodic PT-symmetric tight-binding lattices,” Phys. Rev. A 87, 022119 (2013).
[Crossref]

Y. N. Joglekar, R. Marathe, P. Durganandini, and R. K. Pathak, “PT spectroscopy of the Rabi problem,” Phys. Rev. A 90, 040101 (2014).
[Crossref]

J. Gear, F. Liu, S. T. Chu, S. Rotter, and J. Li, “Parity-time symmetry from stacking purely dielectric and magnetic slabs,” Phys. Rev. A 91, 033825 (2015).
[Crossref]

S. Longhi, “PT-symmetric laser absorber,” Phys. Rev. A 82, 031801 (2010).
[Crossref]

F. J. Shu, C. L. Zou, X. B. Zou, and L. Yang, “Chiral symmetry breaking in a microring optical cavity by engineered dissipation,” Phys. Rev. A 94, 013848 (2016).
[Crossref]

Phys. Rev. E (1)

C. Dembowski, B. Dietz, H. D. Graf, H. L. Harney, A. Heine, W. D. Heiss, and A. Richter, “Encircling an exceptional point,” Phys. Rev. E 69, 056216 (2004).
[Crossref]

Phys. Rev. Lett. (11)

M. Chitsazi, H. Li, F. M. Ellis, and T. Kottos, “Experimental realization of Floquet PT-symmetric systems,” Phys. Rev. Lett. 119, 093901 (2017).
[Crossref]

T. Liu, X. Zhu, F. Chen, S. Liang, and J. Zhu, “Unidirectional wave vector manipulation in two-dimensional space with an all passive acoustic parity-time-symmetric metamaterials crystal,” Phys. Rev. Lett. 120, 124502 (2018).
[Crossref] [PubMed]

W. Wang, L. Q. Wang, R. D. Xue, H. L. Chen, R. P. Guo, Y. Liu, and J. Chen, “Unidirectional excitation of radiative-loss-free surface plasmon polaritons in PT-symmetric systems,” Phys. Rev. Lett. 119, 077401 (2017).
[Crossref]

Z. Lin, A. Pick, M. Loncar, and A. W. Rodriguez, “Enhanced spontaneous emission at third-order Dirac exceptional points in inverse-designed photonic crystals,” Phys. Rev. Lett. 117, 107402 (2016).
[Crossref] [PubMed]

T. Goldzak, A. A. Mailybaev, and N. Moiseyev, “Light stops at exceptional points,” Phys. Rev. Lett. 120, 013901 (2018).
[Crossref] [PubMed]

M. Lawrence, N. Xu, X. Zhang, L. Cong, J. Han, W. Zhang, and S. Zhang, “Manifestation of PT symmetry breaking in polarization space with terahertz metasurfaces,” Phys. Rev. Lett. 113, 093901 (2014).
[Crossref]

X. Luo, J. Huang, H. Zhong, X. Qin, Q. Xie, Y. S. Kivshar, and C. Lee, “Pseudo-parity-time symmetry in optical systems,” Phys. Rev. Lett. 110, 243902 (2013).
[Crossref] [PubMed]

C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80, 5243–5246 (1998).
[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, 103904 (2008).
[Crossref]

S. Klaiman, U. Gunther, and N. Moiseyev, “Visualization of branch points in PT-symmetric waveguides,” Phys. Rev. Lett. 101, 080402 (2008).
[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, 093902 (2009).
[Crossref]

Phys. Rev. X (2)

L. Ge and A. D. Stone, “Parity-time symmetry breaking beyond one dimension: the role of degeneracy,” Phys. Rev. X 4, 031011 (2014).

X. F. Zhu, H. Ramezani, C. Shi, J. Zhu, and X. Zhang, “PT-symmetric acoustics,” Phys. Rev. X 4, 031042 (2014).

Rev. Mod. Phys. (1)

H. Cao and J. Wiersig, “Dielectric microcavities: model systems for wave chaos and non-Hermitian physics,” Rev. Mod. Phys. 87, 61–111 (2015).
[Crossref]

Science (2)

H. Hodaei, M. A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346, 975–978 (2014).
[Crossref] [PubMed]

W. Wan, Y. Chong, L. Ge, H. Noh, A. D. Stone, and H. Cao, “Time-reversed lasing and interferometric control of absorption,” Science 331, 889–892 (2011).
[Crossref] [PubMed]

Other (1)

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University, 2008).

Supplementary Material (1)

NameDescription
» Visualization 1       Variation of field versus distance d0 (from COMSOL simulation). When d0=0.537lambda_0 an EP is achieved and the field becomes very strong.

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Figures (8)

Fig. 1
Fig. 1 Two WGs are curved periodically in order to modulate the distance d(x) and the mutual coupling strength δ(x) between them. WG1 (WG2) has a gain (loss) coefficient of g. EP is realized locally at the critical distance dc where δ = g. At the region of d < dc (filled by light green) the local P T-phase is unbroken because δ > g. At the region of d > dc (filled by light yellow) the local P T-phase is broken. The whole structure is globally P T symmetric with respect to the line between the two WGs.
Fig. 2
Fig. 2 A 3D schematic of the (a) real and (b) imaginary parts of β P T versus wavevector k and δ0 about the BZ edge Λ/2. Parameters are β0 = Λ, α = 0.04Λ, δ1 = 0.001Λ2, and v0 = 0.25. Value of δ0 varies from 0.018Λ2 to 0.022Λ2.
Fig. 3
Fig. 3 Dispersions of β P T near the BZ edge when δ0 equals to (a) 0.0215Λ2, (b) 0.0200Λ2 and (c) 0.0185Λ2, respectively. Usually four curves can be obtained. At the BZ edge either real or complex solutions are possible. Positions of EPs are labeled out by red dots.
Fig. 4
Fig. 4 Field distributions of the eigenstates near the BZ edge when δ0 = 0.020Λ2 and k = 0.49Λ. Values of β P T on the top raw are real, as (a) 0.9902Λ and (b) 1.0056Λ, respectively. The lower two plots have complex β P T of (c) (1.0021-i0.0041)Λ and (d) (1.0021+i0.0041)Λ, respectively. In the plot a schematic distribution of δ(x) is provided to help understand the relative distribution of field in the curved geometry.
Fig. 5
Fig. 5 (a) Real and (b) imaginary parts of β P T when δ0 = 0.021Λ2 (δ0δ1 = geff), and (c) the distribution of fields at EP.
Fig. 6
Fig. 6 (a) Real and (b) imaginary parts of β P T when δ0 = 0.019Λ2 (δ0 + δ1 = geff), and (c) the distribution of fields at EP.
Fig. 7
Fig. 7 (a) Schematic of the structure design in the COMSOL simulation. Width of the supercell is 2a. Two magnetic currents of opposite directions are put into WG1 and displaced by a to enforce the excitation of modes at the BZ edge. Distribution of field Hz indeed varies with d0. Examples can be seen in (b) d0 = 0.516λ0 and (c) d0 = 0.607λ0, respectively.
Fig. 8
Fig. 8 Distribution of (a,c) field amplitude and (b,d) phase (unit π) of Hz at EPs, when (a,b) d0 = 0.537λ0 and (c,d) d0 = 0.587λ0, respectively. Visualization 1 shows the variation of field versus distance d0.

Equations (16)

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( ω 0 i g δ δ ω 0 + i g ) ( E 1 E 2 ) = ω P T ( E 1 E 2 ) ,
ω P T ± = ω 0 ± δ 2 g 2
d ( x ) = d 0 2 d 1 cos ( Λ x ) ,
δ ( x ) = δ 0 + 2 δ 1 cos ( Λ x ) .
δ 0 2 δ 1 < g < δ 0 + 2 δ 1 .
( M k Λ N N M k ) ( E k Λ E k ) = β P T 2 ( E k Λ E k ) ,
M k ( β k 2 i α | k | δ 0 δ 0 β k 2 + i α | k | ) ,
N = ( 0 δ 1 δ 1 0 ) .
β k = β 0 + v 0 ( k Λ 2 ) ,
g e f f = α Λ 2 .
β P T 2 = β 0 2 ± ( δ 0 ± δ 1 ) 2 g e f f 2
{ g e f f < δ 0 δ 1 , Case ( i ) δ 0 δ 1 < g e f f < δ 0 + δ 1 , Case ( ii ) δ 0 + δ 1 < g e f f , Case ( iii )
δ 0 δ 1 = g e f f ,
Ψ E P ( 1 ) = 1 2 [ 1 , i , 1 , i ] T .
δ 0 + δ 1 = g e f f ,
Ψ E P ( 2 ) = 1 2 [ 1 , i , 1 , i ] T .

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