Abstract

In a single magneto-optical (MO) waveguide, the dispersion of guided bulk wave is reciprocal in the Voigt configuration. Here we show that the parity-time (PT) phase in two coupled MO waveguides can be nonreciprocal if the waveguides are properly biased. The nonreciprocal PT phase is closely related to the asymmetric field profile induced by the MO effect that modifies the coupling strength between adjacent waveguides. We show that it is feasible to switch between broken and conserved PT phases by simply reversing the magnetic bias or the propagating direction of wave. Theoretical analysis and numerical calculation prove our theory. This investigation highlights a flexible method in manipulating the field dynamics of waveguide arrays by using the novel properties of PT phase especially the exceptional points.

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

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References

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

2019 (4)

M.-A. Miri and A. Alù, “Exceptional points in optics and photonics,” Science 363, eaar7709 (2019).
[Crossref] [PubMed]

A. Y. Song, Y. Shi, Q. Lin, and S. Fan, “Direction-dependent parity-time phase transition and nonreciprocal amplification with dynamic gain-loss modulation,” Phys. Rev. A 99, 013824 (2019).
[Crossref]

Y. R. Zhang, Z. Z. Zhang, J. Q. Yuan, M. Kang, and J. Chen, “High-order exceptional points in non-Hermitian Moire lattices,” Front. Physics 14, 53603 (2019).
[Crossref]

Y. Li and C. Argyropoulos, “Exceptional points and spectral singularities in active epsilon-near-zero plasmonic waveguides,” Phys. Rev. B 99, 075413 (2019).
[Crossref]

2018 (5)

S. A. H. Gangaraj and F. Monticone, “Topological waveguiding near an exceptional point: defect-immune, slow-light, and loss-immune propagation,” Phys. Rev. Lett. 121, 093901 (2018).
[Crossref]

S. A. H. Gangaraj and F. Monticone, “Coupled topological surface modes in gyrotropic structures: Green’s function analysis,” IEEE Ant. Wireless Propag. Lett. 17, 1993–1997 (2018).
[Crossref]

Y. R. Zhang, Z. Z. Zhang, J. Q. Yuan, W. Wang, L. Q. Wang, Z. X. Li, R. D. Xue, and J. Chen, “Parity-time symmetry in periodically curved optical waveguides,” Opt. Express 26, 027141 (2018).
[Crossref]

T. Goldzak, A. A. Mailybaev, and N. Moiseyev, “Light stops at exceptional points,” Phys. Rev. Lett. 120, 013901 (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, 11–19 (2018).
[Crossref]

2017 (8)

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

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]

D. G. Baranov, A. Krasnok, T. Shegai, A. Alù, and Y. D. Chong, “Coherent perfect absorbers: linear control of light with light,” Nat. Rev. Mater. 2, 17064 (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]

W. Chen, S. K. Ozdemir, G. Zhao, J. Wiersig, and L. Yang, “Exceptional points enhanced sensing in an optical microcavity,” Nature 548, 192–196 (2017).
[Crossref] [PubMed]

S. Assawaworrarit, X. Yu, and S. Fan, “Robust wireless power transfer using a nonlinear parity-time-symmetric circuit,” Nature 546, 387–390 (2017).
[Crossref] [PubMed]

I. Liberal and N. Engheta, "Near-zero refractive index photonics,” Nat. Photo. 11, 149–158 (2017).
[Crossref]

A. Pick, B. Zhen, O. D. Miller, C. W. Hsu, F. Hernandez, A. W. Rodriguez, M. Soljacic, and S. G. Johnson, “General theory of spontaneous emission near exceptional points,” Opt. Express 25, 012325 (2017).
[Crossref]

2016 (5)

R. Thomas, H. Li, F. M. Ellis, and T. Kottos, “Giant nonreciprocity near exceptional-point degeneracies,” Phys. Rev. A 94, 043829 (2016).
[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]

L. Ge and R. El-Ganainy, “Nonlinear modal interactions in parity-time (PT) symmetric lasers,” Sci. Rep. 6, 24889 (2016).
[Crossref]

X. Zhou and Y. D. Chong, “PT symmetry breaking and nonlinear optical isolation in coupled microcavities,” Opt. Express 24, 6916–6930 (2016).
[Crossref] [PubMed]

V. V. Konotop, J. Yang, and D. A. Zezyulin, “Nonlinear waves in PT-symmetric systems,” Rev. Mod. Phys. 88, 035002 (2016).
[Crossref]

2015 (5)

L. Shen, Y. You, Z. Wang, and X. Deng, “Backscattering-immune one-way surface magnetoplasmons at terahertz frequencies,” Opt. Express 23, 950–962 (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]

J. B. Khurgin, “How to deal with the loss in plasmonics and metamaterials,” Nat. Nanotech. 10, 2–6 (2015).
[Crossref]

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

R. P. Guo, L. T. Wu, M. Yang, T. J. Guo, H. X. Cui, X. W. Cao, and J. Chen, “Nonreciprocal propagating electromagnetic modes without phase gradients,” Phys. Rev. A 91, 023808 (2015).
[Crossref]

2014 (3)

A. A. Zyablovsky, A. P. Vinogradov, A. V. Dorofeenko, A. A. Pukhov, and A. A. Lisyansky, “Causality and phase transitions in PT-symmetric optical systems,” Phys. Rev. A 89, 033808 (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]

F. Nazari, N. Bender, H. Ramezani, M. K. Moravvej-Farshi, D. N. Christodoulides, and T. Kottos, “Optical isolation via PT-symmetric nonlinear Fano resonances,” Opt. Express 22, 9574 (2014).
[Crossref] [PubMed]

2013 (3)

T. F. Li, T. J. Guo, H. X. Cui, M. Yang, M. Kang, Q. H. Guo, and J. Chen, “Guided modes in magneto-optical waveguides and the role in resonant transmission,” Opt. Express 21, 9563–9572 (2013).
[Crossref] [PubMed]

Y. Lumer, Y. Plotnik, M. C. Rechtsman, and M. Segev, “Nonlinearly induced PT transition in photonic systems,” Phys. Rev. Lett. 111, 263901 (2013).
[Crossref]

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]

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 (2)

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

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, 093902 (2009).
[Crossref]

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

2008 (3)

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

W. D. Heiss, “Chirality of wavefunctions for three coalescing levels,” J. Phys. A 41, 244010 (2008).
[Crossref]

B. Edwards, A. Alù, M. E. Young, M. Silveirinha, and N. Engheta, “Experimental verification of epsilon-near-zero metamaterial coupling and energy squeezing using a microwave waveguide,” Phys. Rev. Lett. 100, 033903 (2008).
[Crossref] [PubMed]

2007 (2)

2003 (1)

A. B. Yakovlev and G. W. Hanson, “Fundamental modal phenomena on isotropic and anisotropic planar slab dielectric waveguides,” IEEE Trans. Antennas Propag. 51, 888–897 (2003).
[Crossref]

1999 (1)

G. W. Hanson and A. B. Yakovlev, “Investigation of mode interaction on planar dielectric waveguides with loss and gain,” Radio Sci. 34, 1349–1359 (1999).
[Crossref]

1998 (2)

G. W. Hanson and A. B. Yakovlev, “An analysis of leaky wave dispersion phenomena in the vicinity of cutoff using complex frequency plane singularities,” Radio Sci. 33, 803–819 (1998).
[Crossref]

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

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]

Alù, A.

M.-A. Miri and A. Alù, “Exceptional points in optics and photonics,” Science 363, eaar7709 (2019).
[Crossref] [PubMed]

D. G. Baranov, A. Krasnok, T. Shegai, A. Alù, and Y. D. Chong, “Coherent perfect absorbers: linear control of light with light,” Nat. Rev. Mater. 2, 17064 (2017).
[Crossref]

B. Edwards, A. Alù, M. E. Young, M. Silveirinha, and N. Engheta, “Experimental verification of epsilon-near-zero metamaterial coupling and energy squeezing using a microwave waveguide,” Phys. Rev. Lett. 100, 033903 (2008).
[Crossref] [PubMed]

Argyropoulos, C.

Y. Li and C. Argyropoulos, “Exceptional points and spectral singularities in active epsilon-near-zero plasmonic waveguides,” Phys. Rev. B 99, 075413 (2019).
[Crossref]

Assawaworrarit, S.

S. Assawaworrarit, X. Yu, and S. Fan, “Robust wireless power transfer using a nonlinear parity-time-symmetric circuit,” Nature 546, 387–390 (2017).
[Crossref] [PubMed]

Baranov, D. G.

D. G. Baranov, A. Krasnok, T. Shegai, A. Alù, and Y. D. Chong, “Coherent perfect absorbers: linear control of light with light,” Nat. Rev. Mater. 2, 17064 (2017).
[Crossref]

Bender, C. M.

C. M. Bender, “Making sense of non-Hermitian Hamiltonians,” Rep. Prog. Phys. 70, 947–1018 (2007).
[Crossref]

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

Bender, N.

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]

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.

R. P. Guo, L. T. Wu, M. Yang, T. J. Guo, H. X. Cui, X. W. Cao, and J. Chen, “Nonreciprocal propagating electromagnetic modes without phase gradients,” Phys. Rev. A 91, 023808 (2015).
[Crossref]

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.

Y. R. Zhang, Z. Z. Zhang, J. Q. Yuan, M. Kang, and J. Chen, “High-order exceptional points in non-Hermitian Moire lattices,” Front. Physics 14, 53603 (2019).
[Crossref]

Y. R. Zhang, Z. Z. Zhang, J. Q. Yuan, W. Wang, L. Q. Wang, Z. X. Li, R. D. Xue, and J. Chen, “Parity-time symmetry in periodically curved optical waveguides,” Opt. Express 26, 027141 (2018).
[Crossref]

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]

R. P. Guo, L. T. Wu, M. Yang, T. J. Guo, H. X. Cui, X. W. Cao, and J. Chen, “Nonreciprocal propagating electromagnetic modes without phase gradients,” Phys. Rev. A 91, 023808 (2015).
[Crossref]

T. F. Li, T. J. Guo, H. X. Cui, M. Yang, M. Kang, Q. H. Guo, and J. Chen, “Guided modes in magneto-optical waveguides and the role in resonant transmission,” Opt. Express 21, 9563–9572 (2013).
[Crossref] [PubMed]

Chen, W.

W. Chen, S. K. Ozdemir, G. Zhao, J. Wiersig, and L. Yang, “Exceptional points enhanced sensing in an optical microcavity,” Nature 548, 192–196 (2017).
[Crossref] [PubMed]

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]

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]

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[Crossref]

Shegai, T.

D. G. Baranov, A. Krasnok, T. Shegai, A. Alù, and Y. D. Chong, “Coherent perfect absorbers: linear control of light with light,” Nat. Rev. Mater. 2, 17064 (2017).
[Crossref]

Shen, L.

Shi, Y.

A. Y. Song, Y. Shi, Q. Lin, and S. Fan, “Direction-dependent parity-time phase transition and nonreciprocal amplification with dynamic gain-loss modulation,” Phys. Rev. A 99, 013824 (2019).
[Crossref]

Silveirinha, M.

B. Edwards, A. Alù, M. E. Young, M. Silveirinha, and N. Engheta, “Experimental verification of epsilon-near-zero metamaterial coupling and energy squeezing using a microwave waveguide,” Phys. Rev. Lett. 100, 033903 (2008).
[Crossref] [PubMed]

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. Soljacic, and S. G. Johnson, “General theory of spontaneous emission near exceptional points,” Opt. Express 25, 012325 (2017).
[Crossref]

Song, A. Y.

A. Y. Song, Y. Shi, Q. Lin, and S. Fan, “Direction-dependent parity-time phase transition and nonreciprocal amplification with dynamic gain-loss modulation,” Phys. Rev. A 99, 013824 (2019).
[Crossref]

Stone, A. D.

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]

Thomas, R.

R. Thomas, H. Li, F. M. Ellis, and T. Kottos, “Giant nonreciprocity near exceptional-point degeneracies,” Phys. Rev. A 94, 043829 (2016).
[Crossref]

Vinogradov, A. P.

A. A. Zyablovsky, A. P. Vinogradov, A. V. Dorofeenko, A. A. Pukhov, and A. A. Lisyansky, “Causality and phase transitions in PT-symmetric optical systems,” Phys. Rev. A 89, 033808 (2014).
[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, L. Q.

Y. R. Zhang, Z. Z. Zhang, J. Q. Yuan, W. Wang, L. Q. Wang, Z. X. Li, R. D. Xue, and J. Chen, “Parity-time symmetry in periodically curved optical waveguides,” Opt. Express 26, 027141 (2018).
[Crossref]

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, W.

Y. R. Zhang, Z. Z. Zhang, J. Q. Yuan, W. Wang, L. Q. Wang, Z. X. Li, R. D. Xue, and J. Chen, “Parity-time symmetry in periodically curved optical waveguides,” Opt. Express 26, 027141 (2018).
[Crossref]

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, Z.

Wiersig, J.

W. Chen, S. K. Ozdemir, G. Zhao, J. Wiersig, and L. Yang, “Exceptional points enhanced sensing in an optical microcavity,” Nature 548, 192–196 (2017).
[Crossref] [PubMed]

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

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).
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Wu, L. T.

R. P. Guo, L. T. Wu, M. Yang, T. J. Guo, H. X. Cui, X. W. Cao, and J. Chen, “Nonreciprocal propagating electromagnetic modes without phase gradients,” Phys. Rev. A 91, 023808 (2015).
[Crossref]

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).
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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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Xue, R. D.

Y. R. Zhang, Z. Z. Zhang, J. Q. Yuan, W. Wang, L. Q. Wang, Z. X. Li, R. D. Xue, and J. Chen, “Parity-time symmetry in periodically curved optical waveguides,” Opt. Express 26, 027141 (2018).
[Crossref]

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]

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A. B. Yakovlev and G. W. Hanson, “Fundamental modal phenomena on isotropic and anisotropic planar slab dielectric waveguides,” IEEE Trans. Antennas Propag. 51, 888–897 (2003).
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G. W. Hanson and A. B. Yakovlev, “Investigation of mode interaction on planar dielectric waveguides with loss and gain,” Radio Sci. 34, 1349–1359 (1999).
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G. W. Hanson and A. B. Yakovlev, “An analysis of leaky wave dispersion phenomena in the vicinity of cutoff using complex frequency plane singularities,” Radio Sci. 33, 803–819 (1998).
[Crossref]

Yang, J.

V. V. Konotop, J. Yang, and D. A. Zezyulin, “Nonlinear waves in PT-symmetric systems,” Rev. Mod. Phys. 88, 035002 (2016).
[Crossref]

Yang, L.

W. Chen, S. K. Ozdemir, G. Zhao, J. Wiersig, and L. Yang, “Exceptional points enhanced sensing in an optical microcavity,” Nature 548, 192–196 (2017).
[Crossref] [PubMed]

Yang, M.

R. P. Guo, L. T. Wu, M. Yang, T. J. Guo, H. X. Cui, X. W. Cao, and J. Chen, “Nonreciprocal propagating electromagnetic modes without phase gradients,” Phys. Rev. A 91, 023808 (2015).
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T. F. Li, T. J. Guo, H. X. Cui, M. Yang, M. Kang, Q. H. Guo, and J. Chen, “Guided modes in magneto-optical waveguides and the role in resonant transmission,” Opt. Express 21, 9563–9572 (2013).
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B. Edwards, A. Alù, M. E. Young, M. Silveirinha, and N. Engheta, “Experimental verification of epsilon-near-zero metamaterial coupling and energy squeezing using a microwave waveguide,” Phys. Rev. Lett. 100, 033903 (2008).
[Crossref] [PubMed]

Yu, X.

S. Assawaworrarit, X. Yu, and S. Fan, “Robust wireless power transfer using a nonlinear parity-time-symmetric circuit,” Nature 546, 387–390 (2017).
[Crossref] [PubMed]

Yuan, J. Q.

Y. R. Zhang, Z. Z. Zhang, J. Q. Yuan, M. Kang, and J. Chen, “High-order exceptional points in non-Hermitian Moire lattices,” Front. Physics 14, 53603 (2019).
[Crossref]

Y. R. Zhang, Z. Z. Zhang, J. Q. Yuan, W. Wang, L. Q. Wang, Z. X. Li, R. D. Xue, and J. Chen, “Parity-time symmetry in periodically curved optical waveguides,” Opt. Express 26, 027141 (2018).
[Crossref]

Zezyulin, D. A.

V. V. Konotop, J. Yang, and D. A. Zezyulin, “Nonlinear waves in PT-symmetric systems,” Rev. Mod. Phys. 88, 035002 (2016).
[Crossref]

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]

Zhang, Y. R.

Y. R. Zhang, Z. Z. Zhang, J. Q. Yuan, M. Kang, and J. Chen, “High-order exceptional points in non-Hermitian Moire lattices,” Front. Physics 14, 53603 (2019).
[Crossref]

Y. R. Zhang, Z. Z. Zhang, J. Q. Yuan, W. Wang, L. Q. Wang, Z. X. Li, R. D. Xue, and J. Chen, “Parity-time symmetry in periodically curved optical waveguides,” Opt. Express 26, 027141 (2018).
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Zhang, Z. Z.

Y. R. Zhang, Z. Z. Zhang, J. Q. Yuan, M. Kang, and J. Chen, “High-order exceptional points in non-Hermitian Moire lattices,” Front. Physics 14, 53603 (2019).
[Crossref]

Y. R. Zhang, Z. Z. Zhang, J. Q. Yuan, W. Wang, L. Q. Wang, Z. X. Li, R. D. Xue, and J. Chen, “Parity-time symmetry in periodically curved optical waveguides,” Opt. Express 26, 027141 (2018).
[Crossref]

Zhao, G.

W. Chen, S. K. Ozdemir, G. Zhao, J. Wiersig, and L. Yang, “Exceptional points enhanced sensing in an optical microcavity,” Nature 548, 192–196 (2017).
[Crossref] [PubMed]

Zhen, B.

A. Pick, B. Zhen, O. D. Miller, C. W. Hsu, F. Hernandez, A. W. Rodriguez, M. Soljacic, and S. G. Johnson, “General theory of spontaneous emission near exceptional points,” Opt. Express 25, 012325 (2017).
[Crossref]

Zhou, X.

Zyablovsky, A. A.

A. A. Zyablovsky, A. P. Vinogradov, A. V. Dorofeenko, A. A. Pukhov, and A. A. Lisyansky, “Causality and phase transitions in PT-symmetric optical systems,” Phys. Rev. A 89, 033808 (2014).
[Crossref]

Front. Physics (1)

Y. R. Zhang, Z. Z. Zhang, J. Q. Yuan, M. Kang, and J. Chen, “High-order exceptional points in non-Hermitian Moire lattices,” Front. Physics 14, 53603 (2019).
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W. D. Heiss, “Chirality of wavefunctions for three coalescing levels,” J. Phys. A 41, 244010 (2008).
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Nat. Mater. (1)

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J. B. Khurgin, “How to deal with the loss in plasmonics and metamaterials,” Nat. Nanotech. 10, 2–6 (2015).
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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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Nat. Phys. (2)

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, 11–19 (2018).
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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, 192–195 (2010).
[Crossref]

Nat. Rev. Mater. (1)

D. G. Baranov, A. Krasnok, T. Shegai, A. Alù, and Y. D. Chong, “Coherent perfect absorbers: linear control of light with light,” Nat. Rev. Mater. 2, 17064 (2017).
[Crossref]

Nature (3)

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]

W. Chen, S. K. Ozdemir, G. Zhao, J. Wiersig, and L. Yang, “Exceptional points enhanced sensing in an optical microcavity,” Nature 548, 192–196 (2017).
[Crossref] [PubMed]

S. Assawaworrarit, X. Yu, and S. Fan, “Robust wireless power transfer using a nonlinear parity-time-symmetric circuit,” Nature 546, 387–390 (2017).
[Crossref] [PubMed]

Opt. Express (6)

Opt. Lett. (1)

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A. A. Zyablovsky, A. P. Vinogradov, A. V. Dorofeenko, A. A. Pukhov, and A. A. Lisyansky, “Causality and phase transitions in PT-symmetric optical systems,” Phys. Rev. A 89, 033808 (2014).
[Crossref]

A. Y. Song, Y. Shi, Q. Lin, and S. Fan, “Direction-dependent parity-time phase transition and nonreciprocal amplification with dynamic gain-loss modulation,” Phys. Rev. A 99, 013824 (2019).
[Crossref]

R. P. Guo, L. T. Wu, M. Yang, T. J. Guo, H. X. Cui, X. W. Cao, and J. Chen, “Nonreciprocal propagating electromagnetic modes without phase gradients,” Phys. Rev. A 91, 023808 (2015).
[Crossref]

R. Thomas, H. Li, F. M. Ellis, and T. Kottos, “Giant nonreciprocity near exceptional-point degeneracies,” Phys. Rev. A 94, 043829 (2016).
[Crossref]

Phys. Rev. B (1)

Y. Li and C. Argyropoulos, “Exceptional points and spectral singularities in active epsilon-near-zero plasmonic waveguides,” Phys. Rev. B 99, 075413 (2019).
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Phys. Rev. Lett. (10)

B. Edwards, A. Alù, M. E. Young, M. Silveirinha, and N. Engheta, “Experimental verification of epsilon-near-zero metamaterial coupling and energy squeezing using a microwave waveguide,” Phys. Rev. Lett. 100, 033903 (2008).
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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).
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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).
[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]

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]

Radio Sci. (2)

G. W. Hanson and A. B. Yakovlev, “Investigation of mode interaction on planar dielectric waveguides with loss and gain,” Radio Sci. 34, 1349–1359 (1999).
[Crossref]

G. W. Hanson and A. B. Yakovlev, “An analysis of leaky wave dispersion phenomena in the vicinity of cutoff using complex frequency plane singularities,” Radio Sci. 33, 803–819 (1998).
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C. M. Bender, “Making sense of non-Hermitian Hamiltonians,” Rep. Prog. Phys. 70, 947–1018 (2007).
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Rev. Mod. Phys. (2)

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

V. V. Konotop, J. Yang, and D. A. Zezyulin, “Nonlinear waves in PT-symmetric systems,” Rev. Mod. Phys. 88, 035002 (2016).
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Sci. Rep. (1)

L. Ge and R. El-Ganainy, “Nonlinear modal interactions in parity-time (PT) symmetric lasers,” Sci. Rep. 6, 24889 (2016).
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Science (2)

M.-A. Miri and A. Alù, “Exceptional points in optics and photonics,” Science 363, eaar7709 (2019).
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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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Figures (7)

Fig. 1
Fig. 1 A schematic of the structure under investigation and the corresponding simple mechanism responsible to the nonreciprocal P T phase. (a) Type-I configuration in which the forward propagating fields in the two MO waveguides shift toward each other, and the effective coupling rate increases to κ+, greater than κ at B = 0. (b) Type-II configuration, that when incident from the backward direction the opposite effect in the field profile takes place, and the coupling rate κ is weaker than κ. Note that the Type-II configuration can be also achieved from the Type-I configuration by simply reversing the bias fields.
Fig. 2
Fig. 2 Dispersions (k0 = ω/c,k) of the guided bulk mode when γ1 = −γ2 = −1, ε1,2 = 4±j0.5, a = 0.3 mm, and b = 3 mm. EPs are labeled out by red stars.
Fig. 3
Fig. 3 Variation of Ω versus k0 when k = 0.785 mm−1 (blue line) and k = −0.785 mm−1 (red line), respectively. In each case two eigenmodes characterized by dips can be observed.
Fig. 4
Fig. 4 Distributions of Hy (dark) and phase (red, in the unit of π) in the structure at the four eigenmodes of Fig. 3.
Fig. 5
Fig. 5 Variation of Ω versus the distance a and wavevector k at a fixed angular frequency of ω = 2πc/λ0 with λ0 = 8 mm. EPs are labeled out by red stars.
Fig. 6
Fig. 6 Variation of Ω versus k when a = 0.655 mm. Inset shows the distributions of field and phase for the EP in the Type-II configuration. Here δ = 0.1 is assumed.
Fig. 7
Fig. 7 COMSOL simulation of the field inside the structure with λ0 = 8 mm and a = 0.655 mm. (a) For a forward propagation the P T phase is conserved and the field switches periodically between the two adjacent MO waveguides. (b) For a backward propagation the field is distributed uniformly in the two waveguides because an EP is reached.

Equations (12)

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ε ¯ ¯ 1 , 2 = ε 0 ( ε 1 , 2 0 j γ 1 , 2 0 ε | | 0 j γ 1 , 2 0 ε 1 , 2 ) ,
γ 1 = γ 2 .
ε 1 , 2 = ε 0 ( ε m ± j δ ) .
H y = e j k z + j ω t { H 1 e α ( x a b ) , x > a + b H 2 e j β 1 ( x a ) + H 3 e + j β 1 ( x a ) , a + b > x > a H 4 e α x + H 5 e + α x , + a > x > a H 6 e j β 2 ( x + a ) + H 7 e + j β 2 ( x + a ) , a > x > a b H 8 e + α ( x + a + b ) , x < a b
α 2 = k 2 k 0 2 ,
β 1 , 2 2 = ε 1 , 2 2 γ 1 , 2 2 ε 1 , 2 k 0 2 k 2 ,
( 1 A A 1 0 0 0 0 0 0 1 1 B B 1 0 0 0 0 0 0 B 1 B 1 1 0 0 0 0 0 0 C 1 C 1 1 M 1 a A M 1 b A 1 0 0 0 0 0 0 M 1 a M 1 b B B 1 0 0 0 0 0 0 B 1 B M 2 a M 2 b 0 0 0 0 0 0 M 2 a C 1 M 2 b C 1 ) ( H 1 H 2 H 3 H 4 H 5 H 6 H 7 H 8 ) = 0 ,
M 1 , 2 a = γ 1 , 2 k + j β 1 , 2 ε 1 , 2 α ( ε 1 , 2 2 γ 1 , 2 2 ) ,
M 1 , 2 b = γ 1 , 2 k j β 1 , 2 ε 1 , 2 α ( ε 1 , 2 2 γ 1 , 2 2 ) .
( ω 0 + j g κ κ ω 0 j g ) ( ψ 1 ψ 2 ) = ω P T ( ψ 1 ψ 2 ) .
ω P T ± = ω 0 ± κ 2 g 2 ,
Ω = log   | det   { M } |

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