Abstract

The exceptional point (EP), at which the relevant eigenvalues and eigenstates are simultaneously identical, typically exists in non-Hermitian systems with parity-time (PT) symmetric complex potentials, and gives rise to many intriguing behaviors in various physical realms. In this work, we explore the complex band structure of one-dimensional “polariton crystals” that can be constructed in waveguide-resonator coupled systems, with PT-symmetric potential. Analysis based on the transfer matrix and the coupled mode theory shows that the complex band structure is intimately determined by the interaction between the Bragg resonance and the polariton one, the gain/loss coefficients, in addition to the coupling strength. A miniband is induced due to the interaction of these two resonances, which is a defect-like band and appears quite different for the band structure evolution. Furthermore, PT-symmetric phase transition occurs in the momentum space for certain amounts of non-Hermiticity. As the non-Hermiticity increases, the EP formed in the original polariton gap approaches another EP formed at the touch point of the folded Bragg bands (where the thresholdless transition occurs). Then they coalesce at a specific non-Hermiticity, and finally disappear. Subsequently, the transmission spectra of such polariton crystals show intriguing phenomena induced by the EPs. Our results provide a different perspective to understand PT-symmetric polariton crystals and may find applications in gain/loss induced lasing by ‘polaritons’.

© 2017 Optical Society of America

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References

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2017 (1)

2016 (3)

Z. Lin, A. Pick, M. Lončar, and A. W. Rodriguez, “Enhanced spontaneous emission at third-order Dirac exceptional points in inverse-designed photonic crystals,” Phys. Rev. Lett. 117(10), 107402 (2016).
[PubMed]

Z. Liu, Q. Zhang, X. Liu, Y. Yao, and J. J. Xiao, “Absence of exceptional points in square waveguide arrays with apparently balanced gain and loss,” Sci. Rep. 6, 22711 (2016).
[PubMed]

K. Ding, G. Ma, M. Xiao, Z. Q. Zhang, and C. T. Chan, “Emergence, coalescence, and topological properties of multiple exceptional points and their experimental realization,” Phys. Rev. X 6, 021007 (2016).

2015 (4)

K. Ding, Z. Q. Zhang, and C. T. Chan, “Coalescence of exceptional points and phase diagrams for one-dimensional P T-symmetric photonic crystals,” Phys. Rev. B 92, 235310 (2015).

Y. Zhao, C. Qian, K. Qiu, Y. Gao, and X. Xu, “Ultrafast optical switching using photonic molecules in photonic crystal waveguides,” Opt. Express 23(7), 9211–9220 (2015).
[PubMed]

H. Benisty, A. Lupu, and A. Degiron, “Transverse periodic PT symmetry for modal demultiplexing in optical waveguides,” Phys. Rev. A 91, 053825 (2015).

Z. Z. Liu, Q. Zhang, and J. J. Xiao, “EIT-like transmission by interaction between multiple Bragg scattering and local plasmonic resonances,” J. Opt. 18, 015005 (2015).

2014 (10)

K. V. Shanthi and S. Robinson, “Two-dimensional photonic crystal based sensor for pressure sensing,” Photonic Sensors 4, 248 (2014).

B. Peng, S. K. Özdemir, 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 (2014).

R. Fleury, D. L. Sounas, and A. Alù, “Negative refraction and planar focusing based on parity-time symmetric metasurfaces,” Phys. Rev. Lett. 113(2), 023903 (2014).
[PubMed]

H. Alaeian and J. A. Dionne, “Non-Hermitian nanophotonic and plasmonic waveguides,” Phys. Rev. B 89, 075136 (2014).

B. Peng, S. K. Özdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, F. Nori, and L. Yang, “Loss-induced suppression and revival of lasing,” Science 346(6207), 328–332 (2014).
[PubMed]

B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Nonreciprocal light transmission in parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394 (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. Photonics 8, 524 (2014).

L. Feng, Z. J. Wong, R.-M. Ma, Y. Wang, and X. Zhang, “Single-mode laser by parity-time symmetry breaking,” Science 346(6212), 972–975 (2014).
[PubMed]

H. Hodaei, M. A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346(6212), 975–978 (2014).
[PubMed]

Y. Sun, W. Tan, H. Q. Li, J. Li, and H. Chen, “Experimental demonstration of a coherent perfect absorber with PT phase transition,” Phys. Rev. Lett. 112(14), 143903 (2014).
[PubMed]

2013 (1)

M. Kang, F. Liu, and J. Li, “Effective spontaneous PT-symmetry breaking in hybridized metamaterials,” Phys. Rev. A 87, 053824 (2013).

2012 (2)

W. D. Heiss, “The physics of exceptional points,” J. Phys. A Math. Theor. 45, 444016 (2012).

S. V. Suchkov, S. V. Dmitriev, B. A. Malomed, and Y. S. Kivshar, “Wave scattering on a domain wall in a chain of PT-symmetric couplers,” Phys. Rev. A 85, 033825 (2012).

2011 (1)

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106(21), 213901 (2011).
[PubMed]

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, 192 (2010).

2009 (2)

I. J. Rotter, “Non-Hermitian Hamilton operator and the physics of open quantum systems,” Phys. A-Math. Theor. 42, 153001 (2009).

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).
[PubMed]

2008 (3)

J. J. Xiao and K. W. Yu, “Light trapping and releasing in side-coupled microresonator structures with asymmetric cavity–waveguide coupling,” Opt. Commun. 281, 4023 (2008).

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100(3), 030402 (2008).
[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).
[PubMed]

2007 (1)

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

2003 (1)

Y. Akahane, T. Asano, B.-S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature 425(6961), 944–947 (2003).
[PubMed]

2002 (1)

J. Vucković, M. Lončar, H. Mabuchi, and A. Scherer, “Design of photonic crystal microcavities for cavity QED,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 65(1 Pt 2), 016608 (2002).
[PubMed]

2000 (1)

W. D. Heiss, “Repulsion of resonance states and exceptional points,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 61(1), 929–932 (2000).
[PubMed]

1998 (1)

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

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).
[PubMed]

Akahane, Y.

Y. Akahane, T. Asano, B.-S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature 425(6961), 944–947 (2003).
[PubMed]

Alaeian, H.

H. Alaeian and J. A. Dionne, “Non-Hermitian nanophotonic and plasmonic waveguides,” Phys. Rev. B 89, 075136 (2014).

Alù, A.

R. Fleury, D. L. Sounas, and A. Alù, “Negative refraction and planar focusing based on parity-time symmetric metasurfaces,” Phys. Rev. Lett. 113(2), 023903 (2014).
[PubMed]

Asano, T.

Y. Akahane, T. Asano, B.-S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature 425(6961), 944–947 (2003).
[PubMed]

Bender, C. M.

B. Peng, S. K. Özdemir, 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 (2014).

B. Peng, S. K. Özdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, F. Nori, and L. Yang, “Loss-induced suppression and revival of lasing,” Science 346(6207), 328–332 (2014).
[PubMed]

B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Nonreciprocal light transmission in parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394 (2014).

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

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

Benisty, H.

H. Benisty, A. Lupu, and A. Degiron, “Transverse periodic PT symmetry for modal demultiplexing in optical waveguides,” Phys. Rev. A 91, 053825 (2015).

Boettcher, S.

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

Cao, H.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106(21), 213901 (2011).
[PubMed]

Chan, C. T.

K. Ding, G. Ma, M. Xiao, Z. Q. Zhang, and C. T. Chan, “Emergence, coalescence, and topological properties of multiple exceptional points and their experimental realization,” Phys. Rev. X 6, 021007 (2016).

K. Ding, Z. Q. Zhang, and C. T. Chan, “Coalescence of exceptional points and phase diagrams for one-dimensional P T-symmetric photonic crystals,” Phys. Rev. B 92, 235310 (2015).

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. Photonics 8, 524 (2014).

Chen, H.

Y. Sun, W. Tan, H. Q. Li, J. Li, and H. Chen, “Experimental demonstration of a coherent perfect absorber with PT phase transition,” Phys. Rev. Lett. 112(14), 143903 (2014).
[PubMed]

Chen, Y.

Christodoulides, D. N.

H. Hodaei, M. A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346(6212), 975–978 (2014).
[PubMed]

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106(21), 213901 (2011).
[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 (2010).

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).
[PubMed]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100(3), 030402 (2008).
[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).
[PubMed]

Degiron, A.

H. Benisty, A. Lupu, and A. Degiron, “Transverse periodic PT symmetry for modal demultiplexing in optical waveguides,” Phys. Rev. A 91, 053825 (2015).

Ding, K.

K. Ding, G. Ma, M. Xiao, Z. Q. Zhang, and C. T. Chan, “Emergence, coalescence, and topological properties of multiple exceptional points and their experimental realization,” Phys. Rev. X 6, 021007 (2016).

K. Ding, Z. Q. Zhang, and C. T. Chan, “Coalescence of exceptional points and phase diagrams for one-dimensional P T-symmetric photonic crystals,” Phys. Rev. B 92, 235310 (2015).

Dionne, J. A.

H. Alaeian and J. A. Dionne, “Non-Hermitian nanophotonic and plasmonic waveguides,” Phys. Rev. B 89, 075136 (2014).

Dmitriev, S. V.

S. V. Suchkov, S. V. Dmitriev, B. A. Malomed, and Y. S. Kivshar, “Wave scattering on a domain wall in a chain of PT-symmetric couplers,” Phys. Rev. A 85, 033825 (2012).

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).
[PubMed]

Eichelkraut, T.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106(21), 213901 (2011).
[PubMed]

El-Ganainy, R.

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 (2010).

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100(3), 030402 (2008).
[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).
[PubMed]

Fan, S.

B. Peng, S. K. Özdemir, 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 (2014).

B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Nonreciprocal light transmission in parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394 (2014).

Feng, L.

L. Feng, Z. J. Wong, R.-M. Ma, Y. Wang, and X. Zhang, “Single-mode laser by parity-time symmetry breaking,” Science 346(6212), 972–975 (2014).
[PubMed]

Fleury, R.

R. Fleury, D. L. Sounas, and A. Alù, “Negative refraction and planar focusing based on parity-time symmetric metasurfaces,” Phys. Rev. Lett. 113(2), 023903 (2014).
[PubMed]

Gao, Y.

Gianfreda, M.

B. Peng, S. K. Özdemir, 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 (2014).

B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Nonreciprocal light transmission in parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394 (2014).

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).
[PubMed]

Heinrich, M.

H. Hodaei, M. A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346(6212), 975–978 (2014).
[PubMed]

Heiss, W. D.

W. D. Heiss, “The physics of exceptional points,” J. Phys. A Math. Theor. 45, 444016 (2012).

W. D. Heiss, “Repulsion of resonance states and exceptional points,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 61(1), 929–932 (2000).
[PubMed]

Hodaei, H.

H. Hodaei, M. A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346(6212), 975–978 (2014).
[PubMed]

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. Photonics 8, 524 (2014).

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. Photonics 8, 524 (2014).

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. Photonics 8, 524 (2014).

Kang, M.

M. Kang, F. Liu, and J. Li, “Effective spontaneous PT-symmetry breaking in hybridized metamaterials,” Phys. Rev. A 87, 053824 (2013).

Khajavikhan, M.

H. Hodaei, M. A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346(6212), 975–978 (2014).
[PubMed]

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 (2010).

Kivshar, Y. S.

S. V. Suchkov, S. V. Dmitriev, B. A. Malomed, and Y. S. Kivshar, “Wave scattering on a domain wall in a chain of PT-symmetric couplers,” Phys. Rev. A 85, 033825 (2012).

Kottos, T.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106(21), 213901 (2011).
[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, “Nonreciprocal light transmission in parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394 (2014).

B. Peng, S. K. Özdemir, 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 (2014).

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. Photonics 8, 524 (2014).

Li, H. Q.

Y. Sun, W. Tan, H. Q. Li, J. Li, and H. Chen, “Experimental demonstration of a coherent perfect absorber with PT phase transition,” Phys. Rev. Lett. 112(14), 143903 (2014).
[PubMed]

Li, J.

Y. Sun, W. Tan, H. Q. Li, J. Li, and H. Chen, “Experimental demonstration of a coherent perfect absorber with PT phase transition,” Phys. Rev. Lett. 112(14), 143903 (2014).
[PubMed]

M. Kang, F. Liu, and J. Li, “Effective spontaneous PT-symmetry breaking in hybridized metamaterials,” Phys. Rev. A 87, 053824 (2013).

Liertzer, M.

B. Peng, S. K. Özdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, F. Nori, and L. Yang, “Loss-induced suppression and revival of lasing,” Science 346(6207), 328–332 (2014).
[PubMed]

Lin, Z.

Z. Lin, A. Pick, M. Lončar, and A. W. Rodriguez, “Enhanced spontaneous emission at third-order Dirac exceptional points in inverse-designed photonic crystals,” Phys. Rev. Lett. 117(10), 107402 (2016).
[PubMed]

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106(21), 213901 (2011).
[PubMed]

Liu, F.

M. Kang, F. Liu, and J. Li, “Effective spontaneous PT-symmetry breaking in hybridized metamaterials,” Phys. Rev. A 87, 053824 (2013).

Liu, X.

Z. Liu, Q. Zhang, X. Liu, Y. Yao, and J. J. Xiao, “Absence of exceptional points in square waveguide arrays with apparently balanced gain and loss,” Sci. Rep. 6, 22711 (2016).
[PubMed]

Liu, Z.

Z. Liu, Q. Zhang, X. Liu, Y. Yao, and J. J. Xiao, “Absence of exceptional points in square waveguide arrays with apparently balanced gain and loss,” Sci. Rep. 6, 22711 (2016).
[PubMed]

Liu, Z. Z.

Z. Z. Liu, Q. Zhang, Y. Chen, and J. J. Xiao, “General coupled-mode analysis of a geometrically symmetric waveguide array with nonuniform gain and loss,” Photon. Res. 5, 57 (2017).

Z. Z. Liu, Q. Zhang, and J. J. Xiao, “EIT-like transmission by interaction between multiple Bragg scattering and local plasmonic resonances,” J. Opt. 18, 015005 (2015).

Loncar, M.

Z. Lin, A. Pick, M. Lončar, and A. W. Rodriguez, “Enhanced spontaneous emission at third-order Dirac exceptional points in inverse-designed photonic crystals,” Phys. Rev. Lett. 117(10), 107402 (2016).
[PubMed]

J. Vucković, M. Lončar, H. Mabuchi, and A. Scherer, “Design of photonic crystal microcavities for cavity QED,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 65(1 Pt 2), 016608 (2002).
[PubMed]

Long, G. L.

B. Peng, S. K. Özdemir, 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 (2014).

B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Nonreciprocal light transmission in parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394 (2014).

Lupu, A.

H. Benisty, A. Lupu, and A. Degiron, “Transverse periodic PT symmetry for modal demultiplexing in optical waveguides,” Phys. Rev. A 91, 053825 (2015).

Ma, G.

K. Ding, G. Ma, M. Xiao, Z. Q. Zhang, and C. T. Chan, “Emergence, coalescence, and topological properties of multiple exceptional points and their experimental realization,” Phys. Rev. X 6, 021007 (2016).

Ma, R.-M.

L. Feng, Z. J. Wong, R.-M. Ma, Y. Wang, and X. Zhang, “Single-mode laser by parity-time symmetry breaking,” Science 346(6212), 972–975 (2014).
[PubMed]

Mabuchi, H.

J. Vucković, M. Lončar, H. Mabuchi, and A. Scherer, “Design of photonic crystal microcavities for cavity QED,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 65(1 Pt 2), 016608 (2002).
[PubMed]

Makris, K. G.

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 (2010).

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).
[PubMed]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100(3), 030402 (2008).
[PubMed]

Malomed, B. A.

S. V. Suchkov, S. V. Dmitriev, B. A. Malomed, and Y. S. Kivshar, “Wave scattering on a domain wall in a chain of PT-symmetric couplers,” Phys. Rev. A 85, 033825 (2012).

Miri, M. A.

H. Hodaei, M. A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346(6212), 975–978 (2014).
[PubMed]

Monifi, F.

B. Peng, S. K. Özdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, F. Nori, and L. Yang, “Loss-induced suppression and revival of lasing,” Science 346(6207), 328–332 (2014).
[PubMed]

B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Nonreciprocal light transmission in parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394 (2014).

B. Peng, S. K. Özdemir, 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 (2014).

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(9), 093902 (2009).
[PubMed]

Musslimani, Z. H.

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100(3), 030402 (2008).
[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).
[PubMed]

Noda, S.

Y. Akahane, T. Asano, B.-S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature 425(6961), 944–947 (2003).
[PubMed]

Nori, F.

B. Peng, S. K. Özdemir, 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 (2014).

B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Nonreciprocal light transmission in parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394 (2014).

B. Peng, S. K. Özdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, F. Nori, and L. Yang, “Loss-induced suppression and revival of lasing,” Science 346(6207), 328–332 (2014).
[PubMed]

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, “Nonreciprocal light transmission in parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394 (2014).

Özdemir, S. K.

B. Peng, S. K. Özdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, F. Nori, and L. Yang, “Loss-induced suppression and revival of lasing,” Science 346(6207), 328–332 (2014).
[PubMed]

B. Peng, S. K. Özdemir, 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 (2014).

Peng, B.

B. Peng, S. K. Özdemir, 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 (2014).

B. Peng, S. K. Özdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, F. Nori, and L. Yang, “Loss-induced suppression and revival of lasing,” Science 346(6207), 328–332 (2014).
[PubMed]

B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Nonreciprocal light transmission in parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394 (2014).

Pick, A.

Z. Lin, A. Pick, M. Lončar, and A. W. Rodriguez, “Enhanced spontaneous emission at third-order Dirac exceptional points in inverse-designed photonic crystals,” Phys. Rev. Lett. 117(10), 107402 (2016).
[PubMed]

Qian, C.

Qiu, K.

Ramezani, H.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106(21), 213901 (2011).
[PubMed]

Robinson, S.

K. V. Shanthi and S. Robinson, “Two-dimensional photonic crystal based sensor for pressure sensing,” Photonic Sensors 4, 248 (2014).

Rodriguez, A. W.

Z. Lin, A. Pick, M. Lončar, and A. W. Rodriguez, “Enhanced spontaneous emission at third-order Dirac exceptional points in inverse-designed photonic crystals,” Phys. Rev. Lett. 117(10), 107402 (2016).
[PubMed]

Rotter, I. J.

I. J. Rotter, “Non-Hermitian Hamilton operator and the physics of open quantum systems,” Phys. A-Math. Theor. 42, 153001 (2009).

Rotter, S.

B. Peng, S. K. Özdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, F. Nori, and L. Yang, “Loss-induced suppression and revival of lasing,” Science 346(6207), 328–332 (2014).
[PubMed]

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 (2010).

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).
[PubMed]

Scherer, A.

J. Vucković, M. Lončar, H. Mabuchi, and A. Scherer, “Design of photonic crystal microcavities for cavity QED,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 65(1 Pt 2), 016608 (2002).
[PubMed]

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 (2010).

Shanthi, K. V.

K. V. Shanthi and S. Robinson, “Two-dimensional photonic crystal based sensor for pressure sensing,” Photonic Sensors 4, 248 (2014).

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(9), 093902 (2009).
[PubMed]

Song, B.-S.

Y. Akahane, T. Asano, B.-S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature 425(6961), 944–947 (2003).
[PubMed]

Sounas, D. L.

R. Fleury, D. L. Sounas, and A. Alù, “Negative refraction and planar focusing based on parity-time symmetric metasurfaces,” Phys. Rev. Lett. 113(2), 023903 (2014).
[PubMed]

Suchkov, S. V.

S. V. Suchkov, S. V. Dmitriev, B. A. Malomed, and Y. S. Kivshar, “Wave scattering on a domain wall in a chain of PT-symmetric couplers,” Phys. Rev. A 85, 033825 (2012).

Sun, Y.

Y. Sun, W. Tan, H. Q. Li, J. Li, and H. Chen, “Experimental demonstration of a coherent perfect absorber with PT phase transition,” Phys. Rev. Lett. 112(14), 143903 (2014).
[PubMed]

Tan, W.

Y. Sun, W. Tan, H. Q. Li, J. Li, and H. Chen, “Experimental demonstration of a coherent perfect absorber with PT phase transition,” Phys. Rev. Lett. 112(14), 143903 (2014).
[PubMed]

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(9), 093902 (2009).
[PubMed]

Vuckovic, J.

J. Vucković, M. Lončar, H. Mabuchi, and A. Scherer, “Design of photonic crystal microcavities for cavity QED,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 65(1 Pt 2), 016608 (2002).
[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. Photonics 8, 524 (2014).

Wang, Y.

L. Feng, Z. J. Wong, R.-M. Ma, Y. Wang, and X. Zhang, “Single-mode laser by parity-time symmetry breaking,” Science 346(6212), 972–975 (2014).
[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. Photonics 8, 524 (2014).

Wong, Z. J.

L. Feng, Z. J. Wong, R.-M. Ma, Y. Wang, and X. Zhang, “Single-mode laser by parity-time symmetry breaking,” Science 346(6212), 972–975 (2014).
[PubMed]

Xiao, J. J.

Z. Z. Liu, Q. Zhang, Y. Chen, and J. J. Xiao, “General coupled-mode analysis of a geometrically symmetric waveguide array with nonuniform gain and loss,” Photon. Res. 5, 57 (2017).

Z. Liu, Q. Zhang, X. Liu, Y. Yao, and J. J. Xiao, “Absence of exceptional points in square waveguide arrays with apparently balanced gain and loss,” Sci. Rep. 6, 22711 (2016).
[PubMed]

Z. Z. Liu, Q. Zhang, and J. J. Xiao, “EIT-like transmission by interaction between multiple Bragg scattering and local plasmonic resonances,” J. Opt. 18, 015005 (2015).

J. J. Xiao and K. W. Yu, “Light trapping and releasing in side-coupled microresonator structures with asymmetric cavity–waveguide coupling,” Opt. Commun. 281, 4023 (2008).

Xiao, M.

K. Ding, G. Ma, M. Xiao, Z. Q. Zhang, and C. T. Chan, “Emergence, coalescence, and topological properties of multiple exceptional points and their experimental realization,” Phys. Rev. X 6, 021007 (2016).

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. Photonics 8, 524 (2014).

Xu, X.

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. Photonics 8, 524 (2014).

Yang, 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, “Nonreciprocal light transmission in parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394 (2014).

B. Peng, S. K. Özdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, F. Nori, and L. Yang, “Loss-induced suppression and revival of lasing,” Science 346(6207), 328–332 (2014).
[PubMed]

B. Peng, S. K. Özdemir, 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 (2014).

Yao, Y.

Z. Liu, Q. Zhang, X. Liu, Y. Yao, and J. J. Xiao, “Absence of exceptional points in square waveguide arrays with apparently balanced gain and loss,” Sci. Rep. 6, 22711 (2016).
[PubMed]

Yilmaz, H.

B. Peng, S. K. Özdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, F. Nori, and L. Yang, “Loss-induced suppression and revival of lasing,” Science 346(6207), 328–332 (2014).
[PubMed]

Yu, K. W.

J. J. Xiao and K. W. Yu, “Light trapping and releasing in side-coupled microresonator structures with asymmetric cavity–waveguide coupling,” Opt. Commun. 281, 4023 (2008).

Zhang, Q.

Z. Z. Liu, Q. Zhang, Y. Chen, and J. J. Xiao, “General coupled-mode analysis of a geometrically symmetric waveguide array with nonuniform gain and loss,” Photon. Res. 5, 57 (2017).

Z. Liu, Q. Zhang, X. Liu, Y. Yao, and J. J. Xiao, “Absence of exceptional points in square waveguide arrays with apparently balanced gain and loss,” Sci. Rep. 6, 22711 (2016).
[PubMed]

Z. Z. Liu, Q. Zhang, and J. J. Xiao, “EIT-like transmission by interaction between multiple Bragg scattering and local plasmonic resonances,” J. Opt. 18, 015005 (2015).

Zhang, X.

L. Feng, Z. J. Wong, R.-M. Ma, Y. Wang, and X. Zhang, “Single-mode laser by parity-time symmetry breaking,” Science 346(6212), 972–975 (2014).
[PubMed]

Zhang, Z. Q.

K. Ding, G. Ma, M. Xiao, Z. Q. Zhang, and C. T. Chan, “Emergence, coalescence, and topological properties of multiple exceptional points and their experimental realization,” Phys. Rev. X 6, 021007 (2016).

K. Ding, Z. Q. Zhang, and C. T. Chan, “Coalescence of exceptional points and phase diagrams for one-dimensional P T-symmetric photonic crystals,” Phys. Rev. B 92, 235310 (2015).

Zhao, Y.

J. Opt. (1)

Z. Z. Liu, Q. Zhang, and J. J. Xiao, “EIT-like transmission by interaction between multiple Bragg scattering and local plasmonic resonances,” J. Opt. 18, 015005 (2015).

J. Phys. A Math. Theor. (1)

W. D. Heiss, “The physics of exceptional points,” J. Phys. A Math. Theor. 45, 444016 (2012).

Nat. Photonics (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. Photonics 8, 524 (2014).

Nat. Phys. (3)

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 (2010).

B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Nonreciprocal light transmission in parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394 (2014).

B. Peng, S. K. Özdemir, 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 (2014).

Nature (1)

Y. Akahane, T. Asano, B.-S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature 425(6961), 944–947 (2003).
[PubMed]

Opt. Commun. (1)

J. J. Xiao and K. W. Yu, “Light trapping and releasing in side-coupled microresonator structures with asymmetric cavity–waveguide coupling,” Opt. Commun. 281, 4023 (2008).

Opt. Express (1)

Photon. Res. (1)

Photonic Sensors (1)

K. V. Shanthi and S. Robinson, “Two-dimensional photonic crystal based sensor for pressure sensing,” Photonic Sensors 4, 248 (2014).

Phys. A-Math. Theor. (1)

I. J. Rotter, “Non-Hermitian Hamilton operator and the physics of open quantum systems,” Phys. A-Math. Theor. 42, 153001 (2009).

Phys. Rev. A (3)

S. V. Suchkov, S. V. Dmitriev, B. A. Malomed, and Y. S. Kivshar, “Wave scattering on a domain wall in a chain of PT-symmetric couplers,” Phys. Rev. A 85, 033825 (2012).

H. Benisty, A. Lupu, and A. Degiron, “Transverse periodic PT symmetry for modal demultiplexing in optical waveguides,” Phys. Rev. A 91, 053825 (2015).

M. Kang, F. Liu, and J. Li, “Effective spontaneous PT-symmetry breaking in hybridized metamaterials,” Phys. Rev. A 87, 053824 (2013).

Phys. Rev. B (2)

K. Ding, Z. Q. Zhang, and C. T. Chan, “Coalescence of exceptional points and phase diagrams for one-dimensional P T-symmetric photonic crystals,” Phys. Rev. B 92, 235310 (2015).

H. Alaeian and J. A. Dionne, “Non-Hermitian nanophotonic and plasmonic waveguides,” Phys. Rev. B 89, 075136 (2014).

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (1)

J. Vucković, M. Lončar, H. Mabuchi, and A. Scherer, “Design of photonic crystal microcavities for cavity QED,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 65(1 Pt 2), 016608 (2002).
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Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics (1)

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Figures (16)

Fig. 1
Fig. 1 Schematic picture of a unit cell of the polariton crystal. The solid lines mark the dual ports of the unit cell, and the dashed lines mark the reference line of coupling resonator.
Fig. 2
Fig. 2 Complex band structure of the polariton crystal for κ=0.03 ω 0 , γ 1 = γ 2 =0 and ω 1 = ω 2 = ω 0 . The lowest order Bragg resonance is ω 1 = ω 2 = ω 0 (a), ω 0 (b), and 1.9 ω 0 (c). The orange vertical dash-dotted line represents the Brillouin boundary k=π/ 2L corresponding to the unit cell with dual cavities (cf. Figure 1). The grey-shaded zone marks the polariton gap by the localized ‘polariton’ resonance. In panels (a) and (c), the brown-shaded zone denotes the Bragg gap. The dashed lines are the results of the band fold of the solid lines due to lattice constant extended to double, while the different color denotes different band index.
Fig. 3
Fig. 3 Complex band structures at γ=0.01 ω 0 (a), (b), γ=0.04 ω 0 (c), (d), γ=0.09 ω 0 (e), (f), and γ=0.1 ω 0 (g), (h). The Bragg resonance is identical to that of Fig. 1(a). (a), (c), (e), (g) are the real parts, (b), (d), (f), (h) are the imaginary part. (M), (N), (S) denote the EPs.
Fig. 4
Fig. 4 Complex band structures at γ=0.01 ω 0 (a), (b), γ=0.04 ω 0 (c), (d), γ=0.155 ω 0 (e), (f), and γ=0.18 ω 0 (g), (h). The Bragg resonance is identical to that of Fig. 1(c). (a), (c), (e), (g) are the real parts, (b), (d), (f), (h) are the imaginary part. (M), (N), (S) denote the EPs.
Fig. 5
Fig. 5 The respective trajectories of the EPs in ( k,γ) space for the parameters presented in Fig. 3(a) and Fig. 4 (b). The gray region stands for the PT-unbroken phases.
Fig. 6
Fig. 6 Phase rigidity of the eigenstates on the bands shown in Figs. 3(c) and 3(d). The dotted lines correspond to the position of the EPs (M), (N), and (S).
Fig. 7
Fig. 7 Transmission spectra for two different Bragg resonances: 0.77 ω 0 (a), (b), 1.9 ω 0 (c), (d), wherein (a) and (c) express the transmission before the emergence of EP (N), whereas (b) and (d) express the transmission at following cases, including the existence and coalescence of EP (N). The orange dashed arrows specify the tendency of the peaks with the increasing γ.
Fig. 8
Fig. 8 The proposed MDM waveguide structure. The parameters are defined: d=25.nm, h=50nm, l=175nm. The cladding metal is silver (Ag), the embedded dielectric is silica.
Fig. 9
Fig. 9 Transmission spectra obtained by full-wave numerical simulation corresponding to three different L selections: L=161nm (a), L=355nm (b), L=483.8nm (c). The orange dashed arrow is used to label the tendency of the peaks.
Fig. 10
Fig. 10 The transmission spectrum (the colored circle) obtained by CMT analysis [Eq. (6)] for the configuration of Fig. 9(c) corresponding to different non-Hermiticity ε I . The parameters are obtained by curve fitting technique through unit cell numeric simulation. The solid lines are the duplicate of Fig. 9.
Fig. 11
Fig. 11 Schematic figure of the proposed structure consisting of two resonators coupled to a waveguide. a 1 and a 2 are the amplitudes of dual resonators. S 1(2)+ ( S 1(2) ) are the incident (scattering) waveguides mode amplitude.
Fig. 12
Fig. 12 The real (upper panel) and imaginary (lower panel) part of the eigenvalues with the increase of absolute difference between the dual resonant modes’ frequency corresponding to distinguished propagation phase, respectively, θ=π/2 (left column), θ=3π/4 (central column), and θ=π (right column).
Fig. 13
Fig. 13 The real (upper panel) and imaginary (lower panel) part of the eigenvalues with the increase of absolute difference between the gain/loss coefficients embedded in the dual cavities corresponding to distinguished propagation phase, respectively, θ=π/2 (left column), θ=3π/4 (central column), and θ=π (right column).
Fig. 14
Fig. 14 Complex band structure for polariton crystal (Bragg resonance ω 0 ) with different gain/loss coefficients: (a, b) γ=0.01 ω 0 , (c, d) γ=0.04 ω 0 .
Fig. 15
Fig. 15 Transmission spectrum with the parameters consistent with Fig. 14.
Fig. 16
Fig. 16 The transmission spectrum for the case L=483.8nm, obtained by the numeric simulation including left port and right port excitation.

Equations (21)

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d a n dt =j ω n a n ( γ n + κ 2 + κ 2 ) a n + κ S 2n+ + κ S (2n+1) ,
S (2n+1)+ = S 2n+ κ a n ,
S 2n = S (2n+1) κ a n ,
[ S (2n+1)+ S (2n+1) ]=[ 1χ χ χ 1+χ ][ S 2n+ S 2n ]= M n [ S 2n+ S 2n ],
[ S 2n+ S 2n ]=[ e jθ 0 0 e jθ ][ S (2n1)+ S (2n1) ]= M n [ S (2n1)+ S (2n1) ],
M tot = M 1 M 1 M 2 M 2 M 3 .
[ S + S ]=M[ S 1+ S 1 ]=[ M 11 M 12 M 21 M 22 ][ S 1+ S 1 ].
t= 1 M 22 ;r= M 21 M 22 .
d a 1 dt =j ω 1 a 1 ( γ 1 +κ ) a 1 + κ S 1+ + κ ( S 2+ κ a 2 ) e jθ d a 2 dt =j ω 2 a 2 ( γ 2 +κ ) a 2 + κ S 2+ + κ ( S 1+ κ a 1 ) e jθ S 2 =( S 1+ κ a 1 ) e jθ κ a 2 S 1 =( S 2+ κ a 2 ) e jθ κ a 1
j d dt ( a 1 a 2 )=( ω 1 j( γ 1 +κ ) jκ e jθ jκ e jθ ω 2 j( γ 2 +κ ) )( a 1 a 2 )=H( a 1 a 2 )
ω= ( ω 1 + ω 2 )j( γ 1 + γ 2 +2κ )±j 4 e j2θ κ 2 + [ γ 1 γ 2 +j( ω 1 ω 2 ) ] 2 2
( S 6+ S 6 )=( e j2θ 0 0 e j2θ )( S 1+ S 1 )+( e j3θ/2 e j3θ/2 ) κ a 1 +( e jθ/2 e jθ/2 ) κ a 2
( e jkL e j2θ 0 0 e jkL e j2θ )( S 1+ S 1 )= κ ( e j3θ/2 e jθ/2 e j3θ/2 e jθ/2 )( a 1 a 2 )
d dt ( a 1 a 2 )=( j ω 1 ( γ 1 +κ ) κ e jθ κ e jθ j ω 2 ( γ 2 +κ ) )( a 1 a 2 )+( κ e jθ/2 κ e j3θ/2 κ e j3θ/2 κ e jθ/2 )( S 1+ S 6 )
d dt [ a 1 a 2 ]=[ κsin( 2θ ) cos( kL )cos( 2θ ) j ω 1 γ 1 2jκ( 1+ e jkL )sinθ 1+ e j2kL 2 e jkL cos( 2θ ) jκ( 1+ e jkL )sinθ cos( kL )cos( 2θ ) κsin( 2θ ) cos( kL )cos( 2θ ) j ω 2 γ 2 ][ a 1 a 2 ]=H[ a 1 a 2 ]
[ κsin( 2θ ) cos( kL )cos( 2θ ) +jωj ω 1 γ 1 2jκ( 1+ e jkL )sinθ 1+ e j2kL 2 e jkL cos( 2θ ) jκ( 1+ e jkL )sinθ cos( kL )cos( 2θ ) κsin( 2θ ) cos( kL )cos( 2θ ) +jωj ω 2 γ 2 ][ a 1 a 2 ]=0
[ a 1 a 2 ][ κsin( 2θ ) cos( kL )cos( 2θ ) +jωj ω 1 γ 1 2jκ( 1+ e jkL )sinθ 1+ e j2kL 2 e jkL cos( 2θ ) jκ( 1+ e jkL )sinθ cos( kL )cos( 2θ ) κsin( 2θ ) cos( kL )cos( 2θ ) +jωj ω 2 γ 2 ]=0
[ a 1 a 2 ]=[ 2jκ( 1+ e jkL )sinθ 1+ e j2kL 2 e jkL cos( 2θ ) κsin( 2θ ) cos( kL )cos( 2θ ) +jωj ω 1 γ ]
[ a 1 a 2 ] T =[ jκ( 1+ e jkL )sinθ cos( kL )cos( 2θ ) κsin( 2θ ) cos( kL )cos( 2θ ) +jωj ω 1 γ 1 ]
M unit =[ a b c d ].
M unit =[ a c b d ].

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