Abstract

We investigated defect states in band gaps of one-dimensional photonic lattices with delicate modulations of gain and loss that respect parity-time-symmetry (PT-symmetry), viz. n(z) = n*(−z). For the sake of generality, we employ not only periodic structures but also quasiperiodic structures, e.g. Fibonacci sequences, to construct aperiodic PT lattices. Differed from lossless systems for which the defect state is related to only one exceptional point (EP) of the S-matrix, we observed the splitting of one EP into a pair after the introduction of judiciously designed gain and loss in those PT systems, where the defect state enters a non-threshold broken symmetry phase bounded by the EP pair. Some interesting properties associated with defect states and EP splitting are demonstrated, such as enhanced spectral localization, double optical phase abrupt change, and wavelength sensitive reversion of unidirectional transparency.

© 2015 Optical Society of America

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References

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

2014 (8)

X. Zhu, H. Ramezani, C. Shi, J. Zhu, and X. Zhang, “PT-symmetric acoustics,” Phys. Rev. X 4(3), 031042 (2014).
[Crossref]

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).
[Crossref] [PubMed]

B. Peng, S. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. H. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10(5), 394–398 (2014).
[Crossref]

S. Longhi and G. D. Valle, “Optical lattices with exceptional points in the continuum,” Phys. Rev. A 89(5), 052132 (2014).
[Crossref]

H. Ramezani, H.-K. Li, Y. Wang, and X. Zhang, “Unidirectional spectral singularities,” Phys. Rev. Lett. 113(26), 263905 (2014).
[Crossref] [PubMed]

S. Longhi, “Bound states in the continuum in PT-symmetric optical lattices,” Opt. Lett. 39(6), 1697–1700 (2014).
[Crossref] [PubMed]

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(8), 9574–9584 (2014).
[Crossref] [PubMed]

X. F. Zhu, Y. G. Peng, and D. G. Zhao, “Anisotropic reflection oscillation in periodic multilayer structures of parity-time symmetry,” Opt. Express 22(15), 18401–18411 (2014).
[Crossref] [PubMed]

2013 (2)

X. Zhu, L. Feng, P. Zhang, X. Yin, and X. Zhang, “One-way invisible cloak using parity-time symmetric transformation optics,” Opt. Lett. 38(15), 2821–2824 (2013).
[Crossref] [PubMed]

A. Regensburger, M.-A. Miri, C. Bersch, J. Näger, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Observation of defect states in PT-symmetric optical lattices,” Phys. Rev. Lett. 110(22), 223902 (2013).
[Crossref] [PubMed]

2012 (5)

H. X. Ding, Z. H. Shen, X. W. Ni, and X. F. Zhu, “Multi-splitting and self-similarity of band gap structures in quasi-periodic plates of Cantor series,” Appl. Phys. Lett. 100(8), 083501 (2012).
[Crossref]

Z. Chen, P. Han, C. W. Leung, Y. Wang, M. Hu, and Y. Chen, “Study of optical Tamm states based on the phase properties of one-dimensional photonic crystals,” Opt. Express 20(19), 21618–21626 (2012).
[Crossref] [PubMed]

L. Ge, Y. D. Chong, and A. D. Stone, “Conservation relations and anisotropic transmission resonances in one-dimensional PT-symmetric photonic heterostructures,” Phys. Rev. A 85(2), 023802 (2012).
[Crossref]

A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488(7410), 167–171 (2012).
[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(2), 108–113 (2012).
[Crossref] [PubMed]

2011 (2)

Y. D. Chong, L. Ge, and A. D. Stone, “PT-symmetry breaking and laser-absorber modes in optical scattering systems,” Phys. Rev. Lett. 106(9), 093902 (2011).
[Crossref] [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).
[Crossref] [PubMed]

2010 (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(3), 192–195 (2010).
[Crossref]

S. Longhi, “PT-symmetric laser absorber,” Phys. Rev. A 82(3), 031801 (2010).
[Crossref]

F. J. Lawrence, L. C. Botten, K. B. Dossou, R. C. McPhedran, and C. M. de Sterke, “Photonic-crystal surface modes found from impedances,” Phys. Rev. A 82(5), 053840 (2010).
[Crossref]

2009 (2)

X. F. Zhu, T. Xu, S. C. Liu, and J. C. Cheng, “Study of acoustic wave behavior in silicon-based one-dimensional phononic-crystal plates using harmony response analysis,” J. Appl. Phys. 106(10), 104901 (2009).
[Crossref]

S. Longhi, “Bloch oscillations in complex crystals with PT symmetry,” Phys. Rev. Lett. 103(12), 123601 (2009).
[Crossref] [PubMed]

2008 (1)

J. Guo, Y. Sun, Y. Zhang, H. Li, H. Jiang, and H. Chen, “Experimental investigation of interface states in photonic crystal heterostructures,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 78, 026607 (2008).
[Crossref] [PubMed]

2006 (1)

A. P. Vinogradov, A. V. Dorofeenko, S. G. Erokhin, M. Inoue, A. A. Lisyansky, A. M. Merzlikin, and A. B. Granovsky, “Surface state peculiarities in one-dimensional photonic crystal interfaces,” Phys. Rev. B 74(4), 045128 (2006).
[Crossref]

2005 (1)

A. V. Kavokin, I. Shelykh, and G. Malpuech, “Lossless interface modes at the boundary between two periodic dielectric structures,” Phys. Rev. B 72(23), 233102 (2005).
[Crossref]

1977 (1)

1932 (1)

I. E. Tamm, “On the possible bound states of electrons on a crystal surface,” Phys. Z. Sowjetunion 1, 733–735 (1932).

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(2), 108–113 (2012).
[Crossref] [PubMed]

Bender, C. M.

B. Peng, S. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. H. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10(5), 394–398 (2014).
[Crossref]

Bender, N.

Bersch, C.

A. Regensburger, M.-A. Miri, C. Bersch, J. Näger, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Observation of defect states in PT-symmetric optical lattices,” Phys. Rev. Lett. 110(22), 223902 (2013).
[Crossref] [PubMed]

A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488(7410), 167–171 (2012).
[Crossref] [PubMed]

Botten, L. C.

F. J. Lawrence, L. C. Botten, K. B. Dossou, R. C. McPhedran, and C. M. de Sterke, “Photonic-crystal surface modes found from impedances,” Phys. Rev. A 82(5), 053840 (2010).
[Crossref]

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).
[Crossref] [PubMed]

Chen, H.

J. Guo, Y. Sun, Y. Zhang, H. Li, H. Jiang, and H. Chen, “Experimental investigation of interface states in photonic crystal heterostructures,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 78, 026607 (2008).
[Crossref] [PubMed]

Chen, Y.

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(2), 108–113 (2012).
[Crossref] [PubMed]

Chen, Z.

Cheng, J. C.

X. F. Zhu, T. Xu, S. C. Liu, and J. C. Cheng, “Study of acoustic wave behavior in silicon-based one-dimensional phononic-crystal plates using harmony response analysis,” J. Appl. Phys. 106(10), 104901 (2009).
[Crossref]

Chong, Y. D.

L. Ge, Y. D. Chong, and A. D. Stone, “Conservation relations and anisotropic transmission resonances in one-dimensional PT-symmetric photonic heterostructures,” Phys. Rev. A 85(2), 023802 (2012).
[Crossref]

Y. D. Chong, L. Ge, and A. D. Stone, “PT-symmetry breaking and laser-absorber modes in optical scattering systems,” Phys. Rev. Lett. 106(9), 093902 (2011).
[Crossref] [PubMed]

Christodoulides, D. N.

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(8), 9574–9584 (2014).
[Crossref] [PubMed]

A. Regensburger, M.-A. Miri, C. Bersch, J. Näger, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Observation of defect states in PT-symmetric optical lattices,” Phys. Rev. Lett. 110(22), 223902 (2013).
[Crossref] [PubMed]

A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488(7410), 167–171 (2012).
[Crossref] [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).
[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(3), 192–195 (2010).
[Crossref]

de Sterke, C. M.

F. J. Lawrence, L. C. Botten, K. B. Dossou, R. C. McPhedran, and C. M. de Sterke, “Photonic-crystal surface modes found from impedances,” Phys. Rev. A 82(5), 053840 (2010).
[Crossref]

Ding, H. X.

H. X. Ding, Z. H. Shen, X. W. Ni, and X. F. Zhu, “Multi-splitting and self-similarity of band gap structures in quasi-periodic plates of Cantor series,” Appl. Phys. Lett. 100(8), 083501 (2012).
[Crossref]

Dorofeenko, A. V.

A. P. Vinogradov, A. V. Dorofeenko, S. G. Erokhin, M. Inoue, A. A. Lisyansky, A. M. Merzlikin, and A. B. Granovsky, “Surface state peculiarities in one-dimensional photonic crystal interfaces,” Phys. Rev. B 74(4), 045128 (2006).
[Crossref]

Dossou, K. B.

F. J. Lawrence, L. C. Botten, K. B. Dossou, R. C. McPhedran, and C. M. de Sterke, “Photonic-crystal surface modes found from impedances,” Phys. Rev. A 82(5), 053840 (2010).
[Crossref]

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).
[Crossref] [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(3), 192–195 (2010).
[Crossref]

Erokhin, S. G.

A. P. Vinogradov, A. V. Dorofeenko, S. G. Erokhin, M. Inoue, A. A. Lisyansky, A. M. Merzlikin, and A. B. Granovsky, “Surface state peculiarities in one-dimensional photonic crystal interfaces,” Phys. Rev. B 74(4), 045128 (2006).
[Crossref]

Fan, S. H.

B. Peng, S. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. H. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10(5), 394–398 (2014).
[Crossref]

Fegadolli, W. S.

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(2), 108–113 (2012).
[Crossref] [PubMed]

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).
[Crossref] [PubMed]

X. Zhu, L. Feng, P. Zhang, X. Yin, and X. Zhang, “One-way invisible cloak using parity-time symmetric transformation optics,” Opt. Lett. 38(15), 2821–2824 (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(2), 108–113 (2012).
[Crossref] [PubMed]

Ge, L.

L. Ge, Y. D. Chong, and A. D. Stone, “Conservation relations and anisotropic transmission resonances in one-dimensional PT-symmetric photonic heterostructures,” Phys. Rev. A 85(2), 023802 (2012).
[Crossref]

Y. D. Chong, L. Ge, and A. D. Stone, “PT-symmetry breaking and laser-absorber modes in optical scattering systems,” Phys. Rev. Lett. 106(9), 093902 (2011).
[Crossref] [PubMed]

Gianfreda, M.

B. Peng, S. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. H. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10(5), 394–398 (2014).
[Crossref]

Granovsky, A. B.

A. P. Vinogradov, A. V. Dorofeenko, S. G. Erokhin, M. Inoue, A. A. Lisyansky, A. M. Merzlikin, and A. B. Granovsky, “Surface state peculiarities in one-dimensional photonic crystal interfaces,” Phys. Rev. B 74(4), 045128 (2006).
[Crossref]

Guo, J.

J. Guo, Y. Sun, Y. Zhang, H. Li, H. Jiang, and H. Chen, “Experimental investigation of interface states in photonic crystal heterostructures,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 78, 026607 (2008).
[Crossref] [PubMed]

Han, P.

Hong, C.-S.

Hu, M.

Inoue, M.

A. P. Vinogradov, A. V. Dorofeenko, S. G. Erokhin, M. Inoue, A. A. Lisyansky, A. M. Merzlikin, and A. B. Granovsky, “Surface state peculiarities in one-dimensional photonic crystal interfaces,” Phys. Rev. B 74(4), 045128 (2006).
[Crossref]

Jiang, H.

J. Guo, Y. Sun, Y. Zhang, H. Li, H. Jiang, and H. Chen, “Experimental investigation of interface states in photonic crystal heterostructures,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 78, 026607 (2008).
[Crossref] [PubMed]

Kavokin, A. V.

A. V. Kavokin, I. Shelykh, and G. Malpuech, “Lossless interface modes at the boundary between two periodic dielectric structures,” Phys. Rev. B 72(23), 233102 (2005).
[Crossref]

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(3), 192–195 (2010).
[Crossref]

Kottos, T.

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(8), 9574–9584 (2014).
[Crossref] [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).
[Crossref] [PubMed]

Lawrence, F. J.

F. J. Lawrence, L. C. Botten, K. B. Dossou, R. C. McPhedran, and C. M. de Sterke, “Photonic-crystal surface modes found from impedances,” Phys. Rev. A 82(5), 053840 (2010).
[Crossref]

Lei, F.

B. Peng, S. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. H. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10(5), 394–398 (2014).
[Crossref]

Leung, C. W.

Li, H.

J. Guo, Y. Sun, Y. Zhang, H. Li, H. Jiang, and H. Chen, “Experimental investigation of interface states in photonic crystal heterostructures,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 78, 026607 (2008).
[Crossref] [PubMed]

Li, H.-K.

H. Ramezani, H.-K. Li, Y. Wang, and X. Zhang, “Unidirectional spectral singularities,” Phys. Rev. Lett. 113(26), 263905 (2014).
[Crossref] [PubMed]

Lin, Z.

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).
[Crossref] [PubMed]

Lisyansky, A. A.

A. P. Vinogradov, A. V. Dorofeenko, S. G. Erokhin, M. Inoue, A. A. Lisyansky, A. M. Merzlikin, and A. B. Granovsky, “Surface state peculiarities in one-dimensional photonic crystal interfaces,” Phys. Rev. B 74(4), 045128 (2006).
[Crossref]

Liu, S. C.

X. F. Zhu, T. Xu, S. C. Liu, and J. C. Cheng, “Study of acoustic wave behavior in silicon-based one-dimensional phononic-crystal plates using harmony response analysis,” J. Appl. Phys. 106(10), 104901 (2009).
[Crossref]

Long, G. L.

B. Peng, S. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. H. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10(5), 394–398 (2014).
[Crossref]

Longhi, S.

S. Longhi and G. D. Valle, “Optical lattices with exceptional points in the continuum,” Phys. Rev. A 89(5), 052132 (2014).
[Crossref]

S. Longhi, “Bound states in the continuum in PT-symmetric optical lattices,” Opt. Lett. 39(6), 1697–1700 (2014).
[Crossref] [PubMed]

S. Longhi, “PT-symmetric laser absorber,” Phys. Rev. A 82(3), 031801 (2010).
[Crossref]

S. Longhi, “Bloch oscillations in complex crystals with PT symmetry,” Phys. Rev. Lett. 103(12), 123601 (2009).
[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(2), 108–113 (2012).
[Crossref] [PubMed]

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).
[Crossref] [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(3), 192–195 (2010).
[Crossref]

Malpuech, G.

A. V. Kavokin, I. Shelykh, and G. Malpuech, “Lossless interface modes at the boundary between two periodic dielectric structures,” Phys. Rev. B 72(23), 233102 (2005).
[Crossref]

McPhedran, R. C.

F. J. Lawrence, L. C. Botten, K. B. Dossou, R. C. McPhedran, and C. M. de Sterke, “Photonic-crystal surface modes found from impedances,” Phys. Rev. A 82(5), 053840 (2010).
[Crossref]

Merzlikin, A. M.

A. P. Vinogradov, A. V. Dorofeenko, S. G. Erokhin, M. Inoue, A. A. Lisyansky, A. M. Merzlikin, and A. B. Granovsky, “Surface state peculiarities in one-dimensional photonic crystal interfaces,” Phys. Rev. B 74(4), 045128 (2006).
[Crossref]

Miri, M. A.

A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488(7410), 167–171 (2012).
[Crossref] [PubMed]

Miri, M.-A.

A. Regensburger, M.-A. Miri, C. Bersch, J. Näger, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Observation of defect states in PT-symmetric optical lattices,” Phys. Rev. Lett. 110(22), 223902 (2013).
[Crossref] [PubMed]

Monifi, F.

B. Peng, S. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. H. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10(5), 394–398 (2014).
[Crossref]

Moravvej-Farshi, M. K.

Näger, J.

A. Regensburger, M.-A. Miri, C. Bersch, J. Näger, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Observation of defect states in PT-symmetric optical lattices,” Phys. Rev. Lett. 110(22), 223902 (2013).
[Crossref] [PubMed]

Nazari, F.

Ni, X. W.

H. X. Ding, Z. H. Shen, X. W. Ni, and X. F. Zhu, “Multi-splitting and self-similarity of band gap structures in quasi-periodic plates of Cantor series,” Appl. Phys. Lett. 100(8), 083501 (2012).
[Crossref]

Nori, F.

B. Peng, S. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. H. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10(5), 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(2), 108–113 (2012).
[Crossref] [PubMed]

Onishchukov, G.

A. Regensburger, M.-A. Miri, C. Bersch, J. Näger, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Observation of defect states in PT-symmetric optical lattices,” Phys. Rev. Lett. 110(22), 223902 (2013).
[Crossref] [PubMed]

A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488(7410), 167–171 (2012).
[Crossref] [PubMed]

Özdemir, S. K.

B. Peng, S. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. H. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10(5), 394–398 (2014).
[Crossref]

Peng, B.

B. Peng, S. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. H. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10(5), 394–398 (2014).
[Crossref]

Peng, Y. G.

Peschel, U.

A. Regensburger, M.-A. Miri, C. Bersch, J. Näger, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Observation of defect states in PT-symmetric optical lattices,” Phys. Rev. Lett. 110(22), 223902 (2013).
[Crossref] [PubMed]

A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488(7410), 167–171 (2012).
[Crossref] [PubMed]

Ramezani, H.

X. Zhu, H. Ramezani, C. Shi, J. Zhu, and X. Zhang, “PT-symmetric acoustics,” Phys. Rev. X 4(3), 031042 (2014).
[Crossref]

H. Ramezani, H.-K. Li, Y. Wang, and X. Zhang, “Unidirectional spectral singularities,” Phys. Rev. Lett. 113(26), 263905 (2014).
[Crossref] [PubMed]

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(8), 9574–9584 (2014).
[Crossref] [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).
[Crossref] [PubMed]

Regensburger, A.

A. Regensburger, M.-A. Miri, C. Bersch, J. Näger, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Observation of defect states in PT-symmetric optical lattices,” Phys. Rev. Lett. 110(22), 223902 (2013).
[Crossref] [PubMed]

A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488(7410), 167–171 (2012).
[Crossref] [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(3), 192–195 (2010).
[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(2), 108–113 (2012).
[Crossref] [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(3), 192–195 (2010).
[Crossref]

Shelykh, I.

A. V. Kavokin, I. Shelykh, and G. Malpuech, “Lossless interface modes at the boundary between two periodic dielectric structures,” Phys. Rev. B 72(23), 233102 (2005).
[Crossref]

Shen, Z. H.

H. X. Ding, Z. H. Shen, X. W. Ni, and X. F. Zhu, “Multi-splitting and self-similarity of band gap structures in quasi-periodic plates of Cantor series,” Appl. Phys. Lett. 100(8), 083501 (2012).
[Crossref]

Shi, C.

X. Zhu, H. Ramezani, C. Shi, J. Zhu, and X. Zhang, “PT-symmetric acoustics,” Phys. Rev. X 4(3), 031042 (2014).
[Crossref]

Stone, A. D.

L. Ge, Y. D. Chong, and A. D. Stone, “Conservation relations and anisotropic transmission resonances in one-dimensional PT-symmetric photonic heterostructures,” Phys. Rev. A 85(2), 023802 (2012).
[Crossref]

Y. D. Chong, L. Ge, and A. D. Stone, “PT-symmetry breaking and laser-absorber modes in optical scattering systems,” Phys. Rev. Lett. 106(9), 093902 (2011).
[Crossref] [PubMed]

Sun, Y.

J. Guo, Y. Sun, Y. Zhang, H. Li, H. Jiang, and H. Chen, “Experimental investigation of interface states in photonic crystal heterostructures,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 78, 026607 (2008).
[Crossref] [PubMed]

Tamm, I. E.

I. E. Tamm, “On the possible bound states of electrons on a crystal surface,” Phys. Z. Sowjetunion 1, 733–735 (1932).

Valle, G. D.

S. Longhi and G. D. Valle, “Optical lattices with exceptional points in the continuum,” Phys. Rev. A 89(5), 052132 (2014).
[Crossref]

Vinogradov, A. P.

A. P. Vinogradov, A. V. Dorofeenko, S. G. Erokhin, M. Inoue, A. A. Lisyansky, A. M. Merzlikin, and A. B. Granovsky, “Surface state peculiarities in one-dimensional photonic crystal interfaces,” Phys. Rev. B 74(4), 045128 (2006).
[Crossref]

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).
[Crossref] [PubMed]

H. Ramezani, H.-K. Li, Y. Wang, and X. Zhang, “Unidirectional spectral singularities,” Phys. Rev. Lett. 113(26), 263905 (2014).
[Crossref] [PubMed]

Z. Chen, P. Han, C. W. Leung, Y. Wang, M. Hu, and Y. Chen, “Study of optical Tamm states based on the phase properties of one-dimensional photonic crystals,” Opt. Express 20(19), 21618–21626 (2012).
[Crossref] [PubMed]

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).
[Crossref] [PubMed]

Xu, T.

X. F. Zhu, T. Xu, S. C. Liu, and J. C. Cheng, “Study of acoustic wave behavior in silicon-based one-dimensional phononic-crystal plates using harmony response analysis,” J. Appl. Phys. 106(10), 104901 (2009).
[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(2), 108–113 (2012).
[Crossref] [PubMed]

Yang, L.

B. Peng, S. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. H. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10(5), 394–398 (2014).
[Crossref]

Yariv, A.

Yeh, P.

Yin, X.

Zhang, P.

Zhang, X.

H. Ramezani, H.-K. Li, Y. Wang, and X. Zhang, “Unidirectional spectral singularities,” Phys. Rev. Lett. 113(26), 263905 (2014).
[Crossref] [PubMed]

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).
[Crossref] [PubMed]

X. Zhu, H. Ramezani, C. Shi, J. Zhu, and X. Zhang, “PT-symmetric acoustics,” Phys. Rev. X 4(3), 031042 (2014).
[Crossref]

X. Zhu, L. Feng, P. Zhang, X. Yin, and X. Zhang, “One-way invisible cloak using parity-time symmetric transformation optics,” Opt. Lett. 38(15), 2821–2824 (2013).
[Crossref] [PubMed]

Zhang, Y.

J. Guo, Y. Sun, Y. Zhang, H. Li, H. Jiang, and H. Chen, “Experimental investigation of interface states in photonic crystal heterostructures,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 78, 026607 (2008).
[Crossref] [PubMed]

Zhao, D. G.

Zhu, J.

X. Zhu, H. Ramezani, C. Shi, J. Zhu, and X. Zhang, “PT-symmetric acoustics,” Phys. Rev. X 4(3), 031042 (2014).
[Crossref]

Zhu, X.

Zhu, X. F.

X. F. Zhu, Y. G. Peng, and D. G. Zhao, “Anisotropic reflection oscillation in periodic multilayer structures of parity-time symmetry,” Opt. Express 22(15), 18401–18411 (2014).
[Crossref] [PubMed]

H. X. Ding, Z. H. Shen, X. W. Ni, and X. F. Zhu, “Multi-splitting and self-similarity of band gap structures in quasi-periodic plates of Cantor series,” Appl. Phys. Lett. 100(8), 083501 (2012).
[Crossref]

X. F. Zhu, T. Xu, S. C. Liu, and J. C. Cheng, “Study of acoustic wave behavior in silicon-based one-dimensional phononic-crystal plates using harmony response analysis,” J. Appl. Phys. 106(10), 104901 (2009).
[Crossref]

Appl. Phys. Lett. (1)

H. X. Ding, Z. H. Shen, X. W. Ni, and X. F. Zhu, “Multi-splitting and self-similarity of band gap structures in quasi-periodic plates of Cantor series,” Appl. Phys. Lett. 100(8), 083501 (2012).
[Crossref]

J. Appl. Phys. (1)

X. F. Zhu, T. Xu, S. C. Liu, and J. C. Cheng, “Study of acoustic wave behavior in silicon-based one-dimensional phononic-crystal plates using harmony response analysis,” J. Appl. Phys. 106(10), 104901 (2009).
[Crossref]

J. Opt. Soc. Am. (1)

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(2), 108–113 (2012).
[Crossref] [PubMed]

Nat. Phys. (2)

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

B. Peng, S. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. H. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10(5), 394–398 (2014).
[Crossref]

Nature (1)

A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488(7410), 167–171 (2012).
[Crossref] [PubMed]

Opt. Express (3)

Opt. Lett. (2)

Phys. Rev. A (4)

L. Ge, Y. D. Chong, and A. D. Stone, “Conservation relations and anisotropic transmission resonances in one-dimensional PT-symmetric photonic heterostructures,” Phys. Rev. A 85(2), 023802 (2012).
[Crossref]

S. Longhi, “PT-symmetric laser absorber,” Phys. Rev. A 82(3), 031801 (2010).
[Crossref]

S. Longhi and G. D. Valle, “Optical lattices with exceptional points in the continuum,” Phys. Rev. A 89(5), 052132 (2014).
[Crossref]

F. J. Lawrence, L. C. Botten, K. B. Dossou, R. C. McPhedran, and C. M. de Sterke, “Photonic-crystal surface modes found from impedances,” Phys. Rev. A 82(5), 053840 (2010).
[Crossref]

Phys. Rev. B (2)

A. V. Kavokin, I. Shelykh, and G. Malpuech, “Lossless interface modes at the boundary between two periodic dielectric structures,” Phys. Rev. B 72(23), 233102 (2005).
[Crossref]

A. P. Vinogradov, A. V. Dorofeenko, S. G. Erokhin, M. Inoue, A. A. Lisyansky, A. M. Merzlikin, and A. B. Granovsky, “Surface state peculiarities in one-dimensional photonic crystal interfaces,” Phys. Rev. B 74(4), 045128 (2006).
[Crossref]

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

J. Guo, Y. Sun, Y. Zhang, H. Li, H. Jiang, and H. Chen, “Experimental investigation of interface states in photonic crystal heterostructures,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 78, 026607 (2008).
[Crossref] [PubMed]

Phys. Rev. Lett. (5)

Y. D. Chong, L. Ge, and A. D. Stone, “PT-symmetry breaking and laser-absorber modes in optical scattering systems,” Phys. Rev. Lett. 106(9), 093902 (2011).
[Crossref] [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).
[Crossref] [PubMed]

S. Longhi, “Bloch oscillations in complex crystals with PT symmetry,” Phys. Rev. Lett. 103(12), 123601 (2009).
[Crossref] [PubMed]

H. Ramezani, H.-K. Li, Y. Wang, and X. Zhang, “Unidirectional spectral singularities,” Phys. Rev. Lett. 113(26), 263905 (2014).
[Crossref] [PubMed]

A. Regensburger, M.-A. Miri, C. Bersch, J. Näger, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Observation of defect states in PT-symmetric optical lattices,” Phys. Rev. Lett. 110(22), 223902 (2013).
[Crossref] [PubMed]

Phys. Rev. X (1)

X. Zhu, H. Ramezani, C. Shi, J. Zhu, and X. Zhang, “PT-symmetric acoustics,” Phys. Rev. X 4(3), 031042 (2014).
[Crossref]

Phys. Z. Sowjetunion (1)

I. E. Tamm, “On the possible bound states of electrons on a crystal surface,” Phys. Z. Sowjetunion 1, 733–735 (1932).

Science (1)

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).
[Crossref] [PubMed]

Other (2)

T. Kato, Perturbation Theory of Linear Operators (Springer, 1966).

M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon Press, 1970).

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Figures (7)

Fig. 1
Fig. 1 Schematic of two PT lattices with periodic sequences and Fibonacci sequences, where the complex refractive indices are distributed in n(z) = n*(−z) about z = 0. For the primitive unit-cell layer A, B, C, and D, the thickness is L and the complex refractive indices are n + Δn + 0.01τi, n−Δn−0.01τi, n−Δn + 0.01τi, and n + Δn−0.01τi, respectively. The imaginary part is positive for loss material and negative for gain material.
Fig. 2
Fig. 2 (a) The transmission spectrum of the PT lattice with periodic sequences shown in Fig. 1. The refractive indices of A, B, C, and D layers are 3.2 + 0.01i, 2.8−0.01i, 2.8 + 0.01i, and 3.2−0.01i, respectively. The layer thickness L is 50 nm. (b), (c) The electric field intensity (|E(z)|2) distributions of defect states at the frequencies of 0.08385c/L and 0.2515c/L [labeled by blue stars in (a)], when the light is incident from the left hand side. c is the light speed in vacuum. (d) Curve of equation f(ω) = tan[Re(k1)L]/Re(k1) + tan[Re(k2)L]/Re(k2) versus the frequency ωL/(2πc). The spectral positions of defect states are in excellent agreement with zero points of f(ω) as labeled by blue stars in (d).
Fig. 3
Fig. 3 (a), (c), (e) The calculated spectra of Rf, Rb, T, (ϕf + ϕb)/2, λ1 and λ2, as a function of the normalized frequency ωL/(2πc), when the system involves only real modulation (τ = 0) of index of refraction. (b), (d), (f) The calculated spectra of Rf, Rb, T, (ϕf + ϕb)/2, λ1 and λ2, as a function of the normalized frequency ωL/(2πc), when a weak PT modulation (τ = 1) is introduced. Rf and ϕf represents the reflectance and phase of the forward propagating light. Rb and ϕb are the reflectance and phase of the backward propagating light. T is the transmittance. λ1 and λ2 are the eigenvalues of the S-matrix.
Fig. 4
Fig. 4 (a) Phase transition and (b) Q factor (normalized by the Q at τ = 0) as the strength of gain-loss modulation increases from zero. The broken symmetry phase is indicated by the darken area, which is thresholdless. Exceptional points (or phase transition points) are located at blue and red lines separating symmetric and broken symmetry phases. Defect states are located at the green line in the broken symmetry phase.
Fig. 5
Fig. 5 Fibonacci sequence produced by the rules B→BA, A→B with F1 = B or Fj = Fj−1|Fj−2 for j≥3 with F1 = B and F2 = BA.
Fig. 6
Fig. 6 (a), (c), (e) The calculated spectra of Rf, Rb, T, (ϕf + ϕb)/2, λ1 and λ2, as a function of the normalized frequency ωL/(2πc), when the system involves only real modulation (τ = 0) of index of refraction. (b), (d), (f) The calculated spectra of Rf, Rb, T, (ϕf + ϕb)/2, λ1 and λ2, as a function of the normalized frequency ωL/(2πc), when a weak PT modulation (τ = 1) is introduced. Rf and ϕf represents the reflectance and phase of the forward propagating light. Rb and ϕb are the reflectance and phase of the backward propagating light. T is the transmittance. λ1 and λ2 are the eigenvalues of the S-matrix.
Fig. 7
Fig. 7 Electric field intensity (|E(z)|2) distributions of two defect states in the PT Fibonacci lattice at the frequencies of (a) 0.06246c/L and (b) 0.073c/L [labeled by blue stars in Fig. 6(b)], when the light is incident from the left hand side.

Equations (6)

Equations on this page are rendered with MathJax. Learn more.

[ a 0 b 0 ]=[ C 1 U N1 U N2 C 2 U N1 C 3 U N1 C 4 U N1 U N2 ][ a N b N ].
C 1 = e i k 1 L [ cos( k 2 L ) 1 2 i( k 2 k 1 + k 1 k 2 )sin( k 2 L ) ], C 2 = e i k 1 L [ 1 2 i( k 2 k 1 k 1 k 2 )sin( k 2 L ) ], C 3 = e i k 1 L [ 1 2 i( k 2 k 1 k 1 k 2 )sin( k 2 L ) ], C 4 = e i k 1 L [ cos( k 2 L )+ 1 2 i( k 2 k 1 + k 1 k 2 )sin( k 2 L ) ],
r= 1 2 i( k 2 k 1 k 1 k 2 )sin( k 2 L ) cos( k 2 L )ξcos( k 1 L )i[ 1 2 ( k 2 k 1 + k 1 k 2 )sin( k 2 L )+ξsin( k 1 L ) ] ,
cos( k 2 L )ξcos( k 1 L )=0, k 1 k 2 sin( k 2 L )+ξsin( k 1 L )=0.
tan( k 1 L) k 1 + tan( k 2 L) k 2 =0.
k 1 tan( k 1 L)+ k 2 tan( k 2 L)=0,

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