Abstract

In this work we investigate the extraordinary characteristics of one-dimensional (1D) finite periodic parity-time (PT) symmetric network. On the basis of the transfer matrix method, three simple expressions are analytically obtained for transmission, left reflection and right reflection coefficients. For this periodic structure, we provide new criteria for the PT-symmetry breaking transition in terms of the elements of the transfer matrix and the scattering matrix. These criteria indicate that the exceptional points are related only to the cell structure, but not to the cell number. Utilizing these criteria and expressions, the relationships between the transmittances (reflectances) and the cell number are considered in detail. Furthermore, the conditions for ultrastrong transmission are analytically derived. We also show how a PT-symmetric network can become unidirectionally and bidirectionally transparent at specific frequencies. The conditions and related properties of unidirectional and bidirectional transparencies are also examined. Finally, we find that the finite periodic PT-symmetric network with certain cell number can be viewed as a unidirectionally invisible structure at the exceptional points. Our work may pave the way for designing a diversefamily of optical structures and networks with new properties and functionalities.

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

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    [Crossref]
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2018 (1)

Z. Zhang, P. Miao, J. Sun, S. Longhi, N. M. Litchinitser, and L. Feng, “Elimination of Spatial Hole Burning in Microlasers for Stability and Efficiency Enhancement,” ACS Photonics 5, 3016–3022 (2018).
[Crossref]

2015 (1)

M. Wimmer, M.-A. Miri, D. Christodoulides, and U. Peschel, “Observation of Bloch oscillations in complex PT-symmetric photonic lattices,” Sci. Rep. 5, 17760 (2015).
[Crossref] [PubMed]

2014 (5)

C. Yidong, “Nonlinear Optics: Asymmetry from Symmetry,” Nat. Phys. 10, 336 (2014).
[Crossref]

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

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

L. Feng, X. Zhu, S. Yang, H. Zhu, P. Zhang, X. Yin, Y. Wang, and X. Zhang, “Demonstration of a large-scale optical exceptional point structure,” Opt. Express 22, 1760–1767 (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, 9574–9584 (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, 2821–2824 (2013).
[Crossref] [PubMed]

N. Bender, S. Factor, J. D. Bodyfelt, H. Ramezani, D. N. Christodoulides, F. M. Ellis, and T. Kottos, “Observation of Asymmetric Transport in Structures with Active Nonlinearities,” Phys. Rev. Lett. 110, 234101 (2013).
[Crossref] [PubMed]

2012 (5)

S. Bittner, B. Dietz, U. Günther, H. L. Harney, M. Miski-Oglu, A. Richter, and F. Schäfer, “PT Symmetry and Spontaneous Symmetry Breaking in a Microwave Billiard,” Phys. Rev. Lett. 108, 024101 (2012).
[Crossref]

H. Ramezani, D. N. Christodoulides, V. Kovanis, I. Vitebskiy, and T. Kottos, “PT-Symmetric Talbot Effects,” Phys. Rev. Lett. 109, 033902 (2012).
[Crossref]

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, 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, 167–171 (2012).
[Crossref] [PubMed]

S. Ding and G. P. Wang, “Nonreciprocal optical Bloch-Zener oscillations in ternary parity-time-symmetric waveguide lattices,” Appl. Phys. Lett. 100, 151913 (2012).
[Crossref]

2011 (4)

Y. D. Chong, L. Ge, and A. D. Stone, “PT-Symmetry Breaking and Laser-Absorber Modes in Optical Scattering Systems,” Phys. Rev. Lett. 106, 093902 (2011).
[Crossref]

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, 213901 (2011).
[Crossref]

P. Berini and I. De Leon, “Surface plasmon–polariton amplifiers and lasers,” Nat. Photonics 6, 16 (2011).
[Crossref]

J. Schindler, A. Li, M. C. Zheng, F. M. Ellis, and T. Kottos, “Experimental study of active LRC circuits with PT symmetries,” Phys. Rev. A 84, 040101 (2011).
[Crossref]

2010 (5)

H. Ramezani, T. Kottos, R. El-Ganainy, and D. N. Christodoulides, “Unidirectional nonlinear PT-symmetric optical structures,” Phys. Rev. A 82, 043803 (2010).
[Crossref]

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

S. Longhi, “Spectral singularities and Bragg scattering in complex crystals,” Phys. Rev. A 81, 022102 (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 (2010).
[Crossref]

M. C. Zheng, D. N. Christodoulides, R. Fleischmann, and T. Kottos, “PT optical lattices and universality in beam dynamics,” Phys. Rev. A 82, 010103 (2010).
[Crossref]

2009 (4)

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, “Bloch Oscillations in Complex Crystals with PT Symmetry,” Phys. Rev. Lett. 103, 123601 (2009).
[Crossref]

M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460, 1110 (2009).
[Crossref] [PubMed]

A. Mostafazadeh, “Spectral Singularities of Complex Scattering Potentials and Infinite Reflection and Transmission Coefficients at Real Energies,” Phys. Rev. Lett. 102, 220402 (2009).
[Crossref] [PubMed]

2008 (1)

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

2007 (1)

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

2006 (2)

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling Electromagnetic Fields,” Science 312, 1780 (2006).
[Crossref] [PubMed]

U. Leonhardt, “Optical Conformal Mapping,” Science 312, 1777 (2006).
[Crossref] [PubMed]

2000 (1)

J. B. Pendry, “Negative Refraction Makes a Perfect Lens,” Phys. Rev. Lett. 85, 3966–3969 (2000).
[Crossref] [PubMed]

1998 (3)

C. M. Bender and S. Boettcher, “Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry,” Phys. Rev. Lett. 80, 5243–5246 (1998).
[Crossref]

J. C. Knight, J. Broeng, T. A. Birks, and P. S. J. Russell, “Photonic Band Gap Guidance in Optical Fibers,” Science 282, 1476 (1998).
[Crossref] [PubMed]

Z.-Q. Zhang, C. C. Wong, K. K. Fung, Y. L. Ho, W. L. Chan, S. C. Kan, T. L. Chan, and N. Cheung, “Observation of Localized Electromagnetic Waves in Three-Dimensional Networks Of Waveguides,” Phys. Rev. B 81, 5540–5543 (1998).

1997 (1)

J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, “Photonic crystals: putting a new twist on light,” Nature 386, 143 (1997).
[Crossref]

1983 (1)

S. Alexander, “Superconductivity of networks. A percolation approach to the effects of disorder,” Phys. Rev. B 27, 1541–1557 (1983).
[Crossref]

1977 (1)

Aimez, V.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-Symmetry Breaking in Complex Optical Potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref]

Alexander, S.

S. Alexander, “Superconductivity of networks. A percolation approach to the effects of disorder,” Phys. Rev. B 27, 1541–1557 (1983).
[Crossref]

Bakker, R.

M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460, 1110 (2009).
[Crossref] [PubMed]

Belgrave, A. M.

M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460, 1110 (2009).
[Crossref] [PubMed]

Bender, C. M.

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

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

N. Bender, S. Factor, J. D. Bodyfelt, H. Ramezani, D. N. Christodoulides, F. M. Ellis, and T. Kottos, “Observation of Asymmetric Transport in Structures with Active Nonlinearities,” Phys. Rev. Lett. 110, 234101 (2013).
[Crossref] [PubMed]

Berini, P.

P. Berini and I. De Leon, “Surface plasmon–polariton amplifiers and lasers,” Nat. Photonics 6, 16 (2011).
[Crossref]

Bersch, C.

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

Birks, T. A.

J. C. Knight, J. Broeng, T. A. Birks, and P. S. J. Russell, “Photonic Band Gap Guidance in Optical Fibers,” Science 282, 1476 (1998).
[Crossref] [PubMed]

Bittner, S.

S. Bittner, B. Dietz, U. Günther, H. L. Harney, M. Miski-Oglu, A. Richter, and F. Schäfer, “PT Symmetry and Spontaneous Symmetry Breaking in a Microwave Billiard,” Phys. Rev. Lett. 108, 024101 (2012).
[Crossref]

Bodyfelt, J. D.

N. Bender, S. Factor, J. D. Bodyfelt, H. Ramezani, D. N. Christodoulides, F. M. Ellis, and T. Kottos, “Observation of Asymmetric Transport in Structures with Active Nonlinearities,” Phys. Rev. Lett. 110, 234101 (2013).
[Crossref] [PubMed]

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]

Broeng, J.

J. C. Knight, J. Broeng, T. A. Birks, and P. S. J. Russell, “Photonic Band Gap Guidance in Optical Fibers,” Science 282, 1476 (1998).
[Crossref] [PubMed]

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, 213901 (2011).
[Crossref]

Chan, T. L.

Z.-Q. Zhang, C. C. Wong, K. K. Fung, Y. L. Ho, W. L. Chan, S. C. Kan, T. L. Chan, and N. Cheung, “Observation of Localized Electromagnetic Waves in Three-Dimensional Networks Of Waveguides,” Phys. Rev. B 81, 5540–5543 (1998).

Chan, W. L.

Z.-Q. Zhang, C. C. Wong, K. K. Fung, Y. L. Ho, W. L. Chan, S. C. Kan, T. L. Chan, and N. Cheung, “Observation of Localized Electromagnetic Waves in Three-Dimensional Networks Of Waveguides,” Phys. Rev. B 81, 5540–5543 (1998).

Cheung, N.

Z.-Q. Zhang, C. C. Wong, K. K. Fung, Y. L. Ho, W. L. Chan, S. C. Kan, T. L. Chan, and N. Cheung, “Observation of Localized Electromagnetic Waves in Three-Dimensional Networks Of Waveguides,” Phys. Rev. B 81, 5540–5543 (1998).

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, 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, 093902 (2011).
[Crossref]

Christodoulides, D.

M. Wimmer, M.-A. Miri, D. Christodoulides, and U. Peschel, “Observation of Bloch oscillations in complex PT-symmetric photonic lattices,” Sci. Rep. 5, 17760 (2015).
[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, 9574–9584 (2014).
[Crossref] [PubMed]

N. Bender, S. Factor, J. D. Bodyfelt, H. Ramezani, D. N. Christodoulides, F. M. Ellis, and T. Kottos, “Observation of Asymmetric Transport in Structures with Active Nonlinearities,” Phys. Rev. Lett. 110, 234101 (2013).
[Crossref] [PubMed]

H. Ramezani, D. N. Christodoulides, V. Kovanis, I. Vitebskiy, and T. Kottos, “PT-Symmetric Talbot Effects,” Phys. Rev. Lett. 109, 033902 (2012).
[Crossref]

A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488, 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, 213901 (2011).
[Crossref]

H. Ramezani, T. Kottos, R. El-Ganainy, and D. N. Christodoulides, “Unidirectional nonlinear PT-symmetric optical structures,” Phys. Rev. A 82, 043803 (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 (2010).
[Crossref]

M. C. Zheng, D. N. Christodoulides, R. Fleischmann, and T. Kottos, “PT optical lattices and universality in beam dynamics,” Phys. Rev. A 82, 010103 (2010).
[Crossref]

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-Symmetry Breaking in Complex Optical Potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref]

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

Dietz, B.

S. Bittner, B. Dietz, U. Günther, H. L. Harney, M. Miski-Oglu, A. Richter, and F. Schäfer, “PT Symmetry and Spontaneous Symmetry Breaking in a Microwave Billiard,” Phys. Rev. Lett. 108, 024101 (2012).
[Crossref]

Ding, S.

S. Ding and G. P. Wang, “Nonreciprocal optical Bloch-Zener oscillations in ternary parity-time-symmetric waveguide lattices,” Appl. Phys. Lett. 100, 151913 (2012).
[Crossref]

Duchesne, D.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-Symmetry Breaking in Complex Optical Potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[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, 213901 (2011).
[Crossref]

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).
[Crossref]

H. Ramezani, T. Kottos, R. El-Ganainy, and D. N. Christodoulides, “Unidirectional nonlinear PT-symmetric optical structures,” Phys. Rev. A 82, 043803 (2010).
[Crossref]

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

Ellis, F. M.

N. Bender, S. Factor, J. D. Bodyfelt, H. Ramezani, D. N. Christodoulides, F. M. Ellis, and T. Kottos, “Observation of Asymmetric Transport in Structures with Active Nonlinearities,” Phys. Rev. Lett. 110, 234101 (2013).
[Crossref] [PubMed]

J. Schindler, A. Li, M. C. Zheng, F. M. Ellis, and T. Kottos, “Experimental study of active LRC circuits with PT symmetries,” Phys. Rev. A 84, 040101 (2011).
[Crossref]

Factor, S.

N. Bender, S. Factor, J. D. Bodyfelt, H. Ramezani, D. N. Christodoulides, F. M. Ellis, and T. Kottos, “Observation of Asymmetric Transport in Structures with Active Nonlinearities,” Phys. Rev. Lett. 110, 234101 (2013).
[Crossref] [PubMed]

Fan, S.

J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, “Photonic crystals: putting a new twist on light,” Nature 386, 143 (1997).
[Crossref]

Feng, L.

Fleischmann, R.

M. C. Zheng, D. N. Christodoulides, R. Fleischmann, and T. Kottos, “PT optical lattices and universality in beam dynamics,” Phys. Rev. A 82, 010103 (2010).
[Crossref]

Fung, K. K.

Z.-Q. Zhang, C. C. Wong, K. K. Fung, Y. L. Ho, W. L. Chan, S. C. Kan, T. L. Chan, and N. Cheung, “Observation of Localized Electromagnetic Waves in Three-Dimensional Networks Of Waveguides,” Phys. Rev. B 81, 5540–5543 (1998).

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, 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, 093902 (2011).
[Crossref]

Günther, U.

S. Bittner, B. Dietz, U. Günther, H. L. Harney, M. Miski-Oglu, A. Richter, and F. Schäfer, “PT Symmetry and Spontaneous Symmetry Breaking in a Microwave Billiard,” Phys. Rev. Lett. 108, 024101 (2012).
[Crossref]

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, 093902 (2009).
[Crossref]

Harney, H. L.

S. Bittner, B. Dietz, U. Günther, H. L. Harney, M. Miski-Oglu, A. Richter, and F. Schäfer, “PT Symmetry and Spontaneous Symmetry Breaking in a Microwave Billiard,” Phys. Rev. Lett. 108, 024101 (2012).
[Crossref]

Herz, E.

M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460, 1110 (2009).
[Crossref] [PubMed]

Ho, Y. L.

Z.-Q. Zhang, C. C. Wong, K. K. Fung, Y. L. Ho, W. L. Chan, S. C. Kan, T. L. Chan, and N. Cheung, “Observation of Localized Electromagnetic Waves in Three-Dimensional Networks Of Waveguides,” Phys. Rev. B 81, 5540–5543 (1998).

Hong, C. S.

Joannopoulos, J. D.

J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, “Photonic crystals: putting a new twist on light,” Nature 386, 143 (1997).
[Crossref]

Kan, S. C.

Z.-Q. Zhang, C. C. Wong, K. K. Fung, Y. L. Ho, W. L. Chan, S. C. Kan, T. L. Chan, and N. Cheung, “Observation of Localized Electromagnetic Waves in Three-Dimensional Networks Of Waveguides,” Phys. Rev. B 81, 5540–5543 (1998).

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).
[Crossref]

Knight, J. C.

J. C. Knight, J. Broeng, T. A. Birks, and P. S. J. Russell, “Photonic Band Gap Guidance in Optical Fibers,” Science 282, 1476 (1998).
[Crossref] [PubMed]

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

N. Bender, S. Factor, J. D. Bodyfelt, H. Ramezani, D. N. Christodoulides, F. M. Ellis, and T. Kottos, “Observation of Asymmetric Transport in Structures with Active Nonlinearities,” Phys. Rev. Lett. 110, 234101 (2013).
[Crossref] [PubMed]

H. Ramezani, D. N. Christodoulides, V. Kovanis, I. Vitebskiy, and T. Kottos, “PT-Symmetric Talbot Effects,” Phys. Rev. Lett. 109, 033902 (2012).
[Crossref]

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, 213901 (2011).
[Crossref]

J. Schindler, A. Li, M. C. Zheng, F. M. Ellis, and T. Kottos, “Experimental study of active LRC circuits with PT symmetries,” Phys. Rev. A 84, 040101 (2011).
[Crossref]

H. Ramezani, T. Kottos, R. El-Ganainy, and D. N. Christodoulides, “Unidirectional nonlinear PT-symmetric optical structures,” Phys. Rev. A 82, 043803 (2010).
[Crossref]

M. C. Zheng, D. N. Christodoulides, R. Fleischmann, and T. Kottos, “PT optical lattices and universality in beam dynamics,” Phys. Rev. A 82, 010103 (2010).
[Crossref]

Kovanis, V.

H. Ramezani, D. N. Christodoulides, V. Kovanis, I. Vitebskiy, and T. Kottos, “PT-Symmetric Talbot Effects,” Phys. Rev. Lett. 109, 033902 (2012).
[Crossref]

Leon, I. De

P. Berini and I. De Leon, “Surface plasmon–polariton amplifiers and lasers,” Nat. Photonics 6, 16 (2011).
[Crossref]

Leonhardt, U.

U. Leonhardt, “Optical Conformal Mapping,” Science 312, 1777 (2006).
[Crossref] [PubMed]

Li, A.

J. Schindler, A. Li, M. C. Zheng, F. M. Ellis, and T. Kottos, “Experimental study of active LRC circuits with PT symmetries,” Phys. Rev. A 84, 040101 (2011).
[Crossref]

Li, H.-K.

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

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, 213901 (2011).
[Crossref]

Litchinitser, N. M.

Z. Zhang, P. Miao, J. Sun, S. Longhi, N. M. Litchinitser, and L. Feng, “Elimination of Spatial Hole Burning in Microlasers for Stability and Efficiency Enhancement,” ACS Photonics 5, 3016–3022 (2018).
[Crossref]

Longhi, S.

Z. Zhang, P. Miao, J. Sun, S. Longhi, N. M. Litchinitser, and L. Feng, “Elimination of Spatial Hole Burning in Microlasers for Stability and Efficiency Enhancement,” ACS Photonics 5, 3016–3022 (2018).
[Crossref]

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

S. Longhi, “Spectral singularities and Bragg scattering in complex crystals,” Phys. Rev. A 81, 022102 (2010).
[Crossref]

S. Longhi, “Bloch Oscillations in Complex Crystals with PT Symmetry,” Phys. Rev. Lett. 103, 123601 (2009).
[Crossref]

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).
[Crossref]

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

Miao, P.

Z. Zhang, P. Miao, J. Sun, S. Longhi, N. M. Litchinitser, and L. Feng, “Elimination of Spatial Hole Burning in Microlasers for Stability and Efficiency Enhancement,” ACS Photonics 5, 3016–3022 (2018).
[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, 167–171 (2012).
[Crossref] [PubMed]

Miri, M.-A.

M. Wimmer, M.-A. Miri, D. Christodoulides, and U. Peschel, “Observation of Bloch oscillations in complex PT-symmetric photonic lattices,” Sci. Rep. 5, 17760 (2015).
[Crossref] [PubMed]

Miski-Oglu, M.

S. Bittner, B. Dietz, U. Günther, H. L. Harney, M. Miski-Oglu, A. Richter, and F. Schäfer, “PT Symmetry and Spontaneous Symmetry Breaking in a Microwave Billiard,” Phys. Rev. Lett. 108, 024101 (2012).
[Crossref]

Morandotti, R.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-Symmetry Breaking in Complex Optical Potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref]

Moravvej-Farshi, M. K.

Mostafazadeh, A.

A. Mostafazadeh, “Spectral Singularities of Complex Scattering Potentials and Infinite Reflection and Transmission Coefficients at Real Energies,” Phys. Rev. Lett. 102, 220402 (2009).
[Crossref] [PubMed]

Musslimani, Z. H.

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

Narimanov, E. E.

M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460, 1110 (2009).
[Crossref] [PubMed]

Nazari, F.

Noginov, M. A.

M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460, 1110 (2009).
[Crossref] [PubMed]

Onishchukov, G.

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

Pendry, J. B.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling Electromagnetic Fields,” Science 312, 1780 (2006).
[Crossref] [PubMed]

J. B. Pendry, “Negative Refraction Makes a Perfect Lens,” Phys. Rev. Lett. 85, 3966–3969 (2000).
[Crossref] [PubMed]

Peschel, U.

M. Wimmer, M.-A. Miri, D. Christodoulides, and U. Peschel, “Observation of Bloch oscillations in complex PT-symmetric photonic lattices,” Sci. Rep. 5, 17760 (2015).
[Crossref] [PubMed]

A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488, 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, 031042 (2014).

H. Ramezani, H.-K. Li, Y. Wang, and X. Zhang, “Unidirectional Spectral Singularities,” Phys. Rev. Lett. 113, 263905 (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, 9574–9584 (2014).
[Crossref] [PubMed]

N. Bender, S. Factor, J. D. Bodyfelt, H. Ramezani, D. N. Christodoulides, F. M. Ellis, and T. Kottos, “Observation of Asymmetric Transport in Structures with Active Nonlinearities,” Phys. Rev. Lett. 110, 234101 (2013).
[Crossref] [PubMed]

H. Ramezani, D. N. Christodoulides, V. Kovanis, I. Vitebskiy, and T. Kottos, “PT-Symmetric Talbot Effects,” Phys. Rev. Lett. 109, 033902 (2012).
[Crossref]

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, 213901 (2011).
[Crossref]

H. Ramezani, T. Kottos, R. El-Ganainy, and D. N. Christodoulides, “Unidirectional nonlinear PT-symmetric optical structures,” Phys. Rev. A 82, 043803 (2010).
[Crossref]

Regensburger, A.

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

Richter, A.

S. Bittner, B. Dietz, U. Günther, H. L. Harney, M. Miski-Oglu, A. Richter, and F. Schäfer, “PT Symmetry and Spontaneous Symmetry Breaking in a Microwave Billiard,” Phys. Rev. Lett. 108, 024101 (2012).
[Crossref]

Russell, P. S. J.

J. C. Knight, J. Broeng, T. A. Birks, and P. S. J. Russell, “Photonic Band Gap Guidance in Optical Fibers,” Science 282, 1476 (1998).
[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, 192 (2010).
[Crossref]

Salamo, G. J.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-Symmetry Breaking in Complex Optical Potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref]

Schäfer, F.

S. Bittner, B. Dietz, U. Günther, H. L. Harney, M. Miski-Oglu, A. Richter, and F. Schäfer, “PT Symmetry and Spontaneous Symmetry Breaking in a Microwave Billiard,” Phys. Rev. Lett. 108, 024101 (2012).
[Crossref]

Schindler, J.

J. Schindler, A. Li, M. C. Zheng, F. M. Ellis, and T. Kottos, “Experimental study of active LRC circuits with PT symmetries,” Phys. Rev. A 84, 040101 (2011).
[Crossref]

Schurig, D.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling Electromagnetic Fields,” Science 312, 1780 (2006).
[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, 192 (2010).
[Crossref]

Shalaev, V. M.

M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460, 1110 (2009).
[Crossref] [PubMed]

Shi, C.

X. Zhu, H. Ramezani, C. Shi, J. Zhu, and X. Zhang, “PT-Symmetric Acoustics,” Phys. Rev. X 4, 031042 (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, 093902 (2009).
[Crossref]

Smith, D. R.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling Electromagnetic Fields,” Science 312, 1780 (2006).
[Crossref] [PubMed]

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, 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, 093902 (2011).
[Crossref]

Stout, S.

M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460, 1110 (2009).
[Crossref] [PubMed]

Sun, J.

Z. Zhang, P. Miao, J. Sun, S. Longhi, N. M. Litchinitser, and L. Feng, “Elimination of Spatial Hole Burning in Microlasers for Stability and Efficiency Enhancement,” ACS Photonics 5, 3016–3022 (2018).
[Crossref]

Suteewong, T.

M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460, 1110 (2009).
[Crossref] [PubMed]

Villeneuve, P. R.

J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, “Photonic crystals: putting a new twist on light,” Nature 386, 143 (1997).
[Crossref]

Vitebskiy, I.

H. Ramezani, D. N. Christodoulides, V. Kovanis, I. Vitebskiy, and T. Kottos, “PT-Symmetric Talbot Effects,” Phys. Rev. Lett. 109, 033902 (2012).
[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]

Wang, G. P.

S. Ding and G. P. Wang, “Nonreciprocal optical Bloch-Zener oscillations in ternary parity-time-symmetric waveguide lattices,” Appl. Phys. Lett. 100, 151913 (2012).
[Crossref]

Wang, Y.

Wiesner, U.

M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460, 1110 (2009).
[Crossref] [PubMed]

Wimmer, M.

M. Wimmer, M.-A. Miri, D. Christodoulides, and U. Peschel, “Observation of Bloch oscillations in complex PT-symmetric photonic lattices,” Sci. Rep. 5, 17760 (2015).
[Crossref] [PubMed]

Wong, C. C.

Z.-Q. Zhang, C. C. Wong, K. K. Fung, Y. L. Ho, W. L. Chan, S. C. Kan, T. L. Chan, and N. Cheung, “Observation of Localized Electromagnetic Waves in Three-Dimensional Networks Of Waveguides,” Phys. Rev. B 81, 5540–5543 (1998).

Yang, S.

Yariv, A.

Yeh, P.

Yidong, C.

C. Yidong, “Nonlinear Optics: Asymmetry from Symmetry,” Nat. Phys. 10, 336 (2014).
[Crossref]

Yin, X.

Zhang, P.

Zhang, X.

Zhang, Z.

Z. Zhang, P. Miao, J. Sun, S. Longhi, N. M. Litchinitser, and L. Feng, “Elimination of Spatial Hole Burning in Microlasers for Stability and Efficiency Enhancement,” ACS Photonics 5, 3016–3022 (2018).
[Crossref]

Zhang, Z.-Q.

Z.-Q. Zhang, C. C. Wong, K. K. Fung, Y. L. Ho, W. L. Chan, S. C. Kan, T. L. Chan, and N. Cheung, “Observation of Localized Electromagnetic Waves in Three-Dimensional Networks Of Waveguides,” Phys. Rev. B 81, 5540–5543 (1998).

Zheng, M. C.

J. Schindler, A. Li, M. C. Zheng, F. M. Ellis, and T. Kottos, “Experimental study of active LRC circuits with PT symmetries,” Phys. Rev. A 84, 040101 (2011).
[Crossref]

M. C. Zheng, D. N. Christodoulides, R. Fleischmann, and T. Kottos, “PT optical lattices and universality in beam dynamics,” Phys. Rev. A 82, 010103 (2010).
[Crossref]

Zhu, G.

M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460, 1110 (2009).
[Crossref] [PubMed]

Zhu, H.

Zhu, J.

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

Zhu, X.

ACS Photonics (1)

Z. Zhang, P. Miao, J. Sun, S. Longhi, N. M. Litchinitser, and L. Feng, “Elimination of Spatial Hole Burning in Microlasers for Stability and Efficiency Enhancement,” ACS Photonics 5, 3016–3022 (2018).
[Crossref]

Appl. Phys. Lett. (1)

S. Ding and G. P. Wang, “Nonreciprocal optical Bloch-Zener oscillations in ternary parity-time-symmetric waveguide lattices,” Appl. Phys. Lett. 100, 151913 (2012).
[Crossref]

J. Opt. Soc. Am. (1)

Nat. Photonics (1)

P. Berini and I. De Leon, “Surface plasmon–polariton amplifiers and lasers,” Nat. Photonics 6, 16 (2011).
[Crossref]

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, 192 (2010).
[Crossref]

C. Yidong, “Nonlinear Optics: Asymmetry from Symmetry,” Nat. Phys. 10, 336 (2014).
[Crossref]

Nature (3)

J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, “Photonic crystals: putting a new twist on light,” Nature 386, 143 (1997).
[Crossref]

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

M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460, 1110 (2009).
[Crossref] [PubMed]

Opt. Express (2)

Opt. Lett. (1)

Phys. Rev. A (6)

S. Longhi, “Spectral singularities and Bragg scattering in complex crystals,” Phys. Rev. A 81, 022102 (2010).
[Crossref]

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

M. C. Zheng, D. N. Christodoulides, R. Fleischmann, and T. Kottos, “PT optical lattices and universality in beam dynamics,” Phys. Rev. A 82, 010103 (2010).
[Crossref]

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, 023802 (2012).
[Crossref]

J. Schindler, A. Li, M. C. Zheng, F. M. Ellis, and T. Kottos, “Experimental study of active LRC circuits with PT symmetries,” Phys. Rev. A 84, 040101 (2011).
[Crossref]

H. Ramezani, T. Kottos, R. El-Ganainy, and D. N. Christodoulides, “Unidirectional nonlinear PT-symmetric optical structures,” Phys. Rev. A 82, 043803 (2010).
[Crossref]

Phys. Rev. B (2)

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[Crossref]

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Phys. Rev. Lett. (12)

H. Ramezani, D. N. Christodoulides, V. Kovanis, I. Vitebskiy, and T. Kottos, “PT-Symmetric Talbot Effects,” Phys. Rev. Lett. 109, 033902 (2012).
[Crossref]

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

N. Bender, S. Factor, J. D. Bodyfelt, H. Ramezani, D. N. Christodoulides, F. M. Ellis, and T. Kottos, “Observation of Asymmetric Transport in Structures with Active Nonlinearities,” Phys. Rev. Lett. 110, 234101 (2013).
[Crossref] [PubMed]

S. Bittner, B. Dietz, U. Günther, H. L. Harney, M. Miski-Oglu, A. Richter, and F. Schäfer, “PT Symmetry and Spontaneous Symmetry Breaking in a Microwave Billiard,” Phys. Rev. Lett. 108, 024101 (2012).
[Crossref]

A. Mostafazadeh, “Spectral Singularities of Complex Scattering Potentials and Infinite Reflection and Transmission Coefficients at Real Energies,” Phys. Rev. Lett. 102, 220402 (2009).
[Crossref] [PubMed]

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[Crossref]

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, 213901 (2011).
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K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam Dynamics in PT Symmetric Optical Lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[Crossref]

Phys. Rev. X (1)

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

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

Sci. Rep. (1)

M. Wimmer, M.-A. Miri, D. Christodoulides, and U. Peschel, “Observation of Bloch oscillations in complex PT-symmetric photonic lattices,” Sci. Rep. 5, 17760 (2015).
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Figures (9)

Fig. 1
Fig. 1 Schematic of the 1D PT-symmetric periodic optical waveguide network containing N cells. Red lines labeled n1 are the loss segments ( n 1 = 2 + 0.005 ι ), green lines labeled n2 are the gain segments ( n 2 = 2 0.005 ι ), and black lines labeled n0 are the passive segments ( n 0 = 2 ). The lengths of the two loss (gain) segments between adjacent nodes (blue dots) are d1 and d2, and their length ratio is d 1 : d 2 = 1 : 2. The lengths of the two passive segments are d.
Fig. 2
Fig. 2 Logarithm of the absolute values of the eigenvalues of the corresponding scattering matrix for the finite periodic PT-symmetric network with different cell numbers. Red solid lines correspond to N = 1, blue dashed lines correspond to N = 4, and black dashed lines correspond to N = 8.
Fig. 3
Fig. 3 Logarithm of the absolute values of the eigenvalues of the corresponding scattering matrix for the PT-symmetric network with different length ratios. Here the PT-symmetric network contains only one cell. (a) Green solid lines correspond to d 1 : d 2 = 1 : 1, magenta solid lines correspond to d 1 : d 2 = 1 : 3, and blue solid lines correspond to d 1 : d 2 = 1 : 4. (b) Cyan solid lines correspond to d 1 : d 2 = 1 : 5, black solid lines correspond to d 1 : d 2 = 1 : 8, and red dashed lines correspond to d 1 : d 2 = 1 : 17.
Fig. 4
Fig. 4 (a) Transmittance (TN), (b) left reflectance ( R N ( L )), and (c) right reflectance ( R N ( R )) for the finite periodic PT-symmetric network as a function of cell number N. The black dot-solid lines correspond to the PT-symmetric phase ( T 1 < 1 ), the red circle-solid lines correspond to the exceptional point ( T 1 = 1 ), the blue dot-solid lines correspond to the broken phase ( T 1 > 1 ), and the cyan dot-dashed lines correspond to the degeneracy point ( T 1 = 1, not the exceptional point).
Fig. 5
Fig. 5 (a) The imaginary part of 1 / t 1 between frequency 62.10 π c / d and 62.40 π c / d, (b) the values of sin N K with N = 34 in the same frequency range, and (c) the transmittance of the periodic PT-symmetric network with two different cell numbers, where the green line corresponds to N = 1, and the red line corresponds to N = 34
Fig. 6
Fig. 6 Unidirectional transparency characteristics of the finite periodic PT-symmetric network with four cells. (a) Logarithm of the absolute values of the eigenvalues of the corresponding scattering matrix. (b) Transmittance, left reflectance and right reflectance. Phases and delay times of transmitted, left-reflected, and right-reflected waves of the finite periodic PT-symmetric are shown in (c) and (d).
Fig. 7
Fig. 7 Bidirectional transparency characteristics of the finite periodic PT-symmetric network with four cells in the PT-symmetric phase. (a) Logarithm of the absolute values of the eigenvalues of the corresponding scattering matrix. (b) Transmittance, left reflectance and right reflectance. Phases and delay times of transmitted, left-reflected, and right-reflected waves of the PT-symmetric are shown in (c) and (d).
Fig. 8
Fig. 8 Bidirectional transparency characteristics of the finite periodic PT-symmetric network with four cells in the broken phase. (a) Logarithm of the absolute values of the eigenvalues of the corresponding scattering matrix. (b) Transmittance, left reflectance and right reflectance. Phases and delay times of the transmitted, left-reflected, and right-reflected waves of the PT-symmetric are shown in (c) and (d).
Fig. 9
Fig. 9 The absolute value of the phase differences for transmitted waves traveling through the PT-symmetric network ( ϕ PT ) and those traveling through a homogeneous background medium (ϕ) vary with cell number N at ω = 62.1895 π c / d .

Equations (21)

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2 φ i j ( x ) x 2 + ( n i j ω c ) 2 φ i j ( x ) = 0 ,
{ φ 1 , 0 ( x ) = α 1 e ι k ( x d ) + β 1 e ι k ( x d ) , φ 2 N , 2 N + 1 ( x ) = α 2 e ι k x + β 2 e ι k x ,
( α 2 β 2 ) = M N ( α 1 β 1 ) = ( m N ( 11 ) m N ( 12 ) m N ( 21 ) m N ( 22 ) ) ( α 1 β 1 ) .
t N ( L ) = t N ( R ) = t N = 1 m N ( 22 ) , r N ( L ) = m N ( 21 ) m N ( 22 ) , r N ( R ) = m N ( 12 ) m N ( 22 ) .
φ i j ( x ) = α i j e z i j x + β i j e z i j x ,
φ i j ( x ) = φ i sinh  [ z i j ( l i j x ) ] sinh  ( z i j l i j ) + φ j sinh  ( z i j x ) sinh  ( z i j l i j ) ,
j φ i j ( x ) x | x = 0 = 0 .
φ i ε i + φ i 1 κ i 1 , i + φ i + 1 κ i , i + 1 = 0 ,
{ ε i = q coth  ( z i 1 , i d q ) + q coth  ( z i , i + 1 d q ) , κ i 1 , i = q csch  ( z i 1 , i d q ) , κ i , i + 1 = q csch  ( z i , i + 1 d q ) ,
( φ i + 1 φ i ) = ( ε i κ i , i + 1 κ i 1 , i κ i , i + 1 1 0 ) ( φ i φ i 1 ) = Γ i ( φ i φ i 1 ) .
( φ 2 N + 1 φ 2 N ) = ( e ι k d e ι k d 1 1 ) ( α 2 β 2 ) = Q R ( α 2 β 2 ) .
M N = Q R 1 Γ 2 N Γ 2 N 1 Γ 2 N 2 Γ 3 Γ 2 Γ 1 Γ 0 Q L ,
Γ 2 N 1 Γ 2 N 2 = Γ 2 N 3 Γ 2 N 4 = = Γ 3 Γ 2 = Γ = ( Γ 11 Γ 12 Γ 21 Γ 22 ) ,
Γ N 1 = ( Γ 11 U N 2 ( u ) U N 3 ( u ) Γ 12 U N 2 ( u ) Γ 21 U N 2 ( u ) Γ 22 U N 2 ( u ) U N 3 ( u ) ) ,
U N ( u ) = sin  [ ( N + 1 ) arccos  u ] 1 u 2 .
M N = ( m 11 sin  N K sin  K sin  ( N 1 ) K sin  K m 12 sin  N K sin  K     m 21 sin  N K sin  K m 22 sin  N K sin  K sin  ( N 1 ) K sin  K ) ,
{ t N = sin  K ( 1 t 1 sin  N K sin  ( N 1 ) K ) 1 ,   r N ( L ) = r 1 ( L ) ( 1 t 1 sin  ( N 1 ) K sin  N K ) 1 ,   r N ( R ) = r 1 ( R ) ( 1 t 1 sin  ( N 1 ) K sin  N K ) 1 .
( α 2 β 1 ) = S ( ω ) ( α 1 β 2 ) = ( t N r N ( R ) r N ( L ) t N ) ( α 1 β 2 ) ,
s 1 , 2 = sin  K 1 / t 1 × sin  N K sin  ( N 1 ) K ( 1 ± ι sin  N K sin  K 1 T 1 1 ) .
t N = 1 cos  N K + ι Im ( 1 / t 1 ) sin  N K sin  K .
ϕ t N = 1 ι ln  ( | 1 / t 1 sin  N K sin  ( N 1 ) K | 1 / t 1 sin  N K sin  ( N 1 ) K ) .

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