Abstract

Light propagation through a finite-width periodically modulated layer obeying parity-time (PT) symmetry is considered. We consider the configuration when the resonant conditions of mode coupling by the grating are satisfied. It is shown that the dependence of the transmission and reflection coefficients on the slab width has resonant character featuring strong amplification of reflected and transmitted waves with negative angles. The dependence of the scattering data on the gain-and-loss intensity also feature strong resonances near the PT-symmetry breaking point, when the slab strongly amplifies waves reflected and transmitted with negative angles, provided the incident wave has a positive angle of incidence.

© 2016 Optical Society of America

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References

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  1. C. Keller, M. K. Oberthaler, R. Abfalterer, S. Bernet, J. Schmiedmayer, and A. Zeilinger, “Tailored Complex Potentials and Friedel’s Law in Atom Optics,” Phys. Rev. Lett. 79, 3327–3330 (1997).
    [Crossref]
  2. C. M. Bender and S. Boettcher, “Real spectra in Non-Hermitian Hamiltonians having PT-symmetry,” Phys. Rev. Lett. 805243 (1998).
    [Crossref]
  3. C. M. Bender, “Making sense of non-Hermitian Hamiltonians,” Rep. Prog. Phys. 70, 947 (2007).
    [Crossref]
  4. M. V. Berry, “Lop-sided diffraction by absorbing crystals,” J. Phys. A 31, 3943 (1998).
    [Crossref]
  5. K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100103904 (2008).
    [Crossref] [PubMed]
  6. Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100030402 (2008).
    [Crossref]
  7. M. V. Berry, “Optical lattices with PT symmetry are not transparent,” J. Phys. A 41, 244007 (2008).
    [Crossref]
  8. S. Longhi, “Spectral singularities and Bragg scattering in complex crystals,” Phys. Rev. A, 81, 022102 (2010).
    [Crossref]
  9. Y. V. Kartashov, V. A. Vysloukh, V. V. Konotop, and L. Torner, “Diffraction control in PT-symmetric photonic lattices: From beam rectification to dynamic localization,” Phys. Rev. A 93, 013841 (2016).
    [Crossref]
  10. M. Kulishov, J. M. Laniel, N. Bélanger, J. Azaña, and D. V. Plant, “Nonreciprocal waveguide Bragg gratings,” Opt. Expr. 13, 3068–30678 (2005).
    [Crossref]
  11. 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] [PubMed]
  12. L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M.-H. Lu, Y.-F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333, 729 (2011).
    [Crossref] [PubMed]
  13. 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. Materials 12, 108 (2013).
    [Crossref]
  14. A. Yariv and P. Yeh, Optical Waves in Crystlas (John Wiley and Sons, 1984).
  15. A. Mostafazadeh, “Exact PT-symmetry is equivalent to Hermiticity,” J. Phys. A: Math. Gen. 36, 7081–7091 (2003).
    [Crossref]
  16. V. V. Konotop, J. Yang, and D. A. Zezyulin, “Nonlinear waves in PT-symmetric systems,” Rev. Mod. Phys. 88, 035002 (2016).
    [Crossref]
  17. A. A. Skorynin, V. A. Bushuev, and B. I. Mantsyzov, “Dynamical Bragg diffraction of optical pulses in photonic crystals in the Laue geometry: diffraction induced splitting, selective compression, and focusing of pulses,” J. Exp. Theor. Phys. 112, 56 (2012).
    [Crossref]

2016 (2)

Y. V. Kartashov, V. A. Vysloukh, V. V. Konotop, and L. Torner, “Diffraction control in PT-symmetric photonic lattices: From beam rectification to dynamic localization,” Phys. Rev. A 93, 013841 (2016).
[Crossref]

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

2013 (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. Materials 12, 108 (2013).
[Crossref]

2012 (1)

A. A. Skorynin, V. A. Bushuev, and B. I. Mantsyzov, “Dynamical Bragg diffraction of optical pulses in photonic crystals in the Laue geometry: diffraction induced splitting, selective compression, and focusing of pulses,” J. Exp. Theor. Phys. 112, 56 (2012).
[Crossref]

2011 (2)

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] [PubMed]

L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M.-H. Lu, Y.-F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333, 729 (2011).
[Crossref] [PubMed]

2010 (1)

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

2008 (3)

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100103904 (2008).
[Crossref] [PubMed]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100030402 (2008).
[Crossref]

M. V. Berry, “Optical lattices with PT symmetry are not transparent,” J. Phys. A 41, 244007 (2008).
[Crossref]

2007 (1)

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

2005 (1)

M. Kulishov, J. M. Laniel, N. Bélanger, J. Azaña, and D. V. Plant, “Nonreciprocal waveguide Bragg gratings,” Opt. Expr. 13, 3068–30678 (2005).
[Crossref]

2003 (1)

A. Mostafazadeh, “Exact PT-symmetry is equivalent to Hermiticity,” J. Phys. A: Math. Gen. 36, 7081–7091 (2003).
[Crossref]

1998 (2)

M. V. Berry, “Lop-sided diffraction by absorbing crystals,” J. Phys. A 31, 3943 (1998).
[Crossref]

C. M. Bender and S. Boettcher, “Real spectra in Non-Hermitian Hamiltonians having PT-symmetry,” Phys. Rev. Lett. 805243 (1998).
[Crossref]

1997 (1)

C. Keller, M. K. Oberthaler, R. Abfalterer, S. Bernet, J. Schmiedmayer, and A. Zeilinger, “Tailored Complex Potentials and Friedel’s Law in Atom Optics,” Phys. Rev. Lett. 79, 3327–3330 (1997).
[Crossref]

Abfalterer, R.

C. Keller, M. K. Oberthaler, R. Abfalterer, S. Bernet, J. Schmiedmayer, and A. Zeilinger, “Tailored Complex Potentials and Friedel’s Law in Atom Optics,” Phys. Rev. Lett. 79, 3327–3330 (1997).
[Crossref]

Almeida, V. R.

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

Ayache, M.

L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M.-H. Lu, Y.-F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333, 729 (2011).
[Crossref] [PubMed]

Azaña, J.

M. Kulishov, J. M. Laniel, N. Bélanger, J. Azaña, and D. V. Plant, “Nonreciprocal waveguide Bragg gratings,” Opt. Expr. 13, 3068–30678 (2005).
[Crossref]

Bélanger, N.

M. Kulishov, J. M. Laniel, N. Bélanger, J. Azaña, and D. V. Plant, “Nonreciprocal waveguide Bragg gratings,” Opt. Expr. 13, 3068–30678 (2005).
[Crossref]

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. 805243 (1998).
[Crossref]

Bernet, S.

C. Keller, M. K. Oberthaler, R. Abfalterer, S. Bernet, J. Schmiedmayer, and A. Zeilinger, “Tailored Complex Potentials and Friedel’s Law in Atom Optics,” Phys. Rev. Lett. 79, 3327–3330 (1997).
[Crossref]

Berry, M. V.

M. V. Berry, “Optical lattices with PT symmetry are not transparent,” J. Phys. A 41, 244007 (2008).
[Crossref]

M. V. Berry, “Lop-sided diffraction by absorbing crystals,” J. Phys. A 31, 3943 (1998).
[Crossref]

Boettcher, S.

C. M. Bender and S. Boettcher, “Real spectra in Non-Hermitian Hamiltonians having PT-symmetry,” Phys. Rev. Lett. 805243 (1998).
[Crossref]

Bushuev, V. A.

A. A. Skorynin, V. A. Bushuev, and B. I. Mantsyzov, “Dynamical Bragg diffraction of optical pulses in photonic crystals in the Laue geometry: diffraction induced splitting, selective compression, and focusing of pulses,” J. Exp. Theor. Phys. 112, 56 (2012).
[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, 213901 (2011).
[Crossref] [PubMed]

Chen, Y.-F.

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

L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M.-H. Lu, Y.-F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333, 729 (2011).
[Crossref] [PubMed]

Christodoulides, D. N.

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] [PubMed]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100103904 (2008).
[Crossref] [PubMed]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100030402 (2008).
[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] [PubMed]

El-Ganainy, R.

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100030402 (2008).
[Crossref]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100103904 (2008).
[Crossref] [PubMed]

Fainman, Y.

L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M.-H. Lu, Y.-F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333, 729 (2011).
[Crossref] [PubMed]

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. Materials 12, 108 (2013).
[Crossref]

Feng, 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. Materials 12, 108 (2013).
[Crossref]

L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M.-H. Lu, Y.-F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333, 729 (2011).
[Crossref] [PubMed]

Huang, J.

L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M.-H. Lu, Y.-F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333, 729 (2011).
[Crossref] [PubMed]

Kartashov, Y. V.

Y. V. Kartashov, V. A. Vysloukh, V. V. Konotop, and L. Torner, “Diffraction control in PT-symmetric photonic lattices: From beam rectification to dynamic localization,” Phys. Rev. A 93, 013841 (2016).
[Crossref]

Keller, C.

C. Keller, M. K. Oberthaler, R. Abfalterer, S. Bernet, J. Schmiedmayer, and A. Zeilinger, “Tailored Complex Potentials and Friedel’s Law in Atom Optics,” Phys. Rev. Lett. 79, 3327–3330 (1997).
[Crossref]

Konotop, V. V.

Y. V. Kartashov, V. A. Vysloukh, V. V. Konotop, and L. Torner, “Diffraction control in PT-symmetric photonic lattices: From beam rectification to dynamic localization,” Phys. Rev. A 93, 013841 (2016).
[Crossref]

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

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, 213901 (2011).
[Crossref] [PubMed]

Kulishov, M.

M. Kulishov, J. M. Laniel, N. Bélanger, J. Azaña, and D. V. Plant, “Nonreciprocal waveguide Bragg gratings,” Opt. Expr. 13, 3068–30678 (2005).
[Crossref]

Laniel, J. M.

M. Kulishov, J. M. Laniel, N. Bélanger, J. Azaña, and D. V. Plant, “Nonreciprocal waveguide Bragg gratings,” Opt. Expr. 13, 3068–30678 (2005).
[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] [PubMed]

Longhi, S.

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

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. Materials 12, 108 (2013).
[Crossref]

L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M.-H. Lu, Y.-F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333, 729 (2011).
[Crossref] [PubMed]

Makris, K. G.

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100030402 (2008).
[Crossref]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100103904 (2008).
[Crossref] [PubMed]

Mantsyzov, B. I.

A. A. Skorynin, V. A. Bushuev, and B. I. Mantsyzov, “Dynamical Bragg diffraction of optical pulses in photonic crystals in the Laue geometry: diffraction induced splitting, selective compression, and focusing of pulses,” J. Exp. Theor. Phys. 112, 56 (2012).
[Crossref]

Mostafazadeh, A.

A. Mostafazadeh, “Exact PT-symmetry is equivalent to Hermiticity,” J. Phys. A: Math. Gen. 36, 7081–7091 (2003).
[Crossref]

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. 100103904 (2008).
[Crossref] [PubMed]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100030402 (2008).
[Crossref]

Oberthaler, M. K.

C. Keller, M. K. Oberthaler, R. Abfalterer, S. Bernet, J. Schmiedmayer, and A. Zeilinger, “Tailored Complex Potentials and Friedel’s Law in Atom Optics,” Phys. Rev. Lett. 79, 3327–3330 (1997).
[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. Materials 12, 108 (2013).
[Crossref]

Plant, D. V.

M. Kulishov, J. M. Laniel, N. Bélanger, J. Azaña, and D. V. Plant, “Nonreciprocal waveguide Bragg gratings,” Opt. Expr. 13, 3068–30678 (2005).
[Crossref]

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, 213901 (2011).
[Crossref] [PubMed]

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. Materials 12, 108 (2013).
[Crossref]

L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M.-H. Lu, Y.-F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333, 729 (2011).
[Crossref] [PubMed]

Schmiedmayer, J.

C. Keller, M. K. Oberthaler, R. Abfalterer, S. Bernet, J. Schmiedmayer, and A. Zeilinger, “Tailored Complex Potentials and Friedel’s Law in Atom Optics,” Phys. Rev. Lett. 79, 3327–3330 (1997).
[Crossref]

Skorynin, A. A.

A. A. Skorynin, V. A. Bushuev, and B. I. Mantsyzov, “Dynamical Bragg diffraction of optical pulses in photonic crystals in the Laue geometry: diffraction induced splitting, selective compression, and focusing of pulses,” J. Exp. Theor. Phys. 112, 56 (2012).
[Crossref]

Torner, L.

Y. V. Kartashov, V. A. Vysloukh, V. V. Konotop, and L. Torner, “Diffraction control in PT-symmetric photonic lattices: From beam rectification to dynamic localization,” Phys. Rev. A 93, 013841 (2016).
[Crossref]

Vysloukh, V. A.

Y. V. Kartashov, V. A. Vysloukh, V. V. Konotop, and L. Torner, “Diffraction control in PT-symmetric photonic lattices: From beam rectification to dynamic localization,” Phys. Rev. A 93, 013841 (2016).
[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. Materials 12, 108 (2013).
[Crossref]

L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M.-H. Lu, Y.-F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333, 729 (2011).
[Crossref] [PubMed]

Yang, J.

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

Yariv, A.

A. Yariv and P. Yeh, Optical Waves in Crystlas (John Wiley and Sons, 1984).

Yeh, P.

A. Yariv and P. Yeh, Optical Waves in Crystlas (John Wiley and Sons, 1984).

Zeilinger, A.

C. Keller, M. K. Oberthaler, R. Abfalterer, S. Bernet, J. Schmiedmayer, and A. Zeilinger, “Tailored Complex Potentials and Friedel’s Law in Atom Optics,” Phys. Rev. Lett. 79, 3327–3330 (1997).
[Crossref]

Zezyulin, D. A.

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

J. Exp. Theor. Phys. (1)

A. A. Skorynin, V. A. Bushuev, and B. I. Mantsyzov, “Dynamical Bragg diffraction of optical pulses in photonic crystals in the Laue geometry: diffraction induced splitting, selective compression, and focusing of pulses,” J. Exp. Theor. Phys. 112, 56 (2012).
[Crossref]

J. Phys. A (2)

M. V. Berry, “Lop-sided diffraction by absorbing crystals,” J. Phys. A 31, 3943 (1998).
[Crossref]

M. V. Berry, “Optical lattices with PT symmetry are not transparent,” J. Phys. A 41, 244007 (2008).
[Crossref]

J. Phys. A: Math. Gen. (1)

A. Mostafazadeh, “Exact PT-symmetry is equivalent to Hermiticity,” J. Phys. A: Math. Gen. 36, 7081–7091 (2003).
[Crossref]

Nat. Materials (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. Materials 12, 108 (2013).
[Crossref]

Opt. Expr. (1)

M. Kulishov, J. M. Laniel, N. Bélanger, J. Azaña, and D. V. Plant, “Nonreciprocal waveguide Bragg gratings,” Opt. Expr. 13, 3068–30678 (2005).
[Crossref]

Phys. Rev. A (1)

Y. V. Kartashov, V. A. Vysloukh, V. V. Konotop, and L. Torner, “Diffraction control in PT-symmetric photonic lattices: From beam rectification to dynamic localization,” Phys. Rev. A 93, 013841 (2016).
[Crossref]

Phys. Rev. A, (1)

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

Phys. Rev. Lett. (5)

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] [PubMed]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100103904 (2008).
[Crossref] [PubMed]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100030402 (2008).
[Crossref]

C. Keller, M. K. Oberthaler, R. Abfalterer, S. Bernet, J. Schmiedmayer, and A. Zeilinger, “Tailored Complex Potentials and Friedel’s Law in Atom Optics,” Phys. Rev. Lett. 79, 3327–3330 (1997).
[Crossref]

C. M. Bender and S. Boettcher, “Real spectra in Non-Hermitian Hamiltonians having PT-symmetry,” Phys. Rev. Lett. 805243 (1998).
[Crossref]

Rep. Prog. Phys. (1)

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

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

Science (1)

L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M.-H. Lu, Y.-F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333, 729 (2011).
[Crossref] [PubMed]

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A. Yariv and P. Yeh, Optical Waves in Crystlas (John Wiley and Sons, 1984).

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Figures (3)

Fig. 1
Fig. 1 Illustration of waves incident and reflected at a resonant Bragg angle on a periodically modulated layer with the permittivity εpc(x) embedded in the medium with the permittivity εex. The parity symmetry axis in z-direction is defined by L/2.
Fig. 2
Fig. 2 Reflection and transmission coefficients (a) the resonance conservative case εr = 16δ2/(17k2), εi = 0 and δ2 = 0.1k2 (corresponding to p1 = 1, p2 = 3, see the text). In the rest of panels the same εr is used while in (b) εi = 0.8εr, and in (c) εi = 1.1εr. (d) Split of the transmission resonance at εi = 0.24ϵr. Black, red, blue, and green line correspond to |t11|2, |r12|2, |t13|2, and |r14|2. The cyan lines show the total transmitted and reflected intensity normalized to the intensity of the incident wave a 1 ( l )
Fig. 3
Fig. 3 Reflection and transmission coefficients vs width the gain/loss parameter for L = 12.3λ (a) and L = 18.1λ (b). Other parameters are δ2 = 0.1 and εr = 1.6/17 ≈ 0.094. (c) The dependence of the scattering data on the slab width in the exceptional point εi = εr. Black, red, blue, and green line correspond to |t11|2, |r12|2, |t13|2, and |r14|2. In (c) blue and green lines are indistinguishable on the figure scale.

Equations (6)

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d 2 A + / d z 2 + δ 2 A + + ϵ 1 A = 0 , d 2 A / d z 2 + δ 2 A + ϵ 2 A + = 0 . at 0 < z < L .
E j = ( c 1 j e i q j z + c 2 j e i q j z ) ( ϵ 2 e i κ x ( 1 ) j ϵ 1 e i κ x )
E l , r = a 1 l , r e i ( κ x + q z ) + a 2 l , r e i ( κ x q z ) + a 3 l , r e i ( q z κ x ) + a 4 l , r e i ( κ x + q z ) .
M = ( m + e i q L m ^ + e i q L m ˜ + η e i q L m ^ η e i q L m ^ + e i q L m e i q L m ^ η e i q L m ˜ η e i q L m ˜ + η e i q L m ˜ η e i q L m + e i q L m ^ + e i q L m ˜ η e i q L m ˜ η e i q L m ^ + e i q L m e i q L )
m ± = [ cos ( q 2 L ) + cos ( q 1 L ) ] / 2 ± i [ sin ( q 2 L ) ( q 2 / q + q / q 2 ) + sin ( q 1 L ) ( q 1 / q + q / q 1 ) ] / 4 , m ˜ ± = [ cos ( q 2 L ) cos ( q 1 L ) ] / 2 ± i [ sin ( q 2 L ) ( q 2 / q + q / q 2 ) sin ( q 1 L ) ( q 1 / q + q / q 1 ) ] / 4 .
μ 11 = ( cos ( L δ ) + i ( 3 k / 4 δ ) sin ( L δ ) ) e i 3 k L / 2 , μ 12 = ( 3 k / 4 i δ ) e i 3 k L / 2 sin ( L δ ) μ 21 = ( 3 i ϵ 2 / 8 δ 2 ) ( k L cos ( L δ ) ( k / δ 4 L δ ) sin ( L δ ) ) e i 3 k L / 2 μ 22 = ( 3 i ϵ 2 / 8 δ 2 ) ( k / δ sin ( L δ ) k L cos ( L δ ) ) e i 3 k L / 2

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