Abstract

The effect of material dispersion on the optical properties of one-dimensional non-Hermitian scattering systems is investigated in detail. In particular, multilayer heterostructures with gain and loss (parity-time symmetric or not) are examined by taking into account the dispersion of each layer. The exceptional points and phase transitions are characterized based on the spectrum of the corresponding scattering matrix. We demonstrate that an on-average lossy heterostructure can amplify an incident plane wave in the frequency range associated with the emission frequency of the layer with gain.

© 2018 Chinese Laser Press

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    [Crossref]
  2. 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]
  3. Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100, 030402 (2008).
    [Crossref]
  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]
  5. C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
    [Crossref]
  6. 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]
  7. S. Longhi, “PT-symmetric laser absorber,” Phys. Rev. A 82, 031801(R) (2010).
    [Crossref]
  8. 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]
  9. 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, 729–733 (2011).
    [Crossref]
  10. L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. B. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12, 108–113 (2012).
    [Crossref]
  11. N. Lazarides and G. P. Tsironis, “Gain-driven discrete breathers in PT-symmetric nonlinear metamaterials,” Phys. Rev. Lett. 110, 053901 (2013).
    [Crossref]
  12. S. Savoia, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “Tunneling of obliquely incident waves through PT-symmetric epsilon-near-zero bilayers,” Phys. Rev. B 89, 085105 (2014).
    [Crossref]
  13. D. L. Sounas, R. Fleury, and A. Alu, “Unidirectional cloaking based on metasurfaces with balanced loss and gain,” Phys. Rev. Appl. 4, 014005 (2015).
    [Crossref]
  14. H. Alaeian and J. A. Dionne, “Parity-time-symmetric plasmonic metamaterials,” Phys. Rev. A 89, 033829 (2014).
    [Crossref]
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  16. O. V. Shramkova and G. P. Tsironis, “Propagation of electromagnetic waves in PT-symmetric hyperbolic structures,” Phys. Rev. B 94, 035141 (2016).
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  17. O. V. Shramkova and G. P. Tsironis, “Scattering properties of PT-symmetric layered periodic structures,” J. Opt. 18, 105101 (2016).
    [Crossref]
  18. P. Ambichl, K. G. Makris, L. Ge, Y. Chong, A. D. Stone, and S. Rotter, “Breaking of PT-symmetry in bounded and unbounded scattering systems,” Phys. Rev. X 3, 041030 (2013).
    [Crossref]
  19. K. G. Makris, L. Ge, and H. E. Türeci, “Anomalous transient amplification of waves in non-normal photonic media,” Phys. Rev. X 4, 041044 (2014).
    [Crossref]
  20. M. Sarısaman, “Unidirectional reflectionlessness and invisibility in the TE and TM modes of a PT-symmetric slab system,” Phys. Rev. A 95, 013806 (2017).
    [Crossref]
  21. S. Phang, A. Vukovic, S. Creagh, T. M. Benson, P. Sewell, and G. Gradoni, “Parity-time symmetric coupled microresonators with a dispersive gain/loss,” Opt. Express 23, 11493–11507 (2015).
    [Crossref]
  22. A. A. Zyablovsky, A. P. Vinogradov, A. A. Pukhov, A. V. Dorofeenko, and A. A. Lisyansky, “PT-symmetry in optics,” Phys. Usp. 57, 1063–1082 (2014).
    [Crossref]
  23. A. Fang, T. Koschny, and C. M. Soukoulis, “Lasing in metamaterials nanostructures,” J. Opt. 12, 024013 (2010).
    [Crossref]
  24. A. E. Siegman, Lasers (University Science Books, 1986).
  25. A. V. Dorofeenko, A. A. Zyablovsky, A. A. Pukhov, A. A. Lisyansky, and A. P. Vinogradov, “Light propagation in composite materials with gain layers,” Phys. Usp. 55, 1080–1097 (2012).
    [Crossref]

2017 (1)

M. Sarısaman, “Unidirectional reflectionlessness and invisibility in the TE and TM modes of a PT-symmetric slab system,” Phys. Rev. A 95, 013806 (2017).
[Crossref]

2016 (3)

O. V. Shramkova and G. P. Tsironis, “Resonant combinatorial frequency generation induced by a PT-symmetric periodic layered stack,” IEEE J. Sel. Top. Quantum Electron. 22, 82–88 (2016).
[Crossref]

O. V. Shramkova and G. P. Tsironis, “Propagation of electromagnetic waves in PT-symmetric hyperbolic structures,” Phys. Rev. B 94, 035141 (2016).
[Crossref]

O. V. Shramkova and G. P. Tsironis, “Scattering properties of PT-symmetric layered periodic structures,” J. Opt. 18, 105101 (2016).
[Crossref]

2015 (2)

D. L. Sounas, R. Fleury, and A. Alu, “Unidirectional cloaking based on metasurfaces with balanced loss and gain,” Phys. Rev. Appl. 4, 014005 (2015).
[Crossref]

S. Phang, A. Vukovic, S. Creagh, T. M. Benson, P. Sewell, and G. Gradoni, “Parity-time symmetric coupled microresonators with a dispersive gain/loss,” Opt. Express 23, 11493–11507 (2015).
[Crossref]

2014 (4)

K. G. Makris, L. Ge, and H. E. Türeci, “Anomalous transient amplification of waves in non-normal photonic media,” Phys. Rev. X 4, 041044 (2014).
[Crossref]

H. Alaeian and J. A. Dionne, “Parity-time-symmetric plasmonic metamaterials,” Phys. Rev. A 89, 033829 (2014).
[Crossref]

A. A. Zyablovsky, A. P. Vinogradov, A. A. Pukhov, A. V. Dorofeenko, and A. A. Lisyansky, “PT-symmetry in optics,” Phys. Usp. 57, 1063–1082 (2014).
[Crossref]

S. Savoia, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “Tunneling of obliquely incident waves through PT-symmetric epsilon-near-zero bilayers,” Phys. Rev. B 89, 085105 (2014).
[Crossref]

2013 (2)

N. Lazarides and G. P. Tsironis, “Gain-driven discrete breathers in PT-symmetric nonlinear metamaterials,” Phys. Rev. Lett. 110, 053901 (2013).
[Crossref]

P. Ambichl, K. G. Makris, L. Ge, Y. Chong, A. D. Stone, and S. Rotter, “Breaking of PT-symmetry in bounded and unbounded scattering systems,” Phys. Rev. X 3, 041030 (2013).
[Crossref]

2012 (3)

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. B. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12, 108–113 (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. V. Dorofeenko, A. A. Zyablovsky, A. A. Pukhov, A. A. Lisyansky, and A. P. Vinogradov, “Light propagation in composite materials with gain layers,” Phys. Usp. 55, 1080–1097 (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]

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, 729–733 (2011).
[Crossref]

2010 (3)

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

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

A. Fang, T. Koschny, and C. M. Soukoulis, “Lasing in metamaterials nanostructures,” J. Opt. 12, 024013 (2010).
[Crossref]

2009 (1)

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]

2008 (2)

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]

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

1998 (1)

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

Aimez, V.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref]

Alaeian, H.

H. Alaeian and J. A. Dionne, “Parity-time-symmetric plasmonic metamaterials,” Phys. Rev. A 89, 033829 (2014).
[Crossref]

Almeida, V. R.

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. B. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12, 108–113 (2012).
[Crossref]

Alu, A.

D. L. Sounas, R. Fleury, and A. Alu, “Unidirectional cloaking based on metasurfaces with balanced loss and gain,” Phys. Rev. Appl. 4, 014005 (2015).
[Crossref]

Alù, A.

S. Savoia, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “Tunneling of obliquely incident waves through PT-symmetric epsilon-near-zero bilayers,” Phys. Rev. B 89, 085105 (2014).
[Crossref]

Ambichl, P.

P. Ambichl, K. G. Makris, L. Ge, Y. Chong, A. D. Stone, and S. Rotter, “Breaking of PT-symmetry in bounded and unbounded scattering systems,” Phys. Rev. X 3, 041030 (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, 729–733 (2011).
[Crossref]

Bender, C. M.

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

Benson, T. M.

Boettcher, S.

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

Cao, H.

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]

Castaldi, G.

S. Savoia, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “Tunneling of obliquely incident waves through PT-symmetric epsilon-near-zero bilayers,” Phys. Rev. B 89, 085105 (2014).
[Crossref]

Chen, Y. F.

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, 729–733 (2011).
[Crossref]

Chen, Y.-F.

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. B. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12, 108–113 (2012).
[Crossref]

Chong, Y.

P. Ambichl, K. G. Makris, L. Ge, Y. Chong, A. D. Stone, and S. Rotter, “Breaking of PT-symmetry in bounded and unbounded scattering systems,” Phys. Rev. X 3, 041030 (2013).
[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, 023802 (2012).
[Crossref]

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]

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

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]

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

Creagh, S.

Dionne, J. A.

H. Alaeian and J. A. Dionne, “Parity-time-symmetric plasmonic metamaterials,” Phys. Rev. A 89, 033829 (2014).
[Crossref]

Dorofeenko, A. V.

A. A. Zyablovsky, A. P. Vinogradov, A. A. Pukhov, A. V. Dorofeenko, and A. A. Lisyansky, “PT-symmetry in optics,” Phys. Usp. 57, 1063–1082 (2014).
[Crossref]

A. V. Dorofeenko, A. A. Zyablovsky, A. A. Pukhov, A. A. Lisyansky, and A. P. Vinogradov, “Light propagation in composite materials with gain layers,” Phys. Usp. 55, 1080–1097 (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, 192–195 (2010).
[Crossref]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100, 030402 (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. 100, 103904 (2008).
[Crossref]

Engheta, N.

S. Savoia, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “Tunneling of obliquely incident waves through PT-symmetric epsilon-near-zero bilayers,” Phys. Rev. B 89, 085105 (2014).
[Crossref]

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, 729–733 (2011).
[Crossref]

Fang, A.

A. Fang, T. Koschny, and C. M. Soukoulis, “Lasing in metamaterials nanostructures,” J. Opt. 12, 024013 (2010).
[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, 108–113 (2012).
[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. Mater. 12, 108–113 (2012).
[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, 729–733 (2011).
[Crossref]

Fleury, R.

D. L. Sounas, R. Fleury, and A. Alu, “Unidirectional cloaking based on metasurfaces with balanced loss and gain,” Phys. Rev. Appl. 4, 014005 (2015).
[Crossref]

Galdi, V.

S. Savoia, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “Tunneling of obliquely incident waves through PT-symmetric epsilon-near-zero bilayers,” Phys. Rev. B 89, 085105 (2014).
[Crossref]

Ge, L.

K. G. Makris, L. Ge, and H. E. Türeci, “Anomalous transient amplification of waves in non-normal photonic media,” Phys. Rev. X 4, 041044 (2014).
[Crossref]

P. Ambichl, K. G. Makris, L. Ge, Y. Chong, A. D. Stone, and S. Rotter, “Breaking of PT-symmetry in bounded and unbounded scattering systems,” Phys. Rev. X 3, 041030 (2013).
[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]

Gradoni, G.

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]

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, 729–733 (2011).
[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, 192–195 (2010).
[Crossref]

Koschny, T.

A. Fang, T. Koschny, and C. M. Soukoulis, “Lasing in metamaterials nanostructures,” J. Opt. 12, 024013 (2010).
[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]

Lazarides, N.

N. Lazarides and G. P. Tsironis, “Gain-driven discrete breathers in PT-symmetric nonlinear metamaterials,” Phys. Rev. Lett. 110, 053901 (2013).
[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]

Lisyansky, A. A.

A. A. Zyablovsky, A. P. Vinogradov, A. A. Pukhov, A. V. Dorofeenko, and A. A. Lisyansky, “PT-symmetry in optics,” Phys. Usp. 57, 1063–1082 (2014).
[Crossref]

A. V. Dorofeenko, A. A. Zyablovsky, A. A. Pukhov, A. A. Lisyansky, and A. P. Vinogradov, “Light propagation in composite materials with gain layers,” Phys. Usp. 55, 1080–1097 (2012).
[Crossref]

Longhi, S.

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

Lu, M. H.

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, 729–733 (2011).
[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. Mater. 12, 108–113 (2012).
[Crossref]

Makris, K. G.

K. G. Makris, L. Ge, and H. E. Türeci, “Anomalous transient amplification of waves in non-normal photonic media,” Phys. Rev. X 4, 041044 (2014).
[Crossref]

P. Ambichl, K. G. Makris, L. Ge, Y. Chong, A. D. Stone, and S. Rotter, “Breaking of PT-symmetry in bounded and unbounded scattering systems,” Phys. Rev. X 3, 041030 (2013).
[Crossref]

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

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100, 030402 (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. 100, 103904 (2008).
[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]

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]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100, 030402 (2008).
[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, 108–113 (2012).
[Crossref]

Phang, S.

Pukhov, A. A.

A. A. Zyablovsky, A. P. Vinogradov, A. A. Pukhov, A. V. Dorofeenko, and A. A. Lisyansky, “PT-symmetry in optics,” Phys. Usp. 57, 1063–1082 (2014).
[Crossref]

A. V. Dorofeenko, A. A. Zyablovsky, A. A. Pukhov, A. A. Lisyansky, and A. P. Vinogradov, “Light propagation in composite materials with gain layers,” Phys. Usp. 55, 1080–1097 (2012).
[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]

Rotter, S.

P. Ambichl, K. G. Makris, L. Ge, Y. Chong, A. D. Stone, and S. Rotter, “Breaking of PT-symmetry in bounded and unbounded scattering systems,” Phys. Rev. X 3, 041030 (2013).
[Crossref]

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, 192–195 (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]

Sarisaman, M.

M. Sarısaman, “Unidirectional reflectionlessness and invisibility in the TE and TM modes of a PT-symmetric slab system,” Phys. Rev. A 95, 013806 (2017).
[Crossref]

Savoia, S.

S. Savoia, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “Tunneling of obliquely incident waves through PT-symmetric epsilon-near-zero bilayers,” Phys. Rev. B 89, 085105 (2014).
[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, 108–113 (2012).
[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, 729–733 (2011).
[Crossref]

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, 192–195 (2010).
[Crossref]

Sewell, P.

Shramkova, O. V.

O. V. Shramkova and G. P. Tsironis, “Resonant combinatorial frequency generation induced by a PT-symmetric periodic layered stack,” IEEE J. Sel. Top. Quantum Electron. 22, 82–88 (2016).
[Crossref]

O. V. Shramkova and G. P. Tsironis, “Propagation of electromagnetic waves in PT-symmetric hyperbolic structures,” Phys. Rev. B 94, 035141 (2016).
[Crossref]

O. V. Shramkova and G. P. Tsironis, “Scattering properties of PT-symmetric layered periodic structures,” J. Opt. 18, 105101 (2016).
[Crossref]

Siegman, A. E.

A. E. Siegman, Lasers (University Science Books, 1986).

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]

Soukoulis, C. M.

A. Fang, T. Koschny, and C. M. Soukoulis, “Lasing in metamaterials nanostructures,” J. Opt. 12, 024013 (2010).
[Crossref]

Sounas, D. L.

D. L. Sounas, R. Fleury, and A. Alu, “Unidirectional cloaking based on metasurfaces with balanced loss and gain,” Phys. Rev. Appl. 4, 014005 (2015).
[Crossref]

Stone, A. D.

P. Ambichl, K. G. Makris, L. Ge, Y. Chong, A. D. Stone, and S. Rotter, “Breaking of PT-symmetry in bounded and unbounded scattering systems,” Phys. Rev. X 3, 041030 (2013).
[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]

Tsironis, G. P.

O. V. Shramkova and G. P. Tsironis, “Scattering properties of PT-symmetric layered periodic structures,” J. Opt. 18, 105101 (2016).
[Crossref]

O. V. Shramkova and G. P. Tsironis, “Propagation of electromagnetic waves in PT-symmetric hyperbolic structures,” Phys. Rev. B 94, 035141 (2016).
[Crossref]

O. V. Shramkova and G. P. Tsironis, “Resonant combinatorial frequency generation induced by a PT-symmetric periodic layered stack,” IEEE J. Sel. Top. Quantum Electron. 22, 82–88 (2016).
[Crossref]

N. Lazarides and G. P. Tsironis, “Gain-driven discrete breathers in PT-symmetric nonlinear metamaterials,” Phys. Rev. Lett. 110, 053901 (2013).
[Crossref]

Türeci, H. E.

K. G. Makris, L. Ge, and H. E. Türeci, “Anomalous transient amplification of waves in non-normal photonic media,” Phys. Rev. X 4, 041044 (2014).
[Crossref]

Vinogradov, A. P.

A. A. Zyablovsky, A. P. Vinogradov, A. A. Pukhov, A. V. Dorofeenko, and A. A. Lisyansky, “PT-symmetry in optics,” Phys. Usp. 57, 1063–1082 (2014).
[Crossref]

A. V. Dorofeenko, A. A. Zyablovsky, A. A. Pukhov, A. A. Lisyansky, and A. P. Vinogradov, “Light propagation in composite materials with gain layers,” Phys. Usp. 55, 1080–1097 (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]

Vukovic, A.

Xu, Y. L.

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, 729–733 (2011).
[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, 108–113 (2012).
[Crossref]

Zyablovsky, A. A.

A. A. Zyablovsky, A. P. Vinogradov, A. A. Pukhov, A. V. Dorofeenko, and A. A. Lisyansky, “PT-symmetry in optics,” Phys. Usp. 57, 1063–1082 (2014).
[Crossref]

A. V. Dorofeenko, A. A. Zyablovsky, A. A. Pukhov, A. A. Lisyansky, and A. P. Vinogradov, “Light propagation in composite materials with gain layers,” Phys. Usp. 55, 1080–1097 (2012).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (1)

O. V. Shramkova and G. P. Tsironis, “Resonant combinatorial frequency generation induced by a PT-symmetric periodic layered stack,” IEEE J. Sel. Top. Quantum Electron. 22, 82–88 (2016).
[Crossref]

J. Opt. (2)

A. Fang, T. Koschny, and C. M. Soukoulis, “Lasing in metamaterials nanostructures,” J. Opt. 12, 024013 (2010).
[Crossref]

O. V. Shramkova and G. P. Tsironis, “Scattering properties of PT-symmetric layered periodic structures,” J. Opt. 18, 105101 (2016).
[Crossref]

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, 108–113 (2012).
[Crossref]

Nat. Phys. (1)

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

Opt. Express (1)

Phys. Rev. A (4)

M. Sarısaman, “Unidirectional reflectionlessness and invisibility in the TE and TM modes of a PT-symmetric slab system,” Phys. Rev. A 95, 013806 (2017).
[Crossref]

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

H. Alaeian and J. A. Dionne, “Parity-time-symmetric plasmonic metamaterials,” Phys. Rev. A 89, 033829 (2014).
[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]

Phys. Rev. Appl. (1)

D. L. Sounas, R. Fleury, and A. Alu, “Unidirectional cloaking based on metasurfaces with balanced loss and gain,” Phys. Rev. Appl. 4, 014005 (2015).
[Crossref]

Phys. Rev. B (2)

O. V. Shramkova and G. P. Tsironis, “Propagation of electromagnetic waves in PT-symmetric hyperbolic structures,” Phys. Rev. B 94, 035141 (2016).
[Crossref]

S. Savoia, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “Tunneling of obliquely incident waves through PT-symmetric epsilon-near-zero bilayers,” Phys. Rev. B 89, 085105 (2014).
[Crossref]

Phys. Rev. Lett. (6)

N. Lazarides and G. P. Tsironis, “Gain-driven discrete breathers in PT-symmetric nonlinear metamaterials,” Phys. Rev. Lett. 110, 053901 (2013).
[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]

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]

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]

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

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

Phys. Rev. X (2)

P. Ambichl, K. G. Makris, L. Ge, Y. Chong, A. D. Stone, and S. Rotter, “Breaking of PT-symmetry in bounded and unbounded scattering systems,” Phys. Rev. X 3, 041030 (2013).
[Crossref]

K. G. Makris, L. Ge, and H. E. Türeci, “Anomalous transient amplification of waves in non-normal photonic media,” Phys. Rev. X 4, 041044 (2014).
[Crossref]

Phys. Usp. (2)

A. V. Dorofeenko, A. A. Zyablovsky, A. A. Pukhov, A. A. Lisyansky, and A. P. Vinogradov, “Light propagation in composite materials with gain layers,” Phys. Usp. 55, 1080–1097 (2012).
[Crossref]

A. A. Zyablovsky, A. P. Vinogradov, A. A. Pukhov, A. V. Dorofeenko, and A. A. Lisyansky, “PT-symmetry in optics,” Phys. Usp. 57, 1063–1082 (2014).
[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, 729–733 (2011).
[Crossref]

Other (1)

A. E. Siegman, Lasers (University Science Books, 1986).

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Figures (6)

Fig. 1.
Fig. 1.

Geometry of the problem.

Fig. 2.
Fig. 2.

Absorption coefficient |α1| and parameter Δϵ as the functions of frequency at α2=0.2 (solid red line), α2=2 (dashed black line), and α2=5 (dash-dot blue line).

Fig. 3.
Fig. 3.

(a) Reflectance [R(L)(ω)- solid red line and R(R)(ω)- dashed blue line] and transmittance (solid black line) of TM wave incident at Θ=15° on 1D system with ω01=1  PHz, ω02=1.2  PHz, γ1=0.067  PHz, γ2=0.14  PHz, |α1|=20.86, α2=2, ϵ01=2, ϵ02=3.22, and dg=dl=500  nm. (b) Real (solid lines) and imaginary (dashed lines) parts of dielectric permittivities of the layers near emission frequencies of the gain (red lines) and loss (black lines) regions. (c) Modulus of eigenvalues of S-matrix for 1D system as a function of frequency; red and blue lines correspond to the different eigenvalues of the same scattering matrix. The range of frequencies in (a) and (c) corresponding to the loss dominated system is gray shaded.

Fig. 4.
Fig. 4.

(a) Reflectance [R(L)(ω)- solid red line and R(R)(ω)- dashed blue line] and transmittance (solid black line) of TM wave incident at Θ=0° and ω=0.92  PHz on non-Hermitian system with ω01=1  PHz, ω02=2  PHz, γ1=0.067  PHz, γ2=0.14  PHz, α1=2, ϵ01=2, ϵ02=2.2, and dg=dl=500  nm as a function of absorption coefficient. (b) Distribution of the energy flux density of an electromagnetic field in the system with the default parameters and α2=92.3 (solid green line) and α2=156.5 (dashed brown line); the area corresponding to gain material is red shaded, the area corresponding to loss material is green shaded. (c), (d) Transmittance of TM wave incident at Θ=0° (solid black line), Θ=30° (dash-dot green line), Θ=60° (dashed brown line), and dg=500  nm. The range of frequencies in (a), (c), and (d) corresponding to loss dominated system is gray shaded.

Fig. 5.
Fig. 5.

(a) Geometry of non-Hermitian periodic stack. (b), (c) Geometries of non-Hermitian random stacks. (d), (e) Transmittance of TM wave through the stacks incident at ω=0.92  PHz and (d) Θ=0°, (e) Θ=60°; dash-dot black curve, four layers with dg=dl=250  nm, geometry of the system is presented in (a); dashed blue curves, six layers with thicknesses dg1=dg3=dl1=dl3=200  nm and dg2=dl2=100  nm, geometry of the stack is presented in (b); solid green curves, six layers with thicknesses dg1=dg2=200  nm, dl1=dl2=dg3=100  nm, and dl3=300  nm, geometry of the stack is presented in (c). The relevant layer parameters for the stacks are the same as in Fig. 4. The range of frequencies in (d) and (e) corresponding to loss dominated system is gray shaded.

Fig. 6.
Fig. 6.

Eigenvalues of the S-matrix as a function of absorption coefficient for (a) PT-symmetric bilayer with ω01=ω02=ω=1  PHz, γ1=γ2=0.067  PHz, ϵ01=ϵ02=2, and dg=dl=500  nm at Θ=0° (solid and dashed lines of the same color correspond to the real and imaginary parts of eigenvalues) and (b) non-Hermitian bilayer with ω01=1  PHz, ω02=2  PHz, γ1=0.067  PHz, γ2=0.14  PHz, α1=2.4, ϵ01=2, ϵ02=2.2, and dg=dl=500  nm at ω=2.65  PHz and Θ=0°. Red and blue lines correspond to the eigenvalues λ1 and λ2. Vertical dashed lines indicate the position of EPs.

Equations (6)

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ϵj(ω)=ϵ0jαjω0jγjω2ω0j2+iωγj,
Reϵ1(ω)=Reϵ2(ω),Imϵ1(ω)=Imϵ2(ω).
|α1|=α2ω02γ22[(ω2ω012)2+γ12ω2]ω01γ12[(ω2ω022)2+γ22ω2].
Δϵ=ϵ02ϵ01=α2ω02γ2[ω2ω022+γ2γ1(ω2ω012)](ω2ω022)2+γ22ω2.
(BC)=(AD)=(R(L)(ω)T(ω)T(ω)R(R)(ω))(AD),
λ1,2=R(L)(ω)+R(R)(ω)±[R(L)(ω)R(R)(ω)]2+4T(ω)22.