Abstract

We study light propagation through cyclic arrays, composed by copies of a given PT-symmetric dimer, using a group theoretical approach and finite element modeling. The theoretical mode-coupling analysis suggests the use of these devices as output port replicators where the dynamics is controlled by the impinging light field. This is confirmed in good agreement with finite element propagation in an experimentally feasible necklace of passive PT-symmetric dimers constructed from lossy and lossless waveguides.

© 2018 Chinese Laser Press

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References

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    [Crossref]
  2. Y. Chen, A. W. Snyder, and D. N. Payne, “Twin core nonlinear couplers with gain and loss,” IEEE J. Quantum Electron. 28, 239–245 (1992).
    [Crossref]
  3. C. M. Bender, “Real spectra in non-Hermitian Hamiltonians having PT-symmetry,” Phys. Rev. Lett. 80, 5243–5246 (1998).
    [Crossref]
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    [Crossref]
  5. A. Ruschhaupt, F. Delgado, and J. G. Muga, “Physical realization of PT-symmetric potential scattering in a planar slab waveguide,” J. Phys. A 38, L171–L176 (2005).
    [Crossref]
  6. R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical PT-symmetric structures,” Opt. Lett. 32, 2632–2634 (2007).
    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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  23. N. V. Alexeeva, I. V. Barashenkov, and Y. S. Kivshar, “Solitons in PT-symmetric ladders of optical waveguides,” New J. Phys. 19, 113032 (2017).
    [Crossref]
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    [Crossref]
  25. T. Kato, Perturbation Theory for Linear Operators (Springer-Verlag, 1995).
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    [Crossref]
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2018 (2)

R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. D. Musslimani, S. Rotter, and D. N. Christodoulides, “Non-Hermitian physics and PT symmetry,” Nat. Phys. 14, 11–19 (2018).
[Crossref]

B. M. Rodríguez-Lara, R. El-Ganainy, and J. Guerrero, “Symmetry in optics and photonics: a group theory approach,” Sci. Bull. 63, 244–251 (2018).
[Crossref]

2017 (2)

N. V. Alexeeva, I. V. Barashenkov, and Y. S. Kivshar, “Solitons in PT-symmetric ladders of optical waveguides,” New J. Phys. 19, 113032 (2017).
[Crossref]

J. D. Huerta Morales and B. M. Rodríguez-Lara, “Photon propagation through linearly active dimers,” Appl. Sci. 7, 587 (2017).
[Crossref]

2016 (3)

2015 (1)

2014 (3)

M. Ornigotti and A. Szameit, “Quasi PT-symmetry in passive photonic lattices,” J. Opt. 16, 065501 (2014).
[Crossref]

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

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

2013 (1)

I. V. Barashenkov, L. Baker, and N. V. Alexeeva, “PT-symmetry breaking in a necklace of coupled optical waveguides,” Phys. Rev. A 87, 033819 (2013).
[Crossref]

2012 (2)

D. A. Zezyulin and V. V. Konotop, “Nonlinear modes in finite-dimensional PT-symmetric systems,” Phys. Rev. Lett. 108, 213906 (2012).
[Crossref]

K. Li, P. G. Kevrekidis, B. A. Malomed, and U. Günther, “Nonlinear PT-symmetric plaquettes,” J. Phys. A 45, 444021 (2012).
[Crossref]

2011 (1)

K. Li and P. G. Kevrekidis, “PT-symmetric oligomers: analytical solutions, linear stability, and nonlinear dynamics,” Phys. Rev. E 83, 066608 (2011).
[Crossref]

2010 (1)

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (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-symmetric breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref]

2007 (1)

2005 (3)

A. H. Nejadmalayeri, P. R. Herman, J. Burghoff, M. Will, S. Nolte, and A. Tünnermann, “Inscription of optical waveguides in crystalline silicon by mid-infrared femtosecond laser pulses,” Opt. Lett. 30, 964–966 (2005).
[Crossref]

C. M. Bender, “Introduction to PT-symmetric quantum theory,” Contemp. Phys. 46, 277–292 (2005).
[Crossref]

A. Ruschhaupt, F. Delgado, and J. G. Muga, “Physical realization of PT-symmetric potential scattering in a planar slab waveguide,” J. Phys. A 38, L171–L176 (2005).
[Crossref]

1998 (1)

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

1992 (1)

Y. Chen, A. W. Snyder, and D. N. Payne, “Twin core nonlinear couplers with gain and loss,” IEEE J. Quantum Electron. 28, 239–245 (1992).
[Crossref]

1973 (1)

S. Somekh, E. Garmire, A. Yariv, H. L. Garvin, and R. G. Hunsperger, “Channel optical waveguide directional couplers,” Appl. Phys. Lett. 22, 46–47 (1973).
[Crossref]

1867 (1)

J. J. Sylvester, “LX. Thoughts on inverse orthogonal matrices, simultaneous sign successions, and tessellated pavements in two or more colours, with applications to Newtons rule, ornamental tile-work, and the theory of numbers,” Philos. Mag. 34, 461–475 (1867).
[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-symmetric breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref]

Alexeeva, N. V.

N. V. Alexeeva, I. V. Barashenkov, and Y. S. Kivshar, “Solitons in PT-symmetric ladders of optical waveguides,” New J. Phys. 19, 113032 (2017).
[Crossref]

I. V. Barashenkov, L. Baker, and N. V. Alexeeva, “PT-symmetry breaking in a necklace of coupled optical waveguides,” Phys. Rev. A 87, 033819 (2013).
[Crossref]

Baker, L.

I. V. Barashenkov, L. Baker, and N. V. Alexeeva, “PT-symmetry breaking in a necklace of coupled optical waveguides,” Phys. Rev. A 87, 033819 (2013).
[Crossref]

Barashenkov, I. V.

N. V. Alexeeva, I. V. Barashenkov, and Y. S. Kivshar, “Solitons in PT-symmetric ladders of optical waveguides,” New J. Phys. 19, 113032 (2017).
[Crossref]

I. V. Barashenkov, L. Baker, and N. V. Alexeeva, “PT-symmetry breaking in a necklace of coupled optical waveguides,” Phys. Rev. A 87, 033819 (2013).
[Crossref]

Bender, C.

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

Bender, C. M.

C. M. Bender, “Introduction to PT-symmetric quantum theory,” Contemp. Phys. 46, 277–292 (2005).
[Crossref]

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

Burghoff, J.

Chambonneau, M.

Chanal, M.

Chen, Y.

Y. Chen, A. W. Snyder, and D. N. Payne, “Twin core nonlinear couplers with gain and loss,” IEEE J. Quantum Electron. 28, 239–245 (1992).
[Crossref]

Christodoulides, D. N.

R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. D. Musslimani, S. Rotter, and D. N. Christodoulides, “Non-Hermitian physics and PT symmetry,” Nat. Phys. 14, 11–19 (2018).
[Crossref]

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

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, 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-symmetric breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref]

R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical PT-symmetric structures,” Opt. Lett. 32, 2632–2634 (2007).
[Crossref]

Delgado, F.

A. Ruschhaupt, F. Delgado, and J. G. Muga, “Physical realization of PT-symmetric potential scattering in a planar slab waveguide,” J. Phys. A 38, L171–L176 (2005).
[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-symmetric breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref]

El-Ganainy, R.

R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. D. Musslimani, S. Rotter, and D. N. Christodoulides, “Non-Hermitian physics and PT symmetry,” Nat. Phys. 14, 11–19 (2018).
[Crossref]

B. M. Rodríguez-Lara, R. El-Ganainy, and J. Guerrero, “Symmetry in optics and photonics: a group theory approach,” Sci. Bull. 63, 244–251 (2018).
[Crossref]

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

R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical PT-symmetric structures,” Opt. Lett. 32, 2632–2634 (2007).
[Crossref]

Fan, S.

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

Garmire, E.

S. Somekh, E. Garmire, A. Yariv, H. L. Garvin, and R. G. Hunsperger, “Channel optical waveguide directional couplers,” Appl. Phys. Lett. 22, 46–47 (1973).
[Crossref]

Garvin, H. L.

S. Somekh, E. Garmire, A. Yariv, H. L. Garvin, and R. G. Hunsperger, “Channel optical waveguide directional couplers,” Appl. Phys. Lett. 22, 46–47 (1973).
[Crossref]

Gianfreda, M.

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

Grojo, D.

Guerrero, J.

B. M. Rodríguez-Lara, R. El-Ganainy, and J. Guerrero, “Symmetry in optics and photonics: a group theory approach,” Sci. Bull. 63, 244–251 (2018).
[Crossref]

B. M. Rodríguez-Lara and J. Guerrero, “Optical finite representation of the Lorentz group,” Opt. Lett. 40, 5682–5685 (2015).
[Crossref]

J. D. Huerta Morales, J. Guerrero, S. Lopez-Aguayo, and B. M. Rodríguez-Lara, “Revisiting the optical PT-symmetric dimer,” arXiv: 1607.02782 (2016).

Günther, U.

K. Li, P. G. Kevrekidis, B. A. Malomed, and U. Günther, “Nonlinear PT-symmetric plaquettes,” J. Phys. A 45, 444021 (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-symmetric breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref]

Heinrich, M.

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

Herman, P. R.

Hodaei, H.

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

Huerta Morales, J. D.

J. D. Huerta Morales and B. M. Rodríguez-Lara, “Photon propagation through linearly active dimers,” Appl. Sci. 7, 587 (2017).
[Crossref]

J. D. Huerta Morales, J. Guerrero, S. Lopez-Aguayo, and B. M. Rodríguez-Lara, “Revisiting the optical PT-symmetric dimer,” arXiv: 1607.02782 (2016).

Hunsperger, R. G.

S. Somekh, E. Garmire, A. Yariv, H. L. Garvin, and R. G. Hunsperger, “Channel optical waveguide directional couplers,” Appl. Phys. Lett. 22, 46–47 (1973).
[Crossref]

Kato, T.

T. Kato, Perturbation Theory for Linear Operators (Springer-Verlag, 1995).

Kevrekidis, P. G.

K. Li, P. G. Kevrekidis, B. A. Malomed, and U. Günther, “Nonlinear PT-symmetric plaquettes,” J. Phys. A 45, 444021 (2012).
[Crossref]

K. Li and P. G. Kevrekidis, “PT-symmetric oligomers: analytical solutions, linear stability, and nonlinear dynamics,” Phys. Rev. E 83, 066608 (2011).
[Crossref]

Khajavikhan, M.

R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. D. Musslimani, S. Rotter, and D. N. Christodoulides, “Non-Hermitian physics and PT symmetry,” Nat. Phys. 14, 11–19 (2018).
[Crossref]

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

Kip, D.

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

Kivshar, Y. S.

N. V. Alexeeva, I. V. Barashenkov, and Y. S. Kivshar, “Solitons in PT-symmetric ladders of optical waveguides,” New J. Phys. 19, 113032 (2017).
[Crossref]

Konotop, V. V.

D. A. Zezyulin and V. V. Konotop, “Nonlinear modes in finite-dimensional PT-symmetric systems,” Phys. Rev. Lett. 108, 213906 (2012).
[Crossref]

Lei, F.

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

Li, K.

K. Li, P. G. Kevrekidis, B. A. Malomed, and U. Günther, “Nonlinear PT-symmetric plaquettes,” J. Phys. A 45, 444021 (2012).
[Crossref]

K. Li and P. G. Kevrekidis, “PT-symmetric oligomers: analytical solutions, linear stability, and nonlinear dynamics,” Phys. Rev. E 83, 066608 (2011).
[Crossref]

Li, Q.

Liu, X.

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

Liu, Z.

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

Long, G.

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

Longhi, S.

Lopez-Aguayo, S.

J. D. Huerta Morales, J. Guerrero, S. Lopez-Aguayo, and B. M. Rodríguez-Lara, “Revisiting the optical PT-symmetric dimer,” arXiv: 1607.02782 (2016).

Makris, K. G.

R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. D. Musslimani, S. Rotter, and D. N. Christodoulides, “Non-Hermitian physics and PT symmetry,” Nat. Phys. 14, 11–19 (2018).
[Crossref]

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

R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical PT-symmetric structures,” Opt. Lett. 32, 2632–2634 (2007).
[Crossref]

Malomed, B. A.

K. Li, P. G. Kevrekidis, B. A. Malomed, and U. Günther, “Nonlinear PT-symmetric plaquettes,” J. Phys. A 45, 444021 (2012).
[Crossref]

Miri, M.-A.

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

Monifi, F.

B. Peng, S. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. Long, S. Fan, F. Nori, C. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
[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-symmetric breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref]

Muga, J. G.

A. Ruschhaupt, F. Delgado, and J. G. Muga, “Physical realization of PT-symmetric potential scattering in a planar slab waveguide,” J. Phys. A 38, L171–L176 (2005).
[Crossref]

Musslimani, Z. D.

R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. D. Musslimani, S. Rotter, and D. N. Christodoulides, “Non-Hermitian physics and PT symmetry,” Nat. Phys. 14, 11–19 (2018).
[Crossref]

Musslimani, Z. H.

Nejadmalayeri, A. H.

Nolte, S.

Nori, F.

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

Ornigotti, M.

M. Ornigotti and A. Szameit, “Quasi PT-symmetry in passive photonic lattices,” J. Opt. 16, 065501 (2014).
[Crossref]

Ozdemir, S.

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

Payne, D. N.

Y. Chen, A. W. Snyder, and D. N. Payne, “Twin core nonlinear couplers with gain and loss,” IEEE J. Quantum Electron. 28, 239–245 (1992).
[Crossref]

Peng, B.

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

Rodríguez-Lara, B. M.

B. M. Rodríguez-Lara, R. El-Ganainy, and J. Guerrero, “Symmetry in optics and photonics: a group theory approach,” Sci. Bull. 63, 244–251 (2018).
[Crossref]

J. D. Huerta Morales and B. M. Rodríguez-Lara, “Photon propagation through linearly active dimers,” Appl. Sci. 7, 587 (2017).
[Crossref]

B. M. Rodríguez-Lara and J. Guerrero, “Optical finite representation of the Lorentz group,” Opt. Lett. 40, 5682–5685 (2015).
[Crossref]

J. D. Huerta Morales, J. Guerrero, S. Lopez-Aguayo, and B. M. Rodríguez-Lara, “Revisiting the optical PT-symmetric dimer,” arXiv: 1607.02782 (2016).

Rotter, S.

R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. D. Musslimani, S. Rotter, and D. N. Christodoulides, “Non-Hermitian physics and PT symmetry,” Nat. Phys. 14, 11–19 (2018).
[Crossref]

Ruschhaupt, A.

A. Ruschhaupt, F. Delgado, and J. G. Muga, “Physical realization of PT-symmetric potential scattering in a planar slab waveguide,” J. Phys. A 38, L171–L176 (2005).
[Crossref]

Rüter, C. E.

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, 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-symmetric breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref]

Sanner, N.

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-symmetric breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref]

Snyder, A. W.

Y. Chen, A. W. Snyder, and D. N. Payne, “Twin core nonlinear couplers with gain and loss,” IEEE J. Quantum Electron. 28, 239–245 (1992).
[Crossref]

Somekh, S.

S. Somekh, E. Garmire, A. Yariv, H. L. Garvin, and R. G. Hunsperger, “Channel optical waveguide directional couplers,” Appl. Phys. Lett. 22, 46–47 (1973).
[Crossref]

Sylvester, J. J.

J. J. Sylvester, “LX. Thoughts on inverse orthogonal matrices, simultaneous sign successions, and tessellated pavements in two or more colours, with applications to Newtons rule, ornamental tile-work, and the theory of numbers,” Philos. Mag. 34, 461–475 (1867).
[Crossref]

Szameit, A.

M. Ornigotti and A. Szameit, “Quasi PT-symmetry in passive photonic lattices,” J. Opt. 16, 065501 (2014).
[Crossref]

Tünnermann, A.

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-symmetric breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref]

Will, M.

Xiao, J.-J.

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

Yang, L.

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

Yao, Y.

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

Yariv, A.

S. Somekh, E. Garmire, A. Yariv, H. L. Garvin, and R. G. Hunsperger, “Channel optical waveguide directional couplers,” Appl. Phys. Lett. 22, 46–47 (1973).
[Crossref]

Zezyulin, D. A.

D. A. Zezyulin and V. V. Konotop, “Nonlinear modes in finite-dimensional PT-symmetric systems,” Phys. Rev. Lett. 108, 213906 (2012).
[Crossref]

Zhang, Q.

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

Appl. Phys. Lett. (1)

S. Somekh, E. Garmire, A. Yariv, H. L. Garvin, and R. G. Hunsperger, “Channel optical waveguide directional couplers,” Appl. Phys. Lett. 22, 46–47 (1973).
[Crossref]

Appl. Sci. (1)

J. D. Huerta Morales and B. M. Rodríguez-Lara, “Photon propagation through linearly active dimers,” Appl. Sci. 7, 587 (2017).
[Crossref]

Contemp. Phys. (1)

C. M. Bender, “Introduction to PT-symmetric quantum theory,” Contemp. Phys. 46, 277–292 (2005).
[Crossref]

IEEE J. Quantum Electron. (1)

Y. Chen, A. W. Snyder, and D. N. Payne, “Twin core nonlinear couplers with gain and loss,” IEEE J. Quantum Electron. 28, 239–245 (1992).
[Crossref]

J. Opt. (1)

M. Ornigotti and A. Szameit, “Quasi PT-symmetry in passive photonic lattices,” J. Opt. 16, 065501 (2014).
[Crossref]

J. Phys. A (2)

K. Li, P. G. Kevrekidis, B. A. Malomed, and U. Günther, “Nonlinear PT-symmetric plaquettes,” J. Phys. A 45, 444021 (2012).
[Crossref]

A. Ruschhaupt, F. Delgado, and J. G. Muga, “Physical realization of PT-symmetric potential scattering in a planar slab waveguide,” J. Phys. A 38, L171–L176 (2005).
[Crossref]

Nat. Phys. (3)

R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. D. Musslimani, S. Rotter, and D. N. Christodoulides, “Non-Hermitian physics and PT symmetry,” Nat. Phys. 14, 11–19 (2018).
[Crossref]

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

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

New J. Phys. (1)

N. V. Alexeeva, I. V. Barashenkov, and Y. S. Kivshar, “Solitons in PT-symmetric ladders of optical waveguides,” New J. Phys. 19, 113032 (2017).
[Crossref]

Opt. Lett. (5)

Philos. Mag. (1)

J. J. Sylvester, “LX. Thoughts on inverse orthogonal matrices, simultaneous sign successions, and tessellated pavements in two or more colours, with applications to Newtons rule, ornamental tile-work, and the theory of numbers,” Philos. Mag. 34, 461–475 (1867).
[Crossref]

Phys. Rev. A (1)

I. V. Barashenkov, L. Baker, and N. V. Alexeeva, “PT-symmetry breaking in a necklace of coupled optical waveguides,” Phys. Rev. A 87, 033819 (2013).
[Crossref]

Phys. Rev. E (1)

K. Li and P. G. Kevrekidis, “PT-symmetric oligomers: analytical solutions, linear stability, and nonlinear dynamics,” Phys. Rev. E 83, 066608 (2011).
[Crossref]

Phys. Rev. Lett. (3)

D. A. Zezyulin and V. V. Konotop, “Nonlinear modes in finite-dimensional PT-symmetric systems,” Phys. Rev. Lett. 108, 213906 (2012).
[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-symmetric breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref]

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

Sci. Bull. (1)

B. M. Rodríguez-Lara, R. El-Ganainy, and J. Guerrero, “Symmetry in optics and photonics: a group theory approach,” Sci. Bull. 63, 244–251 (2018).
[Crossref]

Sci. Rep. (1)

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

Science (1)

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

Other (2)

J. D. Huerta Morales, J. Guerrero, S. Lopez-Aguayo, and B. M. Rodríguez-Lara, “Revisiting the optical PT-symmetric dimer,” arXiv: 1607.02782 (2016).

T. Kato, Perturbation Theory for Linear Operators (Springer-Verlag, 1995).

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Figures (6)

Fig. 1.
Fig. 1. Sketch for necklaces formed by the repetition of identical PT-symmetric dimers. (a) Necklace where intradimer couplings are larger than interdimer couplings, gd>gN, and repetition factor N. (b) Standard PT-symmetric dimer, N=1; (c) PT-symmetric plaquette, N=2; and (d) necklace with repetition factor N=3.
Fig. 2.
Fig. 2. Real (solid lines) and imaginary (dashed lines) parts of eigenvalues as a function of the gain to interdimer coupling ratio, γ/gN, for arrays of coupled PT-symmetric dimers with homogeneous intra- and inter-dimer couplings, gd=gN with repetition factor (a) N=2 and (b) N=3, and for inhomogeneous couplings, gd=gN/2, with repetition factor (c) N=2 and (d) N=3.
Fig. 3.
Fig. 3. (a) Waveguide geometry used in the finite element simulation and electric field input for replication of the (b) first and (c) second effective dimers. In (b) and (c), red corresponds to an in-phase beam and blue corresponds to a π-dephased beam.
Fig. 4.
Fig. 4. Renormalized power propagation, Pj,k(z), showing output replication for (a), (b) homogeneous and (c), (d) inhomogeneous couplings. The dots represent finite element propagation and the lines the analytic coupled mode approximation. See text for details.
Fig. 5.
Fig. 5. (a) Waveguide geometry used in the finite element simulation and electric field input for replication of the (b) first and (c) second effective dimers. In (b) and (c), red corresponds to an in-phase beam, orange corresponds to a (2π/3)-dephased beam, and yellow corresponds to a (4π/3)-dephased one.
Fig. 6.
Fig. 6. Renormalized power propagation, Pj,k(z), showing output replication for (a), (b) homogeneous and (c), (d) inhomogeneous couplings. The dots represent finite element propagation and the lines the analytic coupled mode approximation. See text for details.

Equations (33)

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izE0(z)=iγE0(z)+gdeiϕdE1(z)+gNeiϕNE2N1(z),izE1(z)=iγE1(z)+gdeiϕdE0(z)+gNeiϕNE2(z),izE2j(z)=iγE2j(z)+gdeiϕdE2j+1(z)+gNeiϕNE2j1(z),izE2j+1(z)=iγE2j+1(z)+gdeiϕdE2j(z)+gNeiϕNE2j+2(z),izE2N2(z)=iγE2N2(z)+gdeiϕdE2N1(z)+gNeiϕNE2N3(z),izE2N1(z)=iγE2N1(z)+gdeiϕdE2N2(z)+gNeiϕNE0(z),
iz|E(z)=H^|E(z),
|E(z)=(E0(z)E1(z)E2(z)E3(z)E2N2(z)E2N1(z))}(j=0)th  dimerk=0k=1}(j=1)th  dimerk=0k=1}(j=N1)thdimerk=0k=1.
H^=(H^dH^0^20^H^+H^+H^dH^0^0^0^H^+H^d0^0^0^0^0^H^dH^H^0^0^H^+H^d),
H^d=(iγgdeiϕdgdeiϕdiγ),H^+=(0gNeiϕN00),H^=(00gNeiϕN0),
H^d=γiσ^z+gdeiϕdσ^++gdeiϕdσ^,H^±=gNe±iϕN,
σ^z=(1001),σ^+=(0100),σ^=(0010).
C^N=(0000110000010000000000010),C^N=(0100000100000000000110000).
H^=1^NH^d+C^NH^++C^NH^,
|E(z)=j=0N1k=01E2j+k(z)|jN|k2,
F^N=1Nj,k=0N1ei2πNjk|jk|,
Λ^N=F^NC^NF^N=j=0N1ei2πNj|jj|.
|E(z)=(F^N1^2)|A(z),
iz|A(z)=H^D|A(z)
H^D=j=0N1|jj|H^j,
H^j=iγσ^z+[Γjσ^++Γj*σ^],
Γj=gdeiϕd+gNeiϕNei2πNj.
λ±j=±|Γj|2γ2,=±gd2+gN2+2gdgNcos(ϕd+ϕN+2πNj)γ2,
γ2g=cos2πNj,j=0,1,2,N2(N21),
|E(z)=U^(z)|E(0),
U^(z)=j=0N1F^N|jj|F^NeiH^jz,
eiH^jz=1^2cosΩjz+izH^jsinc  Ωjz,
Ωj=|Γj|2γ2,
eiH^jz=1^2cosΩjz+iΩjH^jsinΩjz,ΩjR.
eiH^jz=1^2+izH^j,Ωj=0.
eiH^jz=1^2cosh|Ωj|z+i|Ωj|H^jsinh|Ωj|z,ΩjC.
|E(0)=F^N|j|ψ(0),=1Na=0N1ei2πNaj|a|ψ(0);
|E(z)=1Np=0N1ei2πNjp|peiH^jz|ψ(0),
Pj,k(z)=|k|eiH^jz|ψ(0)|2k=01|k|eiH^jz|ψ(0)|2,
limzPj,0(z)=γ|Ωj|2γ,limzPj,1(z)=γ+|Ωj|2γ,
iz(E1(z)E2(z)E3(z)E4(z))=(βR+iβIgd0gNgdβRgN00gNβR+iβIgdgN0gdβR)(E1(z)E2(z)E3(z)E4(z)),
iz(E1(z)E2(z)E3(z)E4(z))=(iγgd0gNgdiγgN00gNiγgdgN0gdiγ)(E1(z)E2(z)E3(z)E4(z)),
Pj,k(z)=|Ey[dx(1)j+k,dy(1)j,z]|2|Ey[dx(1)j,dy(1)j,z]|2+|Ey[dx(1)j+1,dy(1)j,z]|2.

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