Abstract

We analyze the operation of 2 × 2 switches composed of two coupled waveguides operating on the basis of parity-time (PT) symmetry: the two waveguides differ through their gain or loss factors and not through the real part of their propagation constant. Plasmonics constitutes a preferred application for such systems, since combination of plasmonics with gain is increasingly mastered. The exact PT-symmetric case (gain and loss of identical absolute value) is considered as well as various unbalanced cases, thanks to their respective switching diagrams. Although perfect signal-conserving cross and bar states are not always possible in the latter cases, they can nevertheless form the basis of very good switches if precise design rules are followed. We draw from the analysis what the optimal configurations are in terms of, e.g., guide gain or gain-length product to operate the switch. Many analytical or semi-analytical results are pointed out. A practical example based on the coupling of a long-range surface-plasmon-polariton and a polymeric waveguide having gain is provided.

© 2013 OSA

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    [CrossRef]
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    [CrossRef] [PubMed]
  43. M. A.  Noginov, G.  Zhu, A. M.  Belgrave, R.  Bakker, V. M.  Shalaev, E. E.  Narimanov, S.  Stout, E.  Herz, T.  Suteewong, U.  Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460(7259), 1110–1112 (2009).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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2013 (7)

I. V.  Barashenkov, L.  Baker, N. V.  Alexeeva, “PT-symmetry breaking in a necklace of coupled optical waveguides,” Phys. Rev. A 87(3), 033819 (2013).
[CrossRef]

C.  Hang, G.  Huang, V. V.  Konotop, “PT Symmetry with a system of three-level atoms,” Phys. Rev. Lett. 110(8), 083604 (2013).
[CrossRef] [PubMed]

M.  Kang, F.  Liu, J.  Li, “Effective spontaneous PT-symmetry breaking in hybridized metamaterials,” Phys. Rev. A 87(5), 053824 (2013).
[CrossRef]

N.  Lazarides, G. P.  Tsironis, “Gain-driven discrete breathers in PT-symmetric nonlinear metamaterials,” Phys. Rev. Lett. 110(5), 053901 (2013).
[CrossRef] [PubMed]

G.  Castaldi, S.  Savoia, V.  Galdi, A.  Alù, N.  Engheta, “PT metamaterials via complex-coordinate transformation optics,” Phys. Rev. Lett. 110(17), 173901 (2013).
[CrossRef] [PubMed]

Y. V.  Bludov, R.  Driben, V. V.  Konotop, B. A.  Malomed, “Instabilities, solitons and rogue waves in PT-coupled nonlinear waveguides,” J. Opt. 15(6), 064010 (2013).
[CrossRef]

M.  Kulishov, B.  Kress, R.  Slavík, “Resonant cavities based on Parity-Time-symmetric diffractive gratings,” Opt. Express 21(8), 9473–9483 (2013).
[CrossRef] [PubMed]

2012 (12)

M.-A.  Miri, A.  Regensburger, U.  Peschel, D. N.  Christodoulides, “Optical mesh lattices with PT –symmetry,” Phys. Rev. A 86(2), 023807 (2012).
[CrossRef]

H.  Ramezani, T.  Kottos, V.  Kovanis, D. N.  Christodoulides, “Exceptional-point dynamics in photonic honeycomb lattices with PT symmetry,” Phys. Rev. A 85(1), 013818 (2012).
[CrossRef]

S.  Longhi, G.  Della Valle, “Photonic realization of PT-symmetric quantum field theories,” Phys. Rev. A 85(1), 012112 (2012).
[CrossRef]

V. V.  Konotop, V. S.  Shchesnovich, D. A.  Zezyulin, “Giant amplification of modes in parity-time symmetric waveguides,” Phys. Lett. A 376(42-43), 2750–2753 (2012).
[CrossRef]

D. A.  Zezyulin, V. V.  Konotop, “Nonlinear Modes in finite-dimensional PT-symmetric systems,” Phys. Rev. Lett. 108(21), 213906 (2012).
[CrossRef] [PubMed]

A. A.  Sukhorukov, S. V.  Dmitriev, S. V.  Suchkov, Y. S.  Kivshar, “Nonlocality in PT-symmetric waveguide arrays with gain and loss,” Opt. Lett. 37(11), 2148–2150 (2012).
[CrossRef] [PubMed]

H.  Ramezani, J.  Schindler, F. M.  Ellis, U.  Günther, T.  Kottos, “Bypassing the bandwidth theorem with PT symmetry,” Phys. Rev. A 85(6), 062122 (2012).
[CrossRef]

M.  Kulishov, B.  Kress, “Free space diffraction on active gratings with balanced phase and gain/loss modulations,” Opt. Express 20(28), 29319–29328 (2012).
[CrossRef] [PubMed]

L.  Feng, Y. L.  Xu, W. S.  Fegadolli, M. H.  Lu, J. E.  Oliveira, V. R.  Almeida, Y. F.  Chen, A.  Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12(2), 108–113 (2012).
[CrossRef] [PubMed]

A.  Castanié, D.  Felbacq, “Confined plasmonic modes in a nanocavity,” Opt. Commun. 285(16), 3353–3357 (2012).
[CrossRef]

H.  Benisty, M.  Besbes, “Confinement and optical properties of the plasmonic inverse-rib waveguide,” J. Opt. Soc. Am. B 29(4), 818–826 (2012).
[CrossRef]

H.  Benisty, C.  Yan, A.  Degiron, A. T.  Lupu, “Healing near-PT-symmetric structures to restore their characteristic singularities: Analysis and examples,” J. Lightwave Technol. 30(16), 2675–2683 (2012).
[CrossRef]

2011 (5)

H.  Benisty, A.  Degiron, A.  Lupu, A.  De Lustrac, S.  Chénais, S.  Forget, M.  Besbes, G.  Barbillon, A.  Bruyant, S.  Blaize, G.  Lérondel, “Implementation of PT symmetric devices using plasmonics: principle and applications,” Opt. Express 19(19), 18004–18019 (2011).
[CrossRef] [PubMed]

R.-M.  Ma, R. F.  Oulton, V. J.  Sorger, G.  Bartal, X.  Zhang, “Room-temperature sub-diffraction-limited plasmon laser by total internal reflection,” Nat. Mater. 10(2), 110–113 (2011).
[CrossRef] [PubMed]

S.  Longhi, “Invisibility in PT -symmetric complex crystals,” J. Phys. A: Math. Theor. 44(48), 485302 (2011).
[CrossRef]

A. E.  Miroshnichenko, B. A.  Malomed, Yu. S.  Kivshar, “Nonlinearly-PT -symmetric systems: spontaneous symmetry breaking and transmission resonances,” Phys. Rev. A 84(1), 012123 (2011).
[CrossRef]

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

2010 (6)

T.  Kottos, “Broken symmetry makes light work,” Nat. Phys. 6(3), 166–167 (2010).
[CrossRef]

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

M. C.  Zheng, D. N.  Christodoulides, R.  Fleischmann, T.  Kottos, “PT optical lattices and universality in beam dynamics,” Phys. Rev. A 82(1), 010103 (2010).
[CrossRef]

J.  Čtyroký, V.  Kuzmiak, S.  Eyderman, “Waveguide structures with antisymmetric gain/loss profile,” Opt. Express 18(21), 21585–21593 (2010).
[CrossRef] [PubMed]

I.  De Leon, P.  Berini, “Amplification of long-range surface plasmons by a dipolar gain medium,” Nat. Photonics 4(6), 382–387 (2010).
[CrossRef]

H.  Benisty, M.  Besbes, “Plasmonic inverse rib waveguiding for tight confinement and smooth interface definition,” J. Appl. Phys. 108(6), 063108 (2010).
[CrossRef]

2009 (5)

R. F.  Oulton, V. J.  Sorger, T.  Zentgraf, R.-M.  Ma, C.  Gladden, L.  Dai, G.  Bartal, X.  Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461(7264), 629–632 (2009).
[CrossRef] [PubMed]

M. A.  Noginov, G.  Zhu, A. M.  Belgrave, R.  Bakker, V. M.  Shalaev, E. E.  Narimanov, S.  Stout, E.  Herz, T.  Suteewong, U.  Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460(7259), 1110–1112 (2009).
[CrossRef] [PubMed]

A.  Degiron, S.-Y.  Cho, T.  Tyler, N. M.  Jokerst, D. R.  Smith, “Directional coupling between dielectric and long-range plasmon waveguides,” New J. Phys. 11(1), 015002 (2009).
[CrossRef]

A.  Guo, G. J.  Salamo, M.  Volatier-Ravat, V.  Aimez, G. A.  Siviloglou, D. N.  Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103(9), 093902 (2009).
[CrossRef] [PubMed]

O.  Bendix, R.  Fleischmann, T.  Kottos, B.  Shapiro, “Exponentially fragile PT symmetry in lattices with localized eigenmodes,” Phys. Rev. Lett. 103(3), 030402 (2009).
[CrossRef] [PubMed]

2008 (4)

2007 (3)

C. M.  Bender, D. C.  Brody, H. F.  Jones, B. K.  Meister, “Faster than Hermitian Quantum Mechanics,” Phys. Rev. Lett. 98(4), 040403 (2007).
[CrossRef] [PubMed]

R.  El-Ganainy, K. G.  Makris, D. N.  Christodoulides, Z. H.  Musslimani, “Theory of coupled optical PT-symmetric structures,” Opt. Lett. 32(17), 2632–2634 (2007).
[CrossRef] [PubMed]

C. M.  Bender, “Making Sense of non-Hermitian Hamiltonians,” Rep. Prog. Phys. 70(6), 947–1018 (2007).
[CrossRef]

2005 (3)

2004 (2)

2003 (1)

D. J.  Bergman, M. I.  Stockman, “Surface plasmon amplification by stimulated emission of radiation: quantum generation of coherent surface plasmons in nanosystems,” Phys. Rev. Lett. 90(2), 027402 (2003).
[CrossRef] [PubMed]

2001 (1)

P.  Berini, “Plasmon-polariton waves guided by thin lossy metal films of finite width: Bound modes of asymmetric structures,” Phys. Rev. B 63(12), 125417 (2001).
[CrossRef]

2000 (1)

P.  Berini, “Plasmon-polariton waves guided by thin lossy metal films of finite width: Bound modes of symmetric structures,” Phys. Rev. B 61(15), 10484–10503 (2000).
[CrossRef]

1998 (1)

C. M.  Bender, S.  Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80(24), 5243–5246 (1998).
[CrossRef]

1976 (1)

H.  Kogelnik, R. V.  Schmidt, “Switched directional couplers with alternating Δβ,” IEEE J. Quantum Electron. 12(7), 396–401 (1976).
[CrossRef]

Adegoke, J. A.

Aimez, V.

A.  Guo, G. J.  Salamo, M.  Volatier-Ravat, V.  Aimez, G. A.  Siviloglou, D. N.  Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103(9), 093902 (2009).
[CrossRef] [PubMed]

Alexeeva, N. V.

I. V.  Barashenkov, L.  Baker, N. V.  Alexeeva, “PT-symmetry breaking in a necklace of coupled optical waveguides,” Phys. Rev. A 87(3), 033819 (2013).
[CrossRef]

Almeida, V. R.

L.  Feng, Y. L.  Xu, W. S.  Fegadolli, M. H.  Lu, J. E.  Oliveira, V. R.  Almeida, Y. F.  Chen, A.  Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12(2), 108–113 (2012).
[CrossRef] [PubMed]

Alù, A.

G.  Castaldi, S.  Savoia, V.  Galdi, A.  Alù, N.  Engheta, “PT metamaterials via complex-coordinate transformation optics,” Phys. Rev. Lett. 110(17), 173901 (2013).
[CrossRef] [PubMed]

Azaña, J.

Bahoura, M.

Baker, L.

I. V.  Barashenkov, L.  Baker, N. V.  Alexeeva, “PT-symmetry breaking in a necklace of coupled optical waveguides,” Phys. Rev. A 87(3), 033819 (2013).
[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, U.  Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460(7259), 1110–1112 (2009).
[CrossRef] [PubMed]

Barashenkov, I. V.

I. V.  Barashenkov, L.  Baker, N. V.  Alexeeva, “PT-symmetry breaking in a necklace of coupled optical waveguides,” Phys. Rev. A 87(3), 033819 (2013).
[CrossRef]

Barbillon, G.

Bartal, G.

R.-M.  Ma, R. F.  Oulton, V. J.  Sorger, G.  Bartal, X.  Zhang, “Room-temperature sub-diffraction-limited plasmon laser by total internal reflection,” Nat. Mater. 10(2), 110–113 (2011).
[CrossRef] [PubMed]

R. F.  Oulton, V. J.  Sorger, T.  Zentgraf, R.-M.  Ma, C.  Gladden, L.  Dai, G.  Bartal, X.  Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461(7264), 629–632 (2009).
[CrossRef] [PubMed]

Bélanger, N.

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, U.  Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460(7259), 1110–1112 (2009).
[CrossRef] [PubMed]

Bender, C. M.

C. M.  Bender, D. C.  Brody, H. F.  Jones, B. K.  Meister, “Faster than Hermitian Quantum Mechanics,” Phys. Rev. Lett. 98(4), 040403 (2007).
[CrossRef] [PubMed]

C. M.  Bender, “Making Sense of non-Hermitian Hamiltonians,” Rep. Prog. Phys. 70(6), 947–1018 (2007).
[CrossRef]

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C. E.  Rüter, K. G.  Makris, R.  El-Ganainy, D. N.  Christodoulides, M.  Segev, D.  Kip, “Observation of parity–time symmetry in optics,” Nat. Phys. 6(3), 192–195 (2010).
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K. G.  Makris, R.  El-Ganainy, D. N.  Christodoulides, Z. H.  Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100(10), 103904 (2008).
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Y. V.  Bludov, R.  Driben, V. V.  Konotop, B. A.  Malomed, “Instabilities, solitons and rogue waves in PT-coupled nonlinear waveguides,” J. Opt. 15(6), 064010 (2013).
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C. M.  Bender, D. C.  Brody, H. F.  Jones, B. K.  Meister, “Faster than Hermitian Quantum Mechanics,” Phys. Rev. Lett. 98(4), 040403 (2007).
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M.-A.  Miri, A.  Regensburger, U.  Peschel, D. N.  Christodoulides, “Optical mesh lattices with PT –symmetry,” Phys. Rev. A 86(2), 023807 (2012).
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A. E.  Miroshnichenko, B. A.  Malomed, Yu. S.  Kivshar, “Nonlinearly-PT -symmetric systems: spontaneous symmetry breaking and transmission resonances,” Phys. Rev. A 84(1), 012123 (2011).
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Muga, J. G.

A.  Ruschhaupt, F.  Delgado, J. G.  Muga, “Physical realization of PT -symmetric potential scattering in a planar slab waveguide,” J. Phys. A 38(9), L171–L176 (2005).
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K. G.  Makris, R.  El-Ganainy, D. N.  Christodoulides, Z. H.  Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100(10), 103904 (2008).
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M. A.  Noginov, G.  Zhu, A. M.  Belgrave, R.  Bakker, V. M.  Shalaev, E. E.  Narimanov, S.  Stout, E.  Herz, T.  Suteewong, U.  Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460(7259), 1110–1112 (2009).
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Oulton, R. F.

R.-M.  Ma, R. F.  Oulton, V. J.  Sorger, G.  Bartal, X.  Zhang, “Room-temperature sub-diffraction-limited plasmon laser by total internal reflection,” Nat. Mater. 10(2), 110–113 (2011).
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M.-A.  Miri, A.  Regensburger, U.  Peschel, D. N.  Christodoulides, “Optical mesh lattices with PT –symmetry,” Phys. Rev. A 86(2), 023807 (2012).
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H.  Ramezani, J.  Schindler, F. M.  Ellis, U.  Günther, T.  Kottos, “Bypassing the bandwidth theorem with PT symmetry,” Phys. Rev. A 85(6), 062122 (2012).
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G.  Castaldi, S.  Savoia, V.  Galdi, A.  Alù, N.  Engheta, “PT metamaterials via complex-coordinate transformation optics,” Phys. Rev. Lett. 110(17), 173901 (2013).
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L.  Feng, Y. L.  Xu, W. S.  Fegadolli, M. H.  Lu, J. E.  Oliveira, V. R.  Almeida, Y. F.  Chen, A.  Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12(2), 108–113 (2012).
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H.  Kogelnik, R. V.  Schmidt, “Switched directional couplers with alternating Δβ,” IEEE J. Quantum Electron. 12(7), 396–401 (1976).
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Segev, M.

C. E.  Rüter, K. G.  Makris, R.  El-Ganainy, D. N.  Christodoulides, M.  Segev, D.  Kip, “Observation of parity–time symmetry in optics,” Nat. Phys. 6(3), 192–195 (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, U.  Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460(7259), 1110–1112 (2009).
[CrossRef] [PubMed]

Shapiro, B.

O.  Bendix, R.  Fleischmann, T.  Kottos, B.  Shapiro, “Exponentially fragile PT symmetry in lattices with localized eigenmodes,” Phys. Rev. Lett. 103(3), 030402 (2009).
[CrossRef] [PubMed]

Shchesnovich, V. S.

V. V.  Konotop, V. S.  Shchesnovich, D. A.  Zezyulin, “Giant amplification of modes in parity-time symmetric waveguides,” Phys. Lett. A 376(42-43), 2750–2753 (2012).
[CrossRef]

Siviloglou, G. A.

A.  Guo, G. J.  Salamo, M.  Volatier-Ravat, V.  Aimez, G. A.  Siviloglou, D. N.  Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103(9), 093902 (2009).
[CrossRef] [PubMed]

Slavík, R.

Smith, D. R.

A.  Degiron, S.-Y.  Cho, T.  Tyler, N. M.  Jokerst, D. R.  Smith, “Directional coupling between dielectric and long-range plasmon waveguides,” New J. Phys. 11(1), 015002 (2009).
[CrossRef]

Sorger, V. J.

R.-M.  Ma, R. F.  Oulton, V. J.  Sorger, G.  Bartal, X.  Zhang, “Room-temperature sub-diffraction-limited plasmon laser by total internal reflection,” Nat. Mater. 10(2), 110–113 (2011).
[CrossRef] [PubMed]

R. F.  Oulton, V. J.  Sorger, T.  Zentgraf, R.-M.  Ma, C.  Gladden, L.  Dai, G.  Bartal, X.  Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461(7264), 629–632 (2009).
[CrossRef] [PubMed]

Stockman, M. I.

D. J.  Bergman, M. I.  Stockman, “Surface plasmon amplification by stimulated emission of radiation: quantum generation of coherent surface plasmons in nanosystems,” Phys. Rev. Lett. 90(2), 027402 (2003).
[CrossRef] [PubMed]

Stone, A. D.

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

Stout, S.

M. A.  Noginov, G.  Zhu, A. M.  Belgrave, R.  Bakker, V. M.  Shalaev, E. E.  Narimanov, S.  Stout, E.  Herz, T.  Suteewong, U.  Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460(7259), 1110–1112 (2009).
[CrossRef] [PubMed]

Suchkov, S. V.

Sukhorukov, A. A.

Suteewong, T.

M. A.  Noginov, G.  Zhu, A. M.  Belgrave, R.  Bakker, V. M.  Shalaev, E. E.  Narimanov, S.  Stout, E.  Herz, T.  Suteewong, U.  Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460(7259), 1110–1112 (2009).
[CrossRef] [PubMed]

Tetz, K.

Tsironis, G. P.

N.  Lazarides, G. P.  Tsironis, “Gain-driven discrete breathers in PT-symmetric nonlinear metamaterials,” Phys. Rev. Lett. 110(5), 053901 (2013).
[CrossRef] [PubMed]

Tyler, T.

A.  Degiron, S.-Y.  Cho, T.  Tyler, N. M.  Jokerst, D. R.  Smith, “Directional coupling between dielectric and long-range plasmon waveguides,” New J. Phys. 11(1), 015002 (2009).
[CrossRef]

Volatier-Ravat, M.

A.  Guo, G. J.  Salamo, M.  Volatier-Ravat, V.  Aimez, G. A.  Siviloglou, D. N.  Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103(9), 093902 (2009).
[CrossRef] [PubMed]

Volkov, V. S.

Wiesner, U.

M. A.  Noginov, G.  Zhu, A. M.  Belgrave, R.  Bakker, V. M.  Shalaev, E. E.  Narimanov, S.  Stout, E.  Herz, T.  Suteewong, U.  Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460(7259), 1110–1112 (2009).
[CrossRef] [PubMed]

Xu, Y. L.

L.  Feng, Y. L.  Xu, W. S.  Fegadolli, M. H.  Lu, J. E.  Oliveira, V. R.  Almeida, Y. F.  Chen, A.  Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12(2), 108–113 (2012).
[CrossRef] [PubMed]

Yan, C.

Zentgraf, T.

R. F.  Oulton, V. J.  Sorger, T.  Zentgraf, R.-M.  Ma, C.  Gladden, L.  Dai, G.  Bartal, X.  Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461(7264), 629–632 (2009).
[CrossRef] [PubMed]

Zezyulin, D. A.

V. V.  Konotop, V. S.  Shchesnovich, D. A.  Zezyulin, “Giant amplification of modes in parity-time symmetric waveguides,” Phys. Lett. A 376(42-43), 2750–2753 (2012).
[CrossRef]

D. A.  Zezyulin, V. V.  Konotop, “Nonlinear Modes in finite-dimensional PT-symmetric systems,” Phys. Rev. Lett. 108(21), 213906 (2012).
[CrossRef] [PubMed]

Zhang, X.

R.-M.  Ma, R. F.  Oulton, V. J.  Sorger, G.  Bartal, X.  Zhang, “Room-temperature sub-diffraction-limited plasmon laser by total internal reflection,” Nat. Mater. 10(2), 110–113 (2011).
[CrossRef] [PubMed]

R. F.  Oulton, V. J.  Sorger, T.  Zentgraf, R.-M.  Ma, C.  Gladden, L.  Dai, G.  Bartal, X.  Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461(7264), 629–632 (2009).
[CrossRef] [PubMed]

Zheng, M. C.

M. C.  Zheng, D. N.  Christodoulides, R.  Fleischmann, T.  Kottos, “PT optical lattices and universality in beam dynamics,” Phys. Rev. A 82(1), 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, U.  Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460(7259), 1110–1112 (2009).
[CrossRef] [PubMed]

M. A.  Noginov, V. A.  Podolskiy, G.  Zhu, M.  Mayy, M.  Bahoura, J. A.  Adegoke, B. A.  Ritzo, K.  Reynolds, “Compensation of loss in propagating surface plasmon polariton by gain in adjacent dielectric medium,” Opt. Express 16(2), 1385–1392 (2008).
[CrossRef] [PubMed]

IEEE J. Quantum Electron. (1)

H.  Kogelnik, R. V.  Schmidt, “Switched directional couplers with alternating Δβ,” IEEE J. Quantum Electron. 12(7), 396–401 (1976).
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H.  Benisty, M.  Besbes, “Plasmonic inverse rib waveguiding for tight confinement and smooth interface definition,” J. Appl. Phys. 108(6), 063108 (2010).
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Y. V.  Bludov, R.  Driben, V. V.  Konotop, B. A.  Malomed, “Instabilities, solitons and rogue waves in PT-coupled nonlinear waveguides,” J. Opt. 15(6), 064010 (2013).
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J. Opt. Soc. Am. B (1)

J. Phys. A (1)

A.  Ruschhaupt, F.  Delgado, J. G.  Muga, “Physical realization of PT -symmetric potential scattering in a planar slab waveguide,” J. Phys. A 38(9), L171–L176 (2005).
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J. Phys. A: Math. Theor. (1)

S.  Longhi, “Invisibility in PT -symmetric complex crystals,” J. Phys. A: Math. Theor. 44(48), 485302 (2011).
[CrossRef]

Nat. Mater. (2)

L.  Feng, Y. L.  Xu, W. S.  Fegadolli, M. H.  Lu, J. E.  Oliveira, V. R.  Almeida, Y. F.  Chen, A.  Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12(2), 108–113 (2012).
[CrossRef] [PubMed]

R.-M.  Ma, R. F.  Oulton, V. J.  Sorger, G.  Bartal, X.  Zhang, “Room-temperature sub-diffraction-limited plasmon laser by total internal reflection,” Nat. Mater. 10(2), 110–113 (2011).
[CrossRef] [PubMed]

Nat. Photonics (1)

I.  De Leon, P.  Berini, “Amplification of long-range surface plasmons by a dipolar gain medium,” Nat. Photonics 4(6), 382–387 (2010).
[CrossRef]

Nat. Phys. (2)

T.  Kottos, “Broken symmetry makes light work,” Nat. Phys. 6(3), 166–167 (2010).
[CrossRef]

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

Nature (2)

R. F.  Oulton, V. J.  Sorger, T.  Zentgraf, R.-M.  Ma, C.  Gladden, L.  Dai, G.  Bartal, X.  Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461(7264), 629–632 (2009).
[CrossRef] [PubMed]

M. A.  Noginov, G.  Zhu, A. M.  Belgrave, R.  Bakker, V. M.  Shalaev, E. E.  Narimanov, S.  Stout, E.  Herz, T.  Suteewong, U.  Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460(7259), 1110–1112 (2009).
[CrossRef] [PubMed]

New J. Phys. (1)

A.  Degiron, S.-Y.  Cho, T.  Tyler, N. M.  Jokerst, D. R.  Smith, “Directional coupling between dielectric and long-range plasmon waveguides,” New J. Phys. 11(1), 015002 (2009).
[CrossRef]

Opt. Commun. (1)

A.  Castanié, D.  Felbacq, “Confined plasmonic modes in a nanocavity,” Opt. Commun. 285(16), 3353–3357 (2012).
[CrossRef]

Opt. Express (9)

H.  Benisty, A.  Degiron, A.  Lupu, A.  De Lustrac, S.  Chénais, S.  Forget, M.  Besbes, G.  Barbillon, A.  Bruyant, S.  Blaize, G.  Lérondel, “Implementation of PT symmetric devices using plasmonics: principle and applications,” Opt. Express 19(19), 18004–18019 (2011).
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M.  Kulishov, J. M.  Laniel, N.  Bélanger, J.  Azaña, D. V.  Plant, “Nonreciprocal waveguide Bragg gratings,” Opt. Express 13(8), 3068–3078 (2005).
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M.  Kulishov, J. M.  Laniel, N.  Bélanger, D. V.  Plant, “Trapping light in a ring resonator using a grating-assisted coupler with asymmetric transmission,” Opt. Express 13(9), 3567–3578 (2005).
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J.  Čtyroký, V.  Kuzmiak, S.  Eyderman, “Waveguide structures with antisymmetric gain/loss profile,” Opt. Express 18(21), 21585–21593 (2010).
[CrossRef] [PubMed]

M.  Kulishov, B.  Kress, “Free space diffraction on active gratings with balanced phase and gain/loss modulations,” Opt. Express 20(28), 29319–29328 (2012).
[CrossRef] [PubMed]

M.  Kulishov, B.  Kress, R.  Slavík, “Resonant cavities based on Parity-Time-symmetric diffractive gratings,” Opt. Express 21(8), 9473–9483 (2013).
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M. P.  Nezhad, K.  Tetz, Y.  Fainman, “Gain assisted propagation of surface plasmon polaritons on planar metallic waveguides,” Opt. Express 12(17), 4072–4079 (2004).
[CrossRef] [PubMed]

M. A.  Noginov, V. A.  Podolskiy, G.  Zhu, M.  Mayy, M.  Bahoura, J. A.  Adegoke, B. A.  Ritzo, K.  Reynolds, “Compensation of loss in propagating surface plasmon polariton by gain in adjacent dielectric medium,” Opt. Express 16(2), 1385–1392 (2008).
[CrossRef] [PubMed]

A.  Boltasseva, V. S.  Volkov, R. B.  Nielsen, E.  Moreno, S. G.  Rodrigo, S. I.  Bozhevolnyi, “Triangular metal wedges for subwavelength plasmon-polariton guiding at telecom wavelengths,” Opt. Express 16(8), 5252–5260 (2008).
[CrossRef] [PubMed]

Opt. Lett. (3)

Phys. Lett. A (1)

V. V.  Konotop, V. S.  Shchesnovich, D. A.  Zezyulin, “Giant amplification of modes in parity-time symmetric waveguides,” Phys. Lett. A 376(42-43), 2750–2753 (2012).
[CrossRef]

Phys. Rev. A (8)

H.  Ramezani, J.  Schindler, F. M.  Ellis, U.  Günther, T.  Kottos, “Bypassing the bandwidth theorem with PT symmetry,” Phys. Rev. A 85(6), 062122 (2012).
[CrossRef]

I. V.  Barashenkov, L.  Baker, N. V.  Alexeeva, “PT-symmetry breaking in a necklace of coupled optical waveguides,” Phys. Rev. A 87(3), 033819 (2013).
[CrossRef]

M.  Kang, F.  Liu, J.  Li, “Effective spontaneous PT-symmetry breaking in hybridized metamaterials,” Phys. Rev. A 87(5), 053824 (2013).
[CrossRef]

M. C.  Zheng, D. N.  Christodoulides, R.  Fleischmann, T.  Kottos, “PT optical lattices and universality in beam dynamics,” Phys. Rev. A 82(1), 010103 (2010).
[CrossRef]

A. E.  Miroshnichenko, B. A.  Malomed, Yu. S.  Kivshar, “Nonlinearly-PT -symmetric systems: spontaneous symmetry breaking and transmission resonances,” Phys. Rev. A 84(1), 012123 (2011).
[CrossRef]

M.-A.  Miri, A.  Regensburger, U.  Peschel, D. N.  Christodoulides, “Optical mesh lattices with PT –symmetry,” Phys. Rev. A 86(2), 023807 (2012).
[CrossRef]

H.  Ramezani, T.  Kottos, V.  Kovanis, D. N.  Christodoulides, “Exceptional-point dynamics in photonic honeycomb lattices with PT symmetry,” Phys. Rev. A 85(1), 013818 (2012).
[CrossRef]

S.  Longhi, G.  Della Valle, “Photonic realization of PT-symmetric quantum field theories,” Phys. Rev. A 85(1), 012112 (2012).
[CrossRef]

Phys. Rev. B (2)

P.  Berini, “Plasmon-polariton waves guided by thin lossy metal films of finite width: Bound modes of symmetric structures,” Phys. Rev. B 61(15), 10484–10503 (2000).
[CrossRef]

P.  Berini, “Plasmon-polariton waves guided by thin lossy metal films of finite width: Bound modes of asymmetric structures,” Phys. Rev. B 63(12), 125417 (2001).
[CrossRef]

Phys. Rev. Lett. (12)

D. J.  Bergman, M. I.  Stockman, “Surface plasmon amplification by stimulated emission of radiation: quantum generation of coherent surface plasmons in nanosystems,” Phys. Rev. Lett. 90(2), 027402 (2003).
[CrossRef] [PubMed]

N.  Lazarides, G. P.  Tsironis, “Gain-driven discrete breathers in PT-symmetric nonlinear metamaterials,” Phys. Rev. Lett. 110(5), 053901 (2013).
[CrossRef] [PubMed]

G.  Castaldi, S.  Savoia, V.  Galdi, A.  Alù, N.  Engheta, “PT metamaterials via complex-coordinate transformation optics,” Phys. Rev. Lett. 110(17), 173901 (2013).
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C.  Hang, G.  Huang, V. V.  Konotop, “PT Symmetry with a system of three-level atoms,” Phys. Rev. Lett. 110(8), 083604 (2013).
[CrossRef] [PubMed]

D. A.  Zezyulin, V. V.  Konotop, “Nonlinear Modes in finite-dimensional PT-symmetric systems,” Phys. Rev. Lett. 108(21), 213906 (2012).
[CrossRef] [PubMed]

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

C. M.  Bender, D. C.  Brody, H. F.  Jones, B. K.  Meister, “Faster than Hermitian Quantum Mechanics,” Phys. Rev. Lett. 98(4), 040403 (2007).
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S.  Klaiman, U.  Günther, N.  Moiseyev, “Visualization of branch points in PT-symmetric waveguides,” Phys. Rev. Lett. 101(8), 080402 (2008).
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K. G.  Makris, R.  El-Ganainy, D. N.  Christodoulides, Z. H.  Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100(10), 103904 (2008).
[CrossRef] [PubMed]

A.  Guo, G. J.  Salamo, M.  Volatier-Ravat, V.  Aimez, G. A.  Siviloglou, D. N.  Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103(9), 093902 (2009).
[CrossRef] [PubMed]

O.  Bendix, R.  Fleischmann, T.  Kottos, B.  Shapiro, “Exponentially fragile PT symmetry in lattices with localized eigenmodes,” Phys. Rev. Lett. 103(3), 030402 (2009).
[CrossRef] [PubMed]

C. M.  Bender, S.  Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80(24), 5243–5246 (1998).
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Rep. Prog. Phys. (1)

C. M.  Bender, “Making Sense of non-Hermitian Hamiltonians,” Rep. Prog. Phys. 70(6), 947–1018 (2007).
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Other (1)

A. Snyder and J. D. Love, Optical Waveguide Theory (Kluwer, 2000).

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Figures (9)

Fig. 1
Fig. 1

(a) Sketch of a PT symmetry directional coupler; (b) Real parts (“Re”, solid lines) and Imaginary parts (“Im”, dashed lines) of the two normalized eigenvalues βeff1 and βeff2 of a coupled system; (c) Same diagram presented on the whole algebraic gain axis. “GT*” refers to the gauge transform that realigns the real axis with the dotted line; (d) “Uncalibrated” case with g12 = 1.5 using the same gain axis; (e) “Uncalibrated case with g12=0.5; (f) “Biased” case with relatively small fixed losses; (g) “Biased” case with stronger fixed losses. The motion of the centre of the “Re” and “Im” patterns to the left and/or to the top is pointed out in the two last cases.

Fig. 2
Fig. 2

Color map of a transmission Tij(L,g1) in a dB scale. The thin black solid lines on the maps correspond to the iso-level curves Tij = 1. The ordinate axes points corresponding to a complete power crossover are marked by a Ө, and those indicating the bar state with zero net crossover are marked by a ⊗. The green and magenta color arrows correspond to a perfect switching operation from Tii = 1, Tij = 0 to Tii = 0, Tij = 1, where i≠j. Note that arrows from the two top panels (injection in the “gainy” guide 1) are identical, and differ from arrows of the two bottom ones (injection in the “lossy” guide).

Fig. 3
Fig. 3

(a) Trigonometric picture illustrating Eq. (9). (b) Cross and bar states intensity variation with gain g12 = g1).

Fig. 4
Fig. 4

Color maps of switching operation with injection in the gain waveguide, for two “uncalibrated” configurations: (a), (b) ξ = χ2/g1 = 0.5, maps of T11 and T12 as indicated ; (c), (d) same for ξ = χ2/g1 = 3.0; (e) T12,cross gain penalty in dB as a function of loss/gain ratio ξ (log scale).

Fig. 5
Fig. 5

Color maps of switching operation corresponding to the typical plasmonic setting with fixed loss χ2 = 0.5κ and variable gain. The thin black solid lines on the maps correspond to the iso-level curves Tij = 1. The violet arrows indicate a switching from an initial bar state Ө to a final cross state ⊗, green arrows correspond to a switching from an initial cross state ⊗ to a final bar state Ө. The lengths for the two top T1j diagrams are denoted Lp. The vertical dotted line corresponds to the “exact PT symmetry” operation point.

Fig. 6
Fig. 6

Analysis of the six first switching configurations S1 to S6 of length L1 to L6 (a) Gain g1 required for switching. (b) PTSC length L1 to L6; the two extremes are labeled for clarity. (c) Switching amplification level. (d) Transmitted signal amplification level. The vertical dashed line points relate to Fig. 5, χ2 = 0.5κ case. The red dot further signals the S1 configuration of Fig. 5.

Fig. 7
Fig. 7

(a) Cross and bar states intensity variation with gain g1 for a fixed loss level χ2 = 0.42κ. (b) Cross-sectional view of the hybrid plasmonic/dielectric coupler. The modes propagate along the third dimension (the z-axis). The total length L of the device along the z axis, not shown here, is 5 mm. (c) Cross and bar states intensity vs. g = Im(εSU8). The coupler is fed by the SU8 waveguide. To compute these curves, we assumed that all the electromagnetic power contained in the left semi-infinite xy plane is carried by the SU8 waveguide while all the electromagnetic power contained in the right semi-infinite xy plane is carried by the metallic stripe.

Fig. 8
Fig. 8

Cross and bar states intensity variation with optimized fixed gain (g1 = 0.71κ) and variable loss χ2.

Fig. 9
Fig. 9

Color maps of switching operation with fixed gain g1 = 0.8κ and variable loss (χ2 variable negative gain). The thin black solid lines on the maps correspond to the iso-level curves Tij = 1. The violet arrows indicate a switching from an initial bar state Ө to a final cross state ⊗. The vertical dotted line corresponds to the “exact PT symmetry” operation point. In the top maps, the switching shown is the only one with final unity transmission, whereas in the top map, there is not even a single such possibility.

Equations (29)

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δ= β 1 +i g 1 ( β 2 i χ 2 ) 2 = β 1 β 2 2 +i Δ im
Δ im = g 1 + χ 2 2
M( z )=( cos( Ωz ) iδ Ω sin( Ωz ) iκ Ω sin( Ωz ) iκ Ω sin( Ωz ) cos( Ωz )+ iδ Ω sin( Ωz ) )exp( g 1 χ 2 2 z )
Ω= δ 2 + κ 2
T ij = | M ij | 2
κL=( 2m+1 ) π 2 ,m=1,2,3,...
κL=mπ,m=0,1,2,3,...
{ | cos( ΩL )+ Δ im Ω 1 sin( ΩL ) |=1 κ Ω 1 sin( ΩL )=0
ΩL= κ 2 Δ im 2 L
{ | κ Ω 1 sin( ΩL ) |=1 cos( ΩL )+ Δ im Ω 1 sin( ΩL )=0
{ | sin( ΩL ) |= Ω κ cos( ΩL )= Δ im κ
tan( m 2 π 2 Δ im 2 L 2 )= m 2 π 2 Δ im 2 L 2 1
π sinψ= π ψ
γ= L 2 L c = mκ κ 2 χ 2 2
T 12,cross (dB)=10[log (10) 1 ]2πcosψ 1ξ 1+ξ =18.4 1ξ 1+ξ
{ | cos( Ω Θ L )+ Δ im Θ Ω Θ sin( Ω Θ L ) |=1 | cos( Ω L )+ Δ im Ω sin( Ω L ) |=0
κ 2 ( g 1 Θ + χ 2 2 ) 2 L=mπ
L= mπ κ 2 χ 2
tan( ( κL ) 2 ( Δ im L ) 2 )= ( κL ) 2 ( Δ im L ) 2 1
sinψ= mπψ γπ
g 1 =2κcosψ χ 2
{ cos( Ω L )= Δ im κ κ Ω Θ sin( Ω Θ L )=0
L= (m+1)πacos( χ 2 /κ ) κ 2 χ 2 2
Δ im Θ = κ 2 ( mπ L ) 2
g 1 Θ =2 Δ im Θ χ 2
{ κ Ω Θ sin( Ω Θ L )=0 cos( Ω L )= Δ im κ
( κL ) 2 = ( g 1 L ) 2 + ( mπ ) 2
cos( ( κL ) 2 4 ( g 1 L ) 2 )= 2 g 1 L κL
cos( ( mπ ) 2 3 ( g 1 L ) 2 )= 2 g 1 L ( mπ ) 2 + ( g 1 L ) 2

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