Abstract

We design a non-parity-time-symmetric plasmonic waveguide-cavity system, consisting of two metal-dielectric-metal stub resonators side coupled to a metal-dielectric-metal waveguide, to form an exceptional point, and realize unidirectional reflectionless propagation at the optical communication wavelength. The contrast ratio between the forward and backward reflection almost reaches unity. We show that the presence of material loss in the metal is critical for the realization of the unidirectional reflectionlessness in this plasmonic system. We investigate the realized exceptional point, as well as the associated physical effects of level repulsion, crossing and phase transition. We also show that, by periodically cascading the unidirectional reflectionless plasmonic waveguide-cavity system, we can design a wavelength-scale unidirectional plasmonic waveguide perfect absorber. Our results could be potentially important for developing a new generation of highly compact unidirectional integrated nanoplasmonic devices.

© 2015 Optical Society of America

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References

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  32. Y. Huang, C. Min, P. Dastmalchi, and G. Veronis, “Slow-light enhanced subwavelength plasmonic waveguide refractive index sensors,” Opt. Express 23, 14922–14936 (2015).
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  37. J. A. Dionne, H. J. Lezec, and H. A. Atwater, “Highly confined photon transport in subwavelength metallic slot waveguides,” Nano Lett. 6, 1928–1932 (2006).
    [Crossref] [PubMed]
  38. P. Neutens, P. Van Dorpe, I. De Vlaminck, L. Lagae, and G. Borghs, “Electrical detection of confined gap plasmons in metal-insulator-metal waveguides,” Nat. Photonics 3, 283–286 (2009).
    [Crossref]
  39. C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80, 5243–5246 (1998).
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    [Crossref] [PubMed]
  44. S. E. Kocabas, G. Veronis, D. A. B. Miller, and S. Fan, “Transmission line and equivalent circuit models for plasmonic waveguide components,” IEEE J. Sel. Topics Quantum Electron. 14, 1462–1472 (2008).
    [Crossref]
  45. Y. Choi, S. Kang, S. Lim, W. Kim, J. R. Kim, J. H. Lee, and K. An, “Quasieigenstate coalescence in an atom-cavity quantum composite,” Phys. Rev. Lett. 104, 153601 (2010).
    [Crossref] [PubMed]
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  51. W. Shin, W. Cai, P. B. Catrysse, G. Veronis, M. L. Brongersma, and S. Fan, “Broadband sharp 90-degree bends and T-splitters in plasmonic coaxial waveguides,” Nano Lett. 13, 4753–4758 (2013).
    [Crossref] [PubMed]
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2015 (9)

J. Gear, F. Liu, S. T. Chu, S. Rotter, and J. Li, “Parity-time symmetry from stacking purely dielectric and magnetic slabs,” Phys. Rev. A 91, 033825 (2015).
[Crossref]

S. A. R. Horsley, M. Artoni, and G. C. La Rocca, “Spatial Kramer-Kronig relations and the reflection of waves,” Nat. Photonics 9, 436–439 (2015).
[Crossref]

S. Yu, X. Piao, J. Hong, and N. Park, “Progress toward high-Q perfect absorption: A Fano antilaser,” Phys. Rev. A 92, 011802 (2015).
[Crossref]

S. Longhi, “Non-reciprocal transmission in photonic lattices based on unidirectional coherent perfect absorption,” Opt. Lett. 40, 1278–1281 (2015).
[Crossref] [PubMed]

Y. Huang, C. Min, P. Dastmalchi, and G. Veronis, “Slow-light enhanced subwavelength plasmonic waveguide refractive index sensors,” Opt. Express 23, 14922–14936 (2015).
[Crossref] [PubMed]

S. Zhan, H. Li, Z. He, B. Li, Z. Chen, and H. Xu, “Sensing analysis based on plasmon induced transparency in nanocavity-coupled waveguide,” Opt. Express 23, 20313–20320 (2015).
[Crossref] [PubMed]

J. Nath, S. Modak, I. Rezadad, D. Panjwani, F. Rezaie, J. W. Cleary, and R. E. Peale, “Far-infrared absorber based on standing-wave resonances in metal-dielectric-metal cavity,” Opt. Express 23, 20366–20380 (2015).
[Crossref] [PubMed]

A. Mahigir, P. Dastmalchi, W. Shin, S. Fan, and G. Veronis, “Plasmonic coaxial waveguide-cavity devices,” Opt. Express 23, 20549–20562 (2015).
[Crossref] [PubMed]

S. Yu, X. Piao, K.W. Yoo, J. Shin, and N. Park, “One-way optical modal transition based on causality in momentum space,” Opt. Express 23, 24997–25008 (2015).
[Crossref] [PubMed]

2014 (8)

L. Feng, X. Zhu, S. Yang, H. Zhu, P. Zhang, X. Yin, Y. Wang, and X. Zhang, “Demonstration of a large-scale optical exceptional point structure,” Opt. Express 22, 1760–1767 (2014).
[Crossref] [PubMed]

Y. Shen, X. Hua Deng, and L. Chen, “Unidirectional invisibility in a two-layer non-PT-symmetric slab,” Opt. Express 22, 19440–19447 (2014).
[Crossref] [PubMed]

G. Cao, H. Li, Y. Deng, S. Zhan, Z. He, and B. Li, “Plasmon-induced transparency in a single multimode stub resonator,” Opt. Express 22, 25215–25223 (2014).
[Crossref] [PubMed]

J. R. Piper and S. Fan, “Total absorption in a graphene monolayer in the optical regime by critical coupling with a photonic crystal guided resonance,” ACS Photon. 1, 347–353 (2014).
[Crossref]

J. Wu, M. Artoni, and G. C. La Rocca, “Non-Hermitian degeneracies and unidirectional reflectionless atomic lattices,” Phys. Rev. Lett. 113, 123004 (2014).
[Crossref] [PubMed]

Y. Sun, W. Tan, H. Li, J. Li, and H. Chen, “Experimental demonstration of a coherent perfect absorber with PT phase transition,” Phys. Rev. Lett. 112, 143903 (2014).
[Crossref] [PubMed]

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8, 524–529 (2014).
[Crossref]

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

2013 (4)

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

X. Yin and X. Zhang, “Unidirectional light propagation at exceptional points,” Nat. Mater. 12, 175–177 (2013).
[Crossref] [PubMed]

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 (2013).
[Crossref]

W. Shin, W. Cai, P. B. Catrysse, G. Veronis, M. L. Brongersma, and S. Fan, “Broadband sharp 90-degree bends and T-splitters in plasmonic coaxial waveguides,” Nano Lett. 13, 4753–4758 (2013).
[Crossref] [PubMed]

2012 (5)

S. Y. Lee, J. W. Ryu, S. W. Kim, and Y. Chung, “Geometric phase around multiple exceptional points,” Phys. Rev. A 85, 064103 (2012).
[Crossref]

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

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]

I. Zand, A. Mahigir, T. Pakizeh, and M. S. Abrishamian, “Selective-mode optical nanofilters based on plasmonic complementary split-ring resonators,” Opt. Express 20, 7516–7525 (2012).
[Crossref] [PubMed]

Y. Huang, C. Min, and G. Veronis, “Compact slit-based couplers for metal-dielectric-metal plasmonic waveguides,” Opt. Express 20, 22233–22244 (2012).
[Crossref] [PubMed]

2011 (3)

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref] [PubMed]

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

Y. Huang, C. Min, and G. Veronis, “Subwavelength slow-light waveguides based on a plasmonic analogue of electromagnetically induced transparency,” Appl. Phys. Lett. 99, 143117 (2011).
[Crossref]

2010 (6)

Q. H. Song and H. Cao, “Improving optical confinement in nanostructures via external mode coupling,” Phys. Rev. Lett. 105, 053902 (2010).
[Crossref] [PubMed]

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

C. E. Ruter, 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]

Y. Choi, S. Kang, S. Lim, W. Kim, J. R. Kim, J. H. Lee, and K. An, “Quasieigenstate coalescence in an atom-cavity quantum composite,” Phys. Rev. Lett. 104, 153601 (2010).
[Crossref] [PubMed]

S. V. Dmitriev, A. A. Sukhorukov, and Y. S. Kivshar, “Binary parity-time-symmetric nonlinear lattices with balanced gain and loss,” Opt. Lett. 35, 2976–2978 (2010).
[Crossref] [PubMed]

L. Yang, C. Min, and G. Veronis, “Guided subwavelength slow-light mode supported by a plasmonic waveguide system,” Opt. Lett. 35, 4184–4186 (2010).
[Crossref] [PubMed]

2009 (3)

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] [PubMed]

C. Min and G. Veronis, “Absorption switches in metal-dielectric-metal plasmonic waveguides,” Opt. Express 16, 10757–10766 (2009).
[Crossref]

P. Neutens, P. Van Dorpe, I. De Vlaminck, L. Lagae, and G. Borghs, “Electrical detection of confined gap plasmons in metal-insulator-metal waveguides,” Nat. Photonics 3, 283–286 (2009).
[Crossref]

2008 (3)

S. E. Kocabas, G. Veronis, D. A. B. Miller, and S. Fan, “Transmission line and equivalent circuit models for plasmonic waveguide components,” IEEE J. Sel. Topics Quantum Electron. 14, 1462–1472 (2008).
[Crossref]

K. G. Makris, R. EI-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[Crossref] [PubMed]

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

2007 (1)

H. Cartarius, J. Main, and G. Wunner, “Exceptional points in atomic spectra,” Phys. Rev. Lett. 99, 173003 (2007).
[Crossref] [PubMed]

2006 (1)

J. A. Dionne, H. J. Lezec, and H. A. Atwater, “Highly confined photon transport in subwavelength metallic slot waveguides,” Nano Lett. 6, 1928–1932 (2006).
[Crossref] [PubMed]

2004 (2)

2003 (1)

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature (London) 424, 824–830 (2003).
[Crossref]

2000 (1)

W. D. Heiss, “Repulsion of resonance states and exceptional points,” Phys. Rev. E 61, 929–932 (2000).
[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]

1969 (1)

E. N. Economou, “Surface plasmons in thin films,” Phys. Rev. 182, 539–554 (1969).
[Crossref]

Abrishamian, M. S.

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] [PubMed]

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 (2013).
[Crossref]

An, K.

Y. Choi, S. Kang, S. Lim, W. Kim, J. R. Kim, J. H. Lee, and K. An, “Quasieigenstate coalescence in an atom-cavity quantum composite,” Phys. Rev. Lett. 104, 153601 (2010).
[Crossref] [PubMed]

Artoni, M.

S. A. R. Horsley, M. Artoni, and G. C. La Rocca, “Spatial Kramer-Kronig relations and the reflection of waves,” Nat. Photonics 9, 436–439 (2015).
[Crossref]

J. Wu, M. Artoni, and G. C. La Rocca, “Non-Hermitian degeneracies and unidirectional reflectionless atomic lattices,” Phys. Rev. Lett. 113, 123004 (2014).
[Crossref] [PubMed]

Atwater, H. A.

J. A. Dionne, H. J. Lezec, and H. A. Atwater, “Highly confined photon transport in subwavelength metallic slot waveguides,” Nano Lett. 6, 1928–1932 (2006).
[Crossref] [PubMed]

Barnes, W. L.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature (London) 424, 824–830 (2003).
[Crossref]

Bender, C. M.

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

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

Bersch, C.

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

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]

Borghs, G.

P. Neutens, P. Van Dorpe, I. De Vlaminck, L. Lagae, and G. Borghs, “Electrical detection of confined gap plasmons in metal-insulator-metal waveguides,” Nat. Photonics 3, 283–286 (2009).
[Crossref]

Brongersma, M. L.

W. Shin, W. Cai, P. B. Catrysse, G. Veronis, M. L. Brongersma, and S. Fan, “Broadband sharp 90-degree bends and T-splitters in plasmonic coaxial waveguides,” Nano Lett. 13, 4753–4758 (2013).
[Crossref] [PubMed]

Cai, W.

W. Shin, W. Cai, P. B. Catrysse, G. Veronis, M. L. Brongersma, and S. Fan, “Broadband sharp 90-degree bends and T-splitters in plasmonic coaxial waveguides,” Nano Lett. 13, 4753–4758 (2013).
[Crossref] [PubMed]

Cao, G.

Cao, H.

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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 (2013).
[Crossref]

Segev, M.

C. E. Ruter, 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]

Shen, Y.

Shin, J.

Shin, W.

A. Mahigir, P. Dastmalchi, W. Shin, S. Fan, and G. Veronis, “Plasmonic coaxial waveguide-cavity devices,” Opt. Express 23, 20549–20562 (2015).
[Crossref] [PubMed]

W. Shin, W. Cai, P. B. Catrysse, G. Veronis, M. L. Brongersma, and S. Fan, “Broadband sharp 90-degree bends and T-splitters in plasmonic coaxial waveguides,” Nano Lett. 13, 4753–4758 (2013).
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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] [PubMed]

Song, Q. H.

Q. H. Song and H. Cao, “Improving optical confinement in nanostructures via external mode coupling,” Phys. Rev. Lett. 105, 053902 (2010).
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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).
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Y. D. Chong, L. Ge, and A. D. Stone, “PT-symmetry breaking and laser-absorber modes in optical scattering systems,” Phys. Rev. Lett. 106, 093902 (2011).
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Sun, Y.

Y. Sun, W. Tan, H. Li, J. Li, and H. Chen, “Experimental demonstration of a coherent perfect absorber with PT phase transition,” Phys. Rev. Lett. 112, 143903 (2014).
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Y. Sun, W. Tan, H. Li, J. Li, and H. Chen, “Experimental demonstration of a coherent perfect absorber with PT phase transition,” Phys. Rev. Lett. 112, 143903 (2014).
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Tsironis, G. P.

N. Lazarides and G. P. Tsironis, “Gain-driven discrete breathers in PT-symmetric nonlinear metamaterials,” Phys. Rev. Lett. 110, 053901 (2013).
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A. Mahigir, P. Dastmalchi, W. Shin, S. Fan, and G. Veronis, “Plasmonic coaxial waveguide-cavity devices,” Opt. Express 23, 20549–20562 (2015).
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Y. Huang, C. Min, P. Dastmalchi, and G. Veronis, “Slow-light enhanced subwavelength plasmonic waveguide refractive index sensors,” Opt. Express 23, 14922–14936 (2015).
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W. Shin, W. Cai, P. B. Catrysse, G. Veronis, M. L. Brongersma, and S. Fan, “Broadband sharp 90-degree bends and T-splitters in plasmonic coaxial waveguides,” Nano Lett. 13, 4753–4758 (2013).
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Y. Huang, C. Min, and G. Veronis, “Compact slit-based couplers for metal-dielectric-metal plasmonic waveguides,” Opt. Express 20, 22233–22244 (2012).
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Y. Huang, C. Min, and G. Veronis, “Subwavelength slow-light waveguides based on a plasmonic analogue of electromagnetically induced transparency,” Appl. Phys. Lett. 99, 143117 (2011).
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L. Yang, C. Min, and G. Veronis, “Guided subwavelength slow-light mode supported by a plasmonic waveguide system,” Opt. Lett. 35, 4184–4186 (2010).
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L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8, 524–529 (2014).
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L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8, 524–529 (2014).
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S. Yu, X. Piao, K.W. Yoo, J. Shin, and N. Park, “One-way optical modal transition based on causality in momentum space,” Opt. Express 23, 24997–25008 (2015).
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Nat. Mater. (2)

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 (2013).
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[Crossref]

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8, 524–529 (2014).
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Nat. Phys. (2)

C. E. Ruter, 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).
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Opt. Express (11)

C. Min and G. Veronis, “Absorption switches in metal-dielectric-metal plasmonic waveguides,” Opt. Express 16, 10757–10766 (2009).
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L. Feng, X. Zhu, S. Yang, H. Zhu, P. Zhang, X. Yin, Y. Wang, and X. Zhang, “Demonstration of a large-scale optical exceptional point structure,” Opt. Express 22, 1760–1767 (2014).
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Y. Shen, X. Hua Deng, and L. Chen, “Unidirectional invisibility in a two-layer non-PT-symmetric slab,” Opt. Express 22, 19440–19447 (2014).
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S. Yu, X. Piao, K.W. Yoo, J. Shin, and N. Park, “One-way optical modal transition based on causality in momentum space,” Opt. Express 23, 24997–25008 (2015).
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J. Nath, S. Modak, I. Rezadad, D. Panjwani, F. Rezaie, J. W. Cleary, and R. E. Peale, “Far-infrared absorber based on standing-wave resonances in metal-dielectric-metal cavity,” Opt. Express 23, 20366–20380 (2015).
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Y. Huang, C. Min, P. Dastmalchi, and G. Veronis, “Slow-light enhanced subwavelength plasmonic waveguide refractive index sensors,” Opt. Express 23, 14922–14936 (2015).
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S. Zhan, H. Li, Z. He, B. Li, Z. Chen, and H. Xu, “Sensing analysis based on plasmon induced transparency in nanocavity-coupled waveguide,” Opt. Express 23, 20313–20320 (2015).
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A. Mahigir, P. Dastmalchi, W. Shin, S. Fan, and G. Veronis, “Plasmonic coaxial waveguide-cavity devices,” Opt. Express 23, 20549–20562 (2015).
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Y. Huang, C. Min, and G. Veronis, “Compact slit-based couplers for metal-dielectric-metal plasmonic waveguides,” Opt. Express 20, 22233–22244 (2012).
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N. Lazarides and G. P. Tsironis, “Gain-driven discrete breathers in PT-symmetric nonlinear metamaterials,” Phys. Rev. Lett. 110, 053901 (2013).
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Figures (6)

Fig. 1
Fig. 1 (a) Schematic of a MDM plasmonic waveguide side coupled to two MDM stub resonators. (b) Scattering matrix S of the entire two-port plasmonic waveguide system of Fig. 1(a). H L +, and H R + are the complex magnetic field amplitudes of the incoming modes at the left and right ports, respectively. Similarly, H L ,, and H R are the complex magnetic field amplitudes of the outgoing modes from the left and right ports, respectively.
Fig. 2
Fig. 2 (a) Schematic defining the reflection coefficient r1i, and transmission coefficients t1i,t2i, when the fundamental TM mode of a MDM waveguide with width w is incident at a junction with a MDM waveguide with width wi. (b) Schematic defining the reflection coefficient r2i, and transmission coefficient t3i when the fundamental TM mode of a MDM waveguide with width wi is incident at a junction with a MDM waveguide with width w. (c) Schematic defining the reflection coefficient rsi of the fundamental TM mode of a MDM waveguide with width wi at the boundary of a short-circuited MDM waveguide. (d) Schematic defining the reflection coefficient r3i when the fundamental TM mode of a MDM waveguide with width wi is incident at a junction with a MDM waveguide with width w which is terminated by perfect electric conductor (PEC) boundary conditions.
Fig. 3
Fig. 3 (a) Reflection spectra for the structure of Fig. 1(a) calculated for both forward and backward directions using FDFD (solid lines) and scattering matrix theory (circles). Results are shown for w = 50 nm, w1 = 20 nm, w2 = 100 nm, L1 = 175 nm, L2 = 365 nm, and L = 561 nm. Also shown are the reflection spectra calculated using FDFD for lossless metal (blue solid line). (b) Contrast ratio spectra for the structure of Fig. 1(a). All parameters are as in Fig. 3(a). (c) and (d) Magnetic field amplitude profiles for the structure of Fig. 1(a) at f = 193.4 THz (λ0 = 1.55μm), when the fundamental TM mode of the MDM waveguide is incident from the left and right, respectively. All parameters are as in Fig. 3(a). (e) and (f) Magnetic field amplitude in the middle of the MDM waveguide, normalized with respect to the field amplitude of the incident fundamental TM waveguide mode in the middle of the waveguide, when the mode is incident from the left and right, respectively. The two vertical dashed lines indicate the left boundary of the left stub, and the right boundary of the right stub. All parameters are as in Fig. 3(a).
Fig. 4
Fig. 4 (a) and (b) Real and imaginary parts of the eigenvalues of the scattering matrix S as a function of the distance L between the two MDM stub resonators [Fig. 1(a)]. The black and red lines correspond to eigenvalues λ s + = t + r f r band λ s = t r f r b, respectively. All other parameters are as in Fig. 3(a). (c), (d), and (e) Reflection in the forward direction Rf = |rf|2 as a function of the real and imaginary parts of L at f = 192.4 THz, 194.4 THz, and 193.4 THz, respectively. All other parameters are as in Fig. 3(a). The white circle indicates the location of the exceptional point, and the white vertical line indicates the real L-axis.
Fig. 5
Fig. 5 (a) Spectra of the generalized power T + R f R b (black), and of the differential generalized power (red), defined as the derivative of the generalized power with respect to frequency d [ T + R f R b ] / d f, calculated using FDFD. All parameters are as in Fig. 3(a). (b) Phase spectra of the reflection coefficients in the forward (rf, black) and backward (rb, red) directions. All parameters are as in Fig. 3(a).
Fig. 6
Fig. 6 (a) Schematic of a plasmonic waveguide system consisting of an array of two MDM stub resonators side-coupled to a MDM waveguide. The system is obtained by cascading the side-coupled resonator structure of Fig. 1(a). (b) Absorption spectra for the structure of Fig. 6(a) calculated for both forward (black curves) and backward (red curves) directions using FDFD. Results are shown for h = 520 nm for single (dots), double (solid line), and triple (open circles) unit cell structures. All other parameters are as in Fig. 3(a). (c) Reflection in the backward direction as a function of the distance h between two adjacent unit cells for the structure of Fig. 6(a) at f = 193.4 THz (λ0 = 1.55μm). All other parameters are as in Fig. 3(a). Results are shown for the double unit cell structure.

Equations (8)

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( H R H L ) = S ( H L + H R + ) = ( t r b r f t ) ( H L + H R + ) ,
( H R H R + ) = T ( H L + H L ) .
M i = ( b i 2 a i 2 b i a i b i a i b i 1 b i ) ,
T = M 1 ( e γ L 0 0 e γ L ) M 2 .
t = H R H L + | H R + = 0 = H L H R + | H L + = 0 = b 1 b 2 a 1 a 2 e γ L + e y L ,
r f = H L H L + | H R + = 0 = a 2 ( b 1 2 a 1 2 ) e γ L + a 1 e γ L a 1 a 2 e γ L + e γ L ,
r b = H R H R + | H L + = 0 = a 1 ( b 2 2 a 2 2 ) e γ L + a 2 e γ L a 1 a 2 e γ L + e γ L .
t 2 i t 3 i = r 3 i r 2 i 2 e 2 γ d 1 r 1 i e 2 γ d + t 3 i e 2 γ d ( 1 r 1 i e 2 γ d ) 2 ( t 1 i e 2 γ d ) 2 ,

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