Abstract

We present a semi-analytical model of the resonance phenomena occurring in a hybrid system made of a 1D array of periodic subwavelength slits deposited on an insulator/graphene layer. We show that the spectral response of this hybrid system can be fully explained by a simple semi-analytical model based on weak and strong couplings between two elementary sub-systems. The first elementary sub-system consists of a 1D array of periodic subwavelength slits viewed as a homogeneous medium. In this medium lives a metal-insulator-metal lattice mode interacting with surface and cavity plasmon modes. A weak coupling with surface plasmon modes on both faces of the perforated metal film leads to a broadband spectrum while a strong coupling between this first sub-system and a second one made of a graphene-insulator-metal gap leads to a narrow band spectrum. We provide a semi-analytical model based on these two interactions thus allowing efficient access of the full spectrum of the hybrid system.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Full Article  |  PDF Article
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References

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    [Crossref]
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    [Crossref]
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    [Crossref]

2018 (3)

2017 (3)

B. Zhao, B. Guizal, Z. M. Zhang, S. Fan, and M. Antezza, “Near-field heat transfer between graphene/hBN multilayers,” Phys. Rev. B 95(24), 245437 (2017).
[Crossref]

P. Doyeux, S. A. H. Gangaraj, G. W. Hanson, and M. Antezza, “Giant interatomic energy-transport amplification with nonreciprocal photonic topological insulators,” Phys. Rev. Lett. 119(17), 173901 (2017).
[Crossref]

D. Rodrigo, A. Tittl, O. Limaj, F. J. García de Abajo, V. Pruneri, and H. Altug, “Double-layer graphene for enhanced tunable infrared plasmonics,” Light: Sci. Appl. 6(6), e16277 (2017).
[Crossref]

2016 (3)

2015 (6)

K. Edee and J.-P. Plumey, “Numerical scheme for the modal method based on subsectional Gegenbauer polynomial expansion: application to biperiodic binary grating,” J. Opt. Soc. Am. A 32(3), 402–410 (2015).
[Crossref]

S. Yi, M. Zhou, X. Shi, Q. Gan, J. Zi, and Z. Yu, “A Multiple- Resonator Approach for Broadband Light Absorption in a Single Layer of Nanostructured Graphene,” Opt. Express 23(8), 10081–10090 (2015).
[Crossref]

R. Messina, P. A. M. Neto, B. Guizal, and M. Antezza, “Casimir Interaction between a sphere and a grating,” Phys. Rev. A 92(6), 062504 (2015).
[Crossref]

L. Tang, H. Shi, J. Yang, C. Du, F. Gao, J. Zhu, and J. Dua, “Complete optical absorption in graphene by using nano-gratings to excite graphene surface plasmons,” Microelectron. Eng. 145, 58–61 (2015).
[Crossref]

T. Zhang, L. Chen, B. Wang, and X. Li, “Tunable broadband plasmonic field enhancement on a graphene surface using a normal-incidence plane wave at mid-infrared frequencies,” Sci. Rep. 5(1), 11195 (2015).
[Crossref]

B. Zhao and Z. M. Zhang, “Strong Plasmonic Coupling between Graphene Ribbon Array and Metal Gratings,” ACS Photonics 2(11), 1611–1618 (2015).
[Crossref]

2014 (1)

Z. Fang, Y. Wang, A. E. Schlather, Z. Liu, P. M. Ajayan, F. J. García de Abajo, P. Nordlander, X. Zhu, and N. J. Halas, “Active Tunable Absorption Enhancement with Graphene Nanodisk Arrays,” Nano Lett. 14(1), 299–304 (2014).
[Crossref]

2013 (6)

M. Amin, M. Farhat, and H. Bagci, “A dynamically reconfigurable Fano metamaterial through graphene tuning for switching and sensing applications,” Sci. Rep. 3(1), 2105 (2013).
[Crossref]

V. W. Brar, M. S. Jang, M. Sherrott, J. J. Lopez, and H. A. Atwater, “Highly Confined Tunable Mid-Infrared Plasmonics in Graphene Nanoresonators,” Nano Lett. 13(6), 2541–2547 (2013).
[Crossref]

J. H. Strait, P. Nene, W.-M. Chan, C. Manolatou, S. Tiwari, F. Rana, J. W. Kevek, and P. L. McEuen, “Confined Plasmons in Graphene Microstructures: Experiments and Theory,” Phys. Rev. B: Condens. Matter Mater. Phys. 87(24), 241410 (2013).
[Crossref]

S. Thongrattanasiri and F. J. García de Agajo, “Optical field enhancement by strong plasmon interaction in graphene nanostructures,” Phys. Rev. Lett. 110(18), 187401 (2013).
[Crossref]

H. G. Yan, T. Low, W. J. Zhu, Y. Q. Wu, and M. Freitag et al., “Damping pathways of mid-infrared plasmons in graphene nanostructures,” Nat. Photonics 7(5), 394–399 (2013).
[Crossref]

K. Edee, I. Fenniche, G. Granet, and B. Guizal, “Modal method based on subsectional Gegenbauer polynomial expansion for lamellar gratings: weighting function, convergence and stability,” Prog. Electromagn. Res. 133, 17–35 (2013).
[Crossref]

2012 (4)

W. Gao, J. Shu, C. Qiu, and Q. Xu, “Excitation of Plasmonic Waves in Graphene by Guided-Mode Resonances,” ACS Nano 6(9), 7806–7813 (2012).
[Crossref]

H. Yan, X. Li, B. Chandra, G. Tulevski, Y. Wu, M. Freitag, W. Zhu, P. Avouris, and F. Xia, “Tunable Infrared Plasmonic Devices Using Graphene/Insulator Stacks,” Nat. Nanotechnol. 7(5), 330–334 (2012).
[Crossref]

A. Y. Nikitin, F. Guinea, F. J. Garcia-Vidal, and L. Martin-Moreno, “Surface plasmon enhanced absorption and suppressed transmission in periodic arrays of graphene ribbons,” Phys. Rev. B. 85(8), 081405 (2012).
[Crossref]

A. Fallahi and J. Perruisseau-Carrier, “Design of Tunable Biperiodic Graphene Metasurfaces,” Phys. Rev. B: Condens. Matter Mater. Phys. 86(19), 195408 (2012).
[Crossref]

2011 (2)

F. H. L. Koppens, D. E. Chang, and F. J. García de Agajo, “Graphene plasmonics: a platform for strong light-matter interactions,” Nano Lett. 11(8), 3370–3377 (2011).
[Crossref]

K. Edee, “Modal method based on subsectional Gegenbauer polynomial expansion for lamellar gratings,” J. Opt. Soc. Am. A 28(10), 2006–2013 (2011).
[Crossref]

2009 (1)

A. Y. Nikitin, S. G. Rodrigo, F. J. Garcia-Vidal, and L. Martin-Moreno, “In the diffraction shadow: Norton waves versus surface plasmon polaritons in the optical region,” New J. Phys. 11(12), 123020 (2009).
[Crossref]

2008 (1)

H. T. Liu and P. Lalanne, “Microscopic theory of the extraordinary optical transmission,” Nature 452(7188), 728–731 (2008).
[Crossref]

2007 (1)

L. Aigouy, P. Lalanne, J. P. Hugonin, G. Julie, V. Mathet, and M. Mortier, “Near field analysis of surface waves launched at nano-slit apertures,” Phys. Rev. Lett. 98(15), 153902 (2007).
[Crossref]

2001 (1)

L. Martin-Moreno, F. J. Garcia-Vidal, H. J. Lezec, K. M. Pel- lerin, T. Thio, J. B. Pendry, and T. W. Ebbesen, “Theory of Extraordinary Optical Transmission through Subwavelength Hole Arrays,” Phys. Rev. Lett. 86(6), 1114–1117 (2001).
[Crossref]

1998 (2)

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelenght hole arrays,” Nature 391(6668), 667–669 (1998).
[Crossref]

A. D. Rakiá, A. B. Djuriš ić, J. M. Elazar, and M. L. Majewski, “Optical properties of metallic films for vertical-cavity optoelectronic devices,” Appl. Opt. 37(22), 5271–5283 (1998).
[Crossref]

1992 (1)

R. Brendel and D. Bormann, “An infrared dielectric function model for amorphous solids,” J. Appl. Phys. 71(1), 1–6 (1992).
[Crossref]

Aigouy, L.

L. Aigouy, P. Lalanne, J. P. Hugonin, G. Julie, V. Mathet, and M. Mortier, “Near field analysis of surface waves launched at nano-slit apertures,” Phys. Rev. Lett. 98(15), 153902 (2007).
[Crossref]

Ajayan, P. M.

Z. Fang, Y. Wang, A. E. Schlather, Z. Liu, P. M. Ajayan, F. J. García de Abajo, P. Nordlander, X. Zhu, and N. J. Halas, “Active Tunable Absorption Enhancement with Graphene Nanodisk Arrays,” Nano Lett. 14(1), 299–304 (2014).
[Crossref]

Ali Hassani Gangaraj, S.

M. G. Silveirinha, S. Ali Hassani Gangaraj, George W. Hanson, and Mauro Antezza, “Fluctuation-induced forces on an atom near a photonic topological material,” Phys. Rev. A 97(2), 022509 (2018).
[Crossref]

Altug, H.

D. Rodrigo, A. Tittl, O. Limaj, F. J. García de Abajo, V. Pruneri, and H. Altug, “Double-layer graphene for enhanced tunable infrared plasmonics,” Light: Sci. Appl. 6(6), e16277 (2017).
[Crossref]

Amin, M.

M. Amin, M. Farhat, and H. Bagci, “A dynamically reconfigurable Fano metamaterial through graphene tuning for switching and sensing applications,” Sci. Rep. 3(1), 2105 (2013).
[Crossref]

Antezza, M.

P. Doyeux, S. A. H. Gangaraj, G. W. Hanson, and M. Antezza, “Giant interatomic energy-transport amplification with nonreciprocal photonic topological insulators,” Phys. Rev. Lett. 119(17), 173901 (2017).
[Crossref]

B. Zhao, B. Guizal, Z. M. Zhang, S. Fan, and M. Antezza, “Near-field heat transfer between graphene/hBN multilayers,” Phys. Rev. B 95(24), 245437 (2017).
[Crossref]

R. Messina, P. A. M. Neto, B. Guizal, and M. Antezza, “Casimir Interaction between a sphere and a grating,” Phys. Rev. A 92(6), 062504 (2015).
[Crossref]

Antezza, Mauro

M. G. Silveirinha, S. Ali Hassani Gangaraj, George W. Hanson, and Mauro Antezza, “Fluctuation-induced forces on an atom near a photonic topological material,” Phys. Rev. A 97(2), 022509 (2018).
[Crossref]

Atwater, H. A.

V. W. Brar, M. S. Jang, M. Sherrott, J. J. Lopez, and H. A. Atwater, “Highly Confined Tunable Mid-Infrared Plasmonics in Graphene Nanoresonators,” Nano Lett. 13(6), 2541–2547 (2013).
[Crossref]

Avouris, P.

H. Yan, X. Li, B. Chandra, G. Tulevski, Y. Wu, M. Freitag, W. Zhu, P. Avouris, and F. Xia, “Tunable Infrared Plasmonic Devices Using Graphene/Insulator Stacks,” Nat. Nanotechnol. 7(5), 330–334 (2012).
[Crossref]

Aydin, K.

Bagci, H.

M. Amin, M. Farhat, and H. Bagci, “A dynamically reconfigurable Fano metamaterial through graphene tuning for switching and sensing applications,” Sci. Rep. 3(1), 2105 (2013).
[Crossref]

Bormann, D.

R. Brendel and D. Bormann, “An infrared dielectric function model for amorphous solids,” J. Appl. Phys. 71(1), 1–6 (1992).
[Crossref]

Brar, V. W.

V. W. Brar, M. S. Jang, M. Sherrott, J. J. Lopez, and H. A. Atwater, “Highly Confined Tunable Mid-Infrared Plasmonics in Graphene Nanoresonators,” Nano Lett. 13(6), 2541–2547 (2013).
[Crossref]

Brendel, R.

R. Brendel and D. Bormann, “An infrared dielectric function model for amorphous solids,” J. Appl. Phys. 71(1), 1–6 (1992).
[Crossref]

Chan, W.-M.

J. H. Strait, P. Nene, W.-M. Chan, C. Manolatou, S. Tiwari, F. Rana, J. W. Kevek, and P. L. McEuen, “Confined Plasmons in Graphene Microstructures: Experiments and Theory,” Phys. Rev. B: Condens. Matter Mater. Phys. 87(24), 241410 (2013).
[Crossref]

Chandra, B.

H. Yan, X. Li, B. Chandra, G. Tulevski, Y. Wu, M. Freitag, W. Zhu, P. Avouris, and F. Xia, “Tunable Infrared Plasmonic Devices Using Graphene/Insulator Stacks,” Nat. Nanotechnol. 7(5), 330–334 (2012).
[Crossref]

Chang, D. E.

F. H. L. Koppens, D. E. Chang, and F. J. García de Agajo, “Graphene plasmonics: a platform for strong light-matter interactions,” Nano Lett. 11(8), 3370–3377 (2011).
[Crossref]

Chen, L.

T. Zhang, L. Chen, B. Wang, and X. Li, “Tunable broadband plasmonic field enhancement on a graphene surface using a normal-incidence plane wave at mid-infrared frequencies,” Sci. Rep. 5(1), 11195 (2015).
[Crossref]

Djuriš ic, A. B.

Doyeux, P.

P. Doyeux, S. A. H. Gangaraj, G. W. Hanson, and M. Antezza, “Giant interatomic energy-transport amplification with nonreciprocal photonic topological insulators,” Phys. Rev. Lett. 119(17), 173901 (2017).
[Crossref]

Du, C.

L. Tang, H. Shi, J. Yang, C. Du, F. Gao, J. Zhu, and J. Dua, “Complete optical absorption in graphene by using nano-gratings to excite graphene surface plasmons,” Microelectron. Eng. 145, 58–61 (2015).
[Crossref]

Dua, J.

L. Tang, H. Shi, J. Yang, C. Du, F. Gao, J. Zhu, and J. Dua, “Complete optical absorption in graphene by using nano-gratings to excite graphene surface plasmons,” Microelectron. Eng. 145, 58–61 (2015).
[Crossref]

Ebbesen, T. W.

L. Martin-Moreno, F. J. Garcia-Vidal, H. J. Lezec, K. M. Pel- lerin, T. Thio, J. B. Pendry, and T. W. Ebbesen, “Theory of Extraordinary Optical Transmission through Subwavelength Hole Arrays,” Phys. Rev. Lett. 86(6), 1114–1117 (2001).
[Crossref]

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelenght hole arrays,” Nature 391(6668), 667–669 (1998).
[Crossref]

Edee, K.

Elazar, J. M.

Engheta, N.

A. Vakil and N. Engheta, “One-Atom-Thick IR Metamaterials and Transformation Optics Using Graphene,” arXiv:1101.3585v1 (2011).

Fallahi, A.

A. Fallahi and J. Perruisseau-Carrier, “Design of Tunable Biperiodic Graphene Metasurfaces,” Phys. Rev. B: Condens. Matter Mater. Phys. 86(19), 195408 (2012).
[Crossref]

Fan, S.

B. Zhao, B. Guizal, Z. M. Zhang, S. Fan, and M. Antezza, “Near-field heat transfer between graphene/hBN multilayers,” Phys. Rev. B 95(24), 245437 (2017).
[Crossref]

Fang, Z.

Z. Fang, Y. Wang, A. E. Schlather, Z. Liu, P. M. Ajayan, F. J. García de Abajo, P. Nordlander, X. Zhu, and N. J. Halas, “Active Tunable Absorption Enhancement with Graphene Nanodisk Arrays,” Nano Lett. 14(1), 299–304 (2014).
[Crossref]

Farhat, M.

M. Amin, M. Farhat, and H. Bagci, “A dynamically reconfigurable Fano metamaterial through graphene tuning for switching and sensing applications,” Sci. Rep. 3(1), 2105 (2013).
[Crossref]

Fenniche, I.

K. Edee, I. Fenniche, G. Granet, and B. Guizal, “Modal method based on subsectional Gegenbauer polynomial expansion for lamellar gratings: weighting function, convergence and stability,” Prog. Electromagn. Res. 133, 17–35 (2013).
[Crossref]

Freitag, M.

H. Yan, X. Li, B. Chandra, G. Tulevski, Y. Wu, M. Freitag, W. Zhu, P. Avouris, and F. Xia, “Tunable Infrared Plasmonic Devices Using Graphene/Insulator Stacks,” Nat. Nanotechnol. 7(5), 330–334 (2012).
[Crossref]

Freitag et al., M.

H. G. Yan, T. Low, W. J. Zhu, Y. Q. Wu, and M. Freitag et al., “Damping pathways of mid-infrared plasmons in graphene nanostructures,” Nat. Photonics 7(5), 394–399 (2013).
[Crossref]

Gan, Q.

Gangaraj, S. A. H.

P. Doyeux, S. A. H. Gangaraj, G. W. Hanson, and M. Antezza, “Giant interatomic energy-transport amplification with nonreciprocal photonic topological insulators,” Phys. Rev. Lett. 119(17), 173901 (2017).
[Crossref]

Gao, F.

L. Tang, H. Shi, J. Yang, C. Du, F. Gao, J. Zhu, and J. Dua, “Complete optical absorption in graphene by using nano-gratings to excite graphene surface plasmons,” Microelectron. Eng. 145, 58–61 (2015).
[Crossref]

Gao, W.

W. Gao, J. Shu, C. Qiu, and Q. Xu, “Excitation of Plasmonic Waves in Graphene by Guided-Mode Resonances,” ACS Nano 6(9), 7806–7813 (2012).
[Crossref]

García de Abajo, F. J.

D. Rodrigo, A. Tittl, O. Limaj, F. J. García de Abajo, V. Pruneri, and H. Altug, “Double-layer graphene for enhanced tunable infrared plasmonics,” Light: Sci. Appl. 6(6), e16277 (2017).
[Crossref]

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W. Gao, J. Shu, C. Qiu, and Q. Xu, “Excitation of Plasmonic Waves in Graphene by Guided-Mode Resonances,” ACS Nano 6(9), 7806–7813 (2012).
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Rana, F.

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D. Rodrigo, A. Tittl, O. Limaj, F. J. García de Abajo, V. Pruneri, and H. Altug, “Double-layer graphene for enhanced tunable infrared plasmonics,” Light: Sci. Appl. 6(6), e16277 (2017).
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A. Y. Nikitin, S. G. Rodrigo, F. J. Garcia-Vidal, and L. Martin-Moreno, “In the diffraction shadow: Norton waves versus surface plasmon polaritons in the optical region,” New J. Phys. 11(12), 123020 (2009).
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Z. Fang, Y. Wang, A. E. Schlather, Z. Liu, P. M. Ajayan, F. J. García de Abajo, P. Nordlander, X. Zhu, and N. J. Halas, “Active Tunable Absorption Enhancement with Graphene Nanodisk Arrays,” Nano Lett. 14(1), 299–304 (2014).
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V. W. Brar, M. S. Jang, M. Sherrott, J. J. Lopez, and H. A. Atwater, “Highly Confined Tunable Mid-Infrared Plasmonics in Graphene Nanoresonators,” Nano Lett. 13(6), 2541–2547 (2013).
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W. Gao, J. Shu, C. Qiu, and Q. Xu, “Excitation of Plasmonic Waves in Graphene by Guided-Mode Resonances,” ACS Nano 6(9), 7806–7813 (2012).
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M. G. Silveirinha, S. Ali Hassani Gangaraj, George W. Hanson, and Mauro Antezza, “Fluctuation-induced forces on an atom near a photonic topological material,” Phys. Rev. A 97(2), 022509 (2018).
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J. H. Strait, P. Nene, W.-M. Chan, C. Manolatou, S. Tiwari, F. Rana, J. W. Kevek, and P. L. McEuen, “Confined Plasmons in Graphene Microstructures: Experiments and Theory,” Phys. Rev. B: Condens. Matter Mater. Phys. 87(24), 241410 (2013).
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D. Rodrigo, A. Tittl, O. Limaj, F. J. García de Abajo, V. Pruneri, and H. Altug, “Double-layer graphene for enhanced tunable infrared plasmonics,” Light: Sci. Appl. 6(6), e16277 (2017).
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J. H. Strait, P. Nene, W.-M. Chan, C. Manolatou, S. Tiwari, F. Rana, J. W. Kevek, and P. L. McEuen, “Confined Plasmons in Graphene Microstructures: Experiments and Theory,” Phys. Rev. B: Condens. Matter Mater. Phys. 87(24), 241410 (2013).
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H. Yan, X. Li, B. Chandra, G. Tulevski, Y. Wu, M. Freitag, W. Zhu, P. Avouris, and F. Xia, “Tunable Infrared Plasmonic Devices Using Graphene/Insulator Stacks,” Nat. Nanotechnol. 7(5), 330–334 (2012).
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Z. Fang, Y. Wang, A. E. Schlather, Z. Liu, P. M. Ajayan, F. J. García de Abajo, P. Nordlander, X. Zhu, and N. J. Halas, “Active Tunable Absorption Enhancement with Graphene Nanodisk Arrays,” Nano Lett. 14(1), 299–304 (2014).
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T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelenght hole arrays,” Nature 391(6668), 667–669 (1998).
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H. Yan, X. Li, B. Chandra, G. Tulevski, Y. Wu, M. Freitag, W. Zhu, P. Avouris, and F. Xia, “Tunable Infrared Plasmonic Devices Using Graphene/Insulator Stacks,” Nat. Nanotechnol. 7(5), 330–334 (2012).
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H. G. Yan, T. Low, W. J. Zhu, Y. Q. Wu, and M. Freitag et al., “Damping pathways of mid-infrared plasmons in graphene nanostructures,” Nat. Photonics 7(5), 394–399 (2013).
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Xu, Q.

W. Gao, J. Shu, C. Qiu, and Q. Xu, “Excitation of Plasmonic Waves in Graphene by Guided-Mode Resonances,” ACS Nano 6(9), 7806–7813 (2012).
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H. Yan, X. Li, B. Chandra, G. Tulevski, Y. Wu, M. Freitag, W. Zhu, P. Avouris, and F. Xia, “Tunable Infrared Plasmonic Devices Using Graphene/Insulator Stacks,” Nat. Nanotechnol. 7(5), 330–334 (2012).
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H. G. Yan, T. Low, W. J. Zhu, Y. Q. Wu, and M. Freitag et al., “Damping pathways of mid-infrared plasmons in graphene nanostructures,” Nat. Photonics 7(5), 394–399 (2013).
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L. Tang, H. Shi, J. Yang, C. Du, F. Gao, J. Zhu, and J. Dua, “Complete optical absorption in graphene by using nano-gratings to excite graphene surface plasmons,” Microelectron. Eng. 145, 58–61 (2015).
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T. Zhang, L. Chen, B. Wang, and X. Li, “Tunable broadband plasmonic field enhancement on a graphene surface using a normal-incidence plane wave at mid-infrared frequencies,” Sci. Rep. 5(1), 11195 (2015).
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B. Zhao, B. Guizal, Z. M. Zhang, S. Fan, and M. Antezza, “Near-field heat transfer between graphene/hBN multilayers,” Phys. Rev. B 95(24), 245437 (2017).
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B. Zhao and Z. M. Zhang, “Strong Plasmonic Coupling between Graphene Ribbon Array and Metal Gratings,” ACS Photonics 2(11), 1611–1618 (2015).
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B. Zhao, B. Guizal, Z. M. Zhang, S. Fan, and M. Antezza, “Near-field heat transfer between graphene/hBN multilayers,” Phys. Rev. B 95(24), 245437 (2017).
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Zhu, W.

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ACS Nano (1)

W. Gao, J. Shu, C. Qiu, and Q. Xu, “Excitation of Plasmonic Waves in Graphene by Guided-Mode Resonances,” ACS Nano 6(9), 7806–7813 (2012).
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ACS Photonics (1)

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Appl. Opt. (1)

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J. Opt. Soc. Am. A (3)

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Microelectron. Eng. (1)

L. Tang, H. Shi, J. Yang, C. Du, F. Gao, J. Zhu, and J. Dua, “Complete optical absorption in graphene by using nano-gratings to excite graphene surface plasmons,” Microelectron. Eng. 145, 58–61 (2015).
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F. H. L. Koppens, D. E. Chang, and F. J. García de Agajo, “Graphene plasmonics: a platform for strong light-matter interactions,” Nano Lett. 11(8), 3370–3377 (2011).
[Crossref]

Z. Fang, Y. Wang, A. E. Schlather, Z. Liu, P. M. Ajayan, F. J. García de Abajo, P. Nordlander, X. Zhu, and N. J. Halas, “Active Tunable Absorption Enhancement with Graphene Nanodisk Arrays,” Nano Lett. 14(1), 299–304 (2014).
[Crossref]

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H. G. Yan, T. Low, W. J. Zhu, Y. Q. Wu, and M. Freitag et al., “Damping pathways of mid-infrared plasmons in graphene nanostructures,” Nat. Photonics 7(5), 394–399 (2013).
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Figures (12)

Fig. 1.
Fig. 1. Sketch of hybrid structure made of a dispersive metal film perforated with a subwavelength periodic array of 1D nano-slits deposited on a dielectric spacer ended by a continuous graphene sheet.
Fig. 2.
Fig. 2. Reflection, transmission and absorption spectra of the hybrid system for $\mu _c=1eV$ (Fig. 2(a)) and $\mu _c=1.5eV$ (Fig. 2(b)). The hybrid structure exhibits both broadband and tunable narrow band resonances with respect to the chemical potential. Parameters: $\varepsilon ^{1}=\varepsilon ^{3}=\varepsilon ^{slit}=1$, incidence angle= $0^{o}$, $h=800nm$, $d=165nm$, $a=15nm$.
Fig. 3.
Fig. 3. Real part of the magnetic field $H_x(x,z)$ at $\lambda =4.17 \mu m$ (Fig. 3(a)) and at $\lambda =7.30 \mu m$ (Fig. 3(b)). Parameters: $\varepsilon ^{1}=\varepsilon ^{3}=\varepsilon ^{slit}=1$, incidence angle= $0^{o}$, $h=800nm$, $d=165nm$, $a=15nm$.
Fig. 4.
Fig. 4. Sketch of the mechanism of the coupling between cavity lattice modes of the periodic array of nano-slits and the metal/insulator/graphene gap plasmon modes. Strong and weak couplings between three modes are responsible of the resonance phenomena of the hybrid structure.
Fig. 5.
Fig. 5. Configurations used for the computation of the required effective indices (eigenvalues of Eq. (5). $config.1$ is used for the computation of the modes of periodic arrays of nano-slits in general and in particular for the computation of the cavity lattice mode effective index $\gamma _0^{(1)}$. $config.2$ is used for the computation of the effective index $\alpha _0^{(2)}$ of the plasmon mode. The gap plasmon mode effective index $\alpha _0^{(3)}$ is computed thanks to $config.3$.
Fig. 6.
Fig. 6. Sketch of the weak coupling sub-system consisting of a periodic array of nano-slits encapsulated between $\varepsilon ^{(0)}$ and $\varepsilon ^{(3)}$ media. The lattice mode $\gamma _0^{(1)}$ is assumed to live in an $\sqrt {\varepsilon ^{(1)}}$ effective homogeneous medium. Two plasmon modes $\alpha _{sp}^{(0)}$ and $\alpha _0^{(2)}$ ensure the phase matching with the plane waves in media $\varepsilon ^{(0)}$ and $\varepsilon ^{(3)}$.
Fig. 7.
Fig. 7. The sketch of $\alpha _0^{(2)}$ plasmon mode computation.
Fig. 8.
Fig. 8. Comparison between the reflection spectrum of the hybrid structure and the responses of the weakly coupled sub-system (a) and the strongly coupled sub-system (b). As expected, the weakly coupled sub-system reflection spectrum $|R_{12}(\lambda )|^2$ perfectly matches the broadband resonances of the hybrid structure. On the other hand, the strongly coupled sub-system spectrum characteristic function $|S_{11}(\lambda )+S_{12}(\lambda )|^2$ perfectly matches the narrow band resonances of the hybrid structure. Parameters: $\lambda \in [2,10]\mu m$, $\varepsilon ^{(1)}=\varepsilon ^{(3)}=\varepsilon ^{(s)}=1$, $\varepsilon ^{(2)}=1.54^2$, incidence angle= $0^{o}$, $\mu _c=1eV$.
Fig. 9.
Fig. 9. Sketch showing the strong coupling between the gap plasmon mode $\alpha _0^{(3)}$ living in an $\sqrt {\varepsilon ^{(2)}}$ homogeneous medium and $\alpha _0^{(1)}$ lattice mode in an $\sqrt {\varepsilon ^{(1)}}$ effective homogeneous medium.
Fig. 10.
Fig. 10. The sketch of $\alpha _0^{(3)}$ plasmon mode computation.
Fig. 11.
Fig. 11. Comparison between the spectra of the hybrid-structure with the reflection and transmission curves obtained from the PMM for two values of the chemical potential $\mu _c$. The chemical potential is set to $\mu _c=1eV$, in Figs. 11(a) and (b), while $\mu _c=1.5eV$, in Figs. 11(c) and (d). All these results fit very well with the rigorous numerical simulations obtained with the PMM. Our model captures very well all resonances occurring in the hybrid system namely Lorentz and Fano ones. Parameters: $\lambda \in [2,10]\mu m$, $\varepsilon ^{(1)}=\varepsilon ^{(3)}=\varepsilon ^{(s)}=1$, $\varepsilon ^{(2)}=1.54^2$, incidence angle= $0^{o}$.
Fig. 12.
Fig. 12. Dispersion curves of the effective index $\alpha _0^{(3)}$ for different values of $h_2$, ($\mu _c=1eV$) (Fig. 12(a)) and for different values of $\mu _c$ ($h_2=10 nm$) (Fig. 12(b)). Increasing the chemical potential $\mu _c$ or the spacer width $h_2$, the real part of $\alpha _0^{(3)}$ decreases. Parameters: $\varepsilon ^{2}=1.54^2$.

Equations (25)

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ε ( ω ) = 1 + i σ ( ω ) ε 0 ω Δ
σ ( ω ) = σ i n t e r + σ i n t r a
{ σ i n t e r = σ 0 [ 1 + 1 π a t a n ( ω 2 μ c Γ ) 1 π a t a n ( ω + 2 μ c Γ ) ] σ i n t e r = σ 0 2 π l n [ 2 μ c ω 2 μ c + ω ] .
σ i n t r a = σ 0 4 μ c π 1 Γ i ω .
L ( k ) ( ω ) | H q ( k ) ( ω ) = ( γ q ( k ) ( ω ) ) 2 | H q ( k ) ( ω )
L ( k ) ( x , ω ) = ( c ω ) 2 ε ( k ) ( x , ω ) x 1 ε ( k ) ( x , ω ) x + ε ( k ) ( x , ω ) .
R 12 = r 1 + ϕ 1 r 2 ϕ 2 1 + r 1 ϕ 1 r 2 ϕ 2
T 12 = t 1 t 2 ϕ 2 1 + r 1 ϕ 1 r 2 ϕ 2
r 1 = 1 n 01 ( ω ) 1 + n 01 ( ω ) , r 2 = 1 n 13 ( ω ) 1 + n 13 ( ω ) ,
t 1 = 2 1 + n 01 ( ω ) , t 2 = 2 1 + n 13 ( ω ) .
n 01 ( ω ) = γ 0 ( 1 ) ( ω ) / ε ( 1 ) ( ω ) γ 0 ( 0 ) ( ω ) / ε ( 0 ) ( ω ) ,  and  n 13 ( ω ) = γ 0 ( 3 ) ( ω ) / ε ( 3 ) ( ω ) γ 0 ( 1 ) ( ω ) / ε ( 1 ) ( ω ) ,
ϕ 1 = e i k 0 γ 0 ( 1 ) h 1 ϕ c ( 0 ) , ϕ 2 = e i k 0 γ 0 ( 1 ) h 1 ϕ c ( 2 )
ϕ c ( 0 ) = e i k 0 α s p ( 0 ) a ( 0 ) , ϕ c ( 2 ) = e i k 0 α 0 ( 2 ) a ( 2 ) .
α 0 ( 1 ) = ε ( 1 ) γ 0 ( 1 ) 2 ,
[ S 11 S 12 S 21 S 22 ] [ a 1 a 2 ] = [ b 1 b 2 ]
{ S 11 ( ω ) = S 22 ( ω ) = [ 1 n 2 ( ω ) ] [ 1 ϕ 2 ( ω ) ] [ 1 + n ( ω ) ] 2 [ 1 n ( ω ) ] 2 ϕ 2 ( ω ) S 12 ( ω ) = S 21 ( ω ) = 4 n ( ω ) ϕ ( ω ) [ 1 + n ( ω ) ] 2 [ 1 n ( ω ) ] 2 ϕ 2 ( ω ) ,
{ n ( ω ) = α 0 ( 3 ) ( ω ) / ε ( 3 ) ( ω ) α 0 ( 1 ) ( ω ) / ε ( 1 ) ( ω ) ϕ = e i k 0 α 0 ( 3 ) d . ,
Δ ( ω ) = S 11 ( ω ) S 22 ( ω ) S 12 ( ω ) S 21 ( ω ) = [ S 11 ( ω ) S 12 ( ω ) ] [ S 11 ( ω ) + S 12 ( ω ) ] = 0.
{ S 11 ( ω ) S 12 ( ω ) = 0 or S 11 ( ω ) + S 12 ( ω ) = 0 .
r 13 ( ω ) = ( S 11 ( ω ) + S 12 ( ω ) ) .
R = r 1 + ϕ 1 r 13 r 2 ϕ 2 1 + r 1 ϕ 1 r 13 r 2 ϕ 2
T = t 1 r 13 t 2 ϕ 2 1 + r 1 ϕ 1 r 13 r 2 ϕ 2 .
ϕ 1 r 2 ϕ 3 r 1  and  1 + r 1 ϕ 1 r 2 ϕ 2 0 ,
r 13 ( ω ) 0 ,
r 13 ϕ 1 r 2 ϕ 3 0

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