Abstract

Graphene-based and metal-based surface plasmon polariton (SPP) waveguides have attracted intense research interest because they can be used as basic components to propagate electromagnetic (EM) waves in future optical integrated systems. We propose a directional coupler, which can couple EM energy from a multilayer-graphene-based cylindrical long-range SPP waveguide to a metal-based cylindrical hybrid SPP waveguide in the mid-infrared range. This coupler exhibits relatively low coupling length, high coupling efficiency, low insertion loss, and high extinction ratio after adjustment of the wave vector mismatch of the two waveguides. Moreover, this coupler is tolerant to practical fabrication errors like misalignment of graphene layres, and can effectively work in the range of Fermi energy Ef > 0.6 eV when the mobility of graphene varies from 10000 to 800 cm2/Vs. Hence, the coupler offers potential applications in signal routing and information exchange between graphene-based and metal-based SPP waveguides in photonic integrated circuits.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2018 (1)

2017 (1)

2016 (4)

2015 (2)

M. Kim, C. Y. Jeong, H. Heo, and S. Kim, “Optical reflection modulation using surface plasmon resonance in a graphene-embedded hybrid plasmonic waveguide at an optical communication wavelength,” Opt. Lett. 40(6), 871–874 (2015).
[Crossref] [PubMed]

B. F. Zhu, G. B. Ren, Y. Yang, Y. X. Gao, B. L. Wu, Y. D. Lian, J. Wang, and S. S. Jian, “Field enhancement and gradient force in the graphene-coated nanowire pairs,” Plasmonics 10(4), 839–848 (2015).
[Crossref]

2014 (6)

W. Li, B. Chen, C. Meng, W. Fang, Y. Xiao, X. Li, Z. Hu, Y. Xu, L. Tong, H. Wang, W. Liu, J. Bao, and Y. R. Shen, “Ultrafast all-optical graphene modulator,” Nano Lett. 14(2), 955–959 (2014).
[Crossref] [PubMed]

Y. Wu, B. Yao, A. Zhang, Y. Rao, Z. Wang, Y. Cheng, Y. Gong, W. Zhang, Y. Chen, and K. S. Chiang, “Graphene-coated microfiber Bragg grating for high-sensitivity gas sensing,” Opt. Lett. 39(5), 1235–1237 (2014).
[Crossref] [PubMed]

Z. Fang, Y. Wang, A. E. Schlather, Z. Liu, P. M. Ajayan, F. J. 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] [PubMed]

Y. Gao, G. Ren, B. Zhu, H. Liu, Y. Lian, and S. Jian, “Analytical model for plasmon modes in graphene-coated nanowire,” Opt. Express 22(20), 24322–24331 (2014).
[Crossref] [PubMed]

D. A. Smirnova, I. V. Iorsh, I. V. Shadrivov, and Y. S. Kivshai, “Multilayer graphene waveguides,” JETP Lett. 99(8), 456–460 (2014).
[Crossref]

W. Li, B. Chen, C. Meng, W. Fang, Y. Xiao, X. Li, Z. Hu, Y. Xu, L. Tong, H. Wang, W. Liu, J. Bao, and Y. R. Shen, “Ultrafast all-optical graphene modulator,” Nano Lett. 14(2), 955–959 (2014).
[Crossref] [PubMed]

2013 (6)

L. Chen, X. Li, and D. S. Gao, “An efficient directional coupling from dielectric waveguide to hybrid long-range plasmonic waveguide on a silicon platform,” Appl. Phys. B 111(1), 11–19 (2013).
[Crossref]

P. Liu, X. Zhang, Z. Ma, W. Cai, L. Wang, and J. Xu, “Surface plasmon modes in graphene wedge and groove waveguides,” Opt. Express 21(26), 32432–32440 (2013).
[Crossref] [PubMed]

H. J. Li, L. L. Wang, J. Q. Liu, Z. R. Huang, B. Sun, and X. Zhai, “Investigation of the graphene based planar plasmonic filters,” Appl. Phys. Lett. 103(21), 211104 (2013).
[Crossref]

X. Zhu, W. Yan, N. A. Mortensen, and S. Xiao, “Bends and splitters in graphene nanoribbon waveguides,” Opt. Express 21(3), 3486–3491 (2013).
[Crossref] [PubMed]

B. Zhu, G. Ren, S. Zheng, Z. Lin, and S. Jian, “Nanoscale dielectric-graphene-dielectric tunable infrared waveguide with ultrahigh refractive indices,” Opt. Express 21(14), 17089–17096 (2013).
[Crossref] [PubMed]

X. He, Z. Liu, D. N. Wang, M. Yang, T. Y. Hu, and J. G. Tian, “Saturable absorber based on graphene-covered microfiber,” IEEE Photonics Technol. Lett. 25(14), 1392–1394 (2013).
[Crossref]

2012 (5)

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] [PubMed]

Z. Fei, A. S. Rodin, G. O. Andreev, W. Bao, A. S. McLeod, M. Wagner, L. M. Zhang, Z. Zhao, M. Thiemens, G. Dominguez, M. M. Fogler, A. H. Castro Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature 487(7405), 82–85 (2012).
[Crossref] [PubMed]

L. Chen, X. Li, L. Chen, X. Li, G. P. Wang, W. Li, S. H. Chen, L. Xiao, and D. S. Gao, “A silicon-based 3-D hybrid long-range plasmonic waveguide for nanophotonic integration,” J. Lightwave Technol. 30(1), 163–167 (2012).
[Crossref]

J. Christensen, A. Manjavacas, S. Thongrattanasiri, F. H. L. Koppens, and F. J. de Abajo, “Graphene plasmon waveguiding and hybridization in individual and paired nanoribbons,” ACS Nano 6(1), 431–440 (2012).
[Crossref] [PubMed]

B. Wang, X. Zhang, F. J. García-Vidal, X. Yuan, and J. Teng, “Strong Coupling of Surface Plasmon Polaritons in Monolayer Graphene Sheet Arrays,” Phys. Rev. Lett. 109(7), 073901 (2012).
[Crossref] [PubMed]

2011 (5)

F. Liu, Y. X. Li, R. Y. Wan, Y. D. Huang, X. Feng, and W. Zhang, “Hybrid coupling between long-range surface plasmon polariton mode and dielectric waveguide mode,” J. Lightwave Technol. 29(9), 1265–1273 (2011).
[Crossref]

A. Y. Nikitin, F. Guinea, F. J. García-Vidal, and L. Martía-Moren, “Edge and waveguide terahertz surface plasmon modes in graphene microribbons,” Phys. Rev. B Condens. Matter Mater. Phys. 84(16), 161407 (2011).
[Crossref]

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

D. Dai, Y. Shi, S. He, L. Wosinski, and L. Thylen, “Gain enhancement in a hybrid plasmonic nano-waveguide with a low-index or high-index gain medium,” Opt. Express 19(14), 12925–12936 (2011).
[Crossref] [PubMed]

C. F. Chen, C. H. Park, B. W. Boudouris, J. Horng, B. Geng, C. Girit, A. Zettl, M. F. Crommie, R. A. Segalman, S. G. Louie, and F. Wang, “Controlling inelastic light scattering quantum pathways in graphene,” Nature 471(7340), 617–620 (2011).
[Crossref] [PubMed]

2010 (5)

C. R. Dean, A. F. Young, I. Meric, C. Lee, L. Wang, S. Sorgenfrei, K. Watanabe, T. Taniguchi, P. Kim, K. L. Shepard, and J. Hone, “Boron nitride substrates for high-quality graphene electronics,” Nat. Nanotechnol. 5(10), 722–726 (2010).
[Crossref] [PubMed]

D. Chen, “Cylindrical hybrid plasmonic waveguide for subwavelength confinement of light,” Appl. Opt. 49(36), 6868–6871 (2010).
[Crossref] [PubMed]

P. R. West, S. Ishii, G. V. Naik, N. K. Emani, V. M. Shalaev, and A. Boltasseva, “Searching for better plasmonic materials,” Laser Photonics Rev. 4(6), 795–808 (2010).
[Crossref]

A. V. Krasavin and A. V. Zayats, “Numerical analysis of long-range surface plasmon polariton modes in nanoscale plasmonic waveguides,” Opt. Lett. 35(13), 2118–2120 (2010).
[Crossref] [PubMed]

Q. Li, Y. Song, G. Zhou, Y. Su, and M. Qiu, “Asymmetric plasmonic-dielectric coupler with short coupling length, high extinction ratio, and low insertion loss,” Opt. Lett. 35(19), 3153–3155 (2010).
[Crossref] [PubMed]

2009 (3)

X. Guo, M. Qiu, J. Bao, B. J. Wiley, Q. Yang, X. Zhang, Y. Ma, H. Yu, and L. Tong, “Direct coupling of plasmonic and photonic nanowires for hybrid nanophotonic components and circuits,” Nano Lett. 9(12), 4515–4519 (2009).
[Crossref] [PubMed]

S. E. Kocabaş, G. Veronis, D. A. B. Miller, and S. H. Fan, “Modal analysis and coupling in metal-insulator-metal waveguides,” Phys. Rev. B Condens. Matter Mater. Phys. 79(3), 035120 (2009).
[Crossref]

D. Dai and S. He, “A silicon-based hybrid plasmonic waveguide with a metal cap for a nano-scale light confinement,” Opt. Express 17(19), 16646–16653 (2009).
[Crossref] [PubMed]

2008 (2)

G. W. Hanson, “Quasi-transverse electromagnetic modes supported by a graphene parallel- plate waveguide,” J. Appl. Phys. 104(8), 084314 (2008).
[Crossref]

R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, “A hybrid plasmonic waveguide for subwavelength confinement and long-range propagation,” Nat. Photonics 2(8), 496–500 (2008).
[Crossref]

2007 (3)

S. Lal, S. Link, and N. J. Halas, “Nano-optics from sensing to waveguiding,” Nat. Photonics 1(11), 641–648 (2007).
[Crossref]

G. Veronis and S. Fan, “Theoretical investigation of compact couplers between dielectric slab waveguides and two-dimensional metal-dielectric-metal plasmonic waveguides,” Opt. Express 15(3), 1211–1221 (2007).
[Crossref] [PubMed]

L. A. Falkovsky and A. A. Varlamov, “Space-time dispersion of graphene conductivity,” Eur. Phys. J. B 56(4), 281–284 (2007).
[Crossref]

2005 (2)

A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep. 408(3), 131–314 (2005).
[Crossref]

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, and T. W. Ebbesen, “Channel plasmon-polariton guiding by subwavelength metal grooves,” Phys. Rev. Lett. 95(4), 046802 (2005).
[Crossref] [PubMed]

2004 (2)

V. R. Almeida, Q. Xu, C. A. Barrios, M. Lipson, and M. Lipson, “Guiding and confining light in void nanostructure,” Opt. Lett. 29(11), 1209–1211 (2004).
[Crossref] [PubMed]

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science 306(5696), 666–669 (2004).
[Crossref] [PubMed]

2003 (2)

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003).
[Crossref] [PubMed]

L. Tong, R. R. Gattass, J. B. Ashcom, S. He, J. Lou, M. Shen, I. Maxwell, and E. Mazur, “Subwavelength-diameter silica wires for low-loss optical wave guiding,” Nature 426(6968), 816–819 (2003).
[Crossref] [PubMed]

1998 (1)

Ajayan, P. M.

Z. Fang, Y. Wang, A. E. Schlather, Z. Liu, P. M. Ajayan, F. J. 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] [PubMed]

Almeida, V. R.

Andreev, G. O.

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F. H. L. Koppens, D. E. Chang, and F. J. García de Abajo, “Graphene plasmonics: A platform for strong light-matter interactions,” Nano Lett. 11(8), 3370–3377 (2011).
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W. Li, B. Chen, C. Meng, W. Fang, Y. Xiao, X. Li, Z. Hu, Y. Xu, L. Tong, H. Wang, W. Liu, J. Bao, and Y. R. Shen, “Ultrafast all-optical graphene modulator,” Nano Lett. 14(2), 955–959 (2014).
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C. R. Dean, A. F. Young, I. Meric, C. Lee, L. Wang, S. Sorgenfrei, K. Watanabe, T. Taniguchi, P. Kim, K. L. Shepard, and J. Hone, “Boron nitride substrates for high-quality graphene electronics,” Nat. Nanotechnol. 5(10), 722–726 (2010).
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W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003).
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S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, and T. W. Ebbesen, “Channel plasmon-polariton guiding by subwavelength metal grooves,” Phys. Rev. Lett. 95(4), 046802 (2005).
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Dominguez, G.

Z. Fei, A. S. Rodin, G. O. Andreev, W. Bao, A. S. McLeod, M. Wagner, L. M. Zhang, Z. Zhao, M. Thiemens, G. Dominguez, M. M. Fogler, A. H. Castro Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature 487(7405), 82–85 (2012).
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K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science 306(5696), 666–669 (2004).
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S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, and T. W. Ebbesen, “Channel plasmon-polariton guiding by subwavelength metal grooves,” Phys. Rev. Lett. 95(4), 046802 (2005).
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Emani, N. K.

P. R. West, S. Ishii, G. V. Naik, N. K. Emani, V. M. Shalaev, and A. Boltasseva, “Searching for better plasmonic materials,” Laser Photonics Rev. 4(6), 795–808 (2010).
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W. Li, B. Chen, C. Meng, W. Fang, Y. Xiao, X. Li, Z. Hu, Y. Xu, L. Tong, H. Wang, W. Liu, J. Bao, and Y. R. Shen, “Ultrafast all-optical graphene modulator,” Nano Lett. 14(2), 955–959 (2014).
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Z. Fang, Y. Wang, A. E. Schlather, Z. Liu, P. M. Ajayan, F. J. 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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Z. Fei, A. S. Rodin, G. O. Andreev, W. Bao, A. S. McLeod, M. Wagner, L. M. Zhang, Z. Zhao, M. Thiemens, G. Dominguez, M. M. Fogler, A. H. Castro Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature 487(7405), 82–85 (2012).
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Firsov, A. A.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science 306(5696), 666–669 (2004).
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Z. Fei, A. S. Rodin, G. O. Andreev, W. Bao, A. S. McLeod, M. Wagner, L. M. Zhang, Z. Zhao, M. Thiemens, G. Dominguez, M. M. Fogler, A. H. Castro Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature 487(7405), 82–85 (2012).
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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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Gao, Y. X.

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F. H. L. Koppens, D. E. Chang, and F. J. García de Abajo, “Graphene plasmonics: A platform for strong light-matter interactions,” Nano Lett. 11(8), 3370–3377 (2011).
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L. Tong, R. R. Gattass, J. B. Ashcom, S. He, J. Lou, M. Shen, I. Maxwell, and E. Mazur, “Subwavelength-diameter silica wires for low-loss optical wave guiding,” Nature 426(6968), 816–819 (2003).
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K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science 306(5696), 666–669 (2004).
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C. F. Chen, C. H. Park, B. W. Boudouris, J. Horng, B. Geng, C. Girit, A. Zettl, M. F. Crommie, R. A. Segalman, S. G. Louie, and F. Wang, “Controlling inelastic light scattering quantum pathways in graphene,” Nature 471(7340), 617–620 (2011).
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R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, “A hybrid plasmonic waveguide for subwavelength confinement and long-range propagation,” Nat. Photonics 2(8), 496–500 (2008).
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C. F. Chen, C. H. Park, B. W. Boudouris, J. Horng, B. Geng, C. Girit, A. Zettl, M. F. Crommie, R. A. Segalman, S. G. Louie, and F. Wang, “Controlling inelastic light scattering quantum pathways in graphene,” Nature 471(7340), 617–620 (2011).
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Grigorieva, I. V.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science 306(5696), 666–669 (2004).
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A. Y. Nikitin, F. Guinea, F. J. García-Vidal, and L. Martía-Moren, “Edge and waveguide terahertz surface plasmon modes in graphene microribbons,” Phys. Rev. B Condens. Matter Mater. Phys. 84(16), 161407 (2011).
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X. Guo, M. Qiu, J. Bao, B. J. Wiley, Q. Yang, X. Zhang, Y. Ma, H. Yu, and L. Tong, “Direct coupling of plasmonic and photonic nanowires for hybrid nanophotonic components and circuits,” Nano Lett. 9(12), 4515–4519 (2009).
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Z. Fang, Y. Wang, A. E. Schlather, Z. Liu, P. M. Ajayan, F. J. 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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C. F. Chen, C. H. Park, B. W. Boudouris, J. Horng, B. Geng, C. Girit, A. Zettl, M. F. Crommie, R. A. Segalman, S. G. Louie, and F. Wang, “Controlling inelastic light scattering quantum pathways in graphene,” Nature 471(7340), 617–620 (2011).
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X. He, Z. Liu, D. N. Wang, M. Yang, T. Y. Hu, and J. G. Tian, “Saturable absorber based on graphene-covered microfiber,” IEEE Photonics Technol. Lett. 25(14), 1392–1394 (2013).
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W. Li, B. Chen, C. Meng, W. Fang, Y. Xiao, X. Li, Z. Hu, Y. Xu, L. Tong, H. Wang, W. Liu, J. Bao, and Y. R. Shen, “Ultrafast all-optical graphene modulator,” Nano Lett. 14(2), 955–959 (2014).
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W. Li, B. Chen, C. Meng, W. Fang, Y. Xiao, X. Li, Z. Hu, Y. Xu, L. Tong, H. Wang, W. Liu, J. Bao, and Y. R. Shen, “Ultrafast all-optical graphene modulator,” Nano Lett. 14(2), 955–959 (2014).
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Huang, Z. R.

H. J. Li, L. L. Wang, J. Q. Liu, Z. R. Huang, B. Sun, and X. Zhai, “Investigation of the graphene based planar plasmonic filters,” Appl. Phys. Lett. 103(21), 211104 (2013).
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P. R. West, S. Ishii, G. V. Naik, N. K. Emani, V. M. Shalaev, and A. Boltasseva, “Searching for better plasmonic materials,” Laser Photonics Rev. 4(6), 795–808 (2010).
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B. F. Zhu, G. B. Ren, Y. Yang, Y. X. Gao, B. L. Wu, Y. D. Lian, J. Wang, and S. S. Jian, “Field enhancement and gradient force in the graphene-coated nanowire pairs,” Plasmonics 10(4), 839–848 (2015).
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K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science 306(5696), 666–669 (2004).
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Z. Fei, A. S. Rodin, G. O. Andreev, W. Bao, A. S. McLeod, M. Wagner, L. M. Zhang, Z. Zhao, M. Thiemens, G. Dominguez, M. M. Fogler, A. H. Castro Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature 487(7405), 82–85 (2012).
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Kim, P.

C. R. Dean, A. F. Young, I. Meric, C. Lee, L. Wang, S. Sorgenfrei, K. Watanabe, T. Taniguchi, P. Kim, K. L. Shepard, and J. Hone, “Boron nitride substrates for high-quality graphene electronics,” Nat. Nanotechnol. 5(10), 722–726 (2010).
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F. H. L. Koppens, D. E. Chang, and F. J. García de Abajo, “Graphene plasmonics: A platform for strong light-matter interactions,” Nano Lett. 11(8), 3370–3377 (2011).
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S. Lal, S. Link, and N. J. Halas, “Nano-optics from sensing to waveguiding,” Nat. Photonics 1(11), 641–648 (2007).
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Z. Fei, A. S. Rodin, G. O. Andreev, W. Bao, A. S. McLeod, M. Wagner, L. M. Zhang, Z. Zhao, M. Thiemens, G. Dominguez, M. M. Fogler, A. H. Castro Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature 487(7405), 82–85 (2012).
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C. R. Dean, A. F. Young, I. Meric, C. Lee, L. Wang, S. Sorgenfrei, K. Watanabe, T. Taniguchi, P. Kim, K. L. Shepard, and J. Hone, “Boron nitride substrates for high-quality graphene electronics,” Nat. Nanotechnol. 5(10), 722–726 (2010).
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Li, Q.

Li, W.

W. Li, B. Chen, C. Meng, W. Fang, Y. Xiao, X. Li, Z. Hu, Y. Xu, L. Tong, H. Wang, W. Liu, J. Bao, and Y. R. Shen, “Ultrafast all-optical graphene modulator,” Nano Lett. 14(2), 955–959 (2014).
[Crossref] [PubMed]

W. Li, B. Chen, C. Meng, W. Fang, Y. Xiao, X. Li, Z. Hu, Y. Xu, L. Tong, H. Wang, W. Liu, J. Bao, and Y. R. Shen, “Ultrafast all-optical graphene modulator,” Nano Lett. 14(2), 955–959 (2014).
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L. Chen, X. Li, L. Chen, X. Li, G. P. Wang, W. Li, S. H. Chen, L. Xiao, and D. S. Gao, “A silicon-based 3-D hybrid long-range plasmonic waveguide for nanophotonic integration,” J. Lightwave Technol. 30(1), 163–167 (2012).
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Li, X.

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W. Li, B. Chen, C. Meng, W. Fang, Y. Xiao, X. Li, Z. Hu, Y. Xu, L. Tong, H. Wang, W. Liu, J. Bao, and Y. R. Shen, “Ultrafast all-optical graphene modulator,” Nano Lett. 14(2), 955–959 (2014).
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Yan, W.

Yang, M.

X. He, Z. Liu, D. N. Wang, M. Yang, T. Y. Hu, and J. G. Tian, “Saturable absorber based on graphene-covered microfiber,” IEEE Photonics Technol. Lett. 25(14), 1392–1394 (2013).
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X. Guo, M. Qiu, J. Bao, B. J. Wiley, Q. Yang, X. Zhang, Y. Ma, H. Yu, and L. Tong, “Direct coupling of plasmonic and photonic nanowires for hybrid nanophotonic components and circuits,” Nano Lett. 9(12), 4515–4519 (2009).
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B. F. Zhu, G. B. Ren, Y. Yang, Y. X. Gao, B. L. Wu, Y. D. Lian, J. Wang, and S. S. Jian, “Field enhancement and gradient force in the graphene-coated nanowire pairs,” Plasmonics 10(4), 839–848 (2015).
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X. Guo, M. Qiu, J. Bao, B. J. Wiley, Q. Yang, X. Zhang, Y. Ma, H. Yu, and L. Tong, “Direct coupling of plasmonic and photonic nanowires for hybrid nanophotonic components and circuits,” Nano Lett. 9(12), 4515–4519 (2009).
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B. Wang, X. Zhang, F. J. García-Vidal, X. Yuan, and J. Teng, “Strong Coupling of Surface Plasmon Polaritons in Monolayer Graphene Sheet Arrays,” Phys. Rev. Lett. 109(7), 073901 (2012).
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Zhang, W.

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P. Liu, X. Zhang, Z. Ma, W. Cai, L. Wang, and J. Xu, “Surface plasmon modes in graphene wedge and groove waveguides,” Opt. Express 21(26), 32432–32440 (2013).
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B. Wang, X. Zhang, F. J. García-Vidal, X. Yuan, and J. Teng, “Strong Coupling of Surface Plasmon Polaritons in Monolayer Graphene Sheet Arrays,” Phys. Rev. Lett. 109(7), 073901 (2012).
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X. Guo, M. Qiu, J. Bao, B. J. Wiley, Q. Yang, X. Zhang, Y. Ma, H. Yu, and L. Tong, “Direct coupling of plasmonic and photonic nanowires for hybrid nanophotonic components and circuits,” Nano Lett. 9(12), 4515–4519 (2009).
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K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science 306(5696), 666–669 (2004).
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Z. Fei, A. S. Rodin, G. O. Andreev, W. Bao, A. S. McLeod, M. Wagner, L. M. Zhang, Z. Zhao, M. Thiemens, G. Dominguez, M. M. Fogler, A. H. Castro Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature 487(7405), 82–85 (2012).
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B. F. Zhu, G. B. Ren, Y. Yang, Y. X. Gao, B. L. Wu, Y. D. Lian, J. Wang, and S. S. Jian, “Field enhancement and gradient force in the graphene-coated nanowire pairs,” Plasmonics 10(4), 839–848 (2015).
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ACS Nano (2)

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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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Appl. Opt. (2)

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H. J. Li, L. L. Wang, J. Q. Liu, Z. R. Huang, B. Sun, and X. Zhai, “Investigation of the graphene based planar plasmonic filters,” Appl. Phys. Lett. 103(21), 211104 (2013).
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X. He, Z. Liu, D. N. Wang, M. Yang, T. Y. Hu, and J. G. Tian, “Saturable absorber based on graphene-covered microfiber,” IEEE Photonics Technol. Lett. 25(14), 1392–1394 (2013).
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F. H. L. Koppens, D. E. Chang, and F. J. García de Abajo, “Graphene plasmonics: A platform for strong light-matter interactions,” Nano Lett. 11(8), 3370–3377 (2011).
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X. Guo, M. Qiu, J. Bao, B. J. Wiley, Q. Yang, X. Zhang, Y. Ma, H. Yu, and L. Tong, “Direct coupling of plasmonic and photonic nanowires for hybrid nanophotonic components and circuits,” Nano Lett. 9(12), 4515–4519 (2009).
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W. Li, B. Chen, C. Meng, W. Fang, Y. Xiao, X. Li, Z. Hu, Y. Xu, L. Tong, H. Wang, W. Liu, J. Bao, and Y. R. Shen, “Ultrafast all-optical graphene modulator,” Nano Lett. 14(2), 955–959 (2014).
[Crossref] [PubMed]

Z. Fang, Y. Wang, A. E. Schlather, Z. Liu, P. M. Ajayan, F. J. 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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Nat. Nanotechnol. (1)

C. R. Dean, A. F. Young, I. Meric, C. Lee, L. Wang, S. Sorgenfrei, K. Watanabe, T. Taniguchi, P. Kim, K. L. Shepard, and J. Hone, “Boron nitride substrates for high-quality graphene electronics,” Nat. Nanotechnol. 5(10), 722–726 (2010).
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C. F. Chen, C. H. Park, B. W. Boudouris, J. Horng, B. Geng, C. Girit, A. Zettl, M. F. Crommie, R. A. Segalman, S. G. Louie, and F. Wang, “Controlling inelastic light scattering quantum pathways in graphene,” Nature 471(7340), 617–620 (2011).
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Opt. Express (9)

D. Dai, Y. Shi, S. He, L. Wosinski, and L. Thylen, “Gain enhancement in a hybrid plasmonic nano-waveguide with a low-index or high-index gain medium,” Opt. Express 19(14), 12925–12936 (2011).
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S. X. Xia, X. Zhai, L. L. Wang, B. Sun, J. Q. Liu, and S. C. Wen, “Dynamically tunable plasmonically induced transparency in sinusoidally curved and planar graphene layers,” Opt. Express 24(16), 17886–17899 (2016).
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Opt. Lett. (5)

Photon. Res. (1)

Phys. Rep. (1)

A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep. 408(3), 131–314 (2005).
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B. Wang, X. Zhang, F. J. García-Vidal, X. Yuan, and J. Teng, “Strong Coupling of Surface Plasmon Polaritons in Monolayer Graphene Sheet Arrays,” Phys. Rev. Lett. 109(7), 073901 (2012).
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Plasmonics (2)

B. F. Zhu, G. B. Ren, Y. Yang, Y. X. Gao, B. L. Wu, Y. D. Lian, J. Wang, and S. S. Jian, “Field enhancement and gradient force in the graphene-coated nanowire pairs,” Plasmonics 10(4), 839–848 (2015).
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J. P. Liu, X. Zhai, L. L. Wang, H. J. Li, F. Xie, Q. Lin, and S. X. Xia, “Analysis of mid-infrared surface plasmon modes in a graphene-based cylindrical hybrid waveguide,” Plasmonics 11(3), 703–711 (2016).
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Science (1)

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science 306(5696), 666–669 (2004).
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Other (1)

S. S. Yu, Physical Basis of Guided Wave Optics (Northern Jiaotong University, 2002), Chap. 5.

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Figures (8)

Fig. 1
Fig. 1 Schematic diagram of the proposed multilayer-graphene-based LRSPP waveguide. (a) Cross-section of the waveguide. (b) 3D structure of the waveguide.
Fig. 2
Fig. 2 (a), (b), and (c) show the |E|, Ey, and Ey phase distribution of the S0 mode on the cross section of the waveguide, respectively. (d), (e), and (f) show the |E|, Ey, and the Ey phase distribution of the L0 mode on the cross section of the waveguide, respectively. (g) and (h) show the |E|, Ey and charge distribution of the S0 mode on the x-z plane of the waveguide, respectively. (i) and (j) show the |E|, Ey and charge distribution of the L0 mode on the x-z plane of the waveguide, respectively. The black curves are the electric field lines of the L0 mode in Fig. 2(j). The “+” and “-” symbols denote the positive and negative charges in Figs. 2(h) and 2(j), respectively.
Fig. 3
Fig. 3 (a), (b) nr and Lprop as functions of N for both L0 and S0 modes (Ef = 0.8 eV, R = 10 nm, and t = 5 nm). (c), (d) nr and Lprop as functions of Ef for L0 mode (N = 6, R = 10 nm, and t = 5 nm). (e) nr as a function of R for L0 mode (N = 4, t = 5 nm, and Ef = 0.8 eV). (f) nr as a function of t for L0 mode (N = 4, R = 10 nm, and Ef = 0.8 eV).
Fig. 4
Fig. 4 (a) Schematic diagram of the proposed metal-based hybrid waveguide. (b) |E| distribution of the H0 mode on the cross section of the waveguide. (c) nr and Lprop as functions of RAg for the H0 mode (tSiO2 = 6 nm, tSi = 25 nm). (d) nr as a function of tSi for the H0 mode (RAg = 10 nm, tSiO2 = 6 nm). (e) nr as a function of tSiO2 for the H0 mode (RAg = 10 nm, tSi = 25 nm).
Fig. 5
Fig. 5 (a), (b), and (c) show the |E|, Ey, and Ey phase distribution of the s0 mode on the cross section of the coupler, respectively. (d), (e), and (f) show the |E|, Ey, and Ey phase distribution of the a0 mode on the cross section of the waveguide, respectively. (g) nr as a function of d for the s0 and a0 modes. (h) The normalized coupling length Lc /(Lprop)L0 versus d.
Fig. 6
Fig. 6 (a) Schematic diagram of the directional coupler and the |E| distribution of the excited coupling mode on the x-z plane. (b) Normalized output power P/PG0 from the two output arms versus the normalized propagation length z/Lc at d = 16 nm when the EM field is launched from the multilayer-graphene-based LRSPP waveguide at z = –0.9 μm (RAg = 10 nm, tSiO2 = 6 nm, and tSi = 26 nm). (c–e) Coupling efficient (CE), insertion loss (IL), and excited ratio (ER) versus d for different RAg, respectively.
Fig. 7
Fig. 7 (a) shows the |E| distribution of L0 mode on the cross section of the graphene waveguide for the rate of error tolerance of misalignment η = 0.5. (b) shows the |E| distribution of the excited coupling mode on the xy-plane (η = 0.5). (c) Lc and CE as functions of η for different RAg. (d) IL and ER as functions of η for different RAg.
Fig. 8
Fig. 8 (a) (nr)L0 versus Ef for different mobilities μ of graphene (R = 16 nm, t = 4 nm, and N = 6). Point A presents the (nr)H0 value as RAg = 6 nm, tSi = 30 nm, and tSiO2 = 6 nm for the metal-based hybrid waveguide. (b) Coupling efficient (CE), (c) insertion loss (IL) and coupling length Lc, and (d) extinction ratio (ER) versus Ef for different μ, respectively. (e) show the |E| distribution on the x-z plane of the directional coupler (Ef = 0.45 eV and μ = 2000 cm2/Vs).

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