Abstract

In this paper, we theoretically propose a novel graphene-based hybrid plasmonic waveguide (GHPW) consisting of a low-index rectangle waveguide between a high-index cylindrical dielectric waveguide and the substrate with coated graphene on the surface. The geometric dependence of the mode characteristics on the proposed structure is analyzed in detail, showing that the proposed GHPW has a low loss and consequently a relatively long propagation distance. For TM polarization, highly confined modes guided in the low-index gap region between the graphene and the high-index GaAs and the normalized modal area is as small as 0.0018 (λ2/4) at 3 THz. In addition to enabling the building of high-density integration of the proposed structure are examined by analyzing crosstalk in a directional coupler composed of two GHPWs. This structure also exhibits ultra-low crosstalk when a center-to-center separation between adjacent GHPWs is 32μm, which shows great promise for constructing various terahertz integrated devices.

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

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Hybrid wedge plasmon polariton waveguide with good fabrication-error-tolerance for ultra-deep-subwavelength mode confinement

Yusheng Bian, Zheng Zheng, Ya Liu, Jiansheng Liu, Jinsong Zhu, and Tao Zhou
Opt. Express 19(23) 22417-22422 (2011)

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2017 (2)

R. Li, B. Zheng, X. Lin, R. Hao, S. Lin, W. Yin, E. Li, and H. Chen, “Design of Ultracompact Graphene-Based Superscatterers,” IEEE J. Sel. Top. Quant. 23(1), 4600208 (2017).
[Crossref]

X. He, T. Ning, R. Li, L. Pei, J. Zheng, and J. Li, “Dynamical manipulation of Cosine-Gauss beams in a graphene plasmonic waveguide,” Opt. Express 25(12), 13923–13932 (2017).
[Crossref] [PubMed]

2016 (1)

R. Li, X. Lin, S. Lin, X. Zhang, E. Li, and H. Chen, “Graphene induced mode bifurcation at low input power,” Carbon 98, 463–467 (2016).
[Crossref]

2015 (2)

R. J. Li, X. Lin, S. S. Lin, X. Liu, and H. S. Chen, “Tunable deep-subwavelength superscattering using graphene monolayers,” Opt. Lett. 40(8), 1651–1654 (2015).
[Crossref] [PubMed]

R. Li, X. Lin, S. Lin, X. Liu, and H. Chen, “Atomically thin spherical shell-shaped superscatterers based on a Bohr model,” Nanotechnology 26(50), 505201 (2015).
[Crossref] [PubMed]

2014 (1)

2013 (2)

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]

Z. Han and S. I. Bozhevolnyi, “Radiation guiding with surface plasmon polaritons,” Rep. Prog. Phys. 76(1), 016402 (2013).
[Crossref] [PubMed]

2012 (3)

C. H. Gan, “Analysis of surface plasmon excitation at terahertz frequencies with highly doped graphene sheets via attenuated total reflection,” Appl. Phys. Lett. 101(11), 111609 (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] [PubMed]

L. Y. M. Tobing, L. Tjahjana, and D. H. Zhang, “Demonstration of low-loss on-chip integrated plasmonic waveguide based on simple fabrication steps on silicon-on-insulator platform,” Appl. Phys. Lett. 101(4), 041117 (2012).
[Crossref]

2011 (4)

Y. Bian, Z. Zheng, Y. Liu, J. Liu, J. Zhu, and T. Zhou, “Hybrid wedge plasmon polariton waveguide with good fabrication-error-tolerance for ultra-deep-subwavelength mode confinement,” Opt. Express 19(23), 22417–22422 (2011).
[Crossref] [PubMed]

M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang, “A graphene-based broadband optical modulator,” Nature 474(7349), 64–67 (2011).
[Crossref] [PubMed]

A. Vakil and N. Engheta, “Transformation optics using graphene,” Science 332(6035), 1291–1294 (2011).
[Crossref] [PubMed]

P. Y. Chen and A. Alù, “Atomically thin surface cloak using graphene monolayers,” ACS Nano 5(7), 5855–5863 (2011).
[Crossref] [PubMed]

2010 (4)

2009 (5)

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

A. Y. Elezzabi, Z. Han, S. Sederberg, and V. Van, “Ultrafast all-optical modulation in silicon-based nanoplasmonic devices,” Opt. Express 17(13), 11045–11056 (2009).
[Crossref] [PubMed]

A. Ishikawa, S. Zhang, D. A. Genov, G. Bartal, and X. Zhang, “Deep Subwavelength Terahertz Waveguides Using Gap Magnetic Plasmon,” Phys. Rev. Lett. 102(4), 043904 (2009).
[Crossref] [PubMed]

S. H. Nam, A. J. Taylor, and A. Efimov, “Subwavelength hybrid terahertz waveguides,” Opt. Express 17(25), 22890–22897 (2009).
[Crossref] [PubMed]

Y. J. Yu, Y. Zhao, S. Ryu, L. E. Brus, K. S. Kim, and P. Kim, “Tuning the graphene work function by electric field effect,” Nano Lett. 9(10), 3430–3434 (2009).
[Crossref] [PubMed]

2008 (6)

R. L. Olmon, P. M. Krenz, A. C. Jones, G. D. Boreman, and M. B. Raschke, “Near-field imaging of optical antenna modes in the mid-infrared,” Opt. Express 16(25), 20295–20305 (2008).
[Crossref] [PubMed]

F. Wang, Y. Zhang, C. Tian, C. Girit, A. Zettl, M. Crommie, and Y. R. Shen, “Gate-Variable Optical Transitions in Graphene,” Science 320(5873), 206–209 (2008).
[Crossref] [PubMed]

G. W. Hanson, “Dyadic Green’s functions and guided surface waves for a surface conductivity model of graphene,” J. Appl. Phys. 103(6), 064302 (2008).
[Crossref]

E. Moreno, S. G. Rodrigo, S. I. Bozhevolnyi, L. Martín-Moreno, and F. J. García-Vidal, “Guiding and Focusing of Electromagnetic Fields with Wedge Plasmon Polaritons,” Phys. Rev. Lett. 100(2), 023901 (2008).
[Crossref] [PubMed]

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

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 (2)

2006 (3)

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J. Y. Laluet, and T. W. Ebbesen, “Channel plasmon subwavelength waveguide components including interferometers and ring resonators,” Nature 440(7083), 508–511 (2006).
[Crossref] [PubMed]

J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Plasmon slot waveguides: Towards chip-scale propagation with subwavelength-scale localization,” Phys. Rev. B 73(3), 035407 (2006).
[Crossref]

S. A. Maier, S. R. Andrews, L. Martín-Moreno, and F. J. García-Vidal, “Terahertz Surface Plasmon-Polariton Propagation and Focusing on Periodically Corrugated Metal Wires,” Phys. Rev. Lett. 97(17), 176805 (2006).
[Crossref] [PubMed]

2005 (2)

D. F. P. Pile, T. Ogawa, D. K. Gramotnev, T. Okamoto, M. Haraguchi, M. Fukui, and S. Matsuo, “Theoretical and experimental investigation of strongly localized plasmons on triangular metal wedges for subwavelength waveguiding,” Appl. Phys. Lett. 87(6), 061106 (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 (1)

2001 (1)

H. G. Lee, H. C. Jeon, T. W. Kang, and T. W. Kim, “Gallium arsenide crystalline nanorods grown by molecular-beam epitaxy,” Appl. Phys. Lett. 78(21), 3319–3321 (2001).
[Crossref]

1996 (1)

A. D. Berry, R. J. Tonucci, and M. Fatemi, “Fabrication of GaAs and InAs wires in nanochannel glass,” Appl. Phys. Lett. 69(19), 2846–2848 (1996).
[Crossref]

1974 (1)

C. A. Pfieffer, E. N. Economou, and K. L. Ngai, “Surface polaritons in a circularly cylindrical interface: Surface plasmons,” Phys. Rev. B 10(8), 3038–3051 (1974).
[Crossref]

Alù, A.

P. Y. Chen and A. Alù, “Atomically thin surface cloak using graphene monolayers,” ACS Nano 5(7), 5855–5863 (2011).
[Crossref] [PubMed]

Andrews, S. R.

S. A. Maier, S. R. Andrews, L. Martín-Moreno, and F. J. García-Vidal, “Terahertz Surface Plasmon-Polariton Propagation and Focusing on Periodically Corrugated Metal Wires,” Phys. Rev. Lett. 97(17), 176805 (2006).
[Crossref] [PubMed]

Atwater, H. A.

J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Plasmon slot waveguides: Towards chip-scale propagation with subwavelength-scale localization,” Phys. Rev. B 73(3), 035407 (2006).
[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] [PubMed]

Bartal, G.

A. Ishikawa, S. Zhang, D. A. Genov, G. Bartal, and X. Zhang, “Deep Subwavelength Terahertz Waveguides Using Gap Magnetic Plasmon,” Phys. Rev. Lett. 102(4), 043904 (2009).
[Crossref] [PubMed]

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

Benisty, H.

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

Berry, A. D.

A. D. Berry, R. J. Tonucci, and M. Fatemi, “Fabrication of GaAs and InAs wires in nanochannel glass,” Appl. Phys. Lett. 69(19), 2846–2848 (1996).
[Crossref]

Besbes, M.

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

Bian, Y.

Boltasseva, A.

Boreman, G. D.

Bozhevolnyi, S. I.

Z. Han and S. I. Bozhevolnyi, “Radiation guiding with surface plasmon polaritons,” Rep. Prog. Phys. 76(1), 016402 (2013).
[Crossref] [PubMed]

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

E. Moreno, S. G. Rodrigo, S. I. Bozhevolnyi, L. Martín-Moreno, and F. J. García-Vidal, “Guiding and Focusing of Electromagnetic Fields with Wedge Plasmon Polaritons,” Phys. Rev. Lett. 100(2), 023901 (2008).
[Crossref] [PubMed]

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J. Y. Laluet, and T. W. Ebbesen, “Channel plasmon subwavelength waveguide components including interferometers and ring resonators,” Nature 440(7083), 508–511 (2006).
[Crossref] [PubMed]

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]

Brus, L. E.

Y. J. Yu, Y. Zhao, S. Ryu, L. E. Brus, K. S. Kim, and P. Kim, “Tuning the graphene work function by electric field effect,” Nano Lett. 9(10), 3430–3434 (2009).
[Crossref] [PubMed]

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] [PubMed]

Chen, H.

R. Li, B. Zheng, X. Lin, R. Hao, S. Lin, W. Yin, E. Li, and H. Chen, “Design of Ultracompact Graphene-Based Superscatterers,” IEEE J. Sel. Top. Quant. 23(1), 4600208 (2017).
[Crossref]

R. Li, X. Lin, S. Lin, X. Zhang, E. Li, and H. Chen, “Graphene induced mode bifurcation at low input power,” Carbon 98, 463–467 (2016).
[Crossref]

R. Li, X. Lin, S. Lin, X. Liu, and H. Chen, “Atomically thin spherical shell-shaped superscatterers based on a Bohr model,” Nanotechnology 26(50), 505201 (2015).
[Crossref] [PubMed]

Chen, H. S.

Chen, L.

Chen, P. Y.

P. Y. Chen and A. Alù, “Atomically thin surface cloak using graphene monolayers,” ACS Nano 5(7), 5855–5863 (2011).
[Crossref] [PubMed]

Crommie, M.

F. Wang, Y. Zhang, C. Tian, C. Girit, A. Zettl, M. Crommie, and Y. R. Shen, “Gate-Variable Optical Transitions in Graphene,” Science 320(5873), 206–209 (2008).
[Crossref] [PubMed]

Dai, D.

Dai, L.

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

Devaux, E.

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J. Y. Laluet, and T. W. Ebbesen, “Channel plasmon subwavelength waveguide components including interferometers and ring resonators,” Nature 440(7083), 508–511 (2006).
[Crossref] [PubMed]

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]

Dionne, J. A.

J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Plasmon slot waveguides: Towards chip-scale propagation with subwavelength-scale localization,” Phys. Rev. B 73(3), 035407 (2006).
[Crossref]

Duley, W. W.

Ebbesen, T. W.

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J. Y. Laluet, and T. W. Ebbesen, “Channel plasmon subwavelength waveguide components including interferometers and ring resonators,” Nature 440(7083), 508–511 (2006).
[Crossref] [PubMed]

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]

Economou, E. N.

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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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D. F. P. Pile and D. K. Gramotnev, “Channel plasmon-polariton in a triangular groove on a metal surface,” Opt. Lett. 29(10), 1069–1071 (2004).
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Y. J. Yu, Y. Zhao, S. Ryu, L. E. Brus, K. S. Kim, and P. Kim, “Tuning the graphene work function by electric field effect,” Nano Lett. 9(10), 3430–3434 (2009).
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Shen, Y. R.

F. Wang, Y. Zhang, C. Tian, C. Girit, A. Zettl, M. Crommie, and Y. R. Shen, “Gate-Variable Optical Transitions in Graphene,” Science 320(5873), 206–209 (2008).
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Sorger, V. J.

R. F. Oulton, V. J. Sorger, T. Zentgraf, R. M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461(7264), 629–632 (2009).
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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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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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J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Plasmon slot waveguides: Towards chip-scale propagation with subwavelength-scale localization,” Phys. Rev. B 73(3), 035407 (2006).
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F. Wang, Y. Zhang, C. Tian, C. Girit, A. Zettl, M. Crommie, and Y. R. Shen, “Gate-Variable Optical Transitions in Graphene,” Science 320(5873), 206–209 (2008).
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L. Y. M. Tobing, L. Tjahjana, and D. H. Zhang, “Demonstration of low-loss on-chip integrated plasmonic waveguide based on simple fabrication steps on silicon-on-insulator platform,” Appl. Phys. Lett. 101(4), 041117 (2012).
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A. D. Berry, R. J. Tonucci, and M. Fatemi, “Fabrication of GaAs and InAs wires in nanochannel glass,” Appl. Phys. Lett. 69(19), 2846–2848 (1996).
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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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M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang, “A graphene-based broadband optical modulator,” Nature 474(7349), 64–67 (2011).
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[Crossref] [PubMed]

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J. Y. Laluet, and T. W. Ebbesen, “Channel plasmon subwavelength waveguide components including interferometers and ring resonators,” Nature 440(7083), 508–511 (2006).
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[Crossref] [PubMed]

F. Wang, Y. Zhang, C. Tian, C. Girit, A. Zettl, M. Crommie, and Y. R. Shen, “Gate-Variable Optical Transitions in Graphene,” Science 320(5873), 206–209 (2008).
[Crossref] [PubMed]

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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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Yan, H.

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M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang, “A graphene-based broadband optical modulator,” Nature 474(7349), 64–67 (2011).
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Y. J. Yu, Y. Zhao, S. Ryu, L. E. Brus, K. S. Kim, and P. Kim, “Tuning the graphene work function by electric field effect,” Nano Lett. 9(10), 3430–3434 (2009).
[Crossref] [PubMed]

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M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang, “A graphene-based broadband optical modulator,” Nature 474(7349), 64–67 (2011).
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F. Wang, Y. Zhang, C. Tian, C. Girit, A. Zettl, M. Crommie, and Y. R. Shen, “Gate-Variable Optical Transitions in Graphene,” Science 320(5873), 206–209 (2008).
[Crossref] [PubMed]

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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).
[Crossref]

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L. Y. M. Tobing, L. Tjahjana, and D. H. Zhang, “Demonstration of low-loss on-chip integrated plasmonic waveguide based on simple fabrication steps on silicon-on-insulator platform,” Appl. Phys. Lett. 101(4), 041117 (2012).
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Zhang, X. Y.

Zhang, Y.

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Carbon (1)

R. Li, X. Lin, S. Lin, X. Zhang, E. Li, and H. Chen, “Graphene induced mode bifurcation at low input power,” Carbon 98, 463–467 (2016).
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IEEE J. Sel. Top. Quant. (1)

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Nanotechnology (1)

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Nature (3)

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Figures (7)

Fig. 1
Fig. 1 Schematic diagram of the proposed GHPW. (a) Cross section of the waveguide. (b) 3D structure of the waveguide.
Fig. 2
Fig. 2 The distributions of electromagnetic energy density of the GHPW for (a) [d, g] = [30, 7] μm, (b) [d, g] = [30, 0.5] μm, (c) [d, g] = [20, 7] μm, (d) [d, g] = [20, 0.5] μm.
Fig. 3
Fig. 3 The energy flux density along x = 0 for (a) g = 7μm, (b) g = 5μm, (c) g = 3μm, (d) g = 1μm and (e) g = 0.5μm; (f) The energy flux density along y = -(d + g)/2 for different gap distance g when d = dgap = 30μm. Red dotted line denotes the boundary between the GaAs waveguide and gap. Black dotted line denotes the boundary between the graphene sheets and gap.
Fig. 4
Fig. 4 (a) Dependence of the real part of the mode effective refractive index on diameter d of the GaAs waveguide for different gap thickness g. (b) Normalized modal area Am versus the cylinder diameter d for different gap distance g. (c) The hybrid mode’s propagation distance versus the cylinder diameter d for different gap distance g. (d) the hybrid mode’s propagation distance versus the gap distance g as cylinder diameter d varies for the cases of d = 35μm, 30μm, 25μm, 20μm and 15μm.
Fig. 5
Fig. 5 (a) The normalized modal area Am versus the width dgap of the rectangle gap for different gap thickness g. (b) The hybrid mode’s propagation distance versus the width dgap of the rectangle gap for different gap thickness g. Here, we fixed the d = 30μm.
Fig. 6
Fig. 6 (a) Cross section of two parallel GHPW with a separation of S. Field distributions of Ey for the (b) even and (c) odd modes of the directional coupler with dimensions d = dgap = 30μm, g = 0.5μm and S = 35μm.
Fig. 7
Fig. 7 (a) Real parts of the effective refractive index for even and odd modes of the directional coupler composed of two GHPWs. (b) Normalized coupling length Lc/Lprop versus the normalized separation (S-d)/d for different gap g. The other parameters are d = dgap = 30μm.

Equations (5)

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σ g = σ intra (ω,T,τ, μ c )+ σ inter (ω,T,τ, μ c ) ,
σ intra (ω,T,τ, μ c )= i e 2 k B T π 2 ( ω+i τ 1 ) [ μ c k B T +2ln( exp( μ c k B T )+1 ) ]
σ inter (ω,T,τ, μ c )= i e 2 4π ln( 2| μ c |( ω+i τ 1 ) 2| μ c |+( ω+i τ 1 ) )
ε g =1+ i σ g η 0 k 0 Δ ,
A eff = P(r)ds / max[P(r)] ,

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