Abstract

Electronics circuits keep shrinking in dimensions, as requested by Moore’s law. In contrast, photonic waveguides and circuit elements still have lateral dimensions on the order of the wavelength. A key to make photonics have a microelectronics-like development is a drastic reduction of size. To achieve this, we need a low-loss nanoscale waveguide with a drastically reduced mode area and an ultra-high effective refractive index. For this purpose, we propose here several low-loss waveguide structures based on graphene nano-ribbons. An extremely small mode area (~10−7λ02, one order smaller than the smallest mode area of any waveguide that has ever been reported in the literature; here λ0 is the operating wavelength in vacuum) and an extremely large effective refractive index (several hundreds) are achieved. As a device example, a nano-ring cavity of ultra-small size (with a diameter of ~10−2λ0) is designed. Our study paves the way for future VLSI (very-large-scale integration) optoelectronics.

© 2013 Optical Society of America

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  1. L.  Ju, B.  Geng, J.  Horng, C.  Girit, M.  Martin, Z.  Hao, H. A.  Bechtel, X.  Liang, A.  Zettl, Y. R.  Shen, F.  Wang, “Graphene plasmonics for tunable terahertz metamaterials,” Nat. Nanotechnol. 6(10), 630–634 (2011).
    [CrossRef] [PubMed]
  2. A. K.  Geim, K. S.  Novoselov, “The rise of graphene,” Nat. Mater. 6(3), 183–191 (2007).
    [CrossRef] [PubMed]
  3. K. S.  Novoselov, A. K.  Geim, S. V.  Morozov, D.  Jiang, Y.  Zhang, S. V.  Dubonos, I. V.  Grigorieva, A. A.  Firsov, “Electric field effect in atomically thin carbon films,” Science 306(5696), 666–669 (2004).
    [CrossRef] [PubMed]
  4. A.  Vakil, N.  Engheta, “Transformation optics using graphene,” Science 332(6035), 1291–1294 (2011).
    [CrossRef] [PubMed]
  5. S. A.  Mikhailov, K.  Ziegler, “New electromagnetic mode in graphene,” Phys. Rev. Lett. 99(1), 016803 (2007).
    [CrossRef] [PubMed]
  6. M.  Jablan, H.  Buljan, M.  Soljacic, “Plasmonics in graphene at infrared frequencies,” Phys. Rev. B 80(24), 245435 (2009).
    [CrossRef]
  7. F. H. L.  Koppens, D. E.  Chang, F. J.  García de Abajo, “Graphene plasmonics: a platform for strong light-matter interactions,” Nano Lett. 11(8), 3370–3377 (2011).
    [CrossRef] [PubMed]
  8. 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]
  9. A. Y.  Nikitin, F.  Guinea, F. J.  García-Vidal, L.  Martín-Moreno, “Edge and waveguide terahertz surface plasmon modes in graphene microribbons,” Phys. Rev. B 84(16), 161407 (2011).
    [CrossRef]
  10. P. R.  West, S.  Ishii, G. V.  Naik, N. K.  Emani, V. M.  Shalaev, A.  Boltasseva, “Searching for better plasmonic materials,” Laser Photon. Rev. 4(6), 795–808 (2010).
    [CrossRef]
  11. S.  Thongrattanasiri, F. H. L.  Koppens, F. J.  García de Abajo, “Complete optical absorption in periodically patterned graphene,” Phys. Rev. Lett. 108(4), 047401 (2012).
    [CrossRef] [PubMed]
  12. L.  Wu, H. S.  Chu, W. S.  Koh, E. P.  Li, “Highly sensitive graphene biosensors based on surface plasmon resonance,” Opt. Express 18(14), 14395–14400 (2010).
    [CrossRef] [PubMed]
  13. E. H.  Hwang, S.  Das Sarma, “Dielectric function, screening, and plasmons in two-dimensional graphene,” Phys. Rev. B 75(20), 205418 (2007).
    [CrossRef]
  14. B.  Wunsch, T.  Stauber, F.  Sols, F.  Guinea, “Dynamical polarization of graphene at finite doping,” New J. Phys. 8(12), 318 (2006).
    [CrossRef]
  15. G.  Eda, G.  Fanchini, M.  Chhowalla, “Large-area ultrathin films of reduced graphene oxide as a transparent and flexible electronic material,” Nat. Nanotechnol. 3(5), 270–274 (2008).
    [CrossRef] [PubMed]
  16. E.  Verhagen, M.  Spasenović, A.  Polman, L. K.  Kuipers, “Nanowire plasmon excitation by adiabatic mode transformation,” Phys. Rev. Lett. 102(20), 203904 (2009).
    [CrossRef] [PubMed]
  17. S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, 2007).
  18. P.  Berini, “Figures of merit for surface plasmon waveguides,” Opt. Express 14(26), 13030–13042 (2006).
    [CrossRef] [PubMed]
  19. Q.  Huang, F.  Bao, S.  He, “Nonlocal effects in a hybrid plasmonic waveguide for nanoscale confinement,” Opt. Express 21(2), 1430–1439 (2013).
    [CrossRef] [PubMed]
  20. An early version of the present paper was put online in May, 2013, at ArXiv. S. He, X. Zhang, and Y. He, “Graphene nano-ribbon waveguides,” ArXiv preprint 2013, DOI: arXiv: 1305.6500, http://arxiv.org/abs/1305.6500 .
  21. Y.  Francescato, V.  Giannini, S. A.  Maier, “Strongly confined gap plasmon modes in graphene sandwiches and graphene-on-silicon,” New J. Phys. 15(6), 063020 (2013).
    [CrossRef]
  22. R.  Buckley, P.  Berini, “Figures of merit for 2D surface plasmon waveguides and application to metal stripes,” Opt. Express 15(19), 12174–12182 (2007).
    [CrossRef] [PubMed]

2013 (2)

Y.  Francescato, V.  Giannini, S. A.  Maier, “Strongly confined gap plasmon modes in graphene sandwiches and graphene-on-silicon,” New J. Phys. 15(6), 063020 (2013).
[CrossRef]

Q.  Huang, F.  Bao, S.  He, “Nonlocal effects in a hybrid plasmonic waveguide for nanoscale confinement,” Opt. Express 21(2), 1430–1439 (2013).
[CrossRef] [PubMed]

2012 (1)

S.  Thongrattanasiri, F. H. L.  Koppens, F. J.  García de Abajo, “Complete optical absorption in periodically patterned graphene,” Phys. Rev. Lett. 108(4), 047401 (2012).
[CrossRef] [PubMed]

2011 (4)

L.  Ju, B.  Geng, J.  Horng, C.  Girit, M.  Martin, Z.  Hao, H. A.  Bechtel, X.  Liang, A.  Zettl, Y. R.  Shen, F.  Wang, “Graphene plasmonics for tunable terahertz metamaterials,” Nat. Nanotechnol. 6(10), 630–634 (2011).
[CrossRef] [PubMed]

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

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

A. Y.  Nikitin, F.  Guinea, F. J.  García-Vidal, L.  Martín-Moreno, “Edge and waveguide terahertz surface plasmon modes in graphene microribbons,” Phys. Rev. B 84(16), 161407 (2011).
[CrossRef]

2010 (2)

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

L.  Wu, H. S.  Chu, W. S.  Koh, E. P.  Li, “Highly sensitive graphene biosensors based on surface plasmon resonance,” Opt. Express 18(14), 14395–14400 (2010).
[CrossRef] [PubMed]

2009 (2)

M.  Jablan, H.  Buljan, M.  Soljacic, “Plasmonics in graphene at infrared frequencies,” Phys. Rev. B 80(24), 245435 (2009).
[CrossRef]

E.  Verhagen, M.  Spasenović, A.  Polman, L. K.  Kuipers, “Nanowire plasmon excitation by adiabatic mode transformation,” Phys. Rev. Lett. 102(20), 203904 (2009).
[CrossRef] [PubMed]

2008 (2)

G.  Eda, G.  Fanchini, M.  Chhowalla, “Large-area ultrathin films of reduced graphene oxide as a transparent and flexible electronic material,” Nat. Nanotechnol. 3(5), 270–274 (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]

2007 (4)

S. A.  Mikhailov, K.  Ziegler, “New electromagnetic mode in graphene,” Phys. Rev. Lett. 99(1), 016803 (2007).
[CrossRef] [PubMed]

A. K.  Geim, K. S.  Novoselov, “The rise of graphene,” Nat. Mater. 6(3), 183–191 (2007).
[CrossRef] [PubMed]

E. H.  Hwang, S.  Das Sarma, “Dielectric function, screening, and plasmons in two-dimensional graphene,” Phys. Rev. B 75(20), 205418 (2007).
[CrossRef]

R.  Buckley, P.  Berini, “Figures of merit for 2D surface plasmon waveguides and application to metal stripes,” Opt. Express 15(19), 12174–12182 (2007).
[CrossRef] [PubMed]

2006 (2)

P.  Berini, “Figures of merit for surface plasmon waveguides,” Opt. Express 14(26), 13030–13042 (2006).
[CrossRef] [PubMed]

B.  Wunsch, T.  Stauber, F.  Sols, F.  Guinea, “Dynamical polarization of graphene at finite doping,” New J. Phys. 8(12), 318 (2006).
[CrossRef]

2004 (1)

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

Bao, F.

Bechtel, H. A.

L.  Ju, B.  Geng, J.  Horng, C.  Girit, M.  Martin, Z.  Hao, H. A.  Bechtel, X.  Liang, A.  Zettl, Y. R.  Shen, F.  Wang, “Graphene plasmonics for tunable terahertz metamaterials,” Nat. Nanotechnol. 6(10), 630–634 (2011).
[CrossRef] [PubMed]

Berini, P.

Boltasseva, A.

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

Buckley, R.

Buljan, H.

M.  Jablan, H.  Buljan, M.  Soljacic, “Plasmonics in graphene at infrared frequencies,” Phys. Rev. B 80(24), 245435 (2009).
[CrossRef]

Chang, D. E.

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

Chhowalla, M.

G.  Eda, G.  Fanchini, M.  Chhowalla, “Large-area ultrathin films of reduced graphene oxide as a transparent and flexible electronic material,” Nat. Nanotechnol. 3(5), 270–274 (2008).
[CrossRef] [PubMed]

Chu, H. S.

Das Sarma, S.

E. H.  Hwang, S.  Das Sarma, “Dielectric function, screening, and plasmons in two-dimensional graphene,” Phys. Rev. B 75(20), 205418 (2007).
[CrossRef]

Dubonos, S. V.

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

Eda, G.

G.  Eda, G.  Fanchini, M.  Chhowalla, “Large-area ultrathin films of reduced graphene oxide as a transparent and flexible electronic material,” Nat. Nanotechnol. 3(5), 270–274 (2008).
[CrossRef] [PubMed]

Emani, N. K.

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

Engheta, N.

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

Fanchini, G.

G.  Eda, G.  Fanchini, M.  Chhowalla, “Large-area ultrathin films of reduced graphene oxide as a transparent and flexible electronic material,” Nat. Nanotechnol. 3(5), 270–274 (2008).
[CrossRef] [PubMed]

Firsov, A. A.

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

Francescato, Y.

Y.  Francescato, V.  Giannini, S. A.  Maier, “Strongly confined gap plasmon modes in graphene sandwiches and graphene-on-silicon,” New J. Phys. 15(6), 063020 (2013).
[CrossRef]

García de Abajo, F. J.

S.  Thongrattanasiri, F. H. L.  Koppens, F. J.  García de Abajo, “Complete optical absorption in periodically patterned graphene,” Phys. Rev. Lett. 108(4), 047401 (2012).
[CrossRef] [PubMed]

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

García-Vidal, F. J.

A. Y.  Nikitin, F.  Guinea, F. J.  García-Vidal, L.  Martín-Moreno, “Edge and waveguide terahertz surface plasmon modes in graphene microribbons,” Phys. Rev. B 84(16), 161407 (2011).
[CrossRef]

Geim, A. K.

A. K.  Geim, K. S.  Novoselov, “The rise of graphene,” Nat. Mater. 6(3), 183–191 (2007).
[CrossRef] [PubMed]

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

Geng, B.

L.  Ju, B.  Geng, J.  Horng, C.  Girit, M.  Martin, Z.  Hao, H. A.  Bechtel, X.  Liang, A.  Zettl, Y. R.  Shen, F.  Wang, “Graphene plasmonics for tunable terahertz metamaterials,” Nat. Nanotechnol. 6(10), 630–634 (2011).
[CrossRef] [PubMed]

Giannini, V.

Y.  Francescato, V.  Giannini, S. A.  Maier, “Strongly confined gap plasmon modes in graphene sandwiches and graphene-on-silicon,” New J. Phys. 15(6), 063020 (2013).
[CrossRef]

Girit, C.

L.  Ju, B.  Geng, J.  Horng, C.  Girit, M.  Martin, Z.  Hao, H. A.  Bechtel, X.  Liang, A.  Zettl, Y. R.  Shen, F.  Wang, “Graphene plasmonics for tunable terahertz metamaterials,” Nat. Nanotechnol. 6(10), 630–634 (2011).
[CrossRef] [PubMed]

Grigorieva, I. V.

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

Guinea, F.

A. Y.  Nikitin, F.  Guinea, F. J.  García-Vidal, L.  Martín-Moreno, “Edge and waveguide terahertz surface plasmon modes in graphene microribbons,” Phys. Rev. B 84(16), 161407 (2011).
[CrossRef]

B.  Wunsch, T.  Stauber, F.  Sols, F.  Guinea, “Dynamical polarization of graphene at finite doping,” New J. Phys. 8(12), 318 (2006).
[CrossRef]

Hanson, G. W.

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]

Hao, Z.

L.  Ju, B.  Geng, J.  Horng, C.  Girit, M.  Martin, Z.  Hao, H. A.  Bechtel, X.  Liang, A.  Zettl, Y. R.  Shen, F.  Wang, “Graphene plasmonics for tunable terahertz metamaterials,” Nat. Nanotechnol. 6(10), 630–634 (2011).
[CrossRef] [PubMed]

He, S.

Horng, J.

L.  Ju, B.  Geng, J.  Horng, C.  Girit, M.  Martin, Z.  Hao, H. A.  Bechtel, X.  Liang, A.  Zettl, Y. R.  Shen, F.  Wang, “Graphene plasmonics for tunable terahertz metamaterials,” Nat. Nanotechnol. 6(10), 630–634 (2011).
[CrossRef] [PubMed]

Huang, Q.

Hwang, E. H.

E. H.  Hwang, S.  Das Sarma, “Dielectric function, screening, and plasmons in two-dimensional graphene,” Phys. Rev. B 75(20), 205418 (2007).
[CrossRef]

Ishii, S.

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

Jablan, M.

M.  Jablan, H.  Buljan, M.  Soljacic, “Plasmonics in graphene at infrared frequencies,” Phys. Rev. B 80(24), 245435 (2009).
[CrossRef]

Jiang, D.

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

Ju, L.

L.  Ju, B.  Geng, J.  Horng, C.  Girit, M.  Martin, Z.  Hao, H. A.  Bechtel, X.  Liang, A.  Zettl, Y. R.  Shen, F.  Wang, “Graphene plasmonics for tunable terahertz metamaterials,” Nat. Nanotechnol. 6(10), 630–634 (2011).
[CrossRef] [PubMed]

Koh, W. S.

Koppens, F. H. L.

S.  Thongrattanasiri, F. H. L.  Koppens, F. J.  García de Abajo, “Complete optical absorption in periodically patterned graphene,” Phys. Rev. Lett. 108(4), 047401 (2012).
[CrossRef] [PubMed]

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

Kuipers, L. K.

E.  Verhagen, M.  Spasenović, A.  Polman, L. K.  Kuipers, “Nanowire plasmon excitation by adiabatic mode transformation,” Phys. Rev. Lett. 102(20), 203904 (2009).
[CrossRef] [PubMed]

Li, E. P.

Liang, X.

L.  Ju, B.  Geng, J.  Horng, C.  Girit, M.  Martin, Z.  Hao, H. A.  Bechtel, X.  Liang, A.  Zettl, Y. R.  Shen, F.  Wang, “Graphene plasmonics for tunable terahertz metamaterials,” Nat. Nanotechnol. 6(10), 630–634 (2011).
[CrossRef] [PubMed]

Maier, S. A.

Y.  Francescato, V.  Giannini, S. A.  Maier, “Strongly confined gap plasmon modes in graphene sandwiches and graphene-on-silicon,” New J. Phys. 15(6), 063020 (2013).
[CrossRef]

Martin, M.

L.  Ju, B.  Geng, J.  Horng, C.  Girit, M.  Martin, Z.  Hao, H. A.  Bechtel, X.  Liang, A.  Zettl, Y. R.  Shen, F.  Wang, “Graphene plasmonics for tunable terahertz metamaterials,” Nat. Nanotechnol. 6(10), 630–634 (2011).
[CrossRef] [PubMed]

Martín-Moreno, L.

A. Y.  Nikitin, F.  Guinea, F. J.  García-Vidal, L.  Martín-Moreno, “Edge and waveguide terahertz surface plasmon modes in graphene microribbons,” Phys. Rev. B 84(16), 161407 (2011).
[CrossRef]

Mikhailov, S. A.

S. A.  Mikhailov, K.  Ziegler, “New electromagnetic mode in graphene,” Phys. Rev. Lett. 99(1), 016803 (2007).
[CrossRef] [PubMed]

Morozov, S. V.

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

Naik, G. V.

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

Nikitin, A. Y.

A. Y.  Nikitin, F.  Guinea, F. J.  García-Vidal, L.  Martín-Moreno, “Edge and waveguide terahertz surface plasmon modes in graphene microribbons,” Phys. Rev. B 84(16), 161407 (2011).
[CrossRef]

Novoselov, K. S.

A. K.  Geim, K. S.  Novoselov, “The rise of graphene,” Nat. Mater. 6(3), 183–191 (2007).
[CrossRef] [PubMed]

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

Polman, A.

E.  Verhagen, M.  Spasenović, A.  Polman, L. K.  Kuipers, “Nanowire plasmon excitation by adiabatic mode transformation,” Phys. Rev. Lett. 102(20), 203904 (2009).
[CrossRef] [PubMed]

Shalaev, V. M.

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

Shen, Y. R.

L.  Ju, B.  Geng, J.  Horng, C.  Girit, M.  Martin, Z.  Hao, H. A.  Bechtel, X.  Liang, A.  Zettl, Y. R.  Shen, F.  Wang, “Graphene plasmonics for tunable terahertz metamaterials,” Nat. Nanotechnol. 6(10), 630–634 (2011).
[CrossRef] [PubMed]

Soljacic, M.

M.  Jablan, H.  Buljan, M.  Soljacic, “Plasmonics in graphene at infrared frequencies,” Phys. Rev. B 80(24), 245435 (2009).
[CrossRef]

Sols, F.

B.  Wunsch, T.  Stauber, F.  Sols, F.  Guinea, “Dynamical polarization of graphene at finite doping,” New J. Phys. 8(12), 318 (2006).
[CrossRef]

Spasenovic, M.

E.  Verhagen, M.  Spasenović, A.  Polman, L. K.  Kuipers, “Nanowire plasmon excitation by adiabatic mode transformation,” Phys. Rev. Lett. 102(20), 203904 (2009).
[CrossRef] [PubMed]

Stauber, T.

B.  Wunsch, T.  Stauber, F.  Sols, F.  Guinea, “Dynamical polarization of graphene at finite doping,” New J. Phys. 8(12), 318 (2006).
[CrossRef]

Thongrattanasiri, S.

S.  Thongrattanasiri, F. H. L.  Koppens, F. J.  García de Abajo, “Complete optical absorption in periodically patterned graphene,” Phys. Rev. Lett. 108(4), 047401 (2012).
[CrossRef] [PubMed]

Vakil, A.

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

Verhagen, E.

E.  Verhagen, M.  Spasenović, A.  Polman, L. K.  Kuipers, “Nanowire plasmon excitation by adiabatic mode transformation,” Phys. Rev. Lett. 102(20), 203904 (2009).
[CrossRef] [PubMed]

Wang, F.

L.  Ju, B.  Geng, J.  Horng, C.  Girit, M.  Martin, Z.  Hao, H. A.  Bechtel, X.  Liang, A.  Zettl, Y. R.  Shen, F.  Wang, “Graphene plasmonics for tunable terahertz metamaterials,” Nat. Nanotechnol. 6(10), 630–634 (2011).
[CrossRef] [PubMed]

West, P. R.

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

Wu, L.

Wunsch, B.

B.  Wunsch, T.  Stauber, F.  Sols, F.  Guinea, “Dynamical polarization of graphene at finite doping,” New J. Phys. 8(12), 318 (2006).
[CrossRef]

Zettl, A.

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An early version of the present paper was put online in May, 2013, at ArXiv. S. He, X. Zhang, and Y. He, “Graphene nano-ribbon waveguides,” ArXiv preprint 2013, DOI: arXiv: 1305.6500, http://arxiv.org/abs/1305.6500 .

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Figures (8)

Fig. 1
Fig. 1

(a) Plasmonic mode supported by a single freestanding graphene ribbon. Here the Ey distribution of the guided wave is calculated with software CST 2009. The boundary of the graphene ribbon is indicated by the red line. The width of the graphene ribbon is w = 40 nm. The free-space wavelength used in the simulation is 10 μm. A discrete port (a dipole source) is used for the excitation of the waveguide mode. q is the propagation vector of the waveguide mode. (b) The distribution of the energy density of the waveguide mode. This figure, as well as all the mode properties of graphene ribbons throughout the paper, is calculated using the software COMSOL 3.5. One clearly sees that the energy is tightly confined inside the graphene ribbon.

Fig. 2
Fig. 2

(a) The dependence of the effective refractive index on the width of the graphene ribbon. The dark blue lines show the variation of three SPP waveguide modes when the width of the ribbon varies. The green line marked as “2DGSP” is for the SPP waveguide mode in an infinitely-extended graphene sheet. The light blue line marked as “EGSP” is for the SPP waveguide mode in a semi-infinite graphene sheet [9]. We find that modes 1 and 2 originate from the hybridization of EGSPs. Mode 1 has an even parity of Ey with respect to the ribbon axis, and its effective index neff > neffEGSP. In contrast, mode 2 has an odd parity of Ey with respect to the ribbon axis, and its effective index neff < neffEGSP. Mode 2 will be cut off if the width < 50 nm. Thus, there is a single-mode region (red shaded region in (a)) for small width w. The effective index neff of mode 3 will increase as w increases and finally approach to neff2DGSP [which is not shown in (a)]. (b) The electric fields of the three modes. (c) The mode area (Aeff /λ02) of the SPP waveguide in a freestanding graphene ribbon when width w is in the single-mode region.

Fig. 3
Fig. 3

(a) A low-low waveguide structure: a graphene buffered by a silica layer on silicon. (b) The distribution of the energy density in the waveguide structure. Most of the energy is in the graphene and silica layer. Inset of (b) shows the energy field at the edge of the graphene. It shows the energy is tightly confined in the graphene and Wmax is located at the edge of the graphene. (c) and (d) are the distributions of the electric field and the magnetic field in the waveguide structure. In (c), it is found that the electric field is strong in the graphene layer and the silica (the electric field in silicon is very weak), especially at the corners of the graphene ribbon. In (d), the magnetic field is strong in all the three layers. (e) shows the effective index neff of the low-loss waveguide structure when hsio2 varies. The blue solid line is Re(neff) and the green dashed line is Im(neff). (f) The solid green line is the FOM of the SPP waveguide for this structure with different hSiO2.The dashed green line is the FOM for a freestanding graphene waveguide with the same width (20 nm). As w gets smaller, the FOM will first increase for hSiO2 > 3.0 nm and then decreases quickly for hSiO2 < 3.0 nm, leading to a maximum FOM = 145 and it is bigger than the FOM for a freestanding graphene waveguide with the same width. The solid blue line is the propagation length for this structure with different hSiO2, and the dashed blue line is the propagation for a freestanding graphene waveguide with the same width. The propagation length increases as hSiO2 increases.

Fig. 4
Fig. 4

(a) is the side-side configuration and (b) is the top-bottom configuration. d is the gap size between the two ribbons. The wave propagates in the direction of the vector q.

Fig. 5
Fig. 5

(a) is for the symmetric mode and (b) is for the anti-symmetric mode. In (a), the energy is mainly in the gap between the two ribbons, and in (b) the highest energy density is at the left and right corners. The insets in (a) and (b) show the distribution of the electric field of the modes. (c), the dashed blue solid line is for Re(neff) of the symmetric mode, the green solid line is for Re(neff) of the anti-symmetric mode, the dashed red line is the figure of merit of the symmetric mode, and the light blue dashed line is the figure of merit of the anti-symmetric mode. (d) The mode area of the symmetric mode. The mode area decreases rapidly as the gap d decreases.

Fig. 6
Fig. 6

(a)- (f) is the distribution of the energy intensity, electric field, and magnetic field for the two modes in the top-bottom configuration. In (g), the blue solid line is for Re(neff) of the symmetric mode, the green solid line is for Re(neff) of the anti-symmetric mode, the dashed red line is the figure of merit of the symmetric mode, and the light blue dashed line is the figure of merit of the anti-symmetric mode.

Fig. 7
Fig. 7

(a) and (c) are for the mode (which originally is the symmetric mode) that is formed through the coupling/hybridization of the two ribbons. (b) is a high order mode in the x direction.

Fig. 8
Fig. 8

(a) The proposed ring cavity with graphene ribbon width w = 20 nm and inner radius r = 38 nm. (b) The electric field distribution of the cavity mode at f = 30 THz, which is excited by a discrete port in CST.

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