Abstract

We show theoretically that the optical conductivity of doped graphene can exhibit a spatially periodic modulation by a diffraction grating. Doped graphene placed above the grating exhibits a periodic electrostatic potential distribution, resulting in a periodic charge-density redistribution inside the graphene. The optical conductivity of doped graphene depends linearly on the Fermi energy, which is proportional to the square root of the charge density. Therefore, the optical conductivity exhibits a spatial-periodic modulation. The periodicity implies a band structure formation of the graphene plasmon polariton. Band engineering of the graphene plasmon polariton is also discussed.

© 2015 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, and F. Wang, “Graphene plasmonics for tunable terahertz metamaterials,” Nat. Nanotech. 6, 630–634 (2011).
    [Crossref]
  2. 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, 3370–3377 (2011).
    [Crossref]
  3. S. Thongrattanasiri, F. H. L. Koppens, and F. J. García de Abajo, “Complete optical absorption in periodically patterned graphene,” Phys. Rev. Lett. 108, 047401 (2012).
    [Crossref]
  4. T. M. Slipchenko, M. L. Nesterov, L. Martin-Moreno, and A. Y. Nikitin, “Analytical solution for the diffraction of an electromagnetic wave by a graphene grating,” J. Opt. 15, 114008 (2013).
    [Crossref]
  5. A. Y. Nikitin, F. Guinea, and L. Martin-Moreno, “Resonant plasmonic effects in periodic graphene antidot arrays,” Appl. Phys. Lett. 101, 151119 (2012).
    [Crossref]
  6. X. Zhu, W. Wang, W. Yan, M. B. Larsen, P. Bøggild, T. G. Pedersen, S. Xiao, J. Zi, and N. A. Mortensen, “Plasmon–phonon coupling in large-area graphene dot and antidot arrays fabricated by nanosphere lithography,” Nano Lett. 14, 2907–2913 (2014).
    [Crossref]
  7. J. Horng, C.-F. Chen, B. Geng, C. Girit, Y. Zhang, Z. Hao, H. A. Bechtel, M. Martin, A. Zettl, M. F. Crommie, Y. R. Shen, and F. Wang, “Drude conductivity of Dirac fermions in graphene,” Phys. Rev. B 83, 165113 (2011).
    [Crossref]
  8. Here, we assume the period of the grating is on the order of several hundreds of nanometers, so that it is much larger than the atomic period. The Dirac spectrum of graphene is assumed to be unchanged by the grating.
  9. W. Gao, J. Shu, C. Qiu, and Q. Xu, “Excitation of plasmonic waves in graphene by guided-mode resonances,” ACS Nano 6, 7806–7813 (2012).
    [Crossref]
  10. T. R. Zhan, F. Y. Zhao, X. H. Hu, X. H. Liu, and J. Zi, “Band structure of plasmons and optical absorption enhancement in graphene on subwavelength dielectric gratings at infrared frequencies,” Phys. Rev. B 86, 165416 (2012).
    [Crossref]
  11. X. Zhu, W. Yan, P. U. Jepsen, O. Hansen, N. A. Mortensen, and S. Xiao, “Experimental observation of plasmons in a graphene monolayer resting on a two-dimensional subwavelength silicon grating,” Appl. Phys. Lett. 102, 131101 (2013).
    [Crossref]
  12. N. Peres, Y. V. Bludov, A. Ferreira, and M. I. Vasilevskiy, “Exact solution for square-wave grating covered with graphene: surface plasmon-polaritons in the terahertz range,” J. Phys. 25, 125303 (2013).
    [Crossref]
  13. J. B. Pendry, Low Energy Electron Diffraction (Academic, 1974).
  14. E. Noponen and J. Turunen, “Eigenmode method for electromagnetic synthesis of diffractive elements with three-dimensional profiles,” J. Opt. Soc. Am. A 11, 2494–2502 (1994).
    [Crossref]
  15. L. Li, “New formulation of the fourier modal method for crossed surface-relief gratings,” J. Opt. Soc. Am. A 14, 2758–2767 (1997).
    [Crossref]
  16. S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66, 045102 (2002).
    [Crossref]
  17. T. Ochiai and K. Sakoda, “Dispersion relation and optical transmittance of a hexagonal photonic crystal slab,” Phys. Rev. B 63, 125107 (2001).
    [Crossref]
  18. M. Jablan, H. Buljan, and M. Soljačić, “Plasmonics in graphene at infrared frequencies,” Phys. Rev. B 80, 245435 (2009).
    [Crossref]
  19. K. Ohtaka, J. Inoue, and S. Yamaguti, “Derivation of the density of states of leaky photonic bands,” Phys. Rev. B 70, 035109 (2004).
    [Crossref]
  20. T. Weiss, N. A. Gippius, S. G. Tikhodeev, G. Granet, and H. Giessen, “Derivation of plasmonic resonances in the fourier modal method with adaptive spatial resolution and matched coordinates,” J. Opt. Soc. Am. A 28, 238–244 (2011).
    [Crossref]
  21. D. Bykov and L. Doskolovich, “Numerical methods for calculating poles of the scattering matrix with applications in grating theory,” J. Lightwave Technol. 31, 793–801 (2013).
    [Crossref]
  22. S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B 60, 5751–5758 (1999).
    [Crossref]
  23. E. Chow, S. Y. Lin, S. G. Johnson, P. R. Villeneuve, J. D. Joannopoulos, J. R. Wendt, G. A. Vawter, W. Zubrzycki, H. Hou, and A. Alleman, “Three-dimensional control of light in a two-dimensional photonic crystal slab,” Nature 407, 983–986 (2000).
    [Crossref]

2014 (1)

X. Zhu, W. Wang, W. Yan, M. B. Larsen, P. Bøggild, T. G. Pedersen, S. Xiao, J. Zi, and N. A. Mortensen, “Plasmon–phonon coupling in large-area graphene dot and antidot arrays fabricated by nanosphere lithography,” Nano Lett. 14, 2907–2913 (2014).
[Crossref]

2013 (4)

T. M. Slipchenko, M. L. Nesterov, L. Martin-Moreno, and A. Y. Nikitin, “Analytical solution for the diffraction of an electromagnetic wave by a graphene grating,” J. Opt. 15, 114008 (2013).
[Crossref]

X. Zhu, W. Yan, P. U. Jepsen, O. Hansen, N. A. Mortensen, and S. Xiao, “Experimental observation of plasmons in a graphene monolayer resting on a two-dimensional subwavelength silicon grating,” Appl. Phys. Lett. 102, 131101 (2013).
[Crossref]

N. Peres, Y. V. Bludov, A. Ferreira, and M. I. Vasilevskiy, “Exact solution for square-wave grating covered with graphene: surface plasmon-polaritons in the terahertz range,” J. Phys. 25, 125303 (2013).
[Crossref]

D. Bykov and L. Doskolovich, “Numerical methods for calculating poles of the scattering matrix with applications in grating theory,” J. Lightwave Technol. 31, 793–801 (2013).
[Crossref]

2012 (4)

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

A. Y. Nikitin, F. Guinea, and L. Martin-Moreno, “Resonant plasmonic effects in periodic graphene antidot arrays,” Appl. Phys. Lett. 101, 151119 (2012).
[Crossref]

W. Gao, J. Shu, C. Qiu, and Q. Xu, “Excitation of plasmonic waves in graphene by guided-mode resonances,” ACS Nano 6, 7806–7813 (2012).
[Crossref]

T. R. Zhan, F. Y. Zhao, X. H. Hu, X. H. Liu, and J. Zi, “Band structure of plasmons and optical absorption enhancement in graphene on subwavelength dielectric gratings at infrared frequencies,” Phys. Rev. B 86, 165416 (2012).
[Crossref]

2011 (4)

J. Horng, C.-F. Chen, B. Geng, C. Girit, Y. Zhang, Z. Hao, H. A. Bechtel, M. Martin, A. Zettl, M. F. Crommie, Y. R. Shen, and F. Wang, “Drude conductivity of Dirac fermions in graphene,” Phys. Rev. B 83, 165113 (2011).
[Crossref]

L. Ju, B. Geng, J. Horng, C. Girit, M. Martin, Z. Hao, H. A. Bechtel, X. Liang, A. Zettl, Y. R. Shen, and F. Wang, “Graphene plasmonics for tunable terahertz metamaterials,” Nat. Nanotech. 6, 630–634 (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, 3370–3377 (2011).
[Crossref]

T. Weiss, N. A. Gippius, S. G. Tikhodeev, G. Granet, and H. Giessen, “Derivation of plasmonic resonances in the fourier modal method with adaptive spatial resolution and matched coordinates,” J. Opt. Soc. Am. A 28, 238–244 (2011).
[Crossref]

2009 (1)

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

2004 (1)

K. Ohtaka, J. Inoue, and S. Yamaguti, “Derivation of the density of states of leaky photonic bands,” Phys. Rev. B 70, 035109 (2004).
[Crossref]

2002 (1)

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66, 045102 (2002).
[Crossref]

2001 (1)

T. Ochiai and K. Sakoda, “Dispersion relation and optical transmittance of a hexagonal photonic crystal slab,” Phys. Rev. B 63, 125107 (2001).
[Crossref]

2000 (1)

E. Chow, S. Y. Lin, S. G. Johnson, P. R. Villeneuve, J. D. Joannopoulos, J. R. Wendt, G. A. Vawter, W. Zubrzycki, H. Hou, and A. Alleman, “Three-dimensional control of light in a two-dimensional photonic crystal slab,” Nature 407, 983–986 (2000).
[Crossref]

1999 (1)

S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B 60, 5751–5758 (1999).
[Crossref]

1997 (1)

1994 (1)

Alleman, A.

E. Chow, S. Y. Lin, S. G. Johnson, P. R. Villeneuve, J. D. Joannopoulos, J. R. Wendt, G. A. Vawter, W. Zubrzycki, H. Hou, and A. Alleman, “Three-dimensional control of light in a two-dimensional photonic crystal slab,” Nature 407, 983–986 (2000).
[Crossref]

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, and F. Wang, “Graphene plasmonics for tunable terahertz metamaterials,” Nat. Nanotech. 6, 630–634 (2011).
[Crossref]

J. Horng, C.-F. Chen, B. Geng, C. Girit, Y. Zhang, Z. Hao, H. A. Bechtel, M. Martin, A. Zettl, M. F. Crommie, Y. R. Shen, and F. Wang, “Drude conductivity of Dirac fermions in graphene,” Phys. Rev. B 83, 165113 (2011).
[Crossref]

Bludov, Y. V.

N. Peres, Y. V. Bludov, A. Ferreira, and M. I. Vasilevskiy, “Exact solution for square-wave grating covered with graphene: surface plasmon-polaritons in the terahertz range,” J. Phys. 25, 125303 (2013).
[Crossref]

Bøggild, P.

X. Zhu, W. Wang, W. Yan, M. B. Larsen, P. Bøggild, T. G. Pedersen, S. Xiao, J. Zi, and N. A. Mortensen, “Plasmon–phonon coupling in large-area graphene dot and antidot arrays fabricated by nanosphere lithography,” Nano Lett. 14, 2907–2913 (2014).
[Crossref]

Buljan, H.

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

Bykov, D.

Chang, D. E.

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, 3370–3377 (2011).
[Crossref]

Chen, C.-F.

J. Horng, C.-F. Chen, B. Geng, C. Girit, Y. Zhang, Z. Hao, H. A. Bechtel, M. Martin, A. Zettl, M. F. Crommie, Y. R. Shen, and F. Wang, “Drude conductivity of Dirac fermions in graphene,” Phys. Rev. B 83, 165113 (2011).
[Crossref]

Chow, E.

E. Chow, S. Y. Lin, S. G. Johnson, P. R. Villeneuve, J. D. Joannopoulos, J. R. Wendt, G. A. Vawter, W. Zubrzycki, H. Hou, and A. Alleman, “Three-dimensional control of light in a two-dimensional photonic crystal slab,” Nature 407, 983–986 (2000).
[Crossref]

Crommie, M. F.

J. Horng, C.-F. Chen, B. Geng, C. Girit, Y. Zhang, Z. Hao, H. A. Bechtel, M. Martin, A. Zettl, M. F. Crommie, Y. R. Shen, and F. Wang, “Drude conductivity of Dirac fermions in graphene,” Phys. Rev. B 83, 165113 (2011).
[Crossref]

Doskolovich, L.

Fan, S.

S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B 60, 5751–5758 (1999).
[Crossref]

Ferreira, A.

N. Peres, Y. V. Bludov, A. Ferreira, and M. I. Vasilevskiy, “Exact solution for square-wave grating covered with graphene: surface plasmon-polaritons in the terahertz range,” J. Phys. 25, 125303 (2013).
[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, 7806–7813 (2012).
[Crossref]

García de Abajo, F. J.

S. Thongrattanasiri, F. H. L. Koppens, and F. J. García de Abajo, “Complete optical absorption in periodically patterned graphene,” Phys. Rev. Lett. 108, 047401 (2012).
[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, 3370–3377 (2011).
[Crossref]

Geng, B.

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

J. Horng, C.-F. Chen, B. Geng, C. Girit, Y. Zhang, Z. Hao, H. A. Bechtel, M. Martin, A. Zettl, M. F. Crommie, Y. R. Shen, and F. Wang, “Drude conductivity of Dirac fermions in graphene,” Phys. Rev. B 83, 165113 (2011).
[Crossref]

Giessen, H.

Gippius, N. A.

T. Weiss, N. A. Gippius, S. G. Tikhodeev, G. Granet, and H. Giessen, “Derivation of plasmonic resonances in the fourier modal method with adaptive spatial resolution and matched coordinates,” J. Opt. Soc. Am. A 28, 238–244 (2011).
[Crossref]

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66, 045102 (2002).
[Crossref]

Girit, C.

J. Horng, C.-F. Chen, B. Geng, C. Girit, Y. Zhang, Z. Hao, H. A. Bechtel, M. Martin, A. Zettl, M. F. Crommie, Y. R. Shen, and F. Wang, “Drude conductivity of Dirac fermions in graphene,” Phys. Rev. B 83, 165113 (2011).
[Crossref]

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

Granet, G.

Guinea, F.

A. Y. Nikitin, F. Guinea, and L. Martin-Moreno, “Resonant plasmonic effects in periodic graphene antidot arrays,” Appl. Phys. Lett. 101, 151119 (2012).
[Crossref]

Hansen, O.

X. Zhu, W. Yan, P. U. Jepsen, O. Hansen, N. A. Mortensen, and S. Xiao, “Experimental observation of plasmons in a graphene monolayer resting on a two-dimensional subwavelength silicon grating,” Appl. Phys. Lett. 102, 131101 (2013).
[Crossref]

Hao, Z.

J. Horng, C.-F. Chen, B. Geng, C. Girit, Y. Zhang, Z. Hao, H. A. Bechtel, M. Martin, A. Zettl, M. F. Crommie, Y. R. Shen, and F. Wang, “Drude conductivity of Dirac fermions in graphene,” Phys. Rev. B 83, 165113 (2011).
[Crossref]

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

Horng, J.

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

J. Horng, C.-F. Chen, B. Geng, C. Girit, Y. Zhang, Z. Hao, H. A. Bechtel, M. Martin, A. Zettl, M. F. Crommie, Y. R. Shen, and F. Wang, “Drude conductivity of Dirac fermions in graphene,” Phys. Rev. B 83, 165113 (2011).
[Crossref]

Hou, H.

E. Chow, S. Y. Lin, S. G. Johnson, P. R. Villeneuve, J. D. Joannopoulos, J. R. Wendt, G. A. Vawter, W. Zubrzycki, H. Hou, and A. Alleman, “Three-dimensional control of light in a two-dimensional photonic crystal slab,” Nature 407, 983–986 (2000).
[Crossref]

Hu, X. H.

T. R. Zhan, F. Y. Zhao, X. H. Hu, X. H. Liu, and J. Zi, “Band structure of plasmons and optical absorption enhancement in graphene on subwavelength dielectric gratings at infrared frequencies,” Phys. Rev. B 86, 165416 (2012).
[Crossref]

Inoue, J.

K. Ohtaka, J. Inoue, and S. Yamaguti, “Derivation of the density of states of leaky photonic bands,” Phys. Rev. B 70, 035109 (2004).
[Crossref]

Ishihara, T.

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66, 045102 (2002).
[Crossref]

Jablan, M.

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

Jepsen, P. U.

X. Zhu, W. Yan, P. U. Jepsen, O. Hansen, N. A. Mortensen, and S. Xiao, “Experimental observation of plasmons in a graphene monolayer resting on a two-dimensional subwavelength silicon grating,” Appl. Phys. Lett. 102, 131101 (2013).
[Crossref]

Joannopoulos, J. D.

E. Chow, S. Y. Lin, S. G. Johnson, P. R. Villeneuve, J. D. Joannopoulos, J. R. Wendt, G. A. Vawter, W. Zubrzycki, H. Hou, and A. Alleman, “Three-dimensional control of light in a two-dimensional photonic crystal slab,” Nature 407, 983–986 (2000).
[Crossref]

S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B 60, 5751–5758 (1999).
[Crossref]

Johnson, S. G.

E. Chow, S. Y. Lin, S. G. Johnson, P. R. Villeneuve, J. D. Joannopoulos, J. R. Wendt, G. A. Vawter, W. Zubrzycki, H. Hou, and A. Alleman, “Three-dimensional control of light in a two-dimensional photonic crystal slab,” Nature 407, 983–986 (2000).
[Crossref]

S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B 60, 5751–5758 (1999).
[Crossref]

Ju, L.

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

Kolodziejski, L. A.

S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B 60, 5751–5758 (1999).
[Crossref]

Koppens, F. H. L.

S. Thongrattanasiri, F. H. L. Koppens, and F. J. García de Abajo, “Complete optical absorption in periodically patterned graphene,” Phys. Rev. Lett. 108, 047401 (2012).
[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, 3370–3377 (2011).
[Crossref]

Larsen, M. B.

X. Zhu, W. Wang, W. Yan, M. B. Larsen, P. Bøggild, T. G. Pedersen, S. Xiao, J. Zi, and N. A. Mortensen, “Plasmon–phonon coupling in large-area graphene dot and antidot arrays fabricated by nanosphere lithography,” Nano Lett. 14, 2907–2913 (2014).
[Crossref]

Li, L.

Liang, X.

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

Lin, S. Y.

E. Chow, S. Y. Lin, S. G. Johnson, P. R. Villeneuve, J. D. Joannopoulos, J. R. Wendt, G. A. Vawter, W. Zubrzycki, H. Hou, and A. Alleman, “Three-dimensional control of light in a two-dimensional photonic crystal slab,” Nature 407, 983–986 (2000).
[Crossref]

Liu, X. H.

T. R. Zhan, F. Y. Zhao, X. H. Hu, X. H. Liu, and J. Zi, “Band structure of plasmons and optical absorption enhancement in graphene on subwavelength dielectric gratings at infrared frequencies,” Phys. Rev. B 86, 165416 (2012).
[Crossref]

Martin, M.

J. Horng, C.-F. Chen, B. Geng, C. Girit, Y. Zhang, Z. Hao, H. A. Bechtel, M. Martin, A. Zettl, M. F. Crommie, Y. R. Shen, and F. Wang, “Drude conductivity of Dirac fermions in graphene,” Phys. Rev. B 83, 165113 (2011).
[Crossref]

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

Martin-Moreno, L.

T. M. Slipchenko, M. L. Nesterov, L. Martin-Moreno, and A. Y. Nikitin, “Analytical solution for the diffraction of an electromagnetic wave by a graphene grating,” J. Opt. 15, 114008 (2013).
[Crossref]

A. Y. Nikitin, F. Guinea, and L. Martin-Moreno, “Resonant plasmonic effects in periodic graphene antidot arrays,” Appl. Phys. Lett. 101, 151119 (2012).
[Crossref]

Mortensen, N. A.

X. Zhu, W. Wang, W. Yan, M. B. Larsen, P. Bøggild, T. G. Pedersen, S. Xiao, J. Zi, and N. A. Mortensen, “Plasmon–phonon coupling in large-area graphene dot and antidot arrays fabricated by nanosphere lithography,” Nano Lett. 14, 2907–2913 (2014).
[Crossref]

X. Zhu, W. Yan, P. U. Jepsen, O. Hansen, N. A. Mortensen, and S. Xiao, “Experimental observation of plasmons in a graphene monolayer resting on a two-dimensional subwavelength silicon grating,” Appl. Phys. Lett. 102, 131101 (2013).
[Crossref]

Muljarov, E. A.

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66, 045102 (2002).
[Crossref]

Nesterov, M. L.

T. M. Slipchenko, M. L. Nesterov, L. Martin-Moreno, and A. Y. Nikitin, “Analytical solution for the diffraction of an electromagnetic wave by a graphene grating,” J. Opt. 15, 114008 (2013).
[Crossref]

Nikitin, A. Y.

T. M. Slipchenko, M. L. Nesterov, L. Martin-Moreno, and A. Y. Nikitin, “Analytical solution for the diffraction of an electromagnetic wave by a graphene grating,” J. Opt. 15, 114008 (2013).
[Crossref]

A. Y. Nikitin, F. Guinea, and L. Martin-Moreno, “Resonant plasmonic effects in periodic graphene antidot arrays,” Appl. Phys. Lett. 101, 151119 (2012).
[Crossref]

Noponen, E.

Ochiai, T.

T. Ochiai and K. Sakoda, “Dispersion relation and optical transmittance of a hexagonal photonic crystal slab,” Phys. Rev. B 63, 125107 (2001).
[Crossref]

Ohtaka, K.

K. Ohtaka, J. Inoue, and S. Yamaguti, “Derivation of the density of states of leaky photonic bands,” Phys. Rev. B 70, 035109 (2004).
[Crossref]

Pedersen, T. G.

X. Zhu, W. Wang, W. Yan, M. B. Larsen, P. Bøggild, T. G. Pedersen, S. Xiao, J. Zi, and N. A. Mortensen, “Plasmon–phonon coupling in large-area graphene dot and antidot arrays fabricated by nanosphere lithography,” Nano Lett. 14, 2907–2913 (2014).
[Crossref]

Pendry, J. B.

J. B. Pendry, Low Energy Electron Diffraction (Academic, 1974).

Peres, N.

N. Peres, Y. V. Bludov, A. Ferreira, and M. I. Vasilevskiy, “Exact solution for square-wave grating covered with graphene: surface plasmon-polaritons in the terahertz range,” J. Phys. 25, 125303 (2013).
[Crossref]

Qiu, C.

W. Gao, J. Shu, C. Qiu, and Q. Xu, “Excitation of plasmonic waves in graphene by guided-mode resonances,” ACS Nano 6, 7806–7813 (2012).
[Crossref]

Sakoda, K.

T. Ochiai and K. Sakoda, “Dispersion relation and optical transmittance of a hexagonal photonic crystal slab,” Phys. Rev. B 63, 125107 (2001).
[Crossref]

Shen, Y. R.

J. Horng, C.-F. Chen, B. Geng, C. Girit, Y. Zhang, Z. Hao, H. A. Bechtel, M. Martin, A. Zettl, M. F. Crommie, Y. R. Shen, and F. Wang, “Drude conductivity of Dirac fermions in graphene,” Phys. Rev. B 83, 165113 (2011).
[Crossref]

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

Shu, J.

W. Gao, J. Shu, C. Qiu, and Q. Xu, “Excitation of plasmonic waves in graphene by guided-mode resonances,” ACS Nano 6, 7806–7813 (2012).
[Crossref]

Slipchenko, T. M.

T. M. Slipchenko, M. L. Nesterov, L. Martin-Moreno, and A. Y. Nikitin, “Analytical solution for the diffraction of an electromagnetic wave by a graphene grating,” J. Opt. 15, 114008 (2013).
[Crossref]

Soljacic, M.

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

Thongrattanasiri, S.

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

Tikhodeev, S. G.

T. Weiss, N. A. Gippius, S. G. Tikhodeev, G. Granet, and H. Giessen, “Derivation of plasmonic resonances in the fourier modal method with adaptive spatial resolution and matched coordinates,” J. Opt. Soc. Am. A 28, 238–244 (2011).
[Crossref]

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66, 045102 (2002).
[Crossref]

Turunen, J.

Vasilevskiy, M. I.

N. Peres, Y. V. Bludov, A. Ferreira, and M. I. Vasilevskiy, “Exact solution for square-wave grating covered with graphene: surface plasmon-polaritons in the terahertz range,” J. Phys. 25, 125303 (2013).
[Crossref]

Vawter, G. A.

E. Chow, S. Y. Lin, S. G. Johnson, P. R. Villeneuve, J. D. Joannopoulos, J. R. Wendt, G. A. Vawter, W. Zubrzycki, H. Hou, and A. Alleman, “Three-dimensional control of light in a two-dimensional photonic crystal slab,” Nature 407, 983–986 (2000).
[Crossref]

Villeneuve, P. R.

E. Chow, S. Y. Lin, S. G. Johnson, P. R. Villeneuve, J. D. Joannopoulos, J. R. Wendt, G. A. Vawter, W. Zubrzycki, H. Hou, and A. Alleman, “Three-dimensional control of light in a two-dimensional photonic crystal slab,” Nature 407, 983–986 (2000).
[Crossref]

S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B 60, 5751–5758 (1999).
[Crossref]

Wang, F.

J. Horng, C.-F. Chen, B. Geng, C. Girit, Y. Zhang, Z. Hao, H. A. Bechtel, M. Martin, A. Zettl, M. F. Crommie, Y. R. Shen, and F. Wang, “Drude conductivity of Dirac fermions in graphene,” Phys. Rev. B 83, 165113 (2011).
[Crossref]

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

Wang, W.

X. Zhu, W. Wang, W. Yan, M. B. Larsen, P. Bøggild, T. G. Pedersen, S. Xiao, J. Zi, and N. A. Mortensen, “Plasmon–phonon coupling in large-area graphene dot and antidot arrays fabricated by nanosphere lithography,” Nano Lett. 14, 2907–2913 (2014).
[Crossref]

Weiss, T.

Wendt, J. R.

E. Chow, S. Y. Lin, S. G. Johnson, P. R. Villeneuve, J. D. Joannopoulos, J. R. Wendt, G. A. Vawter, W. Zubrzycki, H. Hou, and A. Alleman, “Three-dimensional control of light in a two-dimensional photonic crystal slab,” Nature 407, 983–986 (2000).
[Crossref]

Xiao, S.

X. Zhu, W. Wang, W. Yan, M. B. Larsen, P. Bøggild, T. G. Pedersen, S. Xiao, J. Zi, and N. A. Mortensen, “Plasmon–phonon coupling in large-area graphene dot and antidot arrays fabricated by nanosphere lithography,” Nano Lett. 14, 2907–2913 (2014).
[Crossref]

X. Zhu, W. Yan, P. U. Jepsen, O. Hansen, N. A. Mortensen, and S. Xiao, “Experimental observation of plasmons in a graphene monolayer resting on a two-dimensional subwavelength silicon grating,” Appl. Phys. Lett. 102, 131101 (2013).
[Crossref]

Xu, Q.

W. Gao, J. Shu, C. Qiu, and Q. Xu, “Excitation of plasmonic waves in graphene by guided-mode resonances,” ACS Nano 6, 7806–7813 (2012).
[Crossref]

Yablonskii, A. L.

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66, 045102 (2002).
[Crossref]

Yamaguti, S.

K. Ohtaka, J. Inoue, and S. Yamaguti, “Derivation of the density of states of leaky photonic bands,” Phys. Rev. B 70, 035109 (2004).
[Crossref]

Yan, W.

X. Zhu, W. Wang, W. Yan, M. B. Larsen, P. Bøggild, T. G. Pedersen, S. Xiao, J. Zi, and N. A. Mortensen, “Plasmon–phonon coupling in large-area graphene dot and antidot arrays fabricated by nanosphere lithography,” Nano Lett. 14, 2907–2913 (2014).
[Crossref]

X. Zhu, W. Yan, P. U. Jepsen, O. Hansen, N. A. Mortensen, and S. Xiao, “Experimental observation of plasmons in a graphene monolayer resting on a two-dimensional subwavelength silicon grating,” Appl. Phys. Lett. 102, 131101 (2013).
[Crossref]

Zettl, A.

J. Horng, C.-F. Chen, B. Geng, C. Girit, Y. Zhang, Z. Hao, H. A. Bechtel, M. Martin, A. Zettl, M. F. Crommie, Y. R. Shen, and F. Wang, “Drude conductivity of Dirac fermions in graphene,” Phys. Rev. B 83, 165113 (2011).
[Crossref]

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

Zhan, T. R.

T. R. Zhan, F. Y. Zhao, X. H. Hu, X. H. Liu, and J. Zi, “Band structure of plasmons and optical absorption enhancement in graphene on subwavelength dielectric gratings at infrared frequencies,” Phys. Rev. B 86, 165416 (2012).
[Crossref]

Zhang, Y.

J. Horng, C.-F. Chen, B. Geng, C. Girit, Y. Zhang, Z. Hao, H. A. Bechtel, M. Martin, A. Zettl, M. F. Crommie, Y. R. Shen, and F. Wang, “Drude conductivity of Dirac fermions in graphene,” Phys. Rev. B 83, 165113 (2011).
[Crossref]

Zhao, F. Y.

T. R. Zhan, F. Y. Zhao, X. H. Hu, X. H. Liu, and J. Zi, “Band structure of plasmons and optical absorption enhancement in graphene on subwavelength dielectric gratings at infrared frequencies,” Phys. Rev. B 86, 165416 (2012).
[Crossref]

Zhu, X.

X. Zhu, W. Wang, W. Yan, M. B. Larsen, P. Bøggild, T. G. Pedersen, S. Xiao, J. Zi, and N. A. Mortensen, “Plasmon–phonon coupling in large-area graphene dot and antidot arrays fabricated by nanosphere lithography,” Nano Lett. 14, 2907–2913 (2014).
[Crossref]

X. Zhu, W. Yan, P. U. Jepsen, O. Hansen, N. A. Mortensen, and S. Xiao, “Experimental observation of plasmons in a graphene monolayer resting on a two-dimensional subwavelength silicon grating,” Appl. Phys. Lett. 102, 131101 (2013).
[Crossref]

Zi, J.

X. Zhu, W. Wang, W. Yan, M. B. Larsen, P. Bøggild, T. G. Pedersen, S. Xiao, J. Zi, and N. A. Mortensen, “Plasmon–phonon coupling in large-area graphene dot and antidot arrays fabricated by nanosphere lithography,” Nano Lett. 14, 2907–2913 (2014).
[Crossref]

T. R. Zhan, F. Y. Zhao, X. H. Hu, X. H. Liu, and J. Zi, “Band structure of plasmons and optical absorption enhancement in graphene on subwavelength dielectric gratings at infrared frequencies,” Phys. Rev. B 86, 165416 (2012).
[Crossref]

Zubrzycki, W.

E. Chow, S. Y. Lin, S. G. Johnson, P. R. Villeneuve, J. D. Joannopoulos, J. R. Wendt, G. A. Vawter, W. Zubrzycki, H. Hou, and A. Alleman, “Three-dimensional control of light in a two-dimensional photonic crystal slab,” Nature 407, 983–986 (2000).
[Crossref]

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, 7806–7813 (2012).
[Crossref]

Appl. Phys. Lett. (2)

X. Zhu, W. Yan, P. U. Jepsen, O. Hansen, N. A. Mortensen, and S. Xiao, “Experimental observation of plasmons in a graphene monolayer resting on a two-dimensional subwavelength silicon grating,” Appl. Phys. Lett. 102, 131101 (2013).
[Crossref]

A. Y. Nikitin, F. Guinea, and L. Martin-Moreno, “Resonant plasmonic effects in periodic graphene antidot arrays,” Appl. Phys. Lett. 101, 151119 (2012).
[Crossref]

J. Lightwave Technol. (1)

J. Opt. (1)

T. M. Slipchenko, M. L. Nesterov, L. Martin-Moreno, and A. Y. Nikitin, “Analytical solution for the diffraction of an electromagnetic wave by a graphene grating,” J. Opt. 15, 114008 (2013).
[Crossref]

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

J. Phys. (1)

N. Peres, Y. V. Bludov, A. Ferreira, and M. I. Vasilevskiy, “Exact solution for square-wave grating covered with graphene: surface plasmon-polaritons in the terahertz range,” J. Phys. 25, 125303 (2013).
[Crossref]

Nano Lett. (2)

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, 3370–3377 (2011).
[Crossref]

X. Zhu, W. Wang, W. Yan, M. B. Larsen, P. Bøggild, T. G. Pedersen, S. Xiao, J. Zi, and N. A. Mortensen, “Plasmon–phonon coupling in large-area graphene dot and antidot arrays fabricated by nanosphere lithography,” Nano Lett. 14, 2907–2913 (2014).
[Crossref]

Nat. Nanotech. (1)

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

Nature (1)

E. Chow, S. Y. Lin, S. G. Johnson, P. R. Villeneuve, J. D. Joannopoulos, J. R. Wendt, G. A. Vawter, W. Zubrzycki, H. Hou, and A. Alleman, “Three-dimensional control of light in a two-dimensional photonic crystal slab,” Nature 407, 983–986 (2000).
[Crossref]

Phys. Rev. B (7)

J. Horng, C.-F. Chen, B. Geng, C. Girit, Y. Zhang, Z. Hao, H. A. Bechtel, M. Martin, A. Zettl, M. F. Crommie, Y. R. Shen, and F. Wang, “Drude conductivity of Dirac fermions in graphene,” Phys. Rev. B 83, 165113 (2011).
[Crossref]

T. R. Zhan, F. Y. Zhao, X. H. Hu, X. H. Liu, and J. Zi, “Band structure of plasmons and optical absorption enhancement in graphene on subwavelength dielectric gratings at infrared frequencies,” Phys. Rev. B 86, 165416 (2012).
[Crossref]

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66, 045102 (2002).
[Crossref]

T. Ochiai and K. Sakoda, “Dispersion relation and optical transmittance of a hexagonal photonic crystal slab,” Phys. Rev. B 63, 125107 (2001).
[Crossref]

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

K. Ohtaka, J. Inoue, and S. Yamaguti, “Derivation of the density of states of leaky photonic bands,” Phys. Rev. B 70, 035109 (2004).
[Crossref]

S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B 60, 5751–5758 (1999).
[Crossref]

Phys. Rev. Lett. (1)

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

Other (2)

Here, we assume the period of the grating is on the order of several hundreds of nanometers, so that it is much larger than the atomic period. The Dirac spectrum of graphene is assumed to be unchanged by the grating.

J. B. Pendry, Low Energy Electron Diffraction (Academic, 1974).

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Figures (8)

Fig. 1.
Fig. 1. Schematic illustration of the system under study. The doped (or gated) graphene sheet is placed above a diffraction grating. The electrostatic potential induced by the excess charge in the graphene is modulated by the grating, showing the self-consistent charge-density distribution in the graphene.
Fig. 2.
Fig. 2. Schematic illustration of the Fermi energy and charge-density modulation.
Fig. 3.
Fig. 3. Electrostatic potential ψ in the x z plane ( y = 0 ) for a graphene-grating system. The grating consists of the triangular array of circular rods with radius r a = 0.3 a and permittivity ϵ a = 12 embedded in air ( ϵ b = 1 ). The grating has a thickness of t = 0.6 a , supported by a semi-infinite high-index substrate with permittivity ϵ l = 12 . The region above the grating is air ( ϵ u = 1 ). The lattice constant a is taken to be 1 μm. The doped graphene, whose average Fermi energy is 0.4 eV, is placed above the grating with distance d = 0.01 a . The lower panel shows the cross-sectional profiles at ( x , y ) = ( 0,0 ) (inside a rod) and ( x , y ) = ( 0.5 a , 0 ) (outside the rods). The potential at the graphene sheet is taken to be zero.
Fig. 4.
Fig. 4. Charge density ρ 2 d in the graphene sheet, induced by the diffraction grating. The charge density is normalized by | ρ 0 | , the absolute value of the average charge density. The same parameters as in Fig. 3 are employed.
Fig. 5.
Fig. 5. Maximum and minimum charge densities in the graphene sheet, as a function of the distance d between the graphene and grating. The same parameters as in Fig. 3, except for d (distance between the graphene and grating), are employed.
Fig. 6.
Fig. 6. Photonic band structure of the graphene-grating system. The same parameters as in Fig. 3 are employed. The solid and dashed lines are the light lines of air and substrate, respectively. The photonic band structure is evaluated in the dissipation-less limit τ . For comparison, the imaginary part in the eigenfrequency, caused by finite τ of 4 × 10 13 s , is also indicated by the error bars, for some representative points.
Fig. 7.
Fig. 7. Gap map of the photonic band structure in the graphene-grating system, as a function of rod radius r a . The same parameters as in Fig. 3, except for r a , are employed.
Fig. 8.
Fig. 8. Photonic band structure of the doped graphene on a diffraction grating of a hole-type honeycomb lattice. The following parameters are employed: ϵ a = 1 , ϵ b = 12 , ϵ u = 1 , ϵ l = 12 , r a = 0.28 a , t = 0.6 a , d = 0.01 a , E F = 0.4 eV , and a = 1 μm . The solid and dashed lines are the light lines of air and substrate, respectively.

Equations (43)

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ϵ ( x ) = g e i g · x ϵ g ,
ϵ g = ϵ b δ g , 0 + ( ϵ a ϵ b ) 2 f J 1 ( | g | r a ) | g | r a ,
· [ ϵ ( x ) ψ ( x ) ] = 0 .
ψ ( x ) = α s + β s z + α 0 g e i g · x ψ g α ( c α e κ α z + d α e κ α z ) ,
g ϵ g g g · g ψ g α = κ α g ϵ g g ψ g α .
ψ ( x ) = α + β z + g 0 e i g · x ( a g e | g | z + b g e | g | z ) .
ψ ( x ) = { ρ 0 2 ϵ 0 ϵ u ( z z c ) ( z > z c ) ρ 0 2 ϵ 0 ϵ u ( z z c ) + g 0 e i g · x a g u ( e | g | ( z t 2 ) e 2 | g | ( z c t 2 ) e | g | ( z t 2 ) ) ( t 2 < z < z c ) α s + β s z + α 0 g e i g · x ψ g α ( c α e κ α z + d α e κ α z ) ( t 2 < z < t 2 ) α l + β l ( z + t 2 ) + g 0 e i g · x b g l e | g | ( z + t 2 ) ( z < t 2 ) .
ρ 2 d ( x ) = ρ 0 + g 0 e i g · x ρ g ,
ρ g = 2 ϵ 0 ϵ u | g | a g u e | g | ( z c t 2 ) .
σ ( ω ) = i e 2 h ω c ω + i τ 1 ,
E F = v F π n ,
σ ( x ; ω ) = g σ g ( ω ) e i g · x .
( a g σ + a g σ ) = g σ = p , s ( S g g σ σ ++ S gg σ σ + S gg σ σ + S gg σ σ -- ) ( a g σ + a g σ ) ,
E ( x ) = { g ( a g p + p g + + a g s + s g ) e i K g + · x + ( a g p p g + a g s s g ) e i K g · x ( z > 0 ) g ( a g p + p g + + a g s + s g ) e i K g + · x + ( a g p p g + a g s s g ) e i K g · x ( z < 0 )
p g ± = ± γ g q ( k ^ g ) | k g | q z ^ , s g = ( k ^ g ) ,
K g ± = k g ± γ g z ^ , γ g = q 2 k g 2 ,
q = ω c ϵ u , k g = k + g ,
( k ^ ) = ( k x | k | , k y | k | ) , ( k ^ ) = ( k y | k | , k x | k | ) .
Δ E = 0 ,
Δ H = ( j y 2 d , j x 2 d ) ,
Δ D z = ρ 2 d ,
Δ B z = 0 ,
j i 2 d ( x ) = σ ( x ; ω ) E i ( x , 0 ) ( i = x , y ) ,
S ++ = A 1 B , S + = A 1 C ,
S + = diag ( 1 , 1 ) diag ( 1 , 1 ) S ++ ,
S = diag ( 1,1 ) diag ( 1 , 1 ) S + ,
A gg ; p p = 2 q δ gg + μ 0 ω γ g q ( k ^ g ) ( k ^ g ) σ g g ,
A gg ; p s = μ 0 ω ( k ^ g ) ( k ^ g ) σ g g ,
A gg ; s p = μ 0 ω γ g q ( k ^ g ) ( k ^ g ) σ g g ,
A gg ; s s = 2 γ g δ gg + μ 0 ω ( k ^ g ) ( k ^ g ) σ g g ,
B gg ; p p = 2 q δ gg ,
B gg ; p s = 0 ,
B gg ; s p = 0 ,
B gg ; s s = 2 γ g δ gg ,
C gg ; p p = μ 0 ω γ g q ( k ^ g ) ( k ^ g ) σ g g ,
C gg ; p s = μ 0 ω ( k ^ g ) ( k ^ g ) σ g g ,
C gg ; s p = μ 0 ω γ g q ( k ^ g ) ( k ^ g ) σ g g ,
C gg ; s s = μ 0 ω ( k ^ g ) ( k ^ g ) σ g g .
2 q 2 q 2 k 2 + μ 0 ω σ ( ω ) = 0 .
ω = α e c | k | ω c ϵ u ,
det ( 1 S + grating Λ S + graphene ) = 0 ,
Λ = diag ( e 2 i γ g z c ) ,
ρ 0 = e π ( E F v F ) 2 .

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