Abstract

We investigate the features of additional waves that arise in the graphene layered medium, within the framework of nonlocal effective medium model. The additional wave is manifest on the biquadratic dispersion relation of the medium and represents as a distinctive nonlocal character at long wavelength. In particular, the reflection and transmission coefficients for the nonlocal medium are underdetermined by Maxwell’s boundary conditions. An additional boundary condition based on modal expansions is proposed to derive the generalized Fresnel equations, based on which the additional wave in the graphene layered medium is determined. The additional wave tends to be significant near the effective plasma frequency, near which the graphene plasmons are excited inside the medium.

© 2014 Optical Society of America

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References

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  2. K. Novoselov, A. K. Geim, S. Morozov, D. Jiang, M. K. I. Grigorieva, S. Dubonos, and A. Firsov, “Two-dimensional gas of massless Dirac fermions in graphene,” Nature 438, 197–200 (2005).
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  4. K. I. Bolotin, K. J. Sikes, Z. Jiang, M. Klima, G. Fudenberg, J. Hone, P. Kim, and H. L. Stormer, “Ultrahigh electron mobility in suspended graphene,” Solid State Commun. 146, 351–355 (2008).
    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
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2014 (1)

2013 (1)

T. Zhan, X. Shi, Y. Dai, X. Liu, and J. Zi, “Transfer matrix method for optics in graphene layers,” J. Phys.: Condens. Matter 25, 215301 (2013).

2010 (1)

R. E. V. Profumo, M. Polini, R. Asgari, R. Fazio, and A. H. MacDonald, “Electron-electron interactions in decoupled graphene layers,” Phys. Rev. B 82, 085443 (2010).
[Crossref]

2009 (1)

M. G. Silveirinha, “Additional boundary conditions for nonconnected wire media,” New J. Phys. 11, 113016 (2009).
[Crossref]

2008 (4)

F. Richter, M. Florian, and K. Henneberger, “Poynting’s theorem and energy conservation in the propagation of light in bounded media,” EPL 81, 67005 (2008).
[Crossref]

M. G. Silveirinha, C. A. Fernandes, and J. R. Costa, “Additional boundary condition for a wire medium connected to a metallic surface,” New J. Phys. 10, 053011 (2008).
[Crossref]

K. I. Bolotin, K. J. Sikes, Z. Jiang, M. Klima, G. Fudenberg, J. Hone, P. Kim, and H. L. Stormer, “Ultrahigh electron mobility in suspended graphene,” Solid State Commun. 146, 351–355 (2008).
[Crossref]

A. A. Balandin, S. Ghosh, W. Bao, I. Calizo, D. Teweldebrhan, F. Miao, and C. N. Lau, “Superior thermal conductivity of single-layer graphene,” Nano Lett. 8, 902–907 (2008).
[Crossref] [PubMed]

2007 (4)

L. A. Falkovsky and S. S. Pershoguba, “Optical far-infrared properties of a graphene monolayer and multilayer,” Phys. Rev. B 76, 153410 (2007).
[Crossref]

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

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

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

2006 (3)

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

V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, “Unusual microwave response of Dirac quasiparticles in graphene,” Phys. Rev. Lett. 96, 256802 (2006).
[Crossref] [PubMed]

M. G. Silveirinha, “Additional boundary condition for the wire medium,” IEEE Trans. Antennas Propag. 54, 1766–1780 (2006).
[Crossref]

2005 (2)

K. Novoselov, A. K. Geim, S. Morozov, D. Jiang, M. K. I. Grigorieva, S. Dubonos, and A. Firsov, “Two-dimensional gas of massless Dirac fermions in graphene,” Nature 438, 197–200 (2005).
[Crossref] [PubMed]

Y. Zhang, Y.-W. Tan, H. L. Stormer, and P. Kim, “Experimental observation of the quantum Hall effect and Berry’s phase in graphene,” Nature 438, 201–204 (2005).
[Crossref] [PubMed]

2004 (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, 666–669 (2004).
[Crossref] [PubMed]

2003 (2)

S. Foteinopoulou, E. N. Economou, and C. M. Soukoulis, “Refraction in media with a negative refractive index,” Phys. Rev. Lett. 90, 107402 (2003).
[Crossref] [PubMed]

P. A. Belov, “Backward waves and negative refraction in uniaxial dielectrics with negative dielectric permittivity along the anisotropy axis,” Microw. Opt. Technol. Lett. 37, 259–263 (2003).
[Crossref]

1998 (1)

K. Henneberger, “Additional boundary conditions: an historical mistake,” Phys. Rev. Lett. 80, 2889–2892 (1998).
[Crossref]

1988 (1)

W. A. Davis and C. M. Krowne, “The effects of drift and diffusion in semiconductors on plane wave interaction at interfaces,” IEEE Trans. Antennas Propag. 36, 97–103 (1988).
[Crossref]

1982 (1)

P. Sheng, R. S. Stepleman, and P. N. Sanda, “Exact eigenfunctions for square-wave gratings: application to diffraction and surface-plasmon calculations,” Phys. Rev. B 26, 2907–2916 (1982).
[Crossref]

1981 (1)

A. D. Boardman and R. Ruppin, “The boundary conditions between spatially dispersive media,” Surf. Sci. 112, 153–167 (1981).
[Crossref]

1975 (1)

C. S. Ting, M. J. Frankel, and J. L. Birman, “Electrodynamics of bounded spatially dispersive media: the additional boundary conditions,” Solid State Commun. 17, 1285–1289 (1975).
[Crossref]

1974 (1)

G. S. Agarwal, D. N. Pattanayak, and E. Wolf, “Electromagnetic fields in spatially dispersive media,” Phys. Rev. B 10, 1447–1475 (1974).
[Crossref]

1971 (1)

1969 (1)

G. D. Mahan and G. Obermair, “Polaritons at surfaces,” Phys. Rev. 183, 834–841 (1969).
[Crossref]

1963 (1)

J. J. Hopfield and D. G. Thomas, “Theoretical and experimental effects of spatial dispersion on the optical properties of crystals,” Phys. Rev. 132, 563–572 (1963).
[Crossref]

1958 (1)

S. I. Pekar, “Theory of electromagnetic waves in a crystal with excitons,” J. Phys. Chem. Solids 5, 11–22 (1958).
[Crossref]

Agarwal, G. S.

G. S. Agarwal, D. N. Pattanayak, and E. Wolf, “Electromagnetic fields in spatially dispersive media,” Phys. Rev. B 10, 1447–1475 (1974).
[Crossref]

Asgari, R.

R. E. V. Profumo, M. Polini, R. Asgari, R. Fazio, and A. H. MacDonald, “Electron-electron interactions in decoupled graphene layers,” Phys. Rev. B 82, 085443 (2010).
[Crossref]

Balandin, A. A.

A. A. Balandin, S. Ghosh, W. Bao, I. Calizo, D. Teweldebrhan, F. Miao, and C. N. Lau, “Superior thermal conductivity of single-layer graphene,” Nano Lett. 8, 902–907 (2008).
[Crossref] [PubMed]

Bao, W.

A. A. Balandin, S. Ghosh, W. Bao, I. Calizo, D. Teweldebrhan, F. Miao, and C. N. Lau, “Superior thermal conductivity of single-layer graphene,” Nano Lett. 8, 902–907 (2008).
[Crossref] [PubMed]

Belov, P. A.

P. A. Belov, “Backward waves and negative refraction in uniaxial dielectrics with negative dielectric permittivity along the anisotropy axis,” Microw. Opt. Technol. Lett. 37, 259–263 (2003).
[Crossref]

Birman, J. L.

C. S. Ting, M. J. Frankel, and J. L. Birman, “Electrodynamics of bounded spatially dispersive media: the additional boundary conditions,” Solid State Commun. 17, 1285–1289 (1975).
[Crossref]

Boardman, A. D.

A. D. Boardman and R. Ruppin, “The boundary conditions between spatially dispersive media,” Surf. Sci. 112, 153–167 (1981).
[Crossref]

Bolotin, K. I.

K. I. Bolotin, K. J. Sikes, Z. Jiang, M. Klima, G. Fudenberg, J. Hone, P. Kim, and H. L. Stormer, “Ultrahigh electron mobility in suspended graphene,” Solid State Commun. 146, 351–355 (2008).
[Crossref]

Calizo, I.

A. A. Balandin, S. Ghosh, W. Bao, I. Calizo, D. Teweldebrhan, F. Miao, and C. N. Lau, “Superior thermal conductivity of single-layer graphene,” Nano Lett. 8, 902–907 (2008).
[Crossref] [PubMed]

Carbotte, J. P.

V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, “Unusual microwave response of Dirac quasiparticles in graphene,” Phys. Rev. Lett. 96, 256802 (2006).
[Crossref] [PubMed]

Chern, R.-L.

Costa, J. R.

M. G. Silveirinha, C. A. Fernandes, and J. R. Costa, “Additional boundary condition for a wire medium connected to a metallic surface,” New J. Phys. 10, 053011 (2008).
[Crossref]

Dai, Y.

T. Zhan, X. Shi, Y. Dai, X. Liu, and J. Zi, “Transfer matrix method for optics in graphene layers,” J. Phys.: Condens. Matter 25, 215301 (2013).

Das Sarma, S.

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

Davis, W. A.

W. A. Davis and C. M. Krowne, “The effects of drift and diffusion in semiconductors on plane wave interaction at interfaces,” IEEE Trans. Antennas Propag. 36, 97–103 (1988).
[Crossref]

Dubonos, S.

K. Novoselov, A. K. Geim, S. Morozov, D. Jiang, M. K. I. Grigorieva, S. Dubonos, and A. Firsov, “Two-dimensional gas of massless Dirac fermions in graphene,” Nature 438, 197–200 (2005).
[Crossref] [PubMed]

Dubonos, S. 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, 666–669 (2004).
[Crossref] [PubMed]

Economou, E. N.

S. Foteinopoulou, E. N. Economou, and C. M. Soukoulis, “Refraction in media with a negative refractive index,” Phys. Rev. Lett. 90, 107402 (2003).
[Crossref] [PubMed]

Falkovsky, L. A.

L. A. Falkovsky and S. S. Pershoguba, “Optical far-infrared properties of a graphene monolayer and multilayer,” Phys. Rev. B 76, 153410 (2007).
[Crossref]

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

Fazio, R.

R. E. V. Profumo, M. Polini, R. Asgari, R. Fazio, and A. H. MacDonald, “Electron-electron interactions in decoupled graphene layers,” Phys. Rev. B 82, 085443 (2010).
[Crossref]

Fernandes, C. A.

M. G. Silveirinha, C. A. Fernandes, and J. R. Costa, “Additional boundary condition for a wire medium connected to a metallic surface,” New J. Phys. 10, 053011 (2008).
[Crossref]

Firsov, A.

K. Novoselov, A. K. Geim, S. Morozov, D. Jiang, M. K. I. Grigorieva, S. Dubonos, and A. Firsov, “Two-dimensional gas of massless Dirac fermions in graphene,” Nature 438, 197–200 (2005).
[Crossref] [PubMed]

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, 666–669 (2004).
[Crossref] [PubMed]

Florian, M.

F. Richter, M. Florian, and K. Henneberger, “Poynting’s theorem and energy conservation in the propagation of light in bounded media,” EPL 81, 67005 (2008).
[Crossref]

Foteinopoulou, S.

S. Foteinopoulou, E. N. Economou, and C. M. Soukoulis, “Refraction in media with a negative refractive index,” Phys. Rev. Lett. 90, 107402 (2003).
[Crossref] [PubMed]

Frankel, M. J.

C. S. Ting, M. J. Frankel, and J. L. Birman, “Electrodynamics of bounded spatially dispersive media: the additional boundary conditions,” Solid State Commun. 17, 1285–1289 (1975).
[Crossref]

Fudenberg, G.

K. I. Bolotin, K. J. Sikes, Z. Jiang, M. Klima, G. Fudenberg, J. Hone, P. Kim, and H. L. Stormer, “Ultrahigh electron mobility in suspended graphene,” Solid State Commun. 146, 351–355 (2008).
[Crossref]

Geim, A. K.

K. Novoselov, A. K. Geim, S. Morozov, D. Jiang, M. K. I. Grigorieva, S. Dubonos, and A. Firsov, “Two-dimensional gas of massless Dirac fermions in graphene,” Nature 438, 197–200 (2005).
[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, 666–669 (2004).
[Crossref] [PubMed]

Ghosh, S.

A. A. Balandin, S. Ghosh, W. Bao, I. Calizo, D. Teweldebrhan, F. Miao, and C. N. Lau, “Superior thermal conductivity of single-layer graphene,” Nano Lett. 8, 902–907 (2008).
[Crossref] [PubMed]

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, 666–669 (2004).
[Crossref] [PubMed]

Grigorieva, M. K. I.

K. Novoselov, A. K. Geim, S. Morozov, D. Jiang, M. K. I. Grigorieva, S. Dubonos, and A. Firsov, “Two-dimensional gas of massless Dirac fermions in graphene,” Nature 438, 197–200 (2005).
[Crossref] [PubMed]

Guinea, F.

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

Gusynin, V. P.

V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, “Unusual microwave response of Dirac quasiparticles in graphene,” Phys. Rev. Lett. 96, 256802 (2006).
[Crossref] [PubMed]

Han, D.

Henneberger, K.

F. Richter, M. Florian, and K. Henneberger, “Poynting’s theorem and energy conservation in the propagation of light in bounded media,” EPL 81, 67005 (2008).
[Crossref]

K. Henneberger, “Additional boundary conditions: an historical mistake,” Phys. Rev. Lett. 80, 2889–2892 (1998).
[Crossref]

Hone, J.

K. I. Bolotin, K. J. Sikes, Z. Jiang, M. Klima, G. Fudenberg, J. Hone, P. Kim, and H. L. Stormer, “Ultrahigh electron mobility in suspended graphene,” Solid State Commun. 146, 351–355 (2008).
[Crossref]

Hopfield, J. J.

J. J. Hopfield and D. G. Thomas, “Theoretical and experimental effects of spatial dispersion on the optical properties of crystals,” Phys. Rev. 132, 563–572 (1963).
[Crossref]

Horowitz, B. R.

Hwang, E. H.

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

Jiang, D.

K. Novoselov, A. K. Geim, S. Morozov, D. Jiang, M. K. I. Grigorieva, S. Dubonos, and A. Firsov, “Two-dimensional gas of massless Dirac fermions in graphene,” Nature 438, 197–200 (2005).
[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, 666–669 (2004).
[Crossref] [PubMed]

Jiang, Z.

K. I. Bolotin, K. J. Sikes, Z. Jiang, M. Klima, G. Fudenberg, J. Hone, P. Kim, and H. L. Stormer, “Ultrahigh electron mobility in suspended graphene,” Solid State Commun. 146, 351–355 (2008).
[Crossref]

Kim, P.

K. I. Bolotin, K. J. Sikes, Z. Jiang, M. Klima, G. Fudenberg, J. Hone, P. Kim, and H. L. Stormer, “Ultrahigh electron mobility in suspended graphene,” Solid State Commun. 146, 351–355 (2008).
[Crossref]

Y. Zhang, Y.-W. Tan, H. L. Stormer, and P. Kim, “Experimental observation of the quantum Hall effect and Berry’s phase in graphene,” Nature 438, 201–204 (2005).
[Crossref] [PubMed]

Klima, M.

K. I. Bolotin, K. J. Sikes, Z. Jiang, M. Klima, G. Fudenberg, J. Hone, P. Kim, and H. L. Stormer, “Ultrahigh electron mobility in suspended graphene,” Solid State Commun. 146, 351–355 (2008).
[Crossref]

Krowne, C. M.

W. A. Davis and C. M. Krowne, “The effects of drift and diffusion in semiconductors on plane wave interaction at interfaces,” IEEE Trans. Antennas Propag. 36, 97–103 (1988).
[Crossref]

Lau, C. N.

A. A. Balandin, S. Ghosh, W. Bao, I. Calizo, D. Teweldebrhan, F. Miao, and C. N. Lau, “Superior thermal conductivity of single-layer graphene,” Nano Lett. 8, 902–907 (2008).
[Crossref] [PubMed]

Liu, X.

T. Zhan, X. Shi, Y. Dai, X. Liu, and J. Zi, “Transfer matrix method for optics in graphene layers,” J. Phys.: Condens. Matter 25, 215301 (2013).

MacDonald, A. H.

R. E. V. Profumo, M. Polini, R. Asgari, R. Fazio, and A. H. MacDonald, “Electron-electron interactions in decoupled graphene layers,” Phys. Rev. B 82, 085443 (2010).
[Crossref]

Mahan, G. D.

G. D. Mahan and G. Obermair, “Polaritons at surfaces,” Phys. Rev. 183, 834–841 (1969).
[Crossref]

Miao, F.

A. A. Balandin, S. Ghosh, W. Bao, I. Calizo, D. Teweldebrhan, F. Miao, and C. N. Lau, “Superior thermal conductivity of single-layer graphene,” Nano Lett. 8, 902–907 (2008).
[Crossref] [PubMed]

Mikhailov, S. A.

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

Morozov, S.

K. Novoselov, A. K. Geim, S. Morozov, D. Jiang, M. K. I. Grigorieva, S. Dubonos, and A. Firsov, “Two-dimensional gas of massless Dirac fermions in graphene,” Nature 438, 197–200 (2005).
[Crossref] [PubMed]

Morozov, S. 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, 666–669 (2004).
[Crossref] [PubMed]

Novoselov, K.

K. Novoselov, A. K. Geim, S. Morozov, D. Jiang, M. K. I. Grigorieva, S. Dubonos, and A. Firsov, “Two-dimensional gas of massless Dirac fermions in graphene,” Nature 438, 197–200 (2005).
[Crossref] [PubMed]

Novoselov, K. S.

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, 666–669 (2004).
[Crossref] [PubMed]

Obermair, G.

G. D. Mahan and G. Obermair, “Polaritons at surfaces,” Phys. Rev. 183, 834–841 (1969).
[Crossref]

Pattanayak, D. N.

G. S. Agarwal, D. N. Pattanayak, and E. Wolf, “Electromagnetic fields in spatially dispersive media,” Phys. Rev. B 10, 1447–1475 (1974).
[Crossref]

Pekar, S. I.

S. I. Pekar, “Theory of electromagnetic waves in a crystal with excitons,” J. Phys. Chem. Solids 5, 11–22 (1958).
[Crossref]

Pershoguba, S. S.

L. A. Falkovsky and S. S. Pershoguba, “Optical far-infrared properties of a graphene monolayer and multilayer,” Phys. Rev. B 76, 153410 (2007).
[Crossref]

Polini, M.

R. E. V. Profumo, M. Polini, R. Asgari, R. Fazio, and A. H. MacDonald, “Electron-electron interactions in decoupled graphene layers,” Phys. Rev. B 82, 085443 (2010).
[Crossref]

Profumo, R. E. V.

R. E. V. Profumo, M. Polini, R. Asgari, R. Fazio, and A. H. MacDonald, “Electron-electron interactions in decoupled graphene layers,” Phys. Rev. B 82, 085443 (2010).
[Crossref]

Richter, F.

F. Richter, M. Florian, and K. Henneberger, “Poynting’s theorem and energy conservation in the propagation of light in bounded media,” EPL 81, 67005 (2008).
[Crossref]

Ruppin, R.

A. D. Boardman and R. Ruppin, “The boundary conditions between spatially dispersive media,” Surf. Sci. 112, 153–167 (1981).
[Crossref]

Sanda, P. N.

P. Sheng, R. S. Stepleman, and P. N. Sanda, “Exact eigenfunctions for square-wave gratings: application to diffraction and surface-plasmon calculations,” Phys. Rev. B 26, 2907–2916 (1982).
[Crossref]

Sharapov, S. G.

V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, “Unusual microwave response of Dirac quasiparticles in graphene,” Phys. Rev. Lett. 96, 256802 (2006).
[Crossref] [PubMed]

Sheng, P.

P. Sheng, R. S. Stepleman, and P. N. Sanda, “Exact eigenfunctions for square-wave gratings: application to diffraction and surface-plasmon calculations,” Phys. Rev. B 26, 2907–2916 (1982).
[Crossref]

Shi, X.

T. Zhan, X. Shi, Y. Dai, X. Liu, and J. Zi, “Transfer matrix method for optics in graphene layers,” J. Phys.: Condens. Matter 25, 215301 (2013).

Sikes, K. J.

K. I. Bolotin, K. J. Sikes, Z. Jiang, M. Klima, G. Fudenberg, J. Hone, P. Kim, and H. L. Stormer, “Ultrahigh electron mobility in suspended graphene,” Solid State Commun. 146, 351–355 (2008).
[Crossref]

Silveirinha, M. G.

M. G. Silveirinha, “Additional boundary conditions for nonconnected wire media,” New J. Phys. 11, 113016 (2009).
[Crossref]

M. G. Silveirinha, C. A. Fernandes, and J. R. Costa, “Additional boundary condition for a wire medium connected to a metallic surface,” New J. Phys. 10, 053011 (2008).
[Crossref]

M. G. Silveirinha, “Additional boundary condition for the wire medium,” IEEE Trans. Antennas Propag. 54, 1766–1780 (2006).
[Crossref]

Sols, F.

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

Soukoulis, C. M.

S. Foteinopoulou, E. N. Economou, and C. M. Soukoulis, “Refraction in media with a negative refractive index,” Phys. Rev. Lett. 90, 107402 (2003).
[Crossref] [PubMed]

Stauber, T.

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

Stepleman, R. S.

P. Sheng, R. S. Stepleman, and P. N. Sanda, “Exact eigenfunctions for square-wave gratings: application to diffraction and surface-plasmon calculations,” Phys. Rev. B 26, 2907–2916 (1982).
[Crossref]

Stormer, H. L.

K. I. Bolotin, K. J. Sikes, Z. Jiang, M. Klima, G. Fudenberg, J. Hone, P. Kim, and H. L. Stormer, “Ultrahigh electron mobility in suspended graphene,” Solid State Commun. 146, 351–355 (2008).
[Crossref]

Y. Zhang, Y.-W. Tan, H. L. Stormer, and P. Kim, “Experimental observation of the quantum Hall effect and Berry’s phase in graphene,” Nature 438, 201–204 (2005).
[Crossref] [PubMed]

Tamir, T.

Tan, Y.-W.

Y. Zhang, Y.-W. Tan, H. L. Stormer, and P. Kim, “Experimental observation of the quantum Hall effect and Berry’s phase in graphene,” Nature 438, 201–204 (2005).
[Crossref] [PubMed]

Teweldebrhan, D.

A. A. Balandin, S. Ghosh, W. Bao, I. Calizo, D. Teweldebrhan, F. Miao, and C. N. Lau, “Superior thermal conductivity of single-layer graphene,” Nano Lett. 8, 902–907 (2008).
[Crossref] [PubMed]

Thomas, D. G.

J. J. Hopfield and D. G. Thomas, “Theoretical and experimental effects of spatial dispersion on the optical properties of crystals,” Phys. Rev. 132, 563–572 (1963).
[Crossref]

Ting, C. S.

C. S. Ting, M. J. Frankel, and J. L. Birman, “Electrodynamics of bounded spatially dispersive media: the additional boundary conditions,” Solid State Commun. 17, 1285–1289 (1975).
[Crossref]

Varlamov, A. A.

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

Wolf, E.

G. S. Agarwal, D. N. Pattanayak, and E. Wolf, “Electromagnetic fields in spatially dispersive media,” Phys. Rev. B 10, 1447–1475 (1974).
[Crossref]

Wunsch, B.

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

Zhan, T.

T. Zhan, X. Shi, Y. Dai, X. Liu, and J. Zi, “Transfer matrix method for optics in graphene layers,” J. Phys.: Condens. Matter 25, 215301 (2013).

Zhang, Y.

Y. Zhang, Y.-W. Tan, H. L. Stormer, and P. Kim, “Experimental observation of the quantum Hall effect and Berry’s phase in graphene,” Nature 438, 201–204 (2005).
[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, 666–669 (2004).
[Crossref] [PubMed]

Zi, J.

T. Zhan, X. Shi, Y. Dai, X. Liu, and J. Zi, “Transfer matrix method for optics in graphene layers,” J. Phys.: Condens. Matter 25, 215301 (2013).

Ziegler, K.

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

EPL (1)

F. Richter, M. Florian, and K. Henneberger, “Poynting’s theorem and energy conservation in the propagation of light in bounded media,” EPL 81, 67005 (2008).
[Crossref]

Eur. Phys. J. B (1)

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

IEEE Trans. Antennas Propag. (2)

W. A. Davis and C. M. Krowne, “The effects of drift and diffusion in semiconductors on plane wave interaction at interfaces,” IEEE Trans. Antennas Propag. 36, 97–103 (1988).
[Crossref]

M. G. Silveirinha, “Additional boundary condition for the wire medium,” IEEE Trans. Antennas Propag. 54, 1766–1780 (2006).
[Crossref]

J. Opt. Soc. Am. (1)

J. Phys. Chem. Solids (1)

S. I. Pekar, “Theory of electromagnetic waves in a crystal with excitons,” J. Phys. Chem. Solids 5, 11–22 (1958).
[Crossref]

J. Phys.: Condens. Matter (1)

T. Zhan, X. Shi, Y. Dai, X. Liu, and J. Zi, “Transfer matrix method for optics in graphene layers,” J. Phys.: Condens. Matter 25, 215301 (2013).

Microw. Opt. Technol. Lett. (1)

P. A. Belov, “Backward waves and negative refraction in uniaxial dielectrics with negative dielectric permittivity along the anisotropy axis,” Microw. Opt. Technol. Lett. 37, 259–263 (2003).
[Crossref]

Nano Lett. (1)

A. A. Balandin, S. Ghosh, W. Bao, I. Calizo, D. Teweldebrhan, F. Miao, and C. N. Lau, “Superior thermal conductivity of single-layer graphene,” Nano Lett. 8, 902–907 (2008).
[Crossref] [PubMed]

Nature (2)

K. Novoselov, A. K. Geim, S. Morozov, D. Jiang, M. K. I. Grigorieva, S. Dubonos, and A. Firsov, “Two-dimensional gas of massless Dirac fermions in graphene,” Nature 438, 197–200 (2005).
[Crossref] [PubMed]

Y. Zhang, Y.-W. Tan, H. L. Stormer, and P. Kim, “Experimental observation of the quantum Hall effect and Berry’s phase in graphene,” Nature 438, 201–204 (2005).
[Crossref] [PubMed]

New J. Phys. (3)

M. G. Silveirinha, C. A. Fernandes, and J. R. Costa, “Additional boundary condition for a wire medium connected to a metallic surface,” New J. Phys. 10, 053011 (2008).
[Crossref]

M. G. Silveirinha, “Additional boundary conditions for nonconnected wire media,” New J. Phys. 11, 113016 (2009).
[Crossref]

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

Opt. Express (1)

Phys. Rev. (2)

J. J. Hopfield and D. G. Thomas, “Theoretical and experimental effects of spatial dispersion on the optical properties of crystals,” Phys. Rev. 132, 563–572 (1963).
[Crossref]

G. D. Mahan and G. Obermair, “Polaritons at surfaces,” Phys. Rev. 183, 834–841 (1969).
[Crossref]

Phys. Rev. B (5)

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

P. Sheng, R. S. Stepleman, and P. N. Sanda, “Exact eigenfunctions for square-wave gratings: application to diffraction and surface-plasmon calculations,” Phys. Rev. B 26, 2907–2916 (1982).
[Crossref]

G. S. Agarwal, D. N. Pattanayak, and E. Wolf, “Electromagnetic fields in spatially dispersive media,” Phys. Rev. B 10, 1447–1475 (1974).
[Crossref]

L. A. Falkovsky and S. S. Pershoguba, “Optical far-infrared properties of a graphene monolayer and multilayer,” Phys. Rev. B 76, 153410 (2007).
[Crossref]

R. E. V. Profumo, M. Polini, R. Asgari, R. Fazio, and A. H. MacDonald, “Electron-electron interactions in decoupled graphene layers,” Phys. Rev. B 82, 085443 (2010).
[Crossref]

Phys. Rev. Lett. (4)

S. Foteinopoulou, E. N. Economou, and C. M. Soukoulis, “Refraction in media with a negative refractive index,” Phys. Rev. Lett. 90, 107402 (2003).
[Crossref] [PubMed]

K. Henneberger, “Additional boundary conditions: an historical mistake,” Phys. Rev. Lett. 80, 2889–2892 (1998).
[Crossref]

V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, “Unusual microwave response of Dirac quasiparticles in graphene,” Phys. Rev. Lett. 96, 256802 (2006).
[Crossref] [PubMed]

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

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, 666–669 (2004).
[Crossref] [PubMed]

Solid State Commun. (2)

K. I. Bolotin, K. J. Sikes, Z. Jiang, M. Klima, G. Fudenberg, J. Hone, P. Kim, and H. L. Stormer, “Ultrahigh electron mobility in suspended graphene,” Solid State Commun. 146, 351–355 (2008).
[Crossref]

C. S. Ting, M. J. Frankel, and J. L. Birman, “Electrodynamics of bounded spatially dispersive media: the additional boundary conditions,” Solid State Commun. 17, 1285–1289 (1975).
[Crossref]

Surf. Sci. (1)

A. D. Boardman and R. Ruppin, “The boundary conditions between spatially dispersive media,” Surf. Sci. 112, 153–167 (1981).
[Crossref]

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Figures (5)

Fig. 1
Fig. 1 (a) Schematic diagram of the graphene layered medium: a periodic lattice of graphene layers with period a embedded in a background with dielectric constant ε. In this study, a = 2 h̄c/μ (≈ 989 nm) with μ = 0.4 eV and ε = 1.5 are used as the parameters. A small area of the graphene feature is shown on the top layer for illustration. (b) Schematic diagram of the incidence of a Gaussian beam onto the graphene layered medium.
Fig. 2
Fig. 2 (a) Equifrequency surface and (b) equifrequency curves at Ω = 0.06 and Ω = 0.1 of the TM-polarized dispersion relation for the graphene layered medium with the parameters in Fig. 1.
Fig. 3
Fig. 3 (a) Wave number Kz and (b) effective permittivity ε x eff as the functions of Ω at θ = 5° (Kx = K0 sinθ) for the graphene layered medium with the parameters in Fig. 1. P and PL modes correspond to photon and polariton modes, respectively. Dashed line in (b) stands for the permittivity of the background material.
Fig. 4
Fig. 4 (a) Fresnel coefficients and (b) ratio of t2/t1 at θ = 5° as the functions of Ω for the graphene layered medium with the parameters in Fig. 1.
Fig. 5
Fig. 5 (a) Dispersion curve on the wave vector domain at Ω = 0.0989 for the graphene layered medium with the parameters in Fig. 1. Black and gray contours are equifrequency curves for vacuum and the layered medium, respectively. Dashed lines indicate the continuity of Kx across the interface at θ = 5°. (b) Power intensity of a Gaussian beam incident from vacuum onto the graphene layered medium at the same Ω and θ. The intensity is normalized to have a maximum value of unity.

Equations (60)

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cos ( k x a ) = cos ( q a ) i σ q 2 ω ε 0 ε sin ( q a ) ,
σ ε 0 c = 4 α i Ω + π α [ θ ( Ω 2 ) + i π ln | Ω 2 Ω + 2 | ] ,
k x 2 ε z eff + k z 2 ε x eff = k 0 2 ,
ε z eff = ε z 0 γ 12 K 0 2 1 1 12 K x 2 , ε x eff = ε ( 1 γ 12 ε z 0 K 0 2 ) 1 γ 6 ε z 0 ( K 0 2 1 2 ε K z 2 ) ,
x 2 ( 1 ε x 2 ) a 2 + y 2 b 2 = 1 ,
1 cosh Q + η Q sinh Q = 0 ,
K z ε Ω a ˜ ,
K z [ ε ( Ω a ˜ ) 2 + 12 ( 1 2 η ) ] 1 / 2 .
Ω 0 2 ( 1 + ε a ˜ α ) 1 / 2 ,
K z [ ε ( Ω a ˜ ) 2 + 48 α ε a ˜ ( 1 Ω 0 2 1 Ω 2 ) ] 1 / 2 .
1 cos Q η Q sin Q = 0 ,
K z i [ ( 2 n π ) 2 ε ( Ω a ˜ ) 2 ] 1 / 2 ,
K z i { [ ( 2 n + 1 ) π 2 ( 2 n + 1 ) π η ] 2 ε ( Ω a ˜ ) 2 } 1 / 2 ,
cos K x cos Q + η Q sin Q = 0 ,
K x K 0 [ 4 α a ˜ ( 1 Ω 0 2 1 Ω 2 ) ] 1 / 2 .
θ c ArcSin [ 4 α a ˜ ( 1 Ω 0 2 1 Ω 2 ) ] 1 / 2 ,
1 + r = t 1 + t 2 ,
1 r = α 1 t 1 + α 2 t 2 ,
r = f 1 t 1 + f 2 t 2 ,
r = ρ 1 f 1 t 1 + ρ 2 f 1 t 2 ,
β 1 t 1 + β 2 t 2 = 0 ,
r = ( 1 α 1 ) β 2 ( 1 α 2 ) β 1 ( 1 + α 1 ) β 2 ( 1 + α 2 ) β 1 ,
t 1 = 2 β 2 ( 1 + α 1 ) β 2 ( 1 + α 2 ) β 1 ,
t 2 = 2 β 1 ( 1 + α 1 ) β 2 ( 1 + α 2 ) β 1 ,
H y = e i k 0 ( γ 0 x + 1 γ 0 2 z ) + m = R m e i k 0 ( γ m x + 1 γ m 2 z ) ,
H y = n T n X n ( x ) e i k z , n z ,
D + R = χ _ T ,
Π _ ( D R ) = Ω _ T ,
R = ( Ω _ 1 Π _ + χ _ 1 ) 1 ( Ω _ 1 Π _ χ _ 1 ) D ,
T = 2 ( Π _ 1 Ω _ + χ _ ) 1 D .
1 + R 0 = χ 01 T 1 + χ 02 T 2 ,
1 R 0 = ( Ω 01 / Π 00 ) T 1 + ( Ω 02 / Π 00 ) T 2 ,
R 1 = χ 11 T 1 + χ 12 T 2 ,
R 1 = ( Ω 1 / Π 11 ) T 1 + ( Ω 2 / Π 11 ) T 2 .
[ χ 11 + ( Ω 11 / Π 11 ) T 1 ] + [ χ 12 + ( Ω 12 / Π 11 ) ] T 2 = 0 .
β 1 h 0 , 1 T 1 + β 2 h 0 , 2 T 2 = 0 ,
β 1 t 1 + β 2 t 2 = 0 ,
r = 1 α 1 1 + α 1 , t 1 = 2 1 + α 1 , t 2 = 0 ,
1 + r = t 1 + t 2 ,
1 r = t 1 n 1 + t 2 n 2 ,
( n 1 2 ε ) t 1 + ( n 2 2 ε ) t 2 = 0 ,
r = n * + 1 n * + 1 , n * = n 1 n 2 ( n 1 + n 2 ) n 1 2 + n 1 n 2 + n 2 2 ε .
1 | r | 2 Re [ ( t 1 + t 2 ) * ( α 1 t 1 + α 2 t 2 ) ] = 0 ,
f ( x , z ) = ψ ( k x ) e i k x x + i k z z d k x ,
ψ ( k x ) = w 0 2 cos θ π exp [ w 0 2 4 cos 2 θ ( k x k 0 sin θ ) 2 i k x x 0 + i k z h ] ,
H i = h i ψ ( k x ) e i k x x + i k 0 z z d k x ,
E i = η 0 e i ψ ( k x ) e i k x x + i k 0 z z d k x ,
H r = h r r ( k x ) ψ ( k x ) e i k x x i k 0 z z d k x ,
E r = η 0 e r r ( k x ) ψ ( k x ) e i k x x i k 0 z z d k x ,
H t n = h t n t n ( k x ) ψ ( k x ) e i k x x + i k z n z d k x ,
E t n = η 0 e t n t n ( k x ) ψ ( k x ) e i k x x + i k z n z d k x ,
k z 1 [ ε k 0 2 ε γ a 2 ( 6 ε z 0 36 ( ε z 0 ) 2 12 γ k x 2 a 2 + γ k x 4 a 4 ) ] 1 / 2 ,
k z 2 [ ε k 0 2 ε γ a 2 ( 6 ε z 0 + 36 ( ε z 0 ) 2 12 γ k x 2 a 2 + γ k x 4 a 4 ) ] 1 / 2 .
X n ( x ) = { A n e i q x + B n e i q x , 0 x ξ < a C n e i q x + D n e i q x , a < ξ a x 0 ,
M = [ 1 + σ q ω ε 0 ε 1 σ q ω ε 0 ε 1 1 1 1 1 1 e i q ξ e i q ξ e i k x a e i q ( ξ a ) e i k x a e i q ( ξ a ) e i q ξ e i q ξ e i k x a e i q ( ξ a ) e i k x a e i q ( ξ a ) ] ,
cos ( k x a ) = cos ( q a ) i σ q 2 ω ε 0 ε sin ( q a ) .
A n = 2 e i Q ( e i ( Q + K x ) 1 ) ε K 0 e i K x ( e 2 i Q + e 2 i Q ξ ) Q σ 2 e 2 i Q ξ ( e i Q e i K x ) ε K 0 e i Q ( 1 + e 2 i Q ξ ) Q σ ,
B n = e i Q { e i K x [ Q σ + e 2 i Q ( ξ 1 ) ( 2 ε K 0 + Q σ ) ] 2 e i Q ( 2 ξ 1 ) ε K 0 } Q σ + e 2 i Q ξ [ 2 ( e i ( Q K x ) 1 ) ε K 0 + Q σ ] ,
C n = ( 1 + e 2 i Q ξ ) Q σ 2 ( e i ( Q + K x ) 1 ) ε K 0 ( 1 + e 2 i Q ξ ) Q σ 2 e 2 i Q ξ ( 1 e i ( Q K x ) ) ε K 0 ,
( χ _ ) m n = 4 e 1 2 i ( Q + K x ) Q { 2 ( 2 m π + K x ) cos Q 2 sin K x 2 σ e i m π [ 2 i ε K 0 ( cos Q cos K x ) + Q sin Q σ ] } ( 2 m π Q + K x ) ( 2 m π + Q + K x ) [ ( 1 + e i Q ) Q σ 2 ( e i Q e i K x ) ε K 0 ]

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