Abstract

Based on the effective medium model, nonlocal optical properties in periodic lattice of graphene layers with the period much less than the wavelength are investigated. Strong nonlocal effects are found in a broad frequency range for TM polarization, where the effective permittivity tensor exhibits the Lorentzian resonance. The resonance frequency varies with the wave vector and coincides well with the polaritonic mode. Nonlocal features are manifest on the emergence of additional wave and the occurrence of negative refraction. By examining the characters of the eigenmode, the nonlocal optical properties are attributed to the excitation of plasmons on the graphene surfaces.

© 2014 Optical Society of America

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2013 (2)

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

R.-L. Chern, “Spatial dispersion and nonlocal effective permittivity for periodic layered metamaterials,” Opt. Express 21, 16514–16527 (2013).
[CrossRef] [PubMed]

2012 (3)

A. V. Chebykin, A. A. Orlov, C. R. Simovski, Y. S. Kivshar, P. A. Belov, “Nonlocal effective parameters of multilayered metal-dielectric metamaterials,” Phys. Rev. B 86, 115420 (2012).
[CrossRef]

T. Stauber, G. Gómez-Santos, “Plasmons and near-field amplification in double-layer graphene,” Phys. Rev. B 85, 075410 (2012).
[CrossRef]

B. Wang, X. Zhang, F. J. García-Vidal, X. Yuan, J. Teng, “Strong coupling of surface plasmon polaritons in monolayer graphene sheet arrays,” Phys. Rev. Lett. 109, 073901 (2012).
[CrossRef] [PubMed]

2011 (6)

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

L. Ju, B. Geng, J. Horng, C. Girit, M. Martin, Z. Hao, H. A. Bechtel, X. Liang, A. Zettl, Y. R. Shen, “Graphene plasmonics for tunable terahertz metamaterials,” Nat. Nanotechnol. 6, 630–634 (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, 161407 (2011).
[CrossRef]

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

P.-Y. Chen, A. Alù, “Atomically thin surface cloak using graphene mnolayers,” ACS Nano 5, 5855–5863 (2011).
[CrossRef] [PubMed]

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

2010 (2)

Z. Sun, T. Hasan, F. Torrisi, D. Popa, G. Privitera, F. Wang, F. Bonaccorso, D. M. Basko, A. C. Ferrari, “Graphene mode-locked ultrafast laser,” ACS Nano 4, 803–810 (2010).
[CrossRef] [PubMed]

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

2009 (3)

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

K. S. Kim, Y. Zhao, H. Jang, S. Y. Lee, J. M. Kim, K. S. Kim, J.-H. Ahn, P. Kim, J.-Y. Choi, B. H. Hong, “Large-scale pattern growth of graphene films for stretchable transparent electrodes,” Nature 457, 706–710 (2009).
[CrossRef] [PubMed]

F. Xia, T. Mueller, Y.-m. Lin, A. Valdes-Garcia, P. Avouris, “Ultrafast graphene photodetector,” Nat. Nanotechnol. 4, 839–843 (2009).
[CrossRef] [PubMed]

2008 (9)

K. I. Bolotin, K. J. Sikes, Z. Jiang, M. Klima, G. Fudenberg, J. Hone, P. Kim, 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, C. N. Lau, “Superior thermal conductivity of single-layer graphene,” Nano Lett. 8, 902–907 (2008).
[CrossRef] [PubMed]

R. R. Nair, P. Blake, A. N. Grigorenko, K. S. Novoselov, T. J. Booth, T. Stauber, N. M. R. Peres, A. K. Geim, “Fine structure constant defines visual transparency of graphene,” Science 320, 1308 (2008).
[CrossRef] [PubMed]

K. F. Mak, M. Y. Sfeir, Y. Wu, C. H. Lui, J. A. Misewich, T. F. Heinz, “Measurement of the optical conductivity of graphene,” Phys. Rev. Lett. 101, 196405 (2008).
[CrossRef] [PubMed]

A. B. Kuzmenko, E. van Heumen, F. Carbone, D. van der Marel, “Universal optical conductance of graphite,” Phys. Rev. Lett. 100, 117401 (2008).
[CrossRef] [PubMed]

T. Stauber, N. M. R. Peres, A. K. Geim, “Optical conductivity of graphene in the visible region of the spectrum,” Phys. Rev. B 78, 085432 (2008).
[CrossRef]

Z. Li, E. A. Henriksen, Z. Jiang, Z. Hao, M. C. Martin, P. Kim, H. Stormer, D. N. Basov, “Dirac charge dynamics in graphene by infrared spectroscopy,” Nat. Phys. 4, 532–535 (2008).
[CrossRef]

F. Wang, Y. Zhang, C. Tian, C. Girit, A. Zettl, M. Crommie, Y. R. Shen, “Gate-variable optical transitions in graphene,” Science 320, 206–209 (2008).
[CrossRef] [PubMed]

Y. Liu, R. F. Willis, K. V. Emtsev, T. Seyller, “Plasmon dispersion and damping in electrically isolated two-dimensional charge sheets,” Phys. Rev. B 78, 201403 (2008).
[CrossRef]

2007 (5)

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

J. Elser, V. A. Podolskiy, I. Salakhutdinov, I. Avrutsky, “Nonlocal effects in effective-medium response of nanolayered metamaterials,” Appl. Phys. Lett. 90, 191109 (2007).
[CrossRef]

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

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

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

2006 (3)

V. M. Agranovich, Y. N. Gartstein, “Spatial dispersion and negative refraction of light,” Phys. Usp. 49, 1029 (2006).
[CrossRef]

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

V. P. Gusynin, S. G. Sharapov, J. P. Carbotte, “Unusual Microwave Response of Dirac Quasiparticles in Graphene,” Phys. Rev. Lett. 96, 256802 (2006).
[CrossRef] [PubMed]

2005 (3)

K. Novoselov, A. K. Geim, S. Morozov, D. Jiang, M. K. I. Grigorieva, S. Dubonos, 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, P. Kim, “Experimental observation of the quantum Hall effect and Berry’s phase in graphene,” Nature 438, 201–204 (2005).
[CrossRef] [PubMed]

X. Xu, Y. Xi, D. Han, X. Liu, J. Zi, Z. Zhu, “Effective plasma frequency in one-dimensional metallic-dielectric photonic crystals,” Appl. Phys.Lett. 86, 091112 (2005).

2004 (2)

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, 666–669 (2004).
[CrossRef] [PubMed]

C. R. Simovski, P. A. Belov, “Low-frequency spatial dispersion in wire media,” Phys. Rev. E 70, 046616 (2004).
[CrossRef]

2003 (3)

P. A. Belov, R. Marques, S. I. Maslovski, I. S. Nefedov, M. Silveirinha, C. R. Simovski, S. A. Tretyakov, “Strong spatial dispersion in wire media in the very large wavelength limit,” Phys. Rev. B 67, 113103 (2003).
[CrossRef]

S. Foteinopoulou, E. N. Economou, 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]

1976 (1)

M. F. Bishop, A. A. Maradudin, “Energy flow in a semi-infinite spatially dispersive absorbing dielectric,” Phys. Rev. B 14, 3384–3393 (1976).
[CrossRef]

1969 (1)

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

1968 (1)

R. M. Hornreich, S. Shtrikman, “Theory of gyrotropic birefringence,” Phys. Rev. 171, 1065–1074 (1968).
[CrossRef]

1963 (1)

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

1962 (1)

V. L. Ginzburg, A. A. Rukhadze, V. P. Silin, “The electrodynamics of crystals and the theory of excitons,” J. Phys. Chem. Solids 23, 85–97 (1962).
[CrossRef]

1958 (1)

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

Agranovich, V. M.

V. M. Agranovich, Y. N. Gartstein, “Spatial dispersion and negative refraction of light,” Phys. Usp. 49, 1029 (2006).
[CrossRef]

V. M. Agranovich, V. L. Ginzburg, Crystal Optics with Spatial Dispersion, and Excitons (Springer-Verlag, 1984).
[CrossRef]

Ahn, J.-H.

K. S. Kim, Y. Zhao, H. Jang, S. Y. Lee, J. M. Kim, K. S. Kim, J.-H. Ahn, P. Kim, J.-Y. Choi, B. H. Hong, “Large-scale pattern growth of graphene films for stretchable transparent electrodes,” Nature 457, 706–710 (2009).
[CrossRef] [PubMed]

Alù, A.

P.-Y. Chen, A. Alù, “Atomically thin surface cloak using graphene mnolayers,” ACS Nano 5, 5855–5863 (2011).
[CrossRef] [PubMed]

Asgari, R.

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

Avouris, P.

F. Xia, T. Mueller, Y.-m. Lin, A. Valdes-Garcia, P. Avouris, “Ultrafast graphene photodetector,” Nat. Nanotechnol. 4, 839–843 (2009).
[CrossRef] [PubMed]

Avrutsky, I.

J. Elser, V. A. Podolskiy, I. Salakhutdinov, I. Avrutsky, “Nonlocal effects in effective-medium response of nanolayered metamaterials,” Appl. Phys. Lett. 90, 191109 (2007).
[CrossRef]

Balandin, A. A.

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

Basko, D. M.

Z. Sun, T. Hasan, F. Torrisi, D. Popa, G. Privitera, F. Wang, F. Bonaccorso, D. M. Basko, A. C. Ferrari, “Graphene mode-locked ultrafast laser,” ACS Nano 4, 803–810 (2010).
[CrossRef] [PubMed]

Basov, D. N.

Z. Li, E. A. Henriksen, Z. Jiang, Z. Hao, M. C. Martin, P. Kim, H. Stormer, D. N. Basov, “Dirac charge dynamics in graphene by infrared spectroscopy,” Nat. Phys. 4, 532–535 (2008).
[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, “Graphene plasmonics for tunable terahertz metamaterials,” Nat. Nanotechnol. 6, 630–634 (2011).
[CrossRef] [PubMed]

Belov, P. A.

A. V. Chebykin, A. A. Orlov, C. R. Simovski, Y. S. Kivshar, P. A. Belov, “Nonlocal effective parameters of multilayered metal-dielectric metamaterials,” Phys. Rev. B 86, 115420 (2012).
[CrossRef]

C. R. Simovski, P. A. Belov, “Low-frequency spatial dispersion in wire media,” Phys. Rev. E 70, 046616 (2004).
[CrossRef]

P. A. Belov, R. Marques, S. I. Maslovski, I. S. Nefedov, M. Silveirinha, C. R. Simovski, S. A. Tretyakov, “Strong spatial dispersion in wire media in the very large wavelength limit,” Phys. Rev. B 67, 113103 (2003).
[CrossRef]

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]

Bishop, M. F.

M. F. Bishop, A. A. Maradudin, “Energy flow in a semi-infinite spatially dispersive absorbing dielectric,” Phys. Rev. B 14, 3384–3393 (1976).
[CrossRef]

Blake, P.

R. R. Nair, P. Blake, A. N. Grigorenko, K. S. Novoselov, T. J. Booth, T. Stauber, N. M. R. Peres, A. K. Geim, “Fine structure constant defines visual transparency of graphene,” Science 320, 1308 (2008).
[CrossRef] [PubMed]

Bolotin, K. I.

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

Bonaccorso, F.

Z. Sun, T. Hasan, F. Torrisi, D. Popa, G. Privitera, F. Wang, F. Bonaccorso, D. M. Basko, A. C. Ferrari, “Graphene mode-locked ultrafast laser,” ACS Nano 4, 803–810 (2010).
[CrossRef] [PubMed]

Booth, T. J.

R. R. Nair, P. Blake, A. N. Grigorenko, K. S. Novoselov, T. J. Booth, T. Stauber, N. M. R. Peres, A. K. Geim, “Fine structure constant defines visual transparency of graphene,” Science 320, 1308 (2008).
[CrossRef] [PubMed]

Buljan, H.

M. Jablan, H. Buljan, M. Soljačić, “Plasmonics in graphene at infrared frequencies,” Phys. Rev. B 80, 245435 (2009).
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Nature (4)

M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, X. Zhang, “A graphene-based broadband optical modulator,” Nature 474, 64–67 (2011).
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Phys. Rev. Lett. (1)

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Figures (9)

Fig. 1
Fig. 1

Schematic diagram of the periodic lattice of graphene layers with period a embedded in a background with dielectric constant ε. In this study, a = 0.1 h̄c/μ (≈ 98.9 nm) with μ = 0.2 eV and ε = 1.5 will be used as the parameters. A small area of the graphene feature is shown on the top layer for illustration.

Fig. 2
Fig. 2

(a) In-plane effective permittivity components ε z eff and ε x eff as the functions of Ω for the same periodic lattice of graphene layers as in Fig. 1, where Ω = h̄ω/μ, Kx = kxa, Kz = kza, and ã = μa/(h̄c). (b) Effective plasma frequency Ω0 and its approximate formula [Eq. (8)] as the functions of ã.

Fig. 3
Fig. 3

Out-of-plane effective permittivity component ε y eff for the same periodic lattice of graphene layers as in Fig. 1, where (a) Kx = 0 and (b) Kz = 0. Red and green lines correspond to ε y eff = 0 and ε y eff = ε , respectively.

Fig. 4
Fig. 4

Equifrequency surfaces of (a) TM and (b) TE dispersion relations for the same periodic lattice of graphene layers as in Fig. 1. Black circle in (b) is the section of light cone at Ω = 2.

Fig. 5
Fig. 5

(a) TM frequency bands at Kx = 0, 0.03 and (b) TE frequency band at Kx = 0 for the same periodic lattice of graphene layers as in Fig. 1. Insets in (a) are typical magnetic field patterns of photonic mode (P mode) and polaritonic mode (PL mode). Red dot in (b) corresponds to the critical frequency Ω* ≈ 1.667 that separates P mode and N mode.

Fig. 6
Fig. 6

Pole frequency Ωp for ε x eff and its approximate formula [Eq. (9)] as the functions of Kz for the same periodic lattice of graphene layers as in Fig. 1. Black solid lines are photonic and polaritonic modes.

Fig. 7
Fig. 7

Dispersion curves on the wave vector domain at (a) Ω = 0.435 and (b) Ω = 0.35 for the same periodic lattice of graphene layers as in Fig. 1. Black and gray contours are equifrequency curves for vacuum and graphene layers, respectively. Dashed lines indicate the continuity of Kx at the interface.

Fig. 8
Fig. 8

Magnetic field (Hy) (a) on the xz plane and (b) along the x axis for the surface mode at the polaritonic band for the same periodic lattice of graphene layers as in Fig. 1, where Kx = 0, Kz = 3.919, and Ω = 0.6. In (a), red and green colors correspond to positive and negative values of Hy, respectively, gray arrows are the electric field vectors (Ez, Ex), and black lines are the locations of graphene layers. In (b), Hy is normalized by its maximum value (at the graphene surface). The profile for Ω = 0.9 is also shown for comparison.

Fig. 9
Fig. 9

Plasmon relation of a single graphene layer with the same chemical potential and dielectric background as in Fig. 1. Photonic and polaritonic modes for the periodic lattice of graphene layers are shown for comparison.

Equations (22)

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cos ( k x a ) = cos ( q a ) i σ q 2 ω ε 0 ε sin ( q a ) ,
cos ( k x a ) = cos ( q a ) i σ ω μ 0 2 q 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 ,
k x 2 + k z 2 = ε y eff k 0 2
ε z eff = ε z 0 γ 12 k 0 2 a 2 1 1 12 k x 2 a 2 , ε x eff = ε ( 1 γ 12 ε z 0 k 0 2 a 2 ) 1 γ 6 ε z 0 ( k 0 2 a 2 1 2 ε k z 2 a 2 ) ,
ε y eff = ε y 0 ( 1 + 1 6 k z 2 a 2 ) δ 6 ε 2 k 0 2 a 2 + a 2 12 k 0 2 ( k x 4 k z 4 ) ,
Ω 0 2 ( 1 + ε a ˜ α ) 1 / 2 ,
Ω p 2 [ 1 + ε a ˜ ( 12 + K z 2 ) 2 α ( 6 + K z 2 ) ] 1 / 2 ,
S = 1 2 Re [ E × H * ] ω 4 ε i j k E i E j * .
x 2 ( 1 ε x 2 ) a 2 + y 2 b 2 = 1 ,
k z 2 ε k 0 2 ε = 2 i k 0 σ ˜ .
Ω = 2 α β k z ε ,
H y ( x ) = { A e i q x + B e i q x , 0 < x < ξ C e i q x + D e i q x , ξ a < x < 0
E z ( x ) = { q ω ε 0 ε ( A e i q x B e i q x ) , 0 < x < ξ q ω ε 0 ε ( C e i q x D e i q x ) , ξ a < x < 0
H y ( 0 + ) σ E z ( 0 + ) = H y ( 0 ) ,
E z ( 0 + ) = E z ( 0 ) ,
H y ( ξ ) = e i k x a H y ( ξ a ) ,
E z ( ξ ) = e i k x a E z ( ξ a ) .
| 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 ) | = 0 ,
cos ( k x a ) = cos ( q a ) i σ q 2 ω ε 0 ε sin ( q a ) .
cos ( k x a ) = cos ( q a ) i σ ω μ 0 2 q sin ( q a ) .

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