Abstract

The tunable waveguide properties of the graphene supporting structure SiSiO2–graphene–dielectrics–graphene–SiO2Si (SiSiO2GDGSiO2Si) have been investigated in the terahertz regime by using the finite element method (FEM) and transfer matrix method (TMM). The study shows that the numerical results obtained from FEM and TMM agree well. The contour results show that as the frequency increases, the effective index increases, and the loss shows a peak; with the increase in the Fermi level, the effective index decreases, and the loss decreases. With a smaller effective mode area, the confinement of the SiSiO2GDGSiO2Si structure is much better than that of the Si–dielectrics–graphene–dielectrics–Si structure. The propagation properties of the structure can be modulated by using the applied gate voltage. The modulation depth of the propagation losses can reach more than 90%. The results are helpful to the design of tunable graphene optoelectronic devices, such as polarizers, modulators, and metamaterial devices.

© 2013 Optical Society of America

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2012 (16)

X. Y. He, Q. J. Wang, and S. F. Yu, “Investigation of multilayer subwavelength metallic–dielectric stratified structures,” IEEE J. Quantum Electron. 48, 1554–1559 (2012).
[CrossRef]

X. Y. He, Q. J. Wang, and S. F. Yu, “Numerical study of gain-assisted terahertz hybrid plasmonic waveguide,” Plasmonics 7, 571–577 (2012).
[CrossRef]

A. N. Grigorenko, M. Polini, and K. S. Novoselov, “Graphene plasmonics,” Nat. Photonics 6, 749–758 (2012).
[CrossRef]

S. H. Lee, M. Choi, T. T. Kim, S. Lee, M. Liu, X. Yin, H. K. Choi, S. S. Lee, C. G. Choi, S. Y. Choi, X. Zhang, and B. Min, “Switching terahertz waves with gate-controlled active graphene metamaterials,” Nat. Mater. 11, 936–941 (2012).
[CrossRef]

H. J. Xu, W. B. Lu, W. Zhu, Z. G. Dong, and T. J. Cui, “Efficient manipulation of surface plasmon polariton waves in graphene,” Appl. Phys. Lett. 100, 243110 (2012).
[CrossRef]

J. Kim, H. Son, D. J. Cho, B. Geng, W. Regan, S. Shi, K. Kim, A. Zettl, Y. R. Shen, and F. Wang, “Electrical control of optical plasmon resonance with graphene,” Nano Lett. 12, 5598–5602 (2012).
[CrossRef]

H. Yan, F. Xia, Z. Li, and P. Avouris, “Plasmonics of coupled graphene micro-structures,” New J. Phys. 14, 125001 (2012).
[CrossRef]

J. Christensen, A. Manjavacas, S. Thongrattanasiri, F. H. L. Koppens, and F. J. G. de Abajo, “Graphene plasmon waveguiding and hybridization in individual and paired nanoribbons,” ACS Nano 6, 431–440 (2012).
[CrossRef]

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

J. T. Kim and C. G. Choi, “Graphene-based polymer waveguide polarizer,” Opt. Express 20, 3556–3562 (2012).
[CrossRef]

Z. Lu and W. Zhao, “Nanoscale electro-optic modulators based on graphene-slot waveguides,” J. Opt. Soc. Am. B 29, 1490–1496 (2012).
[CrossRef]

B. Sensale-Rodriguez, R. Yan, S. Rafique, M. Zhu, W. Li, X. Liang, D. Gundlach, V. Protasenko, M. M. Kelly, D. Jena, L. Liu, and H. G. Xing, “Extraordinary control of terahertz beam reflectance in graphene electro-absorption modulators,” Nano Lett. 12, 4518–4522 (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]

J. Chen, M. Badioli, P. Alonso-González, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenovíc, A. Centeno, A. Pesquera, P. Godignon, A. Z. Elorza, N. Camara, F. J. García de Abajo, R. Hillenbrand, and F. H. L. Koppens, “Optical nano-imaging of gate-tunable graphene plasmons,” Nature 487, 77–81 (2012).

Z. Fei, A. S. Rodin, G. O. Andreev, W. Bao, A. S. McLeod, M. Wagner, L. M. Zhang, Z. Zhao, M. Thiemens, G. Dominguez, M. M. Fogler, A. H. C. Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature 487, 82–85 (2012).

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]

2011 (6)

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. Nanotechnol. 6, 630–634 (2011).
[CrossRef]

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

Q. Bao, H. Zhang, B. Wang, Z. Ni, C. H. Y. X. Lim, Y. Wang, D. T. Tang, and K. P. Loh, “Broadband graphene polarizer,” Nat. Photonics 5, 411–415 (2011).
[CrossRef]

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

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

S. Lee and S. Kim, “Long-range channel plasmon polaritons in thin metal film V-grooves,” Opt. Express 19, 9836–9847 (2011).
[CrossRef]

2010 (4)

S. Lee and S. Kim, “Plasmonic mode-gap waveguides using heterometal films,” Opt. Express 18, 2197–2208 (2010).
[CrossRef]

A. R. Wright and C. Zhang, “Dynamic conductivity of graphene with electron-LO-phonon interaction,” Phys. Rev. B 81, 165413 (2010).
[CrossRef]

D. R. Andersen, “Graphene-based long-wave infrared TM surface plasmon modulator,” J. Opt. Soc. Am. B 27, 818–823 (2010).
[CrossRef]

X. G. Xu, S. Sultan, C. Zhang, and J. C. Cao, “Nonlinear optical conductance in a graphene pn junction in the terahertz regime,” Appl. Phys. Lett. 97, 011907 (2010).
[CrossRef]

2009 (3)

A. R. Wright, J. C. Cao, and C. Zhang, “Enhanced optical conductivity of bilayer graphene nanoribbons in the terahertz regime,” Phys. Rev. Lett. 103, 207401 (2009).
[CrossRef]

X. Y. He, “Numerical analysis of the propagation properties of subwavelength semiconductor slit in the terahertz region,” Opt. Express 17, 15359–15371 (2009).
[CrossRef]

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

2008 (4)

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

Z. Q. Li, E. A. Henriksen, Z. Jiang, Z. Hao, M. C. Martin, P. Kim, H. L. Stromer, and D. N. Basov, “Dirac charge dynamics in graphene by infrared spectroscopy,” Nat. Photonics 4, 532–535 (2008).

C. Zhang, L. Chen, and Z. Ma, “Orientation dependence of the optical spectra in graphene at high frequencies,” Phys. Rev. B 77, 241402 (2008).
[CrossRef]

R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, “A hybrid plasmonic waveguide for subwavelength confinement and long-range propagation,” Nat. Photonics 2, 496–500 (2008).
[CrossRef]

2007 (2)

V. P. Gusynin and S. G. Sharapov, “Magneto-optical conductivity in graphene,” J. Phys. Condens. Matter 19, 026222 (2007).
[CrossRef]

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

2005 (1)

J. T. Lü and J. C. Cao, “Confined optical phonon modes and electron–phonon interactions in wurtzite GaN/ZnO quantum wells,” Phys. Rev. B 71, 155304 (2005).
[CrossRef]

2003 (1)

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003).
[CrossRef]

1985 (1)

Alexander, R. W.

Alonso-González, P.

J. Chen, M. Badioli, P. Alonso-González, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenovíc, A. Centeno, A. Pesquera, P. Godignon, A. Z. Elorza, N. Camara, F. J. García de Abajo, R. Hillenbrand, and F. H. L. Koppens, “Optical nano-imaging of gate-tunable graphene plasmons,” Nature 487, 77–81 (2012).

Andersen, D. R.

Andreev, G. O.

Z. Fei, A. S. Rodin, G. O. Andreev, W. Bao, A. S. McLeod, M. Wagner, L. M. Zhang, Z. Zhao, M. Thiemens, G. Dominguez, M. M. Fogler, A. H. C. Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature 487, 82–85 (2012).

Avouris, P.

H. Yan, F. Xia, Z. Li, and P. Avouris, “Plasmonics of coupled graphene micro-structures,” New J. Phys. 14, 125001 (2012).
[CrossRef]

Badioli, M.

J. Chen, M. Badioli, P. Alonso-González, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenovíc, A. Centeno, A. Pesquera, P. Godignon, A. Z. Elorza, N. Camara, F. J. García de Abajo, R. Hillenbrand, and F. H. L. Koppens, “Optical nano-imaging of gate-tunable graphene plasmons,” Nature 487, 77–81 (2012).

Bao, Q.

Q. Bao, H. Zhang, B. Wang, Z. Ni, C. H. Y. X. Lim, Y. Wang, D. T. Tang, and K. P. Loh, “Broadband graphene polarizer,” Nat. Photonics 5, 411–415 (2011).
[CrossRef]

Bao, W.

Z. Fei, A. S. Rodin, G. O. Andreev, W. Bao, A. S. McLeod, M. Wagner, L. M. Zhang, Z. Zhao, M. Thiemens, G. Dominguez, M. M. Fogler, A. H. C. Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature 487, 82–85 (2012).

Barnes, W. L.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003).
[CrossRef]

Basov, D. N.

Z. Fei, A. S. Rodin, G. O. Andreev, W. Bao, A. S. McLeod, M. Wagner, L. M. Zhang, Z. Zhao, M. Thiemens, G. Dominguez, M. M. Fogler, A. H. C. Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature 487, 82–85 (2012).

Z. Q. Li, E. A. Henriksen, Z. Jiang, Z. Hao, M. C. Martin, P. Kim, H. L. Stromer, and D. N. Basov, “Dirac charge dynamics in graphene by infrared spectroscopy,” Nat. Photonics 4, 532–535 (2008).

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. Nanotechnol. 6, 630–634 (2011).
[CrossRef]

Bell, R. J.

Buljan, H.

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

Camara, N.

J. Chen, M. Badioli, P. Alonso-González, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenovíc, A. Centeno, A. Pesquera, P. Godignon, A. Z. Elorza, N. Camara, F. J. García de Abajo, R. Hillenbrand, and F. H. L. Koppens, “Optical nano-imaging of gate-tunable graphene plasmons,” Nature 487, 77–81 (2012).

Cao, J. C.

X. G. Xu, S. Sultan, C. Zhang, and J. C. Cao, “Nonlinear optical conductance in a graphene pn junction in the terahertz regime,” Appl. Phys. Lett. 97, 011907 (2010).
[CrossRef]

A. R. Wright, J. C. Cao, and C. Zhang, “Enhanced optical conductivity of bilayer graphene nanoribbons in the terahertz regime,” Phys. Rev. Lett. 103, 207401 (2009).
[CrossRef]

J. T. Lü and J. C. Cao, “Confined optical phonon modes and electron–phonon interactions in wurtzite GaN/ZnO quantum wells,” Phys. Rev. B 71, 155304 (2005).
[CrossRef]

Centeno, A.

J. Chen, M. Badioli, P. Alonso-González, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenovíc, A. Centeno, A. Pesquera, P. Godignon, A. Z. Elorza, N. Camara, F. J. García de Abajo, R. Hillenbrand, and F. H. L. Koppens, “Optical nano-imaging of gate-tunable graphene plasmons,” Nature 487, 77–81 (2012).

Chang, D. E.

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

Chen, J.

J. Chen, M. Badioli, P. Alonso-González, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenovíc, A. Centeno, A. Pesquera, P. Godignon, A. Z. Elorza, N. Camara, F. J. García de Abajo, R. Hillenbrand, and F. H. L. Koppens, “Optical nano-imaging of gate-tunable graphene plasmons,” Nature 487, 77–81 (2012).

Chen, L.

C. Zhang, L. Chen, and Z. Ma, “Orientation dependence of the optical spectra in graphene at high frequencies,” Phys. Rev. B 77, 241402 (2008).
[CrossRef]

Cho, D. J.

J. Kim, H. Son, D. J. Cho, B. Geng, W. Regan, S. Shi, K. Kim, A. Zettl, Y. R. Shen, and F. Wang, “Electrical control of optical plasmon resonance with graphene,” Nano Lett. 12, 5598–5602 (2012).
[CrossRef]

Choi, C. G.

S. H. Lee, M. Choi, T. T. Kim, S. Lee, M. Liu, X. Yin, H. K. Choi, S. S. Lee, C. G. Choi, S. Y. Choi, X. Zhang, and B. Min, “Switching terahertz waves with gate-controlled active graphene metamaterials,” Nat. Mater. 11, 936–941 (2012).
[CrossRef]

J. T. Kim and C. G. Choi, “Graphene-based polymer waveguide polarizer,” Opt. Express 20, 3556–3562 (2012).
[CrossRef]

Choi, H. K.

S. H. Lee, M. Choi, T. T. Kim, S. Lee, M. Liu, X. Yin, H. K. Choi, S. S. Lee, C. G. Choi, S. Y. Choi, X. Zhang, and B. Min, “Switching terahertz waves with gate-controlled active graphene metamaterials,” Nat. Mater. 11, 936–941 (2012).
[CrossRef]

Choi, M.

S. H. Lee, M. Choi, T. T. Kim, S. Lee, M. Liu, X. Yin, H. K. Choi, S. S. Lee, C. G. Choi, S. Y. Choi, X. Zhang, and B. Min, “Switching terahertz waves with gate-controlled active graphene metamaterials,” Nat. Mater. 11, 936–941 (2012).
[CrossRef]

Choi, S. Y.

S. H. Lee, M. Choi, T. T. Kim, S. Lee, M. Liu, X. Yin, H. K. Choi, S. S. Lee, C. G. Choi, S. Y. Choi, X. Zhang, and B. Min, “Switching terahertz waves with gate-controlled active graphene metamaterials,” Nat. Mater. 11, 936–941 (2012).
[CrossRef]

Christensen, J.

J. Christensen, A. Manjavacas, S. Thongrattanasiri, F. H. L. Koppens, and F. J. G. de Abajo, “Graphene plasmon waveguiding and hybridization in individual and paired nanoribbons,” ACS Nano 6, 431–440 (2012).
[CrossRef]

Crommie, M.

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

Cui, T. J.

H. J. Xu, W. B. Lu, W. Zhu, Z. G. Dong, and T. J. Cui, “Efficient manipulation of surface plasmon polariton waves in graphene,” Appl. Phys. Lett. 100, 243110 (2012).
[CrossRef]

de Abajo, F. J. G.

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V. P. Gusynin and S. G. Sharapov, “Magneto-optical conductivity in graphene,” J. Phys. Condens. Matter 19, 026222 (2007).
[CrossRef]

Shen, Y. R.

J. Kim, H. Son, D. J. Cho, B. Geng, W. Regan, S. Shi, K. Kim, A. Zettl, Y. R. Shen, and F. Wang, “Electrical control of optical plasmon resonance with graphene,” Nano Lett. 12, 5598–5602 (2012).
[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. Nanotechnol. 6, 630–634 (2011).
[CrossRef]

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

Shi, S.

J. Kim, H. Son, D. J. Cho, B. Geng, W. Regan, S. Shi, K. Kim, A. Zettl, Y. R. Shen, and F. Wang, “Electrical control of optical plasmon resonance with graphene,” Nano Lett. 12, 5598–5602 (2012).
[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]

Soljacic, M.

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

Son, H.

J. Kim, H. Son, D. J. Cho, B. Geng, W. Regan, S. Shi, K. Kim, A. Zettl, Y. R. Shen, and F. Wang, “Electrical control of optical plasmon resonance with graphene,” Nano Lett. 12, 5598–5602 (2012).
[CrossRef]

Sorger, V. J.

R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, “A hybrid plasmonic waveguide for subwavelength confinement and long-range propagation,” Nat. Photonics 2, 496–500 (2008).
[CrossRef]

Spasenovíc, M.

J. Chen, M. Badioli, P. Alonso-González, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenovíc, A. Centeno, A. Pesquera, P. Godignon, A. Z. Elorza, N. Camara, F. J. García de Abajo, R. Hillenbrand, and F. H. L. Koppens, “Optical nano-imaging of gate-tunable graphene plasmons,” Nature 487, 77–81 (2012).

Stromer, H. L.

Z. Q. Li, E. A. Henriksen, Z. Jiang, Z. Hao, M. C. Martin, P. Kim, H. L. Stromer, and D. N. Basov, “Dirac charge dynamics in graphene by infrared spectroscopy,” Nat. Photonics 4, 532–535 (2008).

Sultan, S.

X. G. Xu, S. Sultan, C. Zhang, and J. C. Cao, “Nonlinear optical conductance in a graphene pn junction in the terahertz regime,” Appl. Phys. Lett. 97, 011907 (2010).
[CrossRef]

Tang, D. T.

Q. Bao, H. Zhang, B. Wang, Z. Ni, C. H. Y. X. Lim, Y. Wang, D. T. Tang, and K. P. Loh, “Broadband graphene polarizer,” Nat. Photonics 5, 411–415 (2011).
[CrossRef]

Teng, J.

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

Thiemens, M.

Z. Fei, A. S. Rodin, G. O. Andreev, W. Bao, A. S. McLeod, M. Wagner, L. M. Zhang, Z. Zhao, M. Thiemens, G. Dominguez, M. M. Fogler, A. H. C. Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature 487, 82–85 (2012).

Thongrattanasiri, S.

J. Chen, M. Badioli, P. Alonso-González, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenovíc, A. Centeno, A. Pesquera, P. Godignon, A. Z. Elorza, N. Camara, F. J. García de Abajo, R. Hillenbrand, and F. H. L. Koppens, “Optical nano-imaging of gate-tunable graphene plasmons,” Nature 487, 77–81 (2012).

J. Christensen, A. Manjavacas, S. Thongrattanasiri, F. H. L. Koppens, and F. J. G. de Abajo, “Graphene plasmon waveguiding and hybridization in individual and paired nanoribbons,” ACS Nano 6, 431–440 (2012).
[CrossRef]

Tian, C.

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

Ulin-Avila, E.

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

Vakil, A.

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

Wagner, M.

Z. Fei, A. S. Rodin, G. O. Andreev, W. Bao, A. S. McLeod, M. Wagner, L. M. Zhang, Z. Zhao, M. Thiemens, G. Dominguez, M. M. Fogler, A. H. C. Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature 487, 82–85 (2012).

Wang, B.

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

Q. Bao, H. Zhang, B. Wang, Z. Ni, C. H. Y. X. Lim, Y. Wang, D. T. Tang, and K. P. Loh, “Broadband graphene polarizer,” Nat. Photonics 5, 411–415 (2011).
[CrossRef]

Wang, F.

J. Kim, H. Son, D. J. Cho, B. Geng, W. Regan, S. Shi, K. Kim, A. Zettl, Y. R. Shen, and F. Wang, “Electrical control of optical plasmon resonance with graphene,” Nano Lett. 12, 5598–5602 (2012).
[CrossRef]

M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang, “A graphene-based broadband optical modulator,” Nature 474, 64–67 (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. Nanotechnol. 6, 630–634 (2011).
[CrossRef]

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

Wang, Q. J.

X. Y. He, Q. J. Wang, and S. F. Yu, “Numerical study of gain-assisted terahertz hybrid plasmonic waveguide,” Plasmonics 7, 571–577 (2012).
[CrossRef]

X. Y. He, Q. J. Wang, and S. F. Yu, “Investigation of multilayer subwavelength metallic–dielectric stratified structures,” IEEE J. Quantum Electron. 48, 1554–1559 (2012).
[CrossRef]

Wang, Y.

Q. Bao, H. Zhang, B. Wang, Z. Ni, C. H. Y. X. Lim, Y. Wang, D. T. Tang, and K. P. Loh, “Broadband graphene polarizer,” Nat. Photonics 5, 411–415 (2011).
[CrossRef]

Wright, A. R.

A. R. Wright and C. Zhang, “Dynamic conductivity of graphene with electron-LO-phonon interaction,” Phys. Rev. B 81, 165413 (2010).
[CrossRef]

A. R. Wright, J. C. Cao, and C. Zhang, “Enhanced optical conductivity of bilayer graphene nanoribbons in the terahertz regime,” Phys. Rev. Lett. 103, 207401 (2009).
[CrossRef]

Xia, F.

H. Yan, F. Xia, Z. Li, and P. Avouris, “Plasmonics of coupled graphene micro-structures,” New J. Phys. 14, 125001 (2012).
[CrossRef]

Xing, H. G.

B. Sensale-Rodriguez, R. Yan, S. Rafique, M. Zhu, W. Li, X. Liang, D. Gundlach, V. Protasenko, M. M. Kelly, D. Jena, L. Liu, and H. G. Xing, “Extraordinary control of terahertz beam reflectance in graphene electro-absorption modulators,” Nano Lett. 12, 4518–4522 (2012).
[CrossRef]

Xu, H. J.

H. J. Xu, W. B. Lu, W. Zhu, Z. G. Dong, and T. J. Cui, “Efficient manipulation of surface plasmon polariton waves in graphene,” Appl. Phys. Lett. 100, 243110 (2012).
[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]

Xu, X. G.

X. G. Xu, S. Sultan, C. Zhang, and J. C. Cao, “Nonlinear optical conductance in a graphene pn junction in the terahertz regime,” Appl. Phys. Lett. 97, 011907 (2010).
[CrossRef]

Yan, H.

H. Yan, F. Xia, Z. Li, and P. Avouris, “Plasmonics of coupled graphene micro-structures,” New J. Phys. 14, 125001 (2012).
[CrossRef]

Yan, R.

B. Sensale-Rodriguez, R. Yan, S. Rafique, M. Zhu, W. Li, X. Liang, D. Gundlach, V. Protasenko, M. M. Kelly, D. Jena, L. Liu, and H. G. Xing, “Extraordinary control of terahertz beam reflectance in graphene electro-absorption modulators,” Nano Lett. 12, 4518–4522 (2012).
[CrossRef]

Yin, X.

S. H. Lee, M. Choi, T. T. Kim, S. Lee, M. Liu, X. Yin, H. K. Choi, S. S. Lee, C. G. Choi, S. Y. Choi, X. Zhang, and B. Min, “Switching terahertz waves with gate-controlled active graphene metamaterials,” Nat. Mater. 11, 936–941 (2012).
[CrossRef]

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

Yu, S. F.

X. Y. He, Q. J. Wang, and S. F. Yu, “Investigation of multilayer subwavelength metallic–dielectric stratified structures,” IEEE J. Quantum Electron. 48, 1554–1559 (2012).
[CrossRef]

X. Y. He, Q. J. Wang, and S. F. Yu, “Numerical study of gain-assisted terahertz hybrid plasmonic waveguide,” Plasmonics 7, 571–577 (2012).
[CrossRef]

Yuan, X.

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

Zentgraf, T.

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

Zettl, A.

J. Kim, H. Son, D. J. Cho, B. Geng, W. Regan, S. Shi, K. Kim, A. Zettl, Y. R. Shen, and F. Wang, “Electrical control of optical plasmon resonance with graphene,” Nano Lett. 12, 5598–5602 (2012).
[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. Nanotechnol. 6, 630–634 (2011).
[CrossRef]

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

Zhang, C.

A. R. Wright and C. Zhang, “Dynamic conductivity of graphene with electron-LO-phonon interaction,” Phys. Rev. B 81, 165413 (2010).
[CrossRef]

X. G. Xu, S. Sultan, C. Zhang, and J. C. Cao, “Nonlinear optical conductance in a graphene pn junction in the terahertz regime,” Appl. Phys. Lett. 97, 011907 (2010).
[CrossRef]

A. R. Wright, J. C. Cao, and C. Zhang, “Enhanced optical conductivity of bilayer graphene nanoribbons in the terahertz regime,” Phys. Rev. Lett. 103, 207401 (2009).
[CrossRef]

C. Zhang, L. Chen, and Z. Ma, “Orientation dependence of the optical spectra in graphene at high frequencies,” Phys. Rev. B 77, 241402 (2008).
[CrossRef]

Zhang, H.

Q. Bao, H. Zhang, B. Wang, Z. Ni, C. H. Y. X. Lim, Y. Wang, D. T. Tang, and K. P. Loh, “Broadband graphene polarizer,” Nat. Photonics 5, 411–415 (2011).
[CrossRef]

Zhang, L. M.

Z. Fei, A. S. Rodin, G. O. Andreev, W. Bao, A. S. McLeod, M. Wagner, L. M. Zhang, Z. Zhao, M. Thiemens, G. Dominguez, M. M. Fogler, A. H. C. Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature 487, 82–85 (2012).

Zhang, X.

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

S. H. Lee, M. Choi, T. T. Kim, S. Lee, M. Liu, X. Yin, H. K. Choi, S. S. Lee, C. G. Choi, S. Y. Choi, X. Zhang, and B. Min, “Switching terahertz waves with gate-controlled active graphene metamaterials,” Nat. Mater. 11, 936–941 (2012).
[CrossRef]

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

R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, “A hybrid plasmonic waveguide for subwavelength confinement and long-range propagation,” Nat. Photonics 2, 496–500 (2008).
[CrossRef]

Zhang, Y.

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

Zhao, W.

Zhao, Z.

Z. Fei, A. S. Rodin, G. O. Andreev, W. Bao, A. S. McLeod, M. Wagner, L. M. Zhang, Z. Zhao, M. Thiemens, G. Dominguez, M. M. Fogler, A. H. C. Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature 487, 82–85 (2012).

Zhu, M.

B. Sensale-Rodriguez, R. Yan, S. Rafique, M. Zhu, W. Li, X. Liang, D. Gundlach, V. Protasenko, M. M. Kelly, D. Jena, L. Liu, and H. G. Xing, “Extraordinary control of terahertz beam reflectance in graphene electro-absorption modulators,” Nano Lett. 12, 4518–4522 (2012).
[CrossRef]

Zhu, W.

H. J. Xu, W. B. Lu, W. Zhu, Z. G. Dong, and T. J. Cui, “Efficient manipulation of surface plasmon polariton waves in graphene,” Appl. Phys. Lett. 100, 243110 (2012).
[CrossRef]

Ziegler, K.

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

ACS Nano (2)

J. Christensen, A. Manjavacas, S. Thongrattanasiri, F. H. L. Koppens, and F. J. G. de Abajo, “Graphene plasmon waveguiding and hybridization in individual and paired nanoribbons,” ACS Nano 6, 431–440 (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]

Appl. Opt. (1)

Appl. Phys. Lett. (3)

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

H. J. Xu, W. B. Lu, W. Zhu, Z. G. Dong, and T. J. Cui, “Efficient manipulation of surface plasmon polariton waves in graphene,” Appl. Phys. Lett. 100, 243110 (2012).
[CrossRef]

X. G. Xu, S. Sultan, C. Zhang, and J. C. Cao, “Nonlinear optical conductance in a graphene pn junction in the terahertz regime,” Appl. Phys. Lett. 97, 011907 (2010).
[CrossRef]

IEEE J. Quantum Electron. (1)

X. Y. He, Q. J. Wang, and S. F. Yu, “Investigation of multilayer subwavelength metallic–dielectric stratified structures,” IEEE J. Quantum Electron. 48, 1554–1559 (2012).
[CrossRef]

J. Opt. Soc. Am. B (2)

J. Phys. Condens. Matter (1)

V. P. Gusynin and S. G. Sharapov, “Magneto-optical conductivity in graphene,” J. Phys. Condens. Matter 19, 026222 (2007).
[CrossRef]

Nano Lett. (3)

B. Sensale-Rodriguez, R. Yan, S. Rafique, M. Zhu, W. Li, X. Liang, D. Gundlach, V. Protasenko, M. M. Kelly, D. Jena, L. Liu, and H. G. Xing, “Extraordinary control of terahertz beam reflectance in graphene electro-absorption modulators,” Nano Lett. 12, 4518–4522 (2012).
[CrossRef]

J. Kim, H. Son, D. J. Cho, B. Geng, W. Regan, S. Shi, K. Kim, A. Zettl, Y. R. Shen, and F. Wang, “Electrical control of optical plasmon resonance with graphene,” Nano Lett. 12, 5598–5602 (2012).
[CrossRef]

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

Nat. Mater. (1)

S. H. Lee, M. Choi, T. T. Kim, S. Lee, M. Liu, X. Yin, H. K. Choi, S. S. Lee, C. G. Choi, S. Y. Choi, X. Zhang, and B. Min, “Switching terahertz waves with gate-controlled active graphene metamaterials,” Nat. Mater. 11, 936–941 (2012).
[CrossRef]

Nat. Nanotechnol. (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. Nanotechnol. 6, 630–634 (2011).
[CrossRef]

Nat. Photonics (4)

Q. Bao, H. Zhang, B. Wang, Z. Ni, C. H. Y. X. Lim, Y. Wang, D. T. Tang, and K. P. Loh, “Broadband graphene polarizer,” Nat. Photonics 5, 411–415 (2011).
[CrossRef]

A. N. Grigorenko, M. Polini, and K. S. Novoselov, “Graphene plasmonics,” Nat. Photonics 6, 749–758 (2012).
[CrossRef]

Z. Q. Li, E. A. Henriksen, Z. Jiang, Z. Hao, M. C. Martin, P. Kim, H. L. Stromer, and D. N. Basov, “Dirac charge dynamics in graphene by infrared spectroscopy,” Nat. Photonics 4, 532–535 (2008).

R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, “A hybrid plasmonic waveguide for subwavelength confinement and long-range propagation,” Nat. Photonics 2, 496–500 (2008).
[CrossRef]

Nature (4)

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003).
[CrossRef]

J. Chen, M. Badioli, P. Alonso-González, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenovíc, A. Centeno, A. Pesquera, P. Godignon, A. Z. Elorza, N. Camara, F. J. García de Abajo, R. Hillenbrand, and F. H. L. Koppens, “Optical nano-imaging of gate-tunable graphene plasmons,” Nature 487, 77–81 (2012).

Z. Fei, A. S. Rodin, G. O. Andreev, W. Bao, A. S. McLeod, M. Wagner, L. M. Zhang, Z. Zhao, M. Thiemens, G. Dominguez, M. M. Fogler, A. H. C. Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature 487, 82–85 (2012).

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

New J. Phys. (1)

H. Yan, F. Xia, Z. Li, and P. Avouris, “Plasmonics of coupled graphene micro-structures,” New J. Phys. 14, 125001 (2012).
[CrossRef]

Opt. Express (4)

Phys. Rev. B (4)

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

A. R. Wright and C. Zhang, “Dynamic conductivity of graphene with electron-LO-phonon interaction,” Phys. Rev. B 81, 165413 (2010).
[CrossRef]

C. Zhang, L. Chen, and Z. Ma, “Orientation dependence of the optical spectra in graphene at high frequencies,” Phys. Rev. B 77, 241402 (2008).
[CrossRef]

J. T. Lü and J. C. Cao, “Confined optical phonon modes and electron–phonon interactions in wurtzite GaN/ZnO quantum wells,” Phys. Rev. B 71, 155304 (2005).
[CrossRef]

Phys. Rev. Lett. (3)

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

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

A. R. Wright, J. C. Cao, and C. Zhang, “Enhanced optical conductivity of bilayer graphene nanoribbons in the terahertz regime,” Phys. Rev. Lett. 103, 207401 (2009).
[CrossRef]

Plasmonics (1)

X. Y. He, Q. J. Wang, and S. F. Yu, “Numerical study of gain-assisted terahertz hybrid plasmonic waveguide,” Plasmonics 7, 571–577 (2012).
[CrossRef]

Science (2)

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

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

Other (1)

H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer, 1988).

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Figures (9)

Fig. 1.
Fig. 1.

(a) SiSiO2–graphene–dielectric-stripe–metal multilayer structure. (b) SiSiO2–graphene–dielectric–graphene–SiO2Si multilayer structures. A dielectric stripe with the thickness of d is deposited on the substrate. A thin layer of SiO2 with the thickness of 10 nm is located between the graphene and the Si substrate. The conductivity of the graphene layer is σg, which can be controlled by the applied external gate voltage.

Fig. 2.
Fig. 2.

(a) Effective indices and (b) propagation lengths versus dielectric strip thickness for different waveguide structures. The dielectric stripe material is SiO2. The Fermi level of graphene layer is 0.2 eV. The operation frequency is 3 THz.

Fig. 3.
Fig. 3.

(a) Mode areas and (b) propagation losses versus frequency for different waveguide structures, respectively. The waveguide structures include MDM, MetalDGSiO2Si, SiSiO2GDGSiO2Si, SiDGDSi. The dielectric stripe is SiO2. Cu is adopted as the metal layer. The Fermi level of the graphene layer is 0.2 eV.

Fig. 4.
Fig. 4.

(a) Effective index and (b) Im(β/k0) of SiSiO2GDGSiO2Si waveguides versus frequency at different dielectric strip thicknesses. The dielectric material is a SiO2 layer with thicknesses of 1, 2, and 5 μm, respectively. The Fermi level of the graphene layer is 0.2 eV.

Fig. 5.
Fig. 5.

FEM simulation results of the SiSiO2GDGSiO2Si waveguide structure. The width of the dielectric stripe is 100 μm. The Fermi level of graphene layer is 0.2 eV. The frequencies in (a)–(c) are 4, 5, and 6 THz. The dielectric strips thicknesses in (d)–(f) are 1, 3, and 5 μm. The operation frequency is 5 THz.

Fig. 6.
Fig. 6.

(a) Effective indices and (b) propagation lengths of the SiSiO2GDGSiO2Si waveguide structures versus frequency for different dielectric stripe materials. The thickness of the dielectric materials is 2 μm. The Fermi level is 0.2 eV. The dielectric materials are air, polyethylene, and SiO2, respectively. The insets in (a) and (b) are the simulation results for the polyethylene and SiO2 dielectric layers. The frequency is 5 THz.

Fig. 7.
Fig. 7.

Effective indices and Im(β/k0) of SiSiO2GDGSiO2Si waveguides versus frequency at different Fermi levels. The dielectric material is a SiO2 layer with the thickness of 2 μm. The Fermi levels of the graphene layer are 0.1, 0.5, 0.8, and 1.0 eV. The insets in (a) and (b) show the effective indices and losses contours with respect to the frequency and the Fermi levels, respectively.

Fig. 8.
Fig. 8.

Modulation depths of (a) effective indices and (b) losses of SiSiO2GDGSiO2Si waveguides versus frequency obtained by tuning the gate voltage. The dielectric stripe is an SiO2 layer with the thickness of 2 μm. nmax=max(nEf0.1,nEf1.0), Im(β)max=max(Im(β)Ef0.1,Im(β)Ef1.0). The inset in (a) shows the ratios of Im(β)Ef=0.5,0.8,1.0/Im(β)Ef=0.1.

Fig. 9.
Fig. 9.

(a) Effective indices, propagation losses, and mode areas of the SiSiO2GDGSiO2Si structure at different dielectric stripe thickness. The Fermi level of the graphene layer is 0.2 eV. The frequency is 3 THz. (b) The effective indices, propagation losses, mode areas of the SiGDGSi structure versus Fermi level of the graphene layer. The thickness of the dielectric stripe is 2 μm. The frequency is 3 THz.

Equations (12)

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Rσ(Ω)=e2Nf4π2Ω+dω[[f0(ω)f0(ω)]×R{2BΔ2(ω+iΓ)2[Ξ1(B)Ξ2(B)]+[Ξ1(B)Ξ2(+B)Ξ2(B)Ξ2(+B)]×ψ(Δ2(ω+iΓ)22B)+(ωω,ΓΓ)}],
Ξ1(±B)=(ω+iΓ)(ω+iΓ)Δ2[(ωω)+i(ΓΓ)][(ω+ω)+i(Γ+Γ)]±2B,
Ξ2(±B)=(ω+iΓ)(ωiΓ)Δ2[(ωω)+i(Γ+Γ)][(ω+ω)+i(ΓΓ)]±2B,
Iσ(ω)=2ωπ0Rσ(ω)ω2ω2dω.
εg=1+jσgωε0Δ,
ns=α(Vg+V0)=ε0εdedinsulator(Vg+V0),
γ2γ3γNε1ε2εN1[HN+HN]N,N1=(M11M12M21M22),
(M11M12M21M22)=MN1MN2MiM1.
Mi=(εi1εiεi1εi)(γiγiγi1γi1)(exp(γidi)00exp(γidi)).
Am=Wmmax{W(r)}=1max{W(r)}+W(r)d2r,
W(r)=12(d(ε(r)ω)dω|E(r)|2+μ0|H(r)|2),
ε(ω)=(εωp2ω2+ωτ2)+iωτωp2ω(ω2+ωτ2),

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