Abstract

We propose in this paper a dielectric-graphene-dielectric tunable infrared waveguide based on multilayer metamaterials with ultrahigh refractive indices. The waveguide modes with different orders are systematically analyzed with numerical simulations based on both multilayer structures and effective medium approach. The waveguide shows hyperbolic dispersion properties from mid-infrared to far-infrared wavelength, which means the modes with ultrahigh mode indices could be supported in the waveguide. Furthermore, the optical properties of the waveguide modes could be tuned by the biased voltages on graphene layers. The waveguide may have various promising applications in the quantum cascade lasers and bio-sensing.

© 2013 OSA

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    [CrossRef] [PubMed]
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    [CrossRef]
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2013 (2)

F. V. Iorsh, I. S. Mukhin, I. V. Shadrivov, P. A. Belov, and Y. S. Kivshar, “Hyperbolic metamaterials based on multilayer graphene structures,” Phys. Rev. B87(7), 075416 (2013).
[CrossRef]

M. A. K. Othman, C. Guclu, and F. Capolino, “Graphene-based tunable hyperbolic metamaterials and enhanced near-field absorption,” Opt. Express21(6), 7614–7632 (2013).
[CrossRef] [PubMed]

2012 (8)

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(11), 936–941 (2012).
[CrossRef] [PubMed]

G. V. Naik, J. Liu, A. V. Kildishev, V. M. Shalaev, and A. Boltasseva, “Demonstration of Al:ZnO as a plasmonic component for near-infrared metamaterials,” Proc. Natl. Acad. Sci. U.S.A.109(23), 8834–8838 (2012).
[CrossRef] [PubMed]

Y. Zou, P. Tassin, T. Koschny, and C. M. Soukoulis, “Interaction between graphene and metamaterials: split rings vs. wire pairs,” Opt. Express20(11), 12198–12204 (2012).
[CrossRef] [PubMed]

C. Rizza, A. Ciattoni, E. Spinozzi, and L. Columbo, “Terahertz active spatial filtering through optically tunable hyperbolic metamaterials,” Opt. Lett.37(16), 3345–3347 (2012).
[CrossRef] [PubMed]

Y. He, S. He, J. Gao, and X. Yang, “Nanoscale metamaterial optical waveguides with ultrahigh refractive indices,” J. Opt. Soc. Am. B29(9), 2559–2566 (2012).
[CrossRef]

C. Xu, Y. Jin, L. Yang, J. Yang, and X. Jiang, “Characteristics of electro-refractive modulating based on Graphene-Oxide-Silicon waveguide,” Opt. Express20(20), 22398–22405 (2012).
[CrossRef] [PubMed]

X. Yang, J. Yao, J. Rho, X. Yin, and X. Zhang, “Experimental realization of three-dimensional indefinite cavities at the nanoscale with anomalous scaling laws,” Nat. Photonics6(7), 450–454 (2012).
[CrossRef]

A. N. Grigorenko, M. Polini, and K. S. Novoselov, “Graphene plasmonics,” Nat. Photonics6(11), 749–758 (2012).
[CrossRef]

2011 (4)

A. Vakil and N. Engheta, “Transformation Optics Using Graphene,” Science332(6035), 1291–1294 (2011).
[CrossRef] [PubMed]

M. Choi, S. H. Lee, Y. Kim, S. B. Kang, J. Shin, M. H. Kwak, K. Y. Kang, Y. H. Lee, N. Park, and B. Min, “A terahertz metamaterial with unnaturally high refractive index,” Nature470(7334), 369–373 (2011).
[CrossRef] [PubMed]

G. V. Naik and A. Boltasseva, “A comparative study of semiconductor-based plasmonic metamaterials,” Metamaterials (Amst.)5(1), 1–7 (2011).
[CrossRef]

J. Yao, X. Yang, X. Yin, G. Bartal, and X. Zhang, “Three-dimensional nanometer-scale optical cavities of indefinite medium,” Proc. Natl. Acad. Sci. U.S.A.108(28), 11327–11331 (2011).
[CrossRef] [PubMed]

2010 (1)

R. Soref, “Mid-infrared photonics in silicon and germanium,” Nat. Photonics4(8), 495–497 (2010).
[CrossRef]

2009 (2)

F. Y. Meng, Q. Wu, and L. W. Li, “Transmission characteristics of wave modes in a rectangular waveguide filled with anisotropic metamaterial,” Appl. Phys., A Mater. Sci. Process.94(4), 747–753 (2009).
[CrossRef]

J. Shin, J. T. Shen, and S. H. Fan, “Three-Dimensional Metamaterials with an Ultrahigh Effective Refractive Index over a Broad Bandwidth,” Phys. Rev. Lett.102(9), 093903 (2009).
[CrossRef] [PubMed]

2008 (4)

G. W. Hanson, “Dyadic Green’s functions and guided surface waves for a surface conductivity model of graphene,” J. Appl. Phys.103(6), 064302 (2008).
[CrossRef]

T. Stauber, N. M. R. Peres, and A. K. Geim, “Optical conductivity of graphene in the visible region of the spectrum,” Phys. Rev. B78(8), 085432 (2008).
[CrossRef]

M. Björk, J. Knoch, H. Schmid, H. Riel, and W. Riess, “Silicon nanowire tunneling field-effect transistors,” Appl. Phys. Lett.92(19), 193504 (2008).
[CrossRef]

Y. Liu, G. Bartal, and X. Zhang, “All-angle negative refraction and imaging in a bulk medium made of metallic nanowires in the visible region,” Opt. Express16(20), 15439–15448 (2008).
[CrossRef] [PubMed]

2006 (1)

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science312(5781), 1780–1782 (2006).
[CrossRef] [PubMed]

2005 (2)

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, “Two-dimensional gas of massless Dirac fermions in graphene,” Nature438(7065), 197–200 (2005).
[CrossRef] [PubMed]

A. Satta, E. Simoen, T. Clarysse, T. Janssens, A. Benedetti, B. De Jaeger, M. Meuris, and W. Vandervorst, “Diffusion, activation, and recrystallization of boron implanted in preamorphized and crystalline germanium,” Appl. Phys. Lett.87(17), 172109 (2005).
[CrossRef]

2004 (2)

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science305(5685), 788–792 (2004).
[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,” Science306(5696), 666–669 (2004).
[CrossRef] [PubMed]

2002 (2)

R. Köhler, A. Tredicucci, F. Beltram, H. E. Beere, E. H. Linfield, A. G. Davies, D. A. Ritchie, R. C. Iotti, and F. Rossi, “Terahertz semiconductor-heterostructure laser,” Nature417(6885), 156–159 (2002).
[CrossRef] [PubMed]

M. Beck, D. Hofstetter, T. Aellen, J. Faist, U. Oesterle, M. Ilegems, E. Gini, and H. Melchior, “Continuous Wave Operation of a Mid-Infrared Semiconductor Laser at Room Temperature,” Science295(5553), 301–305 (2002).
[CrossRef] [PubMed]

2001 (1)

R. Colombelli, F. Capasso, C. Gmachl, A. L. Hutchinson, D. L. Sivco, A. Tredicucci, M. C. Wanke, A. M. Sergent, and A. Y. Cho, “Far-infrared surface-plasmon quantum-cascade lasers at 21.5 μm and 24 μm wavelengths,” Appl. Phys. Lett.78(18), 2620 (2001).
[CrossRef]

1994 (1)

C. A. Foss, G. L. Hornyak, J. A. Stockert, and C. R. Martin, “Template-synthesized nanoscopic gold Particles: optical spectra and the effects of particle size and shape,” J. Phys. Chem.98(11), 2963–2971 (1994).
[CrossRef]

Aellen, T.

M. Beck, D. Hofstetter, T. Aellen, J. Faist, U. Oesterle, M. Ilegems, E. Gini, and H. Melchior, “Continuous Wave Operation of a Mid-Infrared Semiconductor Laser at Room Temperature,” Science295(5553), 301–305 (2002).
[CrossRef] [PubMed]

Bartal, G.

J. Yao, X. Yang, X. Yin, G. Bartal, and X. Zhang, “Three-dimensional nanometer-scale optical cavities of indefinite medium,” Proc. Natl. Acad. Sci. U.S.A.108(28), 11327–11331 (2011).
[CrossRef] [PubMed]

Y. Liu, G. Bartal, and X. Zhang, “All-angle negative refraction and imaging in a bulk medium made of metallic nanowires in the visible region,” Opt. Express16(20), 15439–15448 (2008).
[CrossRef] [PubMed]

Beck, M.

M. Beck, D. Hofstetter, T. Aellen, J. Faist, U. Oesterle, M. Ilegems, E. Gini, and H. Melchior, “Continuous Wave Operation of a Mid-Infrared Semiconductor Laser at Room Temperature,” Science295(5553), 301–305 (2002).
[CrossRef] [PubMed]

Beere, H. E.

R. Köhler, A. Tredicucci, F. Beltram, H. E. Beere, E. H. Linfield, A. G. Davies, D. A. Ritchie, R. C. Iotti, and F. Rossi, “Terahertz semiconductor-heterostructure laser,” Nature417(6885), 156–159 (2002).
[CrossRef] [PubMed]

Belov, P. A.

F. V. Iorsh, I. S. Mukhin, I. V. Shadrivov, P. A. Belov, and Y. S. Kivshar, “Hyperbolic metamaterials based on multilayer graphene structures,” Phys. Rev. B87(7), 075416 (2013).
[CrossRef]

Beltram, F.

R. Köhler, A. Tredicucci, F. Beltram, H. E. Beere, E. H. Linfield, A. G. Davies, D. A. Ritchie, R. C. Iotti, and F. Rossi, “Terahertz semiconductor-heterostructure laser,” Nature417(6885), 156–159 (2002).
[CrossRef] [PubMed]

Benedetti, A.

A. Satta, E. Simoen, T. Clarysse, T. Janssens, A. Benedetti, B. De Jaeger, M. Meuris, and W. Vandervorst, “Diffusion, activation, and recrystallization of boron implanted in preamorphized and crystalline germanium,” Appl. Phys. Lett.87(17), 172109 (2005).
[CrossRef]

Björk, M.

M. Björk, J. Knoch, H. Schmid, H. Riel, and W. Riess, “Silicon nanowire tunneling field-effect transistors,” Appl. Phys. Lett.92(19), 193504 (2008).
[CrossRef]

Boltasseva, A.

G. V. Naik, J. Liu, A. V. Kildishev, V. M. Shalaev, and A. Boltasseva, “Demonstration of Al:ZnO as a plasmonic component for near-infrared metamaterials,” Proc. Natl. Acad. Sci. U.S.A.109(23), 8834–8838 (2012).
[CrossRef] [PubMed]

G. V. Naik and A. Boltasseva, “A comparative study of semiconductor-based plasmonic metamaterials,” Metamaterials (Amst.)5(1), 1–7 (2011).
[CrossRef]

Capasso, F.

R. Colombelli, F. Capasso, C. Gmachl, A. L. Hutchinson, D. L. Sivco, A. Tredicucci, M. C. Wanke, A. M. Sergent, and A. Y. Cho, “Far-infrared surface-plasmon quantum-cascade lasers at 21.5 μm and 24 μm wavelengths,” Appl. Phys. Lett.78(18), 2620 (2001).
[CrossRef]

Capolino, F.

Cho, A. Y.

R. Colombelli, F. Capasso, C. Gmachl, A. L. Hutchinson, D. L. Sivco, A. Tredicucci, M. C. Wanke, A. M. Sergent, and A. Y. Cho, “Far-infrared surface-plasmon quantum-cascade lasers at 21.5 μm and 24 μm wavelengths,” Appl. Phys. Lett.78(18), 2620 (2001).
[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(11), 936–941 (2012).
[CrossRef] [PubMed]

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(11), 936–941 (2012).
[CrossRef] [PubMed]

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(11), 936–941 (2012).
[CrossRef] [PubMed]

M. Choi, S. H. Lee, Y. Kim, S. B. Kang, J. Shin, M. H. Kwak, K. Y. Kang, Y. H. Lee, N. Park, and B. Min, “A terahertz metamaterial with unnaturally high refractive index,” Nature470(7334), 369–373 (2011).
[CrossRef] [PubMed]

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(11), 936–941 (2012).
[CrossRef] [PubMed]

Ciattoni, A.

Clarysse, T.

A. Satta, E. Simoen, T. Clarysse, T. Janssens, A. Benedetti, B. De Jaeger, M. Meuris, and W. Vandervorst, “Diffusion, activation, and recrystallization of boron implanted in preamorphized and crystalline germanium,” Appl. Phys. Lett.87(17), 172109 (2005).
[CrossRef]

Colombelli, R.

R. Colombelli, F. Capasso, C. Gmachl, A. L. Hutchinson, D. L. Sivco, A. Tredicucci, M. C. Wanke, A. M. Sergent, and A. Y. Cho, “Far-infrared surface-plasmon quantum-cascade lasers at 21.5 μm and 24 μm wavelengths,” Appl. Phys. Lett.78(18), 2620 (2001).
[CrossRef]

Columbo, L.

Davies, A. G.

R. Köhler, A. Tredicucci, F. Beltram, H. E. Beere, E. H. Linfield, A. G. Davies, D. A. Ritchie, R. C. Iotti, and F. Rossi, “Terahertz semiconductor-heterostructure laser,” Nature417(6885), 156–159 (2002).
[CrossRef] [PubMed]

De Jaeger, B.

A. Satta, E. Simoen, T. Clarysse, T. Janssens, A. Benedetti, B. De Jaeger, M. Meuris, and W. Vandervorst, “Diffusion, activation, and recrystallization of boron implanted in preamorphized and crystalline germanium,” Appl. Phys. Lett.87(17), 172109 (2005).
[CrossRef]

Dubonos, S. V.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, “Two-dimensional gas of massless Dirac fermions in graphene,” Nature438(7065), 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,” Science306(5696), 666–669 (2004).
[CrossRef] [PubMed]

Engheta, N.

A. Vakil and N. Engheta, “Transformation Optics Using Graphene,” Science332(6035), 1291–1294 (2011).
[CrossRef] [PubMed]

Faist, J.

M. Beck, D. Hofstetter, T. Aellen, J. Faist, U. Oesterle, M. Ilegems, E. Gini, and H. Melchior, “Continuous Wave Operation of a Mid-Infrared Semiconductor Laser at Room Temperature,” Science295(5553), 301–305 (2002).
[CrossRef] [PubMed]

Fan, S. H.

J. Shin, J. T. Shen, and S. H. Fan, “Three-Dimensional Metamaterials with an Ultrahigh Effective Refractive Index over a Broad Bandwidth,” Phys. Rev. Lett.102(9), 093903 (2009).
[CrossRef] [PubMed]

Firsov, A. A.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, “Two-dimensional gas of massless Dirac fermions in graphene,” Nature438(7065), 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,” Science306(5696), 666–669 (2004).
[CrossRef] [PubMed]

Foss, C. A.

C. A. Foss, G. L. Hornyak, J. A. Stockert, and C. R. Martin, “Template-synthesized nanoscopic gold Particles: optical spectra and the effects of particle size and shape,” J. Phys. Chem.98(11), 2963–2971 (1994).
[CrossRef]

Gao, J.

Geim, A. K.

T. Stauber, N. M. R. Peres, and A. K. Geim, “Optical conductivity of graphene in the visible region of the spectrum,” Phys. Rev. B78(8), 085432 (2008).
[CrossRef]

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, “Two-dimensional gas of massless Dirac fermions in graphene,” Nature438(7065), 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,” Science306(5696), 666–669 (2004).
[CrossRef] [PubMed]

Gini, E.

M. Beck, D. Hofstetter, T. Aellen, J. Faist, U. Oesterle, M. Ilegems, E. Gini, and H. Melchior, “Continuous Wave Operation of a Mid-Infrared Semiconductor Laser at Room Temperature,” Science295(5553), 301–305 (2002).
[CrossRef] [PubMed]

Gmachl, C.

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K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, “Two-dimensional gas of massless Dirac fermions in graphene,” Nature438(7065), 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,” Science306(5696), 666–669 (2004).
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He, Y.

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M. Beck, D. Hofstetter, T. Aellen, J. Faist, U. Oesterle, M. Ilegems, E. Gini, and H. Melchior, “Continuous Wave Operation of a Mid-Infrared Semiconductor Laser at Room Temperature,” Science295(5553), 301–305 (2002).
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C. A. Foss, G. L. Hornyak, J. A. Stockert, and C. R. Martin, “Template-synthesized nanoscopic gold Particles: optical spectra and the effects of particle size and shape,” J. Phys. Chem.98(11), 2963–2971 (1994).
[CrossRef]

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R. Colombelli, F. Capasso, C. Gmachl, A. L. Hutchinson, D. L. Sivco, A. Tredicucci, M. C. Wanke, A. M. Sergent, and A. Y. Cho, “Far-infrared surface-plasmon quantum-cascade lasers at 21.5 μm and 24 μm wavelengths,” Appl. Phys. Lett.78(18), 2620 (2001).
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M. Beck, D. Hofstetter, T. Aellen, J. Faist, U. Oesterle, M. Ilegems, E. Gini, and H. Melchior, “Continuous Wave Operation of a Mid-Infrared Semiconductor Laser at Room Temperature,” Science295(5553), 301–305 (2002).
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F. V. Iorsh, I. S. Mukhin, I. V. Shadrivov, P. A. Belov, and Y. S. Kivshar, “Hyperbolic metamaterials based on multilayer graphene structures,” Phys. Rev. B87(7), 075416 (2013).
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R. Köhler, A. Tredicucci, F. Beltram, H. E. Beere, E. H. Linfield, A. G. Davies, D. A. Ritchie, R. C. Iotti, and F. Rossi, “Terahertz semiconductor-heterostructure laser,” Nature417(6885), 156–159 (2002).
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A. Satta, E. Simoen, T. Clarysse, T. Janssens, A. Benedetti, B. De Jaeger, M. Meuris, and W. Vandervorst, “Diffusion, activation, and recrystallization of boron implanted in preamorphized and crystalline germanium,” Appl. Phys. Lett.87(17), 172109 (2005).
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Jiang, D.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, “Two-dimensional gas of massless Dirac fermions in graphene,” Nature438(7065), 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,” Science306(5696), 666–669 (2004).
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Jiang, X.

Jin, Y.

Kang, K. Y.

M. Choi, S. H. Lee, Y. Kim, S. B. Kang, J. Shin, M. H. Kwak, K. Y. Kang, Y. H. Lee, N. Park, and B. Min, “A terahertz metamaterial with unnaturally high refractive index,” Nature470(7334), 369–373 (2011).
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M. Choi, S. H. Lee, Y. Kim, S. B. Kang, J. Shin, M. H. Kwak, K. Y. Kang, Y. H. Lee, N. Park, and B. Min, “A terahertz metamaterial with unnaturally high refractive index,” Nature470(7334), 369–373 (2011).
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K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, “Two-dimensional gas of massless Dirac fermions in graphene,” Nature438(7065), 197–200 (2005).
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G. V. Naik, J. Liu, A. V. Kildishev, V. M. Shalaev, and A. Boltasseva, “Demonstration of Al:ZnO as a plasmonic component for near-infrared metamaterials,” Proc. Natl. Acad. Sci. U.S.A.109(23), 8834–8838 (2012).
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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(11), 936–941 (2012).
[CrossRef] [PubMed]

Kim, Y.

M. Choi, S. H. Lee, Y. Kim, S. B. Kang, J. Shin, M. H. Kwak, K. Y. Kang, Y. H. Lee, N. Park, and B. Min, “A terahertz metamaterial with unnaturally high refractive index,” Nature470(7334), 369–373 (2011).
[CrossRef] [PubMed]

Kivshar, Y. S.

F. V. Iorsh, I. S. Mukhin, I. V. Shadrivov, P. A. Belov, and Y. S. Kivshar, “Hyperbolic metamaterials based on multilayer graphene structures,” Phys. Rev. B87(7), 075416 (2013).
[CrossRef]

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M. Björk, J. Knoch, H. Schmid, H. Riel, and W. Riess, “Silicon nanowire tunneling field-effect transistors,” Appl. Phys. Lett.92(19), 193504 (2008).
[CrossRef]

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R. Köhler, A. Tredicucci, F. Beltram, H. E. Beere, E. H. Linfield, A. G. Davies, D. A. Ritchie, R. C. Iotti, and F. Rossi, “Terahertz semiconductor-heterostructure laser,” Nature417(6885), 156–159 (2002).
[CrossRef] [PubMed]

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Kwak, M. H.

M. Choi, S. H. Lee, Y. Kim, S. B. Kang, J. Shin, M. H. Kwak, K. Y. Kang, Y. H. Lee, N. Park, and B. Min, “A terahertz metamaterial with unnaturally high refractive index,” Nature470(7334), 369–373 (2011).
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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(11), 936–941 (2012).
[CrossRef] [PubMed]

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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(11), 936–941 (2012).
[CrossRef] [PubMed]

M. Choi, S. H. Lee, Y. Kim, S. B. Kang, J. Shin, M. H. Kwak, K. Y. Kang, Y. H. Lee, N. Park, and B. Min, “A terahertz metamaterial with unnaturally high refractive index,” Nature470(7334), 369–373 (2011).
[CrossRef] [PubMed]

Lee, S. S.

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(11), 936–941 (2012).
[CrossRef] [PubMed]

Lee, Y. H.

M. Choi, S. H. Lee, Y. Kim, S. B. Kang, J. Shin, M. H. Kwak, K. Y. Kang, Y. H. Lee, N. Park, and B. Min, “A terahertz metamaterial with unnaturally high refractive index,” Nature470(7334), 369–373 (2011).
[CrossRef] [PubMed]

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F. Y. Meng, Q. Wu, and L. W. Li, “Transmission characteristics of wave modes in a rectangular waveguide filled with anisotropic metamaterial,” Appl. Phys., A Mater. Sci. Process.94(4), 747–753 (2009).
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R. Köhler, A. Tredicucci, F. Beltram, H. E. Beere, E. H. Linfield, A. G. Davies, D. A. Ritchie, R. C. Iotti, and F. Rossi, “Terahertz semiconductor-heterostructure laser,” Nature417(6885), 156–159 (2002).
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G. V. Naik, J. Liu, A. V. Kildishev, V. M. Shalaev, and A. Boltasseva, “Demonstration of Al:ZnO as a plasmonic component for near-infrared metamaterials,” Proc. Natl. Acad. Sci. U.S.A.109(23), 8834–8838 (2012).
[CrossRef] [PubMed]

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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(11), 936–941 (2012).
[CrossRef] [PubMed]

Liu, Y.

Martin, C. R.

C. A. Foss, G. L. Hornyak, J. A. Stockert, and C. R. Martin, “Template-synthesized nanoscopic gold Particles: optical spectra and the effects of particle size and shape,” J. Phys. Chem.98(11), 2963–2971 (1994).
[CrossRef]

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M. Beck, D. Hofstetter, T. Aellen, J. Faist, U. Oesterle, M. Ilegems, E. Gini, and H. Melchior, “Continuous Wave Operation of a Mid-Infrared Semiconductor Laser at Room Temperature,” Science295(5553), 301–305 (2002).
[CrossRef] [PubMed]

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F. Y. Meng, Q. Wu, and L. W. Li, “Transmission characteristics of wave modes in a rectangular waveguide filled with anisotropic metamaterial,” Appl. Phys., A Mater. Sci. Process.94(4), 747–753 (2009).
[CrossRef]

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A. Satta, E. Simoen, T. Clarysse, T. Janssens, A. Benedetti, B. De Jaeger, M. Meuris, and W. Vandervorst, “Diffusion, activation, and recrystallization of boron implanted in preamorphized and crystalline germanium,” Appl. Phys. Lett.87(17), 172109 (2005).
[CrossRef]

Min, B.

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(11), 936–941 (2012).
[CrossRef] [PubMed]

M. Choi, S. H. Lee, Y. Kim, S. B. Kang, J. Shin, M. H. Kwak, K. Y. Kang, Y. H. Lee, N. Park, and B. Min, “A terahertz metamaterial with unnaturally high refractive index,” Nature470(7334), 369–373 (2011).
[CrossRef] [PubMed]

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K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, “Two-dimensional gas of massless Dirac fermions in graphene,” Nature438(7065), 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,” Science306(5696), 666–669 (2004).
[CrossRef] [PubMed]

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F. V. Iorsh, I. S. Mukhin, I. V. Shadrivov, P. A. Belov, and Y. S. Kivshar, “Hyperbolic metamaterials based on multilayer graphene structures,” Phys. Rev. B87(7), 075416 (2013).
[CrossRef]

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G. V. Naik, J. Liu, A. V. Kildishev, V. M. Shalaev, and A. Boltasseva, “Demonstration of Al:ZnO as a plasmonic component for near-infrared metamaterials,” Proc. Natl. Acad. Sci. U.S.A.109(23), 8834–8838 (2012).
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G. V. Naik and A. Boltasseva, “A comparative study of semiconductor-based plasmonic metamaterials,” Metamaterials (Amst.)5(1), 1–7 (2011).
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A. N. Grigorenko, M. Polini, and K. S. Novoselov, “Graphene plasmonics,” Nat. Photonics6(11), 749–758 (2012).
[CrossRef]

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, “Two-dimensional gas of massless Dirac fermions in graphene,” Nature438(7065), 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,” Science306(5696), 666–669 (2004).
[CrossRef] [PubMed]

Oesterle, U.

M. Beck, D. Hofstetter, T. Aellen, J. Faist, U. Oesterle, M. Ilegems, E. Gini, and H. Melchior, “Continuous Wave Operation of a Mid-Infrared Semiconductor Laser at Room Temperature,” Science295(5553), 301–305 (2002).
[CrossRef] [PubMed]

Othman, M. A. K.

Park, N.

M. Choi, S. H. Lee, Y. Kim, S. B. Kang, J. Shin, M. H. Kwak, K. Y. Kang, Y. H. Lee, N. Park, and B. Min, “A terahertz metamaterial with unnaturally high refractive index,” Nature470(7334), 369–373 (2011).
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J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science312(5781), 1780–1782 (2006).
[CrossRef] [PubMed]

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science305(5685), 788–792 (2004).
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T. Stauber, N. M. R. Peres, and A. K. Geim, “Optical conductivity of graphene in the visible region of the spectrum,” Phys. Rev. B78(8), 085432 (2008).
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Polini, M.

A. N. Grigorenko, M. Polini, and K. S. Novoselov, “Graphene plasmonics,” Nat. Photonics6(11), 749–758 (2012).
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X. Yang, J. Yao, J. Rho, X. Yin, and X. Zhang, “Experimental realization of three-dimensional indefinite cavities at the nanoscale with anomalous scaling laws,” Nat. Photonics6(7), 450–454 (2012).
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M. Björk, J. Knoch, H. Schmid, H. Riel, and W. Riess, “Silicon nanowire tunneling field-effect transistors,” Appl. Phys. Lett.92(19), 193504 (2008).
[CrossRef]

Riess, W.

M. Björk, J. Knoch, H. Schmid, H. Riel, and W. Riess, “Silicon nanowire tunneling field-effect transistors,” Appl. Phys. Lett.92(19), 193504 (2008).
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Ritchie, D. A.

R. Köhler, A. Tredicucci, F. Beltram, H. E. Beere, E. H. Linfield, A. G. Davies, D. A. Ritchie, R. C. Iotti, and F. Rossi, “Terahertz semiconductor-heterostructure laser,” Nature417(6885), 156–159 (2002).
[CrossRef] [PubMed]

Rizza, C.

Rossi, F.

R. Köhler, A. Tredicucci, F. Beltram, H. E. Beere, E. H. Linfield, A. G. Davies, D. A. Ritchie, R. C. Iotti, and F. Rossi, “Terahertz semiconductor-heterostructure laser,” Nature417(6885), 156–159 (2002).
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Satta, A.

A. Satta, E. Simoen, T. Clarysse, T. Janssens, A. Benedetti, B. De Jaeger, M. Meuris, and W. Vandervorst, “Diffusion, activation, and recrystallization of boron implanted in preamorphized and crystalline germanium,” Appl. Phys. Lett.87(17), 172109 (2005).
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Schmid, H.

M. Björk, J. Knoch, H. Schmid, H. Riel, and W. Riess, “Silicon nanowire tunneling field-effect transistors,” Appl. Phys. Lett.92(19), 193504 (2008).
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J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science312(5781), 1780–1782 (2006).
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Sergent, A. M.

R. Colombelli, F. Capasso, C. Gmachl, A. L. Hutchinson, D. L. Sivco, A. Tredicucci, M. C. Wanke, A. M. Sergent, and A. Y. Cho, “Far-infrared surface-plasmon quantum-cascade lasers at 21.5 μm and 24 μm wavelengths,” Appl. Phys. Lett.78(18), 2620 (2001).
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Shadrivov, I. V.

F. V. Iorsh, I. S. Mukhin, I. V. Shadrivov, P. A. Belov, and Y. S. Kivshar, “Hyperbolic metamaterials based on multilayer graphene structures,” Phys. Rev. B87(7), 075416 (2013).
[CrossRef]

Shalaev, V. M.

G. V. Naik, J. Liu, A. V. Kildishev, V. M. Shalaev, and A. Boltasseva, “Demonstration of Al:ZnO as a plasmonic component for near-infrared metamaterials,” Proc. Natl. Acad. Sci. U.S.A.109(23), 8834–8838 (2012).
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J. Shin, J. T. Shen, and S. H. Fan, “Three-Dimensional Metamaterials with an Ultrahigh Effective Refractive Index over a Broad Bandwidth,” Phys. Rev. Lett.102(9), 093903 (2009).
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M. Choi, S. H. Lee, Y. Kim, S. B. Kang, J. Shin, M. H. Kwak, K. Y. Kang, Y. H. Lee, N. Park, and B. Min, “A terahertz metamaterial with unnaturally high refractive index,” Nature470(7334), 369–373 (2011).
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J. Shin, J. T. Shen, and S. H. Fan, “Three-Dimensional Metamaterials with an Ultrahigh Effective Refractive Index over a Broad Bandwidth,” Phys. Rev. Lett.102(9), 093903 (2009).
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A. Satta, E. Simoen, T. Clarysse, T. Janssens, A. Benedetti, B. De Jaeger, M. Meuris, and W. Vandervorst, “Diffusion, activation, and recrystallization of boron implanted in preamorphized and crystalline germanium,” Appl. Phys. Lett.87(17), 172109 (2005).
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R. Colombelli, F. Capasso, C. Gmachl, A. L. Hutchinson, D. L. Sivco, A. Tredicucci, M. C. Wanke, A. M. Sergent, and A. Y. Cho, “Far-infrared surface-plasmon quantum-cascade lasers at 21.5 μm and 24 μm wavelengths,” Appl. Phys. Lett.78(18), 2620 (2001).
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J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science312(5781), 1780–1782 (2006).
[CrossRef] [PubMed]

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science305(5685), 788–792 (2004).
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R. Soref, “Mid-infrared photonics in silicon and germanium,” Nat. Photonics4(8), 495–497 (2010).
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Spinozzi, E.

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T. Stauber, N. M. R. Peres, and A. K. Geim, “Optical conductivity of graphene in the visible region of the spectrum,” Phys. Rev. B78(8), 085432 (2008).
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C. A. Foss, G. L. Hornyak, J. A. Stockert, and C. R. Martin, “Template-synthesized nanoscopic gold Particles: optical spectra and the effects of particle size and shape,” J. Phys. Chem.98(11), 2963–2971 (1994).
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Tassin, P.

Tredicucci, A.

R. Köhler, A. Tredicucci, F. Beltram, H. E. Beere, E. H. Linfield, A. G. Davies, D. A. Ritchie, R. C. Iotti, and F. Rossi, “Terahertz semiconductor-heterostructure laser,” Nature417(6885), 156–159 (2002).
[CrossRef] [PubMed]

R. Colombelli, F. Capasso, C. Gmachl, A. L. Hutchinson, D. L. Sivco, A. Tredicucci, M. C. Wanke, A. M. Sergent, and A. Y. Cho, “Far-infrared surface-plasmon quantum-cascade lasers at 21.5 μm and 24 μm wavelengths,” Appl. Phys. Lett.78(18), 2620 (2001).
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A. Satta, E. Simoen, T. Clarysse, T. Janssens, A. Benedetti, B. De Jaeger, M. Meuris, and W. Vandervorst, “Diffusion, activation, and recrystallization of boron implanted in preamorphized and crystalline germanium,” Appl. Phys. Lett.87(17), 172109 (2005).
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Wanke, M. C.

R. Colombelli, F. Capasso, C. Gmachl, A. L. Hutchinson, D. L. Sivco, A. Tredicucci, M. C. Wanke, A. M. Sergent, and A. Y. Cho, “Far-infrared surface-plasmon quantum-cascade lasers at 21.5 μm and 24 μm wavelengths,” Appl. Phys. Lett.78(18), 2620 (2001).
[CrossRef]

Wiltshire, M. C. K.

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science305(5685), 788–792 (2004).
[CrossRef] [PubMed]

Wu, Q.

F. Y. Meng, Q. Wu, and L. W. Li, “Transmission characteristics of wave modes in a rectangular waveguide filled with anisotropic metamaterial,” Appl. Phys., A Mater. Sci. Process.94(4), 747–753 (2009).
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Yang, J.

Yang, L.

Yang, X.

Y. He, S. He, J. Gao, and X. Yang, “Nanoscale metamaterial optical waveguides with ultrahigh refractive indices,” J. Opt. Soc. Am. B29(9), 2559–2566 (2012).
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X. Yang, J. Yao, J. Rho, X. Yin, and X. Zhang, “Experimental realization of three-dimensional indefinite cavities at the nanoscale with anomalous scaling laws,” Nat. Photonics6(7), 450–454 (2012).
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J. Yao, X. Yang, X. Yin, G. Bartal, and X. Zhang, “Three-dimensional nanometer-scale optical cavities of indefinite medium,” Proc. Natl. Acad. Sci. U.S.A.108(28), 11327–11331 (2011).
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Yao, J.

X. Yang, J. Yao, J. Rho, X. Yin, and X. Zhang, “Experimental realization of three-dimensional indefinite cavities at the nanoscale with anomalous scaling laws,” Nat. Photonics6(7), 450–454 (2012).
[CrossRef]

J. Yao, X. Yang, X. Yin, G. Bartal, and X. Zhang, “Three-dimensional nanometer-scale optical cavities of indefinite medium,” Proc. Natl. Acad. Sci. U.S.A.108(28), 11327–11331 (2011).
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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(11), 936–941 (2012).
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X. Yang, J. Yao, J. Rho, X. Yin, and X. Zhang, “Experimental realization of three-dimensional indefinite cavities at the nanoscale with anomalous scaling laws,” Nat. Photonics6(7), 450–454 (2012).
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J. Yao, X. Yang, X. Yin, G. Bartal, and X. Zhang, “Three-dimensional nanometer-scale optical cavities of indefinite medium,” Proc. Natl. Acad. Sci. U.S.A.108(28), 11327–11331 (2011).
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Figures (5)

Fig. 1
Fig. 1

The conductivity of graphene as a function of free-space wavelength λ0 (a) and the biased voltage Vbiased (c). The dependences of the effective permittivity εx, εy and εz. of the multilayer metamaterial on the wavelength λ0 (b) and biased voltage Vbiased (d).

Fig. 2
Fig. 2

The schematic of the waveguide. The Lx and Ly represents horizontal and vertical dimensions of the cross section of the waveguide, while Lz is the length of the waveguide.

Fig. 3
Fig. 3

The field distributions of modes with order (1,my), which are calculated from the effective medium approach (a) and the multilayer structure (b). Effective refractive indices along the propagation direction neff,z (c) and the propagation lengths Lm (d) as functions of free-space wavelength λ0. The solid and the dot-dashed lines represent the results calculated from the effective medium approach and the multilayer structure, respectively.

Fig. 4
Fig. 4

The dependences of mode indices real(neff,z) (a),(c) and the propagation lengths Lm (b),(d) on the waveguide transparent dimensions Lx and Ly are calculated. The results are calculated from the multilayer structure (a-b) and effective medium approach (c-d), respectively.

Fig. 5
Fig. 5

The dependences of the mode indices of modes (1, my) (a) and propagation lengths (b) on the biased voltage.

Equations (6)

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σ intra (ω,T,τ, μ c )=j e 2 k T B π 2 (ωj τ 1 ) [ μ c k B T +2ln( e μ c / k B T +1) ]
σ inter (ω,T,τ, μ c )=j e 2 4π ln( 2| μ c |(ωj τ 1 ) 2| μ c |+(ωj τ 1 ) ),
| μ c |= ν F π| a 0 ( V g V Dirac )| ,
ε x = ε z = f g ε g,t +(1 f g ) ε d , ε y = ε g,n ε d f g ε d +(1 f g ) ε g,n ,
ε g,t (K,ω)=1 i σ g (K,ω) ω ε 0 d ,
k x 2 + k z 2 ε y + k y 2 ε x = k 0 2 ,

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