Abstract

We propose deep-subwavelength optical waveguides based on metal–dielectric multilayer indefinite metamaterials with ultrahigh effective refractive indices. Waveguide modes with different mode orders are systematically analyzed with numerical simulations based on both metal–dielectric multilayer structures and the effective medium approach. The dependences of waveguide mode indices, propagation lengths, and mode areas on different mode orders, free-space wavelengths, and sizes of waveguide cross sections are studied. Furthermore, waveguide modes are also illustrated with iso-frequency contours in the wave vector space in order to investigate the mechanism of waveguide mode cutoff for high-order modes. The deep-subwavelength optical waveguide with a size smaller than λ0/50 and a mode area in the order of 104λ02 is realized, and an ultrahigh effective refractive index up to 62.0 is achieved at the telecommunication wavelength. This new type of metamaterial optical waveguide opens up opportunities for various applications in enhanced light–matter interactions.

© 2012 Optical Society of America

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2012 (3)

Y. Cui, K. H. Fung, J. Xu, H. Ma, Y. Jin, S. He, and N. X. Fang, “Ultrabroadband light absorption by a sawtooth anisotropic metamaterial slab,” Nano Lett. 12, 1443–1447 (2012).
[CrossRef]

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. Photon. 6, 450–454 (2012).
[CrossRef]

A. Chandran, E. S. Barnard, J. S. White, and M. L. Brongersma, “Metal-dielectric-metal surface plasmon-polariton resonators,” Phys. Rev. B 85, 085416 (2012).
[CrossRef]

2011 (5)

M. Yan, L. Thylén, and M. Qiu, “Layered metal-dielectric waveguide: subwavelength guidance, leveraged modulation sensitivity in mode index, and reversed mode ordering,” Opt. Express 19, 3818–3824 (2011).
[CrossRef]

J. Sun, J. Zhou, B. Li, and F. Kang, “Indefinite permittivity and negative refraction in natural material: graphite,” Appl. Phys. Lett. 98, 101901 (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. 108, 11327–11331 (2011).
[CrossRef]

X. Yang, Y. Liu, R. F. Oulton, X. Yin, and X. Zhang, “Optical forces in hybrid plasmonic waveguides,” Nano Lett. 11, 321–328 (2011).
[CrossRef]

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,” Nature 470, 369–373 (2011).
[CrossRef]

2010 (7)

Z. Jacob, J. Y. Kim, G. Naik, A. Boltasseva, E. Narimanov, and V. Shalaev, “Engineering photonic density of states using metamaterials,” Appl. Phys. B 100, 215–218 (2010).
[CrossRef]

M. A. Noginov, H. Li, Y. A. Barnakov, D. Dryden, G. Nataraj, G. Zhu, C. E. Bonner, M. Mayy, Z. Jacob, and E. E. Narimanov, “Controlling spontaneous emission with metamaterials,” Opt. Lett. 35, 1863–1865 (2010).
[CrossRef]

C. H. Gan and P. Lalanne, “Well-confined surface plasmon polaritons for sensing applications in the near-infrared,” Opt. Lett. 35, 610–612 (2010).
[CrossRef]

S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H.-K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature 466, 735–738 (2010).
[CrossRef]

T. M. Babinec, J. M. HausmannBirgit, M. Khan, Y. Zhang, J. R. Maze, P. R. Hemmer, and M. Loncar, “A diamond nanowire single-photon source,” Nature Nanotechnol. 5, 195–199 (2010).
[CrossRef]

W. T. Lu and S. Sridhar, “Slow light, open-cavity formation, and large longitudinal electric field on a slab waveguide made of indefinite permittivity metamaterials,” Phys. Rev. A 82, 013811 (2010).
[CrossRef]

W. Chen, M. D. Thoreson, S. Ishii, A. V. Kildishev, and V. M. Shalaev, “Ultra-thin ultra-smooth and low-loss silver films on a germanium wetting layer,” Opt. Express 18, 5124–5134 (2010).
[CrossRef]

2009 (6)

M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460, 1110–1112 (2009).
[CrossRef]

R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461, 629–632 (2009).
[CrossRef]

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 94, 747–753 (2009).
[CrossRef]

Z. Shi, G. Piredda, A. C. Liapis, M. A. Nelson, L. Novotny, and R. W. Boyd, “Surface-plasmon polaritons on metal-dielectric nanocomposite films,” Opt. Lett. 34, 3535–3537 (2009).
[CrossRef]

T. Jiang, J. Zhao, and Y. Feng, “Stopping light by an air waveguide with anisotropic metamaterial cladding,” Opt. Express 17, 170–177 (2009).
[CrossRef]

J. Shin, J.-T. Shen, and S. Fan, “Three-dimensional metamaterials with an ultrahigh effective refractive index over a broad bandwidth,” Phys. Rev. Lett. 102, 093903 (2009).
[CrossRef]

2008 (5)

Y. J. Huang, W. T. Lu, and S. Sridhar, “Nanowire waveguide made from extremely anisotropic metamaterials,” Phys. Rev. A 77, 063836 (2008).
[CrossRef]

S. Han, Y. Xiong, D. Genov, Z. Liu, G. Bartal, and X. Zhang, “Ray optics at a deep-subwavelength scale: a transformation optics approach,” Nano Lett. 8, 4243–4247 (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. Express 16, 15439–15448 (2008).
[CrossRef]

J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,” Science 321, 930 (2008).
[CrossRef]

G.-D. Xu, T. Pan, T.-C. Zang, and J. Sun, “Characteristics of guided waves in indefinite-medium waveguides,” Opt. Commun. 281, 2819–2825 (2008).
[CrossRef]

2007 (5)

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315, 1686 (2007).
[CrossRef]

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat Photon 1, 224–227 (2007).
[CrossRef]

J. Elser, A. A. Govyadinov, I. Avrutsky, I. Salakhutdinov, and V. A. Podolskiy, “Plasmonic nanolayer composites: coupled plasmon polaritons, effective-medium response, and subdiffraction light manipulation,” J. Nanomater. 2007, 79469 (2007).
[CrossRef]

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

Q. Lin, O. J. Painter, and G. P. Agrawal, “Nonlinear optical phenomena in silicon waveguides: modeling and applications,” Opt. Express 15, 16604–16644 (2007).
[CrossRef]

2006 (5)

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[CrossRef]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
[CrossRef]

Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical hyperlens: far-field imaging beyond the diffraction limit,” Opt. Express 14, 8247–8256 (2006).
[CrossRef]

A. Salandrino and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystals: theory and simulations,” Phys. Rev. B 74, 075103–075105 (2006).
[CrossRef]

A. A. Govyadinov and V. A. Podolskiy, “Metamaterial photonic funnels for subdiffraction light compression and propagation,” Phys. Rev. B 73, 155108 (2006).
[CrossRef]

2005 (1)

J. T. Shen, P. B. Catrysse, and S. Fan, “Mechanism for designing metallic metamaterials with a high index of refraction,” Phys. Rev. Lett. 94, 197401 (2005).
[CrossRef]

2003 (1)

D. R. Smith and D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett. 90, 077405 (2003).
[CrossRef]

2001 (1)

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
[CrossRef]

1999 (1)

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech 47, 2075–2084 (1999).
[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, 2963–2971 (1994).
[CrossRef]

1972 (1)

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[CrossRef]

1968 (1)

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of permittivity and permeability,” Sov. Phys. Usp. 10, 509–514 (1968).
[CrossRef]

Agrawal, G. P.

Alekseyev, L. V.

Avrutsky, I.

J. Elser, A. A. Govyadinov, I. Avrutsky, I. Salakhutdinov, and V. A. Podolskiy, “Plasmonic nanolayer composites: coupled plasmon polaritons, effective-medium response, and subdiffraction light manipulation,” J. Nanomater. 2007, 79469 (2007).
[CrossRef]

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

Babinec, T. M.

T. M. Babinec, J. M. HausmannBirgit, M. Khan, Y. Zhang, J. R. Maze, P. R. Hemmer, and M. Loncar, “A diamond nanowire single-photon source,” Nature Nanotechnol. 5, 195–199 (2010).
[CrossRef]

Bakker, R.

M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460, 1110–1112 (2009).
[CrossRef]

Barnakov, Y. A.

Barnard, E. S.

A. Chandran, E. S. Barnard, J. S. White, and M. L. Brongersma, “Metal-dielectric-metal surface plasmon-polariton resonators,” Phys. Rev. B 85, 085416 (2012).
[CrossRef]

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. 108, 11327–11331 (2011).
[CrossRef]

R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461, 629–632 (2009).
[CrossRef]

J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,” Science 321, 930 (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. Express 16, 15439–15448 (2008).
[CrossRef]

S. Han, Y. Xiong, D. Genov, Z. Liu, G. Bartal, and X. Zhang, “Ray optics at a deep-subwavelength scale: a transformation optics approach,” Nano Lett. 8, 4243–4247 (2008).
[CrossRef]

Belgrave, A. M.

M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460, 1110–1112 (2009).
[CrossRef]

Boltasseva, A.

Z. Jacob, J. Y. Kim, G. Naik, A. Boltasseva, E. Narimanov, and V. Shalaev, “Engineering photonic density of states using metamaterials,” Appl. Phys. B 100, 215–218 (2010).
[CrossRef]

Bonner, C. E.

Boyd, R. W.

Brongersma, M. L.

A. Chandran, E. S. Barnard, J. S. White, and M. L. Brongersma, “Metal-dielectric-metal surface plasmon-polariton resonators,” Phys. Rev. B 85, 085416 (2012).
[CrossRef]

Cai, W.

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat Photon 1, 224–227 (2007).
[CrossRef]

Catrysse, P. B.

J. T. Shen, P. B. Catrysse, and S. Fan, “Mechanism for designing metallic metamaterials with a high index of refraction,” Phys. Rev. Lett. 94, 197401 (2005).
[CrossRef]

Chandran, A.

A. Chandran, E. S. Barnard, J. S. White, and M. L. Brongersma, “Metal-dielectric-metal surface plasmon-polariton resonators,” Phys. Rev. B 85, 085416 (2012).
[CrossRef]

Chen, W.

Chettiar, U. K.

S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H.-K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature 466, 735–738 (2010).
[CrossRef]

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat Photon 1, 224–227 (2007).
[CrossRef]

Choi, M.

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,” Nature 470, 369–373 (2011).
[CrossRef]

Christy, R. W.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[CrossRef]

Cui, Y.

Y. Cui, K. H. Fung, J. Xu, H. Ma, Y. Jin, S. He, and N. X. Fang, “Ultrabroadband light absorption by a sawtooth anisotropic metamaterial slab,” Nano Lett. 12, 1443–1447 (2012).
[CrossRef]

Cummer, S. A.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
[CrossRef]

Dai, L.

R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461, 629–632 (2009).
[CrossRef]

Drachev, V. P.

S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H.-K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature 466, 735–738 (2010).
[CrossRef]

Dryden, D.

Elser, J.

J. Elser, A. A. Govyadinov, I. Avrutsky, I. Salakhutdinov, and V. A. Podolskiy, “Plasmonic nanolayer composites: coupled plasmon polaritons, effective-medium response, and subdiffraction light manipulation,” J. Nanomater. 2007, 79469 (2007).
[CrossRef]

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J. Shin, J.-T. Shen, and S. Fan, “Three-dimensional metamaterials with an ultrahigh effective refractive index over a broad bandwidth,” Phys. Rev. Lett. 102, 093903 (2009).
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Y. Cui, K. H. Fung, J. Xu, H. Ma, Y. Jin, S. He, and N. X. Fang, “Ultrabroadband light absorption by a sawtooth anisotropic metamaterial slab,” Nano Lett. 12, 1443–1447 (2012).
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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, 2963–2971 (1994).
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Y. Cui, K. H. Fung, J. Xu, H. Ma, Y. Jin, S. He, and N. X. Fang, “Ultrabroadband light absorption by a sawtooth anisotropic metamaterial slab,” Nano Lett. 12, 1443–1447 (2012).
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J. Elser, A. A. Govyadinov, I. Avrutsky, I. Salakhutdinov, and V. A. Podolskiy, “Plasmonic nanolayer composites: coupled plasmon polaritons, effective-medium response, and subdiffraction light manipulation,” J. Nanomater. 2007, 79469 (2007).
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X. Ni, S. Ishii, M. D. Thoreson, V. M. Shalaev, S. Han, S. Lee, and A. V. Kildishev, “Gain-assisted hyperbolic metamaterials,” in Quantum Electronics and Laser Science Conference (QELS), OSA Technical Digest (Optical Society of America, 2012), paper QTu1G.

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T. M. Babinec, J. M. HausmannBirgit, M. Khan, Y. Zhang, J. R. Maze, P. R. Hemmer, and M. Loncar, “A diamond nanowire single-photon source,” Nature Nanotechnol. 5, 195–199 (2010).
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Y. Cui, K. H. Fung, J. Xu, H. Ma, Y. Jin, S. He, and N. X. Fang, “Ultrabroadband light absorption by a sawtooth anisotropic metamaterial slab,” Nano Lett. 12, 1443–1447 (2012).
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T. M. Babinec, J. M. HausmannBirgit, M. Khan, Y. Zhang, J. R. Maze, P. R. Hemmer, and M. Loncar, “A diamond nanowire single-photon source,” Nature Nanotechnol. 5, 195–199 (2010).
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J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech 47, 2075–2084 (1999).
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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, 2963–2971 (1994).
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W. Chen, M. D. Thoreson, S. Ishii, A. V. Kildishev, and V. M. Shalaev, “Ultra-thin ultra-smooth and low-loss silver films on a germanium wetting layer,” Opt. Express 18, 5124–5134 (2010).
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Jiang, T.

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Y. Cui, K. H. Fung, J. Xu, H. Ma, Y. Jin, S. He, and N. X. Fang, “Ultrabroadband light absorption by a sawtooth anisotropic metamaterial slab,” Nano Lett. 12, 1443–1447 (2012).
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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,” Nature 470, 369–373 (2011).
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T. M. Babinec, J. M. HausmannBirgit, M. Khan, Y. Zhang, J. R. Maze, P. R. Hemmer, and M. Loncar, “A diamond nanowire single-photon source,” Nature Nanotechnol. 5, 195–199 (2010).
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S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H.-K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature 466, 735–738 (2010).
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W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat Photon 1, 224–227 (2007).
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X. Ni, S. Ishii, M. D. Thoreson, V. M. Shalaev, S. Han, S. Lee, and A. V. Kildishev, “Gain-assisted hyperbolic metamaterials,” in Quantum Electronics and Laser Science Conference (QELS), OSA Technical Digest (Optical Society of America, 2012), paper QTu1G.

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Z. Jacob, J. Y. Kim, G. Naik, A. Boltasseva, E. Narimanov, and V. Shalaev, “Engineering photonic density of states using metamaterials,” Appl. Phys. B 100, 215–218 (2010).
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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,” Nature 470, 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,” Nature 470, 369–373 (2011).
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X. Ni, S. Ishii, M. D. Thoreson, V. M. Shalaev, S. Han, S. Lee, and A. V. Kildishev, “Gain-assisted hyperbolic metamaterials,” in Quantum Electronics and Laser Science Conference (QELS), OSA Technical Digest (Optical Society of America, 2012), paper QTu1G.

Lee, S. 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,” Nature 470, 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,” Nature 470, 369–373 (2011).
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J. Sun, J. Zhou, B. Li, and F. Kang, “Indefinite permittivity and negative refraction in natural material: graphite,” Appl. Phys. Lett. 98, 101901 (2011).
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Li, L.-W.

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 94, 747–753 (2009).
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Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315, 1686 (2007).
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T. M. Babinec, J. M. HausmannBirgit, M. Khan, Y. Zhang, J. R. Maze, P. R. Hemmer, and M. Loncar, “A diamond nanowire single-photon source,” Nature Nanotechnol. 5, 195–199 (2010).
[CrossRef]

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W. T. Lu and S. Sridhar, “Slow light, open-cavity formation, and large longitudinal electric field on a slab waveguide made of indefinite permittivity metamaterials,” Phys. Rev. A 82, 013811 (2010).
[CrossRef]

Y. J. Huang, W. T. Lu, and S. Sridhar, “Nanowire waveguide made from extremely anisotropic metamaterials,” Phys. Rev. A 77, 063836 (2008).
[CrossRef]

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Y. Cui, K. H. Fung, J. Xu, H. Ma, Y. Jin, S. He, and N. X. Fang, “Ultrabroadband light absorption by a sawtooth anisotropic metamaterial slab,” Nano Lett. 12, 1443–1447 (2012).
[CrossRef]

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R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461, 629–632 (2009).
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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, 2963–2971 (1994).
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Maze, J. R.

T. M. Babinec, J. M. HausmannBirgit, M. Khan, Y. Zhang, J. R. Maze, P. R. Hemmer, and M. Loncar, “A diamond nanowire single-photon source,” Nature Nanotechnol. 5, 195–199 (2010).
[CrossRef]

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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 94, 747–753 (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,” Nature 470, 369–373 (2011).
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D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
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Z. Jacob, J. Y. Kim, G. Naik, A. Boltasseva, E. Narimanov, and V. Shalaev, “Engineering photonic density of states using metamaterials,” Appl. Phys. B 100, 215–218 (2010).
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Nelson, M. A.

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S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H.-K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature 466, 735–738 (2010).
[CrossRef]

X. Ni, S. Ishii, M. D. Thoreson, V. M. Shalaev, S. Han, S. Lee, and A. V. Kildishev, “Gain-assisted hyperbolic metamaterials,” in Quantum Electronics and Laser Science Conference (QELS), OSA Technical Digest (Optical Society of America, 2012), paper QTu1G.

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M. A. Noginov, H. Li, Y. A. Barnakov, D. Dryden, G. Nataraj, G. Zhu, C. E. Bonner, M. Mayy, Z. Jacob, and E. E. Narimanov, “Controlling spontaneous emission with metamaterials,” Opt. Lett. 35, 1863–1865 (2010).
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M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460, 1110–1112 (2009).
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Oulton, R. F.

X. Yang, Y. Liu, R. F. Oulton, X. Yin, and X. Zhang, “Optical forces in hybrid plasmonic waveguides,” Nano Lett. 11, 321–328 (2011).
[CrossRef]

R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461, 629–632 (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,” Nature 470, 369–373 (2011).
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J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
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D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
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J. Elser, V. A. Podolskiy, I. Salakhutdinov, and I. Avrutsky, “Nonlocal effects in effective-medium response of nanolayered metamaterials,” Appl. Phys. Lett. 90, 191109 (2007).
[CrossRef]

J. Elser, A. A. Govyadinov, I. Avrutsky, I. Salakhutdinov, and V. A. Podolskiy, “Plasmonic nanolayer composites: coupled plasmon polaritons, effective-medium response, and subdiffraction light manipulation,” J. Nanomater. 2007, 79469 (2007).
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A. A. Govyadinov and V. A. Podolskiy, “Metamaterial photonic funnels for subdiffraction light compression and propagation,” Phys. Rev. B 73, 155108 (2006).
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[CrossRef]

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J. Elser, A. A. Govyadinov, I. Avrutsky, I. Salakhutdinov, and V. A. Podolskiy, “Plasmonic nanolayer composites: coupled plasmon polaritons, effective-medium response, and subdiffraction light manipulation,” J. Nanomater. 2007, 79469 (2007).
[CrossRef]

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

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A. Salandrino and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystals: theory and simulations,” Phys. Rev. B 74, 075103–075105 (2006).
[CrossRef]

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R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
[CrossRef]

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J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[CrossRef]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
[CrossRef]

D. R. Smith and D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett. 90, 077405 (2003).
[CrossRef]

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Z. Jacob, J. Y. Kim, G. Naik, A. Boltasseva, E. Narimanov, and V. Shalaev, “Engineering photonic density of states using metamaterials,” Appl. Phys. B 100, 215–218 (2010).
[CrossRef]

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W. Chen, M. D. Thoreson, S. Ishii, A. V. Kildishev, and V. M. Shalaev, “Ultra-thin ultra-smooth and low-loss silver films on a germanium wetting layer,” Opt. Express 18, 5124–5134 (2010).
[CrossRef]

S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H.-K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature 466, 735–738 (2010).
[CrossRef]

M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460, 1110–1112 (2009).
[CrossRef]

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat Photon 1, 224–227 (2007).
[CrossRef]

X. Ni, S. Ishii, M. D. Thoreson, V. M. Shalaev, S. Han, S. Lee, and A. V. Kildishev, “Gain-assisted hyperbolic metamaterials,” in Quantum Electronics and Laser Science Conference (QELS), OSA Technical Digest (Optical Society of America, 2012), paper QTu1G.

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R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
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J. T. Shen, P. B. Catrysse, and S. Fan, “Mechanism for designing metallic metamaterials with a high index of refraction,” Phys. Rev. Lett. 94, 197401 (2005).
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J. Shin, J.-T. Shen, and S. Fan, “Three-dimensional metamaterials with an ultrahigh effective refractive index over a broad bandwidth,” Phys. Rev. Lett. 102, 093903 (2009).
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Shin, J.

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,” Nature 470, 369–373 (2011).
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J. Shin, J.-T. Shen, and S. Fan, “Three-dimensional metamaterials with an ultrahigh effective refractive index over a broad bandwidth,” Phys. Rev. Lett. 102, 093903 (2009).
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J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[CrossRef]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
[CrossRef]

D. R. Smith and D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett. 90, 077405 (2003).
[CrossRef]

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
[CrossRef]

Sorger, V. J.

R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461, 629–632 (2009).
[CrossRef]

Sridhar, S.

W. T. Lu and S. Sridhar, “Slow light, open-cavity formation, and large longitudinal electric field on a slab waveguide made of indefinite permittivity metamaterials,” Phys. Rev. A 82, 013811 (2010).
[CrossRef]

Y. J. Huang, W. T. Lu, and S. Sridhar, “Nanowire waveguide made from extremely anisotropic metamaterials,” Phys. Rev. A 77, 063836 (2008).
[CrossRef]

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J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,” Science 321, 930 (2008).
[CrossRef]

Starr, A. F.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
[CrossRef]

Stewart, W. J.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech 47, 2075–2084 (1999).
[CrossRef]

Stockert, J. 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, 2963–2971 (1994).
[CrossRef]

Stout, S.

M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460, 1110–1112 (2009).
[CrossRef]

Sun, C.

J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,” Science 321, 930 (2008).
[CrossRef]

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315, 1686 (2007).
[CrossRef]

Sun, J.

J. Sun, J. Zhou, B. Li, and F. Kang, “Indefinite permittivity and negative refraction in natural material: graphite,” Appl. Phys. Lett. 98, 101901 (2011).
[CrossRef]

G.-D. Xu, T. Pan, T.-C. Zang, and J. Sun, “Characteristics of guided waves in indefinite-medium waveguides,” Opt. Commun. 281, 2819–2825 (2008).
[CrossRef]

Suteewong, T.

M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460, 1110–1112 (2009).
[CrossRef]

Thoreson, M. D.

W. Chen, M. D. Thoreson, S. Ishii, A. V. Kildishev, and V. M. Shalaev, “Ultra-thin ultra-smooth and low-loss silver films on a germanium wetting layer,” Opt. Express 18, 5124–5134 (2010).
[CrossRef]

X. Ni, S. Ishii, M. D. Thoreson, V. M. Shalaev, S. Han, S. Lee, and A. V. Kildishev, “Gain-assisted hyperbolic metamaterials,” in Quantum Electronics and Laser Science Conference (QELS), OSA Technical Digest (Optical Society of America, 2012), paper QTu1G.

Thylén, L.

Veselago, V. G.

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of permittivity and permeability,” Sov. Phys. Usp. 10, 509–514 (1968).
[CrossRef]

Wang, Y.

J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,” Science 321, 930 (2008).
[CrossRef]

White, J. S.

A. Chandran, E. S. Barnard, J. S. White, and M. L. Brongersma, “Metal-dielectric-metal surface plasmon-polariton resonators,” Phys. Rev. B 85, 085416 (2012).
[CrossRef]

Wiesner, U.

M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460, 1110–1112 (2009).
[CrossRef]

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 94, 747–753 (2009).
[CrossRef]

Xiao, S.

S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H.-K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature 466, 735–738 (2010).
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S. Han, Y. Xiong, D. Genov, Z. Liu, G. Bartal, and X. Zhang, “Ray optics at a deep-subwavelength scale: a transformation optics approach,” Nano Lett. 8, 4243–4247 (2008).
[CrossRef]

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315, 1686 (2007).
[CrossRef]

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G.-D. Xu, T. Pan, T.-C. Zang, and J. Sun, “Characteristics of guided waves in indefinite-medium waveguides,” Opt. Commun. 281, 2819–2825 (2008).
[CrossRef]

Xu, J.

Y. Cui, K. H. Fung, J. Xu, H. Ma, Y. Jin, S. He, and N. X. Fang, “Ultrabroadband light absorption by a sawtooth anisotropic metamaterial slab,” Nano Lett. 12, 1443–1447 (2012).
[CrossRef]

Yan, M.

Yang, X.

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. Photon. 6, 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. 108, 11327–11331 (2011).
[CrossRef]

X. Yang, Y. Liu, R. F. Oulton, X. Yin, and X. Zhang, “Optical forces in hybrid plasmonic waveguides,” Nano Lett. 11, 321–328 (2011).
[CrossRef]

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. Photon. 6, 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. 108, 11327–11331 (2011).
[CrossRef]

J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,” Science 321, 930 (2008).
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Yin, X.

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. Photon. 6, 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. 108, 11327–11331 (2011).
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X. Yang, Y. Liu, R. F. Oulton, X. Yin, and X. Zhang, “Optical forces in hybrid plasmonic waveguides,” Nano Lett. 11, 321–328 (2011).
[CrossRef]

Yuan, H.-K.

S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H.-K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature 466, 735–738 (2010).
[CrossRef]

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G.-D. Xu, T. Pan, T.-C. Zang, and J. Sun, “Characteristics of guided waves in indefinite-medium waveguides,” Opt. Commun. 281, 2819–2825 (2008).
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R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461, 629–632 (2009).
[CrossRef]

Zhang, X.

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. Photon. 6, 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. 108, 11327–11331 (2011).
[CrossRef]

X. Yang, Y. Liu, R. F. Oulton, X. Yin, and X. Zhang, “Optical forces in hybrid plasmonic waveguides,” Nano Lett. 11, 321–328 (2011).
[CrossRef]

R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461, 629–632 (2009).
[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. Express 16, 15439–15448 (2008).
[CrossRef]

J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,” Science 321, 930 (2008).
[CrossRef]

S. Han, Y. Xiong, D. Genov, Z. Liu, G. Bartal, and X. Zhang, “Ray optics at a deep-subwavelength scale: a transformation optics approach,” Nano Lett. 8, 4243–4247 (2008).
[CrossRef]

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315, 1686 (2007).
[CrossRef]

Zhang, Y.

T. M. Babinec, J. M. HausmannBirgit, M. Khan, Y. Zhang, J. R. Maze, P. R. Hemmer, and M. Loncar, “A diamond nanowire single-photon source,” Nature Nanotechnol. 5, 195–199 (2010).
[CrossRef]

Zhao, J.

Zhou, J.

J. Sun, J. Zhou, B. Li, and F. Kang, “Indefinite permittivity and negative refraction in natural material: graphite,” Appl. Phys. Lett. 98, 101901 (2011).
[CrossRef]

Zhu, G.

M. A. Noginov, H. Li, Y. A. Barnakov, D. Dryden, G. Nataraj, G. Zhu, C. E. Bonner, M. Mayy, Z. Jacob, and E. E. Narimanov, “Controlling spontaneous emission with metamaterials,” Opt. Lett. 35, 1863–1865 (2010).
[CrossRef]

M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460, 1110–1112 (2009).
[CrossRef]

Appl. Phys. A (1)

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 94, 747–753 (2009).
[CrossRef]

Appl. Phys. B (1)

Z. Jacob, J. Y. Kim, G. Naik, A. Boltasseva, E. Narimanov, and V. Shalaev, “Engineering photonic density of states using metamaterials,” Appl. Phys. B 100, 215–218 (2010).
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J. Sun, J. Zhou, B. Li, and F. Kang, “Indefinite permittivity and negative refraction in natural material: graphite,” Appl. Phys. Lett. 98, 101901 (2011).
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[CrossRef]

IEEE Trans. Microwave Theory Tech (1)

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech 47, 2075–2084 (1999).
[CrossRef]

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J. Elser, A. A. Govyadinov, I. Avrutsky, I. Salakhutdinov, and V. A. Podolskiy, “Plasmonic nanolayer composites: coupled plasmon polaritons, effective-medium response, and subdiffraction light manipulation,” J. Nanomater. 2007, 79469 (2007).
[CrossRef]

J. Phys. Chem. (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, 2963–2971 (1994).
[CrossRef]

Nano Lett. (3)

X. Yang, Y. Liu, R. F. Oulton, X. Yin, and X. Zhang, “Optical forces in hybrid plasmonic waveguides,” Nano Lett. 11, 321–328 (2011).
[CrossRef]

S. Han, Y. Xiong, D. Genov, Z. Liu, G. Bartal, and X. Zhang, “Ray optics at a deep-subwavelength scale: a transformation optics approach,” Nano Lett. 8, 4243–4247 (2008).
[CrossRef]

Y. Cui, K. H. Fung, J. Xu, H. Ma, Y. Jin, S. He, and N. X. Fang, “Ultrabroadband light absorption by a sawtooth anisotropic metamaterial slab,” Nano Lett. 12, 1443–1447 (2012).
[CrossRef]

Nat Photon (1)

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat Photon 1, 224–227 (2007).
[CrossRef]

Nat. Photon. (1)

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. Photon. 6, 450–454 (2012).
[CrossRef]

Nature (4)

R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461, 629–632 (2009).
[CrossRef]

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,” Nature 470, 369–373 (2011).
[CrossRef]

S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H.-K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature 466, 735–738 (2010).
[CrossRef]

M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460, 1110–1112 (2009).
[CrossRef]

Nature Nanotechnol. (1)

T. M. Babinec, J. M. HausmannBirgit, M. Khan, Y. Zhang, J. R. Maze, P. R. Hemmer, and M. Loncar, “A diamond nanowire single-photon source,” Nature Nanotechnol. 5, 195–199 (2010).
[CrossRef]

Opt. Commun. (1)

G.-D. Xu, T. Pan, T.-C. Zang, and J. Sun, “Characteristics of guided waves in indefinite-medium waveguides,” Opt. Commun. 281, 2819–2825 (2008).
[CrossRef]

Opt. Express (6)

Opt. Lett. (3)

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Proc. Natl. Acad. Sci. (1)

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

Science (5)

J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,” Science 321, 930 (2008).
[CrossRef]

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
[CrossRef]

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315, 1686 (2007).
[CrossRef]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[CrossRef]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
[CrossRef]

Sov. Phys. Usp. (1)

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of permittivity and permeability,” Sov. Phys. Usp. 10, 509–514 (1968).
[CrossRef]

Other (2)

A. Sihvola, Electromagnetic Mixing Formulas and Applications (Institution of Electrical Engineers, 1999).

X. Ni, S. Ishii, M. D. Thoreson, V. M. Shalaev, S. Han, S. Lee, and A. V. Kildishev, “Gain-assisted hyperbolic metamaterials,” in Quantum Electronics and Laser Science Conference (QELS), OSA Technical Digest (Optical Society of America, 2012), paper QTu1G.

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Figures (8)

Fig. 1.
Fig. 1.

(a) Schematic of a nanoscale optical waveguide made of the metal–dielectric multilayer indefinite metamaterial, which is constructed with alternative layers of 4 nm silver and 6 nm germanium. Nanoscale optical waveguides can be created due to the TIR at the metamaterial-air interface. (b) Principal components of the permittivity tensor for the multilayer metamaterial with a silver filling ratio of fm=0.4, calculated from the effective medium theory (Eq. 1), where εy is positive and εz (εx) is negative. For example, εx=εz=39.8+2.1i and εy=29.2+0.1i at λ0=1.55μm.

Fig. 2.
Fig. 2.

Distributions of magnetic field Hx, electric field Ey, and EM energy density W for waveguide modes with different mode orders supported in the metamaterial waveguide with cross section of 100 by 100 nm at λ0=1μm. Waveguide modes with mode orders of (1,my), (2,my), and (3,my) in metal–dielectric multilayer waveguide structures are shown in (a), (c), and (d), respectively. The (1,my) modes are also calculated with the effective medium method, as shown in (b), which agree very well with the multilayer results in (a).

Fig. 3.
Fig. 3.

Dependences of (a) waveguide mode indices along the propagation direction neff,z; (b) propagation lengths Lm, and (c) mode areas Am on wavelength λ0 for waveguide modes with different mode orders in the waveguide with cross section of 100 by 100 nm. All the (3,my) modes and the (2, 1) mode have mode cutoff at certain wavelengths. The propagation lengths depend on both the absorption loss and the mode radiation loss. The mode areas are on the order of 103(λ02), due to the tight photon confinement of waveguide modes.

Fig. 4.
Fig. 4.

Waveguide modes plotted in k-space at different free-space wavelengths λ0 for the waveguide with cross section of 100 by 100 nm. (a) Hyperboloid IFC of the indefinite metamaterial at the wavelength of λ0. Two cutting planes in gray color at kx=0 and kx/k0=λ0/Lx represent mode orders of mx=1 and mx=3, respectively. The crossing curves between the two cut planes and the hyperboloid IFC are shown in (b) and (c), respectively, at different wavelengths of λ0=1μm (blue color) and λ0=1.55μm (red color). The markers in (b) and (c) represent the (1,my) and (3,my) modes calculated from multilayer metamaterial waveguide structures. It is noted that the hyperbolic curves have changed their opening directions from the z direction in (b) to the y direction in (c), which will lead to the waveguide mode cutoff for the (3, 1) and (3, 2) modes at λ0=1.55μm shown in (c). Only the kz>0 part of the IFC is shown.

Fig. 5.
Fig. 5.

Waveguide modes plotted in k-space at different waveguide height Ly for a fixed width Lx=100nm. (a) Hyperboloid IFC of the indefinite metamaterial at λ0=1.55μm. Three cutting planes of I, II, and III represent mode orders of mx=1, mx=2, and mx=3, respectively. The three crossing curves between the cutting planes and the hyperboloid IFC are shown in (b) and (c), for different waveguide heights of Ly=80nm and Ly=150nm, respectively. The markers in (b) and (c) represent the (mx,my) modes calculated from multilayer metamaterial waveguide structures. The horizontal dashed lines show the location my=1, 2, and 3 for a certain waveguide height Ly. It is observed that ky will decrease as Ly gets larger, which will result in the waveguide mode cutoff for high-order modes, such as the (2, 1), (3, 2), and (3, 3) modes, as shown in (c). Only the kz>0 part of the IFC is shown.

Fig. 6.
Fig. 6.

Waveguide modes plotted in k-space at different waveguide widths Lx for a fixed height Ly=100nm. (a) Hyperboloid IFC of the indefinite metamaterial at λ0=1.55μm. Three cutting planes of I, II, and III represent mode orders of my=1, my=2, and my=3, respectively. The three crossing circles between the cut planes and the hyperboloid IFC are shown in (b) and (c), for different waveguide widths of Lx=120nm and Lx=70nm, respectively. The markers in (b) and (c) represent the (mx,my) modes calculated from multilayer metamaterial waveguide structures. The horizontal dashed lines show the location mx=1, 2, and 3 for a certain waveguide width Lx. It is observed that kx will increase as Lx gets smaller, which will result in the waveguide mode cutoff for high-order modes, such as the (2, 1), (3, 2), and (3, 3) modes, as shown in (c). Only the kz>0 part of the IFC is shown.

Fig. 7.
Fig. 7.

(a) Effective refractive indices neff for the (1,my) modes as functions of waveguide sizes L for metamaterial waveguides with square cross sections at λ0=1.55μm. Solid curves are calculated from the effective medium method and markers represent the simulation results from multilayer metamaterial waveguide structures. It is shown that a smaller waveguide size L and a higher mode order my will lead to a larger refractive index; (b) The wave vectors of the waveguide modes shown in (a) are plotted in k-space, which match the effective medium calculation (solid curve) in the low k region (k2π/a). In the high k region (when L50nm), as the wavelength is close to the period of the multilayer a, the wave vectors calculated from the multilayer structure deviate from the predication of effective medium theory due to the nonlocal effect.

Fig. 8.
Fig. 8.

Mode propagation of the (1, 1) modes at λ0=1.55μm inside the metamaterial waveguides with square cross sections of (a) 30 nm by 30 nm and (b) 50 nm by 50 nm, respectively. The distributions of the magnetic field Hx are plotted. The length of the waveguides in z direction is 300 nm for both cases. The waveguide mode properties are listed on the bottom of each figure.

Equations (4)

Equations on this page are rendered with MathJax. Learn more.

εx=εz=fmεm+(1fm)εd,εy=εmεdfmεd+(1fm)εm,
kx2+kz2εy+ky2εz=k02,
kx=(mx1)πLx,ky=myπLy,
neff=neff,x2+neff,y2+neff,z2=(kxk0)2+(kyk0)2+(kzk0)2.

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