Abstract

Surface plasmon modes in structures of metal-insulator-metal (MIM), insulator-insulator-metal (IIM) and insulator-metal-insulator (IMI) are studied theoretically for both lossless and lossy cases. Causality dictates which solutions of Maxwell’s equations we accept for these structures. We find that for both lossless and lossy cases, the negative index modes and positive index modes are independent and should be treated separately. For the lossless case, our results differ from some published papers. By studying in detail the lossy case, we demonstrate how the curves should look like.

© 2010 OSA

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    [CrossRef]
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    [CrossRef] [PubMed]
  3. S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, “Experimental demonstration of near-infrared negative-index metamaterials,” Phys. Rev. Lett. 95(13), 137404 (2005).
    [CrossRef] [PubMed]
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    [CrossRef]
  5. G. Dolling, M. Wegener, C. M. Soukoulis, and S. Linden, “Negative-index metamaterial at 780 nm wavelength,” Opt. Lett. 32(1), 53–55 (2007).
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  6. G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Simultaneous negative phase and group velocity of light in a metamaterial,” Science 312(5775), 892–894 (2006).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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2009 (2)

M. C. Gwinner, E. Koroknay, L. Fu, P. Patoka, W. Kandulski, M. Giersig, and H. Giessen, “Periodic large-area metallic split-ring resonator metamaterial fabrication based on shadow nanosphere lithography,” Small 5(3), 400–406 (2009).
[CrossRef] [PubMed]

E. Feigenbaum and M. Orenstein, “Backward propagating slow light in inverted plasmonic taper,” Opt. Express 17(4), 2465–2469 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-4-2465 .
[CrossRef] [PubMed]

2008 (3)

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature 455(7211), 376–379 (2008).
[CrossRef] [PubMed]

A. Hohenau, A. Drezet, M. Weißenbacher, F. R. Aussenegg, and J. R. Krenn, “Effects of damping on surface-plasmon pulse propagation and refraction,” Phys. Rev. B 78(15), 155405 (2008).
[CrossRef]

J. A. Dionne, E. Verhagen, A. Polman, and H. A. Atwater, “Are negative index materials achievable with surface plasmon waveguides? A case study of three plasmonic geometries,” Opt. Express 16(23), 19001–19017 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-23-19001 .
[CrossRef]

2007 (7)

G. Dolling, M. Wegener, C. M. Soukoulis, and S. Linden, “Negative-index metamaterial at 780 nm wavelength,” Opt. Lett. 32(1), 53–55 (2007).
[CrossRef]

I. I. Smolyaninov, Y. Hung, and C. C. Davis, “Imaging and focusing properties of plasmonic metamaterial devices,” Phys. Rev. B 76(20), 205424 (2007).
[CrossRef]

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6(12), 946–950 (2007).
[CrossRef] [PubMed]

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315(5819), 1686 (2007).
[CrossRef] [PubMed]

I. I. Smolyaninov, Y. J. Hung, and C. C. Davis, “Magnifying superlens in the visible frequency range,” Science 315(5819), 1699–1701 (2007).
[CrossRef] [PubMed]

M. I. Stockman, “Criterion for Negative Refraction with Low Optical Losses from a Fundamental Principle of Causality,” Phys. Rev. Lett. 98(17), 177404 (2007).
[CrossRef]

H. J. Lezec, J. A. Dionne, and H. A. Atwater, “Negative refraction at visible frequencies,” Science 316(5823), 430–432 (2007).
[CrossRef] [PubMed]

2006 (2)

H. Shin and S. Fan, “All-angle negative refraction for surface plasmon waves using a metal-dielectric-metal structure,” Phys. Rev. Lett. 96(7), 073907 (2006).
[CrossRef] [PubMed]

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Simultaneous negative phase and group velocity of light in a metamaterial,” Science 312(5775), 892–894 (2006).
[CrossRef] [PubMed]

2005 (3)

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308(5721), 534–537 (2005).
[CrossRef] [PubMed]

V. M. Shalaev, W. Cai, U. K. Chettiar, H. K. Yuan, A. K. Sarychev, V. P. Drachev, and A. V. Kildishev, “Negative index of refraction in optical metamaterials,” Opt. Lett. 30(24), 3356–3358 (2005).
[CrossRef]

S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, “Experimental demonstration of near-infrared negative-index metamaterials,” Phys. Rev. Lett. 95(13), 137404 (2005).
[CrossRef] [PubMed]

2001 (1)

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292(5514), 77–79 (2001).
[CrossRef] [PubMed]

1968 (1)

V. G. Veselago, “Electrodynamics of substances with simultaneously negative values of sigma and mu,” Sov. Phys. Usp. 10(4), 509–514 (1968).
[CrossRef]

Alekseyev, L.

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6(12), 946–950 (2007).
[CrossRef] [PubMed]

Atwater, H. A.

Aussenegg, F. R.

A. Hohenau, A. Drezet, M. Weißenbacher, F. R. Aussenegg, and J. R. Krenn, “Effects of damping on surface-plasmon pulse propagation and refraction,” Phys. Rev. B 78(15), 155405 (2008).
[CrossRef]

Bartal, G.

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature 455(7211), 376–379 (2008).
[CrossRef] [PubMed]

Brueck, S. R. J.

S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, “Experimental demonstration of near-infrared negative-index metamaterials,” Phys. Rev. Lett. 95(13), 137404 (2005).
[CrossRef] [PubMed]

Cai, W.

Chettiar, U. K.

Davis, C. C.

I. I. Smolyaninov, Y. J. Hung, and C. C. Davis, “Magnifying superlens in the visible frequency range,” Science 315(5819), 1699–1701 (2007).
[CrossRef] [PubMed]

I. I. Smolyaninov, Y. Hung, and C. C. Davis, “Imaging and focusing properties of plasmonic metamaterial devices,” Phys. Rev. B 76(20), 205424 (2007).
[CrossRef]

Dionne, J. A.

Dolling, G.

G. Dolling, M. Wegener, C. M. Soukoulis, and S. Linden, “Negative-index metamaterial at 780 nm wavelength,” Opt. Lett. 32(1), 53–55 (2007).
[CrossRef]

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Simultaneous negative phase and group velocity of light in a metamaterial,” Science 312(5775), 892–894 (2006).
[CrossRef] [PubMed]

Drachev, V. P.

Drezet, A.

A. Hohenau, A. Drezet, M. Weißenbacher, F. R. Aussenegg, and J. R. Krenn, “Effects of damping on surface-plasmon pulse propagation and refraction,” Phys. Rev. B 78(15), 155405 (2008).
[CrossRef]

Enkrich, C.

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Simultaneous negative phase and group velocity of light in a metamaterial,” Science 312(5775), 892–894 (2006).
[CrossRef] [PubMed]

Fan, S.

H. Shin and S. Fan, “All-angle negative refraction for surface plasmon waves using a metal-dielectric-metal structure,” Phys. Rev. Lett. 96(7), 073907 (2006).
[CrossRef] [PubMed]

Fan, W.

S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, “Experimental demonstration of near-infrared negative-index metamaterials,” Phys. Rev. Lett. 95(13), 137404 (2005).
[CrossRef] [PubMed]

Fang, N.

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308(5721), 534–537 (2005).
[CrossRef] [PubMed]

Feigenbaum, E.

Franz, K. J.

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6(12), 946–950 (2007).
[CrossRef] [PubMed]

Fu, L.

M. C. Gwinner, E. Koroknay, L. Fu, P. Patoka, W. Kandulski, M. Giersig, and H. Giessen, “Periodic large-area metallic split-ring resonator metamaterial fabrication based on shadow nanosphere lithography,” Small 5(3), 400–406 (2009).
[CrossRef] [PubMed]

Genov, D. A.

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature 455(7211), 376–379 (2008).
[CrossRef] [PubMed]

Giersig, M.

M. C. Gwinner, E. Koroknay, L. Fu, P. Patoka, W. Kandulski, M. Giersig, and H. Giessen, “Periodic large-area metallic split-ring resonator metamaterial fabrication based on shadow nanosphere lithography,” Small 5(3), 400–406 (2009).
[CrossRef] [PubMed]

Giessen, H.

M. C. Gwinner, E. Koroknay, L. Fu, P. Patoka, W. Kandulski, M. Giersig, and H. Giessen, “Periodic large-area metallic split-ring resonator metamaterial fabrication based on shadow nanosphere lithography,” Small 5(3), 400–406 (2009).
[CrossRef] [PubMed]

Gmachl, C.

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6(12), 946–950 (2007).
[CrossRef] [PubMed]

Gwinner, M. C.

M. C. Gwinner, E. Koroknay, L. Fu, P. Patoka, W. Kandulski, M. Giersig, and H. Giessen, “Periodic large-area metallic split-ring resonator metamaterial fabrication based on shadow nanosphere lithography,” Small 5(3), 400–406 (2009).
[CrossRef] [PubMed]

Hoffman, A. J.

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6(12), 946–950 (2007).
[CrossRef] [PubMed]

Hohenau, A.

A. Hohenau, A. Drezet, M. Weißenbacher, F. R. Aussenegg, and J. R. Krenn, “Effects of damping on surface-plasmon pulse propagation and refraction,” Phys. Rev. B 78(15), 155405 (2008).
[CrossRef]

Howard, S. S.

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6(12), 946–950 (2007).
[CrossRef] [PubMed]

Hung, Y.

I. I. Smolyaninov, Y. Hung, and C. C. Davis, “Imaging and focusing properties of plasmonic metamaterial devices,” Phys. Rev. B 76(20), 205424 (2007).
[CrossRef]

Hung, Y. J.

I. I. Smolyaninov, Y. J. Hung, and C. C. Davis, “Magnifying superlens in the visible frequency range,” Science 315(5819), 1699–1701 (2007).
[CrossRef] [PubMed]

Kandulski, W.

M. C. Gwinner, E. Koroknay, L. Fu, P. Patoka, W. Kandulski, M. Giersig, and H. Giessen, “Periodic large-area metallic split-ring resonator metamaterial fabrication based on shadow nanosphere lithography,” Small 5(3), 400–406 (2009).
[CrossRef] [PubMed]

Kildishev, A. V.

Koroknay, E.

M. C. Gwinner, E. Koroknay, L. Fu, P. Patoka, W. Kandulski, M. Giersig, and H. Giessen, “Periodic large-area metallic split-ring resonator metamaterial fabrication based on shadow nanosphere lithography,” Small 5(3), 400–406 (2009).
[CrossRef] [PubMed]

Krenn, J. R.

A. Hohenau, A. Drezet, M. Weißenbacher, F. R. Aussenegg, and J. R. Krenn, “Effects of damping on surface-plasmon pulse propagation and refraction,” Phys. Rev. B 78(15), 155405 (2008).
[CrossRef]

Lee, H.

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315(5819), 1686 (2007).
[CrossRef] [PubMed]

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308(5721), 534–537 (2005).
[CrossRef] [PubMed]

Lezec, H. J.

H. J. Lezec, J. A. Dionne, and H. A. Atwater, “Negative refraction at visible frequencies,” Science 316(5823), 430–432 (2007).
[CrossRef] [PubMed]

Linden, S.

G. Dolling, M. Wegener, C. M. Soukoulis, and S. Linden, “Negative-index metamaterial at 780 nm wavelength,” Opt. Lett. 32(1), 53–55 (2007).
[CrossRef]

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Simultaneous negative phase and group velocity of light in a metamaterial,” Science 312(5775), 892–894 (2006).
[CrossRef] [PubMed]

Liu, Z.

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315(5819), 1686 (2007).
[CrossRef] [PubMed]

Malloy, K. J.

S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, “Experimental demonstration of near-infrared negative-index metamaterials,” Phys. Rev. Lett. 95(13), 137404 (2005).
[CrossRef] [PubMed]

Narimanov, E. E.

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6(12), 946–950 (2007).
[CrossRef] [PubMed]

Orenstein, M.

Osgood, R. M.

S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, “Experimental demonstration of near-infrared negative-index metamaterials,” Phys. Rev. Lett. 95(13), 137404 (2005).
[CrossRef] [PubMed]

Panoiu, N. C.

S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, “Experimental demonstration of near-infrared negative-index metamaterials,” Phys. Rev. Lett. 95(13), 137404 (2005).
[CrossRef] [PubMed]

Patoka, P.

M. C. Gwinner, E. Koroknay, L. Fu, P. Patoka, W. Kandulski, M. Giersig, and H. Giessen, “Periodic large-area metallic split-ring resonator metamaterial fabrication based on shadow nanosphere lithography,” Small 5(3), 400–406 (2009).
[CrossRef] [PubMed]

Podolskiy, V. A.

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6(12), 946–950 (2007).
[CrossRef] [PubMed]

Polman, A.

Sarychev, A. K.

Schultz, S.

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292(5514), 77–79 (2001).
[CrossRef] [PubMed]

Shalaev, V. M.

Shelby, R. A.

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292(5514), 77–79 (2001).
[CrossRef] [PubMed]

Shin, H.

H. Shin and S. Fan, “All-angle negative refraction for surface plasmon waves using a metal-dielectric-metal structure,” Phys. Rev. Lett. 96(7), 073907 (2006).
[CrossRef] [PubMed]

Sivco, D. L.

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6(12), 946–950 (2007).
[CrossRef] [PubMed]

Smith, D. R.

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292(5514), 77–79 (2001).
[CrossRef] [PubMed]

Smolyaninov, I. I.

I. I. Smolyaninov, Y. Hung, and C. C. Davis, “Imaging and focusing properties of plasmonic metamaterial devices,” Phys. Rev. B 76(20), 205424 (2007).
[CrossRef]

I. I. Smolyaninov, Y. J. Hung, and C. C. Davis, “Magnifying superlens in the visible frequency range,” Science 315(5819), 1699–1701 (2007).
[CrossRef] [PubMed]

Soukoulis, C. M.

G. Dolling, M. Wegener, C. M. Soukoulis, and S. Linden, “Negative-index metamaterial at 780 nm wavelength,” Opt. Lett. 32(1), 53–55 (2007).
[CrossRef]

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Simultaneous negative phase and group velocity of light in a metamaterial,” Science 312(5775), 892–894 (2006).
[CrossRef] [PubMed]

Stockman, M. I.

M. I. Stockman, “Criterion for Negative Refraction with Low Optical Losses from a Fundamental Principle of Causality,” Phys. Rev. Lett. 98(17), 177404 (2007).
[CrossRef]

Sun, C.

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315(5819), 1686 (2007).
[CrossRef] [PubMed]

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308(5721), 534–537 (2005).
[CrossRef] [PubMed]

Ulin-Avila, E.

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature 455(7211), 376–379 (2008).
[CrossRef] [PubMed]

Valentine, J.

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature 455(7211), 376–379 (2008).
[CrossRef] [PubMed]

Verhagen, E.

Veselago, V. G.

V. G. Veselago, “Electrodynamics of substances with simultaneously negative values of sigma and mu,” Sov. Phys. Usp. 10(4), 509–514 (1968).
[CrossRef]

Wasserman, D.

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6(12), 946–950 (2007).
[CrossRef] [PubMed]

Wegener, M.

G. Dolling, M. Wegener, C. M. Soukoulis, and S. Linden, “Negative-index metamaterial at 780 nm wavelength,” Opt. Lett. 32(1), 53–55 (2007).
[CrossRef]

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Simultaneous negative phase and group velocity of light in a metamaterial,” Science 312(5775), 892–894 (2006).
[CrossRef] [PubMed]

Weißenbacher, M.

A. Hohenau, A. Drezet, M. Weißenbacher, F. R. Aussenegg, and J. R. Krenn, “Effects of damping on surface-plasmon pulse propagation and refraction,” Phys. Rev. B 78(15), 155405 (2008).
[CrossRef]

Xiong, Y.

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315(5819), 1686 (2007).
[CrossRef] [PubMed]

Yuan, H. K.

Zentgraf, T.

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature 455(7211), 376–379 (2008).
[CrossRef] [PubMed]

Zhang, S.

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature 455(7211), 376–379 (2008).
[CrossRef] [PubMed]

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[CrossRef] [PubMed]

Zhang, X.

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature 455(7211), 376–379 (2008).
[CrossRef] [PubMed]

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315(5819), 1686 (2007).
[CrossRef] [PubMed]

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308(5721), 534–537 (2005).
[CrossRef] [PubMed]

Nat. Mater. (1)

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Nature (1)

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Phys. Rev. B (2)

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[CrossRef]

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[CrossRef]

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[CrossRef]

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[CrossRef] [PubMed]

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Figures (9)

Fig. 1
Fig. 1

(a) Schematic of an incident wave impinging on a three-layer structure from a normal medium. The direction of the incident wave is confined to the x-y plane. (b) and (c) are schematics where (b) shows the correct phase and energy velocities with negative refraction for normal and oblique incidence, while case (c) violates causality (see direction of energy flow).

Fig. 2
Fig. 2

Dispersion relations of lossless Drude model for three types of layered structures: (a) MIM, (b) IIM (here we use air as the cover layer), and (c) IMI. The black curves are typical of those published by others. The red curves are the branches consistent with causality, which we discuss in this paper.

Fig. 3
Fig. 3

Dispersion relations for the symmetric mode of MIM structure, for both lossless (Г = 0) and lossy (Г = 0.2687 eV) cases. (a) and (b) show the curves of the real and imaginary parts of the wave vector, respectively; (c) shows the curves of frequency vs. x component of time averaged Poynting vector. The horizontal dotted curve stands for ω = ωsp . The curves are colored consistently for all of three figures. The insets show details for clarity.

Fig. 4
Fig. 4

Dispersion relations for the anti-symmetric mode of the MIM structure, for both lossless (Г = 0) and lossy (Г = 0.2687 eV) cases. (a) and (b) show the curves of the real and imaginary parts of the wave vector, respectively; (c) shows the ω-Sx › curves. The curves are consistent in color for all of three figures.

Fig. 5
Fig. 5

Details of the dispersion curves of the lossless case shown in Fig. 4. Solid curves are for the propagating modes, and solutions which violate causality are shown as dashed curves. The dash-dot curves indicate non-propagating modes (the imaginary part shown in the same picture in blue). The curves are consistent in color.

Fig. 6
Fig. 6

Dispersion relations of the IIM structure for both lossless and lossy cases. (a) and (b) show the real and imaginary parts of the wave vector, respectively; (c) shows the ω-Sx › curves. For clarity, we do not show the curves of the lossless case when kx is large. The curves are consistent in color.

Fig. 7
Fig. 7

Details of the dispersion curves for the lossless case shown in Fig. 6. Solid curves indicate the propagating modes, and solutions which violate causality are shown by dashed curves. The dash-dot curves indicate the non-propagating modes (the imaginary part shown in the same picture in blue). Based on causality, we plot positive ‹Sx › for the propagating modes in (b). The curves are consistent in color.

Fig. 8
Fig. 8

The physically meaningful dispersion curves of the IMI structure, all curves corresponding to causal solutions and the pseudo solutions are not shown. The details of large real(kx ) are shown in the inset upper left in (a). All the curves are consistent in color.

Fig. 9
Fig. 9

Dispersion relations for (a) MIM, (b) IIM and (c) IMI structures, when the permittivities of metal layers take the actual parameters for silver. All the curves are causal, and the pseudo solutions are dropped. For the lossless case, the black solid curves show the propagating modes; the dash-dot curves correspond to non-propagating modes. For the lossy case, the red (blue) solid curves indicate the negative (positive) index modes, and the dashed curves are the corresponding imaginary parts. The curves are consistent in color.

Equations (9)

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e 2 k 1 d = k 1 / ε 1 + k 2 / ε 2 k 1 / ε 1 k 2 / ε 2 k 1 / ε 1 + k 3 / ε 3 k 1 / ε 1 k 3 / ε 3
k i 2 = k x 2 ε i k 0 2
ε m e t a l = ε ω p 2 ω 2 + i ω Γ
S x = d / 2 + d / 2 S 1 x d z + d / 2 S 2 x d z + + d / 2 + S 3 x d z
d / 2 + d / 2 S 1 x d z = k x 2 ω ε 0 ε 1 | e i k x x | 2 d / 2 + d / 2 ( | C | 2 | e k 1 z | 2 + | D | 2 | e k 1 z | 2 + C D * e k 1 z ( e k 1 z ) * + D C * e k 1 z ( e k 1 z ) * ) d z
d / 2 S 2 x d z =     k x 2 ω ε 0 ε 2 | B | 2 | e i k x x | 2 d / 2 | e k 2 z | 2 d z
+ d / 2 + S 3 x d z = k x 2 ω ε 0 ε 3 | A | 2 | e i k x x | 2 + a + | e k 3 z | 2 d z
{ A = ( e k 1 a ( k 3 ε 1 + ε 3 k 1 ) ( k 3 ε 1 ε 3 k 1 ) e k 1 a ) e k 3 a C B = ( e k 1 a + ( k 2 ε 1 + ε 2 k 1 ) ( k 2 ε 1 + ε 2 k 1 ) e k 1 a ) e k 2 a C D = ( k 3 ε 1 + ε 3 k 1 ) ( k 3 ε 1 ε 3 k 1 ) e 2 k 1 a C
n = k x / k 0

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