Abstract

We discuss optical constants in artificial metamaterials showing negative magnetic permeability and electric permittivity and suggest a simple formula for the refractive index of a general optical medium. Using the effective-field theory, we calculate the effective permeability and the refractive index of nanofabricated media composed of pairs of identical gold nanopillars with magnetic response in the visible spectrum.

© 2006 Optical Society of America

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References

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  1. M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1980).
  2. L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Oxford, 1960).
  3. The principal square root is defined as the regular branch with the cut along the negative x axis.
  4. V. G. Veselago, Sov. Phys. Usp. 10, 509 (1968).
    [CrossRef]
  5. J. B. Pendry, Phys. Rev. Lett. 85, 3966 (2000).
    [CrossRef] [PubMed]
  6. S. A. Ramakrishna, Rep. Prog. Phys. 68, 449 (2005).
    [CrossRef]
  7. D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, Phys. Rev. Lett. 84, 4184 (2000).
    [CrossRef] [PubMed]
  8. J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, IEEE Trans. Microwave Theory Tech. 47, 2075 (1999).
    [CrossRef]
  9. T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, Science 303, 1494 (2004).
    [CrossRef] [PubMed]
  10. A. N. Grigorenko, A. K. Geim, H. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev, and J. Petrovic, Nature 438, 335 (2005); first reported in Seagate University Research Conclave, Londonderry, UK (April 1, 2004).
    [CrossRef] [PubMed]
  11. V. M. Shalaev, W. Cai, U. K. Chettiar, H.-K. Yuan, A. K. Sarychev, V. P. Drachev, and A. V. Kildishev, Opt. Lett. 30, 3356 (2005) [http://arxiv.org/abs/physics/0504091 (April 13, 2005)].
    [CrossRef]
  12. D. R. Smith and N. Kroll, Phys. Rev. Lett. 85, 2933 (2000).
    [CrossRef] [PubMed]
  13. A. N. Grigorenko, H. F. Gleeson, Y. Zhang, N. W. Roberts, A. R. Sidorov, and A. A. Panteleev, Appl. Phys. Lett. 88, 124103 (2006).
    [CrossRef]
  14. L. V. Panina, A. N. Grigorenko, and D. P. Makhnovskiy, Phys. Rev. B 66, 155411 (2002).
    [CrossRef]
  15. V. A. Podolskiy, A. K. Sarychev, and V. M. Shalaev, J. Nonlinear Opt. Phys. Mater. 11, 65 (2002).
    [CrossRef]
  16. S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. Soukoulis, Science 306, 1351 (2004).
    [CrossRef] [PubMed]

2006 (1)

A. N. Grigorenko, H. F. Gleeson, Y. Zhang, N. W. Roberts, A. R. Sidorov, and A. A. Panteleev, Appl. Phys. Lett. 88, 124103 (2006).
[CrossRef]

2005 (3)

A. N. Grigorenko, A. K. Geim, H. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev, and J. Petrovic, Nature 438, 335 (2005); first reported in Seagate University Research Conclave, Londonderry, UK (April 1, 2004).
[CrossRef] [PubMed]

V. M. Shalaev, W. Cai, U. K. Chettiar, H.-K. Yuan, A. K. Sarychev, V. P. Drachev, and A. V. Kildishev, Opt. Lett. 30, 3356 (2005) [http://arxiv.org/abs/physics/0504091 (April 13, 2005)].
[CrossRef]

S. A. Ramakrishna, Rep. Prog. Phys. 68, 449 (2005).
[CrossRef]

2004 (2)

T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, Science 303, 1494 (2004).
[CrossRef] [PubMed]

S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. Soukoulis, Science 306, 1351 (2004).
[CrossRef] [PubMed]

2002 (2)

L. V. Panina, A. N. Grigorenko, and D. P. Makhnovskiy, Phys. Rev. B 66, 155411 (2002).
[CrossRef]

V. A. Podolskiy, A. K. Sarychev, and V. M. Shalaev, J. Nonlinear Opt. Phys. Mater. 11, 65 (2002).
[CrossRef]

2000 (3)

D. R. Smith and N. Kroll, Phys. Rev. Lett. 85, 2933 (2000).
[CrossRef] [PubMed]

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, Phys. Rev. Lett. 84, 4184 (2000).
[CrossRef] [PubMed]

J. B. Pendry, Phys. Rev. Lett. 85, 3966 (2000).
[CrossRef] [PubMed]

1999 (1)

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, IEEE Trans. Microwave Theory Tech. 47, 2075 (1999).
[CrossRef]

1968 (1)

V. G. Veselago, Sov. Phys. Usp. 10, 509 (1968).
[CrossRef]

Basov, D. N.

T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, Science 303, 1494 (2004).
[CrossRef] [PubMed]

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1980).

Cai, W.

Chettiar, U. K.

Drachev, V. P.

Enkrich, C.

S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. Soukoulis, Science 306, 1351 (2004).
[CrossRef] [PubMed]

Fang, N.

T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, Science 303, 1494 (2004).
[CrossRef] [PubMed]

Firsov, A. A.

A. N. Grigorenko, A. K. Geim, H. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev, and J. Petrovic, Nature 438, 335 (2005); first reported in Seagate University Research Conclave, Londonderry, UK (April 1, 2004).
[CrossRef] [PubMed]

Geim, A. K.

A. N. Grigorenko, A. K. Geim, H. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev, and J. Petrovic, Nature 438, 335 (2005); first reported in Seagate University Research Conclave, Londonderry, UK (April 1, 2004).
[CrossRef] [PubMed]

Gleeson, H. F.

A. N. Grigorenko, H. F. Gleeson, Y. Zhang, N. W. Roberts, A. R. Sidorov, and A. A. Panteleev, Appl. Phys. Lett. 88, 124103 (2006).
[CrossRef]

A. N. Grigorenko, A. K. Geim, H. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev, and J. Petrovic, Nature 438, 335 (2005); first reported in Seagate University Research Conclave, Londonderry, UK (April 1, 2004).
[CrossRef] [PubMed]

Grigorenko, A. N.

A. N. Grigorenko, H. F. Gleeson, Y. Zhang, N. W. Roberts, A. R. Sidorov, and A. A. Panteleev, Appl. Phys. Lett. 88, 124103 (2006).
[CrossRef]

A. N. Grigorenko, A. K. Geim, H. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev, and J. Petrovic, Nature 438, 335 (2005); first reported in Seagate University Research Conclave, Londonderry, UK (April 1, 2004).
[CrossRef] [PubMed]

L. V. Panina, A. N. Grigorenko, and D. P. Makhnovskiy, Phys. Rev. B 66, 155411 (2002).
[CrossRef]

Holden, A. J.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, IEEE Trans. Microwave Theory Tech. 47, 2075 (1999).
[CrossRef]

Khrushchev, I. Y.

A. N. Grigorenko, A. K. Geim, H. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev, and J. Petrovic, Nature 438, 335 (2005); first reported in Seagate University Research Conclave, Londonderry, UK (April 1, 2004).
[CrossRef] [PubMed]

Kildishev, A. V.

Koschny, T.

S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. Soukoulis, Science 306, 1351 (2004).
[CrossRef] [PubMed]

Kroll, N.

D. R. Smith and N. Kroll, Phys. Rev. Lett. 85, 2933 (2000).
[CrossRef] [PubMed]

Landau, L. D.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Oxford, 1960).

Lifshitz, E. M.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Oxford, 1960).

Linden, S.

S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. Soukoulis, Science 306, 1351 (2004).
[CrossRef] [PubMed]

Makhnovskiy, D. P.

L. V. Panina, A. N. Grigorenko, and D. P. Makhnovskiy, Phys. Rev. B 66, 155411 (2002).
[CrossRef]

Nemat-Nasser, S. C.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, Phys. Rev. Lett. 84, 4184 (2000).
[CrossRef] [PubMed]

Padilla, W. J.

T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, Science 303, 1494 (2004).
[CrossRef] [PubMed]

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, Phys. Rev. Lett. 84, 4184 (2000).
[CrossRef] [PubMed]

Panina, L. V.

L. V. Panina, A. N. Grigorenko, and D. P. Makhnovskiy, Phys. Rev. B 66, 155411 (2002).
[CrossRef]

Panteleev, A. A.

A. N. Grigorenko, H. F. Gleeson, Y. Zhang, N. W. Roberts, A. R. Sidorov, and A. A. Panteleev, Appl. Phys. Lett. 88, 124103 (2006).
[CrossRef]

Pendry, J. B.

T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, Science 303, 1494 (2004).
[CrossRef] [PubMed]

J. B. Pendry, Phys. Rev. Lett. 85, 3966 (2000).
[CrossRef] [PubMed]

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, IEEE Trans. Microwave Theory Tech. 47, 2075 (1999).
[CrossRef]

Petrovic, J.

A. N. Grigorenko, A. K. Geim, H. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev, and J. Petrovic, Nature 438, 335 (2005); first reported in Seagate University Research Conclave, Londonderry, UK (April 1, 2004).
[CrossRef] [PubMed]

Podolskiy, V. A.

V. A. Podolskiy, A. K. Sarychev, and V. M. Shalaev, J. Nonlinear Opt. Phys. Mater. 11, 65 (2002).
[CrossRef]

Ramakrishna, S. A.

S. A. Ramakrishna, Rep. Prog. Phys. 68, 449 (2005).
[CrossRef]

Robbins, D. J.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, IEEE Trans. Microwave Theory Tech. 47, 2075 (1999).
[CrossRef]

Roberts, N. W.

A. N. Grigorenko, H. F. Gleeson, Y. Zhang, N. W. Roberts, A. R. Sidorov, and A. A. Panteleev, Appl. Phys. Lett. 88, 124103 (2006).
[CrossRef]

Sarychev, A. K.

Schultz, S.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, Phys. Rev. Lett. 84, 4184 (2000).
[CrossRef] [PubMed]

Shalaev, V. M.

Sidorov, A. R.

A. N. Grigorenko, H. F. Gleeson, Y. Zhang, N. W. Roberts, A. R. Sidorov, and A. A. Panteleev, Appl. Phys. Lett. 88, 124103 (2006).
[CrossRef]

Smith, D. R.

T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, Science 303, 1494 (2004).
[CrossRef] [PubMed]

D. R. Smith and N. Kroll, Phys. Rev. Lett. 85, 2933 (2000).
[CrossRef] [PubMed]

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, Phys. Rev. Lett. 84, 4184 (2000).
[CrossRef] [PubMed]

Soukoulis, C.

S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. Soukoulis, Science 306, 1351 (2004).
[CrossRef] [PubMed]

Stewart, W. J.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, IEEE Trans. Microwave Theory Tech. 47, 2075 (1999).
[CrossRef]

Veselago, V. G.

V. G. Veselago, Sov. Phys. Usp. 10, 509 (1968).
[CrossRef]

Vier, D. C.

T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, Science 303, 1494 (2004).
[CrossRef] [PubMed]

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, Phys. Rev. Lett. 84, 4184 (2000).
[CrossRef] [PubMed]

Wegener, M.

S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. Soukoulis, Science 306, 1351 (2004).
[CrossRef] [PubMed]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1980).

Yen, T. J.

T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, Science 303, 1494 (2004).
[CrossRef] [PubMed]

Yuan, H.-K.

Zhang, X.

T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, Science 303, 1494 (2004).
[CrossRef] [PubMed]

Zhang, Y.

A. N. Grigorenko, H. F. Gleeson, Y. Zhang, N. W. Roberts, A. R. Sidorov, and A. A. Panteleev, Appl. Phys. Lett. 88, 124103 (2006).
[CrossRef]

A. N. Grigorenko, A. K. Geim, H. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev, and J. Petrovic, Nature 438, 335 (2005); first reported in Seagate University Research Conclave, Londonderry, UK (April 1, 2004).
[CrossRef] [PubMed]

Zhou, J.

S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. Soukoulis, Science 306, 1351 (2004).
[CrossRef] [PubMed]

Appl. Phys. Lett. (1)

A. N. Grigorenko, H. F. Gleeson, Y. Zhang, N. W. Roberts, A. R. Sidorov, and A. A. Panteleev, Appl. Phys. Lett. 88, 124103 (2006).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, IEEE Trans. Microwave Theory Tech. 47, 2075 (1999).
[CrossRef]

J. Nonlinear Opt. Phys. Mater. (1)

V. A. Podolskiy, A. K. Sarychev, and V. M. Shalaev, J. Nonlinear Opt. Phys. Mater. 11, 65 (2002).
[CrossRef]

Nature (1)

A. N. Grigorenko, A. K. Geim, H. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev, and J. Petrovic, Nature 438, 335 (2005); first reported in Seagate University Research Conclave, Londonderry, UK (April 1, 2004).
[CrossRef] [PubMed]

Opt. Lett. (1)

Phys. Rev. B (1)

L. V. Panina, A. N. Grigorenko, and D. P. Makhnovskiy, Phys. Rev. B 66, 155411 (2002).
[CrossRef]

Phys. Rev. Lett. (3)

D. R. Smith and N. Kroll, Phys. Rev. Lett. 85, 2933 (2000).
[CrossRef] [PubMed]

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, Phys. Rev. Lett. 84, 4184 (2000).
[CrossRef] [PubMed]

J. B. Pendry, Phys. Rev. Lett. 85, 3966 (2000).
[CrossRef] [PubMed]

Rep. Prog. Phys. (1)

S. A. Ramakrishna, Rep. Prog. Phys. 68, 449 (2005).
[CrossRef]

Science (2)

T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, Science 303, 1494 (2004).
[CrossRef] [PubMed]

S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. Soukoulis, Science 306, 1351 (2004).
[CrossRef] [PubMed]

Sov. Phys. Usp. (1)

V. G. Veselago, Sov. Phys. Usp. 10, 509 (1968).
[CrossRef]

Other (3)

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1980).

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Oxford, 1960).

The principal square root is defined as the regular branch with the cut along the negative x axis.

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Figures (3)

Fig. 1
Fig. 1

Refractive index calculated with (a) principal square root and (b) causal square root. Effective permeability calculated within the effective-field theory with (c) principal square root and (d) causal square root. LHM parameters: λ e = 1 μ m , Δ λ e = 0.1 μ m , F e = 10 , λ m = 0.5 μ m , Δ λ m = 0.04 μ m , F m = 0.22 .

Fig. 2
Fig. 2

Nanofabricated medium with magnetic response in the visible spectrum. (a) Micrograph of the sample. (b) Distribution of electric currents (conical arrows) inside a pair of pillars for the resonant symmetric z mode. (c) Same for the antisymmetric z mode. (d, e) Experimental reflection spectra measured for TM and TE polarizations, respectively (solid curves). Insets, current distribution calculated by solving Maxwell equations for the actual experimental geometry at the resonant wavelengths and the reflection calculated with Fresnel coefficients (squares). Resonance parameters: (d) λ e = 0.69 μ m , Δ λ e = 0.23 μ m , F e = 3.9 , λ m = 0.55 μ m , Δ λ m = 0.082 μ m , F m = 0.1 ; (e) λ e = 0.64 μ m , Δ λ e = 0.18 μ m , F e = 3 .

Fig. 3
Fig. 3

Calculated dispersion of the real part of effective (a) magnetic permeability and (b) index of refraction in the sample of Fig. 2 within the effective-field theory. (1) a = 707 nm , (2) a = 500 nm , (3) a = 353 nm .

Equations (3)

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Q = ω 4 π ϵ E 2 sin ( arg ( ϵ ) + arg ( μ ) 2 ) cos ( arg ( ϵ ) arg ( μ ) 2 ) ,
n = ϵ μ cas ϵ μ exp ( i arg ( ϵ ) + arg ( μ ) 2 ) Sign [ cos ( arg ( ϵ ) arg ( μ ) 2 ) ] ,
n = ϵ μ Sign [ Re ( ϵ μ ) ] ,

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