Abstract

The signs of the imaginary parts of the permittivity and permeability in metamaterials such as split ring resonators and strip wires are investigated. It is shown that the Lorentzian model often used to describe the effective parameters (i.e., the permittivity and permeability) of these metamaterials does not physically allow their imaginary parts to be negative. Moreover, a popular technique used to retrieve the effective parameters of a structure from its S-parameters is also investigated. By comparing the effective parameters for an array of dielectric spheres obtained both from S-parameter simulations and analytical calculations, it is shown that an often observed negative imaginary permittivity obtained from the S-parameters is a result of numerical error in the simulations. This is shown both for the finite element method and finite-difference time-domain simulations.

© 2010 Optical Society of America

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  1. V. G. Veselago, “The electrodynamics of substances with simultaneous negative values of ϵ and μ,” Sov. Phys. Usp. 10, 509–514 (1968).
    [CrossRef]
  2. J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely low frequency plasmons in metallic mesostructures,” Phys. Rev. Lett. 76, 4773 (1996).
    [CrossRef] [PubMed]
  3. 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]
  4. D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permittivity and permittivity,” Phys. Rev. Lett. 84, 4184 (2000).
    [CrossRef] [PubMed]
  5. R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of negative index of refraction,” Science 292, 77–79 (2001).
    [CrossRef] [PubMed]
  6. M. Bayindir, K. Aydin, and E. Ozbay, “Transmission properties of composite metamaterials in free space,” Appl. Phys. Lett. 81, 120 (2002).
    [CrossRef]
  7. C. G. Parazzoli, R. B. Greegor, K. Li, B. E. C. Koltenbah, and M. Tanielian, “Experimental verification and simulation of negative index of refraction using Snell’s law,” Phys. Rev. Lett. 90, 107401 (2003).
    [CrossRef] [PubMed]
  8. J. F. Woodley and M. Mojahedi, “Negative group velocity and group delay in left-handed media,” Phys. Rev. E 70 (2004).
    [CrossRef]
  9. P. V. Parimi, W. T. Lu, P. Vodo, J. Sokoloff, J. S. Derov, and S. Sridhar, “Negative refraction and left-handed electromagnetism in microwave photonic crystals,” Phys. Rev. Lett. 92, 127401 (2004).
    [CrossRef] [PubMed]
  10. P. Markos and C. M. Soukoulis, “Transmission properties and effective electromagnetic parameters of double negative metamaterials,” Opt. Express 11, 649–661 (2003).
    [CrossRef] [PubMed]
  11. D. Seetharamdoo, R. Sauleau, K. Mahdjoubi, and A. C. Tarot, “Effective parameters of resonant negative refractive index metamaterials: interpretation and validity,” J. Appl. Phys. 98, 063505 (2005).
    [CrossRef]
  12. U. K. Chettiar, A. V. Kildishev, T. A. Klar, and V. M. Shalaev, “Negative index metameterials combining magnetic resonators with metal films,” Opt. Express 14, 7872–7877 (2006).
    [CrossRef] [PubMed]
  13. T. Koschny, P. Markos, D. R. Smith, and C. M. Soukoulis, “Resonant and antiresonant frequency dependence of the effective parameters of metamaterials,” Phys. Rev. E 68, 065602 (2003).
    [CrossRef]
  14. D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65, 195104 (2002).
    [CrossRef]
  15. L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media (Butterworth-Heinemann, 2002).
  16. J. D. Jackson, Classical Electrodynamics (Wiley, 1999).
  17. R. Y. Chiao, “Superluminal (but causal) propagation of wave packets in transparent media with inverted atomic populations,” Phys. Rev. A 48, R34–R37 (1993).
    [CrossRef] [PubMed]
  18. P. W. Milonni and J. H. Eberly, Lasers (Wiley-Interscience, 1988).
  19. J.-M. Lerat, N. Mallejac, and O. Acher, “Determination of the effective parameters of a metamaterial by field summation method,” J. Appl. Phys. 100, 084908 (2006).
    [CrossRef]
  20. M. S. Wheeler, J. S. Aitchison, and M. Mojahedi, “Three dimensional array of dielectric spheres with an isotropic negative permeability at infrared frequencies,” Phys. Rev. B 72, 193103 (2005).
    [CrossRef]

2006 (2)

J.-M. Lerat, N. Mallejac, and O. Acher, “Determination of the effective parameters of a metamaterial by field summation method,” J. Appl. Phys. 100, 084908 (2006).
[CrossRef]

U. K. Chettiar, A. V. Kildishev, T. A. Klar, and V. M. Shalaev, “Negative index metameterials combining magnetic resonators with metal films,” Opt. Express 14, 7872–7877 (2006).
[CrossRef] [PubMed]

2005 (2)

M. S. Wheeler, J. S. Aitchison, and M. Mojahedi, “Three dimensional array of dielectric spheres with an isotropic negative permeability at infrared frequencies,” Phys. Rev. B 72, 193103 (2005).
[CrossRef]

D. Seetharamdoo, R. Sauleau, K. Mahdjoubi, and A. C. Tarot, “Effective parameters of resonant negative refractive index metamaterials: interpretation and validity,” J. Appl. Phys. 98, 063505 (2005).
[CrossRef]

2004 (2)

J. F. Woodley and M. Mojahedi, “Negative group velocity and group delay in left-handed media,” Phys. Rev. E 70 (2004).
[CrossRef]

P. V. Parimi, W. T. Lu, P. Vodo, J. Sokoloff, J. S. Derov, and S. Sridhar, “Negative refraction and left-handed electromagnetism in microwave photonic crystals,” Phys. Rev. Lett. 92, 127401 (2004).
[CrossRef] [PubMed]

2003 (3)

T. Koschny, P. Markos, D. R. Smith, and C. M. Soukoulis, “Resonant and antiresonant frequency dependence of the effective parameters of metamaterials,” Phys. Rev. E 68, 065602 (2003).
[CrossRef]

P. Markos and C. M. Soukoulis, “Transmission properties and effective electromagnetic parameters of double negative metamaterials,” Opt. Express 11, 649–661 (2003).
[CrossRef] [PubMed]

C. G. Parazzoli, R. B. Greegor, K. Li, B. E. C. Koltenbah, and M. Tanielian, “Experimental verification and simulation of negative index of refraction using Snell’s law,” Phys. Rev. Lett. 90, 107401 (2003).
[CrossRef] [PubMed]

2002 (2)

M. Bayindir, K. Aydin, and E. Ozbay, “Transmission properties of composite metamaterials in free space,” Appl. Phys. Lett. 81, 120 (2002).
[CrossRef]

D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65, 195104 (2002).
[CrossRef]

2001 (1)

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

2000 (1)

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permittivity and permittivity,” Phys. Rev. Lett. 84, 4184 (2000).
[CrossRef] [PubMed]

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]

1996 (1)

J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely low frequency plasmons in metallic mesostructures,” Phys. Rev. Lett. 76, 4773 (1996).
[CrossRef] [PubMed]

1993 (1)

R. Y. Chiao, “Superluminal (but causal) propagation of wave packets in transparent media with inverted atomic populations,” Phys. Rev. A 48, R34–R37 (1993).
[CrossRef] [PubMed]

1968 (1)

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

Acher, O.

J.-M. Lerat, N. Mallejac, and O. Acher, “Determination of the effective parameters of a metamaterial by field summation method,” J. Appl. Phys. 100, 084908 (2006).
[CrossRef]

Aitchison, J. S.

M. S. Wheeler, J. S. Aitchison, and M. Mojahedi, “Three dimensional array of dielectric spheres with an isotropic negative permeability at infrared frequencies,” Phys. Rev. B 72, 193103 (2005).
[CrossRef]

Aydin, K.

M. Bayindir, K. Aydin, and E. Ozbay, “Transmission properties of composite metamaterials in free space,” Appl. Phys. Lett. 81, 120 (2002).
[CrossRef]

Bayindir, M.

M. Bayindir, K. Aydin, and E. Ozbay, “Transmission properties of composite metamaterials in free space,” Appl. Phys. Lett. 81, 120 (2002).
[CrossRef]

Chettiar, U. K.

Chiao, R. Y.

R. Y. Chiao, “Superluminal (but causal) propagation of wave packets in transparent media with inverted atomic populations,” Phys. Rev. A 48, R34–R37 (1993).
[CrossRef] [PubMed]

Derov, J. S.

P. V. Parimi, W. T. Lu, P. Vodo, J. Sokoloff, J. S. Derov, and S. Sridhar, “Negative refraction and left-handed electromagnetism in microwave photonic crystals,” Phys. Rev. Lett. 92, 127401 (2004).
[CrossRef] [PubMed]

Eberly, J. H.

P. W. Milonni and J. H. Eberly, Lasers (Wiley-Interscience, 1988).

Greegor, R. B.

C. G. Parazzoli, R. B. Greegor, K. Li, B. E. C. Koltenbah, and M. Tanielian, “Experimental verification and simulation of negative index of refraction using Snell’s law,” Phys. Rev. Lett. 90, 107401 (2003).
[CrossRef] [PubMed]

Holden, A. 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]

J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely low frequency plasmons in metallic mesostructures,” Phys. Rev. Lett. 76, 4773 (1996).
[CrossRef] [PubMed]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, 1999).

Kildishev, A. V.

Klar, T. A.

Koltenbah, B. E. C.

C. G. Parazzoli, R. B. Greegor, K. Li, B. E. C. Koltenbah, and M. Tanielian, “Experimental verification and simulation of negative index of refraction using Snell’s law,” Phys. Rev. Lett. 90, 107401 (2003).
[CrossRef] [PubMed]

Koschny, T.

T. Koschny, P. Markos, D. R. Smith, and C. M. Soukoulis, “Resonant and antiresonant frequency dependence of the effective parameters of metamaterials,” Phys. Rev. E 68, 065602 (2003).
[CrossRef]

Landau, L. D.

L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media (Butterworth-Heinemann, 2002).

Lerat, J.-M.

J.-M. Lerat, N. Mallejac, and O. Acher, “Determination of the effective parameters of a metamaterial by field summation method,” J. Appl. Phys. 100, 084908 (2006).
[CrossRef]

Li, K.

C. G. Parazzoli, R. B. Greegor, K. Li, B. E. C. Koltenbah, and M. Tanielian, “Experimental verification and simulation of negative index of refraction using Snell’s law,” Phys. Rev. Lett. 90, 107401 (2003).
[CrossRef] [PubMed]

Lifshitz, E. M.

L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media (Butterworth-Heinemann, 2002).

Lu, W. T.

P. V. Parimi, W. T. Lu, P. Vodo, J. Sokoloff, J. S. Derov, and S. Sridhar, “Negative refraction and left-handed electromagnetism in microwave photonic crystals,” Phys. Rev. Lett. 92, 127401 (2004).
[CrossRef] [PubMed]

Mahdjoubi, K.

D. Seetharamdoo, R. Sauleau, K. Mahdjoubi, and A. C. Tarot, “Effective parameters of resonant negative refractive index metamaterials: interpretation and validity,” J. Appl. Phys. 98, 063505 (2005).
[CrossRef]

Mallejac, N.

J.-M. Lerat, N. Mallejac, and O. Acher, “Determination of the effective parameters of a metamaterial by field summation method,” J. Appl. Phys. 100, 084908 (2006).
[CrossRef]

Markos, P.

T. Koschny, P. Markos, D. R. Smith, and C. M. Soukoulis, “Resonant and antiresonant frequency dependence of the effective parameters of metamaterials,” Phys. Rev. E 68, 065602 (2003).
[CrossRef]

P. Markos and C. M. Soukoulis, “Transmission properties and effective electromagnetic parameters of double negative metamaterials,” Opt. Express 11, 649–661 (2003).
[CrossRef] [PubMed]

D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65, 195104 (2002).
[CrossRef]

Milonni, P. W.

P. W. Milonni and J. H. Eberly, Lasers (Wiley-Interscience, 1988).

Mojahedi, M.

M. S. Wheeler, J. S. Aitchison, and M. Mojahedi, “Three dimensional array of dielectric spheres with an isotropic negative permeability at infrared frequencies,” Phys. Rev. B 72, 193103 (2005).
[CrossRef]

J. F. Woodley and M. Mojahedi, “Negative group velocity and group delay in left-handed media,” Phys. Rev. E 70 (2004).
[CrossRef]

Nemat-Nasser, S. C.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permittivity and permittivity,” Phys. Rev. Lett. 84, 4184 (2000).
[CrossRef] [PubMed]

Ozbay, E.

M. Bayindir, K. Aydin, and E. Ozbay, “Transmission properties of composite metamaterials in free space,” Appl. Phys. Lett. 81, 120 (2002).
[CrossRef]

Padilla, W. J.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permittivity and permittivity,” Phys. Rev. Lett. 84, 4184 (2000).
[CrossRef] [PubMed]

Parazzoli, C. G.

C. G. Parazzoli, R. B. Greegor, K. Li, B. E. C. Koltenbah, and M. Tanielian, “Experimental verification and simulation of negative index of refraction using Snell’s law,” Phys. Rev. Lett. 90, 107401 (2003).
[CrossRef] [PubMed]

Parimi, P. V.

P. V. Parimi, W. T. Lu, P. Vodo, J. Sokoloff, J. S. Derov, and S. Sridhar, “Negative refraction and left-handed electromagnetism in microwave photonic crystals,” Phys. Rev. Lett. 92, 127401 (2004).
[CrossRef] [PubMed]

Pendry, J. B.

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]

J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely low frequency plasmons in metallic mesostructures,” Phys. Rev. Lett. 76, 4773 (1996).
[CrossRef] [PubMed]

Pitaevskii, L. P.

L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media (Butterworth-Heinemann, 2002).

Robbins, D. 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]

Sauleau, R.

D. Seetharamdoo, R. Sauleau, K. Mahdjoubi, and A. C. Tarot, “Effective parameters of resonant negative refractive index metamaterials: interpretation and validity,” J. Appl. Phys. 98, 063505 (2005).
[CrossRef]

Schultz, S.

D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65, 195104 (2002).
[CrossRef]

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

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permittivity and permittivity,” Phys. Rev. Lett. 84, 4184 (2000).
[CrossRef] [PubMed]

Seetharamdoo, D.

D. Seetharamdoo, R. Sauleau, K. Mahdjoubi, and A. C. Tarot, “Effective parameters of resonant negative refractive index metamaterials: interpretation and validity,” J. Appl. Phys. 98, 063505 (2005).
[CrossRef]

Shalaev, V. M.

Shelby, R. A.

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

Smith, D. R.

T. Koschny, P. Markos, D. R. Smith, and C. M. Soukoulis, “Resonant and antiresonant frequency dependence of the effective parameters of metamaterials,” Phys. Rev. E 68, 065602 (2003).
[CrossRef]

D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65, 195104 (2002).
[CrossRef]

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

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permittivity and permittivity,” Phys. Rev. Lett. 84, 4184 (2000).
[CrossRef] [PubMed]

Sokoloff, J.

P. V. Parimi, W. T. Lu, P. Vodo, J. Sokoloff, J. S. Derov, and S. Sridhar, “Negative refraction and left-handed electromagnetism in microwave photonic crystals,” Phys. Rev. Lett. 92, 127401 (2004).
[CrossRef] [PubMed]

Soukoulis, C. M.

T. Koschny, P. Markos, D. R. Smith, and C. M. Soukoulis, “Resonant and antiresonant frequency dependence of the effective parameters of metamaterials,” Phys. Rev. E 68, 065602 (2003).
[CrossRef]

P. Markos and C. M. Soukoulis, “Transmission properties and effective electromagnetic parameters of double negative metamaterials,” Opt. Express 11, 649–661 (2003).
[CrossRef] [PubMed]

D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65, 195104 (2002).
[CrossRef]

Sridhar, S.

P. V. Parimi, W. T. Lu, P. Vodo, J. Sokoloff, J. S. Derov, and S. Sridhar, “Negative refraction and left-handed electromagnetism in microwave photonic crystals,” Phys. Rev. Lett. 92, 127401 (2004).
[CrossRef] [PubMed]

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]

J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely low frequency plasmons in metallic mesostructures,” Phys. Rev. Lett. 76, 4773 (1996).
[CrossRef] [PubMed]

Tanielian, M.

C. G. Parazzoli, R. B. Greegor, K. Li, B. E. C. Koltenbah, and M. Tanielian, “Experimental verification and simulation of negative index of refraction using Snell’s law,” Phys. Rev. Lett. 90, 107401 (2003).
[CrossRef] [PubMed]

Tarot, A. C.

D. Seetharamdoo, R. Sauleau, K. Mahdjoubi, and A. C. Tarot, “Effective parameters of resonant negative refractive index metamaterials: interpretation and validity,” J. Appl. Phys. 98, 063505 (2005).
[CrossRef]

Veselago, V. G.

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

Vier, D. C.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permittivity and permittivity,” Phys. Rev. Lett. 84, 4184 (2000).
[CrossRef] [PubMed]

Vodo, P.

P. V. Parimi, W. T. Lu, P. Vodo, J. Sokoloff, J. S. Derov, and S. Sridhar, “Negative refraction and left-handed electromagnetism in microwave photonic crystals,” Phys. Rev. Lett. 92, 127401 (2004).
[CrossRef] [PubMed]

Wheeler, M. S.

M. S. Wheeler, J. S. Aitchison, and M. Mojahedi, “Three dimensional array of dielectric spheres with an isotropic negative permeability at infrared frequencies,” Phys. Rev. B 72, 193103 (2005).
[CrossRef]

Woodley, J. F.

J. F. Woodley and M. Mojahedi, “Negative group velocity and group delay in left-handed media,” Phys. Rev. E 70 (2004).
[CrossRef]

Youngs, I.

J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely low frequency plasmons in metallic mesostructures,” Phys. Rev. Lett. 76, 4773 (1996).
[CrossRef] [PubMed]

Appl. Phys. Lett. (1)

M. Bayindir, K. Aydin, and E. Ozbay, “Transmission properties of composite metamaterials in free space,” Appl. Phys. Lett. 81, 120 (2002).
[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]

J. Appl. Phys. (2)

J.-M. Lerat, N. Mallejac, and O. Acher, “Determination of the effective parameters of a metamaterial by field summation method,” J. Appl. Phys. 100, 084908 (2006).
[CrossRef]

D. Seetharamdoo, R. Sauleau, K. Mahdjoubi, and A. C. Tarot, “Effective parameters of resonant negative refractive index metamaterials: interpretation and validity,” J. Appl. Phys. 98, 063505 (2005).
[CrossRef]

Opt. Express (2)

Phys. Rev. A (1)

R. Y. Chiao, “Superluminal (but causal) propagation of wave packets in transparent media with inverted atomic populations,” Phys. Rev. A 48, R34–R37 (1993).
[CrossRef] [PubMed]

Phys. Rev. B (2)

D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65, 195104 (2002).
[CrossRef]

M. S. Wheeler, J. S. Aitchison, and M. Mojahedi, “Three dimensional array of dielectric spheres with an isotropic negative permeability at infrared frequencies,” Phys. Rev. B 72, 193103 (2005).
[CrossRef]

Phys. Rev. E (2)

J. F. Woodley and M. Mojahedi, “Negative group velocity and group delay in left-handed media,” Phys. Rev. E 70 (2004).
[CrossRef]

T. Koschny, P. Markos, D. R. Smith, and C. M. Soukoulis, “Resonant and antiresonant frequency dependence of the effective parameters of metamaterials,” Phys. Rev. E 68, 065602 (2003).
[CrossRef]

Phys. Rev. Lett. (4)

P. V. Parimi, W. T. Lu, P. Vodo, J. Sokoloff, J. S. Derov, and S. Sridhar, “Negative refraction and left-handed electromagnetism in microwave photonic crystals,” Phys. Rev. Lett. 92, 127401 (2004).
[CrossRef] [PubMed]

C. G. Parazzoli, R. B. Greegor, K. Li, B. E. C. Koltenbah, and M. Tanielian, “Experimental verification and simulation of negative index of refraction using Snell’s law,” Phys. Rev. Lett. 90, 107401 (2003).
[CrossRef] [PubMed]

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permittivity and permittivity,” Phys. Rev. Lett. 84, 4184 (2000).
[CrossRef] [PubMed]

J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely low frequency plasmons in metallic mesostructures,” Phys. Rev. Lett. 76, 4773 (1996).
[CrossRef] [PubMed]

Science (1)

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

Sov. Phys. Usp. (1)

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

Other (3)

L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media (Butterworth-Heinemann, 2002).

J. D. Jackson, Classical Electrodynamics (Wiley, 1999).

P. W. Milonni and J. H. Eberly, Lasers (Wiley-Interscience, 1988).

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Figures (6)

Fig. 1
Fig. 1

Real (a) and imaginary (b) parts of the effective index of refraction obtained analytically (dotted curve) and from using the retrieval technique on S-parameters obtained from FDTD (solid curve) and FEM (dashed curve) simulations.

Fig. 2
Fig. 2

Real (a) and imaginary (b) parts of the effective impedance obtained analytically (dotted curve) and from using the retrieval technique on S-parameters obtained from FDTD (solid curve) and FEM (dashed curve) simulations.

Fig. 3
Fig. 3

Real (a) and imaginary (b) parts of the effective permittivity obtained analytically (dotted curve) and from using the retrieval technique on S-parameters obtained from FDTD (solid curve) and FEM (dashed curve) simulations.

Fig. 4
Fig. 4

Real (a) and imaginary (b) parts of the effective permeability obtained analytically (dotted curve) and from using the retrieval technique on S-parameters obtained from FDTD (solid curve) and FEM (dashed curve) simulations.

Fig. 5
Fig. 5

(a) S11 and (b) S12 calculated analytically for propagation through a 10 μ m thick slab (dotted curve) and determined from FDTD (solid curves) and FEM (dashed curves) simulations for propagation through a sheet of dielectric spheres infinite in the transverse plane and one unit cell thick in the propagation direction.

Fig. 6
Fig. 6

Magnitude of the terms n z and n z calculated from the analytical expressions and retrieved from the FDTD and FEM simulations.

Equations (17)

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Q = ω ( ϵ | E | 2 + μ | H | 2 )
ϵ = 1 + ω p e 2 ω o e 2 ω 2 i γ e ω ,
μ = 1 + ω p m 2 ω o m 2 ω 2 i γ m ω ,
D ( r , t ) = ϵ 0 { E ( r , t ) + G ( τ ) E ( r , t τ ) d τ } ,
G ( τ ) = 1 2 π ( ϵ ( ω ) ϵ o 1 ) e i ω τ d ω .
G ( τ ) = ω p 2 e γ τ 2 sin ( ν o τ ) ν o Θ ( τ ) ,
ω p e = N e 2 f m e ,
n = 1 k d Cos 1 [ 1 2 S 12 ( 1 S 11 2 + S 12 2 ) ] + 2 l π k d ,
z = ( 1 + S 11 ) 2 + S 12 2 ( 1 S 11 ) 2 + S 12 2 ,
ϵ = n z ,
μ = n z .
ϵ a n = k 0 3 + 4 π i N V a 1 k 0 3 2 π i N V a 1 ,
μ a n = k 0 3 + 4 π i N V b 1 k 0 3 2 π i N V b 1 ,
ϵ + i ϵ = n + i n z + i z = ( n + i n ) ( z i z ) z 2 + z 2 = n z + n z + i ( n z n z ) z 2 + z 2 .
μ + j μ = ( n + j n ) ( z + j z ) = n z n z + j ( n z + n z ) .
ϵ = ( | n z | + | n z | ) z 2 + z 2 ,
μ = | n z | | n z | .

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