Abstract

A homogenization model is applied to describe the wave interaction with finite three-dimensional metamaterial objects composed of periodic arrays of magnetodielectric spheres and is validated with full-wave numerical simulations. The homogenization is based on a dipolar model of the inclusions, which is shown to hold even in the case of densely packed arrays once weak forms of spatial dispersion and the full dynamic array coupling are taken into account. The numerical simulations are based on a fast surface-integral equation solver that enables the analysis of scattering from complex piecewise homogeneous objects. We validate the homogenization model by considering electrically large disk- and cube-shaped arrays and quantify the accuracy of the transition from an array of spheres to a homogeneous object as a function of the array size. Simulation results show that the fields scattered from large arrays with up to one thousand spheres and equivalent homogeneous objects agree well, not only far away from the arrays but also near them.

© 2013 OSA

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  1. R. W. Ziolkowski, “Design, fabrication, and testing of double negative metamaterials,” IEEE Trans. Antennas Propag.51(7), 1516–1529 (2003).
    [CrossRef]
  2. C. Caloz, T. Itoh, and A. Rennings, “CRLH metamaterial leaky-wave and resonant antennas,” IEEE Antennas Propag. Mag.50(5), 25–39 (2008).
    [CrossRef]
  3. E. Lier, D. H. Werner, C. P. Scarborough, Q. Wu, and J. A. Bossard, “An octave-bandwidth negligible-loss radiofrequency metamaterial,” Nat. Mater.10(3), 216–222 (2011).
    [CrossRef] [PubMed]
  4. X.-X. Liu and A. Alù, “Subwavelength leaky-wave optical nanoantennas: Directive radiation from linear arrays of plasmonic nanoparticles,” Phys. Rev. B82(14), 144305 (2010).
    [CrossRef]
  5. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett.85(18), 3966–3969 (2000).
    [CrossRef] [PubMed]
  6. Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science315(5819), 1686 (2007).
    [CrossRef] [PubMed]
  7. X.-X. Liu and A. Alù, “Limitations and potentials of metamaterial lenses,” J. Nanophotonics5(1), 053509 (2011).
    [CrossRef]
  8. N. Kundtz and D. R. Smith, “Extreme-angle broadband metamaterial lens,” Nat. Mater.9(2), 129–132 (2010).
    [CrossRef] [PubMed]
  9. J. C. Maxwell Garnett, “Colours in metal glasses and in metallic films,” Philos. Trans. R. Soc. Lond. A203, 443–445 (1904).
  10. M. Born and K. Huang, Dynamic Theory of Crystal Lattices (Oxford University 1998).
  11. C. R. Simovski and S. A. Tretyakov, “Local constitutive parameters of metamaterials from an effective-medium perspective,” Phys. Rev. B75(19), 195111 (2007).
    [CrossRef]
  12. D. R. Smith, S. Schultz, P. Markoš, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B65(19), 195104 (2002).
    [CrossRef]
  13. X. Chen, T. M. Grzegorczyk, B.-I. Wu, J. Pacheco, and J. A. Kong, “Robust method to retrieve the constitutive effective parameters of metamaterials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.70(1), 016608 (2004).
    [CrossRef] [PubMed]
  14. L. Landau and E. Lifschitz, Electrodynamics of Continuous Media (Butterworth-Heinemann, 1984).
  15. C. R. Simovski, “On electromagnetic characterization and homogenization of nanostructured metamaterials,” J. Opt.13(1), 013001 (2011).
    [CrossRef]
  16. D. R. Smith, D. C. Vier, Th. Koschny, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous metamaterials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.71(33 Pt 2B), 036617 (2005).
    [CrossRef] [PubMed]
  17. P. A. Belov and C. R. Simovski, “Homogenization of electromagnetic crystals formed by uniaxial resonant scatterers,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.72(2), 026615 (2005).
    [CrossRef] [PubMed]
  18. C. R. Simovski, “Bloch material parameters of magneto-dielectric metamaterials and the concept of Bloch lattices,” Metamaterials (Amst.)1(2), 62–80 (2007).
    [CrossRef]
  19. R. A. Shore and A. Yaghjian, “Traveling waves on two-and three-dimensional periodic arrays of lossless scatterers,” Radio Sci.42(6), RS6S21 (2007).
    [CrossRef]
  20. M. G. Silveirinha, “Generalized Lorentz-Lorenz formulas for microstructured materials,” Phys. Rev. B76(24), 245117 (2007).
    [CrossRef]
  21. C. Fietz and G. Shvets, “Current-driven metamaterial homogenization,” Physica B405(14), 2930–2934 (2010).
    [CrossRef]
  22. A. Alù, “First-principles homogenization theory for periodic metamaterials,” Phys. Rev. B84(7), 075153 (2011).
    [CrossRef]
  23. A. Alù, “Restoring the physical meaning of metamaterial constitutive parameters,” Phys. Rev. B.83(8), 081102 (2011).
    [CrossRef]
  24. X.-X. Liu and A. Alù, “Homogenization of quasi-isotropic metamaterials composed of dense arrays of magnetodielectric spheres,” Metamaterials (Amst.)5(2-3), 56–63 (2011).
    [CrossRef]
  25. D. L. Smith, L. N. Medgyesi-Mitschang, and D. W. Forester, “Surface integral equation formulations for left-handed materials,” Prog. Electromagnetics Res.51, 27–48 (2005).
    [CrossRef]
  26. P. Yla-Oijala, Ö. Ergül, L. Gürel, and M. Taskinen, “Efficient surface integral equation methods for the analysis of complex metamaterial structures,” in Proceedings of European conference on antennas and propagation (EuCAP), 1560- 1564, Berlin, Germany (2009).
  27. J. Rivero, J. M. Taboada, L. Landesa, F. Obelleiro, and I. García-Tuñón, “Surface integral equation formulation for the analysis of left-handed metamaterials,” Opt. Express18(15), 15876–15886 (2010).
    [CrossRef] [PubMed]
  28. M.-F. Wu and A. E. Yilmaz, “A well-conditioned multiple-grid AIM accelerated PMCHWT solver for composite structures,” USNC/URSI Rad. Sci. Meet. (2010).
  29. M.-F. Wu, G. Kaur, and A. E. Yilmaz, “A multiple-grid adaptive integral method for multi-region problems,” IEEE Trans. Antenn. Propag.58(5), 1601–1613 (2010).
    [CrossRef]
  30. M.-F. Wu, X.-X. Liu, A. Alu, and A. E. Yilmaz, “A fast surface integral equation solver for composite structures with metamaterial regions,” Proc. IEEE Antennas and Propagation Soc. Int. Symp. 2688–2691 (2011).
  31. A. Alù and N. Engheta, “Dynamical theory of artificial optical magnetism produced by rings of plasmonic nanoparticles,” Phys. Rev. B78(8), 085112 (2008).
    [CrossRef]
  32. F. Shafiei, F. Monticone, K. Q. Le, X. X. Liu, T. Hartsfield, A. Alù, and X. Li, “A subwavelength plasmonic metamolecule exhibiting magnetic-based optical fano resonance,” Nat. Nanotechnol.8(2), 95–99 (2013).
    [CrossRef] [PubMed]
  33. X.-X. Liu, D. A. Powell, and A. Alù, “Correcting the Fabry-Perot artifacts in metamaterial retrieval procedures,” Phys. Rev. B84(23), 235106 (2011).
    [CrossRef]
  34. X.-X. Liu and A. Alù, “First-principle homogenization of magnetodielectric metamaterial arrays,” Proc. IEEE Antennas and Propagation Soc. Int. Symp. 1522–1525 (2011).
  35. X.-X. Liu and A. Alù, “Generalized retrieval method for metamaterial constitutive parameters based on a physically driven homogenization approach,” Phys. Rev. B87(23), 235136 (2013).
    [CrossRef]

2013 (2)

F. Shafiei, F. Monticone, K. Q. Le, X. X. Liu, T. Hartsfield, A. Alù, and X. Li, “A subwavelength plasmonic metamolecule exhibiting magnetic-based optical fano resonance,” Nat. Nanotechnol.8(2), 95–99 (2013).
[CrossRef] [PubMed]

X.-X. Liu and A. Alù, “Generalized retrieval method for metamaterial constitutive parameters based on a physically driven homogenization approach,” Phys. Rev. B87(23), 235136 (2013).
[CrossRef]

2011 (7)

X.-X. Liu, D. A. Powell, and A. Alù, “Correcting the Fabry-Perot artifacts in metamaterial retrieval procedures,” Phys. Rev. B84(23), 235106 (2011).
[CrossRef]

E. Lier, D. H. Werner, C. P. Scarborough, Q. Wu, and J. A. Bossard, “An octave-bandwidth negligible-loss radiofrequency metamaterial,” Nat. Mater.10(3), 216–222 (2011).
[CrossRef] [PubMed]

X.-X. Liu and A. Alù, “Limitations and potentials of metamaterial lenses,” J. Nanophotonics5(1), 053509 (2011).
[CrossRef]

C. R. Simovski, “On electromagnetic characterization and homogenization of nanostructured metamaterials,” J. Opt.13(1), 013001 (2011).
[CrossRef]

A. Alù, “First-principles homogenization theory for periodic metamaterials,” Phys. Rev. B84(7), 075153 (2011).
[CrossRef]

A. Alù, “Restoring the physical meaning of metamaterial constitutive parameters,” Phys. Rev. B.83(8), 081102 (2011).
[CrossRef]

X.-X. Liu and A. Alù, “Homogenization of quasi-isotropic metamaterials composed of dense arrays of magnetodielectric spheres,” Metamaterials (Amst.)5(2-3), 56–63 (2011).
[CrossRef]

2010 (5)

N. Kundtz and D. R. Smith, “Extreme-angle broadband metamaterial lens,” Nat. Mater.9(2), 129–132 (2010).
[CrossRef] [PubMed]

X.-X. Liu and A. Alù, “Subwavelength leaky-wave optical nanoantennas: Directive radiation from linear arrays of plasmonic nanoparticles,” Phys. Rev. B82(14), 144305 (2010).
[CrossRef]

C. Fietz and G. Shvets, “Current-driven metamaterial homogenization,” Physica B405(14), 2930–2934 (2010).
[CrossRef]

J. Rivero, J. M. Taboada, L. Landesa, F. Obelleiro, and I. García-Tuñón, “Surface integral equation formulation for the analysis of left-handed metamaterials,” Opt. Express18(15), 15876–15886 (2010).
[CrossRef] [PubMed]

M.-F. Wu, G. Kaur, and A. E. Yilmaz, “A multiple-grid adaptive integral method for multi-region problems,” IEEE Trans. Antenn. Propag.58(5), 1601–1613 (2010).
[CrossRef]

2008 (2)

A. Alù and N. Engheta, “Dynamical theory of artificial optical magnetism produced by rings of plasmonic nanoparticles,” Phys. Rev. B78(8), 085112 (2008).
[CrossRef]

C. Caloz, T. Itoh, and A. Rennings, “CRLH metamaterial leaky-wave and resonant antennas,” IEEE Antennas Propag. Mag.50(5), 25–39 (2008).
[CrossRef]

2007 (5)

C. R. Simovski and S. A. Tretyakov, “Local constitutive parameters of metamaterials from an effective-medium perspective,” Phys. Rev. B75(19), 195111 (2007).
[CrossRef]

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science315(5819), 1686 (2007).
[CrossRef] [PubMed]

C. R. Simovski, “Bloch material parameters of magneto-dielectric metamaterials and the concept of Bloch lattices,” Metamaterials (Amst.)1(2), 62–80 (2007).
[CrossRef]

R. A. Shore and A. Yaghjian, “Traveling waves on two-and three-dimensional periodic arrays of lossless scatterers,” Radio Sci.42(6), RS6S21 (2007).
[CrossRef]

M. G. Silveirinha, “Generalized Lorentz-Lorenz formulas for microstructured materials,” Phys. Rev. B76(24), 245117 (2007).
[CrossRef]

2005 (3)

D. R. Smith, D. C. Vier, Th. Koschny, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous metamaterials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.71(33 Pt 2B), 036617 (2005).
[CrossRef] [PubMed]

P. A. Belov and C. R. Simovski, “Homogenization of electromagnetic crystals formed by uniaxial resonant scatterers,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.72(2), 026615 (2005).
[CrossRef] [PubMed]

D. L. Smith, L. N. Medgyesi-Mitschang, and D. W. Forester, “Surface integral equation formulations for left-handed materials,” Prog. Electromagnetics Res.51, 27–48 (2005).
[CrossRef]

2004 (1)

X. Chen, T. M. Grzegorczyk, B.-I. Wu, J. Pacheco, and J. A. Kong, “Robust method to retrieve the constitutive effective parameters of metamaterials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.70(1), 016608 (2004).
[CrossRef] [PubMed]

2003 (1)

R. W. Ziolkowski, “Design, fabrication, and testing of double negative metamaterials,” IEEE Trans. Antennas Propag.51(7), 1516–1529 (2003).
[CrossRef]

2002 (1)

D. R. Smith, S. Schultz, P. Markoš, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B65(19), 195104 (2002).
[CrossRef]

2000 (1)

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett.85(18), 3966–3969 (2000).
[CrossRef] [PubMed]

1904 (1)

J. C. Maxwell Garnett, “Colours in metal glasses and in metallic films,” Philos. Trans. R. Soc. Lond. A203, 443–445 (1904).

Alù, A.

F. Shafiei, F. Monticone, K. Q. Le, X. X. Liu, T. Hartsfield, A. Alù, and X. Li, “A subwavelength plasmonic metamolecule exhibiting magnetic-based optical fano resonance,” Nat. Nanotechnol.8(2), 95–99 (2013).
[CrossRef] [PubMed]

X.-X. Liu and A. Alù, “Generalized retrieval method for metamaterial constitutive parameters based on a physically driven homogenization approach,” Phys. Rev. B87(23), 235136 (2013).
[CrossRef]

X.-X. Liu, D. A. Powell, and A. Alù, “Correcting the Fabry-Perot artifacts in metamaterial retrieval procedures,” Phys. Rev. B84(23), 235106 (2011).
[CrossRef]

A. Alù, “First-principles homogenization theory for periodic metamaterials,” Phys. Rev. B84(7), 075153 (2011).
[CrossRef]

A. Alù, “Restoring the physical meaning of metamaterial constitutive parameters,” Phys. Rev. B.83(8), 081102 (2011).
[CrossRef]

X.-X. Liu and A. Alù, “Homogenization of quasi-isotropic metamaterials composed of dense arrays of magnetodielectric spheres,” Metamaterials (Amst.)5(2-3), 56–63 (2011).
[CrossRef]

X.-X. Liu and A. Alù, “Limitations and potentials of metamaterial lenses,” J. Nanophotonics5(1), 053509 (2011).
[CrossRef]

X.-X. Liu and A. Alù, “Subwavelength leaky-wave optical nanoantennas: Directive radiation from linear arrays of plasmonic nanoparticles,” Phys. Rev. B82(14), 144305 (2010).
[CrossRef]

A. Alù and N. Engheta, “Dynamical theory of artificial optical magnetism produced by rings of plasmonic nanoparticles,” Phys. Rev. B78(8), 085112 (2008).
[CrossRef]

Belov, P. A.

P. A. Belov and C. R. Simovski, “Homogenization of electromagnetic crystals formed by uniaxial resonant scatterers,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.72(2), 026615 (2005).
[CrossRef] [PubMed]

Bossard, J. A.

E. Lier, D. H. Werner, C. P. Scarborough, Q. Wu, and J. A. Bossard, “An octave-bandwidth negligible-loss radiofrequency metamaterial,” Nat. Mater.10(3), 216–222 (2011).
[CrossRef] [PubMed]

Caloz, C.

C. Caloz, T. Itoh, and A. Rennings, “CRLH metamaterial leaky-wave and resonant antennas,” IEEE Antennas Propag. Mag.50(5), 25–39 (2008).
[CrossRef]

Chen, X.

X. Chen, T. M. Grzegorczyk, B.-I. Wu, J. Pacheco, and J. A. Kong, “Robust method to retrieve the constitutive effective parameters of metamaterials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.70(1), 016608 (2004).
[CrossRef] [PubMed]

Engheta, N.

A. Alù and N. Engheta, “Dynamical theory of artificial optical magnetism produced by rings of plasmonic nanoparticles,” Phys. Rev. B78(8), 085112 (2008).
[CrossRef]

Fietz, C.

C. Fietz and G. Shvets, “Current-driven metamaterial homogenization,” Physica B405(14), 2930–2934 (2010).
[CrossRef]

Forester, D. W.

D. L. Smith, L. N. Medgyesi-Mitschang, and D. W. Forester, “Surface integral equation formulations for left-handed materials,” Prog. Electromagnetics Res.51, 27–48 (2005).
[CrossRef]

García-Tuñón, I.

Grzegorczyk, T. M.

X. Chen, T. M. Grzegorczyk, B.-I. Wu, J. Pacheco, and J. A. Kong, “Robust method to retrieve the constitutive effective parameters of metamaterials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.70(1), 016608 (2004).
[CrossRef] [PubMed]

Hartsfield, T.

F. Shafiei, F. Monticone, K. Q. Le, X. X. Liu, T. Hartsfield, A. Alù, and X. Li, “A subwavelength plasmonic metamolecule exhibiting magnetic-based optical fano resonance,” Nat. Nanotechnol.8(2), 95–99 (2013).
[CrossRef] [PubMed]

Itoh, T.

C. Caloz, T. Itoh, and A. Rennings, “CRLH metamaterial leaky-wave and resonant antennas,” IEEE Antennas Propag. Mag.50(5), 25–39 (2008).
[CrossRef]

Kaur, G.

M.-F. Wu, G. Kaur, and A. E. Yilmaz, “A multiple-grid adaptive integral method for multi-region problems,” IEEE Trans. Antenn. Propag.58(5), 1601–1613 (2010).
[CrossRef]

Kong, J. A.

X. Chen, T. M. Grzegorczyk, B.-I. Wu, J. Pacheco, and J. A. Kong, “Robust method to retrieve the constitutive effective parameters of metamaterials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.70(1), 016608 (2004).
[CrossRef] [PubMed]

Koschny, Th.

D. R. Smith, D. C. Vier, Th. Koschny, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous metamaterials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.71(33 Pt 2B), 036617 (2005).
[CrossRef] [PubMed]

Kundtz, N.

N. Kundtz and D. R. Smith, “Extreme-angle broadband metamaterial lens,” Nat. Mater.9(2), 129–132 (2010).
[CrossRef] [PubMed]

Landesa, L.

Le, K. Q.

F. Shafiei, F. Monticone, K. Q. Le, X. X. Liu, T. Hartsfield, A. Alù, and X. Li, “A subwavelength plasmonic metamolecule exhibiting magnetic-based optical fano resonance,” Nat. Nanotechnol.8(2), 95–99 (2013).
[CrossRef] [PubMed]

Lee, H.

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science315(5819), 1686 (2007).
[CrossRef] [PubMed]

Li, X.

F. Shafiei, F. Monticone, K. Q. Le, X. X. Liu, T. Hartsfield, A. Alù, and X. Li, “A subwavelength plasmonic metamolecule exhibiting magnetic-based optical fano resonance,” Nat. Nanotechnol.8(2), 95–99 (2013).
[CrossRef] [PubMed]

Lier, E.

E. Lier, D. H. Werner, C. P. Scarborough, Q. Wu, and J. A. Bossard, “An octave-bandwidth negligible-loss radiofrequency metamaterial,” Nat. Mater.10(3), 216–222 (2011).
[CrossRef] [PubMed]

Liu, X. X.

F. Shafiei, F. Monticone, K. Q. Le, X. X. Liu, T. Hartsfield, A. Alù, and X. Li, “A subwavelength plasmonic metamolecule exhibiting magnetic-based optical fano resonance,” Nat. Nanotechnol.8(2), 95–99 (2013).
[CrossRef] [PubMed]

Liu, X.-X.

X.-X. Liu and A. Alù, “Generalized retrieval method for metamaterial constitutive parameters based on a physically driven homogenization approach,” Phys. Rev. B87(23), 235136 (2013).
[CrossRef]

X.-X. Liu, D. A. Powell, and A. Alù, “Correcting the Fabry-Perot artifacts in metamaterial retrieval procedures,” Phys. Rev. B84(23), 235106 (2011).
[CrossRef]

X.-X. Liu and A. Alù, “Homogenization of quasi-isotropic metamaterials composed of dense arrays of magnetodielectric spheres,” Metamaterials (Amst.)5(2-3), 56–63 (2011).
[CrossRef]

X.-X. Liu and A. Alù, “Limitations and potentials of metamaterial lenses,” J. Nanophotonics5(1), 053509 (2011).
[CrossRef]

X.-X. Liu and A. Alù, “Subwavelength leaky-wave optical nanoantennas: Directive radiation from linear arrays of plasmonic nanoparticles,” Phys. Rev. B82(14), 144305 (2010).
[CrossRef]

Liu, Z.

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science315(5819), 1686 (2007).
[CrossRef] [PubMed]

Markoš, P.

D. R. Smith, S. Schultz, P. Markoš, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B65(19), 195104 (2002).
[CrossRef]

Maxwell Garnett, J. C.

J. C. Maxwell Garnett, “Colours in metal glasses and in metallic films,” Philos. Trans. R. Soc. Lond. A203, 443–445 (1904).

Medgyesi-Mitschang, L. N.

D. L. Smith, L. N. Medgyesi-Mitschang, and D. W. Forester, “Surface integral equation formulations for left-handed materials,” Prog. Electromagnetics Res.51, 27–48 (2005).
[CrossRef]

Monticone, F.

F. Shafiei, F. Monticone, K. Q. Le, X. X. Liu, T. Hartsfield, A. Alù, and X. Li, “A subwavelength plasmonic metamolecule exhibiting magnetic-based optical fano resonance,” Nat. Nanotechnol.8(2), 95–99 (2013).
[CrossRef] [PubMed]

Obelleiro, F.

Pacheco, J.

X. Chen, T. M. Grzegorczyk, B.-I. Wu, J. Pacheco, and J. A. Kong, “Robust method to retrieve the constitutive effective parameters of metamaterials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.70(1), 016608 (2004).
[CrossRef] [PubMed]

Pendry, J. B.

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett.85(18), 3966–3969 (2000).
[CrossRef] [PubMed]

Powell, D. A.

X.-X. Liu, D. A. Powell, and A. Alù, “Correcting the Fabry-Perot artifacts in metamaterial retrieval procedures,” Phys. Rev. B84(23), 235106 (2011).
[CrossRef]

Rennings, A.

C. Caloz, T. Itoh, and A. Rennings, “CRLH metamaterial leaky-wave and resonant antennas,” IEEE Antennas Propag. Mag.50(5), 25–39 (2008).
[CrossRef]

Rivero, J.

Scarborough, C. P.

E. Lier, D. H. Werner, C. P. Scarborough, Q. Wu, and J. A. Bossard, “An octave-bandwidth negligible-loss radiofrequency metamaterial,” Nat. Mater.10(3), 216–222 (2011).
[CrossRef] [PubMed]

Schultz, S.

D. R. Smith, S. Schultz, P. Markoš, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B65(19), 195104 (2002).
[CrossRef]

Shafiei, F.

F. Shafiei, F. Monticone, K. Q. Le, X. X. Liu, T. Hartsfield, A. Alù, and X. Li, “A subwavelength plasmonic metamolecule exhibiting magnetic-based optical fano resonance,” Nat. Nanotechnol.8(2), 95–99 (2013).
[CrossRef] [PubMed]

Shore, R. A.

R. A. Shore and A. Yaghjian, “Traveling waves on two-and three-dimensional periodic arrays of lossless scatterers,” Radio Sci.42(6), RS6S21 (2007).
[CrossRef]

Shvets, G.

C. Fietz and G. Shvets, “Current-driven metamaterial homogenization,” Physica B405(14), 2930–2934 (2010).
[CrossRef]

Silveirinha, M. G.

M. G. Silveirinha, “Generalized Lorentz-Lorenz formulas for microstructured materials,” Phys. Rev. B76(24), 245117 (2007).
[CrossRef]

Simovski, C. R.

C. R. Simovski, “On electromagnetic characterization and homogenization of nanostructured metamaterials,” J. Opt.13(1), 013001 (2011).
[CrossRef]

C. R. Simovski and S. A. Tretyakov, “Local constitutive parameters of metamaterials from an effective-medium perspective,” Phys. Rev. B75(19), 195111 (2007).
[CrossRef]

C. R. Simovski, “Bloch material parameters of magneto-dielectric metamaterials and the concept of Bloch lattices,” Metamaterials (Amst.)1(2), 62–80 (2007).
[CrossRef]

P. A. Belov and C. R. Simovski, “Homogenization of electromagnetic crystals formed by uniaxial resonant scatterers,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.72(2), 026615 (2005).
[CrossRef] [PubMed]

Smith, D. L.

D. L. Smith, L. N. Medgyesi-Mitschang, and D. W. Forester, “Surface integral equation formulations for left-handed materials,” Prog. Electromagnetics Res.51, 27–48 (2005).
[CrossRef]

Smith, D. R.

N. Kundtz and D. R. Smith, “Extreme-angle broadband metamaterial lens,” Nat. Mater.9(2), 129–132 (2010).
[CrossRef] [PubMed]

D. R. Smith, D. C. Vier, Th. Koschny, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous metamaterials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.71(33 Pt 2B), 036617 (2005).
[CrossRef] [PubMed]

D. R. Smith, S. Schultz, P. Markoš, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B65(19), 195104 (2002).
[CrossRef]

Soukoulis, C. M.

D. R. Smith, D. C. Vier, Th. Koschny, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous metamaterials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.71(33 Pt 2B), 036617 (2005).
[CrossRef] [PubMed]

D. R. Smith, S. Schultz, P. Markoš, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B65(19), 195104 (2002).
[CrossRef]

Sun, C.

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science315(5819), 1686 (2007).
[CrossRef] [PubMed]

Taboada, J. M.

Tretyakov, S. A.

C. R. Simovski and S. A. Tretyakov, “Local constitutive parameters of metamaterials from an effective-medium perspective,” Phys. Rev. B75(19), 195111 (2007).
[CrossRef]

Vier, D. C.

D. R. Smith, D. C. Vier, Th. Koschny, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous metamaterials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.71(33 Pt 2B), 036617 (2005).
[CrossRef] [PubMed]

Werner, D. H.

E. Lier, D. H. Werner, C. P. Scarborough, Q. Wu, and J. A. Bossard, “An octave-bandwidth negligible-loss radiofrequency metamaterial,” Nat. Mater.10(3), 216–222 (2011).
[CrossRef] [PubMed]

Wu, B.-I.

X. Chen, T. M. Grzegorczyk, B.-I. Wu, J. Pacheco, and J. A. Kong, “Robust method to retrieve the constitutive effective parameters of metamaterials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.70(1), 016608 (2004).
[CrossRef] [PubMed]

Wu, M.-F.

M.-F. Wu, G. Kaur, and A. E. Yilmaz, “A multiple-grid adaptive integral method for multi-region problems,” IEEE Trans. Antenn. Propag.58(5), 1601–1613 (2010).
[CrossRef]

Wu, Q.

E. Lier, D. H. Werner, C. P. Scarborough, Q. Wu, and J. A. Bossard, “An octave-bandwidth negligible-loss radiofrequency metamaterial,” Nat. Mater.10(3), 216–222 (2011).
[CrossRef] [PubMed]

Xiong, Y.

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science315(5819), 1686 (2007).
[CrossRef] [PubMed]

Yaghjian, A.

R. A. Shore and A. Yaghjian, “Traveling waves on two-and three-dimensional periodic arrays of lossless scatterers,” Radio Sci.42(6), RS6S21 (2007).
[CrossRef]

Yilmaz, A. E.

M.-F. Wu, G. Kaur, and A. E. Yilmaz, “A multiple-grid adaptive integral method for multi-region problems,” IEEE Trans. Antenn. Propag.58(5), 1601–1613 (2010).
[CrossRef]

Zhang, X.

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science315(5819), 1686 (2007).
[CrossRef] [PubMed]

Ziolkowski, R. W.

R. W. Ziolkowski, “Design, fabrication, and testing of double negative metamaterials,” IEEE Trans. Antennas Propag.51(7), 1516–1529 (2003).
[CrossRef]

IEEE Antennas Propag. Mag. (1)

C. Caloz, T. Itoh, and A. Rennings, “CRLH metamaterial leaky-wave and resonant antennas,” IEEE Antennas Propag. Mag.50(5), 25–39 (2008).
[CrossRef]

IEEE Trans. Antenn. Propag. (1)

M.-F. Wu, G. Kaur, and A. E. Yilmaz, “A multiple-grid adaptive integral method for multi-region problems,” IEEE Trans. Antenn. Propag.58(5), 1601–1613 (2010).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

R. W. Ziolkowski, “Design, fabrication, and testing of double negative metamaterials,” IEEE Trans. Antennas Propag.51(7), 1516–1529 (2003).
[CrossRef]

J. Nanophotonics (1)

X.-X. Liu and A. Alù, “Limitations and potentials of metamaterial lenses,” J. Nanophotonics5(1), 053509 (2011).
[CrossRef]

J. Opt. (1)

C. R. Simovski, “On electromagnetic characterization and homogenization of nanostructured metamaterials,” J. Opt.13(1), 013001 (2011).
[CrossRef]

Metamaterials (Amst.) (2)

C. R. Simovski, “Bloch material parameters of magneto-dielectric metamaterials and the concept of Bloch lattices,” Metamaterials (Amst.)1(2), 62–80 (2007).
[CrossRef]

X.-X. Liu and A. Alù, “Homogenization of quasi-isotropic metamaterials composed of dense arrays of magnetodielectric spheres,” Metamaterials (Amst.)5(2-3), 56–63 (2011).
[CrossRef]

Nat. Mater. (2)

N. Kundtz and D. R. Smith, “Extreme-angle broadband metamaterial lens,” Nat. Mater.9(2), 129–132 (2010).
[CrossRef] [PubMed]

E. Lier, D. H. Werner, C. P. Scarborough, Q. Wu, and J. A. Bossard, “An octave-bandwidth negligible-loss radiofrequency metamaterial,” Nat. Mater.10(3), 216–222 (2011).
[CrossRef] [PubMed]

Nat. Nanotechnol. (1)

F. Shafiei, F. Monticone, K. Q. Le, X. X. Liu, T. Hartsfield, A. Alù, and X. Li, “A subwavelength plasmonic metamolecule exhibiting magnetic-based optical fano resonance,” Nat. Nanotechnol.8(2), 95–99 (2013).
[CrossRef] [PubMed]

Opt. Express (1)

Philos. Trans. R. Soc. Lond. A (1)

J. C. Maxwell Garnett, “Colours in metal glasses and in metallic films,” Philos. Trans. R. Soc. Lond. A203, 443–445 (1904).

Phys. Rev. B (8)

X.-X. Liu and A. Alù, “Subwavelength leaky-wave optical nanoantennas: Directive radiation from linear arrays of plasmonic nanoparticles,” Phys. Rev. B82(14), 144305 (2010).
[CrossRef]

C. R. Simovski and S. A. Tretyakov, “Local constitutive parameters of metamaterials from an effective-medium perspective,” Phys. Rev. B75(19), 195111 (2007).
[CrossRef]

D. R. Smith, S. Schultz, P. Markoš, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B65(19), 195104 (2002).
[CrossRef]

M. G. Silveirinha, “Generalized Lorentz-Lorenz formulas for microstructured materials,” Phys. Rev. B76(24), 245117 (2007).
[CrossRef]

A. Alù, “First-principles homogenization theory for periodic metamaterials,” Phys. Rev. B84(7), 075153 (2011).
[CrossRef]

X.-X. Liu, D. A. Powell, and A. Alù, “Correcting the Fabry-Perot artifacts in metamaterial retrieval procedures,” Phys. Rev. B84(23), 235106 (2011).
[CrossRef]

X.-X. Liu and A. Alù, “Generalized retrieval method for metamaterial constitutive parameters based on a physically driven homogenization approach,” Phys. Rev. B87(23), 235136 (2013).
[CrossRef]

A. Alù and N. Engheta, “Dynamical theory of artificial optical magnetism produced by rings of plasmonic nanoparticles,” Phys. Rev. B78(8), 085112 (2008).
[CrossRef]

Phys. Rev. B. (1)

A. Alù, “Restoring the physical meaning of metamaterial constitutive parameters,” Phys. Rev. B.83(8), 081102 (2011).
[CrossRef]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (3)

X. Chen, T. M. Grzegorczyk, B.-I. Wu, J. Pacheco, and J. A. Kong, “Robust method to retrieve the constitutive effective parameters of metamaterials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.70(1), 016608 (2004).
[CrossRef] [PubMed]

D. R. Smith, D. C. Vier, Th. Koschny, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous metamaterials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.71(33 Pt 2B), 036617 (2005).
[CrossRef] [PubMed]

P. A. Belov and C. R. Simovski, “Homogenization of electromagnetic crystals formed by uniaxial resonant scatterers,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.72(2), 026615 (2005).
[CrossRef] [PubMed]

Phys. Rev. Lett. (1)

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett.85(18), 3966–3969 (2000).
[CrossRef] [PubMed]

Physica B (1)

C. Fietz and G. Shvets, “Current-driven metamaterial homogenization,” Physica B405(14), 2930–2934 (2010).
[CrossRef]

Prog. Electromagnetics Res. (1)

D. L. Smith, L. N. Medgyesi-Mitschang, and D. W. Forester, “Surface integral equation formulations for left-handed materials,” Prog. Electromagnetics Res.51, 27–48 (2005).
[CrossRef]

Radio Sci. (1)

R. A. Shore and A. Yaghjian, “Traveling waves on two-and three-dimensional periodic arrays of lossless scatterers,” Radio Sci.42(6), RS6S21 (2007).
[CrossRef]

Science (1)

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science315(5819), 1686 (2007).
[CrossRef] [PubMed]

Other (6)

M. Born and K. Huang, Dynamic Theory of Crystal Lattices (Oxford University 1998).

L. Landau and E. Lifschitz, Electrodynamics of Continuous Media (Butterworth-Heinemann, 1984).

P. Yla-Oijala, Ö. Ergül, L. Gürel, and M. Taskinen, “Efficient surface integral equation methods for the analysis of complex metamaterial structures,” in Proceedings of European conference on antennas and propagation (EuCAP), 1560- 1564, Berlin, Germany (2009).

M.-F. Wu and A. E. Yilmaz, “A well-conditioned multiple-grid AIM accelerated PMCHWT solver for composite structures,” USNC/URSI Rad. Sci. Meet. (2010).

M.-F. Wu, X.-X. Liu, A. Alu, and A. E. Yilmaz, “A fast surface integral equation solver for composite structures with metamaterial regions,” Proc. IEEE Antennas and Propagation Soc. Int. Symp. 2688–2691 (2011).

X.-X. Liu and A. Alù, “First-principle homogenization of magnetodielectric metamaterial arrays,” Proc. IEEE Antennas and Propagation Soc. Int. Symp. 1522–1525 (2011).

Supplementary Material (5)

» Media 1: MPG (830 KB)     
» Media 2: MPG (932 KB)     
» Media 3: MPG (796 KB)     
» Media 4: MPG (806 KB)     
» Media 5: MPG (890 KB)     

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Figures (6)

Fig. 1
Fig. 1

The (a) permittivity, (b) permeability, and (c) magneto-electric coefficient that are used to homogenize an infinitely periodic metamaterial array composed of magnetodielectric spheres as specified in Section 2. Effective, equivalent, and NRW retrieval parameters are calculated using Eq. (1), Eq. (2), and the method in [31]. The dashed vertical lines in all plots indicate the frequencies of interest in the following numerical simulation.

Fig. 2
Fig. 2

Disk-shaped finite array in the double-positive regime. The disks are placed in the y-z plane. Electric near-field distribution (in dB scale) due to a plane wave traveling toward z ^ for (a) the finite array and (b) the homogeneous disk with ε 1 =2.752 ε 0 and μ 1 =2.275 μ 0 . (c) Normalized difference between the two fields. (d) Snapshot of the time-domain electric field as it propagates through the disk at y=0 for the disk-shaped array and homogeneous disk. See also Media 1 for time-domain animations of electric fields in the y-z plane.

Fig. 3
Fig. 3

Disk-shaped finite array in the double-negative regime. The disks are placed in the y-z plane. Electric near-field distribution (in dB scale) due to a plane wave traveling toward z ^ for (a) the finite array and (b) the homogeneous disk with ε 2 =0.9936 ε 0 and μ 2 =0.6063 μ 0 . (c) Normalized difference between the two fields. (d) Snapshot of the time-domain electric field as it propagates through the disk at y=0 for the disk-shaped array and homogeneous disk. See also Media 2 for time-domain animations of electric fields in the y-z plane.

Fig. 4
Fig. 4

Far-field (a, b) and near-field (c-h) distributions at fDNG = 1.454 GHz for a 4-element cube-shaped array and homogenized cube in the double-negative regime with ε 2 =0.9936 ε 0 and μ 2 =0.6063 μ 0 . (i) Snapshot in time of the electric field component E x as it propagates toward the z ^ direction through the center of the cubes ( y=0 ). See also Media 3 for time-domain animations in the y-z plane.

Fig. 5
Fig. 5

Similar to Fig. 4, but for a 6-element cubic array and corresponding homogeneous cube. See also Media 4 for time-domain animations in the y-z plane.

Fig. 6
Fig. 6

Similar to Figs. 46, but for a 10-element cubic array and corresponding homogeneous cube. See also Media 5 for time-domain animations in the y-z plane.

Tables (2)

Tables Icon

Table 1 Computational requirements for the simulations in Figs. 2-3.

Tables Icon

Table 2 Computational requirements for the simulations in Figs. 4-6.

Equations (2)

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D av = ε 0 E av + P av = ε eff E av χ eff o β ^ × H av B av = μ 0 H av + M av = μ eff H av + χ eff o β ^ × E av ,
D av = ε 0 E av + P av = ε eq E av B av = μ 0 H av + M av = μ eq H av ,

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