Abstract

While the effective medium treatment of unbounded metamaterials appears to be well established and firmly proven, related phenomena in finite structures have not received sufficient attention. We report on mesoscopic effects associated with the boundaries of finite discrete metamaterial samples, which can invalidate an effective medium description. We show how to avoid such effects by proper choice of boundary configuration. As all metamaterial implementations are naturally finite, we are confident that our findings are crucial for future metamaterial research.

© 2012 OSA

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  1. C. R. Simovski, “Material parameters of metamaterials,” Opt. Spectrosc.107, 726–753 (2009).
    [CrossRef]
  2. S. A. Schelkunoff and H. T. Friis, Antennas Theory and Practice (Wiley, 1966).
  3. J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory Tech.47, 2075–2084 (1999).
    [CrossRef]
  4. M. Gorkunov, M. Lapine, E. Shamonina, and K. H. Ringhofer, “Effective magnetic properties of a composite material with circular conductive elements,” Eur. Phys. J. B28, 263–269 (2002).
    [CrossRef]
  5. R. Marqués, F. Medina, and R. Rafii-El-Idrissi, “Role of bianisotropy in negative permeability and left-handed metamaterials,” Phys. Rev. B65, 144440 (2002).
    [CrossRef]
  6. E. Shamonina, V. A. Kalinin, K. H. Ringhofer, and L. Solymar, “Magnetoinductive waves in one, two, and three dimensions,” J. Appl. Phys.92, 6252–6261 (2002).
    [CrossRef]
  7. R. R. A. Syms, E. Shamonina, V. Kalinin, and L. Solymar, “A theory of metamaterials based on periodically loaded transmission lines: interaction between magnetoinductive and electromagnetic waves,” J. Appl. Phys.97, 064909 (2005).
    [CrossRef]
  8. J. D. Baena, L. Jelinek, R. Marqués, and M. Silveirinha, “Unified homogenization theory for magnetoinductive and electromagnetic waves in split-ring metamaterials,” Phys. Rev. A78, 013842 (2008).
    [CrossRef]
  9. M. Silveirinha, J. Baena, L. Jelinek, and R. Marques, “Nonlocal homogenization of an array of cubic particles made of resonant rings,” Metamaterials3, 115–128 (2009).
    [CrossRef]
  10. V. M. Agranovich and Y. N. Gartstein, “Electrodynamics of metamaterials and the Landau–Lifshitz approach to the magnetic permeability,” Metamaterials3, 1–9 (2009).
    [CrossRef]
  11. C. R. Simovski, “Analytical modelling of double-negative composites,” Metamaterials2, 169–185 (2008).
    [CrossRef]
  12. P. A. Belov and C. R. Simovski, “Boundary conditions for interfaces of electromagnetic crystals and the generalized Ewald-Oseen extinction principle,” Phys. Rev. B73, 045102 (2006).
    [CrossRef]
  13. M. Albooyeh, D. Morits, and C. R. Simovski, “Electromagnetic characterization of substrated metasurfaces,” Metamaterials5, 178–205 (2011).
    [CrossRef]
  14. M. Silveirinha, “Metamaterial homogenization approach with application to the characterization of microstructured composites with negative parameters,” Phys. Rev. B75, 115104 (2007).
    [CrossRef]
  15. A. Alù, “First-principles homogenization theory for periodic metamaterials,” Phys. Rev. B84, 075153 (2011).
    [CrossRef]
  16. M. Gorkunov, S. A. Gredeskul, I. V. Shadrivov, and Y. S. Kivshar, “Effect of microscopic disorder on magnetic properties of metamaterials,” Phys. Rev. E73, 056605 (2006).
    [CrossRef]
  17. J. Rico-García, J. López-Alonso, and A. Aradian, “Toy model to describe the effect of positional blocklike disorder in metamaterials composites,” J. Opt. Soc. Am. B29, 53–67 (2012).
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    [CrossRef]
  19. A. P. Vinogradov, D. P. Makhnovskii, and K. N. Rozanov, “Effective boundary layer in composite materials,” J. Communication Technol. Electron.44, 317–322 (1999).
  20. O. N. Gadomskii and S. V. Sukhov, “Microscopic theory of a transition layer on the ideal surface of semiinfinite dielectric media and the near-field effect,” Opt. Spectrosc.89, 261–266 (2000).
    [CrossRef]
  21. M. Lapine and S. Tretyakov, “Contemporary notes on metamaterials,” IET Microw. Antenn. Propag.1, 3–11 (2007).
    [CrossRef]
  22. O. Zhuromskyy, E. Shamonina, and L. Solymar, “2D metamaterials with hexagonal structure: spatial resonances and near field imaging,” Opt. Express13, 9299–9309 (2005).
    [CrossRef] [PubMed]
  23. R. Marqués, L. Jelinek, M. Freire, J. Baena, and M. Lapine, “Bulk metamaterials made of resonant rings,” Proc. IEEE99, 1660–1668 (2011).
    [CrossRef]
  24. A. M. Nicolson and G. F. Ross, “Measurement of the intrinsic properties of materials by time-domain techniques,” IEEE Trans. Instrum. Meas.IM–19, 377–382 (1970).
    [CrossRef]
  25. W. B. Weir, “Automatic measurement of complex dielectric constant and permeability at microwave frequencies,” Proc. IEEE62, 33–36 (1974).
    [CrossRef]
  26. X.-X. Liu, D. A. Powell, and A. Alù, “Correcting the Fabry-Perot artifacts in metamaterial retrieval procedures,” Phys. Rev. B84, 235106 (2011).
    [CrossRef]
  27. O. Luukkonen, S. I. Maslovski, and S. A. Tretyakov, “A stepwise Nicolson–Ross–Weir–based material parameter extraction method,” IEEE Antenn. Wireless Propag. Lett.10, 1295–1298 (2011).
    [CrossRef]
  28. M. Lapine, L. Jelinek, R. Marqués, and M. Freire, “Exact modelling method for discrete finite metamaterial lens,” IET Microw. Antenn. Propag.4, 1132–1139 (2010).
    [CrossRef]
  29. H. Wallén, H. Kettunen, and A. Sihvola, “Surface modes of negative-parameter interfaces and the importance of rounding sharp corners,” Metamaterials2, 113–121 (2008).
    [CrossRef]

2012 (1)

2011 (6)

R. R. A. Syms, O. Sydoruk, and L. Solymar, “Lossy metamaterials: no effective medium properties without noise,” Phys. Rev. B84, 235150 (2011).
[CrossRef]

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

M. Albooyeh, D. Morits, and C. R. Simovski, “Electromagnetic characterization of substrated metasurfaces,” Metamaterials5, 178–205 (2011).
[CrossRef]

R. Marqués, L. Jelinek, M. Freire, J. Baena, and M. Lapine, “Bulk metamaterials made of resonant rings,” Proc. IEEE99, 1660–1668 (2011).
[CrossRef]

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

O. Luukkonen, S. I. Maslovski, and S. A. Tretyakov, “A stepwise Nicolson–Ross–Weir–based material parameter extraction method,” IEEE Antenn. Wireless Propag. Lett.10, 1295–1298 (2011).
[CrossRef]

2010 (1)

M. Lapine, L. Jelinek, R. Marqués, and M. Freire, “Exact modelling method for discrete finite metamaterial lens,” IET Microw. Antenn. Propag.4, 1132–1139 (2010).
[CrossRef]

2009 (3)

C. R. Simovski, “Material parameters of metamaterials,” Opt. Spectrosc.107, 726–753 (2009).
[CrossRef]

M. Silveirinha, J. Baena, L. Jelinek, and R. Marques, “Nonlocal homogenization of an array of cubic particles made of resonant rings,” Metamaterials3, 115–128 (2009).
[CrossRef]

V. M. Agranovich and Y. N. Gartstein, “Electrodynamics of metamaterials and the Landau–Lifshitz approach to the magnetic permeability,” Metamaterials3, 1–9 (2009).
[CrossRef]

2008 (3)

C. R. Simovski, “Analytical modelling of double-negative composites,” Metamaterials2, 169–185 (2008).
[CrossRef]

J. D. Baena, L. Jelinek, R. Marqués, and M. Silveirinha, “Unified homogenization theory for magnetoinductive and electromagnetic waves in split-ring metamaterials,” Phys. Rev. A78, 013842 (2008).
[CrossRef]

H. Wallén, H. Kettunen, and A. Sihvola, “Surface modes of negative-parameter interfaces and the importance of rounding sharp corners,” Metamaterials2, 113–121 (2008).
[CrossRef]

2007 (2)

M. Lapine and S. Tretyakov, “Contemporary notes on metamaterials,” IET Microw. Antenn. Propag.1, 3–11 (2007).
[CrossRef]

M. Silveirinha, “Metamaterial homogenization approach with application to the characterization of microstructured composites with negative parameters,” Phys. Rev. B75, 115104 (2007).
[CrossRef]

2006 (2)

M. Gorkunov, S. A. Gredeskul, I. V. Shadrivov, and Y. S. Kivshar, “Effect of microscopic disorder on magnetic properties of metamaterials,” Phys. Rev. E73, 056605 (2006).
[CrossRef]

P. A. Belov and C. R. Simovski, “Boundary conditions for interfaces of electromagnetic crystals and the generalized Ewald-Oseen extinction principle,” Phys. Rev. B73, 045102 (2006).
[CrossRef]

2005 (2)

R. R. A. Syms, E. Shamonina, V. Kalinin, and L. Solymar, “A theory of metamaterials based on periodically loaded transmission lines: interaction between magnetoinductive and electromagnetic waves,” J. Appl. Phys.97, 064909 (2005).
[CrossRef]

O. Zhuromskyy, E. Shamonina, and L. Solymar, “2D metamaterials with hexagonal structure: spatial resonances and near field imaging,” Opt. Express13, 9299–9309 (2005).
[CrossRef] [PubMed]

2002 (3)

M. Gorkunov, M. Lapine, E. Shamonina, and K. H. Ringhofer, “Effective magnetic properties of a composite material with circular conductive elements,” Eur. Phys. J. B28, 263–269 (2002).
[CrossRef]

R. Marqués, F. Medina, and R. Rafii-El-Idrissi, “Role of bianisotropy in negative permeability and left-handed metamaterials,” Phys. Rev. B65, 144440 (2002).
[CrossRef]

E. Shamonina, V. A. Kalinin, K. H. Ringhofer, and L. Solymar, “Magnetoinductive waves in one, two, and three dimensions,” J. Appl. Phys.92, 6252–6261 (2002).
[CrossRef]

2000 (1)

O. N. Gadomskii and S. V. Sukhov, “Microscopic theory of a transition layer on the ideal surface of semiinfinite dielectric media and the near-field effect,” Opt. Spectrosc.89, 261–266 (2000).
[CrossRef]

1999 (2)

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

A. P. Vinogradov, D. P. Makhnovskii, and K. N. Rozanov, “Effective boundary layer in composite materials,” J. Communication Technol. Electron.44, 317–322 (1999).

1974 (1)

W. B. Weir, “Automatic measurement of complex dielectric constant and permeability at microwave frequencies,” Proc. IEEE62, 33–36 (1974).
[CrossRef]

1970 (1)

A. M. Nicolson and G. F. Ross, “Measurement of the intrinsic properties of materials by time-domain techniques,” IEEE Trans. Instrum. Meas.IM–19, 377–382 (1970).
[CrossRef]

Agranovich, V. M.

V. M. Agranovich and Y. N. Gartstein, “Electrodynamics of metamaterials and the Landau–Lifshitz approach to the magnetic permeability,” Metamaterials3, 1–9 (2009).
[CrossRef]

Albooyeh, M.

M. Albooyeh, D. Morits, and C. R. Simovski, “Electromagnetic characterization of substrated metasurfaces,” Metamaterials5, 178–205 (2011).
[CrossRef]

Alù, A.

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

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

Aradian, A.

Baena, J.

R. Marqués, L. Jelinek, M. Freire, J. Baena, and M. Lapine, “Bulk metamaterials made of resonant rings,” Proc. IEEE99, 1660–1668 (2011).
[CrossRef]

M. Silveirinha, J. Baena, L. Jelinek, and R. Marques, “Nonlocal homogenization of an array of cubic particles made of resonant rings,” Metamaterials3, 115–128 (2009).
[CrossRef]

Baena, J. D.

J. D. Baena, L. Jelinek, R. Marqués, and M. Silveirinha, “Unified homogenization theory for magnetoinductive and electromagnetic waves in split-ring metamaterials,” Phys. Rev. A78, 013842 (2008).
[CrossRef]

Belov, P. A.

P. A. Belov and C. R. Simovski, “Boundary conditions for interfaces of electromagnetic crystals and the generalized Ewald-Oseen extinction principle,” Phys. Rev. B73, 045102 (2006).
[CrossRef]

Freire, M.

R. Marqués, L. Jelinek, M. Freire, J. Baena, and M. Lapine, “Bulk metamaterials made of resonant rings,” Proc. IEEE99, 1660–1668 (2011).
[CrossRef]

M. Lapine, L. Jelinek, R. Marqués, and M. Freire, “Exact modelling method for discrete finite metamaterial lens,” IET Microw. Antenn. Propag.4, 1132–1139 (2010).
[CrossRef]

Friis, H. T.

S. A. Schelkunoff and H. T. Friis, Antennas Theory and Practice (Wiley, 1966).

Gadomskii, O. N.

O. N. Gadomskii and S. V. Sukhov, “Microscopic theory of a transition layer on the ideal surface of semiinfinite dielectric media and the near-field effect,” Opt. Spectrosc.89, 261–266 (2000).
[CrossRef]

Gartstein, Y. N.

V. M. Agranovich and Y. N. Gartstein, “Electrodynamics of metamaterials and the Landau–Lifshitz approach to the magnetic permeability,” Metamaterials3, 1–9 (2009).
[CrossRef]

Gorkunov, M.

M. Gorkunov, S. A. Gredeskul, I. V. Shadrivov, and Y. S. Kivshar, “Effect of microscopic disorder on magnetic properties of metamaterials,” Phys. Rev. E73, 056605 (2006).
[CrossRef]

M. Gorkunov, M. Lapine, E. Shamonina, and K. H. Ringhofer, “Effective magnetic properties of a composite material with circular conductive elements,” Eur. Phys. J. B28, 263–269 (2002).
[CrossRef]

Gredeskul, S. A.

M. Gorkunov, S. A. Gredeskul, I. V. Shadrivov, and Y. S. Kivshar, “Effect of microscopic disorder on magnetic properties of metamaterials,” Phys. Rev. E73, 056605 (2006).
[CrossRef]

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. Microw. Theory Tech.47, 2075–2084 (1999).
[CrossRef]

Jelinek, L.

R. Marqués, L. Jelinek, M. Freire, J. Baena, and M. Lapine, “Bulk metamaterials made of resonant rings,” Proc. IEEE99, 1660–1668 (2011).
[CrossRef]

M. Lapine, L. Jelinek, R. Marqués, and M. Freire, “Exact modelling method for discrete finite metamaterial lens,” IET Microw. Antenn. Propag.4, 1132–1139 (2010).
[CrossRef]

M. Silveirinha, J. Baena, L. Jelinek, and R. Marques, “Nonlocal homogenization of an array of cubic particles made of resonant rings,” Metamaterials3, 115–128 (2009).
[CrossRef]

J. D. Baena, L. Jelinek, R. Marqués, and M. Silveirinha, “Unified homogenization theory for magnetoinductive and electromagnetic waves in split-ring metamaterials,” Phys. Rev. A78, 013842 (2008).
[CrossRef]

Kalinin, V.

R. R. A. Syms, E. Shamonina, V. Kalinin, and L. Solymar, “A theory of metamaterials based on periodically loaded transmission lines: interaction between magnetoinductive and electromagnetic waves,” J. Appl. Phys.97, 064909 (2005).
[CrossRef]

Kalinin, V. A.

E. Shamonina, V. A. Kalinin, K. H. Ringhofer, and L. Solymar, “Magnetoinductive waves in one, two, and three dimensions,” J. Appl. Phys.92, 6252–6261 (2002).
[CrossRef]

Kettunen, H.

H. Wallén, H. Kettunen, and A. Sihvola, “Surface modes of negative-parameter interfaces and the importance of rounding sharp corners,” Metamaterials2, 113–121 (2008).
[CrossRef]

Kivshar, Y. S.

M. Gorkunov, S. A. Gredeskul, I. V. Shadrivov, and Y. S. Kivshar, “Effect of microscopic disorder on magnetic properties of metamaterials,” Phys. Rev. E73, 056605 (2006).
[CrossRef]

Lapine, M.

R. Marqués, L. Jelinek, M. Freire, J. Baena, and M. Lapine, “Bulk metamaterials made of resonant rings,” Proc. IEEE99, 1660–1668 (2011).
[CrossRef]

M. Lapine, L. Jelinek, R. Marqués, and M. Freire, “Exact modelling method for discrete finite metamaterial lens,” IET Microw. Antenn. Propag.4, 1132–1139 (2010).
[CrossRef]

M. Lapine and S. Tretyakov, “Contemporary notes on metamaterials,” IET Microw. Antenn. Propag.1, 3–11 (2007).
[CrossRef]

M. Gorkunov, M. Lapine, E. Shamonina, and K. H. Ringhofer, “Effective magnetic properties of a composite material with circular conductive elements,” Eur. Phys. J. B28, 263–269 (2002).
[CrossRef]

Liu, X.-X.

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

López-Alonso, J.

Luukkonen, O.

O. Luukkonen, S. I. Maslovski, and S. A. Tretyakov, “A stepwise Nicolson–Ross–Weir–based material parameter extraction method,” IEEE Antenn. Wireless Propag. Lett.10, 1295–1298 (2011).
[CrossRef]

Makhnovskii, D. P.

A. P. Vinogradov, D. P. Makhnovskii, and K. N. Rozanov, “Effective boundary layer in composite materials,” J. Communication Technol. Electron.44, 317–322 (1999).

Marques, R.

M. Silveirinha, J. Baena, L. Jelinek, and R. Marques, “Nonlocal homogenization of an array of cubic particles made of resonant rings,” Metamaterials3, 115–128 (2009).
[CrossRef]

Marqués, R.

R. Marqués, L. Jelinek, M. Freire, J. Baena, and M. Lapine, “Bulk metamaterials made of resonant rings,” Proc. IEEE99, 1660–1668 (2011).
[CrossRef]

M. Lapine, L. Jelinek, R. Marqués, and M. Freire, “Exact modelling method for discrete finite metamaterial lens,” IET Microw. Antenn. Propag.4, 1132–1139 (2010).
[CrossRef]

J. D. Baena, L. Jelinek, R. Marqués, and M. Silveirinha, “Unified homogenization theory for magnetoinductive and electromagnetic waves in split-ring metamaterials,” Phys. Rev. A78, 013842 (2008).
[CrossRef]

R. Marqués, F. Medina, and R. Rafii-El-Idrissi, “Role of bianisotropy in negative permeability and left-handed metamaterials,” Phys. Rev. B65, 144440 (2002).
[CrossRef]

Maslovski, S. I.

O. Luukkonen, S. I. Maslovski, and S. A. Tretyakov, “A stepwise Nicolson–Ross–Weir–based material parameter extraction method,” IEEE Antenn. Wireless Propag. Lett.10, 1295–1298 (2011).
[CrossRef]

Medina, F.

R. Marqués, F. Medina, and R. Rafii-El-Idrissi, “Role of bianisotropy in negative permeability and left-handed metamaterials,” Phys. Rev. B65, 144440 (2002).
[CrossRef]

Morits, D.

M. Albooyeh, D. Morits, and C. R. Simovski, “Electromagnetic characterization of substrated metasurfaces,” Metamaterials5, 178–205 (2011).
[CrossRef]

Nicolson, A. M.

A. M. Nicolson and G. F. Ross, “Measurement of the intrinsic properties of materials by time-domain techniques,” IEEE Trans. Instrum. Meas.IM–19, 377–382 (1970).
[CrossRef]

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. Microw. Theory Tech.47, 2075–2084 (1999).
[CrossRef]

Powell, D. A.

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

Rafii-El-Idrissi, R.

R. Marqués, F. Medina, and R. Rafii-El-Idrissi, “Role of bianisotropy in negative permeability and left-handed metamaterials,” Phys. Rev. B65, 144440 (2002).
[CrossRef]

Rico-García, J.

Ringhofer, K. H.

E. Shamonina, V. A. Kalinin, K. H. Ringhofer, and L. Solymar, “Magnetoinductive waves in one, two, and three dimensions,” J. Appl. Phys.92, 6252–6261 (2002).
[CrossRef]

M. Gorkunov, M. Lapine, E. Shamonina, and K. H. Ringhofer, “Effective magnetic properties of a composite material with circular conductive elements,” Eur. Phys. J. B28, 263–269 (2002).
[CrossRef]

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. Microw. Theory Tech.47, 2075–2084 (1999).
[CrossRef]

Ross, G. F.

A. M. Nicolson and G. F. Ross, “Measurement of the intrinsic properties of materials by time-domain techniques,” IEEE Trans. Instrum. Meas.IM–19, 377–382 (1970).
[CrossRef]

Rozanov, K. N.

A. P. Vinogradov, D. P. Makhnovskii, and K. N. Rozanov, “Effective boundary layer in composite materials,” J. Communication Technol. Electron.44, 317–322 (1999).

Schelkunoff, S. A.

S. A. Schelkunoff and H. T. Friis, Antennas Theory and Practice (Wiley, 1966).

Shadrivov, I. V.

M. Gorkunov, S. A. Gredeskul, I. V. Shadrivov, and Y. S. Kivshar, “Effect of microscopic disorder on magnetic properties of metamaterials,” Phys. Rev. E73, 056605 (2006).
[CrossRef]

Shamonina, E.

O. Zhuromskyy, E. Shamonina, and L. Solymar, “2D metamaterials with hexagonal structure: spatial resonances and near field imaging,” Opt. Express13, 9299–9309 (2005).
[CrossRef] [PubMed]

R. R. A. Syms, E. Shamonina, V. Kalinin, and L. Solymar, “A theory of metamaterials based on periodically loaded transmission lines: interaction between magnetoinductive and electromagnetic waves,” J. Appl. Phys.97, 064909 (2005).
[CrossRef]

M. Gorkunov, M. Lapine, E. Shamonina, and K. H. Ringhofer, “Effective magnetic properties of a composite material with circular conductive elements,” Eur. Phys. J. B28, 263–269 (2002).
[CrossRef]

E. Shamonina, V. A. Kalinin, K. H. Ringhofer, and L. Solymar, “Magnetoinductive waves in one, two, and three dimensions,” J. Appl. Phys.92, 6252–6261 (2002).
[CrossRef]

Sihvola, A.

H. Wallén, H. Kettunen, and A. Sihvola, “Surface modes of negative-parameter interfaces and the importance of rounding sharp corners,” Metamaterials2, 113–121 (2008).
[CrossRef]

Silveirinha, M.

M. Silveirinha, J. Baena, L. Jelinek, and R. Marques, “Nonlocal homogenization of an array of cubic particles made of resonant rings,” Metamaterials3, 115–128 (2009).
[CrossRef]

J. D. Baena, L. Jelinek, R. Marqués, and M. Silveirinha, “Unified homogenization theory for magnetoinductive and electromagnetic waves in split-ring metamaterials,” Phys. Rev. A78, 013842 (2008).
[CrossRef]

M. Silveirinha, “Metamaterial homogenization approach with application to the characterization of microstructured composites with negative parameters,” Phys. Rev. B75, 115104 (2007).
[CrossRef]

Simovski, C. R.

M. Albooyeh, D. Morits, and C. R. Simovski, “Electromagnetic characterization of substrated metasurfaces,” Metamaterials5, 178–205 (2011).
[CrossRef]

C. R. Simovski, “Material parameters of metamaterials,” Opt. Spectrosc.107, 726–753 (2009).
[CrossRef]

C. R. Simovski, “Analytical modelling of double-negative composites,” Metamaterials2, 169–185 (2008).
[CrossRef]

P. A. Belov and C. R. Simovski, “Boundary conditions for interfaces of electromagnetic crystals and the generalized Ewald-Oseen extinction principle,” Phys. Rev. B73, 045102 (2006).
[CrossRef]

Solymar, L.

R. R. A. Syms, O. Sydoruk, and L. Solymar, “Lossy metamaterials: no effective medium properties without noise,” Phys. Rev. B84, 235150 (2011).
[CrossRef]

O. Zhuromskyy, E. Shamonina, and L. Solymar, “2D metamaterials with hexagonal structure: spatial resonances and near field imaging,” Opt. Express13, 9299–9309 (2005).
[CrossRef] [PubMed]

R. R. A. Syms, E. Shamonina, V. Kalinin, and L. Solymar, “A theory of metamaterials based on periodically loaded transmission lines: interaction between magnetoinductive and electromagnetic waves,” J. Appl. Phys.97, 064909 (2005).
[CrossRef]

E. Shamonina, V. A. Kalinin, K. H. Ringhofer, and L. Solymar, “Magnetoinductive waves in one, two, and three dimensions,” J. Appl. Phys.92, 6252–6261 (2002).
[CrossRef]

Stewart, W. J.

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

Sukhov, S. V.

O. N. Gadomskii and S. V. Sukhov, “Microscopic theory of a transition layer on the ideal surface of semiinfinite dielectric media and the near-field effect,” Opt. Spectrosc.89, 261–266 (2000).
[CrossRef]

Sydoruk, O.

R. R. A. Syms, O. Sydoruk, and L. Solymar, “Lossy metamaterials: no effective medium properties without noise,” Phys. Rev. B84, 235150 (2011).
[CrossRef]

Syms, R. R. A.

R. R. A. Syms, O. Sydoruk, and L. Solymar, “Lossy metamaterials: no effective medium properties without noise,” Phys. Rev. B84, 235150 (2011).
[CrossRef]

R. R. A. Syms, E. Shamonina, V. Kalinin, and L. Solymar, “A theory of metamaterials based on periodically loaded transmission lines: interaction between magnetoinductive and electromagnetic waves,” J. Appl. Phys.97, 064909 (2005).
[CrossRef]

Tretyakov, S.

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[CrossRef]

Tretyakov, S. A.

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Vinogradov, A. P.

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[CrossRef]

Weir, W. B.

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[CrossRef]

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Eur. Phys. J. B (1)

M. Gorkunov, M. Lapine, E. Shamonina, and K. H. Ringhofer, “Effective magnetic properties of a composite material with circular conductive elements,” Eur. Phys. J. B28, 263–269 (2002).
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IEEE Antenn. Wireless Propag. Lett. (1)

O. Luukkonen, S. I. Maslovski, and S. A. Tretyakov, “A stepwise Nicolson–Ross–Weir–based material parameter extraction method,” IEEE Antenn. Wireless Propag. Lett.10, 1295–1298 (2011).
[CrossRef]

IEEE Trans. Instrum. Meas. (1)

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IEEE Trans. Microw. Theory Tech. (1)

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IET Microw. Antenn. Propag. (2)

M. Lapine and S. Tretyakov, “Contemporary notes on metamaterials,” IET Microw. Antenn. Propag.1, 3–11 (2007).
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J. Appl. Phys. (2)

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[CrossRef]

R. R. A. Syms, E. Shamonina, V. Kalinin, and L. Solymar, “A theory of metamaterials based on periodically loaded transmission lines: interaction between magnetoinductive and electromagnetic waves,” J. Appl. Phys.97, 064909 (2005).
[CrossRef]

J. Communication Technol. Electron. (1)

A. P. Vinogradov, D. P. Makhnovskii, and K. N. Rozanov, “Effective boundary layer in composite materials,” J. Communication Technol. Electron.44, 317–322 (1999).

J. Opt. Soc. Am. B (1)

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Opt. Express (1)

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R. R. A. Syms, O. Sydoruk, and L. Solymar, “Lossy metamaterials: no effective medium properties without noise,” Phys. Rev. B84, 235150 (2011).
[CrossRef]

M. Silveirinha, “Metamaterial homogenization approach with application to the characterization of microstructured composites with negative parameters,” Phys. Rev. B75, 115104 (2007).
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P. A. Belov and C. R. Simovski, “Boundary conditions for interfaces of electromagnetic crystals and the generalized Ewald-Oseen extinction principle,” Phys. Rev. B73, 045102 (2006).
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M. Gorkunov, S. A. Gredeskul, I. V. Shadrivov, and Y. S. Kivshar, “Effect of microscopic disorder on magnetic properties of metamaterials,” Phys. Rev. E73, 056605 (2006).
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Proc. IEEE (2)

W. B. Weir, “Automatic measurement of complex dielectric constant and permeability at microwave frequencies,” Proc. IEEE62, 33–36 (1974).
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Scheme of a symmetric finite cube with discrete structure, showing two options of surface configuration (for the same unit cell in the bulk): “flat” geometry (a) and “ragged” geometry (b). Note that the actual number of elements is not necessarily reflected here.

Fig. 2
Fig. 2

Real part of the normalised polarisability (arbitrary units) of discrete cubes with 5, 8 or 11 layers of resonators in each direction, having either a “flat” geometry (a) or a “ragged” geometry (b).

Fig. 3
Fig. 3

Real (a) and imaginary (b) parts of the normalised polarisability (arbitrary units) of discrete cubes with 11 layers in each direction, having either a “flat” geometry (dashed lines) or “ragged” geometry (dotted lines), in comparison with the polarizability of a homogeneous cube (solid lines).

Fig. 4
Fig. 4

Frequency dependence of the A and C coefficients calculated with Eq. (3) based on the effective permeability μ of a homogeneous cube Eq. (1) and its polarisability α, obtained from numerical simulations.

Fig. 5
Fig. 5

A comparison between the permeability μ of a homogeneous cube (solid line) and the effective permeability μ# obtained for 11-layer discrete cubes with a “flat” geometry (dashed lines) or a “ragged” geometry (dotted lines); real (a) and imaginary (b) parts.

Equations (4)

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μ = 1 + γ ( k res a ) 2 / ( k 0 a ) 2 1 2 ϰ a 4 ϰ c γ / 3 ,
α = A ( μ 1 ) / ( μ + C ) ,
C = | μ | 2 Re μ Re α Im α Im μ 1 Re μ + Re α Im α Im μ , A = ( Re μ + C ) 2 + ( Im μ ) 2 Im μ ( C + 1 ) Im α .
μ # = ( A + C α # ) / ( A α # ) .

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