Abstract

The prediction and the engineering of the electromagnetic properties of metamaterials are increasingly important issues. Recently, several approaches have been proposed to compute these properties through appropriate averages of local fields within the unit cell. In particular, we proposed a Field Summation method that has been used successfully to determine either analytically or numerically the effective properties of different composites and metamaterials. But this method also provides interesting clues for understanding the behaviour of these materials. It helps chose appropriate planes to visualize the fields using electromagnetic simulation software, and understand behaviours leading to either positive or negative effective parameters, with either small or large values. It helps establish whether the materials can be adequately described by effective parameters.

© 2007 Optical Society of America

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  1. K. Aydin, K. Guven, M. Kafesaki, L. Zhang, C. M. Soukoulis, and E. Ozbay, "Experimental observation of true left-handed transmission peak in metamaterials," Opt. Lett. 29, 2623 (2004).
    [CrossRef] [PubMed]
  2. C.  Enkrich, M.  Wegener, S.  Linden, S.  Burger, L.  Zschiedrich, F.  Schmidt, J. F.  Zhou, Th.  Koschny, and C. M.  Soukoulis, "Magnetic metamaterials at telecommunication and visible frequencies," Phys. Rev. Lett.  95, 203901 (2005).
    [CrossRef] [PubMed]
  3. T. Decoopman, A. Marteau, E. Lheurette, O. Vanbésien and D. Lippens, "Left-Handed Electromagnetic Properties of Split-Ring Resonator and Wire Loaded Transmission Line in a Fin-Line Technology," IEEE Trans. MTT 54, 1451-1457 (2006).
    [CrossRef]
  4. V. M.  Shalaev, W.  Cai, U. K.  Chettiar, H.-K.  Yuan, A. K.  Sarychev, V. P.  Drachev, and A. V.  Kildishev, "Negative index of refraction in optical metamaterials," Opt. Lett.  30, 3356-3358 (2005).
    [CrossRef]
  5. A. N.  Grigorenko, A. K.  Geim, H. F.  Gleeson, Y.  Zhang, A. A.  Firsov, I. Y.  Khrushchev, and J.  Petrovic, "Nanofabricated media with negative permeability at visible frequencies," Nature  438, 335-338 (2005).
    [CrossRef] [PubMed]
  6. 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]
  7. D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "Composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84, 4184 (2000).
    [CrossRef] [PubMed]
  8. V. Lomakin, Y. Fainman, Y. Urzhumov, and G. Shvets, "Doubly negative metamaterials in the near infrared and visible regimes based on thin film nanocomposites," Opt. Express 14, 11164-11177 (2006).
    [CrossRef] [PubMed]
  9. W. J.  Padilla, D. R.  Smith, and D. N.  Basov, "Spectroscopy of metamaterials from infrared to optical frequencies," J. Opt. Soc. Am. B  23, 404 (2006).
    [CrossRef]
  10. O. Reynet, O. Acher, "Voltage controlled metamaterial," Appl. Phys. Lett. 84, 1198-2000 (2004).
    [CrossRef]
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  12. K. N. Rozanov and E. A. Preobrazhenskii, "Synthesis of wideband radar absorbers based on complex media composed from active electric Dipoles," J. Commun. Technol. Electron. 50, 858-864 (2005).
  13. A.-L. Adenot-Engelvin, C. Dudek and O. Acher, "Microwave permeability of metamaterials based on ferromagnetic composites," J. Magn. Magn. Mater. 300, 33-37 (2006).
    [CrossRef]
  14. A. Alù, A. Salandrino, and N. Engheta, "Negative effective permeability and left-handed materials at optical frequencies," Opt. Express 14, 1557-1567 (2006).
    [CrossRef] [PubMed]
  15. R. Ziolkowski, "Pulsed and CW Gaussian beam interactions with double negative metamaterial slabs," Opt. Express 11, 662-681 (2003).
    [CrossRef] [PubMed]
  16. 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]
  17. B. I.  Popa and S. A.  Cummer, "Wave fields measured inside a negative refractive index metamaterial," Appl. Phys. Lett.  85, 4564-4566 (2004).
    [CrossRef]
  18. D. R.  Smith and J. B.  Pendry, "Homogenization of metamaterials by field averaging," J. Opt. Soc. Am. B  23, 391-403 (2006).
    [CrossRef]
  19. J.-M. Lerat, N. Malléjac and O. Acher, "Determination of the effective parameters of a metamaterial by field summation method," J. Appl. Phys. 100, 084908 (2006).
    [CrossRef]
  20. T. Driscoll, D. N. Basov, A. F. Starr, P. M. Rye, S. Nemat-Nasser, D. Schurig and D. R. Smith, "Free-space microwave focusing by a negative-index gradient lens," Appl. Phys. Lett. 88, 081101 (2006).
    [CrossRef]
  21. A. Cho, "Voilà ! Cloak of Invisibility Unveiled," Science 314, 403 (2006).
    [CrossRef] [PubMed]
  22. 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]
  23. O. Acher, A. -L. Adenot, and F. Duverger, "Fresnel coefficients at an interface with a lamellar composite material," Phys. Rev. B 62, 13748 (2000).
    [CrossRef]
  24. S. M. Rytov, "Electromagnetic properties of a finely stratified medium," Sov. Phys. JETP 2, 466-475 (1956).
  25. A. N. Lagarkov, A. K. Sarychev, Y. R. Smychkovich and A. P. Vinogradov, "Effective Medium Theory for microwave dielectric constant and magnetic permeability of conducting stick composites," J. Electromagn. Waves Appl. 6, 1159 (1992).
  26. O. Acher, A. -L. Adenot, F. Lubrano, and F. Duverger, "Low density artificial magnetic composites," J. Appl. Phys.,  85, 4639-4641 (1999).
    [CrossRef]
  27. O. Reynet, A. -L. Adenot, S. Deprot, O. Acher, and M. Latrach, "Effect of the magnetic properties of inclusions on the high-frequency dielectric response of diluted composites," Phys. Rev. B 66, 94412 (2002).
    [CrossRef]
  28. M. V. Gorkunov, S. A. Gredeskul, I. V. Shadrinov and Y. S. Kivshar, "Effect of microscopic disorder on magnetic properties of metamaterials," Phys. Rev. E 73, 056605 (2006).
    [CrossRef]
  29. O. Acher, M. Ledieu, A.-L. Adenot and O. Reynet, "Microwave properties of diluted composites made of magnetic wires with giant magneto-impedance effect," IEEE Trans. Magn.,  39, 3085-3090 (2003).
    [CrossRef]
  30. D. Maystre and S. Enoch, "Perfect lenses made with left-handed materilas: Alice’s mirror," J. Opt. Soc. Am. A 21, 122-131 (2004).
    [CrossRef]
  31. T. Decoopman, G. Tayeb, S. Enoch, D. Maystre and B Gralak, "Photonic Crystla Lens: From Negative Refraction and Negative Index to Negative Permittivity and Permeability," Phys. Rev. Lett. 97, 073905 (2006).
    [CrossRef] [PubMed]

2006 (11)

T. Decoopman, A. Marteau, E. Lheurette, O. Vanbésien and D. Lippens, "Left-Handed Electromagnetic Properties of Split-Ring Resonator and Wire Loaded Transmission Line in a Fin-Line Technology," IEEE Trans. MTT 54, 1451-1457 (2006).
[CrossRef]

V. Lomakin, Y. Fainman, Y. Urzhumov, and G. Shvets, "Doubly negative metamaterials in the near infrared and visible regimes based on thin film nanocomposites," Opt. Express 14, 11164-11177 (2006).
[CrossRef] [PubMed]

W. J.  Padilla, D. R.  Smith, and D. N.  Basov, "Spectroscopy of metamaterials from infrared to optical frequencies," J. Opt. Soc. Am. B  23, 404 (2006).
[CrossRef]

A.-L. Adenot-Engelvin, C. Dudek and O. Acher, "Microwave permeability of metamaterials based on ferromagnetic composites," J. Magn. Magn. Mater. 300, 33-37 (2006).
[CrossRef]

A. Alù, A. Salandrino, and N. Engheta, "Negative effective permeability and left-handed materials at optical frequencies," Opt. Express 14, 1557-1567 (2006).
[CrossRef] [PubMed]

D. R.  Smith and J. B.  Pendry, "Homogenization of metamaterials by field averaging," J. Opt. Soc. Am. B  23, 391-403 (2006).
[CrossRef]

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

T. Driscoll, D. N. Basov, A. F. Starr, P. M. Rye, S. Nemat-Nasser, D. Schurig and D. R. Smith, "Free-space microwave focusing by a negative-index gradient lens," Appl. Phys. Lett. 88, 081101 (2006).
[CrossRef]

A. Cho, "Voilà ! Cloak of Invisibility Unveiled," Science 314, 403 (2006).
[CrossRef] [PubMed]

M. V. Gorkunov, S. A. Gredeskul, I. V. Shadrinov and Y. S. Kivshar, "Effect of microscopic disorder on magnetic properties of metamaterials," Phys. Rev. E 73, 056605 (2006).
[CrossRef]

T. Decoopman, G. Tayeb, S. Enoch, D. Maystre and B Gralak, "Photonic Crystla Lens: From Negative Refraction and Negative Index to Negative Permittivity and Permeability," Phys. Rev. Lett. 97, 073905 (2006).
[CrossRef] [PubMed]

2005 (5)

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]

K. N. Rozanov and E. A. Preobrazhenskii, "Synthesis of wideband radar absorbers based on complex media composed from active electric Dipoles," J. Commun. Technol. Electron. 50, 858-864 (2005).

C.  Enkrich, M.  Wegener, S.  Linden, S.  Burger, L.  Zschiedrich, F.  Schmidt, J. F.  Zhou, Th.  Koschny, and C. M.  Soukoulis, "Magnetic metamaterials at telecommunication and visible frequencies," Phys. Rev. Lett.  95, 203901 (2005).
[CrossRef] [PubMed]

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

A. N.  Grigorenko, A. K.  Geim, H. F.  Gleeson, Y.  Zhang, A. A.  Firsov, I. Y.  Khrushchev, and J.  Petrovic, "Nanofabricated media with negative permeability at visible frequencies," Nature  438, 335-338 (2005).
[CrossRef] [PubMed]

2004 (4)

K. Aydin, K. Guven, M. Kafesaki, L. Zhang, C. M. Soukoulis, and E. Ozbay, "Experimental observation of true left-handed transmission peak in metamaterials," Opt. Lett. 29, 2623 (2004).
[CrossRef] [PubMed]

O. Reynet, O. Acher, "Voltage controlled metamaterial," Appl. Phys. Lett. 84, 1198-2000 (2004).
[CrossRef]

B. I.  Popa and S. A.  Cummer, "Wave fields measured inside a negative refractive index metamaterial," Appl. Phys. Lett.  85, 4564-4566 (2004).
[CrossRef]

D. Maystre and S. Enoch, "Perfect lenses made with left-handed materilas: Alice’s mirror," J. Opt. Soc. Am. A 21, 122-131 (2004).
[CrossRef]

2003 (2)

O. Acher, M. Ledieu, A.-L. Adenot and O. Reynet, "Microwave properties of diluted composites made of magnetic wires with giant magneto-impedance effect," IEEE Trans. Magn.,  39, 3085-3090 (2003).
[CrossRef]

R. Ziolkowski, "Pulsed and CW Gaussian beam interactions with double negative metamaterial slabs," Opt. Express 11, 662-681 (2003).
[CrossRef] [PubMed]

2002 (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]

O. Reynet, A. -L. Adenot, S. Deprot, O. Acher, and M. Latrach, "Effect of the magnetic properties of inclusions on the high-frequency dielectric response of diluted composites," Phys. Rev. B 66, 94412 (2002).
[CrossRef]

2000 (2)

O. Acher, A. -L. Adenot, and F. Duverger, "Fresnel coefficients at an interface with a lamellar composite material," Phys. Rev. B 62, 13748 (2000).
[CrossRef]

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

1999 (2)

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]

O. Acher, A. -L. Adenot, F. Lubrano, and F. Duverger, "Low density artificial magnetic composites," J. Appl. Phys.,  85, 4639-4641 (1999).
[CrossRef]

1992 (1)

A. N. Lagarkov, A. K. Sarychev, Y. R. Smychkovich and A. P. Vinogradov, "Effective Medium Theory for microwave dielectric constant and magnetic permeability of conducting stick composites," J. Electromagn. Waves Appl. 6, 1159 (1992).

1956 (1)

S. M. Rytov, "Electromagnetic properties of a finely stratified medium," Sov. Phys. JETP 2, 466-475 (1956).

Acher, O.

A.-L. Adenot-Engelvin, C. Dudek and O. Acher, "Microwave permeability of metamaterials based on ferromagnetic composites," J. Magn. Magn. Mater. 300, 33-37 (2006).
[CrossRef]

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

O. Reynet, O. Acher, "Voltage controlled metamaterial," Appl. Phys. Lett. 84, 1198-2000 (2004).
[CrossRef]

O. Acher, M. Ledieu, A.-L. Adenot and O. Reynet, "Microwave properties of diluted composites made of magnetic wires with giant magneto-impedance effect," IEEE Trans. Magn.,  39, 3085-3090 (2003).
[CrossRef]

O. Reynet, A. -L. Adenot, S. Deprot, O. Acher, and M. Latrach, "Effect of the magnetic properties of inclusions on the high-frequency dielectric response of diluted composites," Phys. Rev. B 66, 94412 (2002).
[CrossRef]

O. Acher, A. -L. Adenot, and F. Duverger, "Fresnel coefficients at an interface with a lamellar composite material," Phys. Rev. B 62, 13748 (2000).
[CrossRef]

O. Acher, A. -L. Adenot, F. Lubrano, and F. Duverger, "Low density artificial magnetic composites," J. Appl. Phys.,  85, 4639-4641 (1999).
[CrossRef]

Adenot, A. -L.

O. Reynet, A. -L. Adenot, S. Deprot, O. Acher, and M. Latrach, "Effect of the magnetic properties of inclusions on the high-frequency dielectric response of diluted composites," Phys. Rev. B 66, 94412 (2002).
[CrossRef]

O. Acher, A. -L. Adenot, and F. Duverger, "Fresnel coefficients at an interface with a lamellar composite material," Phys. Rev. B 62, 13748 (2000).
[CrossRef]

O. Acher, A. -L. Adenot, F. Lubrano, and F. Duverger, "Low density artificial magnetic composites," J. Appl. Phys.,  85, 4639-4641 (1999).
[CrossRef]

Adenot, A.-L.

O. Acher, M. Ledieu, A.-L. Adenot and O. Reynet, "Microwave properties of diluted composites made of magnetic wires with giant magneto-impedance effect," IEEE Trans. Magn.,  39, 3085-3090 (2003).
[CrossRef]

Adenot-Engelvin, A.-L.

A.-L. Adenot-Engelvin, C. Dudek and O. Acher, "Microwave permeability of metamaterials based on ferromagnetic composites," J. Magn. Magn. Mater. 300, 33-37 (2006).
[CrossRef]

Alù, A.

Aydin, K.

Basov, D. N.

W. J.  Padilla, D. R.  Smith, and D. N.  Basov, "Spectroscopy of metamaterials from infrared to optical frequencies," J. Opt. Soc. Am. B  23, 404 (2006).
[CrossRef]

T. Driscoll, D. N. Basov, A. F. Starr, P. M. Rye, S. Nemat-Nasser, D. Schurig and D. R. Smith, "Free-space microwave focusing by a negative-index gradient lens," Appl. Phys. Lett. 88, 081101 (2006).
[CrossRef]

Burger, S.

C.  Enkrich, M.  Wegener, S.  Linden, S.  Burger, L.  Zschiedrich, F.  Schmidt, J. F.  Zhou, Th.  Koschny, and C. M.  Soukoulis, "Magnetic metamaterials at telecommunication and visible frequencies," Phys. Rev. Lett.  95, 203901 (2005).
[CrossRef] [PubMed]

Cai, W.

Chettiar, U. K.

Cho, A.

A. Cho, "Voilà ! Cloak of Invisibility Unveiled," Science 314, 403 (2006).
[CrossRef] [PubMed]

Cummer, S. A.

B. I.  Popa and S. A.  Cummer, "Wave fields measured inside a negative refractive index metamaterial," Appl. Phys. Lett.  85, 4564-4566 (2004).
[CrossRef]

Decoopman, T.

T. Decoopman, A. Marteau, E. Lheurette, O. Vanbésien and D. Lippens, "Left-Handed Electromagnetic Properties of Split-Ring Resonator and Wire Loaded Transmission Line in a Fin-Line Technology," IEEE Trans. MTT 54, 1451-1457 (2006).
[CrossRef]

T. Decoopman, G. Tayeb, S. Enoch, D. Maystre and B Gralak, "Photonic Crystla Lens: From Negative Refraction and Negative Index to Negative Permittivity and Permeability," Phys. Rev. Lett. 97, 073905 (2006).
[CrossRef] [PubMed]

Deprot, S.

O. Reynet, A. -L. Adenot, S. Deprot, O. Acher, and M. Latrach, "Effect of the magnetic properties of inclusions on the high-frequency dielectric response of diluted composites," Phys. Rev. B 66, 94412 (2002).
[CrossRef]

Drachev, V. P.

Driscoll, T.

T. Driscoll, D. N. Basov, A. F. Starr, P. M. Rye, S. Nemat-Nasser, D. Schurig and D. R. Smith, "Free-space microwave focusing by a negative-index gradient lens," Appl. Phys. Lett. 88, 081101 (2006).
[CrossRef]

Dudek, C.

A.-L. Adenot-Engelvin, C. Dudek and O. Acher, "Microwave permeability of metamaterials based on ferromagnetic composites," J. Magn. Magn. Mater. 300, 33-37 (2006).
[CrossRef]

Duverger, F.

O. Acher, A. -L. Adenot, and F. Duverger, "Fresnel coefficients at an interface with a lamellar composite material," Phys. Rev. B 62, 13748 (2000).
[CrossRef]

O. Acher, A. -L. Adenot, F. Lubrano, and F. Duverger, "Low density artificial magnetic composites," J. Appl. Phys.,  85, 4639-4641 (1999).
[CrossRef]

Engheta, N.

Enkrich, C.

C.  Enkrich, M.  Wegener, S.  Linden, S.  Burger, L.  Zschiedrich, F.  Schmidt, J. F.  Zhou, Th.  Koschny, and C. M.  Soukoulis, "Magnetic metamaterials at telecommunication and visible frequencies," Phys. Rev. Lett.  95, 203901 (2005).
[CrossRef] [PubMed]

Enoch, S.

T. Decoopman, G. Tayeb, S. Enoch, D. Maystre and B Gralak, "Photonic Crystla Lens: From Negative Refraction and Negative Index to Negative Permittivity and Permeability," Phys. Rev. Lett. 97, 073905 (2006).
[CrossRef] [PubMed]

D. Maystre and S. Enoch, "Perfect lenses made with left-handed materilas: Alice’s mirror," J. Opt. Soc. Am. A 21, 122-131 (2004).
[CrossRef]

Fainman, Y.

Firsov, A. A.

A. N.  Grigorenko, A. K.  Geim, H. F.  Gleeson, Y.  Zhang, A. A.  Firsov, I. Y.  Khrushchev, and J.  Petrovic, "Nanofabricated media with negative permeability at visible frequencies," Nature  438, 335-338 (2005).
[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, "Nanofabricated media with negative permeability at visible frequencies," Nature  438, 335-338 (2005).
[CrossRef] [PubMed]

Gleeson, H. F.

A. N.  Grigorenko, A. K.  Geim, H. F.  Gleeson, Y.  Zhang, A. A.  Firsov, I. Y.  Khrushchev, and J.  Petrovic, "Nanofabricated media with negative permeability at visible frequencies," Nature  438, 335-338 (2005).
[CrossRef] [PubMed]

Gorkunov, M. V.

M. V. Gorkunov, S. A. Gredeskul, I. V. Shadrinov and Y. S. Kivshar, "Effect of microscopic disorder on magnetic properties of metamaterials," Phys. Rev. E 73, 056605 (2006).
[CrossRef]

Gralak, B

T. Decoopman, G. Tayeb, S. Enoch, D. Maystre and B Gralak, "Photonic Crystla Lens: From Negative Refraction and Negative Index to Negative Permittivity and Permeability," Phys. Rev. Lett. 97, 073905 (2006).
[CrossRef] [PubMed]

Gredeskul, S. A.

M. V. Gorkunov, S. A. Gredeskul, I. V. Shadrinov and Y. S. Kivshar, "Effect of microscopic disorder on magnetic properties of metamaterials," Phys. Rev. E 73, 056605 (2006).
[CrossRef]

Grigorenko, A. N.

A. N.  Grigorenko, A. K.  Geim, H. F.  Gleeson, Y.  Zhang, A. A.  Firsov, I. Y.  Khrushchev, and J.  Petrovic, "Nanofabricated media with negative permeability at visible frequencies," Nature  438, 335-338 (2005).
[CrossRef] [PubMed]

Guven, K.

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]

Kafesaki, M.

Khrushchev, I. Y.

A. N.  Grigorenko, A. K.  Geim, H. F.  Gleeson, Y.  Zhang, A. A.  Firsov, I. Y.  Khrushchev, and J.  Petrovic, "Nanofabricated media with negative permeability at visible frequencies," Nature  438, 335-338 (2005).
[CrossRef] [PubMed]

Kildishev, A. V.

Kivshar, Y. S.

M. V. Gorkunov, S. A. Gredeskul, I. V. Shadrinov and Y. S. Kivshar, "Effect of microscopic disorder on magnetic properties of metamaterials," Phys. Rev. E 73, 056605 (2006).
[CrossRef]

Koschny, Th.

C.  Enkrich, M.  Wegener, S.  Linden, S.  Burger, L.  Zschiedrich, F.  Schmidt, J. F.  Zhou, Th.  Koschny, and C. M.  Soukoulis, "Magnetic metamaterials at telecommunication and visible frequencies," Phys. Rev. Lett.  95, 203901 (2005).
[CrossRef] [PubMed]

Lagarkov, A. N.

A. N. Lagarkov, A. K. Sarychev, Y. R. Smychkovich and A. P. Vinogradov, "Effective Medium Theory for microwave dielectric constant and magnetic permeability of conducting stick composites," J. Electromagn. Waves Appl. 6, 1159 (1992).

Latrach, M.

O. Reynet, A. -L. Adenot, S. Deprot, O. Acher, and M. Latrach, "Effect of the magnetic properties of inclusions on the high-frequency dielectric response of diluted composites," Phys. Rev. B 66, 94412 (2002).
[CrossRef]

Ledieu, M.

O. Acher, M. Ledieu, A.-L. Adenot and O. Reynet, "Microwave properties of diluted composites made of magnetic wires with giant magneto-impedance effect," IEEE Trans. Magn.,  39, 3085-3090 (2003).
[CrossRef]

Lerat, J.-M.

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

Lheurette, E.

T. Decoopman, A. Marteau, E. Lheurette, O. Vanbésien and D. Lippens, "Left-Handed Electromagnetic Properties of Split-Ring Resonator and Wire Loaded Transmission Line in a Fin-Line Technology," IEEE Trans. MTT 54, 1451-1457 (2006).
[CrossRef]

Linden, S.

C.  Enkrich, M.  Wegener, S.  Linden, S.  Burger, L.  Zschiedrich, F.  Schmidt, J. F.  Zhou, Th.  Koschny, and C. M.  Soukoulis, "Magnetic metamaterials at telecommunication and visible frequencies," Phys. Rev. Lett.  95, 203901 (2005).
[CrossRef] [PubMed]

Lippens, D.

T. Decoopman, A. Marteau, E. Lheurette, O. Vanbésien and D. Lippens, "Left-Handed Electromagnetic Properties of Split-Ring Resonator and Wire Loaded Transmission Line in a Fin-Line Technology," IEEE Trans. MTT 54, 1451-1457 (2006).
[CrossRef]

Lomakin, V.

Lubrano, F.

O. Acher, A. -L. Adenot, F. Lubrano, and F. Duverger, "Low density artificial magnetic composites," J. Appl. Phys.,  85, 4639-4641 (1999).
[CrossRef]

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]

Malléjac, N.

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

Markos, P.

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]

Marteau, A.

T. Decoopman, A. Marteau, E. Lheurette, O. Vanbésien and D. Lippens, "Left-Handed Electromagnetic Properties of Split-Ring Resonator and Wire Loaded Transmission Line in a Fin-Line Technology," IEEE Trans. MTT 54, 1451-1457 (2006).
[CrossRef]

Maystre, D.

T. Decoopman, G. Tayeb, S. Enoch, D. Maystre and B Gralak, "Photonic Crystla Lens: From Negative Refraction and Negative Index to Negative Permittivity and Permeability," Phys. Rev. Lett. 97, 073905 (2006).
[CrossRef] [PubMed]

D. Maystre and S. Enoch, "Perfect lenses made with left-handed materilas: Alice’s mirror," J. Opt. Soc. Am. A 21, 122-131 (2004).
[CrossRef]

Nemat-Nasser, S.

T. Driscoll, D. N. Basov, A. F. Starr, P. M. Rye, S. Nemat-Nasser, D. Schurig and D. R. Smith, "Free-space microwave focusing by a negative-index gradient lens," Appl. Phys. Lett. 88, 081101 (2006).
[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 permeability and permittivity," Phys. Rev. Lett. 84, 4184 (2000).
[CrossRef] [PubMed]

Ozbay, E.

Padilla, W. J.

W. J.  Padilla, D. R.  Smith, and D. N.  Basov, "Spectroscopy of metamaterials from infrared to optical frequencies," J. Opt. Soc. Am. B  23, 404 (2006).
[CrossRef]

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

Pendry, J. B.

D. R.  Smith and J. B.  Pendry, "Homogenization of metamaterials by field averaging," J. Opt. Soc. Am. B  23, 391-403 (2006).
[CrossRef]

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]

Petrovic, J.

A. N.  Grigorenko, A. K.  Geim, H. F.  Gleeson, Y.  Zhang, A. A.  Firsov, I. Y.  Khrushchev, and J.  Petrovic, "Nanofabricated media with negative permeability at visible frequencies," Nature  438, 335-338 (2005).
[CrossRef] [PubMed]

Popa, B. I.

B. I.  Popa and S. A.  Cummer, "Wave fields measured inside a negative refractive index metamaterial," Appl. Phys. Lett.  85, 4564-4566 (2004).
[CrossRef]

Preobrazhenskii, E. A.

K. N. Rozanov and E. A. Preobrazhenskii, "Synthesis of wideband radar absorbers based on complex media composed from active electric Dipoles," J. Commun. Technol. Electron. 50, 858-864 (2005).

Reynet, O.

O. Reynet, O. Acher, "Voltage controlled metamaterial," Appl. Phys. Lett. 84, 1198-2000 (2004).
[CrossRef]

O. Acher, M. Ledieu, A.-L. Adenot and O. Reynet, "Microwave properties of diluted composites made of magnetic wires with giant magneto-impedance effect," IEEE Trans. Magn.,  39, 3085-3090 (2003).
[CrossRef]

O. Reynet, A. -L. Adenot, S. Deprot, O. Acher, and M. Latrach, "Effect of the magnetic properties of inclusions on the high-frequency dielectric response of diluted composites," Phys. Rev. B 66, 94412 (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. Microwave Theory Tech. 47, 2075-2084 (1999).
[CrossRef]

Rozanov, K. N.

K. N. Rozanov and E. A. Preobrazhenskii, "Synthesis of wideband radar absorbers based on complex media composed from active electric Dipoles," J. Commun. Technol. Electron. 50, 858-864 (2005).

Rye, P. M.

T. Driscoll, D. N. Basov, A. F. Starr, P. M. Rye, S. Nemat-Nasser, D. Schurig and D. R. Smith, "Free-space microwave focusing by a negative-index gradient lens," Appl. Phys. Lett. 88, 081101 (2006).
[CrossRef]

Rytov, S. M.

S. M. Rytov, "Electromagnetic properties of a finely stratified medium," Sov. Phys. JETP 2, 466-475 (1956).

Salandrino, A.

Sarychev, A. K.

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

A. N. Lagarkov, A. K. Sarychev, Y. R. Smychkovich and A. P. Vinogradov, "Effective Medium Theory for microwave dielectric constant and magnetic permeability of conducting stick composites," J. Electromagn. Waves Appl. 6, 1159 (1992).

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]

Schmidt, F.

C.  Enkrich, M.  Wegener, S.  Linden, S.  Burger, L.  Zschiedrich, F.  Schmidt, J. F.  Zhou, Th.  Koschny, and C. M.  Soukoulis, "Magnetic metamaterials at telecommunication and visible frequencies," Phys. Rev. Lett.  95, 203901 (2005).
[CrossRef] [PubMed]

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]

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

Schurig, D.

T. Driscoll, D. N. Basov, A. F. Starr, P. M. Rye, S. Nemat-Nasser, D. Schurig and D. R. Smith, "Free-space microwave focusing by a negative-index gradient lens," Appl. Phys. Lett. 88, 081101 (2006).
[CrossRef]

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]

Shadrinov, I. V.

M. V. Gorkunov, S. A. Gredeskul, I. V. Shadrinov and Y. S. Kivshar, "Effect of microscopic disorder on magnetic properties of metamaterials," Phys. Rev. E 73, 056605 (2006).
[CrossRef]

Shalaev, V. M.

Shvets, G.

Smith, D. R.

D. R.  Smith and J. B.  Pendry, "Homogenization of metamaterials by field averaging," J. Opt. Soc. Am. B  23, 391-403 (2006).
[CrossRef]

W. J.  Padilla, D. R.  Smith, and D. N.  Basov, "Spectroscopy of metamaterials from infrared to optical frequencies," J. Opt. Soc. Am. B  23, 404 (2006).
[CrossRef]

T. Driscoll, D. N. Basov, A. F. Starr, P. M. Rye, S. Nemat-Nasser, D. Schurig and D. R. Smith, "Free-space microwave focusing by a negative-index gradient lens," Appl. Phys. Lett. 88, 081101 (2006).
[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]

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

Smychkovich, Y. R.

A. N. Lagarkov, A. K. Sarychev, Y. R. Smychkovich and A. P. Vinogradov, "Effective Medium Theory for microwave dielectric constant and magnetic permeability of conducting stick composites," J. Electromagn. Waves Appl. 6, 1159 (1992).

Soukoulis, C. M.

C.  Enkrich, M.  Wegener, S.  Linden, S.  Burger, L.  Zschiedrich, F.  Schmidt, J. F.  Zhou, Th.  Koschny, and C. M.  Soukoulis, "Magnetic metamaterials at telecommunication and visible frequencies," Phys. Rev. Lett.  95, 203901 (2005).
[CrossRef] [PubMed]

K. Aydin, K. Guven, M. Kafesaki, L. Zhang, C. M. Soukoulis, and E. Ozbay, "Experimental observation of true left-handed transmission peak in metamaterials," Opt. Lett. 29, 2623 (2004).
[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]

Starr, A. F.

T. Driscoll, D. N. Basov, A. F. Starr, P. M. Rye, S. Nemat-Nasser, D. Schurig and D. R. Smith, "Free-space microwave focusing by a negative-index gradient lens," Appl. Phys. Lett. 88, 081101 (2006).
[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. Microwave Theory Tech. 47, 2075-2084 (1999).
[CrossRef]

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]

Tayeb, G.

T. Decoopman, G. Tayeb, S. Enoch, D. Maystre and B Gralak, "Photonic Crystla Lens: From Negative Refraction and Negative Index to Negative Permittivity and Permeability," Phys. Rev. Lett. 97, 073905 (2006).
[CrossRef] [PubMed]

Urzhumov, Y.

Vanbésien, O.

T. Decoopman, A. Marteau, E. Lheurette, O. Vanbésien and D. Lippens, "Left-Handed Electromagnetic Properties of Split-Ring Resonator and Wire Loaded Transmission Line in a Fin-Line Technology," IEEE Trans. MTT 54, 1451-1457 (2006).
[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 permeability and permittivity," Phys. Rev. Lett. 84, 4184 (2000).
[CrossRef] [PubMed]

Vinogradov, A. P.

A. N. Lagarkov, A. K. Sarychev, Y. R. Smychkovich and A. P. Vinogradov, "Effective Medium Theory for microwave dielectric constant and magnetic permeability of conducting stick composites," J. Electromagn. Waves Appl. 6, 1159 (1992).

Wegener, M.

C.  Enkrich, M.  Wegener, S.  Linden, S.  Burger, L.  Zschiedrich, F.  Schmidt, J. F.  Zhou, Th.  Koschny, and C. M.  Soukoulis, "Magnetic metamaterials at telecommunication and visible frequencies," Phys. Rev. Lett.  95, 203901 (2005).
[CrossRef] [PubMed]

Yuan, H.-K.

Zhang, L.

Zhang, Y.

A. N.  Grigorenko, A. K.  Geim, H. F.  Gleeson, Y.  Zhang, A. A.  Firsov, I. Y.  Khrushchev, and J.  Petrovic, "Nanofabricated media with negative permeability at visible frequencies," Nature  438, 335-338 (2005).
[CrossRef] [PubMed]

Zhou, J. F.

C.  Enkrich, M.  Wegener, S.  Linden, S.  Burger, L.  Zschiedrich, F.  Schmidt, J. F.  Zhou, Th.  Koschny, and C. M.  Soukoulis, "Magnetic metamaterials at telecommunication and visible frequencies," Phys. Rev. Lett.  95, 203901 (2005).
[CrossRef] [PubMed]

Ziolkowski, R.

Zschiedrich, L.

C.  Enkrich, M.  Wegener, S.  Linden, S.  Burger, L.  Zschiedrich, F.  Schmidt, J. F.  Zhou, Th.  Koschny, and C. M.  Soukoulis, "Magnetic metamaterials at telecommunication and visible frequencies," Phys. Rev. Lett.  95, 203901 (2005).
[CrossRef] [PubMed]

Appl. Phys. Lett. (3)

O. Reynet, O. Acher, "Voltage controlled metamaterial," Appl. Phys. Lett. 84, 1198-2000 (2004).
[CrossRef]

B. I.  Popa and S. A.  Cummer, "Wave fields measured inside a negative refractive index metamaterial," Appl. Phys. Lett.  85, 4564-4566 (2004).
[CrossRef]

T. Driscoll, D. N. Basov, A. F. Starr, P. M. Rye, S. Nemat-Nasser, D. Schurig and D. R. Smith, "Free-space microwave focusing by a negative-index gradient lens," Appl. Phys. Lett. 88, 081101 (2006).
[CrossRef]

IEEE Trans. Magn. (1)

O. Acher, M. Ledieu, A.-L. Adenot and O. Reynet, "Microwave properties of diluted composites made of magnetic wires with giant magneto-impedance effect," IEEE Trans. Magn.,  39, 3085-3090 (2003).
[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]

IEEE Trans. MTT (1)

T. Decoopman, A. Marteau, E. Lheurette, O. Vanbésien and D. Lippens, "Left-Handed Electromagnetic Properties of Split-Ring Resonator and Wire Loaded Transmission Line in a Fin-Line Technology," IEEE Trans. MTT 54, 1451-1457 (2006).
[CrossRef]

J. Appl. Phys. (3)

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]

O. Acher, A. -L. Adenot, F. Lubrano, and F. Duverger, "Low density artificial magnetic composites," J. Appl. Phys.,  85, 4639-4641 (1999).
[CrossRef]

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

J. Commun. Technol. Electron. (1)

K. N. Rozanov and E. A. Preobrazhenskii, "Synthesis of wideband radar absorbers based on complex media composed from active electric Dipoles," J. Commun. Technol. Electron. 50, 858-864 (2005).

J. Electromagn. Waves Appl. (1)

A. N. Lagarkov, A. K. Sarychev, Y. R. Smychkovich and A. P. Vinogradov, "Effective Medium Theory for microwave dielectric constant and magnetic permeability of conducting stick composites," J. Electromagn. Waves Appl. 6, 1159 (1992).

J. Magn. Magn. Mater. (1)

A.-L. Adenot-Engelvin, C. Dudek and O. Acher, "Microwave permeability of metamaterials based on ferromagnetic composites," J. Magn. Magn. Mater. 300, 33-37 (2006).
[CrossRef]

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

J. Opt. Soc. Am. B (2)

Nature (1)

A. N.  Grigorenko, A. K.  Geim, H. F.  Gleeson, Y.  Zhang, A. A.  Firsov, I. Y.  Khrushchev, and J.  Petrovic, "Nanofabricated media with negative permeability at visible frequencies," Nature  438, 335-338 (2005).
[CrossRef] [PubMed]

Opt. Express (3)

Opt. Lett. (2)

Phys. Rev. B (3)

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]

O. Acher, A. -L. Adenot, and F. Duverger, "Fresnel coefficients at an interface with a lamellar composite material," Phys. Rev. B 62, 13748 (2000).
[CrossRef]

O. Reynet, A. -L. Adenot, S. Deprot, O. Acher, and M. Latrach, "Effect of the magnetic properties of inclusions on the high-frequency dielectric response of diluted composites," Phys. Rev. B 66, 94412 (2002).
[CrossRef]

Phys. Rev. E (1)

M. V. Gorkunov, S. A. Gredeskul, I. V. Shadrinov and Y. S. Kivshar, "Effect of microscopic disorder on magnetic properties of metamaterials," Phys. Rev. E 73, 056605 (2006).
[CrossRef]

Phys. Rev. Lett. (3)

T. Decoopman, G. Tayeb, S. Enoch, D. Maystre and B Gralak, "Photonic Crystla Lens: From Negative Refraction and Negative Index to Negative Permittivity and Permeability," Phys. Rev. Lett. 97, 073905 (2006).
[CrossRef] [PubMed]

C.  Enkrich, M.  Wegener, S.  Linden, S.  Burger, L.  Zschiedrich, F.  Schmidt, J. F.  Zhou, Th.  Koschny, and C. M.  Soukoulis, "Magnetic metamaterials at telecommunication and visible frequencies," Phys. Rev. Lett.  95, 203901 (2005).
[CrossRef] [PubMed]

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

Science (1)

A. Cho, "Voilà ! Cloak of Invisibility Unveiled," Science 314, 403 (2006).
[CrossRef] [PubMed]

Sov. Phys. JETP (1)

S. M. Rytov, "Electromagnetic properties of a finely stratified medium," Sov. Phys. JETP 2, 466-475 (1956).

Other (1)

S. Tretyakov, Analytical modelling in Applied Electromagnetics, (Artech House, 2003).

Supplementary Material (8)

» Media 1: AVI (812 KB)     
» Media 2: AVI (1556 KB)     
» Media 3: AVI (2306 KB)     
» Media 4: AVI (1888 KB)     
» Media 5: AVI (754 KB)     
» Media 6: AVI (2107 KB)     
» Media 7: AVI (2481 KB)     
» Media 8: AVI (2381 KB)     

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Figures (9)

Fig. 1.
Fig. 1.

Slab of periodic medium illuminated by an incoming wave: notations, and averaging zones used in different Field Summation Methods.

Fig. 2.
Fig. 2.

(a). Permeability and permittivity obtained by field averaging on a single scatterer thick metamaterial slab; (b). Reflection and Transmission coefficients for the metamaterial slab, calculated using the effective parameters obtained by Field Summation (full curve); and computed in the numerical experiment (S11, S12). Red markers correspond to frequencies of Figs. 3–6.

Fig. 3.
Fig. 3.

H field as a function of time on the P and M planes in a metamaterial: (a). (0.81 MB movie) at 0.9 GHz, the permeability is slightly larger than unity; [Media 1] (b). (1.56 MB movie) at 0.98 GHz, the permeability is large and positive. [Media 2]

Fig. 4.
Fig. 4.

(2.31 MB movie) H field as a function of time on the P and M planes in a metamaterial at the resonance frequency F=0.99 GHz [Media 3]

Fig. 5.
Fig. 5.

H field as a function of time on the P and M planes in a metamaterial: (a) (1.89 MB movie) at 1 GHz, the permeability is negative; [Media 4] (b) (0.75 MB movie) at 1.1 GHz, the permeability is slightly lower than unity. [Media 5]

Fig. 6.
Fig. 6.

(a). Permeability of a 4-scatterer thick metamaterial made with a resistive metal, computed using the Field Summation method; red markers correspond to frequencies illustrated on Fig. 7; (b). Reflection and Transmission coefficients computed from the effective parameters obtained by the field averaging method, and comparison with S11 and S12 values obtained directly from the simulation software.

Fig. 7.
Fig. 7.

H field in a 4-scaterrer thick metamaterial slab made with a resistive metal; (a) (2.10 MB movie) at 0.85 GHz (μ>1); [Media 6] (b) (2.48 MB movie) at 0.96 GHz (μ is imaginary); [Media 7] (c) (2.38 MB movie) at 1.05 GHz (μ <1). [Media 8]

Fig. 8.
Fig. 8.

Reflection and Transmission coefficients for a 4-scatterrer thick slab consisting of highly resonant patterns, computed using the simulation software.

Fig. 9.
Fig. 9.

H field in a 4-scatterrer thick slab consisting of highly resonant patterns, at a frequency 1.04 GHz; (a) Top view of the P plane; (b) Top view of the M plane.

Equations (5)

Equations on this page are rendered with MathJax. Learn more.

μ 0 μ V V = B x xyz H x xyz ε 0 ε V V = D z xyz E z xyz
μ 0 μ S L = B x xy H x x n y kd sin ( n y kd )
ε 0 ε S L = D z xy E z z n y kd sin ( n y kd )
μ 0 μ V S = B x xyz H x xy ε 0 ε S V = D z xy E z xyz
μ V S = H x xyz H x xy

Metrics