Abstract

Over the past several years, metamaterials have been introduced and rapidly been adopted as a means of achieving unique electromagnetic material response. In metamaterials, artificially structured—often periodically positioned—inclusions replace the atoms and molecules of conventional materials. The scale of these inclusions is smaller than that of the electromagnetic wavelength of interest, so that a homogenized description applies. We present a homogenization technique in which macroscopic fields are determined via averaging the local fields obtained from a full-wave electromagnetic simulation or analytical calculation. The field-averaging method can be applied to homogenize any periodic structure with unit cells having inclusions of arbitrary geometry and material. By analyzing the dispersion diagrams and retrieved parameters found by field averaging, we review the properties of several basic metamaterial structures.

© 2006 Optical Society of America

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  8. D. R. Smith, W. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "A composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4187 (2000).
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    [CrossRef] [PubMed]
  14. P. F. Loschialpo, D. L. Smith, D. W. Forester, F. J. Rachford, and J. Schelleng, "Electromagnetic waves focused by a negative-index planar lens," Phys. Rev. E 67, 025602 (2003).
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    [CrossRef]
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    [CrossRef]
  21. J. B. Pendry, "Calculating photonic band structure," J. Phys. Condens. Matter 8, 1085-1108 (1996).
    [CrossRef]
  22. D. J. Bergman, "The dielectric constant of a composite material—a problem in classical physics," Phys. Rep. 9, 377-407 (1978).
    [CrossRef]
  23. D. R. Smith, D. C. Vier, N. Kroll, and S. Schultz, "Direct calculation of permeability and permittivity for a left-handed metamaterial," Appl. Phys. Lett. 77, 2246-2248 (2000).
    [CrossRef]
  24. J. A. Kong, Electromagnetic Wave Theory, 2nd ed. (Wiley, 1990), Sec. 2.6.
  25. T. Koschny, P. Markos, D. R. Smith, and C. M. Soukoulis, "Resonant and antiresonant frequency dependence of the effective parameters of metamaterials," Phys. Rev. E 68, 065602 (2003).
    [CrossRef]
  26. S. I. Maslovski, S. A. Tretyakov, and P. A. Belov, "Wire media with negative effective permittivity: a quasi-static model," Microwave Opt. Technol. Lett. 35, 47-51 (2002).
    [CrossRef]
  27. P. A. Belov, R. Marqués, S. I. Maslovski, I. S. Nefedov, M. Silveirinha, C. R. Simovski, and S. A. Tretyakov, "Strong spatial dispersion in wire media in the very large wavelength limit," Phys. Rev. B 67, 113103 (2003).
    [CrossRef]
  28. A. L. Pokrovsky, "Analytical and numerical studies of wire-mesh metallic photonic crystals," Phys. Rev. B 69, 195108 (2004).
    [CrossRef]
  29. C. R. Simovski and P. A. Belov, "Low-frequency spatial dispersion in wire media," Phys. Rev. E 70, 046616 (2004).
    [CrossRef]
  30. M. G. Silveirinha and C. A. Fernandes, "Homogenization of 3-D-connected and nonconnected wire metamaterials," IEEE Trans. Microwave Theory Tech. 53, 1418-1430 (2005).
    [CrossRef]
  31. R. Marqués, F. Medina, and R. Rafii-El-Idrissi, "Role of bianisotropy in negative permeability and left-handed metamaterials," Phys. Rev. B 65, 144440 (2002).
    [CrossRef]
  32. R. Marqués, F. Mesa, J. Martel, and F. Medina, "Comparative analysis of edge- and broadside-coupled split ring resonators for metamaterial design—theory and experiments," IEEE Trans. Antennas Propag. 51, 2572-2581 (2003).
    [CrossRef]

2005 (1)

M. G. Silveirinha and C. A. Fernandes, "Homogenization of 3-D-connected and nonconnected wire metamaterials," IEEE Trans. Microwave Theory Tech. 53, 1418-1430 (2005).
[CrossRef]

2004 (4)

A. L. Pokrovsky, "Analytical and numerical studies of wire-mesh metallic photonic crystals," Phys. Rev. B 69, 195108 (2004).
[CrossRef]

C. R. Simovski and P. A. Belov, "Low-frequency spatial dispersion in wire media," Phys. Rev. E 70, 046616 (2004).
[CrossRef]

J. B. Pendry and D. R. Smith, "Reversing light with negative refraction," Phys. Today 57, 37-43 (2004).
[CrossRef]

F. Falcone, T. Lopetegi, M. A. G. Laso, J. D. Baena, J. Bonache, M. Beruete, R. Marques, F. Martin, and M. Sorolla, "Babinet principle applied to the design of metasurfaces and metamaterials," Phys. Rev. Lett. 93, 197401 (2004).
[CrossRef] [PubMed]

2003 (7)

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

P. F. Loschialpo, D. L. Smith, D. W. Forester, F. J. Rachford, and J. Schelleng, "Electromagnetic waves focused by a negative-index planar lens," Phys. Rev. E 67, 025602 (2003).
[CrossRef]

A. A. Houck, J. B. Brock, and I. L. Chuang, "Experimental observations of a left-handed material that obeys Snell's law," Phys. Rev. Lett. 90, 137401 (2003).
[CrossRef] [PubMed]

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

P. A. Belov, R. Marqués, S. I. Maslovski, I. S. Nefedov, M. Silveirinha, C. R. Simovski, and S. A. Tretyakov, "Strong spatial dispersion in wire media in the very large wavelength limit," Phys. Rev. B 67, 113103 (2003).
[CrossRef]

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

R. Marqués, F. Mesa, J. Martel, and F. Medina, "Comparative analysis of edge- and broadside-coupled split ring resonators for metamaterial design—theory and experiments," IEEE Trans. Antennas Propag. 51, 2572-2581 (2003).
[CrossRef]

2002 (5)

R. Marqués, F. Medina, and R. Rafii-El-Idrissi, "Role of bianisotropy in negative permeability and left-handed metamaterials," Phys. Rev. B 65, 144440 (2002).
[CrossRef]

S. I. Maslovski, S. A. Tretyakov, and P. A. Belov, "Wire media with negative effective permittivity: a quasi-static model," Microwave Opt. Technol. Lett. 35, 47-51 (2002).
[CrossRef]

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

P. Markos and C. M. Soukoulis, "Numerical studies of left-handed materials and arrays of split ring resonators," Phys. Rev. E 65, 036622 (2002).
[CrossRef]

M. Bayindir, K. Aydin, E. Ozbay, P. Markos, and C. M. Soukoulis, "Transmission properties of composite metamaterials in free space," Appl. Phys. Lett. 81, 120-122 (2002).
[CrossRef]

2001 (1)

R. M. Walser, "Electromagnetic metamaterials," in Complex Mediums II: Beyond Linear Isotropic Dielectrics, A. Lakhtakia, W. S. Weiglhofer, and I. J. Hodgkinson, eds., Proc. SPIE 4467, 1-15 (2001).
[CrossRef]

2000 (2)

D. R. Smith, W. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "A composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4187 (2000).
[CrossRef] [PubMed]

D. R. Smith, D. C. Vier, N. Kroll, and S. Schultz, "Direct calculation of permeability and permittivity for a left-handed metamaterial," Appl. Phys. Lett. 77, 2246-2248 (2000).
[CrossRef]

1999 (1)

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

1996 (3)

J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, "Extremely low frequency plasmons in metallic mesostructures," Phys. Rev. Lett. 76, 4773-4776 (1996).
[CrossRef] [PubMed]

V. M. Shalaev, "Electromagnetic properties of small-particle composites," Phys. Rep. 272, 61-137 (1996).
[CrossRef]

J. B. Pendry, "Calculating photonic band structure," J. Phys. Condens. Matter 8, 1085-1108 (1996).
[CrossRef]

1992 (1)

M. M. I. Saadoun and N. Engheta, "A reciprocal phase shifter using novel pseudochiral or omega-medium," Microwave Opt. Technol. Lett. 5, 184-188 (1992).
[CrossRef]

1978 (1)

D. J. Bergman, "The dielectric constant of a composite material—a problem in classical physics," Phys. Rep. 9, 377-407 (1978).
[CrossRef]

1968 (1)

V. G. Veselago, "The electrodynamics of substances with simultaneously negative values of epsi and µ," Sov. Phys. Usp. 10, 509-514 (1968).
[CrossRef]

1966 (1)

K. S. Yee, "Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media," IEEE Trans. Antennas Propag. 14, 302-307 (1966).
[CrossRef]

Aydin, K.

M. Bayindir, K. Aydin, E. Ozbay, P. Markos, and C. M. Soukoulis, "Transmission properties of composite metamaterials in free space," Appl. Phys. Lett. 81, 120-122 (2002).
[CrossRef]

Baena, J. D.

F. Falcone, T. Lopetegi, M. A. G. Laso, J. D. Baena, J. Bonache, M. Beruete, R. Marques, F. Martin, and M. Sorolla, "Babinet principle applied to the design of metasurfaces and metamaterials," Phys. Rev. Lett. 93, 197401 (2004).
[CrossRef] [PubMed]

Bayindir, M.

M. Bayindir, K. Aydin, E. Ozbay, P. Markos, and C. M. Soukoulis, "Transmission properties of composite metamaterials in free space," Appl. Phys. Lett. 81, 120-122 (2002).
[CrossRef]

Belov, P. A.

C. R. Simovski and P. A. Belov, "Low-frequency spatial dispersion in wire media," Phys. Rev. E 70, 046616 (2004).
[CrossRef]

P. A. Belov, R. Marqués, S. I. Maslovski, I. S. Nefedov, M. Silveirinha, C. R. Simovski, and S. A. Tretyakov, "Strong spatial dispersion in wire media in the very large wavelength limit," Phys. Rev. B 67, 113103 (2003).
[CrossRef]

S. I. Maslovski, S. A. Tretyakov, and P. A. Belov, "Wire media with negative effective permittivity: a quasi-static model," Microwave Opt. Technol. Lett. 35, 47-51 (2002).
[CrossRef]

Bergman, D. J.

D. J. Bergman, "The dielectric constant of a composite material—a problem in classical physics," Phys. Rep. 9, 377-407 (1978).
[CrossRef]

Beruete, M.

F. Falcone, T. Lopetegi, M. A. G. Laso, J. D. Baena, J. Bonache, M. Beruete, R. Marques, F. Martin, and M. Sorolla, "Babinet principle applied to the design of metasurfaces and metamaterials," Phys. Rev. Lett. 93, 197401 (2004).
[CrossRef] [PubMed]

Bonache, J.

F. Falcone, T. Lopetegi, M. A. G. Laso, J. D. Baena, J. Bonache, M. Beruete, R. Marques, F. Martin, and M. Sorolla, "Babinet principle applied to the design of metasurfaces and metamaterials," Phys. Rev. Lett. 93, 197401 (2004).
[CrossRef] [PubMed]

Brock, J. B.

A. A. Houck, J. B. Brock, and I. L. Chuang, "Experimental observations of a left-handed material that obeys Snell's law," Phys. Rev. Lett. 90, 137401 (2003).
[CrossRef] [PubMed]

Chuang, I. L.

A. A. Houck, J. B. Brock, and I. L. Chuang, "Experimental observations of a left-handed material that obeys Snell's law," Phys. Rev. Lett. 90, 137401 (2003).
[CrossRef] [PubMed]

Engheta, N.

M. M. I. Saadoun and N. Engheta, "A reciprocal phase shifter using novel pseudochiral or omega-medium," Microwave Opt. Technol. Lett. 5, 184-188 (1992).
[CrossRef]

Falcone, F.

F. Falcone, T. Lopetegi, M. A. G. Laso, J. D. Baena, J. Bonache, M. Beruete, R. Marques, F. Martin, and M. Sorolla, "Babinet principle applied to the design of metasurfaces and metamaterials," Phys. Rev. Lett. 93, 197401 (2004).
[CrossRef] [PubMed]

Fernandes, C. A.

M. G. Silveirinha and C. A. Fernandes, "Homogenization of 3-D-connected and nonconnected wire metamaterials," IEEE Trans. Microwave Theory Tech. 53, 1418-1430 (2005).
[CrossRef]

Forester, D. W.

P. F. Loschialpo, D. L. Smith, D. W. Forester, F. J. Rachford, and J. Schelleng, "Electromagnetic waves focused by a negative-index planar lens," Phys. Rev. E 67, 025602 (2003).
[CrossRef]

Greegor, R. B.

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

Holden, A. J.

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

J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, "Extremely low frequency plasmons in metallic mesostructures," Phys. Rev. Lett. 76, 4773-4776 (1996).
[CrossRef] [PubMed]

Houck, A. A.

A. A. Houck, J. B. Brock, and I. L. Chuang, "Experimental observations of a left-handed material that obeys Snell's law," Phys. Rev. Lett. 90, 137401 (2003).
[CrossRef] [PubMed]

Koltenbah, B. E.

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

Kong, J. A.

J. A. Kong, Electromagnetic Wave Theory, 2nd ed. (Wiley, 1990), Sec. 2.6.

Koschny, T.

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

Kroll, N.

D. R. Smith, D. C. Vier, N. Kroll, and S. Schultz, "Direct calculation of permeability and permittivity for a left-handed metamaterial," Appl. Phys. Lett. 77, 2246-2248 (2000).
[CrossRef]

Laso, M. A.

F. Falcone, T. Lopetegi, M. A. G. Laso, J. D. Baena, J. Bonache, M. Beruete, R. Marques, F. Martin, and M. Sorolla, "Babinet principle applied to the design of metasurfaces and metamaterials," Phys. Rev. Lett. 93, 197401 (2004).
[CrossRef] [PubMed]

Li, K.

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

Lindell, I. V.

I. V. Lindell, A. H. Sihvola, S. A. Tretyakov, and A. J. Viitanen, Electromagnetic Waves in Chiral and Bi-Isotropic Media (Artech House, 1994).

Lopetegi, T.

F. Falcone, T. Lopetegi, M. A. G. Laso, J. D. Baena, J. Bonache, M. Beruete, R. Marques, F. Martin, and M. Sorolla, "Babinet principle applied to the design of metasurfaces and metamaterials," Phys. Rev. Lett. 93, 197401 (2004).
[CrossRef] [PubMed]

Loschialpo, P. F.

P. F. Loschialpo, D. L. Smith, D. W. Forester, F. J. Rachford, and J. Schelleng, "Electromagnetic waves focused by a negative-index planar lens," Phys. Rev. E 67, 025602 (2003).
[CrossRef]

Markos, P.

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

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

M. Bayindir, K. Aydin, E. Ozbay, P. Markos, and C. M. Soukoulis, "Transmission properties of composite metamaterials in free space," Appl. Phys. Lett. 81, 120-122 (2002).
[CrossRef]

P. Markos and C. M. Soukoulis, "Numerical studies of left-handed materials and arrays of split ring resonators," Phys. Rev. E 65, 036622 (2002).
[CrossRef]

Marques, R.

F. Falcone, T. Lopetegi, M. A. G. Laso, J. D. Baena, J. Bonache, M. Beruete, R. Marques, F. Martin, and M. Sorolla, "Babinet principle applied to the design of metasurfaces and metamaterials," Phys. Rev. Lett. 93, 197401 (2004).
[CrossRef] [PubMed]

Marqués, R.

R. Marqués, F. Mesa, J. Martel, and F. Medina, "Comparative analysis of edge- and broadside-coupled split ring resonators for metamaterial design—theory and experiments," IEEE Trans. Antennas Propag. 51, 2572-2581 (2003).
[CrossRef]

P. A. Belov, R. Marqués, S. I. Maslovski, I. S. Nefedov, M. Silveirinha, C. R. Simovski, and S. A. Tretyakov, "Strong spatial dispersion in wire media in the very large wavelength limit," Phys. Rev. B 67, 113103 (2003).
[CrossRef]

R. Marqués, F. Medina, and R. Rafii-El-Idrissi, "Role of bianisotropy in negative permeability and left-handed metamaterials," Phys. Rev. B 65, 144440 (2002).
[CrossRef]

Martel, J.

R. Marqués, F. Mesa, J. Martel, and F. Medina, "Comparative analysis of edge- and broadside-coupled split ring resonators for metamaterial design—theory and experiments," IEEE Trans. Antennas Propag. 51, 2572-2581 (2003).
[CrossRef]

Martin, F.

F. Falcone, T. Lopetegi, M. A. G. Laso, J. D. Baena, J. Bonache, M. Beruete, R. Marques, F. Martin, and M. Sorolla, "Babinet principle applied to the design of metasurfaces and metamaterials," Phys. Rev. Lett. 93, 197401 (2004).
[CrossRef] [PubMed]

Maslovski, S. I.

P. A. Belov, R. Marqués, S. I. Maslovski, I. S. Nefedov, M. Silveirinha, C. R. Simovski, and S. A. Tretyakov, "Strong spatial dispersion in wire media in the very large wavelength limit," Phys. Rev. B 67, 113103 (2003).
[CrossRef]

S. I. Maslovski, S. A. Tretyakov, and P. A. Belov, "Wire media with negative effective permittivity: a quasi-static model," Microwave Opt. Technol. Lett. 35, 47-51 (2002).
[CrossRef]

Medina, F.

R. Marqués, F. Mesa, J. Martel, and F. Medina, "Comparative analysis of edge- and broadside-coupled split ring resonators for metamaterial design—theory and experiments," IEEE Trans. Antennas Propag. 51, 2572-2581 (2003).
[CrossRef]

R. Marqués, F. Medina, and R. Rafii-El-Idrissi, "Role of bianisotropy in negative permeability and left-handed metamaterials," Phys. Rev. B 65, 144440 (2002).
[CrossRef]

Mesa, F.

R. Marqués, F. Mesa, J. Martel, and F. Medina, "Comparative analysis of edge- and broadside-coupled split ring resonators for metamaterial design—theory and experiments," IEEE Trans. Antennas Propag. 51, 2572-2581 (2003).
[CrossRef]

Nefedov, I. S.

P. A. Belov, R. Marqués, S. I. Maslovski, I. S. Nefedov, M. Silveirinha, C. R. Simovski, and S. A. Tretyakov, "Strong spatial dispersion in wire media in the very large wavelength limit," Phys. Rev. B 67, 113103 (2003).
[CrossRef]

Nemat-Nasser, S. C.

D. R. Smith, W. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "A composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4187 (2000).
[CrossRef] [PubMed]

Ozbay, E.

M. Bayindir, K. Aydin, E. Ozbay, P. Markos, and C. M. Soukoulis, "Transmission properties of composite metamaterials in free space," Appl. Phys. Lett. 81, 120-122 (2002).
[CrossRef]

Padilla, W.

D. R. Smith, W. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "A composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4187 (2000).
[CrossRef] [PubMed]

Parazzoli, C. G.

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

Pendry, J. B.

J. B. Pendry and D. R. Smith, "Reversing light with negative refraction," Phys. Today 57, 37-43 (2004).
[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]

J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, "Extremely low frequency plasmons in metallic mesostructures," Phys. Rev. Lett. 76, 4773-4776 (1996).
[CrossRef] [PubMed]

J. B. Pendry, "Calculating photonic band structure," J. Phys. Condens. Matter 8, 1085-1108 (1996).
[CrossRef]

Pokrovsky, A. L.

A. L. Pokrovsky, "Analytical and numerical studies of wire-mesh metallic photonic crystals," Phys. Rev. B 69, 195108 (2004).
[CrossRef]

Rachford, F. J.

P. F. Loschialpo, D. L. Smith, D. W. Forester, F. J. Rachford, and J. Schelleng, "Electromagnetic waves focused by a negative-index planar lens," Phys. Rev. E 67, 025602 (2003).
[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. B 65, 144440 (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]

Saadoun, M. M.

M. M. I. Saadoun and N. Engheta, "A reciprocal phase shifter using novel pseudochiral or omega-medium," Microwave Opt. Technol. Lett. 5, 184-188 (1992).
[CrossRef]

Schelleng, J.

P. F. Loschialpo, D. L. Smith, D. W. Forester, F. J. Rachford, and J. Schelleng, "Electromagnetic waves focused by a negative-index planar lens," Phys. Rev. E 67, 025602 (2003).
[CrossRef]

Schultz, S.

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

D. R. Smith, D. C. Vier, N. Kroll, and S. Schultz, "Direct calculation of permeability and permittivity for a left-handed metamaterial," Appl. Phys. Lett. 77, 2246-2248 (2000).
[CrossRef]

D. R. Smith, W. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "A composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4187 (2000).
[CrossRef] [PubMed]

Shalaev, V. M.

V. M. Shalaev, "Electromagnetic properties of small-particle composites," Phys. Rep. 272, 61-137 (1996).
[CrossRef]

Sihvola, A.

A. Sihvola, "Electromagnetic emergence in metamaterials," in Advances in Electromagnetics of Complex Media and Metamaterials, S.Zouhdi, A.Sihvola, and M.Arsalane, eds., Vol. 89 of NATO Science Series II: Mathematics, Physics, and Chemistry (Kluwer Academic, 2003), pp. 3-17.
[CrossRef]

Sihvola, A. H.

I. V. Lindell, A. H. Sihvola, S. A. Tretyakov, and A. J. Viitanen, Electromagnetic Waves in Chiral and Bi-Isotropic Media (Artech House, 1994).

Silveirinha, M.

P. A. Belov, R. Marqués, S. I. Maslovski, I. S. Nefedov, M. Silveirinha, C. R. Simovski, and S. A. Tretyakov, "Strong spatial dispersion in wire media in the very large wavelength limit," Phys. Rev. B 67, 113103 (2003).
[CrossRef]

Silveirinha, M. G.

M. G. Silveirinha and C. A. Fernandes, "Homogenization of 3-D-connected and nonconnected wire metamaterials," IEEE Trans. Microwave Theory Tech. 53, 1418-1430 (2005).
[CrossRef]

Simovski, C. R.

C. R. Simovski and P. A. Belov, "Low-frequency spatial dispersion in wire media," Phys. Rev. E 70, 046616 (2004).
[CrossRef]

P. A. Belov, R. Marqués, S. I. Maslovski, I. S. Nefedov, M. Silveirinha, C. R. Simovski, and S. A. Tretyakov, "Strong spatial dispersion in wire media in the very large wavelength limit," Phys. Rev. B 67, 113103 (2003).
[CrossRef]

Smith, D. L.

P. F. Loschialpo, D. L. Smith, D. W. Forester, F. J. Rachford, and J. Schelleng, "Electromagnetic waves focused by a negative-index planar lens," Phys. Rev. E 67, 025602 (2003).
[CrossRef]

Smith, D. R.

J. B. Pendry and D. R. Smith, "Reversing light with negative refraction," Phys. Today 57, 37-43 (2004).
[CrossRef]

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

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, D. C. Vier, N. Kroll, and S. Schultz, "Direct calculation of permeability and permittivity for a left-handed metamaterial," Appl. Phys. Lett. 77, 2246-2248 (2000).
[CrossRef]

D. R. Smith, W. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "A composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4187 (2000).
[CrossRef] [PubMed]

Sorolla, M.

F. Falcone, T. Lopetegi, M. A. G. Laso, J. D. Baena, J. Bonache, M. Beruete, R. Marques, F. Martin, and M. Sorolla, "Babinet principle applied to the design of metasurfaces and metamaterials," Phys. Rev. Lett. 93, 197401 (2004).
[CrossRef] [PubMed]

Soukoulis, C. M.

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

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

M. Bayindir, K. Aydin, E. Ozbay, P. Markos, and C. M. Soukoulis, "Transmission properties of composite metamaterials in free space," Appl. Phys. Lett. 81, 120-122 (2002).
[CrossRef]

P. Markos and C. M. Soukoulis, "Numerical studies of left-handed materials and arrays of split ring resonators," Phys. Rev. E 65, 036622 (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. Microwave Theory Tech. 47, 2075-2084 (1999).
[CrossRef]

J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, "Extremely low frequency plasmons in metallic mesostructures," Phys. Rev. Lett. 76, 4773-4776 (1996).
[CrossRef] [PubMed]

Tanielian, M.

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

Tretyakov, S. A.

P. A. Belov, R. Marqués, S. I. Maslovski, I. S. Nefedov, M. Silveirinha, C. R. Simovski, and S. A. Tretyakov, "Strong spatial dispersion in wire media in the very large wavelength limit," Phys. Rev. B 67, 113103 (2003).
[CrossRef]

S. I. Maslovski, S. A. Tretyakov, and P. A. Belov, "Wire media with negative effective permittivity: a quasi-static model," Microwave Opt. Technol. Lett. 35, 47-51 (2002).
[CrossRef]

I. V. Lindell, A. H. Sihvola, S. A. Tretyakov, and A. J. Viitanen, Electromagnetic Waves in Chiral and Bi-Isotropic Media (Artech House, 1994).

Veselago, V. G.

V. G. Veselago, "The electrodynamics of substances with simultaneously negative values of epsi and µ," Sov. Phys. Usp. 10, 509-514 (1968).
[CrossRef]

Vier, D. C.

D. R. Smith, W. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "A composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4187 (2000).
[CrossRef] [PubMed]

D. R. Smith, D. C. Vier, N. Kroll, and S. Schultz, "Direct calculation of permeability and permittivity for a left-handed metamaterial," Appl. Phys. Lett. 77, 2246-2248 (2000).
[CrossRef]

Viitanen, A. J.

I. V. Lindell, A. H. Sihvola, S. A. Tretyakov, and A. J. Viitanen, Electromagnetic Waves in Chiral and Bi-Isotropic Media (Artech House, 1994).

Walser, R. M.

R. M. Walser, "Electromagnetic metamaterials," in Complex Mediums II: Beyond Linear Isotropic Dielectrics, A. Lakhtakia, W. S. Weiglhofer, and I. J. Hodgkinson, eds., Proc. SPIE 4467, 1-15 (2001).
[CrossRef]

Yee, K. S.

K. S. Yee, "Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media," IEEE Trans. Antennas Propag. 14, 302-307 (1966).
[CrossRef]

Youngs, I.

J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, "Extremely low frequency plasmons in metallic mesostructures," Phys. Rev. Lett. 76, 4773-4776 (1996).
[CrossRef] [PubMed]

Ziolkowski, R. W.

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

Appl. Phys. Lett. (2)

M. Bayindir, K. Aydin, E. Ozbay, P. Markos, and C. M. Soukoulis, "Transmission properties of composite metamaterials in free space," Appl. Phys. Lett. 81, 120-122 (2002).
[CrossRef]

D. R. Smith, D. C. Vier, N. Kroll, and S. Schultz, "Direct calculation of permeability and permittivity for a left-handed metamaterial," Appl. Phys. Lett. 77, 2246-2248 (2000).
[CrossRef]

IEEE Trans. Antennas Propag. (3)

R. Marqués, F. Mesa, J. Martel, and F. Medina, "Comparative analysis of edge- and broadside-coupled split ring resonators for metamaterial design—theory and experiments," IEEE Trans. Antennas Propag. 51, 2572-2581 (2003).
[CrossRef]

K. S. Yee, "Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media," IEEE Trans. Antennas Propag. 14, 302-307 (1966).
[CrossRef]

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

IEEE Trans. Microwave Theory Tech. (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]

M. G. Silveirinha and C. A. Fernandes, "Homogenization of 3-D-connected and nonconnected wire metamaterials," IEEE Trans. Microwave Theory Tech. 53, 1418-1430 (2005).
[CrossRef]

J. Phys. Condens. Matter (1)

J. B. Pendry, "Calculating photonic band structure," J. Phys. Condens. Matter 8, 1085-1108 (1996).
[CrossRef]

Microwave Opt. Technol. Lett. (2)

S. I. Maslovski, S. A. Tretyakov, and P. A. Belov, "Wire media with negative effective permittivity: a quasi-static model," Microwave Opt. Technol. Lett. 35, 47-51 (2002).
[CrossRef]

M. M. I. Saadoun and N. Engheta, "A reciprocal phase shifter using novel pseudochiral or omega-medium," Microwave Opt. Technol. Lett. 5, 184-188 (1992).
[CrossRef]

Phys. Rep. (2)

D. J. Bergman, "The dielectric constant of a composite material—a problem in classical physics," Phys. Rep. 9, 377-407 (1978).
[CrossRef]

V. M. Shalaev, "Electromagnetic properties of small-particle composites," Phys. Rep. 272, 61-137 (1996).
[CrossRef]

Phys. Rev. B (4)

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]

P. A. Belov, R. Marqués, S. I. Maslovski, I. S. Nefedov, M. Silveirinha, C. R. Simovski, and S. A. Tretyakov, "Strong spatial dispersion in wire media in the very large wavelength limit," Phys. Rev. B 67, 113103 (2003).
[CrossRef]

A. L. Pokrovsky, "Analytical and numerical studies of wire-mesh metallic photonic crystals," Phys. Rev. B 69, 195108 (2004).
[CrossRef]

R. Marqués, F. Medina, and R. Rafii-El-Idrissi, "Role of bianisotropy in negative permeability and left-handed metamaterials," Phys. Rev. B 65, 144440 (2002).
[CrossRef]

Phys. Rev. E (4)

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

C. R. Simovski and P. A. Belov, "Low-frequency spatial dispersion in wire media," Phys. Rev. E 70, 046616 (2004).
[CrossRef]

P. Markos and C. M. Soukoulis, "Numerical studies of left-handed materials and arrays of split ring resonators," Phys. Rev. E 65, 036622 (2002).
[CrossRef]

P. F. Loschialpo, D. L. Smith, D. W. Forester, F. J. Rachford, and J. Schelleng, "Electromagnetic waves focused by a negative-index planar lens," Phys. Rev. E 67, 025602 (2003).
[CrossRef]

Phys. Rev. Lett. (5)

A. A. Houck, J. B. Brock, and I. L. Chuang, "Experimental observations of a left-handed material that obeys Snell's law," Phys. Rev. Lett. 90, 137401 (2003).
[CrossRef] [PubMed]

F. Falcone, T. Lopetegi, M. A. G. Laso, J. D. Baena, J. Bonache, M. Beruete, R. Marques, F. Martin, and M. Sorolla, "Babinet principle applied to the design of metasurfaces and metamaterials," Phys. Rev. Lett. 93, 197401 (2004).
[CrossRef] [PubMed]

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

D. R. Smith, W. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "A composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4187 (2000).
[CrossRef] [PubMed]

J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, "Extremely low frequency plasmons in metallic mesostructures," Phys. Rev. Lett. 76, 4773-4776 (1996).
[CrossRef] [PubMed]

Phys. Today (1)

J. B. Pendry and D. R. Smith, "Reversing light with negative refraction," Phys. Today 57, 37-43 (2004).
[CrossRef]

Proc. SPIE (1)

R. M. Walser, "Electromagnetic metamaterials," in Complex Mediums II: Beyond Linear Isotropic Dielectrics, A. Lakhtakia, W. S. Weiglhofer, and I. J. Hodgkinson, eds., Proc. SPIE 4467, 1-15 (2001).
[CrossRef]

Sov. Phys. Usp. (1)

V. G. Veselago, "The electrodynamics of substances with simultaneously negative values of epsi and µ," Sov. Phys. Usp. 10, 509-514 (1968).
[CrossRef]

Other (4)

A. Sihvola, "Electromagnetic emergence in metamaterials," in Advances in Electromagnetics of Complex Media and Metamaterials, S.Zouhdi, A.Sihvola, and M.Arsalane, eds., Vol. 89 of NATO Science Series II: Mathematics, Physics, and Chemistry (Kluwer Academic, 2003), pp. 3-17.
[CrossRef]

I. V. Lindell, A. H. Sihvola, S. A. Tretyakov, and A. J. Viitanen, Electromagnetic Waves in Chiral and Bi-Isotropic Media (Artech House, 1994).

W.S.Weiglhofer and A.Lakhtakia, eds., Introduction to Complex Mediums for Optics and Electromagnetic (SPIE, 2003).
[CrossRef]

J. A. Kong, Electromagnetic Wave Theory, 2nd ed. (Wiley, 1990), Sec. 2.6.

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Figures (12)

Fig. 1
Fig. 1

Definition of the averaged fields relative to a periodic unit cell. Maxwell’s curl equations in a cubic geometry suggests the averaged electric fields lie on the edges of a cubic lattice, while the averaged magnetic fields lie on the edges of a second interpenetrating cubic lattice, illustrated as the shaded cube.

Fig. 2
Fig. 2

Cubic lattice of dielectric spheres. (right) Close-up of a single unit cell of the array. The dielectric constant of the spheres is ε = 4 .

Fig. 3
Fig. 3

Effective permittivity and permeability of a cubic array of dielectric spheres, as a function of sphere radius. The dielectric constant of the spheres is ε = 4 . The permeability values all start from zero frequency at unity and disperse downward with increasing frequency. The permittivity values start from zero frequency in accordance with the Maxwell–Garnett formula and disperse upward with increasing frequency.

Fig. 4
Fig. 4

Dispersion diagram for the wire lattice. (inset) The wire lattice consists of an array of conducting wires with square cross section. The ratio of the side of the (square) wire to the unit-cell length is 0.03.

Fig. 5
Fig. 5

Retrieved values of permittivity (black curve) and permeability (gray curve) for the wire lattice. Because there are no propagating modes below the plasma frequency, the curves begin at the plasma frequency. The dots correspond to an S-parameter retrieval performed on an S-parameter calculation of the unit cell. The dashed curve corresponds to the ideal form for a plasmonic medium, Eq. (31).

Fig. 6
Fig. 6

(a) Diagram of the SRR and axes used in the simulations. w = 0.8 d , t = g = 0.04 d , and the linewidths are 0.08 d , where d is the unit-cell length. (b) SRRs arranged asymmetrically. A structure composed as shown exhibits electric, magnetic, and magnetoelectric resonant responses. (c) SRRs arranged symmetrically. The restoration of mirror plane symmetry along the z axis eliminates the magnetoelectric coupling. (d) This arrangement of SRRs reduces the resonant electric and magnetoelectric responses.

Fig. 7
Fig. 7

Dispersion curves for the SRR medium, with the SRRs oriented for magnetic response, solid black curve (inset, left); electric response, dashed black curve (inset, middle); and magnetoelectric response, solid gray curve (inset, right).

Fig. 8
Fig. 8

Retrieved permittivity and permeability for the SRR medium, oriented for primarily magnetic response.

Fig. 9
Fig. 9

Permittivity and permeability for the SRR medium, oriented for primarily electric response.

Fig. 10
Fig. 10

Material parameters ε y y and μ x x and the magnetoelectric coupling term κ y x computed as a function of frequency (in units of d c ). The shaded region indicates the onset of the bandgap for the SRR oriented in the bianisotropic configuration (rightmost inset, Fig. 7).

Fig. 11
Fig. 11

Dispersion curves for a combination wire–SRR medium. Shaded regions indicate frequency bandgaps. (inset) Depiction of the unit cell.

Fig. 12
Fig. 12

Retrieved material parameters ( ε z z , μ x x , and n) as a function of normalized frequency for the unit cell depicted in Fig. 11.

Equations (67)

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H ¯ i = d 1 H d x i ,
E ¯ i = d 1 E d x i ,
D ¯ i = d 2 D d s i ,
B ¯ i = d 2 B d s i .
× E = i ω B ,
× H = i ω D .
f ( x ) = u ( x ) exp ( i q x ) ,
f ( x + a ) = f ( x ) exp ( i q a ) .
E z y E y z = i ω B x ,
E x z E z x = i ω B y ,
E x y E y x = i ω B z .
d d E z ( d , d , z ) d z d d E z ( d , d , z ) d z d d E y ( d , y , d ) d y + d d E y ( d , y , d ) d y = i ω d d d y d d d z B x ( d , y , z ) ,
d d E x ( x , d , d ) d x d d E x ( x , d , d ) d x d d E z ( d , d , z ) d z + d d E z ( d , d , z ) d z = i ω d d d x d d d z B y ( x , d , z ) ,
d d E x ( x , d , d ) d x d d E x ( x , d , d ) d x d d E y ( d , y , d ) d y + d d E y ( d , y , d ) d y = i ω d d d x d d d y B z ( x , y , d ) .
E ¯ x ( 0 , ± d , d ) = ( 2 d ) 1 d d E x ( x , ± d , d ) d x ,
E ¯ y ( ± d , 0 , d ) = ( 2 d ) 1 d d E y ( ± d , y , d ) d y ,
B ¯ z ( 0 , 0 , ± d ) = ( 2 d ) 2 d d d x d d d y B z ( x , y , ± d ) .
E ¯ z ( d , d , 0 ) E ¯ z ( d , d , 0 ) E ¯ y ( d , 0 , d ) + E ¯ y ( d , 0 , d ) = i 2 d ω B ¯ x ( d , 0 , 0 ) ,
E ¯ x ( 0 , d , d ) E ¯ x ( 0 , d , d ) E ¯ z ( d , d , 0 ) + E ¯ z ( d , d , 0 ) = i 2 d ω B ¯ y ( 0 , d , 0 ) ,
E ¯ x ( 0 , d , d ) E ¯ x ( 0 , d , d ) E ¯ y ( d , 0 , d ) + E ¯ y ( d , 0 , d ) = i 2 d ω B ¯ z ( 0 , 0 , d ) .
× H = i ω D .
( H z y H y z ) = i ω D x ,
( H x z H z x ) = i ω D y ,
( H x y H y x ) = i ω D z .
0 2 d H z ( 0 , 2 d , z ) d z 0 2 d H z ( 0 , 0 , z ) d z 0 2 d H y ( 0 , y , 2 d ) d y + 0 2 d H y ( 0 , y , 0 ) d y = i ω 0 2 d d y 0 2 d d z D x ( 0 , y , z ) ,
0 2 d H x ( x , 0 , 2 d ) d x 0 2 d H x ( x , 0 , 0 ) d x 0 2 d H z ( 2 d , 0 , z ) d z + 0 2 d H z ( 0 , 0 , z ) d z = i ω 0 2 d d x 0 2 d d z D y ( x , 0 , z ) ,
0 2 d H x ( x , 2 d , 0 ) d x 0 2 d H x ( x , 0 , 0 ) d x 0 2 d H y ( 2 d , y , 0 ) d y + 0 2 d H y ( 0 , y , 0 ) d y = i ω 0 2 d d x 0 2 d d y D z ( x , y , 0 ) .
H ¯ x ( d , d ± d , 0 ) = ( 2 d ) 1 0 2 d H x ( x , d ± d , 0 ) d x ,
H ¯ y ( d ± d , d , 0 ) = ( 2 d ) 1 0 2 d H y ( d ± d , y , 0 ) d y ,
D ¯ z ( d , d , 0 ) = ( 2 d ) 2 0 2 d d x 0 2 d d y D z ( x , y , 0 ) .
H ¯ z ( 0 , 2 d , d ) H ¯ z ( 0 , 0 , d ) H ¯ y ( 0 , d , 2 d ) + H ¯ y ( 0 , d , 0 ) = i 2 d ω D ¯ x ( 0 , d , d ) ,
H ¯ x ( d , 0 , 2 d ) H ¯ x ( d , 0 , 0 ) H ¯ z ( 2 d , 0 , d ) + H ¯ z ( 0 , 0 , d ) = i 2 d ω D ¯ y ( d , 0 , d ) ,
H ¯ x ( d , 2 d , 0 ) H ¯ x ( d , 0 , 0 ) H ¯ y ( 2 d , d , 0 ) + H ¯ y ( 0 , d , 0 ) = i 2 d ω D ¯ z ( d , d , 0 ) .
D ¯ = ε 0 ε ¯ E ¯ i ε 0 μ 0 κ ¯ H ¯ ,
B ¯ = i ε 0 μ 0 κ ¯ T E ¯ + μ 0 μ ¯ H ¯ ,
E ¯ x sin ( q y d ) = d ω B ¯ z ,
H ¯ z sin ( q y d ) = d ω D ¯ x .
sin 2 ( q y d ) = ω 2 d 2 ε ¯ x μ ¯ z .
ε ¯ x = D ¯ x ( 0 , d , d ) E ¯ x ( 0 , d , d ) = ε 0 ( 2 d ) 2 0 2 d d z 0 2 d d y E x ( d , y , z ) ( 2 d ) 1 0 2 d d x E x ( x , d , d ) .
ε ¯ x = ε 0 sin ( q y d ) q y d .
μ ¯ z = μ 0 sin ( q y d ) q y d .
ω = q y ε 0 μ 0 .
E x ( x , y , z ) = E 0 u ( y ) exp ( i q y y ) ,
ε ¯ x = ε 0 0 2 d d y u ( y ) exp ( i q y y ) 2 d .
u ( y ) = m u m exp ( i ξ m y ) ,
ε ¯ x = ε 0 2 d m u m 0 2 d d y exp [ i ( q z + ξ m ) y ] = ε 0 u 0 sin ( q y d ) q y d + ε 0 d m 0 u m sin ( q y + ξ m ) d q y + ξ m ,
ε ¯ x = ε 0 ε ¯ x sin ( q y d ) q y d .
μ ¯ z = μ 0 μ ¯ z sin ( q y d ) q y d .
ω = c q y ε ¯ x μ ¯ z ,
ε ¯ = n z , μ ¯ = n z
ε MG = 1 + 2 ν 1 ( ε 1 ε + 2 ) 1 ν 1 ( ε 1 ε + 2 ) .
ε ( ω ) = 1 ω p 2 ω 2 .
μ ( ω ) = 1 F ω 2 ω 2 ω 0 2 ,
ε z z = α ,
ε y y = α + β ω 2 ( ω m 0 2 ω 2 ) ,
μ x x = 1 + γ ω 2 ( ω m 0 2 ω 2 ) ,
κ x y = δ ω ω m 0 2 ( ω m 0 2 ω 2 ) ,
B ¯ x = μ 0 μ x x H ¯ x + i ε 0 μ 0 κ y x E ¯ y ,
D ¯ z = ε z z E ¯ z .
D ¯ y = ε 0 ε y y E ¯ y i ε 0 μ 0 κ y x H ¯ x ,
B ¯ z = μ z z H ¯ z .
D ¯ y = ε 0 ε y y E ¯ y i ε 0 μ 0 κ y x H ¯ x ,
B ¯ x = μ 0 μ x x H ¯ x + i ε 0 μ 0 κ y x E ¯ y .
n eff = ε y y μ x x κ y x 2 .
κ y x = i z ( 1 ε 0 D ¯ y E ¯ y ε y y ) = i z ( ς ε y y ) ,
κ y x = i 1 z ( 1 μ 0 B ¯ x H ¯ x μ x x ) = i 1 z ( ξ μ x x ) ,
κ y x = i ( ς ξ n eff 2 ) ( z ς ξ z ) ,

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