Abstract

We apply the equivalent theory to orthorhombic anisotropic materials and provide a general unit-cell design criterion for achieving a length-independent retrieval of the effective material parameters from a single layer of unit cells. We introduce a graphical retrieval method and phase unwrapping techniques. The graphical method utilizes the linear regression technique. Our method can reduce the uncertainty of experimental measurements and the ambiguity of phase unwrapping. Moreover, the graphical method can simultaneously determine the bulk values of the six effective material parameters, permittivity and permeability tensors, from a single layer of unit cells.

© 2010 Optical Society of America

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  1. N. Fang, H. Lee, C. Sun, and X. Zhang, "Sub-diffraction-limited optical imaging with a silver superlens," Science 308, 534-537 (2005).
    [CrossRef] [PubMed]
  2. J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, "An optical cloak made of dielectrics," Nat. Mater. 8, 568-571 (2009).
    [CrossRef] [PubMed]
  3. S. Feng and K. Halterman, "Parametrically shielding electromagnetic fields by nonlinear metamaterials," Phys. Rev. Lett. 100, 063901 (2008).
    [CrossRef] [PubMed]
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    [CrossRef]
  5. J. Valentine, S. Zhang, T. Zentgraf, E. U. Avila, D. A. Genov, G. Bartal, and X. Zhang, "Three-dimensional optical metamaterial with a negative refractive index," Nature (London) 455, 376-380 (2008).
    [CrossRef] [PubMed]
  6. N. Liu, H. Guo, L. Fu, S. Kaiser, H. Schweizer, and H. Giessen, "Three-dimensional photonic metamaterials at optical frequencies," Nat. Mater. 7, 31-37 (2008).
    [CrossRef]
  7. S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, "Experimental demonstration of near-infrared negative-index metamaterials," Phys. Rev. Lett. 95, 137404 (2005).
    [CrossRef] [PubMed]
  8. 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]
  9. A. Andryieuski, R. Malureanu, and A. V. Lavrinenko, "Wave propagation retrieval method for metamaterials: Unambiguous restoration of effective parameters," Phys. Rev. B 80, 193101 (2009).
    [CrossRef]
  10. A. Herpin, "Calcul du pouvoir réflecteur d’un système stratifié quelconque," Compt. Rend. 225, 182-183 (1947).
  11. D. R. Smith, S. Schultz, P. Markoš, and C. M. Soukoulis, "Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients," Phys. Rev. B 65, 195104 (2002).
    [CrossRef]
  12. T. Koschny, P. Markoš, E. N. Economou, D. R. Smith, D. C. Vier, and C. M. Soukoulis, "Impact of inherent periodic structure on effective medium description of left-handed and related metamaterials," Phys. Rev. B 71, 245105 (2005).
    [CrossRef]
  13. C. Menzel, C. Rockstuhl, T. Paul, and F. Lederer, "Retrieving effective parameters for metamaterials at oblique incidence," Phys. Rev. B 77, 195328 (2008).
    [CrossRef]
  14. X. Chen, T. M. Grzegorczyk, B.-I. Wu, J. Pacheco, Jr., and J. A. Kong, "Robust method to retrieve the constitutive effective parameters of metamaterials," Phys. Rev. E 70, 016608 (2004).
    [CrossRef]
  15. L. I. Epstein, "The design of optical filters," J. Opt. Soc. Am. 42, 806-810 (1952).
    [CrossRef]
  16. A. Lakhtakia, "Constraints on effective constitutive parameters of certain bianisotropic laminated composite materials," Electromagnetics 29,508-514 (2009).
    [CrossRef]
  17. M. J. Roberts, S. Feng, M. Moran, and L. Johnson, "Effective permittivity near zero in nanolaminates of silver and amorphous polycarbonate," J. Nanophoton. 4, 043511 (2010).
    [CrossRef]
  18. M. W McCall, A. Lakhtakia, and W. S Weiglhofer, "The negative index of refraction demystified," Eur. J. Phys. 23, 353-359 (2002).
    [CrossRef]

2010 (1)

M. J. Roberts, S. Feng, M. Moran, and L. Johnson, "Effective permittivity near zero in nanolaminates of silver and amorphous polycarbonate," J. Nanophoton. 4, 043511 (2010).
[CrossRef]

2009 (3)

A. Lakhtakia, "Constraints on effective constitutive parameters of certain bianisotropic laminated composite materials," Electromagnetics 29,508-514 (2009).
[CrossRef]

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, "An optical cloak made of dielectrics," Nat. Mater. 8, 568-571 (2009).
[CrossRef] [PubMed]

A. Andryieuski, R. Malureanu, and A. V. Lavrinenko, "Wave propagation retrieval method for metamaterials: Unambiguous restoration of effective parameters," Phys. Rev. B 80, 193101 (2009).
[CrossRef]

2008 (4)

S. Feng and K. Halterman, "Parametrically shielding electromagnetic fields by nonlinear metamaterials," Phys. Rev. Lett. 100, 063901 (2008).
[CrossRef] [PubMed]

J. Valentine, S. Zhang, T. Zentgraf, E. U. Avila, D. A. Genov, G. Bartal, and X. Zhang, "Three-dimensional optical metamaterial with a negative refractive index," Nature (London) 455, 376-380 (2008).
[CrossRef] [PubMed]

N. Liu, H. Guo, L. Fu, S. Kaiser, H. Schweizer, and H. Giessen, "Three-dimensional photonic metamaterials at optical frequencies," Nat. Mater. 7, 31-37 (2008).
[CrossRef]

C. Menzel, C. Rockstuhl, T. Paul, and F. Lederer, "Retrieving effective parameters for metamaterials at oblique incidence," Phys. Rev. B 77, 195328 (2008).
[CrossRef]

2005 (4)

T. Koschny, P. Markoš, E. N. Economou, D. R. Smith, D. C. Vier, and C. M. Soukoulis, "Impact of inherent periodic structure on effective medium description of left-handed and related metamaterials," Phys. Rev. B 71, 245105 (2005).
[CrossRef]

S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, "Experimental demonstration of near-infrared negative-index metamaterials," Phys. Rev. Lett. 95, 137404 (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]

N. Fang, H. Lee, C. Sun, and X. Zhang, "Sub-diffraction-limited optical imaging with a silver superlens," Science 308, 534-537 (2005).
[CrossRef] [PubMed]

2004 (1)

X. Chen, T. M. Grzegorczyk, B.-I. Wu, J. Pacheco, Jr., and J. A. Kong, "Robust method to retrieve the constitutive effective parameters of metamaterials," Phys. Rev. E 70, 016608 (2004).
[CrossRef]

2002 (2)

M. W McCall, A. Lakhtakia, and W. S Weiglhofer, "The negative index of refraction demystified," Eur. J. Phys. 23, 353-359 (2002).
[CrossRef]

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

1952 (1)

1947 (1)

A. Herpin, "Calcul du pouvoir réflecteur d’un système stratifié quelconque," Compt. Rend. 225, 182-183 (1947).

Andryieuski, A.

A. Andryieuski, R. Malureanu, and A. V. Lavrinenko, "Wave propagation retrieval method for metamaterials: Unambiguous restoration of effective parameters," Phys. Rev. B 80, 193101 (2009).
[CrossRef]

Avila, E. U.

J. Valentine, S. Zhang, T. Zentgraf, E. U. Avila, D. A. Genov, G. Bartal, and X. Zhang, "Three-dimensional optical metamaterial with a negative refractive index," Nature (London) 455, 376-380 (2008).
[CrossRef] [PubMed]

Bartal, G.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, "An optical cloak made of dielectrics," Nat. Mater. 8, 568-571 (2009).
[CrossRef] [PubMed]

J. Valentine, S. Zhang, T. Zentgraf, E. U. Avila, D. A. Genov, G. Bartal, and X. Zhang, "Three-dimensional optical metamaterial with a negative refractive index," Nature (London) 455, 376-380 (2008).
[CrossRef] [PubMed]

Brueck, S. R. J.

S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, "Experimental demonstration of near-infrared negative-index metamaterials," Phys. Rev. Lett. 95, 137404 (2005).
[CrossRef] [PubMed]

Cai, W.

Chen, X.

X. Chen, T. M. Grzegorczyk, B.-I. Wu, J. Pacheco, Jr., and J. A. Kong, "Robust method to retrieve the constitutive effective parameters of metamaterials," Phys. Rev. E 70, 016608 (2004).
[CrossRef]

Chettiar, U. K.

Drachev, V. P.

Economou, E. N.

T. Koschny, P. Markoš, E. N. Economou, D. R. Smith, D. C. Vier, and C. M. Soukoulis, "Impact of inherent periodic structure on effective medium description of left-handed and related metamaterials," Phys. Rev. B 71, 245105 (2005).
[CrossRef]

Epstein, L. I.

Fan, W.

S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, "Experimental demonstration of near-infrared negative-index metamaterials," Phys. Rev. Lett. 95, 137404 (2005).
[CrossRef] [PubMed]

Fang, N.

N. Fang, H. Lee, C. Sun, and X. Zhang, "Sub-diffraction-limited optical imaging with a silver superlens," Science 308, 534-537 (2005).
[CrossRef] [PubMed]

Feng, S.

M. J. Roberts, S. Feng, M. Moran, and L. Johnson, "Effective permittivity near zero in nanolaminates of silver and amorphous polycarbonate," J. Nanophoton. 4, 043511 (2010).
[CrossRef]

S. Feng and K. Halterman, "Parametrically shielding electromagnetic fields by nonlinear metamaterials," Phys. Rev. Lett. 100, 063901 (2008).
[CrossRef] [PubMed]

Fu, L.

N. Liu, H. Guo, L. Fu, S. Kaiser, H. Schweizer, and H. Giessen, "Three-dimensional photonic metamaterials at optical frequencies," Nat. Mater. 7, 31-37 (2008).
[CrossRef]

Genov, D. A.

J. Valentine, S. Zhang, T. Zentgraf, E. U. Avila, D. A. Genov, G. Bartal, and X. Zhang, "Three-dimensional optical metamaterial with a negative refractive index," Nature (London) 455, 376-380 (2008).
[CrossRef] [PubMed]

Giessen, H.

N. Liu, H. Guo, L. Fu, S. Kaiser, H. Schweizer, and H. Giessen, "Three-dimensional photonic metamaterials at optical frequencies," Nat. Mater. 7, 31-37 (2008).
[CrossRef]

Grzegorczyk, T. M.

X. Chen, T. M. Grzegorczyk, B.-I. Wu, J. Pacheco, Jr., and J. A. Kong, "Robust method to retrieve the constitutive effective parameters of metamaterials," Phys. Rev. E 70, 016608 (2004).
[CrossRef]

Guo, H.

N. Liu, H. Guo, L. Fu, S. Kaiser, H. Schweizer, and H. Giessen, "Three-dimensional photonic metamaterials at optical frequencies," Nat. Mater. 7, 31-37 (2008).
[CrossRef]

Halterman, K.

S. Feng and K. Halterman, "Parametrically shielding electromagnetic fields by nonlinear metamaterials," Phys. Rev. Lett. 100, 063901 (2008).
[CrossRef] [PubMed]

Herpin, A.

A. Herpin, "Calcul du pouvoir réflecteur d’un système stratifié quelconque," Compt. Rend. 225, 182-183 (1947).

Johnson, L.

M. J. Roberts, S. Feng, M. Moran, and L. Johnson, "Effective permittivity near zero in nanolaminates of silver and amorphous polycarbonate," J. Nanophoton. 4, 043511 (2010).
[CrossRef]

Kaiser, S.

N. Liu, H. Guo, L. Fu, S. Kaiser, H. Schweizer, and H. Giessen, "Three-dimensional photonic metamaterials at optical frequencies," Nat. Mater. 7, 31-37 (2008).
[CrossRef]

Kildishev, A. V.

Kong, J. A.

X. Chen, T. M. Grzegorczyk, B.-I. Wu, J. Pacheco, Jr., and J. A. Kong, "Robust method to retrieve the constitutive effective parameters of metamaterials," Phys. Rev. E 70, 016608 (2004).
[CrossRef]

Koschny, T.

T. Koschny, P. Markoš, E. N. Economou, D. R. Smith, D. C. Vier, and C. M. Soukoulis, "Impact of inherent periodic structure on effective medium description of left-handed and related metamaterials," Phys. Rev. B 71, 245105 (2005).
[CrossRef]

Lakhtakia, A.

A. Lakhtakia, "Constraints on effective constitutive parameters of certain bianisotropic laminated composite materials," Electromagnetics 29,508-514 (2009).
[CrossRef]

M. W McCall, A. Lakhtakia, and W. S Weiglhofer, "The negative index of refraction demystified," Eur. J. Phys. 23, 353-359 (2002).
[CrossRef]

Lavrinenko, A. V.

A. Andryieuski, R. Malureanu, and A. V. Lavrinenko, "Wave propagation retrieval method for metamaterials: Unambiguous restoration of effective parameters," Phys. Rev. B 80, 193101 (2009).
[CrossRef]

Lederer, F.

C. Menzel, C. Rockstuhl, T. Paul, and F. Lederer, "Retrieving effective parameters for metamaterials at oblique incidence," Phys. Rev. B 77, 195328 (2008).
[CrossRef]

Lee, H.

N. Fang, H. Lee, C. Sun, and X. Zhang, "Sub-diffraction-limited optical imaging with a silver superlens," Science 308, 534-537 (2005).
[CrossRef] [PubMed]

Li, J.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, "An optical cloak made of dielectrics," Nat. Mater. 8, 568-571 (2009).
[CrossRef] [PubMed]

Liu, N.

N. Liu, H. Guo, L. Fu, S. Kaiser, H. Schweizer, and H. Giessen, "Three-dimensional photonic metamaterials at optical frequencies," Nat. Mater. 7, 31-37 (2008).
[CrossRef]

Malloy, K. J.

S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, "Experimental demonstration of near-infrared negative-index metamaterials," Phys. Rev. Lett. 95, 137404 (2005).
[CrossRef] [PubMed]

Malureanu, R.

A. Andryieuski, R. Malureanu, and A. V. Lavrinenko, "Wave propagation retrieval method for metamaterials: Unambiguous restoration of effective parameters," Phys. Rev. B 80, 193101 (2009).
[CrossRef]

Markoš, P.

T. Koschny, P. Markoš, E. N. Economou, D. R. Smith, D. C. Vier, and C. M. Soukoulis, "Impact of inherent periodic structure on effective medium description of left-handed and related metamaterials," Phys. Rev. B 71, 245105 (2005).
[CrossRef]

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

McCall, M. W

M. W McCall, A. Lakhtakia, and W. S Weiglhofer, "The negative index of refraction demystified," Eur. J. Phys. 23, 353-359 (2002).
[CrossRef]

Menzel, C.

C. Menzel, C. Rockstuhl, T. Paul, and F. Lederer, "Retrieving effective parameters for metamaterials at oblique incidence," Phys. Rev. B 77, 195328 (2008).
[CrossRef]

Moran, M.

M. J. Roberts, S. Feng, M. Moran, and L. Johnson, "Effective permittivity near zero in nanolaminates of silver and amorphous polycarbonate," J. Nanophoton. 4, 043511 (2010).
[CrossRef]

Osgood, R. M.

S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, "Experimental demonstration of near-infrared negative-index metamaterials," Phys. Rev. Lett. 95, 137404 (2005).
[CrossRef] [PubMed]

Pacheco, J.

X. Chen, T. M. Grzegorczyk, B.-I. Wu, J. Pacheco, Jr., and J. A. Kong, "Robust method to retrieve the constitutive effective parameters of metamaterials," Phys. Rev. E 70, 016608 (2004).
[CrossRef]

Panoiu, N. C.

S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, "Experimental demonstration of near-infrared negative-index metamaterials," Phys. Rev. Lett. 95, 137404 (2005).
[CrossRef] [PubMed]

Paul, T.

C. Menzel, C. Rockstuhl, T. Paul, and F. Lederer, "Retrieving effective parameters for metamaterials at oblique incidence," Phys. Rev. B 77, 195328 (2008).
[CrossRef]

Roberts, M. J.

M. J. Roberts, S. Feng, M. Moran, and L. Johnson, "Effective permittivity near zero in nanolaminates of silver and amorphous polycarbonate," J. Nanophoton. 4, 043511 (2010).
[CrossRef]

Rockstuhl, C.

C. Menzel, C. Rockstuhl, T. Paul, and F. Lederer, "Retrieving effective parameters for metamaterials at oblique incidence," Phys. Rev. B 77, 195328 (2008).
[CrossRef]

Sarychev, A. K.

Schultz, S.

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

Schweizer, H.

N. Liu, H. Guo, L. Fu, S. Kaiser, H. Schweizer, and H. Giessen, "Three-dimensional photonic metamaterials at optical frequencies," Nat. Mater. 7, 31-37 (2008).
[CrossRef]

Shalaev, V. M.

Smith, D. R.

T. Koschny, P. Markoš, E. N. Economou, D. R. Smith, D. C. Vier, and C. M. Soukoulis, "Impact of inherent periodic structure on effective medium description of left-handed and related metamaterials," Phys. Rev. B 71, 245105 (2005).
[CrossRef]

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

Soukoulis, C. M.

T. Koschny, P. Markoš, E. N. Economou, D. R. Smith, D. C. Vier, and C. M. Soukoulis, "Impact of inherent periodic structure on effective medium description of left-handed and related metamaterials," Phys. Rev. B 71, 245105 (2005).
[CrossRef]

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

Sun, C.

N. Fang, H. Lee, C. Sun, and X. Zhang, "Sub-diffraction-limited optical imaging with a silver superlens," Science 308, 534-537 (2005).
[CrossRef] [PubMed]

Valentine, J.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, "An optical cloak made of dielectrics," Nat. Mater. 8, 568-571 (2009).
[CrossRef] [PubMed]

J. Valentine, S. Zhang, T. Zentgraf, E. U. Avila, D. A. Genov, G. Bartal, and X. Zhang, "Three-dimensional optical metamaterial with a negative refractive index," Nature (London) 455, 376-380 (2008).
[CrossRef] [PubMed]

Vier, D. C.

T. Koschny, P. Markoš, E. N. Economou, D. R. Smith, D. C. Vier, and C. M. Soukoulis, "Impact of inherent periodic structure on effective medium description of left-handed and related metamaterials," Phys. Rev. B 71, 245105 (2005).
[CrossRef]

Weiglhofer, W. S

M. W McCall, A. Lakhtakia, and W. S Weiglhofer, "The negative index of refraction demystified," Eur. J. Phys. 23, 353-359 (2002).
[CrossRef]

Wu, B.-I.

X. Chen, T. M. Grzegorczyk, B.-I. Wu, J. Pacheco, Jr., and J. A. Kong, "Robust method to retrieve the constitutive effective parameters of metamaterials," Phys. Rev. E 70, 016608 (2004).
[CrossRef]

Yuan, H.-K.

Zentgraf, T.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, "An optical cloak made of dielectrics," Nat. Mater. 8, 568-571 (2009).
[CrossRef] [PubMed]

J. Valentine, S. Zhang, T. Zentgraf, E. U. Avila, D. A. Genov, G. Bartal, and X. Zhang, "Three-dimensional optical metamaterial with a negative refractive index," Nature (London) 455, 376-380 (2008).
[CrossRef] [PubMed]

Zhang, S.

J. Valentine, S. Zhang, T. Zentgraf, E. U. Avila, D. A. Genov, G. Bartal, and X. Zhang, "Three-dimensional optical metamaterial with a negative refractive index," Nature (London) 455, 376-380 (2008).
[CrossRef] [PubMed]

S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, "Experimental demonstration of near-infrared negative-index metamaterials," Phys. Rev. Lett. 95, 137404 (2005).
[CrossRef] [PubMed]

Zhang, X.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, "An optical cloak made of dielectrics," Nat. Mater. 8, 568-571 (2009).
[CrossRef] [PubMed]

J. Valentine, S. Zhang, T. Zentgraf, E. U. Avila, D. A. Genov, G. Bartal, and X. Zhang, "Three-dimensional optical metamaterial with a negative refractive index," Nature (London) 455, 376-380 (2008).
[CrossRef] [PubMed]

N. Fang, H. Lee, C. Sun, and X. Zhang, "Sub-diffraction-limited optical imaging with a silver superlens," Science 308, 534-537 (2005).
[CrossRef] [PubMed]

Compt. Rend. (1)

A. Herpin, "Calcul du pouvoir réflecteur d’un système stratifié quelconque," Compt. Rend. 225, 182-183 (1947).

Electromagnetics (1)

A. Lakhtakia, "Constraints on effective constitutive parameters of certain bianisotropic laminated composite materials," Electromagnetics 29,508-514 (2009).
[CrossRef]

Eur. J. Phys. (1)

M. W McCall, A. Lakhtakia, and W. S Weiglhofer, "The negative index of refraction demystified," Eur. J. Phys. 23, 353-359 (2002).
[CrossRef]

J. Nanophoton. (1)

M. J. Roberts, S. Feng, M. Moran, and L. Johnson, "Effective permittivity near zero in nanolaminates of silver and amorphous polycarbonate," J. Nanophoton. 4, 043511 (2010).
[CrossRef]

J. Opt. Soc. Am. (1)

Nat. Mater. (2)

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, "An optical cloak made of dielectrics," Nat. Mater. 8, 568-571 (2009).
[CrossRef] [PubMed]

N. Liu, H. Guo, L. Fu, S. Kaiser, H. Schweizer, and H. Giessen, "Three-dimensional photonic metamaterials at optical frequencies," Nat. Mater. 7, 31-37 (2008).
[CrossRef]

Nature (London) (1)

J. Valentine, S. Zhang, T. Zentgraf, E. U. Avila, D. A. Genov, G. Bartal, and X. Zhang, "Three-dimensional optical metamaterial with a negative refractive index," Nature (London) 455, 376-380 (2008).
[CrossRef] [PubMed]

Opt. Lett. (1)

Phys. Rev. B (4)

A. Andryieuski, R. Malureanu, and A. V. Lavrinenko, "Wave propagation retrieval method for metamaterials: Unambiguous restoration of effective parameters," Phys. Rev. B 80, 193101 (2009).
[CrossRef]

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

T. Koschny, P. Markoš, E. N. Economou, D. R. Smith, D. C. Vier, and C. M. Soukoulis, "Impact of inherent periodic structure on effective medium description of left-handed and related metamaterials," Phys. Rev. B 71, 245105 (2005).
[CrossRef]

C. Menzel, C. Rockstuhl, T. Paul, and F. Lederer, "Retrieving effective parameters for metamaterials at oblique incidence," Phys. Rev. B 77, 195328 (2008).
[CrossRef]

Phys. Rev. E (1)

X. Chen, T. M. Grzegorczyk, B.-I. Wu, J. Pacheco, Jr., and J. A. Kong, "Robust method to retrieve the constitutive effective parameters of metamaterials," Phys. Rev. E 70, 016608 (2004).
[CrossRef]

Phys. Rev. Lett. (2)

S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, "Experimental demonstration of near-infrared negative-index metamaterials," Phys. Rev. Lett. 95, 137404 (2005).
[CrossRef] [PubMed]

S. Feng and K. Halterman, "Parametrically shielding electromagnetic fields by nonlinear metamaterials," Phys. Rev. Lett. 100, 063901 (2008).
[CrossRef] [PubMed]

Science (1)

N. Fang, H. Lee, C. Sun, and X. Zhang, "Sub-diffraction-limited optical imaging with a silver superlens," Science 308, 534-537 (2005).
[CrossRef] [PubMed]

Other (1)

F. Capolino, ed., Theory and phenomena of metamaterials (CRC Press, Taylor and Francis Group, New York, 2009).
[CrossRef]

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Figures (6)

Fig. 1.
Fig. 1.

A schematic shows how to construct a symmetric unit cell, which is composed of two original asymmetric unit cells.

Fig. 2.
Fig. 2.

Retrieval lines at the frequency 30THz calculated from the scattering matrix of a single layer of unit cells at seven incidence angles (denoted as circles) uniformly distributed from 0 to 30 degree. The horizontal axis is sin2 θ. Background medium is vacuum. Top panels: real part. Bottom panels: imaginary part.

Fig. 3.
Fig. 3.

Branch number predicted by the method-3 [Eq. (20)] vs. frequency. Since dλ, most of the frequencies are in the fundamental branch m = 0. The m = −1 indicates the frequencies of the negative refractive index [n < 0, see Fig. 4(a)]. The m = 1 corresponds to the frequencies of the high valves of the positive refractive index [see Fig. 4(a)]. Top: TM polarization. Bottom: TE polarization.

Fig. 4.
Fig. 4.

Effective refractive index, impedance, and admittance retrieved from a single unitcell layer. Top panels: real part. Bottom panels: imaginary part.

Fig. 5.
Fig. 5.

In-plane values of the retrieved material parameters. Upper panels: real parts. Lower panels: imaginary parts.

Fig. 6.
Fig. 6.

The effective ε ¯ z (right panels) and µ ¯ z (left panels). Upper panels: real parts. Lower panels: imaginary parts.

Equations (35)

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ε = n = ( ε nx 0 0 0 ε ny 0 0 0 ε nz ) , μ = n = ( μ nx 0 0 0 μ ny 0 0 0 μ nz ) ,
× ( μ = n 1 · × E ) = k 0 2 ( ε = n · E ) ,
× ( ε = n 1 · × H ) = k 0 2 ( μ = n · H ) ,
M 11 = M 22 = cos ψ e ,
M 12 = sin ψ e i 𝒵 e , M 21 = i 𝒵 e sin ψ e ,
( cos ψ e 1 i 𝒵 e sin ψ e i 𝒵 e sin ψ e cos ψ e ) N = ( cos ( N ψ e ) 1 i 𝒵 e sin ( N ψ e ) i 𝒵 e sin ( N ψ e ) cos ( N ψ e ) ) .
TM : 𝒵 e = k z k 0 ε x , k z 2 = k 0 2 ε x μ y ε x ε z k x 2 ,
TE : 𝒴 e = k z k 0 μ x , k z 2 = k 0 2 ε y μ x μ x μ z k x 2 ,
( H y i E x i ) = M ( H y o E x o ) = ( M 11 M 12 M 21 M 22 ) ( H y o E x o ) ,
M 11 = M 22 = [ ( 1 + S 21 ) ( 1 S 12 ) + S 11 S 22 ] 2 S 11 ,
M 12 = [ ( 1 + S 21 ) ( 1 + S 12 ) S 11 S 22 ] 2 S 11 𝒵 o ,
M 21 = 𝒵 i 2 S 11 [ ( 1 S 21 ) ( 1 S 12 ) S 11 S 22 ] .
cos ( K e d ) = M 11 = cos ( k 1 z d 1 ) cos ( k 2 z d 2 ) η + sin ( k 1 z d 1 ) sin ( k 2 z d 2 ) ,
𝒵 e 2 = M 21 M 12 = 𝒵 1 2 sin ( k 1 z d 1 ) cos ( k 2 z d 2 ) + η + cos ( k 1 z d 1 ) sin ( k 2 z d 2 ) η sin ( k 2 z d 2 ) sin ( k 1 z d 1 ) cos ( k 2 z d 2 ) + η + cos ( k 1 z d 1 ) sin ( k 2 z d 2 ) + η sin ( k 2 z d 2 ) ,
η ± = 1 2 ( ε 2 x k 1 z ε 1 x k 2 z ± ε 1 x k 2 z ε 2 x k 1 z ) for TM ,
η ± = 1 2 ( μ 2 x k 1 z μ 1 x k 2 z ± μ 1 x k 2 z μ 2 x k 1 z ) for TE .
TM : K e 2 k 0 2 = ε ¯ x μ ¯ y ε ¯ x ε ¯ z k x 2 k 0 2 , 𝒵 e 2 = μ ¯ y ε ¯ x k x 2 k 0 2 ε ¯ x ε ¯ z ,
TE : K e 2 k 0 2 = ε ¯ y μ ¯ x μ ¯ x μ ¯ z k x 2 k 0 2 , 𝒴 e 2 = ε ¯ y μ ¯ x k x 2 k 0 2 μ ¯ x μ ¯ z .
ε ¯ p = d 1 ε 1 p + d 2 ε 2 p d , ε ¯ z = d ε 1 z ε 2 z d 1 ε 2 z + d 2 ε 1 z ,
μ ¯ p = d 1 μ 1 p + d 2 μ 2 p d , μ ¯ z = d μ 1 z μ 2 z d 1 μ 2 z + d 2 μ 1 z ,
TM : Y M = ε ¯ x μ ¯ y ε ¯ x ε ¯ z X , Y m = μ ¯ y ε ¯ x X ε ¯ x ε ¯ z ,
TE : Y E = ε ¯ y μ ¯ x μ ¯ x μ ¯ z X , Y e = ε ¯ y μ ¯ x X μ ¯ x μ ¯ z ,
X ε b μ b sin 2 θ , Y m μ b ε b S cos 2 θ , Y e ε b μ b S cos 2 θ , S ( 1 S 21 ) 2 S 11 2 ( 1 + S 21 ) 2 S 11 2 ,
Y M = Y E { 1 k 0 d [ 2 m π ± cos 1 ( 1 S 21 2 + S 11 2 2 S 11 ) ] } 2 , m = 0 , ± 1 , ± 2 , ,
TM : n m = ± Y M 0 , Z = ± Y m 0 , ε ¯ x = n m Z , μ ¯ y = n m Z , ε ¯ z = ε ¯ x S M ,
TE : n e = ± Y E 0 , Y = ± Y e 0 , ε ¯ y = n e Y , μ ¯ x = n e Y , μ ¯ z = μ ¯ x S E ,
A 1 S 21 2 + S 11 2 2 S 11 = cos φ = cos ( φ r + i φ i ) = cos φ r cosh φ i i sin φ r sinh φ i ,
φ = { cos 1 A if ( A ) < 0 2 π cos 1 A if ( A ) > 0 .
TM : min { Y M 0 ( m ) Y m 0 S M ( m ) S m : m = 0 , ± 1 , ± 2 , } ,
TE : min { Y E 0 ( m ) Y e 0 S E ( m ) S e : m = 0 , ± 1 , ± 2 , } .
TM : min { Y M 0 ( m ) S m Y m 0 S M ( m ) : m = 0 , ± 1 , ± 2 , } ,
TE : min { Y E 0 ( m ) S e Y e 0 S E ( m ) : m = 0 , ± 1 , ± 2 , } .
TM : min { Y M 0 ( m ) S M ( m ) Y m 0 S m : m = 0 , ± 1 , ± 2 , } ,
TE : min { Y E 0 ( m ) S E ( m ) Y e 0 S e : m = 0 , ± 1 , ± 2 , } .
ε = 1 f ep 2 f 2 f er 2 + i γ f , μ = 1 f mp 2 f 2 f mr 2 + i γ f ,

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