Abstract

We studied both experimentally and theoretically the influence of the distance between adjacent cut-wire-pair layers on the magnetic and the electric resonances in the microwave-frequency regime. Besides the dependence on the separation between cut-wire pairs, along the electric-field direction, the electric resonance strongly depends on the distance between cut-wire-pair layers, while the magnetic resonance is almost unchanged. This contrast can be understood by the difference in the distribution of induced-charge density and in the direction of the induced current between the electric and magnetic resonances. A simple model is proposed to simulate our experimental results and the simulation results are in good agreement with the experiment. This result provides important information in obtaining left-handed behavior when the cut-wire pairs are combined with the continuous wire.

© 2008 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
  4. 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]
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    [CrossRef]
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    [CrossRef]
  7. J. Huangfu, L. Ran, H. Chen, X. M. Zhang, K. Chen, T. M. Grzegorczyk, and J. A. Kong, “Experimental confirmation of negative refractive index of a metamaterial composed of Ω-like metallic patterns,” Appl. Phys. Lett. 84, 1537–1539 (2004).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]

2007 (3)

2006 (4)

C. M. Soukoulis, M. Kafesaki, and E. N. Economou, “Negative-Index Materials: new frontiers in optics,” Adv. Mater. 18, 1941–1952 (2006).
[CrossRef]

J. Zhou, L. Zhang, G. Tuttle, T. Koschny, and C. M. Soukoulis, “Negative index materials using simple short wire pairs,” Phys. Rev. B 73, 041101(R) (2006).
[CrossRef]

K. Guven, M. D. Caliskan, and E. Ozbay, “Experimental observation of left-handed transmission in a bilayer metamaterial under normal-to-plane propagation,” Opt. Express 14, 8685–8693 (2006).
[CrossRef] [PubMed]

J. Zhou, E. N. Economon, T. Koschny, and C. M. Soukoulis, “Unifying approach to left-handed material design,” Opt. Lett. 31, 3620–3622 (2006).
[CrossRef] [PubMed]

2005 (3)

2004 (5)

T. Koschny, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Effective medium theory of left-handed materials,” Phys. Rev. Lett. 93, 107402 (2004).
[CrossRef] [PubMed]

N. Katsarahis, T. Koschny, M. Kafesaki, E. N. Economou, E. Ozbay, and C. M. Soukoulis, “Left- and right-handed transmission peaks near the magnetic resonance frequency in composite metamaterials,” Phys. Rev. B 70, 201101, (2004).
[CrossRef]

K. Aydin, K. Guven, N. Katsaraki, C. M. Soukoulis, and E. Ozbay, “Effect of disorder on magnetic resonance band gap of split-ring resonator structures,” Opt. Express 12, 5896–5901 (2004).
[CrossRef] [PubMed]

H. Chen, L. Ran, J. Huangfu, X. Zhang, K. Chen, T. M. Grzegorczyk, and J. A. Kong, “Left-handed materials composed of only S-shaped resonators,” Phys. Rev. E 70, 057605 (2004).
[CrossRef]

J. Huangfu, L. Ran, H. Chen, X. M. Zhang, K. Chen, T. M. Grzegorczyk, and J. A. Kong, “Experimental confirmation of negative refractive index of a metamaterial composed of Ω-like metallic patterns,” Appl. Phys. Lett. 84, 1537–1539 (2004).
[CrossRef]

2002 (2)

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]

P. Gay-Balamz and O. J. F. Martin, “Electromagnetic resonances in individual and coupled split-ring resonators,” J. Appl. Phys. 92, 2929–2936 (2002).
[CrossRef]

2000 (1)

D. R. Smith, W. Padilla, D. Vier, S. Nemat-Nesser, and S. Chultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000).
[CrossRef] [PubMed]

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]

1968 (1)

V. G. Veselago, “The electrodynamics of substances with negative ε and µ,” Sov. Phys. Usp. 10, 509–514 (1968).
[CrossRef]

Aydin, K.

Cai, W.

Caliskan, M. D.

Chen, H.

H. Chen, L. Ran, J. Huangfu, X. Zhang, K. Chen, T. M. Grzegorczyk, and J. A. Kong, “Left-handed materials composed of only S-shaped resonators,” Phys. Rev. E 70, 057605 (2004).
[CrossRef]

J. Huangfu, L. Ran, H. Chen, X. M. Zhang, K. Chen, T. M. Grzegorczyk, and J. A. Kong, “Experimental confirmation of negative refractive index of a metamaterial composed of Ω-like metallic patterns,” Appl. Phys. Lett. 84, 1537–1539 (2004).
[CrossRef]

Chen, K.

H. Chen, L. Ran, J. Huangfu, X. Zhang, K. Chen, T. M. Grzegorczyk, and J. A. Kong, “Left-handed materials composed of only S-shaped resonators,” Phys. Rev. E 70, 057605 (2004).
[CrossRef]

J. Huangfu, L. Ran, H. Chen, X. M. Zhang, K. Chen, T. M. Grzegorczyk, and J. A. Kong, “Experimental confirmation of negative refractive index of a metamaterial composed of Ω-like metallic patterns,” Appl. Phys. Lett. 84, 1537–1539 (2004).
[CrossRef]

Chettiar, U. K.

Chultz, S.

D. R. Smith, W. Padilla, D. Vier, S. Nemat-Nesser, and S. Chultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000).
[CrossRef] [PubMed]

Dolling, G.

Dong, Z. G.

Z. G. Dong, S. Y. Lei, Q. Li, M. X. Xu, H. Liu, T. Li, F. M. Wang, and S. N. Zhu, “Non-left-handed transmission and bianisotropic effect in a π-shaped metallic metamaterial,” Phys. Rev. B 75, 075117 (2007).
[CrossRef]

Drachev, V. P.

Economon, E. N.

Economou, E. N.

C. M. Soukoulis, M. Kafesaki, and E. N. Economou, “Negative-Index Materials: new frontiers in optics,” Adv. Mater. 18, 1941–1952 (2006).
[CrossRef]

Th. 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]

T. Koschny, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Effective medium theory of left-handed materials,” Phys. Rev. Lett. 93, 107402 (2004).
[CrossRef] [PubMed]

N. Katsarahis, T. Koschny, M. Kafesaki, E. N. Economou, E. Ozbay, and C. M. Soukoulis, “Left- and right-handed transmission peaks near the magnetic resonance frequency in composite metamaterials,” Phys. Rev. B 70, 201101, (2004).
[CrossRef]

Enkrich, C.

Feth, N.

Gay-Balamz, P.

P. Gay-Balamz and O. J. F. Martin, “Electromagnetic resonances in individual and coupled split-ring resonators,” J. Appl. Phys. 92, 2929–2936 (2002).
[CrossRef]

Grzegorczyk, T. M.

J. Huangfu, L. Ran, H. Chen, X. M. Zhang, K. Chen, T. M. Grzegorczyk, and J. A. Kong, “Experimental confirmation of negative refractive index of a metamaterial composed of Ω-like metallic patterns,” Appl. Phys. Lett. 84, 1537–1539 (2004).
[CrossRef]

H. Chen, L. Ran, J. Huangfu, X. Zhang, K. Chen, T. M. Grzegorczyk, and J. A. Kong, “Left-handed materials composed of only S-shaped resonators,” Phys. Rev. E 70, 057605 (2004).
[CrossRef]

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]

Huangfu, J.

H. Chen, L. Ran, J. Huangfu, X. Zhang, K. Chen, T. M. Grzegorczyk, and J. A. Kong, “Left-handed materials composed of only S-shaped resonators,” Phys. Rev. E 70, 057605 (2004).
[CrossRef]

J. Huangfu, L. Ran, H. Chen, X. M. Zhang, K. Chen, T. M. Grzegorczyk, and J. A. Kong, “Experimental confirmation of negative refractive index of a metamaterial composed of Ω-like metallic patterns,” Appl. Phys. Lett. 84, 1537–1539 (2004).
[CrossRef]

Kafesaki, M.

C. M. Soukoulis, M. Kafesaki, and E. N. Economou, “Negative-Index Materials: new frontiers in optics,” Adv. Mater. 18, 1941–1952 (2006).
[CrossRef]

T. Koschny, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Effective medium theory of left-handed materials,” Phys. Rev. Lett. 93, 107402 (2004).
[CrossRef] [PubMed]

N. Katsarahis, T. Koschny, M. Kafesaki, E. N. Economou, E. Ozbay, and C. M. Soukoulis, “Left- and right-handed transmission peaks near the magnetic resonance frequency in composite metamaterials,” Phys. Rev. B 70, 201101, (2004).
[CrossRef]

Katsarahis, N.

N. Katsarahis, T. Koschny, M. Kafesaki, E. N. Economou, E. Ozbay, and C. M. Soukoulis, “Left- and right-handed transmission peaks near the magnetic resonance frequency in composite metamaterials,” Phys. Rev. B 70, 201101, (2004).
[CrossRef]

Katsaraki, N.

Kildishev, A. V.

Kim, J. B.

Klein, M. W.

Kong, J. A.

J. Huangfu, L. Ran, H. Chen, X. M. Zhang, K. Chen, T. M. Grzegorczyk, and J. A. Kong, “Experimental confirmation of negative refractive index of a metamaterial composed of Ω-like metallic patterns,” Appl. Phys. Lett. 84, 1537–1539 (2004).
[CrossRef]

H. Chen, L. Ran, J. Huangfu, X. Zhang, K. Chen, T. M. Grzegorczyk, and J. A. Kong, “Left-handed materials composed of only S-shaped resonators,” Phys. Rev. E 70, 057605 (2004).
[CrossRef]

Koschny, T.

J. Zhou, E. N. Economon, T. Koschny, and C. M. Soukoulis, “Unifying approach to left-handed material design,” Opt. Lett. 31, 3620–3622 (2006).
[CrossRef] [PubMed]

J. Zhou, L. Zhang, G. Tuttle, T. Koschny, and C. M. Soukoulis, “Negative index materials using simple short wire pairs,” Phys. Rev. B 73, 041101(R) (2006).
[CrossRef]

T. Koschny, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Effective medium theory of left-handed materials,” Phys. Rev. Lett. 93, 107402 (2004).
[CrossRef] [PubMed]

N. Katsarahis, T. Koschny, M. Kafesaki, E. N. Economou, E. Ozbay, and C. M. Soukoulis, “Left- and right-handed transmission peaks near the magnetic resonance frequency in composite metamaterials,” Phys. Rev. B 70, 201101, (2004).
[CrossRef]

Koschny, Th.

Th. 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]

Lam, V. D.

Lee, S. J.

Lee, Y. P.

Lei, S. Y.

Z. G. Dong, S. Y. Lei, Q. Li, M. X. Xu, H. Liu, T. Li, F. M. Wang, and S. N. Zhu, “Non-left-handed transmission and bianisotropic effect in a π-shaped metallic metamaterial,” Phys. Rev. B 75, 075117 (2007).
[CrossRef]

Li, Q.

Z. G. Dong, S. Y. Lei, Q. Li, M. X. Xu, H. Liu, T. Li, F. M. Wang, and S. N. Zhu, “Non-left-handed transmission and bianisotropic effect in a π-shaped metallic metamaterial,” Phys. Rev. B 75, 075117 (2007).
[CrossRef]

Li, T.

Z. G. Dong, S. Y. Lei, Q. Li, M. X. Xu, H. Liu, T. Li, F. M. Wang, and S. N. Zhu, “Non-left-handed transmission and bianisotropic effect in a π-shaped metallic metamaterial,” Phys. Rev. B 75, 075117 (2007).
[CrossRef]

Linden, S.

Liu, H.

Z. G. Dong, S. Y. Lei, Q. Li, M. X. Xu, H. Liu, T. Li, F. M. Wang, and S. N. Zhu, “Non-left-handed transmission and bianisotropic effect in a π-shaped metallic metamaterial,” Phys. Rev. B 75, 075117 (2007).
[CrossRef]

Markoš, P.

Th. 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]

Martin, O. J. F.

P. Gay-Balamz and O. J. F. Martin, “Electromagnetic resonances in individual and coupled split-ring resonators,” J. Appl. Phys. 92, 2929–2936 (2002).
[CrossRef]

Nemat-Nesser, S.

D. R. Smith, W. Padilla, D. Vier, S. Nemat-Nesser, and S. Chultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000).
[CrossRef] [PubMed]

Ozbay, E.

Padilla, W.

D. R. Smith, W. Padilla, D. Vier, S. Nemat-Nesser, and S. Chultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000).
[CrossRef] [PubMed]

Pendry, J. B.

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

Ran, L.

J. Huangfu, L. Ran, H. Chen, X. M. Zhang, K. Chen, T. M. Grzegorczyk, and J. A. Kong, “Experimental confirmation of negative refractive index of a metamaterial composed of Ω-like metallic patterns,” Appl. Phys. Lett. 84, 1537–1539 (2004).
[CrossRef]

H. Chen, L. Ran, J. Huangfu, X. Zhang, K. Chen, T. M. Grzegorczyk, and J. A. Kong, “Left-handed materials composed of only S-shaped resonators,” Phys. Rev. E 70, 057605 (2004).
[CrossRef]

Rhee, J. Y.

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]

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]

Shalaev, V. M.

Smith, D. R.

Th. 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]

D. R. Smith, W. Padilla, D. Vier, S. Nemat-Nesser, and S. Chultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000).
[CrossRef] [PubMed]

Soukoulis, C. M.

C. M. Soukoulis, M. Kafesaki, and E. N. Economou, “Negative-Index Materials: new frontiers in optics,” Adv. Mater. 18, 1941–1952 (2006).
[CrossRef]

J. Zhou, E. N. Economon, T. Koschny, and C. M. Soukoulis, “Unifying approach to left-handed material design,” Opt. Lett. 31, 3620–3622 (2006).
[CrossRef] [PubMed]

J. Zhou, L. Zhang, G. Tuttle, T. Koschny, and C. M. Soukoulis, “Negative index materials using simple short wire pairs,” Phys. Rev. B 73, 041101(R) (2006).
[CrossRef]

Th. 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]

G. Dolling, C. Enkrich, M. Wegener, J. F. Zhou, C. M. Soukoulis, and S. Linden, “Cut-wire pairs and plate pairs as magnetic atoms for optical metamaterials,” Opt. Lett. 30, 3198–3200 (2005).
[CrossRef] [PubMed]

N. Katsarahis, T. Koschny, M. Kafesaki, E. N. Economou, E. Ozbay, and C. M. Soukoulis, “Left- and right-handed transmission peaks near the magnetic resonance frequency in composite metamaterials,” Phys. Rev. B 70, 201101, (2004).
[CrossRef]

T. Koschny, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Effective medium theory of left-handed materials,” Phys. Rev. Lett. 93, 107402 (2004).
[CrossRef] [PubMed]

K. Aydin, K. Guven, N. Katsaraki, C. M. Soukoulis, and E. Ozbay, “Effect of disorder on magnetic resonance band gap of split-ring resonator structures,” Opt. Express 12, 5896–5901 (2004).
[CrossRef] [PubMed]

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

Tuttle, G.

J. Zhou, L. Zhang, G. Tuttle, T. Koschny, and C. M. Soukoulis, “Negative index materials using simple short wire pairs,” Phys. Rev. B 73, 041101(R) (2006).
[CrossRef]

Veselago, V. G.

V. G. Veselago, “The electrodynamics of substances with negative ε and µ,” Sov. Phys. Usp. 10, 509–514 (1968).
[CrossRef]

Vier, D.

D. R. Smith, W. Padilla, D. Vier, S. Nemat-Nesser, and S. Chultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000).
[CrossRef] [PubMed]

Vier, D. C.

Th. 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]

Wang, F. M.

Z. G. Dong, S. Y. Lei, Q. Li, M. X. Xu, H. Liu, T. Li, F. M. Wang, and S. N. Zhu, “Non-left-handed transmission and bianisotropic effect in a π-shaped metallic metamaterial,” Phys. Rev. B 75, 075117 (2007).
[CrossRef]

Wegener, M.

Xu, M. X.

Z. G. Dong, S. Y. Lei, Q. Li, M. X. Xu, H. Liu, T. Li, F. M. Wang, and S. N. Zhu, “Non-left-handed transmission and bianisotropic effect in a π-shaped metallic metamaterial,” Phys. Rev. B 75, 075117 (2007).
[CrossRef]

Yuan, H–K.

Zhang, L.

J. Zhou, L. Zhang, G. Tuttle, T. Koschny, and C. M. Soukoulis, “Negative index materials using simple short wire pairs,” Phys. Rev. B 73, 041101(R) (2006).
[CrossRef]

Zhang, X.

H. Chen, L. Ran, J. Huangfu, X. Zhang, K. Chen, T. M. Grzegorczyk, and J. A. Kong, “Left-handed materials composed of only S-shaped resonators,” Phys. Rev. E 70, 057605 (2004).
[CrossRef]

Zhang, X. M.

J. Huangfu, L. Ran, H. Chen, X. M. Zhang, K. Chen, T. M. Grzegorczyk, and J. A. Kong, “Experimental confirmation of negative refractive index of a metamaterial composed of Ω-like metallic patterns,” Appl. Phys. Lett. 84, 1537–1539 (2004).
[CrossRef]

Zhou, J.

J. Zhou, L. Zhang, G. Tuttle, T. Koschny, and C. M. Soukoulis, “Negative index materials using simple short wire pairs,” Phys. Rev. B 73, 041101(R) (2006).
[CrossRef]

J. Zhou, E. N. Economon, T. Koschny, and C. M. Soukoulis, “Unifying approach to left-handed material design,” Opt. Lett. 31, 3620–3622 (2006).
[CrossRef] [PubMed]

Zhou, J. F.

Zhu, S. N.

Z. G. Dong, S. Y. Lei, Q. Li, M. X. Xu, H. Liu, T. Li, F. M. Wang, and S. N. Zhu, “Non-left-handed transmission and bianisotropic effect in a π-shaped metallic metamaterial,” Phys. Rev. B 75, 075117 (2007).
[CrossRef]

Adv. Mater. (1)

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Figures (5)

Fig. 1.
Fig. 1.

(Color online) (a) Geometry of the cut-wire pair with the length of cut wire is l=5.5 mm and the width w=1.0 mm. t 1 is the thickness of the PCB board and t 2 that for the CU cut-wire. (b) Periods of cut-wire pairs.

Fig. 2.
Fig. 2.

(Color online) Measured transmission spectra of the cut-wire-pair structure with different numbers of layers in the propagation direction, where the distance between layers is kept 1.0 mm.

Fig. 3.
Fig. 3.

(Color online) Measured transmission spectra of various cut-wire-pair structures; (a) two and (b) three layers. The period of cut-wire pair in the x-y plane is kept constant to be ax =3.5 mm and ay =7.0 mm; while the distance between layers is varied from 1.0 to 4.0 mm.

Fig. 4.
Fig. 4.

(Color online) (a) Schematic of misaligned cut-wire-pair boards. (b) Comparison of the measured transmission spectra between aligned and misaligned cut-wire-pair layers.

Fig. 5.
Fig. 5.

Calculated electric and magnetic resonance frequencies as a function of the distance between cut-wire-pair layers for 3-layer structure. The solid lines are guide to the eyes only.

Equations (12)

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J = J ( y ) y ̂ = σ E o y ̂ α ρ = σ E o y ̂ α ρ y y ̂ ,
· J + ρ t = J y + ρ t = α d 2 ρ d y 2 + ρ t = 0 .
d 2 ρ d y 2 + ω α ρ = 0 .
ρ = ρ o sin ( ω α y )
J = σ E o y ̂ α ρ o ω α cos ( ω α y ) y ̂ ,
r [ · J + ρ t ] d 3 r = J d 3 r + p t = 0 ,
ρ = ρ o sin ( π l y ) ,
J = 4 α ρ o l [ 1 π 4 cos ( π l y ) ] y ̂ .
U E = 1 2 ε E 2 d v = Q 2 2 C eff
U B = 1 2 B 2 μ d v = 1 2 L eff I 2 .
ω e = 1 L e , eff C e , eff
ω m = 1 L m , eff C m , eff ,

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