Abstract

In this paper, the transmission of a Gaussian beam passing through a slab made of a one-dimensional left-handed meta-material (1D LHM) is studied. The analytical solution of the electric and the magnetic fields inside and outside the slab are given. The calculation of the power flow of the beam predicts that in the negative pass band of the 1D LHM, there exist different directions of lateral displacements. Such phenomenon is further verified by experiment.

© 2006 Optical Society of America

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  1. R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental Verification of a Negative Index of Refraction," Science 292,77-79 (2001).
    [CrossRef] [PubMed]
  2. J. A. Kong, B.-I. Wu, and Y. Zhang, "A Unique Lateral Displacement of a Gaussian Beam Transmitted through a Slab with a Negative Permittivity and Permeability," Microwave Opt. Technol. Lett. 33, 136-139 (2002).
    [CrossRef]
  3. L. Ran, J. Huangfu, H. Chen, X. Zhang, K. Chen, T. M. Grzegorczyk, and J. A. Kong, "Beam Shifting Experiment for the Characterization of Left-handed Properties," J. Appl. Phys. 95,2238- 2241 (2004).
    [CrossRef]
  4. D. R. Smith, D. Schurig, "Electromagnetic Wave Propagation in Media with Indefinite Permittivity and Permeability Tensors," Phys. Rev. Lett. 90,077405 (2003).
    [CrossRef] [PubMed]
  5. D. R. Smith, P. Kolinko, D. Schurig, "Negative Refraction in Indefinite Media," J. Opt. Soc. Am. B 21, 1032-1043 (2004).
    [CrossRef]
  6. T. M. Grzegorczyk, M. Nikku, X. Chen, B.-I. Wu, and J. A. Kong, "Refraction Laws for Anisotropic Media and Their Application to Left-handed Metamaterials", IEEE Trans. Microw. Theory and Technol. 53,1443-1450 (2005).
    [CrossRef]
  7. 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]
  8. L. Ran, J. Huangfu, H. Chen, Y. Li, X. Zhang, K. Chen, and J. A. Kong, "Microwave Solid-state Left-handed Material with a Broad Bandwidth and an Ultralow Loss," Phys. Rev. B 70,073102 (2004).
    [CrossRef]

2005 (1)

T. M. Grzegorczyk, M. Nikku, X. Chen, B.-I. Wu, and J. A. Kong, "Refraction Laws for Anisotropic Media and Their Application to Left-handed Metamaterials", IEEE Trans. Microw. Theory and Technol. 53,1443-1450 (2005).
[CrossRef]

2004 (4)

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]

L. Ran, J. Huangfu, H. Chen, Y. Li, X. Zhang, K. Chen, and J. A. Kong, "Microwave Solid-state Left-handed Material with a Broad Bandwidth and an Ultralow Loss," Phys. Rev. B 70,073102 (2004).
[CrossRef]

D. R. Smith, P. Kolinko, D. Schurig, "Negative Refraction in Indefinite Media," J. Opt. Soc. Am. B 21, 1032-1043 (2004).
[CrossRef]

L. Ran, J. Huangfu, H. Chen, X. Zhang, K. Chen, T. M. Grzegorczyk, and J. A. Kong, "Beam Shifting Experiment for the Characterization of Left-handed Properties," J. Appl. Phys. 95,2238- 2241 (2004).
[CrossRef]

2003 (1)

D. R. Smith, D. Schurig, "Electromagnetic Wave Propagation in Media with Indefinite Permittivity and Permeability Tensors," Phys. Rev. Lett. 90,077405 (2003).
[CrossRef] [PubMed]

2002 (1)

J. A. Kong, B.-I. Wu, and Y. Zhang, "A Unique Lateral Displacement of a Gaussian Beam Transmitted through a Slab with a Negative Permittivity and Permeability," Microwave Opt. Technol. Lett. 33, 136-139 (2002).
[CrossRef]

2001 (1)

R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental Verification of a Negative Index of Refraction," Science 292,77-79 (2001).
[CrossRef] [PubMed]

Chen, H.

L. Ran, J. Huangfu, H. Chen, X. Zhang, K. Chen, T. M. Grzegorczyk, and J. A. Kong, "Beam Shifting Experiment for the Characterization of Left-handed Properties," J. Appl. Phys. 95,2238- 2241 (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]

L. Ran, J. Huangfu, H. Chen, Y. Li, X. Zhang, K. Chen, and J. A. Kong, "Microwave Solid-state Left-handed Material with a Broad Bandwidth and an Ultralow Loss," Phys. Rev. B 70,073102 (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]

L. Ran, J. Huangfu, H. Chen, X. Zhang, K. Chen, T. M. Grzegorczyk, and J. A. Kong, "Beam Shifting Experiment for the Characterization of Left-handed Properties," J. Appl. Phys. 95,2238- 2241 (2004).
[CrossRef]

L. Ran, J. Huangfu, H. Chen, Y. Li, X. Zhang, K. Chen, and J. A. Kong, "Microwave Solid-state Left-handed Material with a Broad Bandwidth and an Ultralow Loss," Phys. Rev. B 70,073102 (2004).
[CrossRef]

Chen, X.

T. M. Grzegorczyk, M. Nikku, X. Chen, B.-I. Wu, and J. A. Kong, "Refraction Laws for Anisotropic Media and Their Application to Left-handed Metamaterials", IEEE Trans. Microw. Theory and Technol. 53,1443-1450 (2005).
[CrossRef]

Grzegorczyk, T. M.

T. M. Grzegorczyk, M. Nikku, X. Chen, B.-I. Wu, and J. A. Kong, "Refraction Laws for Anisotropic Media and Their Application to Left-handed Metamaterials", IEEE Trans. Microw. Theory and Technol. 53,1443-1450 (2005).
[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]

L. Ran, J. Huangfu, H. Chen, X. Zhang, K. Chen, T. M. Grzegorczyk, and J. A. Kong, "Beam Shifting Experiment for the Characterization of Left-handed Properties," J. Appl. Phys. 95,2238- 2241 (2004).
[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]

L. Ran, J. Huangfu, H. Chen, X. Zhang, K. Chen, T. M. Grzegorczyk, and J. A. Kong, "Beam Shifting Experiment for the Characterization of Left-handed Properties," J. Appl. Phys. 95,2238- 2241 (2004).
[CrossRef]

L. Ran, J. Huangfu, H. Chen, Y. Li, X. Zhang, K. Chen, and J. A. Kong, "Microwave Solid-state Left-handed Material with a Broad Bandwidth and an Ultralow Loss," Phys. Rev. B 70,073102 (2004).
[CrossRef]

Kolinko, P.

Kong, J. A.

T. M. Grzegorczyk, M. Nikku, X. Chen, B.-I. Wu, and J. A. Kong, "Refraction Laws for Anisotropic Media and Their Application to Left-handed Metamaterials", IEEE Trans. Microw. Theory and Technol. 53,1443-1450 (2005).
[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]

L. Ran, J. Huangfu, H. Chen, X. Zhang, K. Chen, T. M. Grzegorczyk, and J. A. Kong, "Beam Shifting Experiment for the Characterization of Left-handed Properties," J. Appl. Phys. 95,2238- 2241 (2004).
[CrossRef]

L. Ran, J. Huangfu, H. Chen, Y. Li, X. Zhang, K. Chen, and J. A. Kong, "Microwave Solid-state Left-handed Material with a Broad Bandwidth and an Ultralow Loss," Phys. Rev. B 70,073102 (2004).
[CrossRef]

J. A. Kong, B.-I. Wu, and Y. Zhang, "A Unique Lateral Displacement of a Gaussian Beam Transmitted through a Slab with a Negative Permittivity and Permeability," Microwave Opt. Technol. Lett. 33, 136-139 (2002).
[CrossRef]

Li, Y.

L. Ran, J. Huangfu, H. Chen, Y. Li, X. Zhang, K. Chen, and J. A. Kong, "Microwave Solid-state Left-handed Material with a Broad Bandwidth and an Ultralow Loss," Phys. Rev. B 70,073102 (2004).
[CrossRef]

Nikku, M.

T. M. Grzegorczyk, M. Nikku, X. Chen, B.-I. Wu, and J. A. Kong, "Refraction Laws for Anisotropic Media and Their Application to Left-handed Metamaterials", IEEE Trans. Microw. Theory and Technol. 53,1443-1450 (2005).
[CrossRef]

Ran, L.

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]

L. Ran, J. Huangfu, H. Chen, Y. Li, X. Zhang, K. Chen, and J. A. Kong, "Microwave Solid-state Left-handed Material with a Broad Bandwidth and an Ultralow Loss," Phys. Rev. B 70,073102 (2004).
[CrossRef]

L. Ran, J. Huangfu, H. Chen, X. Zhang, K. Chen, T. M. Grzegorczyk, and J. A. Kong, "Beam Shifting Experiment for the Characterization of Left-handed Properties," J. Appl. Phys. 95,2238- 2241 (2004).
[CrossRef]

Schultz, S.

R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental Verification of a Negative Index of Refraction," Science 292,77-79 (2001).
[CrossRef] [PubMed]

Schurig, D.

D. R. Smith, P. Kolinko, D. Schurig, "Negative Refraction in Indefinite Media," J. Opt. Soc. Am. B 21, 1032-1043 (2004).
[CrossRef]

D. R. Smith, D. Schurig, "Electromagnetic Wave Propagation in Media with Indefinite Permittivity and Permeability Tensors," Phys. Rev. Lett. 90,077405 (2003).
[CrossRef] [PubMed]

Shelby, R. A.

R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental Verification of a Negative Index of Refraction," Science 292,77-79 (2001).
[CrossRef] [PubMed]

Smith, D. R.

D. R. Smith, P. Kolinko, D. Schurig, "Negative Refraction in Indefinite Media," J. Opt. Soc. Am. B 21, 1032-1043 (2004).
[CrossRef]

D. R. Smith, D. Schurig, "Electromagnetic Wave Propagation in Media with Indefinite Permittivity and Permeability Tensors," Phys. Rev. Lett. 90,077405 (2003).
[CrossRef] [PubMed]

R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental Verification of a Negative Index of Refraction," Science 292,77-79 (2001).
[CrossRef] [PubMed]

Wu, B.-I.

T. M. Grzegorczyk, M. Nikku, X. Chen, B.-I. Wu, and J. A. Kong, "Refraction Laws for Anisotropic Media and Their Application to Left-handed Metamaterials", IEEE Trans. Microw. Theory and Technol. 53,1443-1450 (2005).
[CrossRef]

J. A. Kong, B.-I. Wu, and Y. Zhang, "A Unique Lateral Displacement of a Gaussian Beam Transmitted through a Slab with a Negative Permittivity and Permeability," Microwave Opt. Technol. Lett. 33, 136-139 (2002).
[CrossRef]

Zhang, X.

L. Ran, J. Huangfu, H. Chen, X. Zhang, K. Chen, T. M. Grzegorczyk, and J. A. Kong, "Beam Shifting Experiment for the Characterization of Left-handed Properties," J. Appl. Phys. 95,2238- 2241 (2004).
[CrossRef]

L. Ran, J. Huangfu, H. Chen, Y. Li, X. Zhang, K. Chen, and J. A. Kong, "Microwave Solid-state Left-handed Material with a Broad Bandwidth and an Ultralow Loss," Phys. Rev. B 70,073102 (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]

Zhang, Y.

J. A. Kong, B.-I. Wu, and Y. Zhang, "A Unique Lateral Displacement of a Gaussian Beam Transmitted through a Slab with a Negative Permittivity and Permeability," Microwave Opt. Technol. Lett. 33, 136-139 (2002).
[CrossRef]

IEEE Trans. Microw. Theory and Technol. (1)

T. M. Grzegorczyk, M. Nikku, X. Chen, B.-I. Wu, and J. A. Kong, "Refraction Laws for Anisotropic Media and Their Application to Left-handed Metamaterials", IEEE Trans. Microw. Theory and Technol. 53,1443-1450 (2005).
[CrossRef]

J. Appl. Phys. (1)

L. Ran, J. Huangfu, H. Chen, X. Zhang, K. Chen, T. M. Grzegorczyk, and J. A. Kong, "Beam Shifting Experiment for the Characterization of Left-handed Properties," J. Appl. Phys. 95,2238- 2241 (2004).
[CrossRef]

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

Microwave Opt. Technol. Lett. (1)

J. A. Kong, B.-I. Wu, and Y. Zhang, "A Unique Lateral Displacement of a Gaussian Beam Transmitted through a Slab with a Negative Permittivity and Permeability," Microwave Opt. Technol. Lett. 33, 136-139 (2002).
[CrossRef]

Phys. Rev. B (1)

L. Ran, J. Huangfu, H. Chen, Y. Li, X. Zhang, K. Chen, and J. A. Kong, "Microwave Solid-state Left-handed Material with a Broad Bandwidth and an Ultralow Loss," Phys. Rev. B 70,073102 (2004).
[CrossRef]

Phys. Rev. E (1)

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]

Phys. Rev. Lett. (1)

D. R. Smith, D. Schurig, "Electromagnetic Wave Propagation in Media with Indefinite Permittivity and Permeability Tensors," Phys. Rev. Lett. 90,077405 (2003).
[CrossRef] [PubMed]

Science (1)

R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental Verification of a Negative Index of Refraction," Science 292,77-79 (2001).
[CrossRef] [PubMed]

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Figures (5)

Fig. 1.
Fig. 1.

Illustration of the lateral displacement of an obliquely incident Gaussian beam.

Fig. 2.
Fig. 2.

K-surfaces for the illustration of the lateral shifts for a Gaussian beam incidence.

Fig. 3.
Fig. 3.

The normalized power flow calculated at the second interface of the 1D LHM slab. (a)θ = 18.4°, (b)θ = 45°

Fig. 4.
Fig. 4.

Schematic of the experimental setup for detecting the lateral shift of the incident Gaussian beam.

Fig. 5.
Fig. 5.

Normalized transmission power detected at the second interface of the S-shaped 1DLHM slab.

Equations (21)

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ε ̿ = [ ε 1 cos 2 θ + ε 3 sin 2 θ 0 sin θ cos θ ( ε 1 ε 3 ) 0 ε 2 0 sin θ cos θ ( ε 1 ε 3 ) 0 ε 1 sin 2 θ + ε 3 cos 2 θ ]
μ ̿ = [ μ 1 cos 2 θ + μ 3 sin 2 θ 0 sin θ cos θ ( μ 1 μ 3 ) 0 μ 2 0 sin θ cos θ ( μ 1 μ 3 ) 0 μ 1 sin 2 θ + μ 3 cos 2 θ ]
α k z 2 + β k x k z + γ k x 2 ω 2 ε 2 μ 1 μ 3 = 0 ,
E iy = dk x exp [ i ( k x x + k 0 z z ) ] ψ ( k x )
E 1 y = dk x ψ ( k x ) ( e ik iz z + Re ik iz z ) e ik x x
H 1 x = 1 ω μ 0 dk x ψ ( k x ) k iz ( e ik iz z + Re ik iz z ) e ik x x
H 1 z = 1 ω μ 0 dk x ψ ( k x ) k x ( e ik iz z + Re ik iz z ) e ik x x
E 2 y = dk x ψ ( k x ) ( ξe ik z 1 z + ζe ik z 2 z ) e ik x x
H 2 x = 1 ω dk x ψ ( k x ) [ a ( ξk z 1 e ik z 1 z + ζk z 2 e ik z 2 z ) + b ( ξ e ik z 1 z + ζ e ik z 2 z ) k x ] e ik x x
H 2 z = 1 ω dk x ψ ( k x ) [ c ( ξk z 1 e ik z 1 z + ζk z 2 e ik z 2 z ) + d ( ξ e ik z 1 z + ζ e ik z 2 z ) k z ] e ik x x
E 3 y = dk x ψ ( k x ) Te ik tz z e ik x x
H 3 x = 1 ω μ 0 dk x ψ ( k x ) T k tz e ik tz z e ik x x
H 3 z = 1 ω μ 0 dk x ψ ( k x ) T k x e ik tz z e ik x x
ξ = 2 p 2 k iz ( p 1 p 4 p 2 p 3 ) μ 0
ζ = 2 p 1 k iz ( p 1 p 4 p 2 p 3 ) μ 0
R = 2 ( p 2 p 1 ) k iz ( p 1 p 4 p 2 p 3 ) μ 0 1
T = 2 ( p 2 e i ( k z 1 k tz ) z 0 p 1 e i ( k z 2 k tz ) z 0 ) k iz ( p 1 p 4 p 2 p 3 ) μ 0
S ¯ nz = 1 2 Re { E ny · H nx * }
S ¯ nx = 1 2 Re { E ny · H nz * }
ε 2 = 1 ω ε p 2 ω ε 0 2 ω 2 ω ε 0 2 + i γ ε ω
μ 1 = 1 ω μ p 2 ω μ o 2 ω 2 ω μ o 2 + i γ μ ω

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