Abstract

We present experimental and numerical studies of localized terahertz surface waves on a subwavelength-thick metamaterial film consisting of in-plane split-ring resonators. A simple and intuitive model is derived that describes the propagation of surface waves as guided modes in a waveguide filled with a Lorentz-like medium. The effective medium model allows us to deduce the dispersion relation of the surface waves, in excellent agreement with the numerical data obtained from 3D full-wave calculations. Both the accuracy of the analytical model and the numerical calculations are confirmed by spectroscopic terahertz time-domain measurements.

© 2010 Optical Society of America

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  1. J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, Science 305, 847 (2004).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  3. C. R. Williams, S. R. Andrews, S. A. Maier, A. I. Fernández-Domínguez, L. Martín-Moreno, and F. J. García-Vidal, Nat. Photonics 2, 175 (2008).
    [CrossRef]
  4. R. Ulrich and M. Tacke, Appl. Phys. Lett. 22, 251 (1973).
    [CrossRef]
  5. J. T. Shen, P. B. Catrysse, and S. Fan, Phys. Rev. Lett. 94, 197401 (2005).
    [CrossRef] [PubMed]
  6. W. Zhu, A. Agrawal, and A. Nahata, Opt. Express 16, 6216 (2008).
    [CrossRef] [PubMed]
  7. M. Navarro-Cía, M. Beruete, S. Agrafiotis, F. Falcone, M. Sorolla, and S. A. Maier, Opt. Express 17, 18184 (2009).
    [CrossRef] [PubMed]
  8. J. N. Gollub, D. R. Smith, D. C. Vier, T. Perram, and J. J. Mock, Phys. Rev. B 71, 195402 (2005).
    [CrossRef]
  9. R. Ruppin, Phys. Lett. A 277, 61 (2000).
    [CrossRef]
  10. T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, Science 303, 1494 (2004).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  13. J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, IEEE Trans. Microwave Theory Tech. 47, 2075 (1999).
    [CrossRef]

2009 (1)

2008 (3)

W. Zhu, A. Agrawal, and A. Nahata, Opt. Express 16, 6216 (2008).
[CrossRef] [PubMed]

O. Paul, C. Imhof, B. Reinhard, R. Zengerle, and R. Beigang, Opt. Express 16, 6736 (2008).
[CrossRef] [PubMed]

C. R. Williams, S. R. Andrews, S. A. Maier, A. I. Fernández-Domínguez, L. Martín-Moreno, and F. J. García-Vidal, Nat. Photonics 2, 175 (2008).
[CrossRef]

2006 (1)

2005 (3)

A. P. Hibbins, B. R. Evans, and J. R. Sambles, Science 308, 670 (2005).
[CrossRef] [PubMed]

J. T. Shen, P. B. Catrysse, and S. Fan, Phys. Rev. Lett. 94, 197401 (2005).
[CrossRef] [PubMed]

J. N. Gollub, D. R. Smith, D. C. Vier, T. Perram, and J. J. Mock, Phys. Rev. B 71, 195402 (2005).
[CrossRef]

2004 (2)

T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, Science 303, 1494 (2004).
[CrossRef] [PubMed]

J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, Science 305, 847 (2004).
[CrossRef] [PubMed]

2000 (1)

R. Ruppin, Phys. Lett. A 277, 61 (2000).
[CrossRef]

1999 (1)

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, IEEE Trans. Microwave Theory Tech. 47, 2075 (1999).
[CrossRef]

1973 (1)

R. Ulrich and M. Tacke, Appl. Phys. Lett. 22, 251 (1973).
[CrossRef]

Agrafiotis, S.

Agrawal, A.

Andrews, S. R.

C. R. Williams, S. R. Andrews, S. A. Maier, A. I. Fernández-Domínguez, L. Martín-Moreno, and F. J. García-Vidal, Nat. Photonics 2, 175 (2008).
[CrossRef]

Azad, A. K.

Basov, D. N.

T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, Science 303, 1494 (2004).
[CrossRef] [PubMed]

Beigang, R.

Beruete, M.

Catrysse, P. B.

J. T. Shen, P. B. Catrysse, and S. Fan, Phys. Rev. Lett. 94, 197401 (2005).
[CrossRef] [PubMed]

Dai, J.

Evans, B. R.

A. P. Hibbins, B. R. Evans, and J. R. Sambles, Science 308, 670 (2005).
[CrossRef] [PubMed]

Falcone, F.

Fan, S.

J. T. Shen, P. B. Catrysse, and S. Fan, Phys. Rev. Lett. 94, 197401 (2005).
[CrossRef] [PubMed]

Fang, N.

T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, Science 303, 1494 (2004).
[CrossRef] [PubMed]

Fernández-Domínguez, A. I.

C. R. Williams, S. R. Andrews, S. A. Maier, A. I. Fernández-Domínguez, L. Martín-Moreno, and F. J. García-Vidal, Nat. Photonics 2, 175 (2008).
[CrossRef]

Garcia-Vidal, F. J.

J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, Science 305, 847 (2004).
[CrossRef] [PubMed]

García-Vidal, F. J.

C. R. Williams, S. R. Andrews, S. A. Maier, A. I. Fernández-Domínguez, L. Martín-Moreno, and F. J. García-Vidal, Nat. Photonics 2, 175 (2008).
[CrossRef]

Gollub, J. N.

J. N. Gollub, D. R. Smith, D. C. Vier, T. Perram, and J. J. Mock, Phys. Rev. B 71, 195402 (2005).
[CrossRef]

Hibbins, A. P.

A. P. Hibbins, B. R. Evans, and J. R. Sambles, Science 308, 670 (2005).
[CrossRef] [PubMed]

Holden, A. J.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, IEEE Trans. Microwave Theory Tech. 47, 2075 (1999).
[CrossRef]

Imhof, C.

Maier, S. A.

M. Navarro-Cía, M. Beruete, S. Agrafiotis, F. Falcone, M. Sorolla, and S. A. Maier, Opt. Express 17, 18184 (2009).
[CrossRef] [PubMed]

C. R. Williams, S. R. Andrews, S. A. Maier, A. I. Fernández-Domínguez, L. Martín-Moreno, and F. J. García-Vidal, Nat. Photonics 2, 175 (2008).
[CrossRef]

Martín-Moreno, L.

C. R. Williams, S. R. Andrews, S. A. Maier, A. I. Fernández-Domínguez, L. Martín-Moreno, and F. J. García-Vidal, Nat. Photonics 2, 175 (2008).
[CrossRef]

J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, Science 305, 847 (2004).
[CrossRef] [PubMed]

Mock, J. J.

J. N. Gollub, D. R. Smith, D. C. Vier, T. Perram, and J. J. Mock, Phys. Rev. B 71, 195402 (2005).
[CrossRef]

Nahata, A.

Navarro-Cía, M.

Padilla, W. J.

T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, Science 303, 1494 (2004).
[CrossRef] [PubMed]

Paul, O.

Pendry, J. B.

T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, Science 303, 1494 (2004).
[CrossRef] [PubMed]

J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, Science 305, 847 (2004).
[CrossRef] [PubMed]

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, IEEE Trans. Microwave Theory Tech. 47, 2075 (1999).
[CrossRef]

Perram, T.

J. N. Gollub, D. R. Smith, D. C. Vier, T. Perram, and J. J. Mock, Phys. Rev. B 71, 195402 (2005).
[CrossRef]

Reinhard, B.

Robbins, D. J.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, IEEE Trans. Microwave Theory Tech. 47, 2075 (1999).
[CrossRef]

Ruppin, R.

R. Ruppin, Phys. Lett. A 277, 61 (2000).
[CrossRef]

Sambles, J. R.

A. P. Hibbins, B. R. Evans, and J. R. Sambles, Science 308, 670 (2005).
[CrossRef] [PubMed]

Shen, J. T.

J. T. Shen, P. B. Catrysse, and S. Fan, Phys. Rev. Lett. 94, 197401 (2005).
[CrossRef] [PubMed]

Smith, D. R.

J. N. Gollub, D. R. Smith, D. C. Vier, T. Perram, and J. J. Mock, Phys. Rev. B 71, 195402 (2005).
[CrossRef]

T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, Science 303, 1494 (2004).
[CrossRef] [PubMed]

Sorolla, M.

Stewart, W. J.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, IEEE Trans. Microwave Theory Tech. 47, 2075 (1999).
[CrossRef]

Tacke, M.

R. Ulrich and M. Tacke, Appl. Phys. Lett. 22, 251 (1973).
[CrossRef]

Ulrich, R.

R. Ulrich and M. Tacke, Appl. Phys. Lett. 22, 251 (1973).
[CrossRef]

Vier, D. C.

J. N. Gollub, D. R. Smith, D. C. Vier, T. Perram, and J. J. Mock, Phys. Rev. B 71, 195402 (2005).
[CrossRef]

T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, Science 303, 1494 (2004).
[CrossRef] [PubMed]

Williams, C. R.

C. R. Williams, S. R. Andrews, S. A. Maier, A. I. Fernández-Domínguez, L. Martín-Moreno, and F. J. García-Vidal, Nat. Photonics 2, 175 (2008).
[CrossRef]

Yen, T. J.

T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, Science 303, 1494 (2004).
[CrossRef] [PubMed]

Zengerle, R.

Zhang, W.

Zhang, X.

T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, Science 303, 1494 (2004).
[CrossRef] [PubMed]

Zhu, W.

Appl. Phys. Lett. (1)

R. Ulrich and M. Tacke, Appl. Phys. Lett. 22, 251 (1973).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, IEEE Trans. Microwave Theory Tech. 47, 2075 (1999).
[CrossRef]

Nat. Photonics (1)

C. R. Williams, S. R. Andrews, S. A. Maier, A. I. Fernández-Domínguez, L. Martín-Moreno, and F. J. García-Vidal, Nat. Photonics 2, 175 (2008).
[CrossRef]

Opt. Express (3)

Opt. Lett. (1)

Phys. Lett. A (1)

R. Ruppin, Phys. Lett. A 277, 61 (2000).
[CrossRef]

Phys. Rev. B (1)

J. N. Gollub, D. R. Smith, D. C. Vier, T. Perram, and J. J. Mock, Phys. Rev. B 71, 195402 (2005).
[CrossRef]

Phys. Rev. Lett. (1)

J. T. Shen, P. B. Catrysse, and S. Fan, Phys. Rev. Lett. 94, 197401 (2005).
[CrossRef] [PubMed]

Science (3)

J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, Science 305, 847 (2004).
[CrossRef] [PubMed]

A. P. Hibbins, B. R. Evans, and J. R. Sambles, Science 308, 670 (2005).
[CrossRef] [PubMed]

T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, Science 303, 1494 (2004).
[CrossRef] [PubMed]

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Figures (3)

Fig. 1
Fig. 1

(a) Schematic of one unit cell of the SRR array, a = 41 μ m , r = 17.25 μ m , w = 7.5 μ m , g = 3.25 μ m . (b) Microscope image of a sample with grating structure.

Fig. 2
Fig. 2

(a) Dispersion relation of the localized wave. (b) Magnetic field normal to the metamaterial surface (z component) in the x z plane and (c) in the x y plane.

Fig. 3
Fig. 3

(a) Measured and (b) calculated spectra of THz transmission through the sample.

Equations (4)

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μ eff = 1 + F 1 ω 2 ω 1 2 ω 2 i Γ 1 ω + F 2 ω 2 ω 2 2 ω 2 i Γ 2 ω
E = E 0 exp [ i ( k x x + k z z ω t ) ]
ρ = μ o k z μ eff k o z μ o k z + μ eff k o z .
arg ( ρ 2 ) + 4 d eff k z = 2 π .

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