Abstract

The excitation of ‘designer’ surface-plasmon-like modes on periodically perforated metals is demonstrated at microwave frequencies using the classical method of prism-coupling. In addition we provide a complete formalism for accurately determining the dispersion of these surface modes. Our findings fully validate the use of metamaterials to give surface plasmon-like behavior at frequencies below the visible.

© 2008 Optical Society of America

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References

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  1. Q1. A. Sommerfeld, "Ueber die fortpflanzung elektrodynamischer wellen längs eines drahtes," Ann. Phys. Chem. 67, 233-290 (1899).
    [CrossRef]
  2. J. Zenneck, "Fortplfanzung ebener elektromagnetischer Wellen längs einer ebenen Leiterfläche," Ann. Phys. 23, 846-866 (1907).
    [CrossRef]
  3. R. W. Wood, "On a remarkable case of uneven distribution of light in a diffraction grating spectrum," Philos. Mag. 4, 396-397 (1902).
  4. U. Fano, "The Theory of Anomalous Diffraction Gratings and of Quasi-Stationary Waves on Metallic Surfaces (Sommerfeld’s Waves)," J. Opt. Soc. Am. 31, 213-222 (1941).
    [CrossRef]
  5. A. Otto, "Excitation of nonradiative surface plasma waves in silver by the method of frustrated total reflection," Zeitschrift fur Physik 216, 398-410 (1968).
  6. H. Raether, Surface Plasmons (Springer-Verlag, Berlin, 1988).
  7. J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, "Low Frequency Plasmons in Thin Wire Structures," J. Phys. Condens. Matter 10, 4785-4809 (1998).
    [CrossRef]
  8. D. R. Smith, R. C. Vier, W. Padilla, S. C. Nemat-Nasser, and S. Schultz, "Loop-wire medium for investigating plasmons at microwave frequencies," Appl. Phys. Lett. 75, 1425-1427 (1999).
    [CrossRef]
  9. J. B. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, "Mimicking surface plasmons with structured surfaces," Science 305, 847-848 (2004).
    [CrossRef] [PubMed]
  10. Q2. F. J. Garcia-Vidal, L. Martin-Moreno, and J. B. Pendry, "Surfaces with holes in them: new plasmonic metamaterials," J. Opt. A 7, S97-S101 (2005)
    [CrossRef]
  11. C. H. Palmer, F. C. Evering, Jr., and F. M. Nelson, "Diffraction Anomalies for Gratings of Rectangular Profile," Appl. Opt. 4, 1271-1274 (1965)
    [CrossRef]
  12. R. Ulrich, and M. Tacke, "Submillimeter waveguiding on periodic metal structure," Appl. Phys. Lett. 22, 251-253 (1973).
    [CrossRef]
  13. A. P. Hibbins, B. R. Evans, and J. R. Sambles, "Experimental verification of designer surface plasmons," Science 308, 670-672 (2005).
    [CrossRef] [PubMed]
  14. F. J. García de Abajo and J. J. Sáenz, "Electromagnetic surface modes in structured perfect-conductor surfaces," Phys. Rev. Lett. 95, 233901 (2005)
    [CrossRef]
  15. H. A. Bethe, "Theory of diffraction by small holes," Phys. Rev. 66, 163-182 (1944)
    [CrossRef]
  16. HFSS, High-frequency structure simulator version 11.1.1 Finite-element package, Ansoft Corporation, Pittsburgh, PA (2008).
  17. E. Hendry, A. P. Hibbins, and J. R. Sambles, Electromagnetic Materials Group, School of Physics, University of Exeter, Exeter EX4 4QL, United Kingdom are preparing a manuscript to be called "The importance of diffraction in determining the dispersion of designer surface plasmons".
  18. K. L. Tsakmakidis, A. D. Boardman, and O. Hess, "‘Trapped rainbow’ storage of light in metamaterials," Nature 450, 397-401 (2007).
    [CrossRef] [PubMed]

2007

K. L. Tsakmakidis, A. D. Boardman, and O. Hess, "‘Trapped rainbow’ storage of light in metamaterials," Nature 450, 397-401 (2007).
[CrossRef] [PubMed]

2005

A. P. Hibbins, B. R. Evans, and J. R. Sambles, "Experimental verification of designer surface plasmons," Science 308, 670-672 (2005).
[CrossRef] [PubMed]

F. J. García de Abajo and J. J. Sáenz, "Electromagnetic surface modes in structured perfect-conductor surfaces," Phys. Rev. Lett. 95, 233901 (2005)
[CrossRef]

Q2. F. J. Garcia-Vidal, L. Martin-Moreno, and J. B. Pendry, "Surfaces with holes in them: new plasmonic metamaterials," J. Opt. A 7, S97-S101 (2005)
[CrossRef]

2004

J. B. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, "Mimicking surface plasmons with structured surfaces," Science 305, 847-848 (2004).
[CrossRef] [PubMed]

1999

D. R. Smith, R. C. Vier, W. Padilla, S. C. Nemat-Nasser, and S. Schultz, "Loop-wire medium for investigating plasmons at microwave frequencies," Appl. Phys. Lett. 75, 1425-1427 (1999).
[CrossRef]

1998

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, "Low Frequency Plasmons in Thin Wire Structures," J. Phys. Condens. Matter 10, 4785-4809 (1998).
[CrossRef]

1973

R. Ulrich, and M. Tacke, "Submillimeter waveguiding on periodic metal structure," Appl. Phys. Lett. 22, 251-253 (1973).
[CrossRef]

1968

A. Otto, "Excitation of nonradiative surface plasma waves in silver by the method of frustrated total reflection," Zeitschrift fur Physik 216, 398-410 (1968).

1965

1944

H. A. Bethe, "Theory of diffraction by small holes," Phys. Rev. 66, 163-182 (1944)
[CrossRef]

1941

1907

J. Zenneck, "Fortplfanzung ebener elektromagnetischer Wellen längs einer ebenen Leiterfläche," Ann. Phys. 23, 846-866 (1907).
[CrossRef]

1902

R. W. Wood, "On a remarkable case of uneven distribution of light in a diffraction grating spectrum," Philos. Mag. 4, 396-397 (1902).

1899

Q1. A. Sommerfeld, "Ueber die fortpflanzung elektrodynamischer wellen längs eines drahtes," Ann. Phys. Chem. 67, 233-290 (1899).
[CrossRef]

Bethe, H. A.

H. A. Bethe, "Theory of diffraction by small holes," Phys. Rev. 66, 163-182 (1944)
[CrossRef]

Boardman, A. D.

K. L. Tsakmakidis, A. D. Boardman, and O. Hess, "‘Trapped rainbow’ storage of light in metamaterials," Nature 450, 397-401 (2007).
[CrossRef] [PubMed]

Evans, B. R.

A. P. Hibbins, B. R. Evans, and J. R. Sambles, "Experimental verification of designer surface plasmons," Science 308, 670-672 (2005).
[CrossRef] [PubMed]

Evering, F. C.

Fano, U.

García de Abajo, F. J.

F. J. García de Abajo and J. J. Sáenz, "Electromagnetic surface modes in structured perfect-conductor surfaces," Phys. Rev. Lett. 95, 233901 (2005)
[CrossRef]

Garcia-Vidal, F. J.

Q2. F. J. Garcia-Vidal, L. Martin-Moreno, and J. B. Pendry, "Surfaces with holes in them: new plasmonic metamaterials," J. Opt. A 7, S97-S101 (2005)
[CrossRef]

J. B. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, "Mimicking surface plasmons with structured surfaces," Science 305, 847-848 (2004).
[CrossRef] [PubMed]

Hess, O.

K. L. Tsakmakidis, A. D. Boardman, and O. Hess, "‘Trapped rainbow’ storage of light in metamaterials," Nature 450, 397-401 (2007).
[CrossRef] [PubMed]

Hibbins, A. P.

A. P. Hibbins, B. R. Evans, and J. R. Sambles, "Experimental verification of designer surface plasmons," Science 308, 670-672 (2005).
[CrossRef] [PubMed]

Holden, A. J.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, "Low Frequency Plasmons in Thin Wire Structures," J. Phys. Condens. Matter 10, 4785-4809 (1998).
[CrossRef]

Martin-Moreno, L.

Q2. F. J. Garcia-Vidal, L. Martin-Moreno, and J. B. Pendry, "Surfaces with holes in them: new plasmonic metamaterials," J. Opt. A 7, S97-S101 (2005)
[CrossRef]

J. B. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, "Mimicking surface plasmons with structured surfaces," Science 305, 847-848 (2004).
[CrossRef] [PubMed]

Nelson, F. M.

Nemat-Nasser, S. C.

D. R. Smith, R. C. Vier, W. Padilla, S. C. Nemat-Nasser, and S. Schultz, "Loop-wire medium for investigating plasmons at microwave frequencies," Appl. Phys. Lett. 75, 1425-1427 (1999).
[CrossRef]

Otto, A.

A. Otto, "Excitation of nonradiative surface plasma waves in silver by the method of frustrated total reflection," Zeitschrift fur Physik 216, 398-410 (1968).

Padilla, W.

D. R. Smith, R. C. Vier, W. Padilla, S. C. Nemat-Nasser, and S. Schultz, "Loop-wire medium for investigating plasmons at microwave frequencies," Appl. Phys. Lett. 75, 1425-1427 (1999).
[CrossRef]

Palmer, C. H.

Pendry, J. B.

Q2. F. J. Garcia-Vidal, L. Martin-Moreno, and J. B. Pendry, "Surfaces with holes in them: new plasmonic metamaterials," J. Opt. A 7, S97-S101 (2005)
[CrossRef]

J. B. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, "Mimicking surface plasmons with structured surfaces," Science 305, 847-848 (2004).
[CrossRef] [PubMed]

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, "Low Frequency Plasmons in Thin Wire Structures," J. Phys. Condens. Matter 10, 4785-4809 (1998).
[CrossRef]

Robbins, D. J.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, "Low Frequency Plasmons in Thin Wire Structures," J. Phys. Condens. Matter 10, 4785-4809 (1998).
[CrossRef]

Sáenz, J. J.

F. J. García de Abajo and J. J. Sáenz, "Electromagnetic surface modes in structured perfect-conductor surfaces," Phys. Rev. Lett. 95, 233901 (2005)
[CrossRef]

Sambles, J. R.

A. P. Hibbins, B. R. Evans, and J. R. Sambles, "Experimental verification of designer surface plasmons," Science 308, 670-672 (2005).
[CrossRef] [PubMed]

Schultz, S.

D. R. Smith, R. C. Vier, W. Padilla, S. C. Nemat-Nasser, and S. Schultz, "Loop-wire medium for investigating plasmons at microwave frequencies," Appl. Phys. Lett. 75, 1425-1427 (1999).
[CrossRef]

Smith, D. R.

D. R. Smith, R. C. Vier, W. Padilla, S. C. Nemat-Nasser, and S. Schultz, "Loop-wire medium for investigating plasmons at microwave frequencies," Appl. Phys. Lett. 75, 1425-1427 (1999).
[CrossRef]

Sommerfeld, A.

Q1. A. Sommerfeld, "Ueber die fortpflanzung elektrodynamischer wellen längs eines drahtes," Ann. Phys. Chem. 67, 233-290 (1899).
[CrossRef]

Stewart, W. J.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, "Low Frequency Plasmons in Thin Wire Structures," J. Phys. Condens. Matter 10, 4785-4809 (1998).
[CrossRef]

Tacke, M.

R. Ulrich, and M. Tacke, "Submillimeter waveguiding on periodic metal structure," Appl. Phys. Lett. 22, 251-253 (1973).
[CrossRef]

Tsakmakidis, K. L.

K. L. Tsakmakidis, A. D. Boardman, and O. Hess, "‘Trapped rainbow’ storage of light in metamaterials," Nature 450, 397-401 (2007).
[CrossRef] [PubMed]

Ulrich, R.

R. Ulrich, and M. Tacke, "Submillimeter waveguiding on periodic metal structure," Appl. Phys. Lett. 22, 251-253 (1973).
[CrossRef]

Vier, R. C.

D. R. Smith, R. C. Vier, W. Padilla, S. C. Nemat-Nasser, and S. Schultz, "Loop-wire medium for investigating plasmons at microwave frequencies," Appl. Phys. Lett. 75, 1425-1427 (1999).
[CrossRef]

Wood, R. W.

R. W. Wood, "On a remarkable case of uneven distribution of light in a diffraction grating spectrum," Philos. Mag. 4, 396-397 (1902).

Zenneck, J.

J. Zenneck, "Fortplfanzung ebener elektromagnetischer Wellen längs einer ebenen Leiterfläche," Ann. Phys. 23, 846-866 (1907).
[CrossRef]

Ann. Phys.

J. Zenneck, "Fortplfanzung ebener elektromagnetischer Wellen längs einer ebenen Leiterfläche," Ann. Phys. 23, 846-866 (1907).
[CrossRef]

Ann. Phys. Chem.

Q1. A. Sommerfeld, "Ueber die fortpflanzung elektrodynamischer wellen längs eines drahtes," Ann. Phys. Chem. 67, 233-290 (1899).
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

D. R. Smith, R. C. Vier, W. Padilla, S. C. Nemat-Nasser, and S. Schultz, "Loop-wire medium for investigating plasmons at microwave frequencies," Appl. Phys. Lett. 75, 1425-1427 (1999).
[CrossRef]

R. Ulrich, and M. Tacke, "Submillimeter waveguiding on periodic metal structure," Appl. Phys. Lett. 22, 251-253 (1973).
[CrossRef]

J. Opt. A

Q2. F. J. Garcia-Vidal, L. Martin-Moreno, and J. B. Pendry, "Surfaces with holes in them: new plasmonic metamaterials," J. Opt. A 7, S97-S101 (2005)
[CrossRef]

J. Opt. Soc. Am.

J. Phys. Condens. Matter

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, "Low Frequency Plasmons in Thin Wire Structures," J. Phys. Condens. Matter 10, 4785-4809 (1998).
[CrossRef]

Nature

K. L. Tsakmakidis, A. D. Boardman, and O. Hess, "‘Trapped rainbow’ storage of light in metamaterials," Nature 450, 397-401 (2007).
[CrossRef] [PubMed]

Philos. Mag.

R. W. Wood, "On a remarkable case of uneven distribution of light in a diffraction grating spectrum," Philos. Mag. 4, 396-397 (1902).

Phys. Rev.

H. A. Bethe, "Theory of diffraction by small holes," Phys. Rev. 66, 163-182 (1944)
[CrossRef]

Phys. Rev. Lett.

F. J. García de Abajo and J. J. Sáenz, "Electromagnetic surface modes in structured perfect-conductor surfaces," Phys. Rev. Lett. 95, 233901 (2005)
[CrossRef]

Science

A. P. Hibbins, B. R. Evans, and J. R. Sambles, "Experimental verification of designer surface plasmons," Science 308, 670-672 (2005).
[CrossRef] [PubMed]

J. B. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, "Mimicking surface plasmons with structured surfaces," Science 305, 847-848 (2004).
[CrossRef] [PubMed]

Zeitschrift fur Physik

A. Otto, "Excitation of nonradiative surface plasma waves in silver by the method of frustrated total reflection," Zeitschrift fur Physik 216, 398-410 (1968).

Other

H. Raether, Surface Plasmons (Springer-Verlag, Berlin, 1988).

HFSS, High-frequency structure simulator version 11.1.1 Finite-element package, Ansoft Corporation, Pittsburgh, PA (2008).

E. Hendry, A. P. Hibbins, and J. R. Sambles, Electromagnetic Materials Group, School of Physics, University of Exeter, Exeter EX4 4QL, United Kingdom are preparing a manuscript to be called "The importance of diffraction in determining the dispersion of designer surface plasmons".

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Figures (3)

Fig. 1.
Fig. 1.

(a) Schematic of the experimental set-up. (b) Photograph of sample surface and experimental coordinate system, where a = 6.96 mm and d = 9.53 mm

Fig. 2.
Fig. 2.

(a) Reflection spectrum from the sample surface for θ int=46.5° and ϕ = 45° (where ϕ is the angle between the plane of incidence and the xz-plane) (squares). The predictions of the FEM model are also shown (line). The solid vertical line corresponds to the cut off frequency of an infinite length guide, Eq. (1), filled with ε h=2.29 and the dimensions previously described. The broken vertical line corresponds to the modified ν SP associated with complete confinement of the waveguide mode to the truncated cavity. (b) EM field predictions on the plane of incidence of the mode at 14.6 GHz. The greyscale on the left shows the electric field at a phase corresponding to maximal field enhancement. The lightest shading indicates field intensities of at least five times the incident field. The plot on the right shows the Poynting vector (magnitude and direction) on resonance. Here the lightest shading corresponds to power enhancements of at least ten times.

Fig. 3.
Fig. 3.

Dispersion curves of the SP-like mode determined from experiment, and numerical and analytical models, when ϕ = 45°. Frequency is plotted against wave vector along the surface of the sample. The data points derived from the experiment (optimum coupling condition) are shown as circles, and the straight line represents the air light line. The dashed and dotted curves correspond to the analytical solutions provided in Refs. 9 (and 10), and 12 respectively. The solid curve represents the solution of the analytical formalism provided in the present study, Eq. (7). The inset provides a comparison between the FEM modelled results on the same axes as the main diagram. The data points from the optimum coupling condition are shown as a solid grey line, whereas the broken line represents the Eigen mode solution in the absence of the prism.

Equations (15)

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ν SP = c 2 a ε h 1 2
k = k 0 ( 1 + [ β ν 2 ν SP 2 ν 2 ] ) 1 2
β = Γ ( 4 π 4 a 4 d 4 ε h )
Γ = ( [ Re ( a 3 α e 1 ) ] 1 + [ Re ( a 3 α m 1 ) ] 1 ) 2
( k z m , n ) 2 = k 0 2 ( k x + 2 m π d ) 2 ( k y + 2 n π d ) 2 ,
( q z s , t ) 2 = ε h k 0 2 ( s π a ) 2 ( t π a ) 2 ,
E x vac = m , n A x m , n ψ 1 ( x , n ) exp ( i k z m , n z ) ,
E y vac = m , n A y m , n ψ 1 ( x , y ) exp ( i k z m , n z ) ,
E x cav = s , t B x s , t ψ 2 ( x , y ) [ exp ( i q z s , t z ) exp ( i q z s , t ( 2 h z ) ) ] ,
E y cav = s , t B y s , t ψ 3 ( x , y ) [ exp ( i q z s , t z ) exp ( i q z s , t ( 2 h z ) ) ] ,
ψ 1 ( x , y ) = exp ( i ( k x + 2 m π d 1 ) x ) · exp ( i ( k y + 2 n π d 2 ) y ) ,
ψ 2 ( x , y ) = cos ( s π x a 1 ) sin ( t π y a 2 ) ,
ψ 3 ( x , y ) = sin ( s π x a 1 ) cos ( t π y a 2 ) .
m , n ( k 0 2 ( k y + 2 n π d ) 2 ) i tan ( h q z 0 , 1 ) ( S m , n ) 2 k z m , n q z 0 , 1 = 1 ,
S m , n = 4 π 2 a 2 d sin ( a ( k x 2 + m π d ) ) cos ( a ( k y 2 + n π d ) ) ( k x + 2 m π d ) ( ( π a ) 2 ( k y + 2 n π d ) 2 ) .

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