Abstract

As interest in plasmonics grows the optical properties of thin metal films becomes increasingly significant. Here we explore the transmissivity of thin metal films at normal incidence, from the ultraviolet to microwaves, and show how, contrary to simplistic treatments, the microwave transmissivity may be much less than the optical transmissivity for films which are well below the skin depth in thickness. This arises because the film is acting as a zero order Fabry-Perot with very high reflectivity at each interface. The skin depth then becomes irrelevant for thin metal films at microwave frequencies. We also note in passing that the expected exponential dependence on thickness at higher thicknesses has an asymptotic limit at zero thickness which may be as high as four times the input intensity.

© 2008 Optical Society of America

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References

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  1. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature 391, 667-669 (1998).
    [CrossRef]
  2. J. B. Pendry, "Negative refractive index makes a perfect lens," Phys. Rev. Lett. 85, 3966-3969 (2000).
    [CrossRef] [PubMed]
  3. D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, "Metamaterials and negative refractive index," Science 305, 788-792 (2004).
    [CrossRef] [PubMed]
  4. D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, "Metamaterial electromagnetic cloak at microwave frequencies," Science 314, 977-980 (2006).
    [CrossRef] [PubMed]
  5. J. A. Stratton, Electromagnetic Theory (McGraw Hill, 1941).
  6. J. R. Reitz, F. J. Milford, and R. W. Christy, Foundations of Electromagnetic Theory, 4th Ed. (Addison-Wesley, 1992).
  7. D. J. Nash and J. R. Sambles, "Surface plasmon study of the optical dielectric function of silver," J. Mod. Opt. 43, 81-91 (1996).
  8. M. J. Tillin and J. R. Sambles, "A surface plasmon polariton study of the dielectric-constants of reactive metals - Aluminium," Thin Solid Films 167, 73-83 (1988).
    [CrossRef]

2006 (1)

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, "Metamaterial electromagnetic cloak at microwave frequencies," Science 314, 977-980 (2006).
[CrossRef] [PubMed]

2004 (1)

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, "Metamaterials and negative refractive index," Science 305, 788-792 (2004).
[CrossRef] [PubMed]

2000 (1)

J. B. Pendry, "Negative refractive index makes a perfect lens," Phys. Rev. Lett. 85, 3966-3969 (2000).
[CrossRef] [PubMed]

1998 (1)

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature 391, 667-669 (1998).
[CrossRef]

1996 (1)

D. J. Nash and J. R. Sambles, "Surface plasmon study of the optical dielectric function of silver," J. Mod. Opt. 43, 81-91 (1996).

1988 (1)

M. J. Tillin and J. R. Sambles, "A surface plasmon polariton study of the dielectric-constants of reactive metals - Aluminium," Thin Solid Films 167, 73-83 (1988).
[CrossRef]

Cummer, S. A.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, "Metamaterial electromagnetic cloak at microwave frequencies," Science 314, 977-980 (2006).
[CrossRef] [PubMed]

Ebbesen, T. W.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature 391, 667-669 (1998).
[CrossRef]

Ghaemi, H. F.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature 391, 667-669 (1998).
[CrossRef]

Justice, B. J.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, "Metamaterial electromagnetic cloak at microwave frequencies," Science 314, 977-980 (2006).
[CrossRef] [PubMed]

Lezec, H. J.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature 391, 667-669 (1998).
[CrossRef]

Mock, J. J.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, "Metamaterial electromagnetic cloak at microwave frequencies," Science 314, 977-980 (2006).
[CrossRef] [PubMed]

Nash, D. J.

D. J. Nash and J. R. Sambles, "Surface plasmon study of the optical dielectric function of silver," J. Mod. Opt. 43, 81-91 (1996).

Pendry, J. B.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, "Metamaterial electromagnetic cloak at microwave frequencies," Science 314, 977-980 (2006).
[CrossRef] [PubMed]

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, "Metamaterials and negative refractive index," Science 305, 788-792 (2004).
[CrossRef] [PubMed]

J. B. Pendry, "Negative refractive index makes a perfect lens," Phys. Rev. Lett. 85, 3966-3969 (2000).
[CrossRef] [PubMed]

Sambles, J. R.

D. J. Nash and J. R. Sambles, "Surface plasmon study of the optical dielectric function of silver," J. Mod. Opt. 43, 81-91 (1996).

M. J. Tillin and J. R. Sambles, "A surface plasmon polariton study of the dielectric-constants of reactive metals - Aluminium," Thin Solid Films 167, 73-83 (1988).
[CrossRef]

Schurig, D.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, "Metamaterial electromagnetic cloak at microwave frequencies," Science 314, 977-980 (2006).
[CrossRef] [PubMed]

Smith, D. R.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, "Metamaterial electromagnetic cloak at microwave frequencies," Science 314, 977-980 (2006).
[CrossRef] [PubMed]

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, "Metamaterials and negative refractive index," Science 305, 788-792 (2004).
[CrossRef] [PubMed]

Starr, A. F.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, "Metamaterial electromagnetic cloak at microwave frequencies," Science 314, 977-980 (2006).
[CrossRef] [PubMed]

Thio, T.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature 391, 667-669 (1998).
[CrossRef]

Tillin, M. J.

M. J. Tillin and J. R. Sambles, "A surface plasmon polariton study of the dielectric-constants of reactive metals - Aluminium," Thin Solid Films 167, 73-83 (1988).
[CrossRef]

Wiltshire, M. C. K.

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, "Metamaterials and negative refractive index," Science 305, 788-792 (2004).
[CrossRef] [PubMed]

Wolff, P. A.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature 391, 667-669 (1998).
[CrossRef]

J. Mod. Opt. (1)

D. J. Nash and J. R. Sambles, "Surface plasmon study of the optical dielectric function of silver," J. Mod. Opt. 43, 81-91 (1996).

Nature (1)

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature 391, 667-669 (1998).
[CrossRef]

Phys. Rev. Lett. (1)

J. B. Pendry, "Negative refractive index makes a perfect lens," Phys. Rev. Lett. 85, 3966-3969 (2000).
[CrossRef] [PubMed]

Science (2)

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, "Metamaterials and negative refractive index," Science 305, 788-792 (2004).
[CrossRef] [PubMed]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, "Metamaterial electromagnetic cloak at microwave frequencies," Science 314, 977-980 (2006).
[CrossRef] [PubMed]

Thin Solid Films (1)

M. J. Tillin and J. R. Sambles, "A surface plasmon polariton study of the dielectric-constants of reactive metals - Aluminium," Thin Solid Films 167, 73-83 (1988).
[CrossRef]

Other (2)

J. A. Stratton, Electromagnetic Theory (McGraw Hill, 1941).

J. R. Reitz, F. J. Milford, and R. W. Christy, Foundations of Electromagnetic Theory, 4th Ed. (Addison-Wesley, 1992).

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Figures (5)

Fig. 1.
Fig. 1.

Plots of the transmissivity T as a function of silver film thickness d for a wide range of frequencies (permittivities in brackets). (a). linear; (b). logarithmic (Note that the line for f=3 ×1015 Hz corresponds to light above the plasma frequency and as such the silver is no longer acting as a metal – it is added for comparison only).

Fig. 2.
Fig. 2.

2kmk0 as a function of frequency with km being calculated using the values of ωp and τ as defined in the text.

Fig. 3.
Fig. 3.

Transmission as a function of film thickness d for silver and aluminium at 632.8 nm. For silver εm =-19+0.5i [7] and for aluminium -37+13i [8].

Fig. 4.
Fig. 4.

First order calculation for transmissivity T, using the Drude model for silver, with ωp =1.32×1016 s-1 and τ=1.45×10-14 s for d=20 nm and d=40nm. The solid line is exact.

Fig. 5.
Fig. 5.

Plot of 1/T against d2 for silver at f=3×1011 and 3×1012Hz (inset: close up for low values of d). All plots for lower frequencies overlie that of the line for f=3×1011.

Equations (1)

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T [ ( 1 A r ) 2 + A i 2 ] e 2 d k m k 0

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