Abstract

Metal surfaces, which are generally regarded as excellent reflectors of electromagnetic radiation, may, at high angles of incidence, become strong absorbers for transverse magnetic radiation. This effect, often referred to as the pseudo-Brewster angle, results in a reflectivity minimum, and is most strongly evident in the microwave domain, where metals are often treated as perfect conductors. A detailed analysis of this reflectivity minimum is presented here and it is shown why, in the limit of very long wavelengths, metals close to grazing incidence have a minimum in reflectance given by (√2-1)2.

© 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 refraction 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. B. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305, 847–848 (2004).
    [Crossref] [PubMed]
  6. A. P. Hibbins, B. R. Evans, and J. R. Sambles, “Experimental verification of designer surface plasmons,” Science 308, 670–672 (2005).
    [Crossref] [PubMed]
  7. J. A. Stratton, Electromagnetic Theory (McGraw Hill, 1941).
  8. S. P. F. Humphreys-Owen, “Comparison of reflection methods for measuring optical constants without polarimetric analysis,” Proc. Phys. Soc. 77, 949–957 (1961).
    [Crossref]
  9. E. D. Palik, Handbook of Optical Constants of Solids (Academic, 1985).
  10. H. Raether, Surface Plasmons on Smooth and Rough surfaces (Springer-Verlag, 1988).
  11. E. Kretschmann and H. Raether, “Radiative decay of non-radiative surface plasmons excited by light,” Z. Naturforsch. Teil A 23, 2135–2136 (1968).
  12. M. C. Hutley and D. Masytre, “The total absorption of light by a diffraction grating,” Opt. Commun. 19, 431–436 (1976).
    [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]

2005 (1)

A. P. Hibbins, B. R. Evans, and J. R. Sambles, “Experimental verification of designer surface plasmons,” Science 308, 670–672 (2005).
[Crossref] [PubMed]

2004 (2)

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, L. Martin-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305, 847–848 (2004).
[Crossref] [PubMed]

2000 (1)

J. B. Pendry, “Negative refraction 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]

1976 (1)

M. C. Hutley and D. Masytre, “The total absorption of light by a diffraction grating,” Opt. Commun. 19, 431–436 (1976).
[Crossref]

1968 (1)

E. Kretschmann and H. Raether, “Radiative decay of non-radiative surface plasmons excited by light,” Z. Naturforsch. Teil A 23, 2135–2136 (1968).

1961 (1)

S. P. F. Humphreys-Owen, “Comparison of reflection methods for measuring optical constants without polarimetric analysis,” Proc. Phys. Soc. 77, 949–957 (1961).
[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]

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]

Garcia-Vidal, F. J.

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

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]

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]

Humphreys-Owen, S. P. F.

S. P. F. Humphreys-Owen, “Comparison of reflection methods for measuring optical constants without polarimetric analysis,” Proc. Phys. Soc. 77, 949–957 (1961).
[Crossref]

Hutley, M. C.

M. C. Hutley and D. Masytre, “The total absorption of light by a diffraction grating,” Opt. Commun. 19, 431–436 (1976).
[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]

Kretschmann, E.

E. Kretschmann and H. Raether, “Radiative decay of non-radiative surface plasmons excited by light,” Z. Naturforsch. Teil A 23, 2135–2136 (1968).

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]

Martin-Moreno, L.

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

Masytre, D.

M. C. Hutley and D. Masytre, “The total absorption of light by a diffraction grating,” Opt. Commun. 19, 431–436 (1976).
[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]

Palik, E. D.

E. D. Palik, Handbook of Optical Constants of Solids (Academic, 1985).

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]

J. B. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305, 847–848 (2004).
[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 refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000).
[Crossref] [PubMed]

Raether, H.

E. Kretschmann and H. Raether, “Radiative decay of non-radiative surface plasmons excited by light,” Z. Naturforsch. Teil A 23, 2135–2136 (1968).

H. Raether, Surface Plasmons on Smooth and Rough surfaces (Springer-Verlag, 1988).

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]

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]

Stratton, J. A.

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

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]

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]

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]

Opt. Commun. (1)

M. C. Hutley and D. Masytre, “The total absorption of light by a diffraction grating,” Opt. Commun. 19, 431–436 (1976).
[Crossref]

Phys. Rev. Lett. (1)

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000).
[Crossref] [PubMed]

Proc. Phys. Soc. (1)

S. P. F. Humphreys-Owen, “Comparison of reflection methods for measuring optical constants without polarimetric analysis,” Proc. Phys. Soc. 77, 949–957 (1961).
[Crossref]

Science (4)

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]

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

A. P. Hibbins, B. R. Evans, and J. R. Sambles, “Experimental verification of designer surface plasmons,” Science 308, 670–672 (2005).
[Crossref] [PubMed]

Z. Naturforsch. Teil A (1)

E. Kretschmann and H. Raether, “Radiative decay of non-radiative surface plasmons excited by light,” Z. Naturforsch. Teil A 23, 2135–2136 (1968).

Other (3)

E. D. Palik, Handbook of Optical Constants of Solids (Academic, 1985).

H. Raether, Surface Plasmons on Smooth and Rough surfaces (Springer-Verlag, 1988).

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

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Figures (4)

Fig. 1.
Fig. 1.

p-polarised reflectivity from a planar surface as a function of incident angle (θ) for an interface between air and silver described by a Drude model with ωp=1.32×1016 rad/s and τ=1.45×1014 s at wavelengths of 600 nm, 60 microns and 6 mm. Also shown are data for a wavelength of 600 nm with permittivity values taken from Palik [9] of εr =13.91, εi =0.93. (Inset: reflectivity as a function of tan(θ) [log scale]).

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Fig. 2.
Fig. 2.

Reflectivity minimum due to surface plasmon (SP) excitation using the Kretschmann-Raether [11] geometry (inset). Light of 600 nm wavelength is incident upon a 50 nm thick silver film (εr=-13.91, εi=0.9255) through a glass prism (n=1.5) with air bounding. The SP is excited at a particular internal angle (measured from the normal to the interface) giving a reflection minimum.

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Fig. 3.
Fig. 3.

The angle (open squares) and reflectivity (solid squares) at the reflectivity minimum as a function of| εr |=εi . The angle of the minimum is observed to rapidly approach grazing incidence, whilst the reflectivity of the minimum asymptotically approaches 44.7%. Inset: The reflection minimum for the case when | εr |=εi =18300 which occurs at a wavelength of 27.3 µm for the Drude model with the parameters used here.

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Fig. 4.
Fig. 4.

Reflectivity curves for various values of εi as functions of tanθ (log scale) with ε 1=1 and εr =-40000 (such that for the curve with εi =1×107 the system approximates that for silver at a wavelength of 1cm). The asymptotic limit of the minimum reflectance of 17.16% when εi ≫|εr |≫ε 1 is clearly evident. Inset: tan(θ) (log scale) as a function of ε i (log scale) demonstrating the validity of the equation obtained for the angle of the reflectance minimum θ ≅ ε i ε 1 in this limit.

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Equations (6)

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r p = ε m ε 1 cos ⁡ θ − [ ε m ε 1 − sin 2 θ ] 1 2 ε m ε 1 cos ⁡ θ + [ ε m ε 1 − sin 2 θ ] 1 2
( 1 − 2 ε r ε 1 ε r 2 + ε i 2 ) ε 1 2 T 3 + 3 ε 1 2 T 2 − ( ε r 2 + ε i 2 ) ( 1 + T ) = 0
[ E z E x ] 2 = A 2 + B 2 = ε 1 sin ⁡ 2 θ ε i
tan ϕ = − k zi k zr = ε i 1 2 ( 1 − ε r 2 ε i ) ε i 1 2 ( 1 + ε r 2 ε i ) ≈ − 1 .
1 − E o = E x cos ⁡ θ .
1 + E o ≅ i ( ε i ε 1 ) 1 2 E x e i π 4 .

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