Abstract

We investigate the scattering properties of a metal surface composed of the sum of a sinusoidal component and a one-dimensional Gaussian random process. In experimental work, such surfaces are produced in gold-coated photoresist by combining speckle-scanning methods with holographic grating fabrication techniques. In the diffusely scattered light, light bands and related effects that arise from the excitation of surface plasmon polaritons are observed; the angular position of the diffuse light bands corresponds to the positions of the resonant absorption anomalies of the unperturbed periodic surface. It is also shown that the measurements are closely consistent with the predictions of rigorous numerical methods based on the reduced Rayleigh equations, in which the diffuse scatter is determined through an average over an ensemble of rough surfaces. With an analytical theoretical method that treats the grating exactly and the roughness as a perturbation, it is shown that an observed enhancement of a diffuse light band in the backscattering configuration results from the coherent interference of scattering contributions from counterpropagating surface plasmon polaritons.

© 1995 Optical Society of America

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References

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  1. H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer-Verlag, Berlin, 1988).
  2. We consider a wave with the electric field vector in the plane of incidence to be ppolarized, with spolarization denoting the orthogonal case.
  3. M. C. Hutley, Diffraction Gratings (Academic, London, 1982), Chap. 6.
  4. M. C. Hutley, D. Maystre, “The total absorption of light by a diffraction grating,” Opt. Commun. 19, 431–436 (1976).
    [CrossRef]
  5. M. C. Hutley, V. M. Bird, “A detailed experimental study of the anomalies of a sinusoidal diffraction grating,” Opt. Acta 20, 771–782 (1973).
    [CrossRef]
  6. J. J. Cowan, “Holography with standing surface plasma waves,” Opt. Commun. 12, 373–378 (1974).
    [CrossRef]
  7. J. M. Simon, J. M. Gonzalvez Pagliere, “Diffuse light in diffraction gratings and surface plasma oscillations,” Opt. Acta 33, 1035–1049 (1986).
    [CrossRef]
  8. J. M. Simon, S. A. Ledesma, “Diffuse light bands from diffraction gratings: a correlation study,” Optik 89, 145–150 (1992).
  9. J. M. Simon, J. M. Gonzalvez Pagliere, “Diffuse light bands from diffraction gratings: polarization dependence and phase behavior,” J. Mod. Opt. 35, 1549–1555 (1988).
    [CrossRef]
  10. R. A. Depine, V. L. Brudny, “Speckle patterns generated by rough surfaces with a periodic component,” J. Mod. Opt. 38, 2281–2293 (1991).
    [CrossRef]
  11. V. L. Brudny, R. A. Depine, “Speckle pattern intensification in highly conducting microrough gratings,” J. Mod. Opt. 40, 427–439 (1993).
    [CrossRef]
  12. See Fig. 1 of A. D. Arsenieva, A. A. Maradudin, J. Q. Lu, A. R. McGurn, “Scattering of light from random surfaces that are periodic on average,” Opt. Lett. 18, 1588–1590 (1993).
    [CrossRef] [PubMed]
  13. Ref. 3, Chap. 4.
  14. M. E. Knotts, T. R. Michel, K. A. O’Donnell, “Comparisons of theory and experiment in light scattering from a randomly rough surface,” J. Opt. Soc. Am. A 10, 928–941 (1993).
    [CrossRef]
  15. M. E. Knotts, K. A. O’Donnell, “Measurements of light scattering by a series of conducting surfaces with one-dimensional roughness,” J. Opt. Soc. Am. A 11, 697–710 (1994).
    [CrossRef]
  16. F. Toigo, A. Marvin, V. Celli, N. R. Hill, “Optical properties of rough surfaces: general theory and the small roughness limit,” Phys. Rev. B 15, 5618–5626 (1977).
    [CrossRef]
  17. T. R. Michel, “Resonant light scattering from weakly rough random surfaces and imperfect gratings,” J. Opt. Soc. Am. A 11, 1874–1885 (1994).
    [CrossRef]
  18. K. A. O’Donnell, M. E. Knotts, “Polarization dependence of scattering from one-dimensional rough surfaces,” J. Opt. Soc. Am. A 8, 1126–1131 (1991).
    [CrossRef]
  19. T. R. Michel, M. E. Knotts, K. A. O’Donnell, “Stokes matrix of a one-dimensional perfectly conducting rough surface,” J. Opt. Soc. Am. A 9, 585–596 (1992).
    [CrossRef]
  20. E. K. Popov, L. V. Tsonev, M. L. Sabeva, “Technological problems in holographic recording of plane gratings,” Opt. Eng. 31, 2168–2173 (1992).
    [CrossRef]
  21. A. A. Maradudin, E. R. Méndez, “Enhanced backscattering of light from weakly rough, random metal surfaces,” Appl. Opt. 32, 3335–3343 (1993), and references therein.
    [CrossRef] [PubMed]

1994 (2)

1993 (4)

1992 (3)

J. M. Simon, S. A. Ledesma, “Diffuse light bands from diffraction gratings: a correlation study,” Optik 89, 145–150 (1992).

T. R. Michel, M. E. Knotts, K. A. O’Donnell, “Stokes matrix of a one-dimensional perfectly conducting rough surface,” J. Opt. Soc. Am. A 9, 585–596 (1992).
[CrossRef]

E. K. Popov, L. V. Tsonev, M. L. Sabeva, “Technological problems in holographic recording of plane gratings,” Opt. Eng. 31, 2168–2173 (1992).
[CrossRef]

1991 (2)

R. A. Depine, V. L. Brudny, “Speckle patterns generated by rough surfaces with a periodic component,” J. Mod. Opt. 38, 2281–2293 (1991).
[CrossRef]

K. A. O’Donnell, M. E. Knotts, “Polarization dependence of scattering from one-dimensional rough surfaces,” J. Opt. Soc. Am. A 8, 1126–1131 (1991).
[CrossRef]

1988 (1)

J. M. Simon, J. M. Gonzalvez Pagliere, “Diffuse light bands from diffraction gratings: polarization dependence and phase behavior,” J. Mod. Opt. 35, 1549–1555 (1988).
[CrossRef]

1986 (1)

J. M. Simon, J. M. Gonzalvez Pagliere, “Diffuse light in diffraction gratings and surface plasma oscillations,” Opt. Acta 33, 1035–1049 (1986).
[CrossRef]

1977 (1)

F. Toigo, A. Marvin, V. Celli, N. R. Hill, “Optical properties of rough surfaces: general theory and the small roughness limit,” Phys. Rev. B 15, 5618–5626 (1977).
[CrossRef]

1976 (1)

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

1974 (1)

J. J. Cowan, “Holography with standing surface plasma waves,” Opt. Commun. 12, 373–378 (1974).
[CrossRef]

1973 (1)

M. C. Hutley, V. M. Bird, “A detailed experimental study of the anomalies of a sinusoidal diffraction grating,” Opt. Acta 20, 771–782 (1973).
[CrossRef]

Arsenieva, A. D.

Bird, V. M.

M. C. Hutley, V. M. Bird, “A detailed experimental study of the anomalies of a sinusoidal diffraction grating,” Opt. Acta 20, 771–782 (1973).
[CrossRef]

Brudny, V. L.

V. L. Brudny, R. A. Depine, “Speckle pattern intensification in highly conducting microrough gratings,” J. Mod. Opt. 40, 427–439 (1993).
[CrossRef]

R. A. Depine, V. L. Brudny, “Speckle patterns generated by rough surfaces with a periodic component,” J. Mod. Opt. 38, 2281–2293 (1991).
[CrossRef]

Celli, V.

F. Toigo, A. Marvin, V. Celli, N. R. Hill, “Optical properties of rough surfaces: general theory and the small roughness limit,” Phys. Rev. B 15, 5618–5626 (1977).
[CrossRef]

Cowan, J. J.

J. J. Cowan, “Holography with standing surface plasma waves,” Opt. Commun. 12, 373–378 (1974).
[CrossRef]

Depine, R. A.

V. L. Brudny, R. A. Depine, “Speckle pattern intensification in highly conducting microrough gratings,” J. Mod. Opt. 40, 427–439 (1993).
[CrossRef]

R. A. Depine, V. L. Brudny, “Speckle patterns generated by rough surfaces with a periodic component,” J. Mod. Opt. 38, 2281–2293 (1991).
[CrossRef]

Gonzalvez Pagliere, J. M.

J. M. Simon, J. M. Gonzalvez Pagliere, “Diffuse light bands from diffraction gratings: polarization dependence and phase behavior,” J. Mod. Opt. 35, 1549–1555 (1988).
[CrossRef]

J. M. Simon, J. M. Gonzalvez Pagliere, “Diffuse light in diffraction gratings and surface plasma oscillations,” Opt. Acta 33, 1035–1049 (1986).
[CrossRef]

Hill, N. R.

F. Toigo, A. Marvin, V. Celli, N. R. Hill, “Optical properties of rough surfaces: general theory and the small roughness limit,” Phys. Rev. B 15, 5618–5626 (1977).
[CrossRef]

Hutley, M. C.

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

M. C. Hutley, V. M. Bird, “A detailed experimental study of the anomalies of a sinusoidal diffraction grating,” Opt. Acta 20, 771–782 (1973).
[CrossRef]

M. C. Hutley, Diffraction Gratings (Academic, London, 1982), Chap. 6.

Knotts, M. E.

Ledesma, S. A.

J. M. Simon, S. A. Ledesma, “Diffuse light bands from diffraction gratings: a correlation study,” Optik 89, 145–150 (1992).

Lu, J. Q.

Maradudin, A. A.

Marvin, A.

F. Toigo, A. Marvin, V. Celli, N. R. Hill, “Optical properties of rough surfaces: general theory and the small roughness limit,” Phys. Rev. B 15, 5618–5626 (1977).
[CrossRef]

Maystre, D.

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

McGurn, A. R.

Méndez, E. R.

Michel, T. R.

O’Donnell, K. A.

Popov, E. K.

E. K. Popov, L. V. Tsonev, M. L. Sabeva, “Technological problems in holographic recording of plane gratings,” Opt. Eng. 31, 2168–2173 (1992).
[CrossRef]

Raether, H.

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

Sabeva, M. L.

E. K. Popov, L. V. Tsonev, M. L. Sabeva, “Technological problems in holographic recording of plane gratings,” Opt. Eng. 31, 2168–2173 (1992).
[CrossRef]

Simon, J. M.

J. M. Simon, S. A. Ledesma, “Diffuse light bands from diffraction gratings: a correlation study,” Optik 89, 145–150 (1992).

J. M. Simon, J. M. Gonzalvez Pagliere, “Diffuse light bands from diffraction gratings: polarization dependence and phase behavior,” J. Mod. Opt. 35, 1549–1555 (1988).
[CrossRef]

J. M. Simon, J. M. Gonzalvez Pagliere, “Diffuse light in diffraction gratings and surface plasma oscillations,” Opt. Acta 33, 1035–1049 (1986).
[CrossRef]

Toigo, F.

F. Toigo, A. Marvin, V. Celli, N. R. Hill, “Optical properties of rough surfaces: general theory and the small roughness limit,” Phys. Rev. B 15, 5618–5626 (1977).
[CrossRef]

Tsonev, L. V.

E. K. Popov, L. V. Tsonev, M. L. Sabeva, “Technological problems in holographic recording of plane gratings,” Opt. Eng. 31, 2168–2173 (1992).
[CrossRef]

Appl. Opt. (1)

J. Mod. Opt. (3)

J. M. Simon, J. M. Gonzalvez Pagliere, “Diffuse light bands from diffraction gratings: polarization dependence and phase behavior,” J. Mod. Opt. 35, 1549–1555 (1988).
[CrossRef]

R. A. Depine, V. L. Brudny, “Speckle patterns generated by rough surfaces with a periodic component,” J. Mod. Opt. 38, 2281–2293 (1991).
[CrossRef]

V. L. Brudny, R. A. Depine, “Speckle pattern intensification in highly conducting microrough gratings,” J. Mod. Opt. 40, 427–439 (1993).
[CrossRef]

J. Opt. Soc. Am. A (5)

Opt. Acta (2)

J. M. Simon, J. M. Gonzalvez Pagliere, “Diffuse light in diffraction gratings and surface plasma oscillations,” Opt. Acta 33, 1035–1049 (1986).
[CrossRef]

M. C. Hutley, V. M. Bird, “A detailed experimental study of the anomalies of a sinusoidal diffraction grating,” Opt. Acta 20, 771–782 (1973).
[CrossRef]

Opt. Commun. (2)

J. J. Cowan, “Holography with standing surface plasma waves,” Opt. Commun. 12, 373–378 (1974).
[CrossRef]

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

Opt. Eng. (1)

E. K. Popov, L. V. Tsonev, M. L. Sabeva, “Technological problems in holographic recording of plane gratings,” Opt. Eng. 31, 2168–2173 (1992).
[CrossRef]

Opt. Lett. (1)

Optik (1)

J. M. Simon, S. A. Ledesma, “Diffuse light bands from diffraction gratings: a correlation study,” Optik 89, 145–150 (1992).

Phys. Rev. B (1)

F. Toigo, A. Marvin, V. Celli, N. R. Hill, “Optical properties of rough surfaces: general theory and the small roughness limit,” Phys. Rev. B 15, 5618–5626 (1977).
[CrossRef]

Other (4)

Ref. 3, Chap. 4.

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

We consider a wave with the electric field vector in the plane of incidence to be ppolarized, with spolarization denoting the orthogonal case.

M. C. Hutley, Diffraction Gratings (Academic, London, 1982), Chap. 6.

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Figures (15)

Fig. 1
Fig. 1

Segments of typical profilometer scans of patches A–E. The vertical scale of each segment is ±60 nm; this scale exaggerates the slope by a factor of 5.

Fig. 2
Fig. 2

Statistical results from surface profilometry. Top, probability densities of surface height for patches A–E; bottom, positive frequency parts of spectral densities S(|q|) for patches A–E; normalization is such that the total surface variance in square nanometers is 1 / 2 π - S ( q ) d q. The peak at 2π/p0 ≅ 0.0113 nm–1 rises typically by a factor of 6 above the scale. The dashed curve is the fit to the roughness spectrum of patch E employed in Sections 4 and 5.

Fig. 3
Fig. 3

Zeroth-order diffracted powers Pp (top) and Ps (bottom) for patches A and C–E as a function of incidence angle θi. The dashed vertical line indicates the emergence of the first diffracted order. Perfect reflectivity is denoted by 0.5; results for patch B are almost identical to those for A and are not shown.

Fig. 4
Fig. 4

Diffuse intensities Ip and Is for incidence angle θi = 0° (patches B–E in order of increasing curve height). Upper right, detail of Ip for |θs| ≤ 12°.

Fig. 5
Fig. 5

Diffuse intensities Ip and Is for incidence angle θi = 4.8° (patches B–E in order of increasing curve height). Upper right, detail of Ip for |θs| ≤ 12°.

Fig. 6
Fig. 6

Diffuse intensities Ip and Is for incidence angle θi = 10° (patches B–E in order of increasing curve height). The diffracted order at θs = −75° is not shown. Upper right, detail of Ip for |θs| ≤ 8°. Although data are not shown at the exact backscattering position, tilting the surface reveals no backscattering enhancement in these results.

Fig. 7
Fig. 7

Diffuse intensity I+ for patches C (dashed curves) and E (solid curves) for the incidence angles shown.

Fig. 8
Fig. 8

Matrix element s34 for patches C (dashed curves) and E (solid curves) for the incidence angles shown.

Fig. 9
Fig. 9

Diffuse intensities Ip and I+ for patch D as a function of the incident and scattering angles θi and θs. The band at θs ≅ −4.8° is enhanced as it passes through the backscattering direction (θs = −θi). Vertical arrows denote the zeroth grating order; the faint structures in Ip along the plot’s diagonals arise from speckle noise.

Fig. 10
Fig. 10

For incidence angle θi = 40°, Ip, Is, I+, and s34 for patches B (lowest curves) through E (highest curves). Ticks on the vertical axis are spaced by 0.005, and the dashed horizontal lines denote zero for the respective quantities. Dashed curve, Ip for unroughened patch A taken under identical conditions (flux collection ±0.06° from the plane of incidence), multiplied by 3000.

Fig. 11
Fig. 11

For θi = 0° (top), θi = 4.8° (center), and θi = 10° (bottom), the diffuse intensity Ip as obtained from numerical simulations (solid curves) and from the perturbation theory treating the grating exactly (dashed curves). Also shown are expanded details for |θs| ≤ 12°. Parameters are λ = 633 nm, = −9.93 + 1.053i, ca/ω = 97.7 nm, cb/ω = 282 nm, δ = 13.0 nm, h = 18.0 nm, p0 = 555 nm, Nk = 700, λ/L = 0.691 × 10−3, Np = 1000, and Ng = 2.

Fig. 12
Fig. 12

For the incidence angles shown, the diffuse intensity I+ as obtained from numerical simulations (solid curves) and from the perturbation theory treating the grating exactly (dashed curves). Parameters are the same as in Fig. 11.

Fig. 13
Fig. 13

For the incidence angles shown, matrix element s34 obtained from numerical simulations (solid curves) and from the perturbation theory treating the grating exactly (dashed curves). Parameters are the same as in Fig. 11.

Fig. 14
Fig. 14

Top, exact calculations (solid curves) of Is compared with lowest-order perturbation theory (dashed curves) for (left to right) θi = 0°, 4.8°, and 10°; bottom, for θi = 4.8°, experimental data (dashed curves) for Is compared with lowest-order perturbation theory (solid curves) for the measured roughness spectra of Fig. 2 (patches B–E in order of increasing height).

Fig. 15
Fig. 15

For θi = 0°, 4.8°, and 10°, contributions to the diffuse intensity Ip from the series of Eqs. (12). Dashed curves, ∑1; dotted–dashed curves, ∑2; solid curves, ∑3.

Tables (2)

Tables Icon

Table 1 Parameters of Patches A–E

Tables Icon

Table 2 Scattered Powers Contained in Diffuse Intensities

Equations (12)

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S = [ s 11 s 12 0 0 s 12 s 11 0 0 0 0 s 33 s 34 0 0 - s 34 s 33 ] ,
s 11 = I p + I s = I + + I - = I R + I L , s 12 = I p - I s , s 33 = I + - I - , - s 34 = I R - I L ,
Φ ν ( x 1 , x 3 ) = exp { i ω c [ k x 1 - α 0 ( k ) x 3 ] } + - d q 2 π R ν ( q k ) exp { i ω c [ q x 1 + α 0 ( q ) x 3 ] } ,
s 11 = f ( θ s , θ i ) [ Δ R p ( q k ) 2 + Δ R s ( q k ) 2 ] , s 12 = f ( θ s , θ i ) [ Δ R p ( q k ) 2 - Δ R s ( q k ) 2 ] , s 33 = 2 f ( θ s , θ i ) Re [ Δ R p * ( q k ) Δ R s ( q k ) ] , s 34 = 2 f ( θ s , θ i ) Im [ Δ R p * ( q k ) Δ R s ( q k ) ] ,
- d q 2 π M ν ( + ) ( p q ) R ν ( q k ) = - M ν ( - ) ( p k ) ,
M ν ( ± ) ( p q ) = I ( α ɛ ( p ) α 0 ( q ) p - q ) × [ p q ± α ɛ ( p ) α 0 ( q ) ] σ α ɛ ( p ) α 0 ( q ) , I ( α k ) = ω c - d x 1 exp { - i ω c [ k x 1 + α s ( x 1 ) ] } ,
R ν ( q k ) = 2 π δ ( q - k ) R ν ( 0 ) ( k ) - 2 i G ν ( 0 ) ( q ) × - d p 2 π V ν ( q p ) G ν ( p k ) α 0 ( k ) ,
G ν ( q k ) = 2 π δ ( q - k ) G ν ( 0 ) ( k ) + G ν ( 0 ) ( q ) × - d p 2 π V ν ( q p ) G ν ( p k ) ,
U ν ( q k ) = ( ɛ - 1 ) [ ɛ q k - α ɛ ( q ) α ɛ ( k ) ɛ 2 ] σ ( ω c ) 2 × - d x 1 exp [ - i ω c ( q - k ) x 1 ] ζ ( x 1 ) .
Δ R ν ( q k ) = - 2 i m = - N g N g n = - N g N g g ν m ( q - m ) U ν ( q - m k n ) × g ν n ( k ) α 0 ( k ) ,
S ( q ) = ω c - d x 1 exp ( - i ω c q x 1 ) W ( x 1 ) = 2 b exp ( - a 2 / b 2 ) [ 1 - erf ( a / b ) ] exp ( - q 2 a 2 ) 1 + q 2 b 2 .
Σ 1 ( q k ) = 4 f ( θ s , θ i ) cos 2 θ i g 0 ( q ) U ( q k ) g 0 ( k ) 2 , Σ 2 ( q k ) = 4 f ( θ s , θ i ) cos 2 θ i × m , n = - 2 ( m , n ) ( 0 , 0 ) 2 g m ( q - m ) U ( q - m k n ) g n ( k ) 2 , Σ 3 ( q k ) = 8 f ( θ s , θ i ) cos 2 θ i × Re m , n = - 2 ( m > n ) 2 [ g m ( q - m ) U ( q - m k n ) g n ( k ) ] * × g n ( q - n ) U ( q - n k m ) g m ( k ) ,

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