Abstract

The distribution of zero crossings for the profile of a statistically rough surface of a silver thin film is determined and compared with a Poisson distribution. It is shown that the density of zero crossings may be related to the autocorrelation length and might be a useful parameter for characterizing spatial information of statistically rough surfaces.

© 1992 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. M. B. Priestley, Spectral Analysis and Time Series (Academic, New York, 1981), p. 113.
  2. G. M. Jenkins, D. G. Watts, Spectral Analysis and Its Applications (Holden-Day, San Francisco, Calif., 1968), p. 5.
  3. S. O. Rice, “Mathematical analysis of random noise,” Bell Syst. Tech. J. 23, 282–332 (1944); Bell Syst. Tech. J. 24, 46–156 (1945).
  4. J. S. Bendat, Principles and Applications of Random Noise Theory (Wiley, New York, 1958), p. 370.
  5. F. Varnier, N. Mayani, G. Rasigni, M. Rasigni, A. Llebaria, “Study of the surface roughness for thin films of LiF and Ag/LiF,” Surf. Sci. 188, 107–122 (1987).
    [CrossRef]
  6. M. Rasigni, G. Rasigni, J. P. Palmari, A. Llebaria, “Study of surface roughness using a microdensitometer analysis of electron micrographs of surface replicas: I. Surface profiles,” J. Opt. Soc. Am. 71, 1124–1133 (1981).
    [CrossRef]
  7. F. Varnier, M. Rasigni, G. Rasigni, J. P. Palmari, A. Llebaria, “Height and slope distributions for surfaces of rough metallic deposits,” Appl. Opt. 21, 3681–3684 (1982).
    [CrossRef] [PubMed]
  8. G. Rasigni, F. Varnier, M. Rasigni, J. P. Palmari, A. Llebaria, “Autocovariance functions for polished optical surfaces,” J. Opt. Soc. Am. 73, 222–233 (1983).
    [CrossRef]
  9. G. Rasigni, F. Varnier, M. Rasigni, J. P. Palmari, A. Llebaria, “Roughness spectrum and surface plasmons for surfaces of silver, copper, gold, and magnesium deposits,” Phys. Rev. B 27, 819–830 (1983).
    [CrossRef]
  10. T. R. Thomas, Rough Surfaces (Longmans, London, 1982), p. 92.
  11. J. M. Elson, J. M. Bennett, “Relation between the angular dependence of scattering and the statistical properties of optical surfaces,” J. Opt. Soc. Am. 69, 31–47 (1979).
    [CrossRef]
  12. Y. W. Lee, Statistical Theory of Communication (Wiley, New York, 1960), p. 177.
  13. A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, New York, 1965), p. 486.

1987 (1)

F. Varnier, N. Mayani, G. Rasigni, M. Rasigni, A. Llebaria, “Study of the surface roughness for thin films of LiF and Ag/LiF,” Surf. Sci. 188, 107–122 (1987).
[CrossRef]

1983 (2)

G. Rasigni, F. Varnier, M. Rasigni, J. P. Palmari, A. Llebaria, “Autocovariance functions for polished optical surfaces,” J. Opt. Soc. Am. 73, 222–233 (1983).
[CrossRef]

G. Rasigni, F. Varnier, M. Rasigni, J. P. Palmari, A. Llebaria, “Roughness spectrum and surface plasmons for surfaces of silver, copper, gold, and magnesium deposits,” Phys. Rev. B 27, 819–830 (1983).
[CrossRef]

1982 (1)

1981 (1)

1979 (1)

1944 (1)

S. O. Rice, “Mathematical analysis of random noise,” Bell Syst. Tech. J. 23, 282–332 (1944); Bell Syst. Tech. J. 24, 46–156 (1945).

Bendat, J. S.

J. S. Bendat, Principles and Applications of Random Noise Theory (Wiley, New York, 1958), p. 370.

Bennett, J. M.

Elson, J. M.

Jenkins, G. M.

G. M. Jenkins, D. G. Watts, Spectral Analysis and Its Applications (Holden-Day, San Francisco, Calif., 1968), p. 5.

Lee, Y. W.

Y. W. Lee, Statistical Theory of Communication (Wiley, New York, 1960), p. 177.

Llebaria, A.

Mayani, N.

F. Varnier, N. Mayani, G. Rasigni, M. Rasigni, A. Llebaria, “Study of the surface roughness for thin films of LiF and Ag/LiF,” Surf. Sci. 188, 107–122 (1987).
[CrossRef]

Palmari, J. P.

Papoulis, A.

A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, New York, 1965), p. 486.

Priestley, M. B.

M. B. Priestley, Spectral Analysis and Time Series (Academic, New York, 1981), p. 113.

Rasigni, G.

Rasigni, M.

Rice, S. O.

S. O. Rice, “Mathematical analysis of random noise,” Bell Syst. Tech. J. 23, 282–332 (1944); Bell Syst. Tech. J. 24, 46–156 (1945).

Thomas, T. R.

T. R. Thomas, Rough Surfaces (Longmans, London, 1982), p. 92.

Varnier, F.

F. Varnier, N. Mayani, G. Rasigni, M. Rasigni, A. Llebaria, “Study of the surface roughness for thin films of LiF and Ag/LiF,” Surf. Sci. 188, 107–122 (1987).
[CrossRef]

G. Rasigni, F. Varnier, M. Rasigni, J. P. Palmari, A. Llebaria, “Autocovariance functions for polished optical surfaces,” J. Opt. Soc. Am. 73, 222–233 (1983).
[CrossRef]

G. Rasigni, F. Varnier, M. Rasigni, J. P. Palmari, A. Llebaria, “Roughness spectrum and surface plasmons for surfaces of silver, copper, gold, and magnesium deposits,” Phys. Rev. B 27, 819–830 (1983).
[CrossRef]

F. Varnier, M. Rasigni, G. Rasigni, J. P. Palmari, A. Llebaria, “Height and slope distributions for surfaces of rough metallic deposits,” Appl. Opt. 21, 3681–3684 (1982).
[CrossRef] [PubMed]

Watts, D. G.

G. M. Jenkins, D. G. Watts, Spectral Analysis and Its Applications (Holden-Day, San Francisco, Calif., 1968), p. 5.

Appl. Opt. (1)

Bell Syst. Tech. J. (1)

S. O. Rice, “Mathematical analysis of random noise,” Bell Syst. Tech. J. 23, 282–332 (1944); Bell Syst. Tech. J. 24, 46–156 (1945).

J. Opt. Soc. Am. (3)

Phys. Rev. B (1)

G. Rasigni, F. Varnier, M. Rasigni, J. P. Palmari, A. Llebaria, “Roughness spectrum and surface plasmons for surfaces of silver, copper, gold, and magnesium deposits,” Phys. Rev. B 27, 819–830 (1983).
[CrossRef]

Surf. Sci. (1)

F. Varnier, N. Mayani, G. Rasigni, M. Rasigni, A. Llebaria, “Study of the surface roughness for thin films of LiF and Ag/LiF,” Surf. Sci. 188, 107–122 (1987).
[CrossRef]

Other (6)

Y. W. Lee, Statistical Theory of Communication (Wiley, New York, 1960), p. 177.

A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, New York, 1965), p. 486.

T. R. Thomas, Rough Surfaces (Longmans, London, 1982), p. 92.

J. S. Bendat, Principles and Applications of Random Noise Theory (Wiley, New York, 1958), p. 370.

M. B. Priestley, Spectral Analysis and Time Series (Academic, New York, 1981), p. 113.

G. M. Jenkins, D. G. Watts, Spectral Analysis and Its Applications (Holden-Day, San Francisco, Calif., 1968), p. 5.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)

Fig. 1
Fig. 1

Electron photomicrograph of a shadowed surface replica of Ag thin film (714-Å thickness) deposited onto an underlayer of LiF (1014-Å thickness). Pt shadow-casting exists at an angle of 50°; the line represents 1 μm.

Fig. 2
Fig. 2

(a) Profiles for three lines of the surface (for convenience only 500 points are represented); (b) corresponding height distribution. The dots represent the Gaussian ( 1 / 2 π δ ) exp ( - h 2 / 2 δ 2 ). The values of δ/Ra (with R a = 1 / L 0 L h d x) are given for each line.

Fig. 3
Fig. 3

Profiles of the rough surface: (a) continuous profile, (b) discrete values of the profile with data points such as hi = 0, (c) discrete values of the profile with a flat zone.

Fig. 4
Fig. 4

Probability P(K = k, τ) for k = 0, 1, 2 versus length τ: (a) related to one line; (b) related to an average of 14 regularly spaced lines; dotted curves, experimental results; solid curves, Poisson distribution.

Fig. 5
Fig. 5

Probability P(K = k, τ) versus k for various values of m (mean number of zero crossings by interval of length τ) related to intervals of pixel lengths 5, 15, 25, 35, 45, and 60; ●, experimental values; ×, Poissonian model.

Fig. 6
Fig. 6

Probability P(K = 1, τ) versus τ. The full line is a least-squares fit to the first five experimental data, namely, P(K = 1, τ) = 0.0649 τ.

Fig. 7
Fig. 7

Autocorrelation function ρ(τ) averaged over 14 regularly spaced lines for the surface studied. The initial portion of ρ(τ) is replotted on an expanded scale, and the dots represent the Gaussian ρ(τ) = exp(−τ22).

Fig. 8
Fig. 8

Autocorrelation function ρ S ˙ (τ) averaged over 14 regularly spaced lines of the surface slopes for the surface studied. The initial portion of ρ S ˙ (τ) is replotted on an expanded scale.

Tables (1)

Tables Icon

Table 1 Characteristics of the Microdensitometer Recording Process

Equations (24)

Equations on this page are rendered with MathJax. Learn more.

R a = 1 L 0 L h d x .
R a = - + h p ( h ) d h
p ( h ) = 1 2 π δ exp ( - h 2 / 2 δ 2 ) ,
R a = 2 π δ .
δ / R a ~ 1.25.
h ( x ) = 0.
h i h i + 1 0 if K = 1. h i h i + 1 > 0 if K = 0.
P ( K = k ; τ ) = m k k ! exp ( - m ) ,
P ( K = 1 ; τ ) = λ τ .
G ( τ ) = h ( x ) h ( x + τ ) ,
ρ ( τ ) = G ( τ ) G ( 0 ) ,
- 1 ρ ( τ ) 1.
cos α = ρ ( τ ) ,
P ( K = odd ; τ ) = α π .
ρ ( τ ) = cos [ π P ( K = odd ; τ ) ] .
P ( K = odd ; τ ) ~ P ( K = 1 ; τ ) .
P ( K = 1 ; τ ) = 1 π { 2 [ 1 - G ( τ ) G ( 0 ) ] } 1 / 2 .
G ( τ ) = G ( 0 ) + [ G ¨ ( τ ) 2 ] τ = 0 τ 2 + .. ,
P ( K = 1 ; τ ) = 1 π - G ¨ ( 0 ) G ( 0 ) = 1 π - ρ ¨ ( 0 ) τ = γ τ .
γ = 1 π - G ¨ ( 0 ) G ( 0 ) = 1 π - ρ ¨ ( 0 ) .
G ¨ ( x ) = - G S ˙ ( x ) ,
P ( K = 1 ; τ ) = 1 π G S ˙ ( 0 ) G ( 0 ) τ = μ τ where μ = 1 π G S ˙ ( 0 ) G ( 0 ) .
ρ ( τ ) = exp ( - τ 2 / σ 2 ) ,
P ( K = 1 ; τ ) = 2 π τ σ .

Metrics