Abstract

Statistical parameters are obtained for an ensemble of specular points at a randomly rough Gaussian statistically isotropic surface at normal incidence. The joint probability density functions (PDFs) of specular point heights and total curvátures are derived separately for maxima, minima, and saddle points. The joint PDFs of brightness and surface elevations of specular points of different types are obtained analytically in an explicit form.

© 2006 Optical Society of America

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  1. C. Zeng, J. Kondev, D. McNamara, and A. A. Middleton, "Statistical topography of glassy interfaces," Phys. Rev. Lett. 80, 109-112 (1998).
    [CrossRef]
  2. A. S. Balankin and O. Sussarey, "A new statistical distribution for self-affine crack roughness parameters," Philos. Mag. Lett. 79, 629-637 (1999).
    [CrossRef]
  3. P. Lehmann, "Surface roughness measurement based on the intensity correlation function of scattered light under speckle-pattern illumination," Appl. Opt. 38, 1144-1152 (1999).
    [CrossRef]
  4. M. Elias and M. Menu, "Experimental characterization of a random metallic rough surface by spectrometric measurements in the visible range," Opt. Commun. 180, 191-198 (2000).
    [CrossRef]
  5. F. P. Quintian, M. A. Rebollo, R. G. Berlasso, and N. G. Gaggiolli, "A valuable method for online wire quality control: light scattering from cylindrical rough surface," in Proc. SPIE 4399, 328-334 (2003).
    [CrossRef]
  6. C. Cox and W. Munk, "Measurement of roughness of the sea surface from photographs of the sun's glitter," J. Opt. Soc. Am. 44, 838-850 (1954).
    [CrossRef]
  7. C. Cox and W. Munk, "Statistics of sea surface derived from the sun's glitter," J. Mar. Res. 13, 198-227 (1954).
  8. O. I. Yordanov and M. A. Michalev, "Statistics of specular points on a non-Gaussian random surface," J. Opt. Soc. Am. A 6, 1578-1583 (1989).
    [CrossRef]
  9. G. L. Stamm and L. A. Harris, "Sea echo measurements made with 1.06-µm laser radiation," Appl. Opt. 13, 2477-2479 (1974).
    [CrossRef] [PubMed]
  10. J. L. Bufton, F. E. Hoge, and R. N. Swift, "Airborne measurements of laser backscatter from the ocean surface," Appl. Opt. 22, 2603-2618 (1983).
    [CrossRef] [PubMed]
  11. J. A. Reagan and D. A. Zielinskie, "Spaceborne lidar remote sensing techniques aided by surface return," Opt. Eng. 30, 96-102 (1991).
    [CrossRef]
  12. J. H. Churnside, S. G. Handson, and J. W. Wilson, "Determination of ocean wave spectra from images of backscattered incoherent light," Appl. Opt. 34, 962-968 (1995).
    [CrossRef] [PubMed]
  13. M. S. Longuet-Higgins, "The statistical analysis of a random, moving surface," Philos. Trans. R. Soc. London, Ser. A 249, 321-387 (1957).
    [CrossRef]
  14. M. S. Longuet-Higgins, "The statistical distribution of the curvature of a random Gaussian surface," Proc. Cambridge Philos. Soc. 54, 439-454 (1958).
    [CrossRef]
  15. M. S. Longuet-Higgins, "The distribution of the sizes of images reflected in a random surface," Proc. Cambridge Philos. Soc. 55, 91-100 (1959).
    [CrossRef]
  16. M. S. Longuet-Higgins, "Reflection and refraction at a random moving surface," J. Opt. Soc. Am. 50, 838-856 (1960).
    [CrossRef]
  17. M. S. Longuet-Higgins, "The statistical geometry of random surfaces," in Hydrodynamic Instability, in Proceedings of the 13th Symposium on Applied Mathematics (American Mathematical Society. 1962), Vol. 13, pp. 105-143
  18. R. J. Adler, The Geometry of Random Fields (Wiley, 1981).
  19. R. G. Gardachov, "The probability density of the total curvature of a uniform random Gaussian sea surface in the specular points," Int. J. Remote Sens. 21, 2917-2926 (2000).
    [CrossRef]

2003

F. P. Quintian, M. A. Rebollo, R. G. Berlasso, and N. G. Gaggiolli, "A valuable method for online wire quality control: light scattering from cylindrical rough surface," in Proc. SPIE 4399, 328-334 (2003).
[CrossRef]

2000

M. Elias and M. Menu, "Experimental characterization of a random metallic rough surface by spectrometric measurements in the visible range," Opt. Commun. 180, 191-198 (2000).
[CrossRef]

R. G. Gardachov, "The probability density of the total curvature of a uniform random Gaussian sea surface in the specular points," Int. J. Remote Sens. 21, 2917-2926 (2000).
[CrossRef]

1999

P. Lehmann, "Surface roughness measurement based on the intensity correlation function of scattered light under speckle-pattern illumination," Appl. Opt. 38, 1144-1152 (1999).
[CrossRef]

A. S. Balankin and O. Sussarey, "A new statistical distribution for self-affine crack roughness parameters," Philos. Mag. Lett. 79, 629-637 (1999).
[CrossRef]

1998

C. Zeng, J. Kondev, D. McNamara, and A. A. Middleton, "Statistical topography of glassy interfaces," Phys. Rev. Lett. 80, 109-112 (1998).
[CrossRef]

1995

1991

J. A. Reagan and D. A. Zielinskie, "Spaceborne lidar remote sensing techniques aided by surface return," Opt. Eng. 30, 96-102 (1991).
[CrossRef]

1989

1983

1974

1960

1959

M. S. Longuet-Higgins, "The distribution of the sizes of images reflected in a random surface," Proc. Cambridge Philos. Soc. 55, 91-100 (1959).
[CrossRef]

1958

M. S. Longuet-Higgins, "The statistical distribution of the curvature of a random Gaussian surface," Proc. Cambridge Philos. Soc. 54, 439-454 (1958).
[CrossRef]

1957

M. S. Longuet-Higgins, "The statistical analysis of a random, moving surface," Philos. Trans. R. Soc. London, Ser. A 249, 321-387 (1957).
[CrossRef]

1954

C. Cox and W. Munk, "Statistics of sea surface derived from the sun's glitter," J. Mar. Res. 13, 198-227 (1954).

C. Cox and W. Munk, "Measurement of roughness of the sea surface from photographs of the sun's glitter," J. Opt. Soc. Am. 44, 838-850 (1954).
[CrossRef]

Adler, R. J.

R. J. Adler, The Geometry of Random Fields (Wiley, 1981).

Balankin, A. S.

A. S. Balankin and O. Sussarey, "A new statistical distribution for self-affine crack roughness parameters," Philos. Mag. Lett. 79, 629-637 (1999).
[CrossRef]

Berlasso, R. G.

F. P. Quintian, M. A. Rebollo, R. G. Berlasso, and N. G. Gaggiolli, "A valuable method for online wire quality control: light scattering from cylindrical rough surface," in Proc. SPIE 4399, 328-334 (2003).
[CrossRef]

Bufton, J. L.

Churnside, J. H.

Cox, C.

C. Cox and W. Munk, "Measurement of roughness of the sea surface from photographs of the sun's glitter," J. Opt. Soc. Am. 44, 838-850 (1954).
[CrossRef]

C. Cox and W. Munk, "Statistics of sea surface derived from the sun's glitter," J. Mar. Res. 13, 198-227 (1954).

Elias, M.

M. Elias and M. Menu, "Experimental characterization of a random metallic rough surface by spectrometric measurements in the visible range," Opt. Commun. 180, 191-198 (2000).
[CrossRef]

Gaggiolli, N. G.

F. P. Quintian, M. A. Rebollo, R. G. Berlasso, and N. G. Gaggiolli, "A valuable method for online wire quality control: light scattering from cylindrical rough surface," in Proc. SPIE 4399, 328-334 (2003).
[CrossRef]

Gardachov, R. G.

R. G. Gardachov, "The probability density of the total curvature of a uniform random Gaussian sea surface in the specular points," Int. J. Remote Sens. 21, 2917-2926 (2000).
[CrossRef]

Handson, S. G.

Harris, L. A.

Hoge, F. E.

Kondev, J.

C. Zeng, J. Kondev, D. McNamara, and A. A. Middleton, "Statistical topography of glassy interfaces," Phys. Rev. Lett. 80, 109-112 (1998).
[CrossRef]

Lehmann, P.

Longuet-Higgins, M. S.

M. S. Longuet-Higgins, "Reflection and refraction at a random moving surface," J. Opt. Soc. Am. 50, 838-856 (1960).
[CrossRef]

M. S. Longuet-Higgins, "The distribution of the sizes of images reflected in a random surface," Proc. Cambridge Philos. Soc. 55, 91-100 (1959).
[CrossRef]

M. S. Longuet-Higgins, "The statistical distribution of the curvature of a random Gaussian surface," Proc. Cambridge Philos. Soc. 54, 439-454 (1958).
[CrossRef]

M. S. Longuet-Higgins, "The statistical analysis of a random, moving surface," Philos. Trans. R. Soc. London, Ser. A 249, 321-387 (1957).
[CrossRef]

M. S. Longuet-Higgins, "The statistical geometry of random surfaces," in Hydrodynamic Instability, in Proceedings of the 13th Symposium on Applied Mathematics (American Mathematical Society. 1962), Vol. 13, pp. 105-143

McNamara, D.

C. Zeng, J. Kondev, D. McNamara, and A. A. Middleton, "Statistical topography of glassy interfaces," Phys. Rev. Lett. 80, 109-112 (1998).
[CrossRef]

Menu, M.

M. Elias and M. Menu, "Experimental characterization of a random metallic rough surface by spectrometric measurements in the visible range," Opt. Commun. 180, 191-198 (2000).
[CrossRef]

Michalev, M. A.

Middleton, A. A.

C. Zeng, J. Kondev, D. McNamara, and A. A. Middleton, "Statistical topography of glassy interfaces," Phys. Rev. Lett. 80, 109-112 (1998).
[CrossRef]

Munk, W.

C. Cox and W. Munk, "Statistics of sea surface derived from the sun's glitter," J. Mar. Res. 13, 198-227 (1954).

C. Cox and W. Munk, "Measurement of roughness of the sea surface from photographs of the sun's glitter," J. Opt. Soc. Am. 44, 838-850 (1954).
[CrossRef]

Quintian, F. P.

F. P. Quintian, M. A. Rebollo, R. G. Berlasso, and N. G. Gaggiolli, "A valuable method for online wire quality control: light scattering from cylindrical rough surface," in Proc. SPIE 4399, 328-334 (2003).
[CrossRef]

Reagan, J. A.

J. A. Reagan and D. A. Zielinskie, "Spaceborne lidar remote sensing techniques aided by surface return," Opt. Eng. 30, 96-102 (1991).
[CrossRef]

Rebollo, M. A.

F. P. Quintian, M. A. Rebollo, R. G. Berlasso, and N. G. Gaggiolli, "A valuable method for online wire quality control: light scattering from cylindrical rough surface," in Proc. SPIE 4399, 328-334 (2003).
[CrossRef]

Stamm, G. L.

Sussarey, O.

A. S. Balankin and O. Sussarey, "A new statistical distribution for self-affine crack roughness parameters," Philos. Mag. Lett. 79, 629-637 (1999).
[CrossRef]

Swift, R. N.

Wilson, J. W.

Yordanov, O. I.

Zeng, C.

C. Zeng, J. Kondev, D. McNamara, and A. A. Middleton, "Statistical topography of glassy interfaces," Phys. Rev. Lett. 80, 109-112 (1998).
[CrossRef]

Zielinskie, D. A.

J. A. Reagan and D. A. Zielinskie, "Spaceborne lidar remote sensing techniques aided by surface return," Opt. Eng. 30, 96-102 (1991).
[CrossRef]

Appl. Opt.

Int. J. Remote Sens.

R. G. Gardachov, "The probability density of the total curvature of a uniform random Gaussian sea surface in the specular points," Int. J. Remote Sens. 21, 2917-2926 (2000).
[CrossRef]

J. Mar. Res.

C. Cox and W. Munk, "Statistics of sea surface derived from the sun's glitter," J. Mar. Res. 13, 198-227 (1954).

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

M. Elias and M. Menu, "Experimental characterization of a random metallic rough surface by spectrometric measurements in the visible range," Opt. Commun. 180, 191-198 (2000).
[CrossRef]

Opt. Eng.

J. A. Reagan and D. A. Zielinskie, "Spaceborne lidar remote sensing techniques aided by surface return," Opt. Eng. 30, 96-102 (1991).
[CrossRef]

Philos. Mag. Lett.

A. S. Balankin and O. Sussarey, "A new statistical distribution for self-affine crack roughness parameters," Philos. Mag. Lett. 79, 629-637 (1999).
[CrossRef]

Philos. Trans. R. Soc. London, Ser. A

M. S. Longuet-Higgins, "The statistical analysis of a random, moving surface," Philos. Trans. R. Soc. London, Ser. A 249, 321-387 (1957).
[CrossRef]

Phys. Rev. Lett.

C. Zeng, J. Kondev, D. McNamara, and A. A. Middleton, "Statistical topography of glassy interfaces," Phys. Rev. Lett. 80, 109-112 (1998).
[CrossRef]

Proc. Cambridge Philos. Soc.

M. S. Longuet-Higgins, "The statistical distribution of the curvature of a random Gaussian surface," Proc. Cambridge Philos. Soc. 54, 439-454 (1958).
[CrossRef]

M. S. Longuet-Higgins, "The distribution of the sizes of images reflected in a random surface," Proc. Cambridge Philos. Soc. 55, 91-100 (1959).
[CrossRef]

Proc. SPIE

F. P. Quintian, M. A. Rebollo, R. G. Berlasso, and N. G. Gaggiolli, "A valuable method for online wire quality control: light scattering from cylindrical rough surface," in Proc. SPIE 4399, 328-334 (2003).
[CrossRef]

Other

M. S. Longuet-Higgins, "The statistical geometry of random surfaces," in Hydrodynamic Instability, in Proceedings of the 13th Symposium on Applied Mathematics (American Mathematical Society. 1962), Vol. 13, pp. 105-143

R. J. Adler, The Geometry of Random Fields (Wiley, 1981).

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Figures (6)

Fig. 1
Fig. 1

Conditional PDF w max ( ζ ω ) = w max ( ζ , ω ) w max ( ζ ) of maxima heights for roughness with Gaussian autocorrelation function (26). Solid curves correspond to Eq. (27) normalized by relation (31), and dashed curves correspond to the asymptotic equations (29).

Fig. 2
Fig. 2

Conditional PDF w e ( ζ ω ) = w e ( ζ , ω ) w e ( ζ ) of surface extrema for roughness with Gaussian autocorrelation function (26). Curves are plotted according to relation (30), normalized by relation (31), for the set of Gaussian curvatures ω shown.

Fig. 3
Fig. 3

Critical point height PDF w c ( ζ ) (long-dashed curve) and maxima height PDF w max ( ζ ) (solid curve) in comparison with Gaussian PDF w G ( ζ ) of surface roughness heights (short-dashed curve).

Fig. 4
Fig. 4

Extremum height PDF w e ( ζ ) (solid curve) and saddle-point height PDF w s ( ζ ) (long-dashed curve) in comparison with Gaussian PDF w G ( ζ ) of surface roughness heights (short-dashed curve).

Fig. 5
Fig. 5

Brightness PDFs of extrema [ w e (long-dashed curve)], saddle points [ w s (short-dashed curve)], and the full set of critical points [ w c (solid curve with maximum)]. The cumulative PDF F c corresponding to the w c is depicted by the monotonically rising solid curve.

Fig. 6
Fig. 6

Conditional PDF w ( ζ J ) of specular point heights for the set of their brightnesses J (solid curves), along with the saddle-point height PDF w s ( ζ ) (dotted curve), extremum height PDF w e ( ζ ) (long-dashed curve), and roughness height PDF w G ( ζ ) (short-dashed curve).

Equations (59)

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P c P { ζ ( ζ 0 , ζ 0 + Δ ζ ) ; γ Δ γ ; ζ i k ( ζ i k 0 , ζ i k 0 + Δ ζ i k ) } = ζ 0 ζ 0 + Δ ζ d ζ Δ γ d γ ζ x x 0 ζ x x 0 + Δ ζ x x d ζ x x ζ x y 0 ζ x y 0 + Δ ζ x y d ζ x y ζ y y 0 ζ y y 0 + Δ ζ y y d ζ y y × w 6 ( ζ , γ , ζ x x , ζ x y , ζ y y ) .
Δ γ Δ γ x Δ γ y = Ω Δ x Δ y ,
n c = Ω w 6 ( ζ , 0 , ζ x x , ζ x y , ζ y y ) Δ ζ Δ ζ x x Δ ζ x y Δ ζ y y .
w c ( ζ , ζ x x , ζ x y , ζ y y ) = Ω w 6 ( ζ , 0 , ζ x x , ζ x y , ζ y y ) N c ,
N c = Ω w 6 ( ζ , 0 , ζ x x , ζ x y , ζ y y ) d ζ d ζ x x d ζ x y d ζ y y .
w 6 ( ζ , 0 , ζ x x , ζ x y , ζ y y ) = w 2 ( 0 ) w 4 ( ζ , ζ x x , ζ x y , ζ y y ) ,
w 2 ( 0 ) = 1 2 π m 1 , m 1 = 2 R ( 1 ) ( 0 ) < 0 ,
R ( n ) ( 0 ) d n R ( t ) d t n t = 0 ,
w 4 ( ζ , ζ x x , ζ x y , ζ y y ) w 4 ( ζ 0 , ζ 1 , ζ 2 , ζ 3 ) = 1 ( 2 π ) 2 Δ 4 exp ( 1 2 i , k = 0 3 M i k ζ i ζ k ) ,
g i k = [ 1 m 1 0 m 1 m 1 3 m 2 0 m 2 0 0 m 2 0 m 1 m 2 0 3 m 2 ] ,
Δ 4 = 4 m 2 2 ( 2 m 2 m 1 2 ) .
M i k = μ i k Δ 4
μ i k = [ 8 m 2 3 2 m 1 m 2 2 0 2 m 1 m 2 2 2 m 1 m 2 2 m 2 ( 3 m 2 m 1 2 ) 0 m 2 ( m 1 2 m 2 ) 0 0 4 m 2 ( 2 m 2 m 1 2 ) 0 2 m 1 m 2 2 m 2 ( m 1 2 m 2 ) 0 m 2 ( 3 m 2 m 1 2 ) ] .
w c ( ζ , ζ x x , ζ x y , ζ y y ) w c ( ζ 0 , ζ 1 , ζ 2 , ζ 3 ) = ζ 1 ζ 3 ζ 2 2 2 π m 1 N c w 4 ( ζ 0 , ζ 1 , ζ 2 , ζ 3 ) ,
w 4 ( ζ 0 , ζ 1 , ζ 2 , ζ 3 ) = 1 ( 2 π ) 2 Δ 4 exp { 1 2 Δ 4 [ 8 m 2 3 ζ 0 2 + m 2 ( 3 m 2 m 1 2 ) ( ζ 1 2 + ζ 3 2 ) 4 m 1 m 2 2 ζ 0 ( ζ 1 + ζ 3 ) + 4 m 2 ( 2 m 2 m 1 2 ) ζ 2 2 + 2 m 2 ( m 1 2 m 2 ) ζ 1 ζ 3 ] } .
N c = 2 m 2 π 3 m 1 .
ζ 1 = x 1 + x 3 , ζ 3 = x 1 x 3 , ζ 2 = x 2 .
Ω = x 1 2 x 2 2 x 3 2 .
x 1 = ± Ω + x 2 2 + x 3 2 x ± ( Ω , x 2 , x 3 ) .
w c ( ζ 0 , Ω , x 2 , x 3 ) = 3 4 m 2 Ω Ω + x 2 2 + x 3 2 ± w 4 ( ζ 0 , x 3 + x ± , x 2 , x ± x 3 ) .
w s ( ζ , Ω ) = A s Ω ( 2 π ) 2 Δ 4 ± R s d x 2 d x 3 Ω + x 2 2 + x 3 2 exp { 2 m 2 Δ 4 [ m 2 ( 2 m 2 ζ 2 2 m 1 ζ Ω + x 2 2 + x 3 2 + Ω + x 2 2 + x 3 2 ) + ( 2 m 2 m 1 2 ) ( x 2 2 + x 3 2 ) ] } .
A s = w 2 ( 0 ) N s = 1 2 π m 1 N s , N s = m 2 π 3 m 1 = 1 2 N c .
w s ( ζ , ω ) = w s ( ω ) w s ( ζ ) ,
w s ( ω ) = ω exp ( ω ) , ω < 0 ,
w s ( ζ ) = 1 σ 2 π exp ( ζ 2 2 σ 2 ) .
σ 2 = 1 1 3 k , k m 2 m 1 2 = R ( 2 ) ( 0 ) [ R ( 1 ) ( 0 ) ] 2 .
r 2 = ζ ( ζ 1 + ζ 3 ) 2 ( ζ 1 + ζ 3 ) 2 ζ 2 = m 1 2 2 m 2 = 1 2 k 1 ,
R ( ρ 2 ) = exp ( ρ 2 l 2 ) ,
{ w max ( ζ , ω ) w min ( ζ , ω ) } = ω σ 2 π exp ( ω ζ 2 2 σ 2 ) × erfc [ ω 3 k 1 2 k 1 ζ σ 6 ( 2 k 1 ) ] ,
erfc ( z ) 2 π z exp ( t 2 ) d t .
{ w max ( ζ ω ) w min ( ζ ω ) } 1 n k π exp [ k n 2 ( ζ ω 2 k ) 2 ] , n 2 = 2 k 1 .
w e ( ζ , ω ) = 1 2 [ w max ( ζ , ω ) + w min ( ζ , ω ) ] .
w e ( ω ) = w max ( ω ) = w min ( ω ) = w min , max ( ζ , ω ) d ζ
= ω e ω erfc ( 3 2 ω ) , ω > 0 .
w e ( ω ) 2 ω 3 π e ω 2 .
w c ( ω ) = 1 2 { w s ( ω ) , ω < 0 w e ( ω ) , ω > 0 } .
{ w max ( ζ ) w min ( ζ ) } = 0 { w max ( ζ , ω ) w min ( ζ , ω ) } d ω = 1 σ 2 π { e ζ 2 2 σ 2 erfc ( ζ σ n 6 ) + σ 3 2 k [ ± ζ n 2 π exp ( ζ 2 k n 2 ) + ( ζ 2 1 ) exp ( ζ 2 2 ) erfc ( ζ n 2 ) ] } .
{ w max ( ζ ) w min ( ζ ) } = 1 2 3 π [ exp ( 3 ζ 2 4 ) erfc ( ζ 2 ) ± ζ π exp ( ζ 2 ) + ζ 2 1 2 exp ( ζ 2 2 ) erfc ( ζ 2 ) ] .
w max ( ζ ) 1 k 3 2 π ζ 2 exp ( ζ 2 2 ) , ζ 1 .
w max ( ζ ) n 3 π ζ exp ( k n 2 ζ 2 ) , ζ 1 .
w e ( ζ ) = 1 2 [ w max ( ζ ) + w min ( ζ ) ] = 1 σ 2 π [ exp ( ζ 2 2 σ 2 ) + σ 3 2 k ( ζ 2 1 ) exp ( ζ 2 2 ) ] ,
w c ( ζ ) = 1 2 [ w s ( ζ ) + w e ( ζ ) ] .
w G ( ζ ) = 1 2 π exp ( ζ 2 2 ) ,
w c ( ζ ) w G ( ζ ) 3 4 k ζ 2 , ζ 1 ;
J I I 0 = 2 m 2 Ω 1 ω .
w c ( ω ) = 1 2 [ w s ( ω ) + w e ( ω ) ] = ω 2 [ e ω + e ω erfc ( 3 2 ω ) ] , ω > 0 ;
w c ( J ) = 1 2 [ w s ( J ) + w e ( J ) ] ,
w s ( J ) = 1 J 3 e 1 J , w e ( J ) = 1 J 3 e 1 J erfc ( 3 2 J ) .
J s = 0 w s ( J ) J d J = 1 ,
J e = J max = J min = 0 w e ( J ) J d J = 3 1 .
J c = 0 J w c ( J ) d J = 1 2 ( J s + J e ) = 3 2 .
F c ( J 0 ) P { J < J 0 } = 0 J 0 w c ( J ) d J = J 0 + 1 2 J 0 e 1 J 0 + J 0 1 2 J 0 e 1 J 0 erfc ( 3 2 J 0 ) + 3 2 π J 0 e 1 2 J 0 ,
w c ( J ) 1 J 3 , J 1 ,
w c ( J ) 1 6 π J 5 2 exp ( 1 2 J ) , J 1 .
w c ( ζ , ω ) = 1 4 [ 2 w s ( ζ , ω ) + w max ( ζ , ω ) + w min ( ζ , ω ) ] ,
w ( ζ , J ) = 1 4 2 π σ J 3 exp ( ζ 2 2 σ 2 ) [ 2 e 1 J + e 1 J ± erfc ( σ n 3 k J ± ζ σ n 6 ) ] .
w ( ζ J ) = w ( ζ , J ) w ( J ) ,
w ( ζ J ) 1 σ 2 π exp ( ζ 2 2 σ 2 ) ,
w ( ζ J ) 1 2 n k π ± exp [ k n 2 ( ζ ± 1 2 k J ) ] .

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