Abstract

The intensity distribution and contrast of image speckle patterns for various objects having different surface profiles are investigated on a computer as a function of the surface roughness properties of the objects and the point spread of an imaging system. The present computer simulation study for the objects having a random surface gives a theoretical background for the experimental results obtained by Fujii and Asakura. The same study for the objects having a periodic surface of two different profiles indicates that the contrast variation takes a complicated form as a function of the point spread of the imaging system, while the intensity distribution of images simply follows a periodic variation. It becomes clear from the simulation study that the maximum value of the image contrast variation is dependent mainly on the rms surface roughness of the objects but not on their surface profile.

© 1976 Optical Society of America

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References

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  1. J. C. Dainty (editor), Laser Speckle and Related Phenomena (Springer-Verlag, Berlin, 1975).
  2. H. Fujii and T. Asakura, “Effect of surface roughness on the statistical distribution of image speckle intensity,” Opt. Commun. 11, 35–38 (1974).
    [Crossref]
  3. H. Fujii and T. Asakura, “Statistical properties of image speckle patterns in partially coherent light,” Nouv. Rev. Opt. 6, 5–14 (1975).
    [Crossref]
  4. H. Fujii and T. Asakura, “A contrast variation of image speckle intensity under illumination of partially coherent light,” Opt. Comm. 12, 32–38 (1974).
    [Crossref]
  5. J. Ohtsubo and T. Asakura, “Statistical properties of speckle intensity variations in the diffraction field under illumination of coherent light,” Opt. Commun. 14, 30–34 (1975).
    [Crossref]
  6. H. Fujii, T. Asakura, and Y. Shindo, “Measurements of surface roughness properties by means of laser speckle techniques,” Opt. Commun. 16, 68–72 (1976).
    [Crossref]
  7. H. Fujii, T. Asakura, and Y. Shindo, “Measurement of surface roughness properties by using image speckle contrast,” J. Opt. Soc. Am. 66, 1217–1222 (1976) (preceding paper).
    [Crossref]
  8. H. M. Pedersen, “Object-roughness dependence of partially devoloped speckle patterns in coherent light,” Opt. Commun. 16, 63–67 (1976).
    [Crossref]
  9. J. Ohtsubo, H. Fujii, and T. Asakura, “Surface roughness measurement by using speckle pattern,” Jpn. J. Appl. Phys. Suppl. 14-1, 293–298 (1975).
  10. N. George and A. Jain, “Space and wavelength dependence of speckle intensity,” Appl. Phys. 4, 201–212 (1974).
    [Crossref]
  11. N. George, A. Jain, and R. D. S. Melville, “Experiments on the space and wavelength dependence of speckle,” Appl. Phys. 7, 157–169 (1975).
    [Crossref]
  12. N. George (private communication).

1976 (3)

H. Fujii, T. Asakura, and Y. Shindo, “Measurements of surface roughness properties by means of laser speckle techniques,” Opt. Commun. 16, 68–72 (1976).
[Crossref]

H. M. Pedersen, “Object-roughness dependence of partially devoloped speckle patterns in coherent light,” Opt. Commun. 16, 63–67 (1976).
[Crossref]

H. Fujii, T. Asakura, and Y. Shindo, “Measurement of surface roughness properties by using image speckle contrast,” J. Opt. Soc. Am. 66, 1217–1222 (1976) (preceding paper).
[Crossref]

1975 (4)

N. George, A. Jain, and R. D. S. Melville, “Experiments on the space and wavelength dependence of speckle,” Appl. Phys. 7, 157–169 (1975).
[Crossref]

J. Ohtsubo, H. Fujii, and T. Asakura, “Surface roughness measurement by using speckle pattern,” Jpn. J. Appl. Phys. Suppl. 14-1, 293–298 (1975).

H. Fujii and T. Asakura, “Statistical properties of image speckle patterns in partially coherent light,” Nouv. Rev. Opt. 6, 5–14 (1975).
[Crossref]

J. Ohtsubo and T. Asakura, “Statistical properties of speckle intensity variations in the diffraction field under illumination of coherent light,” Opt. Commun. 14, 30–34 (1975).
[Crossref]

1974 (3)

H. Fujii and T. Asakura, “Effect of surface roughness on the statistical distribution of image speckle intensity,” Opt. Commun. 11, 35–38 (1974).
[Crossref]

H. Fujii and T. Asakura, “A contrast variation of image speckle intensity under illumination of partially coherent light,” Opt. Comm. 12, 32–38 (1974).
[Crossref]

N. George and A. Jain, “Space and wavelength dependence of speckle intensity,” Appl. Phys. 4, 201–212 (1974).
[Crossref]

Asakura, T.

H. Fujii, T. Asakura, and Y. Shindo, “Measurements of surface roughness properties by means of laser speckle techniques,” Opt. Commun. 16, 68–72 (1976).
[Crossref]

H. Fujii, T. Asakura, and Y. Shindo, “Measurement of surface roughness properties by using image speckle contrast,” J. Opt. Soc. Am. 66, 1217–1222 (1976) (preceding paper).
[Crossref]

J. Ohtsubo, H. Fujii, and T. Asakura, “Surface roughness measurement by using speckle pattern,” Jpn. J. Appl. Phys. Suppl. 14-1, 293–298 (1975).

H. Fujii and T. Asakura, “Statistical properties of image speckle patterns in partially coherent light,” Nouv. Rev. Opt. 6, 5–14 (1975).
[Crossref]

J. Ohtsubo and T. Asakura, “Statistical properties of speckle intensity variations in the diffraction field under illumination of coherent light,” Opt. Commun. 14, 30–34 (1975).
[Crossref]

H. Fujii and T. Asakura, “A contrast variation of image speckle intensity under illumination of partially coherent light,” Opt. Comm. 12, 32–38 (1974).
[Crossref]

H. Fujii and T. Asakura, “Effect of surface roughness on the statistical distribution of image speckle intensity,” Opt. Commun. 11, 35–38 (1974).
[Crossref]

Fujii, H.

H. Fujii, T. Asakura, and Y. Shindo, “Measurements of surface roughness properties by means of laser speckle techniques,” Opt. Commun. 16, 68–72 (1976).
[Crossref]

H. Fujii, T. Asakura, and Y. Shindo, “Measurement of surface roughness properties by using image speckle contrast,” J. Opt. Soc. Am. 66, 1217–1222 (1976) (preceding paper).
[Crossref]

H. Fujii and T. Asakura, “Statistical properties of image speckle patterns in partially coherent light,” Nouv. Rev. Opt. 6, 5–14 (1975).
[Crossref]

J. Ohtsubo, H. Fujii, and T. Asakura, “Surface roughness measurement by using speckle pattern,” Jpn. J. Appl. Phys. Suppl. 14-1, 293–298 (1975).

H. Fujii and T. Asakura, “Effect of surface roughness on the statistical distribution of image speckle intensity,” Opt. Commun. 11, 35–38 (1974).
[Crossref]

H. Fujii and T. Asakura, “A contrast variation of image speckle intensity under illumination of partially coherent light,” Opt. Comm. 12, 32–38 (1974).
[Crossref]

George, N.

N. George, A. Jain, and R. D. S. Melville, “Experiments on the space and wavelength dependence of speckle,” Appl. Phys. 7, 157–169 (1975).
[Crossref]

N. George and A. Jain, “Space and wavelength dependence of speckle intensity,” Appl. Phys. 4, 201–212 (1974).
[Crossref]

N. George (private communication).

Jain, A.

N. George, A. Jain, and R. D. S. Melville, “Experiments on the space and wavelength dependence of speckle,” Appl. Phys. 7, 157–169 (1975).
[Crossref]

N. George and A. Jain, “Space and wavelength dependence of speckle intensity,” Appl. Phys. 4, 201–212 (1974).
[Crossref]

Melville, R. D. S.

N. George, A. Jain, and R. D. S. Melville, “Experiments on the space and wavelength dependence of speckle,” Appl. Phys. 7, 157–169 (1975).
[Crossref]

Ohtsubo, J.

J. Ohtsubo, H. Fujii, and T. Asakura, “Surface roughness measurement by using speckle pattern,” Jpn. J. Appl. Phys. Suppl. 14-1, 293–298 (1975).

J. Ohtsubo and T. Asakura, “Statistical properties of speckle intensity variations in the diffraction field under illumination of coherent light,” Opt. Commun. 14, 30–34 (1975).
[Crossref]

Pedersen, H. M.

H. M. Pedersen, “Object-roughness dependence of partially devoloped speckle patterns in coherent light,” Opt. Commun. 16, 63–67 (1976).
[Crossref]

Shindo, Y.

H. Fujii, T. Asakura, and Y. Shindo, “Measurements of surface roughness properties by means of laser speckle techniques,” Opt. Commun. 16, 68–72 (1976).
[Crossref]

H. Fujii, T. Asakura, and Y. Shindo, “Measurement of surface roughness properties by using image speckle contrast,” J. Opt. Soc. Am. 66, 1217–1222 (1976) (preceding paper).
[Crossref]

Appl. Phys. (2)

N. George and A. Jain, “Space and wavelength dependence of speckle intensity,” Appl. Phys. 4, 201–212 (1974).
[Crossref]

N. George, A. Jain, and R. D. S. Melville, “Experiments on the space and wavelength dependence of speckle,” Appl. Phys. 7, 157–169 (1975).
[Crossref]

J. Opt. Soc. Am. (1)

Jpn. J. Appl. Phys. Suppl. (1)

J. Ohtsubo, H. Fujii, and T. Asakura, “Surface roughness measurement by using speckle pattern,” Jpn. J. Appl. Phys. Suppl. 14-1, 293–298 (1975).

Nouv. Rev. Opt. (1)

H. Fujii and T. Asakura, “Statistical properties of image speckle patterns in partially coherent light,” Nouv. Rev. Opt. 6, 5–14 (1975).
[Crossref]

Opt. Comm. (1)

H. Fujii and T. Asakura, “A contrast variation of image speckle intensity under illumination of partially coherent light,” Opt. Comm. 12, 32–38 (1974).
[Crossref]

Opt. Commun. (4)

J. Ohtsubo and T. Asakura, “Statistical properties of speckle intensity variations in the diffraction field under illumination of coherent light,” Opt. Commun. 14, 30–34 (1975).
[Crossref]

H. Fujii, T. Asakura, and Y. Shindo, “Measurements of surface roughness properties by means of laser speckle techniques,” Opt. Commun. 16, 68–72 (1976).
[Crossref]

H. M. Pedersen, “Object-roughness dependence of partially devoloped speckle patterns in coherent light,” Opt. Commun. 16, 63–67 (1976).
[Crossref]

H. Fujii and T. Asakura, “Effect of surface roughness on the statistical distribution of image speckle intensity,” Opt. Commun. 11, 35–38 (1974).
[Crossref]

Other (2)

J. C. Dainty (editor), Laser Speckle and Related Phenomena (Springer-Verlag, Berlin, 1975).

N. George (private communication).

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Figures (13)

FIG. 1
FIG. 1

Optical system for producing image speckle patterns.

FIG. 2
FIG. 2

Surface height variations (phase variations) of the three different object surfaces. (a) Random waved surface, (b) triangular waved surface, and (c) sinusoidal waved surface.

FIG. 3
FIG. 3

(a) Computer-generated Gaussian random surface ϕr(x), (b) the autocorrelation function Φ(ξ) of ϕr(x), and (c) the probability density distribution P(ϕr) of ϕr(x) (this probability distribution experimentally obtained on a computer is in good agreement with the Gaussian probability distribution theoretically calculated).

FIG. 4
FIG. 4

Computer-generated random waved surface ϕr(x/α) and the intensity distribution I(x′/α) of corresponding image speckle patterns for four objects with different rms roughnesses Rs, produced by using the optical imaging system fixed as a/α = 1.

FIG. 5
FIG. 5

Intensity distributions I(x′/α) of image speckle patterns, for the random surface object with the fixed rms roughness Rs = 0.20 μm [therefore, ϕr(x/α) takes the same form], produced under four different imaging conditions of the point spread denoted by a/α.

FIG. 6
FIG. 6

Average contrast variations V of image speckle patterns as a function of the point spread a/α of the optical imaging system for the random surface objects with five different rms roughnesses Rs.

FIG. 7
FIG. 7

Triangular waved surfaces ϕt(x/α′) and the intensity distribution I(x′/α′) of corresponding image patterns for four objects with different rms roughnesses Rs, produced under the same optical imaging system of a/α′ = 1.

FIG. 8
FIG. 8

Intensity distributions I(x′/α′) of image patterns for the object having a triangular waved surface with the rms roughness Rs = 0.20 μm, under eight different imaging conditions of the point spread denoted by a/α′.

FIG. 9
FIG. 9

Average contrast variations V of image patterns as a function of the point spread a/α′ of the optical imaging system for the objects having a triangular surface with five different rms roughnesses Rs. In this figure the average contrast variations are shown only in the limited region of 0 < a/α′ < 4.0.

FIG. 10
FIG. 10

Sinusoidal waved surfaces ϕs(x/α′) and the intensity distribution I(x′/α′) of corresponding image patterns for four objects with different rms roughnesses Rs, produced under the same optical imaging system of a/α′ = 1.

FIG. 11
FIG. 11

Intensity distributions I(x′/α′) of image patterns, for the object having a sinusoidal waved surface with the rms roughness Rs = 0.20 μm, under eight different imaging conditions of the point spread denoted by a/α′.

FIG. 12
FIG. 12

Average contrast variations of image patterns as a function of the point spread a/α′ of the optical imaging system for the objects having a sinusoidal surface with five different rms roughnesses Rs. In this figure the average contrast variations are shown only in the limited region of 0 < a/α′ < 4.0.

FIG. 13
FIG. 13

Variations of the maximum average contrast Vmax as a function of the rms surface roughness Rs = [〈ΔL2〉]1/2 for three different objects with random, triangular, and sinusoidal waved surfaces.

Tables (2)

Tables Icon

TABLE I Intensity distribution I(x′) for the four cases of the point spread a against the period δ of the triangular wave function.

Tables Icon

TABLE II Mean value I ( x ) ¯ and mean-square value I ( x ) 2 ¯ of the intensity for the four cases of the point spread a against the period δ of the triangular wave function.

Equations (17)

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I ( x ) = | - K ( x - x ) exp { i ϕ ( x ) } d x | 2 ,
K ( x ) = 1 , x a = 0 , x > a .
I ( x ) = C 2 + S 2 ,
C = x - a x + a cos { ϕ ( x ) } d x ,
S = x - a x + a sin { ϕ ( x ) } d x .
V = [ I ( x ) 2 - I ( x ) 2 ] 1 / 2 I ( x ) ,
ϕ t ( x ) = c ( x - 2 n δ ) δ ( 2 n - 1 2 ) < x < δ ( 2 n + 1 2 ) = - c [ x - ( 2 n + 1 ) δ ] δ ( 2 n + 1 2 ) < x < δ ( 2 n + 3 2 )
I ( x + n δ ) = I ( x ) ,             n = + 1 ,             + 2 ,
I ( x ) ¯ = 4 δ c 2 a - ( δ / 2 ) ( δ / 2 ) - a sin 2 ( c a ) d x + 4 δ c 2 ( δ / 2 ) - a ( δ / 2 ) + a { 1 - 2 cos ( c a ) cos [ c ( x - δ 2 ) ] + cos 2 [ c ( x - δ 2 ) ] } d x = 4 δ c 2 [ ( 2 a + δ 2 ) - 3 2 c sin ( 2 a c ) - 1 2 ( δ - 2 a ) cos ( 2 a c ) ] , I ( x ) 2 ¯ = 16 δ c 4 a - ( δ / 2 ) ( δ / 2 ) - a sin 4 ( c a ) d x + 16 δ c 4 ( δ / 2 ) - a ( δ / 2 ) + a { 1 - 2 cos ( c a ) cos [ c ( x - δ 2 ) ] } 2 d x = 16 δ c 4 [ 6 a + 3 δ 8 - 29 6 c sin ( 2 a c ) + ( 3 a - δ 2 ) cos ( 2 a c ) + 11 48 c sin ( 4 a c ) + 1 8 ( δ - 2 a ) cos ( 4 a c ) ] .
ϕ s ( x ) = γ sin ( π x / δ ) ,
cos ( z sin θ ) = J 0 ( z ) + 2 n = 1 cos ( 2 n θ ) J 2 n ( z ) , sin ( z sin θ ) = 2 n = 1 sin [ ( 2 n - 1 ) θ ] J 2 n - 1 ( z ) ,
I ( x ) = C 0 2 + S 0 2
C 0 = 2 a J 0 + 4 δ π n = 1 F ( 2 n ) C 2 n , S 0 = 4 δ π n = 1 F ( 2 n - 1 ) S 2 n - 1 ,
J 0 = J 0 ( γ ) , F ( n ) = ( 1 / n ) J n ( γ ) sin ( n π a / δ ) , C n = cos ( n π x / δ ) , S n = sin ( n π x / δ ) .
I ( x ) ¯ = 4 a 2 J 0 2 + 8 δ 2 π 2 n = 1 F ( n ) 2 , I ( x ) 2 ¯ = 16 a 4 J 0 4 + n = 1 [ Z + m = 1 ( H + 32 δ 4 π 4 l = 1 Θ ) ] ,
Z = 64 a 2 δ 2 π 2 J 0 2 [ F ( n ) 2 + 2 F ( 2 n ) 2 ] , H = 128 a δ 3 π 3 J 0 ( F ( 2 n ) F ( 2 m ) { F [ 2 ( n + m ) ] + F [ 2 ( m - n ) ] + F [ 2 ( n - m ) ] } + F ( 2 n - 1 ) F ( 2 m - 1 ) { F [ 2 ( m - n ) ] + F [ 2 ( n - m ) ] - F [ 2 ( n + m - 1 ) ] } ) , Θ = F ( 2 n ) F ( 2 m ) F ( 2 l ) { F [ 2 ( n + m + l ) ] + F [ 2 ( n + m - l ) ] + F [ 2 ( n + l - m ) ] + F [ 2 ( m + l - n ) ] + F [ 2 ( n - m - l ) ] + F [ 2 ( m - n - l ) ] + F [ 2 ( l - m - n ) ] } + F ( 2 n - 1 ) F ( 2 m - 1 ) F ( 2 l - 1 ) { F [ 2 ( n + m - l ) - 1 ] + F [ 2 ( n - m + l ) - 1 ] + F [ 2 ( l + m - n ) - 1 ] - F [ 2 ( n + m + l ) - 3 ] - F [ 2 ( l - n - m ) + 1 ] - F [ 2 ( m - n - l ) + 1 ] - F [ 2 ( n - m - l ) + 1 ] } + 2 F ( 2 n ) F ( 2 m ) F ( 2 l - 1 ) { F [ 2 ( n + m + l ) - 1 ] + F [ 2 ( l - n - m ) - 1 ] + F [ 2 ( l + n - m ) - 1 ] + F [ 2 ( l + m - n ) - 1 ] - F [ 2 ( n - m - l ) + 1 ] - F [ 2 ( n + m - l ) + 1 ] - F [ 2 ( m - n - l ) + 1 ] } ,
( V α / V max ¯ ) = 0.81 ± 0.01.