Abstract

We provide a comprehensive analysis of the space and wavelength dependence of speckle generated by a thin diffuser and imaged by a 4F optical system. The use of this space-invariant system is shown to lead to the well-known features of speckle patterns in an analytically simple and elegant manner, thereby providing a clear insight into speckle in an optical configuration that includes the Fresnel zone, the optical Fourier transform plane, and the image plane. In our analysis we assume a white-noise diffuser. The spatial variation mainly depends on the imaging system, whereas the wavelength dependence is related to diffuser heights. Motion dynamics of speckle are also included.

© 2008 Optical Society of America

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References

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  1. Lord Rayleigh, "On the resultant of a large number of vibrations of the same pitch and of arbitrary phase," Phil. Mag. 10, 73-78 (1880).
  2. R. K. Erf, Speckle Metrology (Academic, 1979).
  3. J. C. Dainty, Laser Speckle and Related Phenomena (Springer-Verlag, 1984).
  4. J. W. Goodman, Speckle Phenomena in Optics (Roberts and Company, 2006).
  5. M. Franon, Laser Speckle and Applications in Optics (Academic, 1979).
  6. B. Eliason and F. M. Mottier, "Determination of the granular radiance distribution of diffuser and its use for vibration analysis," J. Opt. Soc. Am. 61, 559-565 (1971).
    [CrossRef]
  7. N. George and A. Jain, "Space and wavelength dependence of speckle intensity," Appl. Phys. 4, 201-212 (1974).
    [CrossRef]
  8. N. George, A. Jain, and R. D. S. Melville, "Experiments on the space and wavelength dependence of speckle intensity," Appl. Phys. 7, 157-169 (1975).
    [CrossRef]
  9. N. George, "Speckle at various planes in an optical system," Opt. Eng. 25, 754-764 (1986).
  10. H. Kadono, T. Asakura, and N. Takai, "Roughness and correlation-length measurements of rough-surface object using the speckle contrast in the diffraction field," Optik 80, 115-120 (1988).
  11. L. Leushacke and M. Kirchner, "Three-dimensional correlation coefficient of speckle intensity for rectangular and circular apertures," J. Opt. Soc. Am. A 7, 827-832 (1990).
    [CrossRef]
  12. Q. B. Li and F. P. Chiang, "Three-dimensional dimension of laser speckle," Appl. Opt. 31, 6287-6291 (1992).
    [CrossRef] [PubMed]
  13. C. Cheng, C. Liu, N. Zhang, T. Jia, R. Li, and Z. Xu, "Absolute measurement of roughness and lateral-correlation length of random surfaces by use of the simplified model of image-speckle contrast," Appl. Opt. 41, 4148-4156 (2002).
    [CrossRef] [PubMed]
  14. R. Kingslake, Optics in Photography (SPIE Press, 1992).
  15. N. Takai, T. Iwai, and T. Asakura, "Correlation distance of dynamic speckles," Appl. Opt. 22, 170-177 (1983).
    [CrossRef] [PubMed]
  16. T. Asakura and N. Takai, "Dynamic laser speckles and their application to velocity measurements of the diffuser object," Appl. Opt. 25, 179-194 (1981).
  17. T. Okamoto and T. Asakura, "Effects of imaging properties on dynamic speckles produced by a set of moving phase screens," Waves Random Media 2, 49-65 (1992).
    [CrossRef]
  18. T. Nakamura and T. Asakura, "Cross-correlation function of dynamic speckles from rotating cylindrical objects," Optik 108, 13-19 (1998).
  19. D. C. Sinclair, "Demonstration of chromatic aberration in the eye using coherent light," J. Opt. Soc. Am. 55, 575-576 (1965).
    [CrossRef] [PubMed]
  20. I. S. Reed, "On a moment theorem for complex Gaussian processes," IRE Trans. Inf. Theor. IT-8, 194-195 (1962).
    [CrossRef]
  21. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables (Dover, 1965).
  22. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

2002 (1)

1998 (1)

T. Nakamura and T. Asakura, "Cross-correlation function of dynamic speckles from rotating cylindrical objects," Optik 108, 13-19 (1998).

1992 (2)

T. Okamoto and T. Asakura, "Effects of imaging properties on dynamic speckles produced by a set of moving phase screens," Waves Random Media 2, 49-65 (1992).
[CrossRef]

Q. B. Li and F. P. Chiang, "Three-dimensional dimension of laser speckle," Appl. Opt. 31, 6287-6291 (1992).
[CrossRef] [PubMed]

1990 (1)

1988 (1)

H. Kadono, T. Asakura, and N. Takai, "Roughness and correlation-length measurements of rough-surface object using the speckle contrast in the diffraction field," Optik 80, 115-120 (1988).

1986 (1)

N. George, "Speckle at various planes in an optical system," Opt. Eng. 25, 754-764 (1986).

1983 (1)

1981 (1)

T. Asakura and N. Takai, "Dynamic laser speckles and their application to velocity measurements of the diffuser object," Appl. Opt. 25, 179-194 (1981).

1975 (1)

N. George, A. Jain, and R. D. S. Melville, "Experiments on the space and wavelength dependence of speckle intensity," Appl. Phys. 7, 157-169 (1975).
[CrossRef]

1974 (1)

N. George and A. Jain, "Space and wavelength dependence of speckle intensity," Appl. Phys. 4, 201-212 (1974).
[CrossRef]

1971 (1)

1965 (1)

1962 (1)

I. S. Reed, "On a moment theorem for complex Gaussian processes," IRE Trans. Inf. Theor. IT-8, 194-195 (1962).
[CrossRef]

1880 (1)

Lord Rayleigh, "On the resultant of a large number of vibrations of the same pitch and of arbitrary phase," Phil. Mag. 10, 73-78 (1880).

Appl. Opt. (4)

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 intensity," Appl. Phys. 7, 157-169 (1975).
[CrossRef]

IRE Trans. Inf. Theor. (1)

I. S. Reed, "On a moment theorem for complex Gaussian processes," IRE Trans. Inf. Theor. IT-8, 194-195 (1962).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (1)

Opt. Eng. (1)

N. George, "Speckle at various planes in an optical system," Opt. Eng. 25, 754-764 (1986).

Optik (2)

H. Kadono, T. Asakura, and N. Takai, "Roughness and correlation-length measurements of rough-surface object using the speckle contrast in the diffraction field," Optik 80, 115-120 (1988).

T. Nakamura and T. Asakura, "Cross-correlation function of dynamic speckles from rotating cylindrical objects," Optik 108, 13-19 (1998).

Phil. Mag. (1)

Lord Rayleigh, "On the resultant of a large number of vibrations of the same pitch and of arbitrary phase," Phil. Mag. 10, 73-78 (1880).

Waves Random Media (1)

T. Okamoto and T. Asakura, "Effects of imaging properties on dynamic speckles produced by a set of moving phase screens," Waves Random Media 2, 49-65 (1992).
[CrossRef]

Other (7)

R. Kingslake, Optics in Photography (SPIE Press, 1992).

R. K. Erf, Speckle Metrology (Academic, 1979).

J. C. Dainty, Laser Speckle and Related Phenomena (Springer-Verlag, 1984).

J. W. Goodman, Speckle Phenomena in Optics (Roberts and Company, 2006).

M. Franon, Laser Speckle and Applications in Optics (Academic, 1979).

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables (Dover, 1965).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

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Figures (6)

Fig. 1
Fig. 1

4F optical system. Object is imaged onto the output plane by two lenses with focal length F and an aperture with diameter D.

Fig. 2
Fig. 2

Speckle sizes in two dimensions.

Fig. 3
Fig. 3

Normalized absolute value of the cross-correlation function versus t using a square aperture.

Fig. 4
Fig. 4

Normalized cross-spectral density function versus Δ z and λ 2 using a square aperture with varying diffuser roughness.

Fig. 5
Fig. 5

Motion dynamics of speckle in zones I, II, III, and IV.

Fig. 6
Fig. 6

Area of the diffuser that is contributing to one on-axis point.

Equations (45)

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E y 3 ( x , y , z ; ν ) = ν 2 e i k ( 3 F + z ) c 2 F 2 d u d v d x d y × e i 2 π η h ( x , y ) P ( u , v ) × e i π ν ( z F ) ( u 2 + v 2 ) c F 2 + i 2 π ν [ ( x + x ) u + ( y + y ) v ] c F ,
w ( x 1 , y 1 , z 1 ; ν 1 ) w * ( x 2 , y 2 , z 2 ; ν 2 ) = E y ( x 1 , y 1 , z 1 ; ν 1 ) E y * ( x 1 , y 1 , z 1 ; ν 1 ) × E y ( x 2 , y 2 , z 2 ; ν 2 ) E y * ( x 2 , y 2 , z 2 ; ν 2 ) .
w ( x 1 , y 1 , z 1 ; ν 1 ) w * ( x 2 , y 2 , z 2 ; ν 2 ) = w ( x 1 , y 1 , z 1 ; ν 1 ) w ( x 2 , y 2 , z 2 ; ν 2 ) + | E y ( x 1 , y 1 , z 1 ; ν 1 ) E y * ( x 2 , y 2 , z 2 ; ν 2 ) | 2 .
E y 3 ( x 1 , y 1 , z 1 ; ν 1 ) E y 3 * ( x 2 , y 2 , z 2 ; ν 2 ) = d u d u d v d v d x d x d y d y × e i 2 π [ η 1 h ( x , y ) η 2 h ( x , y ) ] × H ( x 1 , y 1 , z 1 ; u , v ; x , y ; ν 1 ) × H * ( x 2 , y 2 , z 2 ; u , v ; x , y ; ν 2 ) ,
H ( x , y , z ; u , v ; x , y ; ν )  = ν 2 c 2 F 2 P ( u , v ) e i π ν ( z F ) ( u 2 + v 2 ) c F 2 + i 2 π ν [ ( x + x ) u + ( y + y ) v ] c F .
R g ( x x , y y ; η 1 , η 2 ) = exp { i 2 π [ η 1 h ( x , y ) η 2 h ( x , y ) ] } .
R g ( x x , y y ; η 1 , η 2 ) = A ( η 1 , η 2 ) δ ( x x , y y ) × e [ 2 π ( η 1 η 2 ) σ ] 2 2 ,
A ( η 1 , η 2 ) = 2 π l 0 2 α 2 [ 1 e α ( 1 + α ) ] ,
E y 3 ( x 1 , y 1 , z 1 ; ν 1 ) E y 3 * ( x 2 , y 2 , z 2 ; ν 2 ) = A e [ 2 π ( η 1 η 2 ) σ ] 2 2 d u d u d v d v d x d y × H ( x 1 , y 1 , z 1 ; u , v ; x , y ; ν 1 ) × H * ( x 2 , y 2 , z 2 ; u , v ; x , y ; ν 2 ) .
E y 3 ( x 1 , y 1 , z 1 ; ν 1 ) E y 3 * ( x 2 , y 2 , z 2 ; ν 2 ) = A ν 0 2 e [ 2 π ( η 1 η 2 ) σ ] 2 2 c 2 F 2 d u d v P ( u , v ) × e i 2 π ν 0 c F [ ( x 1 x 2 ) u + ( y 1 y 2 ) v ] e i π ν 0 ( z 1 z 2 ) c F 2 ( u 2 + v 2 ) ,
E y 3 ( x 1 , y 1 , z 1 ; ν 1 ) E y 3 * ( x 1 + Δ x , y 1 + Δ y , z ; ν 2 ) = A ν 0 2 e [ 2 π ( η 1 η 2 ) σ ] 2 2 c 2 F 2 d u d v P ( u , v ) × e i 2 π ν 0 c F [ Δ x u + Δ y v ] ,
E y 3 ( x , y , z 1 ; ν 1 ) E y 3 * ( x , y , z 2 ; ν 2 ) = A ν 0 2 e [ 2 π ( η 1 η 2 ) σ ] 2 2 c 2 F 2  × d u d v P ( u , v ) e i π ν 0 Δ z c F 2 ( u 2 + v 2 ) ,
E y 3 ( x 1 , y 1 , z ; ν 1 ) E y 3 * ( x 1 + Δ x , y 1 + Δ y , z ; ν 2 ) = A ν 0 2 D 2 e [ 2 π ( η 1 η 2 ) σ ] 2 2 c 2 F 2  sinc ( D ν 0 Δ x c F ) sinc ( D ν 0 Δ y c F ) .
E y 3 ( x , y , z 1 ; ν 1 ) E y 3 * ( x , y , z 2 ; ν 2 ) = A ν 0 2 D 2 e [ 2 π ( η 1 η 2 ) σ ] 2 2 c 2 F 2 [ C ( t ) t + i S ( t ) t ] 2 ,
C ( ξ ) = 0 ξ cos ( π t 2 2 ) d t ,
S ( ξ ) = 0 ξ sin ( π t 2 2 ) d t .
d 8 c F 2 ν 0 D 2 ,
E y 3 ( x , y , z ; ν 1 ) E y 3 * ( x , y , z ; ν 2 ) = A ν 0 2 e [ 2 π ( η 1 η 2 ) σ ] 2 2 c 2 F 2 .
Δ ν = 2 c σ ( n 1 ) .
w 1 ( x 1 , z 1 ; λ ) = | E y 1 ( x 1 , z 1 ; λ ) | 2 = 1 λ 2 F z 1 | d x d ξ 1 E 0 ( x ) P ( x ) × e i π x 2 λ F + i π ξ 1 2 π z 1 i 2 π λ ( x F + x 1 z 1 ) ξ 1 | 2 ,
w 1 ( x 1 , z 1 ; λ ) = 1 λ F | d x E 0 ( x ) P ( x ) e i π ( F z 1 ) x 2 λ F 2 i 2 π x 1 x λ F | 2 .
w 1 ( x 1 , z 1 ; λ ) = 1 λ F | d x E 0 ( x a ) P 1 ( x a ) × e i π ( F z 1 ) x 2 λ F 2 i 2 π x 1 x λ F | 2 .
w 1 ( x 1 , z 1 ; λ ) = 1 λ F | d x E 0 ( x ) P 1 ( x ) × e i π ( F z 1 ) x 2 λ F 2 i 2 π λ [ x 1 ( 1 z 1 F ) a ] x | 2 .
w 1 ( x 1 , z 1 ; λ ) = w 1 ( x 1 ( 1 z 1 F ) a ; λ ) .
w 2 ( x 2 , z 2 ; λ ) = | E y 2 ( x 2 , z 2 ; λ ) | 2 = 1 λ z 2 | d u E 2 ( u ) P 2 ( u ) e i π λ z 2 ( x 2 u ) 2 | 2 ,
E 2 ( u ) = i λ F d x E 0 ( x ) P 1 ( x ) e i 2 π x u λ F .
w 2 ( x 2 , z 2 ; λ ) = 1 λ 2 F z 2 | d x d u E 0 ( x ) P 1 ( x ) × rect ( u D 2 ) e i π λ z 2 u 2 i 2 π λ ( x F + x 2 z 2 ) u | 2 .
w 2 ( x 2 , z 2 ; λ ) = 1 2 λ F | d x E 0 ( x ) P 1 ( x ) e i π z 2 x 2 λ F 2 i 2 π x x 2 λ F × { C [ D 2 2 ( x 2 + z 2 x F ) 2 λ z 2 ] + C [ D 2 + 2 ( x 2 + z 2 x F ) 2 λ z 2 ] + i S [ D 2 2 ( x 2 + z 2 x F ) 2 λ z 2 ] + i S [ D 2 + 2 ( x 2 + z 2 x F ) 2 λ z 2 ] } | 2 .
w 2 ( x 2 , z 2 ; λ ) = 1 2 λ F | d x E 0 ( x ) P 1 ( x ) × e i π z 2 x 2 λ F 2 i 2 π x ( x 2 + z 2 a F ) λ F × { C [ D 2 2 ( ( x 2 + z 2 a F ) + z 2 x F ) 2 λ z 2 ] + C [ D 2 + 2 ( ( x 2 + z 2 a F ) + z 2 x F ) 2 λ z 2 ] + i S [ D 2 2 ( ( x 2 + z 2 a F ) + z 2 x F ) 2 λ z 2 ] + i S [ D 2 + 2 ( ( x 2 + z 2 a F ) + z 2 x F ) 2 λ z 2 ] } | 2 .
w 2 ( x 2 , z 2 ; λ ) = w 2 ( x 2 + z 2 a F ; λ ) .
w 3 ( x 3 , z 3 ; λ ) = | E y 3 ( x 3 , z 3 ; λ ) | 2 = 1 λ F | d u E 2 ( u ) P 2 ( u ) e i π λ F ( 1 z 3 F ) u 2 i 2 π x 3 u λ F | 2 .
w 3 ( x 3 , z 3 ; λ ) = F 2 ( F z 3 ) | d x E 0 ( x ) P 1 ( x ) × e i π λ ( F z 3 ) x 2 i 2 π x 3 x λ ( F z 3 ) × { C [ F z 3 2 λ F 2 ( D 2 2 F ( x + x 3 ) F z 3 ) ] + C [ F z 3 2 λ F 2 ( D 2 + 2 F ( x + x 3 ) F z 3 ) ] + i S [ F z 3 2 λ F 2 ( D 2 2 F ( x + x 3 ) F z 3 ) ] + i S [ F z 3 2 λ F 2 ( D 2 + 2 F ( x + x 3 ) F z 3 ) ] } | 2 .
w 3 ( x 3 , z 3 ; λ ) = F 2 ( F z 3 ) | d x E 0 ( x ) P 1 ( x ) × e i π λ ( F z 3 ) x 2 i 2 π ( x 3 + a ) x λ ( F z 3 ) × { C [ F z 3 2 λ F 2 ( D 2 2 F ( x + ( x 3 + a ) ) F z 3 ) ] + C [ F z 3 2 λ F 2 ( D 2 + 2 F ( x + ( x 3 + a ) ) F z 3 ) ] + i S [ F z 3 2 λ F 2 ( D 2 2 F ( x + ( x 3 + a ) ) F z 3 ) ] + i S [ F z 3 2 λ F 2 ( D 2 + 2 F ( x + ( x 3 + a ) ) F z 3 ) ] } | 2 .
w 3 ( x 3 , z 3 ; λ ) = w 3 ( x 3 + a ; λ ) .
w 4 ( x 4 , F ; λ ) = D 2 2 λ F | d x E 0 ( x ) P 1 ( x ) × sinc [ D 2 ( x + x 4 ) λ F ] | 2 .
w 4 ( x 4 , F ; λ ) = D 2 2 λ F | d x E 0 ( x ) P 1 ( x ) × sinc [ D 2 ( x + ( x 4 + a ) ) λ F ] | 2 .
w 4 ( x 4 , z 4 ; λ ) = w 4 ( x 4 + a ; λ ) .
E y 2 ( u , v ; ν ) = i ν e i k 2 F c F d x d y E y 1 ( x , y ; ν ) × e i 2 π ν ( u x + v y ) c F ,
E y 4 ( x , y ; ν ) = i ν e i k 2 F c F d u d v P ( u , v ) E y 2 ( u , v ; ν ) × e i 2 π ν ( u x + v y ) c F .
E y 4 ( x , y ; ν ) = ν 2 e i k 4 F c 2 F 2 d u d v d x d y × E y 1 ( x , y ; ν ) P ( u , v ) e i 2 π ν [ u ( x + x ) + v ( y + y ) ] c F .
E y 3 ( x , y ; ν ) = i ν e i k ( F + z ) c F d u d v P ( u , v ) × E y 2 ( u , v ; ν ) e i ν π ( z F ) c F 2 ( u 2 + v 2 ) + i 2 π ν ( u x + v y ) c F .
E y 3 ( x , y ; ν ) = ν 2 e i k ( 3 F + z ) c 2 F 2 d u d v d x d y × E y 1 ( x , y ; ν ) P ( u , v ) × e i ν π ( z F ) c F 2 ( u 2 + v 2 ) + i 2 π ν [ u ( x + x ) + v ( y + y ) ] c F .
E y 3 ( x 1 , y 1 , z 1 ; ν 1 ) E y 3 * ( x 2 , y 2 , z 2 ; ν 2 ) = A ν 1 2 ν 2 2 e [ 2 π ( η 1 η 2 ) σ ] 2 2 c 4 F 4 d u d u d v d v d x d y × P ( u , v ) P * ( u , v ) e i 2 π x c F [ u ν 1 u ν 2 ] + i 2 π y c F [ v ν 1 v ν 2 ] × e i 2 π c F [ x 1 u ν 1 x 2 u ν 2 ] + i 2 π c F [ y 1 v ν 1 y 2 v ν 2 ] × e i π ν 1 ( z 1 F ) ( u 2 + v 2 ) c F 2 i π ν 2 ( z 2 F ) ( u 2 + v 2 ) c F 2 ,
E y 3 ( x 1 , y 1 , z 1 ; ν 1 ) E y 3 * ( x 2 , y 2 , z 2 ; ν 2 ) = A ν 2 2 e [ 2 π ( η 1 η 2 ) σ ] 2 2 c 2 F 2 d u d v P ( ν 2 u ν 1 , ν 2 v ν 1 ) × P * ( u , v ) e i 2 π ν 2 c F [ ( x 1 x 2 ) u + ( y 1 y 2 ) v ] × e i π ν 2 ( u 2 + v 2 ) c F 2 [ ν 2 ν 1 ( z 1 F ) ( z 2 F ) ] .
E y 3 ( x 1 , y 1 , z 1 ; ν 1 ) E y 3 * ( x 2 , y 2 , z 2 ; ν 2 ) = A ν 0 2 e [ 2 π ( η 1 η 2 ) σ ] 2 2 c 2 F 2 d u d v P ( u , v ) × e i 2 π ν 0 c F [ ( x 1 x 2 ) u + ( y 1 y 2 ) v ] e i π ν 0 ( z 1 z 2 ) c F 2 ( u 2 + v 2 ) .

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