Abstract

A theoretical analysis based on statistical optics is presented for the diffraction halo functions of random speckle patterns either generated by a coherent source such as a laser or artificially created and illuminated by an incoherent radiation such as white light. It is shown that for a coherent speckle pattern the halo function is given by the autocorrelation of the squared aperture function of the recording system, whereas for a incoherent speckle pattern it is governed by the squared autocorrelation of the aperture function.

© 1985 Optical Society of America

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References

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  1. R. K. Erf, Ed., Speckle Metrology (Academic, New York, 1978).
  2. F. P. Chiang, “A New Family of 2D and 3D Experimental Stress Analysis Techniques Using Laser Speckles,” Solid Mech. Arch. 3, 1 (1978).
  3. A. Asandi, F. P. Chiang, “Theory and Applications of the White Light Speckle Method for Strain Analysis,” Opt. Eng. 21, 570 (1982).
  4. T. D. Dudderar, P. G. Simpkins, “The Development of Scattered Light Speckle Metrology,” Opt. Eng. 21, 396 (1982).
    [CrossRef]
  5. G. H. Kaufmann, “On the Numerical Processing of Speckle Photograph Fringes,” Opt. Laser Techn. 12, 207 (1980).
    [CrossRef]
  6. C. S. Vikram, “Simple Approach to Process Speckle-Photography Data,” Opt. Lett. 7, 374 (1982).
    [CrossRef] [PubMed]
  7. C. S. Vikram, K. Vedam, “Processing Speckle Photography Data: Circular Imaging Aperture,” Appl. Opt. 22, 653 (1983).
    [CrossRef] [PubMed]
  8. R. P. Khetan, F.-P. Chiang, “Strain Analysis by One-Beam Laser Speckle Interferometry. 1: Single Aperture Method,” Appl. Opt. 15, 2205 (1976).
    [CrossRef] [PubMed]
  9. H. J. Tiziani, “Vibration Analysis of Deformation Measurement,” in Ref. 1, Chap. 8.
  10. I. Yamaguchi, “Fringe Formation in Speckle Photography,” J. Opt. Soc. Am. A 1, 81 (1984).
    [CrossRef]
  11. J. B. Chen, F.-P. Chiang, “Statistical Analysis of Whole Field Filtering of Specklegram and Its Upper Limit of Measurement,” J. Opt. Soc. Am. A 1, 845 (1984).
    [CrossRef]
  12. D. W. Li, J. B. Chen, F.-P. Chiang, “Statistical Analysis of One-Beam Subjective Laser Speckle Interferometry,” J. Opt. Soc. Am. A 2, 657 (1985).
    [CrossRef]
  13. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  14. J. W. Goodman, Laser Speckle and Related Phenomenon, J. C. Dainty, Ed. (Springer, New York, 1975), Chap. 2.
  15. F.-P. Chiang, A. Asundi, “White Light Speckle Method of Experimental Strain Analysis,” Appl. Opt. 18, 409 (1979).
    [CrossRef] [PubMed]
  16. F.-P. Chiang, R. M. Juang, “Laser Speckle Interferometry for Plate Bending Problems,” Appl. Opt, 15, 2199 (1976).
    [CrossRef] [PubMed]
  17. F.-P. Chiang, R. M. Juang, “Vibration Analysis of Plate and Shell by Laser Speckle Interferometry,” Opt. Acta 23, 997 (1976).
    [CrossRef]
  18. F.-P. Chiang, “Multi-Aperture Speckle Interferometry,” in Proceedings, Conference on Speckle Phenomena and Their Applications, Loughborough U, England, (27–28 Mar. 1974).
  19. F.-P. Chiang et al., “Multiaperture White Light Speckle Method Applied to the Strain Analysis of Cylinders with Holes Under Compression,” Opt. Lasers Eng. 2, 151 (1981).
    [CrossRef]
  20. R. Meynart, “Diffraction Halo in Speckle Photography,” Appl. Opt. 23, 2235 (1984).
    [CrossRef] [PubMed]
  21. K. A. Stetson, “Vulnerability of Speckle Photography to Lens Aberrations,” J. Opt. Soc. Am. 67, 1587 (1977).
    [CrossRef]
  22. F.-P. Chiang, D. W. Li, “Diffraction Halo Functions of Coherent and Incoherent Random Speckle Patterns,” State U. of New York at Stony Brook, College of Engineering and Applications Scientific Technical Report 455 (1984).
    [PubMed]

1985

1984

1983

1982

C. S. Vikram, “Simple Approach to Process Speckle-Photography Data,” Opt. Lett. 7, 374 (1982).
[CrossRef] [PubMed]

A. Asandi, F. P. Chiang, “Theory and Applications of the White Light Speckle Method for Strain Analysis,” Opt. Eng. 21, 570 (1982).

T. D. Dudderar, P. G. Simpkins, “The Development of Scattered Light Speckle Metrology,” Opt. Eng. 21, 396 (1982).
[CrossRef]

1981

F.-P. Chiang et al., “Multiaperture White Light Speckle Method Applied to the Strain Analysis of Cylinders with Holes Under Compression,” Opt. Lasers Eng. 2, 151 (1981).
[CrossRef]

1980

G. H. Kaufmann, “On the Numerical Processing of Speckle Photograph Fringes,” Opt. Laser Techn. 12, 207 (1980).
[CrossRef]

1979

1978

F. P. Chiang, “A New Family of 2D and 3D Experimental Stress Analysis Techniques Using Laser Speckles,” Solid Mech. Arch. 3, 1 (1978).

1977

1976

R. P. Khetan, F.-P. Chiang, “Strain Analysis by One-Beam Laser Speckle Interferometry. 1: Single Aperture Method,” Appl. Opt. 15, 2205 (1976).
[CrossRef] [PubMed]

F.-P. Chiang, R. M. Juang, “Laser Speckle Interferometry for Plate Bending Problems,” Appl. Opt, 15, 2199 (1976).
[CrossRef] [PubMed]

F.-P. Chiang, R. M. Juang, “Vibration Analysis of Plate and Shell by Laser Speckle Interferometry,” Opt. Acta 23, 997 (1976).
[CrossRef]

Asandi, A.

A. Asandi, F. P. Chiang, “Theory and Applications of the White Light Speckle Method for Strain Analysis,” Opt. Eng. 21, 570 (1982).

Asundi, A.

Chen, J. B.

Chiang, F. P.

A. Asandi, F. P. Chiang, “Theory and Applications of the White Light Speckle Method for Strain Analysis,” Opt. Eng. 21, 570 (1982).

F. P. Chiang, “A New Family of 2D and 3D Experimental Stress Analysis Techniques Using Laser Speckles,” Solid Mech. Arch. 3, 1 (1978).

Chiang, F.-P.

D. W. Li, J. B. Chen, F.-P. Chiang, “Statistical Analysis of One-Beam Subjective Laser Speckle Interferometry,” J. Opt. Soc. Am. A 2, 657 (1985).
[CrossRef]

J. B. Chen, F.-P. Chiang, “Statistical Analysis of Whole Field Filtering of Specklegram and Its Upper Limit of Measurement,” J. Opt. Soc. Am. A 1, 845 (1984).
[CrossRef]

F.-P. Chiang et al., “Multiaperture White Light Speckle Method Applied to the Strain Analysis of Cylinders with Holes Under Compression,” Opt. Lasers Eng. 2, 151 (1981).
[CrossRef]

F.-P. Chiang, A. Asundi, “White Light Speckle Method of Experimental Strain Analysis,” Appl. Opt. 18, 409 (1979).
[CrossRef] [PubMed]

R. P. Khetan, F.-P. Chiang, “Strain Analysis by One-Beam Laser Speckle Interferometry. 1: Single Aperture Method,” Appl. Opt. 15, 2205 (1976).
[CrossRef] [PubMed]

F.-P. Chiang, R. M. Juang, “Laser Speckle Interferometry for Plate Bending Problems,” Appl. Opt, 15, 2199 (1976).
[CrossRef] [PubMed]

F.-P. Chiang, R. M. Juang, “Vibration Analysis of Plate and Shell by Laser Speckle Interferometry,” Opt. Acta 23, 997 (1976).
[CrossRef]

F.-P. Chiang, D. W. Li, “Diffraction Halo Functions of Coherent and Incoherent Random Speckle Patterns,” State U. of New York at Stony Brook, College of Engineering and Applications Scientific Technical Report 455 (1984).
[PubMed]

F.-P. Chiang, “Multi-Aperture Speckle Interferometry,” in Proceedings, Conference on Speckle Phenomena and Their Applications, Loughborough U, England, (27–28 Mar. 1974).

Dudderar, T. D.

T. D. Dudderar, P. G. Simpkins, “The Development of Scattered Light Speckle Metrology,” Opt. Eng. 21, 396 (1982).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Laser Speckle and Related Phenomenon, J. C. Dainty, Ed. (Springer, New York, 1975), Chap. 2.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Juang, R. M.

F.-P. Chiang, R. M. Juang, “Vibration Analysis of Plate and Shell by Laser Speckle Interferometry,” Opt. Acta 23, 997 (1976).
[CrossRef]

F.-P. Chiang, R. M. Juang, “Laser Speckle Interferometry for Plate Bending Problems,” Appl. Opt, 15, 2199 (1976).
[CrossRef] [PubMed]

Kaufmann, G. H.

G. H. Kaufmann, “On the Numerical Processing of Speckle Photograph Fringes,” Opt. Laser Techn. 12, 207 (1980).
[CrossRef]

Khetan, R. P.

Li, D. W.

D. W. Li, J. B. Chen, F.-P. Chiang, “Statistical Analysis of One-Beam Subjective Laser Speckle Interferometry,” J. Opt. Soc. Am. A 2, 657 (1985).
[CrossRef]

F.-P. Chiang, D. W. Li, “Diffraction Halo Functions of Coherent and Incoherent Random Speckle Patterns,” State U. of New York at Stony Brook, College of Engineering and Applications Scientific Technical Report 455 (1984).
[PubMed]

Meynart, R.

Simpkins, P. G.

T. D. Dudderar, P. G. Simpkins, “The Development of Scattered Light Speckle Metrology,” Opt. Eng. 21, 396 (1982).
[CrossRef]

Stetson, K. A.

Tiziani, H. J.

H. J. Tiziani, “Vibration Analysis of Deformation Measurement,” in Ref. 1, Chap. 8.

Vedam, K.

Vikram, C. S.

Yamaguchi, I.

Appl. Opt

F.-P. Chiang, R. M. Juang, “Laser Speckle Interferometry for Plate Bending Problems,” Appl. Opt, 15, 2199 (1976).
[CrossRef] [PubMed]

Appl. Opt.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Acta

F.-P. Chiang, R. M. Juang, “Vibration Analysis of Plate and Shell by Laser Speckle Interferometry,” Opt. Acta 23, 997 (1976).
[CrossRef]

Opt. Eng.

A. Asandi, F. P. Chiang, “Theory and Applications of the White Light Speckle Method for Strain Analysis,” Opt. Eng. 21, 570 (1982).

T. D. Dudderar, P. G. Simpkins, “The Development of Scattered Light Speckle Metrology,” Opt. Eng. 21, 396 (1982).
[CrossRef]

Opt. Laser Techn.

G. H. Kaufmann, “On the Numerical Processing of Speckle Photograph Fringes,” Opt. Laser Techn. 12, 207 (1980).
[CrossRef]

Opt. Lasers Eng.

F.-P. Chiang et al., “Multiaperture White Light Speckle Method Applied to the Strain Analysis of Cylinders with Holes Under Compression,” Opt. Lasers Eng. 2, 151 (1981).
[CrossRef]

Opt. Lett.

Solid Mech. Arch.

F. P. Chiang, “A New Family of 2D and 3D Experimental Stress Analysis Techniques Using Laser Speckles,” Solid Mech. Arch. 3, 1 (1978).

Other

R. K. Erf, Ed., Speckle Metrology (Academic, New York, 1978).

F.-P. Chiang, “Multi-Aperture Speckle Interferometry,” in Proceedings, Conference on Speckle Phenomena and Their Applications, Loughborough U, England, (27–28 Mar. 1974).

H. J. Tiziani, “Vibration Analysis of Deformation Measurement,” in Ref. 1, Chap. 8.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

J. W. Goodman, Laser Speckle and Related Phenomenon, J. C. Dainty, Ed. (Springer, New York, 1975), Chap. 2.

F.-P. Chiang, D. W. Li, “Diffraction Halo Functions of Coherent and Incoherent Random Speckle Patterns,” State U. of New York at Stony Brook, College of Engineering and Applications Scientific Technical Report 455 (1984).
[PubMed]

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Figures (5)

Fig. 1
Fig. 1

Halo shape as a function of recording aperture for (a) incoherent speckle pattern and (b) coherent speckle pattern.

Fig. 2
Fig. 2

Profiles of coherent and incoherent halo functions for (a) square aperture and (b) circular aperture.

Fig. 3
Fig. 3

Effect of defocus on halo function with increasing amount of defocus to the right from (a) incoherent speckle pattern and (b) coherent speckle pattern.

Fig. 4
Fig. 4

Theoretical diffraction patterns of a conventional four-aperture arrangement (top left) and a modified arrangement (top right) with a random phase term introduced at the right aperture: (a), (b) coherent diffraction functions for the two-aperture sets; (c), and (d) incoherent diffraction functions for the two-aperture sets. The number in circle indicates the relative peak strength of light intensity at the center of the diffraction order.

Fig. 5
Fig. 5

Experimental diffraction pattern of four-aperture recording with (right) and without (left) random phase term: (a), (b) diffraction patterns for coherent speckle field; (c), (d) diffraction patterns for incoherent speckle field.

Equations (29)

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A i ( x ) = h ( x ) * A 0 ( x / M ) ,
t ch ( x ) = | A i ( x ) | 2 ,
I i ( x ) = | h ( x ) | 2 * I 0 ( x / M ) ,
t ih ( x ) = I i ( x ) ,
A h ( u ) = t ( x ) exp ( j 2 π λ f u x ) d 2 x ,
I c ( u ) = | A h ( u ) | 2 = t ch ( x ) t ch * ( x ) × exp [ j 2 π λ f ( x x ) u ] d 2 x d 2 x .
I c ( u ) = R A ( x , x ) exp [ j 2 π λ f ( x x ) u ] d 2 x d 2 x ,
R A ( x , x ) = t ch ( x ) t ch * ( x )
R A ( x , x ) = I 2 + C | | P ( r ) | 2 × exp { j 2 π q [ r ( x x ) ] } d 2 r | 2 ,
Γ ( u ) = I 2 δ ( u ) + C | P ( r ) | 2 | P ( r λ 0 q u ) | 2 d 2 r ,
I c ( u ) = I 2 δ ( u ) + C | P ( r ) | 2 | P ( r λ 0 q u λ f ) | 2 d 2 r .
I ch ( u ) = | P ( r ) | 2 | ( P ( r λ 0 q u λ f ) | 2 d 2 r .
I h ( u ) = | A h | 2 = | [ | h ( x ) | 2 * I 0 ( x M ) ] exp ( j 2 π λ f x u ) d 2 x | 2 = | | h ( x ) | 2 exp ( j 2 π λ f x u ) d 2 x | 2 | I 0 ( x M ) exp ( j 2 π λ f x u ) d 2 x | 2
I h ( u ) = I ihI I ihII ,
I ihI = | | h ( x ) | 2 exp ( j 2 π λ f x u ) d 2 x | 2 ,
I ihII = | I 0 ( x M ) exp ( j 2 π λ f x u ) d 2 x | 2 = I 0 ( x M ) I 0 ( x M ) exp [ j 2 π λ f ( x x ) u ] d 2 x d 2 x .
I ihI ( u ) = | P ( r ) P * ( r + λ 0 qu λ f ) d 2 r | 2
I 0 ( x M ) I 0 ( x M ) = δ ( x x ) ,
I ih = I ih I = | P ( r ) P * ( r + λ 0 q u λ f ) d 2 r | 2 .
P ( r x , r y ) = { 1 for | r x | , | r y | a , 0 otherwise ,
I ch = { ( 1 u x 2 c ) ( 1 u y 2 c ) for | u x | , | u y | 2 c , 0 otherwise ,
c = fa λ / q λ 0 .
P ( r x , r y ) = { 1 for ( r x 2 + r y 2 ) 1 / 2 D / 2 , 0 otherwise ,
I ch = { 2 π { cos 1 ( u c ) u c [ 1 ( u c ) 2 ] 1 / 2 } for u c , 0 otherwise ,
c = fD λ / q λ 0 ,
u = ( u x 2 + u y 2 ) 1 / 2 .
I ih = I ch 2 .
c = λ af / q λ 0 ,
d = λ bf / q λ 0 .

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