Abstract

The formation of Young fringes that are observed in speckle photography is discussed. Both the shape and the contrast of the fringes are derived by starting from the cross-correlation function of the light intensities to be recorded in a double-exposure specklegram. The effects of focusing and defocusing in specklegram recording on speckle displacement affecting the shape, as well as the origin of speckle decorrelation affecting the contrast, are discussed in detail. The degree of the decorrelation is shown to be given by the squared modulation transfer function of the imaging lens for a spatial frequency that is proportional to the speckle displacement at its pupil plane. These theoretical relations are verified by experiments performed for in-plane object translation combined with tilt.

© 1984 Optical Society of America

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References

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  1. W. Fink, P. A. Büger, “Eine Methode zur kontaktlosen Messung kleiner Verschiebungen rauher Oberflächen,” Z. Angew. Phys. 30, 176–178 (1970).
  2. U. Köpf, “Ein kohärent-optisches Verfahren zur Messung mechanischer Schwingungen,” Optik 33, 517–521 (1971).
  3. H. J. Tiziani, “Applications of speckling for in-plane vibration analysis,” Opt. Acta 18, 891–902 (1971).
    [CrossRef]
  4. E. Archbold, A. E. Ennos, “Displacement measurement from double-exposure laser photographs,” Opt. Acta 19, 253–271 (1972).
    [CrossRef]
  5. H. J. Tiziani, “A study of the use of laser speckles to measure small tilts of optically rough surfaces,” Opt. Commun. 5, 271–276 (1972).
    [CrossRef]
  6. R. P. Khetan, F. P. Chiang, “Strain analysis by one-beam laser speckle interferometry. 1: Single aperture method,” Appl. Opt. 15, 2205–2215 (1976).
    [CrossRef] [PubMed]
  7. K. A. Stetson, “Problem of defocusing in speckle photography, its connection to hologram interferometry, and its solutions.” J. Opt. Soc. Am. 66, 1267–1271 (1976).
    [CrossRef]
  8. D. A. Gregory, “Basic physical principles of defocused speckle photography: a tilt topology inspection technique,” Opt. Laser Technol. 8, 201–213 (1976).
    [CrossRef]
  9. I. Yamaguchi, “Speckle displacement and decorrelation in the diffraction and image fields for small object deformation,” Opt. Acta 28, 1359–1376 (1981).
    [CrossRef]
  10. I. Yamaguchi, H. Saito, “Deformation measurement by speckle photography,” in Proceedings of the 13th International Congress on High-Speed Photography and Photonics, S. Hyodo, ed. (Japan Society of Precision Engineering, Tokyo, 1979), pp. 264–267.
    [CrossRef]
  11. I. Yamaguchi, “Fringe loci and visibility in holographic interferometry with diffuse objects. I. Fringes of equal inclination,” Opt. Acta 24, 1011–1025 (1977).
    [CrossRef]
  12. I. Yamaguchi, “Fringe loci and visibility in holographic interferometry. II. Fringes of equal thickness,” Opt. Acta 25, 299–314 (1978).
    [CrossRef]
  13. I. Yamaguchi, S. Komatsu, H. Saito, “Dynamics of speckles produced by a moving object and its applications,” Jpn. J. Appl. Phys. Suppl. 14-1, 301–306 (1975).
  14. A. E. Ennos, M. S. Virdee, “Laser speckle photography as a practical alternative to holographic interferometry for measuring plate deformation,” Opt. Eng. 21, 478–482 (1982).
    [CrossRef]

1982 (1)

A. E. Ennos, M. S. Virdee, “Laser speckle photography as a practical alternative to holographic interferometry for measuring plate deformation,” Opt. Eng. 21, 478–482 (1982).
[CrossRef]

1981 (1)

I. Yamaguchi, “Speckle displacement and decorrelation in the diffraction and image fields for small object deformation,” Opt. Acta 28, 1359–1376 (1981).
[CrossRef]

1978 (1)

I. Yamaguchi, “Fringe loci and visibility in holographic interferometry. II. Fringes of equal thickness,” Opt. Acta 25, 299–314 (1978).
[CrossRef]

1977 (1)

I. Yamaguchi, “Fringe loci and visibility in holographic interferometry with diffuse objects. I. Fringes of equal inclination,” Opt. Acta 24, 1011–1025 (1977).
[CrossRef]

1976 (3)

1975 (1)

I. Yamaguchi, S. Komatsu, H. Saito, “Dynamics of speckles produced by a moving object and its applications,” Jpn. J. Appl. Phys. Suppl. 14-1, 301–306 (1975).

1972 (2)

E. Archbold, A. E. Ennos, “Displacement measurement from double-exposure laser photographs,” Opt. Acta 19, 253–271 (1972).
[CrossRef]

H. J. Tiziani, “A study of the use of laser speckles to measure small tilts of optically rough surfaces,” Opt. Commun. 5, 271–276 (1972).
[CrossRef]

1971 (2)

U. Köpf, “Ein kohärent-optisches Verfahren zur Messung mechanischer Schwingungen,” Optik 33, 517–521 (1971).

H. J. Tiziani, “Applications of speckling for in-plane vibration analysis,” Opt. Acta 18, 891–902 (1971).
[CrossRef]

1970 (1)

W. Fink, P. A. Büger, “Eine Methode zur kontaktlosen Messung kleiner Verschiebungen rauher Oberflächen,” Z. Angew. Phys. 30, 176–178 (1970).

Archbold, E.

E. Archbold, A. E. Ennos, “Displacement measurement from double-exposure laser photographs,” Opt. Acta 19, 253–271 (1972).
[CrossRef]

Büger, P. A.

W. Fink, P. A. Büger, “Eine Methode zur kontaktlosen Messung kleiner Verschiebungen rauher Oberflächen,” Z. Angew. Phys. 30, 176–178 (1970).

Chiang, F. P.

Ennos, A. E.

A. E. Ennos, M. S. Virdee, “Laser speckle photography as a practical alternative to holographic interferometry for measuring plate deformation,” Opt. Eng. 21, 478–482 (1982).
[CrossRef]

E. Archbold, A. E. Ennos, “Displacement measurement from double-exposure laser photographs,” Opt. Acta 19, 253–271 (1972).
[CrossRef]

Fink, W.

W. Fink, P. A. Büger, “Eine Methode zur kontaktlosen Messung kleiner Verschiebungen rauher Oberflächen,” Z. Angew. Phys. 30, 176–178 (1970).

Gregory, D. A.

D. A. Gregory, “Basic physical principles of defocused speckle photography: a tilt topology inspection technique,” Opt. Laser Technol. 8, 201–213 (1976).
[CrossRef]

Khetan, R. P.

Komatsu, S.

I. Yamaguchi, S. Komatsu, H. Saito, “Dynamics of speckles produced by a moving object and its applications,” Jpn. J. Appl. Phys. Suppl. 14-1, 301–306 (1975).

Köpf, U.

U. Köpf, “Ein kohärent-optisches Verfahren zur Messung mechanischer Schwingungen,” Optik 33, 517–521 (1971).

Saito, H.

I. Yamaguchi, S. Komatsu, H. Saito, “Dynamics of speckles produced by a moving object and its applications,” Jpn. J. Appl. Phys. Suppl. 14-1, 301–306 (1975).

I. Yamaguchi, H. Saito, “Deformation measurement by speckle photography,” in Proceedings of the 13th International Congress on High-Speed Photography and Photonics, S. Hyodo, ed. (Japan Society of Precision Engineering, Tokyo, 1979), pp. 264–267.
[CrossRef]

Stetson, K. A.

Tiziani, H. J.

H. J. Tiziani, “A study of the use of laser speckles to measure small tilts of optically rough surfaces,” Opt. Commun. 5, 271–276 (1972).
[CrossRef]

H. J. Tiziani, “Applications of speckling for in-plane vibration analysis,” Opt. Acta 18, 891–902 (1971).
[CrossRef]

Virdee, M. S.

A. E. Ennos, M. S. Virdee, “Laser speckle photography as a practical alternative to holographic interferometry for measuring plate deformation,” Opt. Eng. 21, 478–482 (1982).
[CrossRef]

Yamaguchi, I.

I. Yamaguchi, “Speckle displacement and decorrelation in the diffraction and image fields for small object deformation,” Opt. Acta 28, 1359–1376 (1981).
[CrossRef]

I. Yamaguchi, “Fringe loci and visibility in holographic interferometry. II. Fringes of equal thickness,” Opt. Acta 25, 299–314 (1978).
[CrossRef]

I. Yamaguchi, “Fringe loci and visibility in holographic interferometry with diffuse objects. I. Fringes of equal inclination,” Opt. Acta 24, 1011–1025 (1977).
[CrossRef]

I. Yamaguchi, S. Komatsu, H. Saito, “Dynamics of speckles produced by a moving object and its applications,” Jpn. J. Appl. Phys. Suppl. 14-1, 301–306 (1975).

I. Yamaguchi, H. Saito, “Deformation measurement by speckle photography,” in Proceedings of the 13th International Congress on High-Speed Photography and Photonics, S. Hyodo, ed. (Japan Society of Precision Engineering, Tokyo, 1979), pp. 264–267.
[CrossRef]

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

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

I. Yamaguchi, S. Komatsu, H. Saito, “Dynamics of speckles produced by a moving object and its applications,” Jpn. J. Appl. Phys. Suppl. 14-1, 301–306 (1975).

Opt. Acta (5)

I. Yamaguchi, “Fringe loci and visibility in holographic interferometry with diffuse objects. I. Fringes of equal inclination,” Opt. Acta 24, 1011–1025 (1977).
[CrossRef]

I. Yamaguchi, “Fringe loci and visibility in holographic interferometry. II. Fringes of equal thickness,” Opt. Acta 25, 299–314 (1978).
[CrossRef]

H. J. Tiziani, “Applications of speckling for in-plane vibration analysis,” Opt. Acta 18, 891–902 (1971).
[CrossRef]

E. Archbold, A. E. Ennos, “Displacement measurement from double-exposure laser photographs,” Opt. Acta 19, 253–271 (1972).
[CrossRef]

I. Yamaguchi, “Speckle displacement and decorrelation in the diffraction and image fields for small object deformation,” Opt. Acta 28, 1359–1376 (1981).
[CrossRef]

Opt. Commun. (1)

H. J. Tiziani, “A study of the use of laser speckles to measure small tilts of optically rough surfaces,” Opt. Commun. 5, 271–276 (1972).
[CrossRef]

Opt. Eng. (1)

A. E. Ennos, M. S. Virdee, “Laser speckle photography as a practical alternative to holographic interferometry for measuring plate deformation,” Opt. Eng. 21, 478–482 (1982).
[CrossRef]

Opt. Laser Technol. (1)

D. A. Gregory, “Basic physical principles of defocused speckle photography: a tilt topology inspection technique,” Opt. Laser Technol. 8, 201–213 (1976).
[CrossRef]

Optik (1)

U. Köpf, “Ein kohärent-optisches Verfahren zur Messung mechanischer Schwingungen,” Optik 33, 517–521 (1971).

Z. Angew. Phys. (1)

W. Fink, P. A. Büger, “Eine Methode zur kontaktlosen Messung kleiner Verschiebungen rauher Oberflächen,” Z. Angew. Phys. 30, 176–178 (1970).

Other (1)

I. Yamaguchi, H. Saito, “Deformation measurement by speckle photography,” in Proceedings of the 13th International Congress on High-Speed Photography and Photonics, S. Hyodo, ed. (Japan Society of Precision Engineering, Tokyo, 1979), pp. 264–267.
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Coordinate systems for calculating the intensity cross-correlation function in the image field.

Fig. 2
Fig. 2

Speckle displacements at the aperture plane that affect the decorrelation of speckles in the image field.

Fig. 3
Fig. 3

Coordinate systems for observing the Young fringes.

Fig. 4
Fig. 4

Experimental arrangement for recording specklegrams.

Fig. 5
Fig. 5

An example of (a) the output signals from the linear photodiode array scanning the Young fringes and (b) its Fourier transform displayed on a synchroscope.

Fig. 6
Fig. 6

Spatial frequency of the Young fringes produced from specklegrams that are recorded before and after surface tilt for various amounts of defocus.

Fig. 7
Fig. 7

Contrast of Young fringes obtained from the focused specklegrams against the object translation for various amounts of tilt angle.

Fig. 8
Fig. 8

Contrast of Young fringes obtained from the focused specklegrams against the calculated speckle displacement at the center of the lens aperture (normalized by the aperture diameter). The solid curve represents the theoretical values for an aberration-free imaging system, that is, its squared MTF, whereas dashed curve is the square of the actual lens MTF measured in the same condition as in specklegram recording.

Equations (20)

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I 1 ( q , D ) I 2 ( q + q ¯ , D ) = | P ( s ) 2 d 2 s | 2 + | P ( s ) P * ( s - A P ) exp ( i Φ ) × exp [ - i k s L o + D · ( q ¯ - A ) ] d 2 s | 2 ,
A P = a T - L o m o · a - a z L 0 s - s · a T ,
Φ = k { D L o ( L o + D ) [ a z L o s 2 + s · ( s · a T ) ] + a z z 2 2 L o 2 } ,
A = - 1 M ( a T + a z L o q ) - D [ ( 1 + M ) a T L o - M m o · a ] .
M D = L o - Δ L L o + D = L o - Δ L f - 1.
A = - a T M D - a z M L o q + Δ L M D m o · a .
A x = - a x M D [ 1 + Δ L L s ( 1 - l s x 2 ) ] + a y M D Δ L L s l s x l s y - a z M D ( M D q x L o - Δ L L s l s x l s z ) + Δ L M D { x x l s x + x y l s y - Ω y ( 1 + l s z ) + Ω z l s y ] , A y = a x M D Δ L L s l s x l s y - a y M D [ 1 + Δ L L s ( 1 - l s y 2 ) ] - a z M D ( M D q y L o - Δ L L s l s y l s z ) + Δ L M D [ x y l s x + y y l s y + Ω x ( 1 + l s z ) - Ω z l s x ] ,
A = 1 M ( a T + a z L o q ) .
I 1 ( q , D ) I 2 ( q + A , D ) = | P ( s ) 2 d 2 s | 2 + | P ( s ) P * ( s - A P ) exp ( i Φ ) d 2 s | 2 .
A p x = - a x [ L o L s ( l s x 2 - 1 ) - 1 ] - a y L o L s l s x l s y - a z ( L o L s l s x l s z + s x L o ) - L o [ x x ( l s x + s x L o ) + x y ( l s y + s y L o ) - Ω y ( 1 + l s z ) + Ω z ( l s y + s y L o ) ] , A p y = - a x L o L s l s y l s x - a y [ L o L s ( l s y 2 - 1 ) - 1 ] - a z ( L 0 L s l s y l s z + s y L o ) - L o [ y y ( l s y + s y L o ) L o + x y ( l s x + s x L o ) + Ω x ( 1 + l s z ) - Ω z ( l s x - s x L o ) ] .
T 12 ( q ) = T o - η [ I 1 ( q , D ) + I 2 ( q , D ) ] ,
I F ( p ) = | U B ( q - q o ) T 12 ( q ) exp ( - i 2 π q · p λ o f o ) d 2 q | 2 ,
J Y ( p ) = I F ( p ) .
I 1 ( q , D ) I 2 ( q + q ¯ , D ) = C ( 0 ) + γ C ( q ¯ - A ) ,
C ( q ) = | P ( s ) 2 exp ( - i k s · q L o + D ) d 2 s | 2
γ = | P ( s ) P * ( s - A P ) exp ( i Φ ) d 2 s | 2 / | P ( s ) 2 d 2 s | 2
J y ( p ) = ( I o - 2 η I ) 2 | U B ( q ) exp ( - i 2 π q · p λ o f o ) d 2 q | 2 + 2 η 2 U B ( q ) 2 d 2 q C ˜ ( p λ o f o ) [ 1 + γ cos ( 2 π p · A λ o f o ) ] .
C ˜ ( S ) C ( q ) exp ( i 2 π q · S ) d 2 q = P ( s ) 2 P [ s + λ ( L o + D ) S ] 2 d 2 s
A x = - a x M D - Δ L M D Ω y ( 1 + cos ϑ s ) , A y = 0 ,
A p x = a x + Ω y L o ( 1 + cos ϑ s ) , A p y = 0 ,

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