Abstract

Contrary to what is found in most of the existing scientific literature, where a specific frame is developed, the theory of speckle interferometry is (conveniently) presented here as a particular case of the more general theory of holographic interferometry. In addition to the intellectual benefit of dealing with a single unified theory, this brings about many advantages when it comes to discuss fundamental topics such as the three-dimensional evolution of the complex amplitude of the diffuse optical wavefronts, the degree of approximation of the leading formulas, the loss of fringe contrast, the decorrelation effects, the real influence of the terms generally neglected in out-of-focus regions. In the same way, the statistical properties of the speckle fields, usually treated as a separate subject matter, are also integrated in the theory, thus providing a comprehensive knowledge of the qualitative features of speckle interferometry methods, otherwise difficult to understand.

© 2006 Optical Society of America

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References

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  1. R. Dändliker, "The story of speckles in interferometry," in Interferometry in Speckle Light, Theory and Applications, P.Jacquot and J.M.Fournier, eds. (Springer Verlag, 2000), pp. 3-10.
  2. K. A. Stetson, "Problem of defocusing in speckle photography, its connection to hologram interferometry, and its solutions," J. Opt. Soc. Am. A 66, 1267-1271 (1976).
    [CrossRef]
  3. P. Jacquot and P. K. Rastogi, "Speckle motions induced by rigid-body movements in free-space geometry: an explicit investigation and extension to new cases," Appl. Opt. 18, 2022-2032 (1979).
    [CrossRef] [PubMed]
  4. A. E. Ennos, "Speckle interferometry," in Laser Speckle and Related Phenomena, J.C.Dainty, ed. (Springer Verlag, 1975), pp. 203-253.
    [CrossRef]
  5. C. Forno, "Deformation measurement using high-resolution Moiré photography," Opt. Laser Eng. 8, 189-212 (1988).
    [CrossRef]
  6. A. Andersson, A. Runnemalm, and M. Sjödahl, "Digital speckle-pattern interferometry: fringe retrieval for large in-plane deformations with digital speckle photography," Appl. Opt. 38, 5408-5412 (1999).
    [CrossRef]
  7. J. W. Goodman, "Statistical properties of laser speckle patterns," in Laser Speckle and Related Phenomena, J.C.Dainty, ed. (Springer Verlag, 1975), pp. 9-75.
    [CrossRef]
  8. Q. B. Li and F. P. Chiang, "Three-dimensional dimension of Laser Speckle," Appl. Opt. 31, 6287-6291 (1992).
    [CrossRef] [PubMed]
  9. J. D. Briers, "Holographic, speckle, and Moiré techniques in optical metrology," Prog. Quantum Electron. 17, 167-233 (1993).
    [CrossRef]
  10. M. Lehmann, "Statistical theory of two-wave speckle interferometry and its application to the optimization of deformation measurements," PhD thesis EPFL No. 1797 (Swiss Federal Institute of Technology Lausanne, 1998).
  11. H. Helmers, M. Bischoff, and L. Ehlkes, "ESPI-system with active in-line digital phase phase stabilization," in Fringe '01, Fourth International Workshop on Automatic Processing of Fringe Patterns, W.Jüptner and W.Osten, eds. (Elsevier, 2001), pp. 673-679.
  12. T. Maack and R. Kowarschik, "Camera influence on the accuracy of a phase-shifting speckle interferometer," Appl. Opt. 35, 3514-3524 (1996).
    [CrossRef] [PubMed]
  13. T. Maack, R. Kowarschik, and G. Notni, "Optimum lens aperture in phase-shifting speckle interferometric setups for maximum accuracy of phase measurement," Appl. Opt. 36, 6217-6224 (1997).
    [CrossRef]
  14. D. A. Gregory, "Basic physical principles of defocused speckle photography: a tilt topology inspection technique," Opt. Laser Technol. 8, 201-213 (1976).
    [CrossRef]
  15. C. Froehly, J. Monneret, J. Pasteur, and J. Ch. Viénot, "Étude des faibles déplacements d'objets opaques et de la distorsion optique dans les lasers à solide par interférométrie holographique," Opt. Act. 16, 343-362 (1969).
    [CrossRef]
  16. S. Walles, "On the concept of homologous rays in holographic interferometry of diffusely reflecting surfaces," Opt. Act. 17, 899-913 (1970).
    [CrossRef]
  17. M. Dubas and W. Schumann, "Sur la détermination holographique de l'état de déformation à la surface d'un corps non-transparent," Opt. Act. 21, 547-562 (1974).
    [CrossRef]
  18. J. Svetlík, "Speckle displacement: two related approaches," J. Mod. Opt. 39, 149-157 (1992).
    [CrossRef]
  19. I. Yamaguchi, "Speckle displacement and decorrelation in the diffraction and image fields for small object deformation," Opt. Act. 28, 1359-1376 (1981).
    [CrossRef]
  20. M. Sjödahl, "Calculation of speckle displacement, decorrelation, and object-point location in imaging systems," Appl. Opt. 34, 7998-8009 (1995).
    [CrossRef] [PubMed]
  21. P. Jacquot, "Speckle motions in three-dimensional image fields," presented at the Topical Meeting on Hologram Interferometry and Speckle Metrology, Cape Cod, Massachusetts, 2-4 June 1980.
  22. P. K. Rastogi and P. Jacquot, "Speckle metrology techniques: a parametric examination of the observed fringes," Opt. Eng. 21, 411-426 (1982).
  23. P. Jacquot, "Photographie de speckles: exemples tirés de l'analyse de déformation de corps solides," in Lasers et Applications Industrielles, C.Bonjour and M.Matthey, eds. (Presses Polytechniques Romandes, 1982), pp. 149-204.
  24. W. Schumann and M. Dubas, Holographic Interferometry From the Scope of Deformation Analysis of Opaque Bodies, Vol. 16 of Springer Series in Optical Sciences (Springer, 1979).
  25. W. Schumann, Z. P. Zuercher, and D. Cuche, Holography and Deformation Analysis, Vol. 46 of Springer Series in Optical Sciences (Springer, 1985).
  26. D.W.Robinson and G.T.Reid, eds., Interferogram Analysis, Digital Fringe Pattern Measurement Techniques (Institute of Physics Publishing, 1993).
  27. N. Goudemand, "Application of dynamic phase-shifting with wavelet analysis to electronic speckle contouring," Appl. Opt. 45, 3704-3711 (2006).
    [CrossRef] [PubMed]
  28. M. Dubas, "Sur l'Analyse Expérimentale de l'État de Déformation à la Surface d'un Corps Opaque par Interférométrie Holographique en particulier à l'Aide de la Localisation des Franges," PhD thesis ETHZ No. 5673 (Swiss Federal Institute of Technology Zurich, 1976).
  29. N. E. Molin and K. A. Stetson, "Fringe localization in hologram interferometry of mutually independent and dependent rotations around orthogonal, nonintersecting axes," Optik 33, 399-422 (1971).
  30. M. Lehmann, "Decorrelation-induced phase errors in phase-shifting speckle interferometry," Appl. Opt. 36, 3657-3667 (1997).
    [CrossRef] [PubMed]
  31. R. Thalmann, "Electronic fringe interpolation in holographic interferometry, applied to deformation measurement of solid objects," PhD thesis (Institute of Microtechnology, Neuchatel, Switzerland, 1986).
  32. K. A. Stetson, "Fringe interpretation for hologram interferometry of rigid-body motions and homogeneous deformations," J. Opt. Soc. Am. A 64, 1-10 (1974).
    [CrossRef]
  33. H. C. Lee, "Review of image-blur models in a photographic system using the principles of optics," Opt. Eng. 29, 405-421 (1990).
    [CrossRef]
  34. B. Lutz and W. Schumann, "Approach to extend the domain of visibility of recovered modified fringes when holographic interferometry is applied to large deformations," Opt. Eng. 34, 1879-1886 (1995).
    [CrossRef]
  35. W. Schumann and B. Lutz, "Holographic interferometry applied to large deformations, the role of the second derivative of the optical path difference in real-time technique," in Interferometry `94: Photomechanics, R. J. Pryputniewicz and J. Stupnicki, eds., Proc. SPIE 2342, 219-231 (1994).
    [CrossRef]
  36. R. Dändliker and P. Jacquot, "Holographic interferometry and speckle methods," in Optical Sensors, E.Wagner, R.Dändliker, and K.Spenner, eds. (VCH Verlagsgesellschaft, 1992), Chap. 23, pp. 589-628.
  37. T. M. Kreis, "Frequency analysis of digital holography," Opt. Eng. 41, 771-778 (2002).
    [CrossRef]

2006

2002

T. M. Kreis, "Frequency analysis of digital holography," Opt. Eng. 41, 771-778 (2002).
[CrossRef]

1999

1997

1996

1995

M. Sjödahl, "Calculation of speckle displacement, decorrelation, and object-point location in imaging systems," Appl. Opt. 34, 7998-8009 (1995).
[CrossRef] [PubMed]

B. Lutz and W. Schumann, "Approach to extend the domain of visibility of recovered modified fringes when holographic interferometry is applied to large deformations," Opt. Eng. 34, 1879-1886 (1995).
[CrossRef]

1994

W. Schumann and B. Lutz, "Holographic interferometry applied to large deformations, the role of the second derivative of the optical path difference in real-time technique," in Interferometry `94: Photomechanics, R. J. Pryputniewicz and J. Stupnicki, eds., Proc. SPIE 2342, 219-231 (1994).
[CrossRef]

1993

J. D. Briers, "Holographic, speckle, and Moiré techniques in optical metrology," Prog. Quantum Electron. 17, 167-233 (1993).
[CrossRef]

1992

J. Svetlík, "Speckle displacement: two related approaches," J. Mod. Opt. 39, 149-157 (1992).
[CrossRef]

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

1990

H. C. Lee, "Review of image-blur models in a photographic system using the principles of optics," Opt. Eng. 29, 405-421 (1990).
[CrossRef]

1988

C. Forno, "Deformation measurement using high-resolution Moiré photography," Opt. Laser Eng. 8, 189-212 (1988).
[CrossRef]

1982

P. K. Rastogi and P. Jacquot, "Speckle metrology techniques: a parametric examination of the observed fringes," Opt. Eng. 21, 411-426 (1982).

1981

I. Yamaguchi, "Speckle displacement and decorrelation in the diffraction and image fields for small object deformation," Opt. Act. 28, 1359-1376 (1981).
[CrossRef]

1979

1976

K. A. Stetson, "Problem of defocusing in speckle photography, its connection to hologram interferometry, and its solutions," J. Opt. Soc. Am. A 66, 1267-1271 (1976).
[CrossRef]

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

1974

M. Dubas and W. Schumann, "Sur la détermination holographique de l'état de déformation à la surface d'un corps non-transparent," Opt. Act. 21, 547-562 (1974).
[CrossRef]

K. A. Stetson, "Fringe interpretation for hologram interferometry of rigid-body motions and homogeneous deformations," J. Opt. Soc. Am. A 64, 1-10 (1974).
[CrossRef]

1971

N. E. Molin and K. A. Stetson, "Fringe localization in hologram interferometry of mutually independent and dependent rotations around orthogonal, nonintersecting axes," Optik 33, 399-422 (1971).

1970

S. Walles, "On the concept of homologous rays in holographic interferometry of diffusely reflecting surfaces," Opt. Act. 17, 899-913 (1970).
[CrossRef]

1969

C. Froehly, J. Monneret, J. Pasteur, and J. Ch. Viénot, "Étude des faibles déplacements d'objets opaques et de la distorsion optique dans les lasers à solide par interférométrie holographique," Opt. Act. 16, 343-362 (1969).
[CrossRef]

Andersson, A.

Bischoff, M.

H. Helmers, M. Bischoff, and L. Ehlkes, "ESPI-system with active in-line digital phase phase stabilization," in Fringe '01, Fourth International Workshop on Automatic Processing of Fringe Patterns, W.Jüptner and W.Osten, eds. (Elsevier, 2001), pp. 673-679.

Briers, J. D.

J. D. Briers, "Holographic, speckle, and Moiré techniques in optical metrology," Prog. Quantum Electron. 17, 167-233 (1993).
[CrossRef]

Ch. Viénot, J.

C. Froehly, J. Monneret, J. Pasteur, and J. Ch. Viénot, "Étude des faibles déplacements d'objets opaques et de la distorsion optique dans les lasers à solide par interférométrie holographique," Opt. Act. 16, 343-362 (1969).
[CrossRef]

Chiang, F. P.

Cuche, D.

W. Schumann, Z. P. Zuercher, and D. Cuche, Holography and Deformation Analysis, Vol. 46 of Springer Series in Optical Sciences (Springer, 1985).

Dändliker, R.

R. Dändliker and P. Jacquot, "Holographic interferometry and speckle methods," in Optical Sensors, E.Wagner, R.Dändliker, and K.Spenner, eds. (VCH Verlagsgesellschaft, 1992), Chap. 23, pp. 589-628.

R. Dändliker, "The story of speckles in interferometry," in Interferometry in Speckle Light, Theory and Applications, P.Jacquot and J.M.Fournier, eds. (Springer Verlag, 2000), pp. 3-10.

Dubas, M.

M. Dubas and W. Schumann, "Sur la détermination holographique de l'état de déformation à la surface d'un corps non-transparent," Opt. Act. 21, 547-562 (1974).
[CrossRef]

M. Dubas, "Sur l'Analyse Expérimentale de l'État de Déformation à la Surface d'un Corps Opaque par Interférométrie Holographique en particulier à l'Aide de la Localisation des Franges," PhD thesis ETHZ No. 5673 (Swiss Federal Institute of Technology Zurich, 1976).

W. Schumann and M. Dubas, Holographic Interferometry From the Scope of Deformation Analysis of Opaque Bodies, Vol. 16 of Springer Series in Optical Sciences (Springer, 1979).

Ehlkes, L.

H. Helmers, M. Bischoff, and L. Ehlkes, "ESPI-system with active in-line digital phase phase stabilization," in Fringe '01, Fourth International Workshop on Automatic Processing of Fringe Patterns, W.Jüptner and W.Osten, eds. (Elsevier, 2001), pp. 673-679.

Ennos, A. E.

A. E. Ennos, "Speckle interferometry," in Laser Speckle and Related Phenomena, J.C.Dainty, ed. (Springer Verlag, 1975), pp. 203-253.
[CrossRef]

Forno, C.

C. Forno, "Deformation measurement using high-resolution Moiré photography," Opt. Laser Eng. 8, 189-212 (1988).
[CrossRef]

Froehly, C.

C. Froehly, J. Monneret, J. Pasteur, and J. Ch. Viénot, "Étude des faibles déplacements d'objets opaques et de la distorsion optique dans les lasers à solide par interférométrie holographique," Opt. Act. 16, 343-362 (1969).
[CrossRef]

Goodman, J. W.

J. W. Goodman, "Statistical properties of laser speckle patterns," in Laser Speckle and Related Phenomena, J.C.Dainty, ed. (Springer Verlag, 1975), pp. 9-75.
[CrossRef]

Goudemand, N.

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]

Helmers, H.

H. Helmers, M. Bischoff, and L. Ehlkes, "ESPI-system with active in-line digital phase phase stabilization," in Fringe '01, Fourth International Workshop on Automatic Processing of Fringe Patterns, W.Jüptner and W.Osten, eds. (Elsevier, 2001), pp. 673-679.

Jacquot, P.

P. K. Rastogi and P. Jacquot, "Speckle metrology techniques: a parametric examination of the observed fringes," Opt. Eng. 21, 411-426 (1982).

P. Jacquot and P. K. Rastogi, "Speckle motions induced by rigid-body movements in free-space geometry: an explicit investigation and extension to new cases," Appl. Opt. 18, 2022-2032 (1979).
[CrossRef] [PubMed]

R. Dändliker and P. Jacquot, "Holographic interferometry and speckle methods," in Optical Sensors, E.Wagner, R.Dändliker, and K.Spenner, eds. (VCH Verlagsgesellschaft, 1992), Chap. 23, pp. 589-628.

P. Jacquot, "Speckle motions in three-dimensional image fields," presented at the Topical Meeting on Hologram Interferometry and Speckle Metrology, Cape Cod, Massachusetts, 2-4 June 1980.

P. Jacquot, "Photographie de speckles: exemples tirés de l'analyse de déformation de corps solides," in Lasers et Applications Industrielles, C.Bonjour and M.Matthey, eds. (Presses Polytechniques Romandes, 1982), pp. 149-204.

Kowarschik, R.

Kreis, T. M.

T. M. Kreis, "Frequency analysis of digital holography," Opt. Eng. 41, 771-778 (2002).
[CrossRef]

Lee, H. C.

H. C. Lee, "Review of image-blur models in a photographic system using the principles of optics," Opt. Eng. 29, 405-421 (1990).
[CrossRef]

Lehmann, M.

M. Lehmann, "Decorrelation-induced phase errors in phase-shifting speckle interferometry," Appl. Opt. 36, 3657-3667 (1997).
[CrossRef] [PubMed]

M. Lehmann, "Statistical theory of two-wave speckle interferometry and its application to the optimization of deformation measurements," PhD thesis EPFL No. 1797 (Swiss Federal Institute of Technology Lausanne, 1998).

Li, Q. B.

Lutz, B.

B. Lutz and W. Schumann, "Approach to extend the domain of visibility of recovered modified fringes when holographic interferometry is applied to large deformations," Opt. Eng. 34, 1879-1886 (1995).
[CrossRef]

W. Schumann and B. Lutz, "Holographic interferometry applied to large deformations, the role of the second derivative of the optical path difference in real-time technique," in Interferometry `94: Photomechanics, R. J. Pryputniewicz and J. Stupnicki, eds., Proc. SPIE 2342, 219-231 (1994).
[CrossRef]

Maack, T.

Molin, N. E.

N. E. Molin and K. A. Stetson, "Fringe localization in hologram interferometry of mutually independent and dependent rotations around orthogonal, nonintersecting axes," Optik 33, 399-422 (1971).

Monneret, J.

C. Froehly, J. Monneret, J. Pasteur, and J. Ch. Viénot, "Étude des faibles déplacements d'objets opaques et de la distorsion optique dans les lasers à solide par interférométrie holographique," Opt. Act. 16, 343-362 (1969).
[CrossRef]

Notni, G.

Pasteur, J.

C. Froehly, J. Monneret, J. Pasteur, and J. Ch. Viénot, "Étude des faibles déplacements d'objets opaques et de la distorsion optique dans les lasers à solide par interférométrie holographique," Opt. Act. 16, 343-362 (1969).
[CrossRef]

Rastogi, P. K.

P. K. Rastogi and P. Jacquot, "Speckle metrology techniques: a parametric examination of the observed fringes," Opt. Eng. 21, 411-426 (1982).

P. Jacquot and P. K. Rastogi, "Speckle motions induced by rigid-body movements in free-space geometry: an explicit investigation and extension to new cases," Appl. Opt. 18, 2022-2032 (1979).
[CrossRef] [PubMed]

Runnemalm, A.

Schumann, W.

B. Lutz and W. Schumann, "Approach to extend the domain of visibility of recovered modified fringes when holographic interferometry is applied to large deformations," Opt. Eng. 34, 1879-1886 (1995).
[CrossRef]

W. Schumann and B. Lutz, "Holographic interferometry applied to large deformations, the role of the second derivative of the optical path difference in real-time technique," in Interferometry `94: Photomechanics, R. J. Pryputniewicz and J. Stupnicki, eds., Proc. SPIE 2342, 219-231 (1994).
[CrossRef]

M. Dubas and W. Schumann, "Sur la détermination holographique de l'état de déformation à la surface d'un corps non-transparent," Opt. Act. 21, 547-562 (1974).
[CrossRef]

W. Schumann, Z. P. Zuercher, and D. Cuche, Holography and Deformation Analysis, Vol. 46 of Springer Series in Optical Sciences (Springer, 1985).

W. Schumann and M. Dubas, Holographic Interferometry From the Scope of Deformation Analysis of Opaque Bodies, Vol. 16 of Springer Series in Optical Sciences (Springer, 1979).

Sjödahl, M.

Stetson, K. A.

K. A. Stetson, "Problem of defocusing in speckle photography, its connection to hologram interferometry, and its solutions," J. Opt. Soc. Am. A 66, 1267-1271 (1976).
[CrossRef]

K. A. Stetson, "Fringe interpretation for hologram interferometry of rigid-body motions and homogeneous deformations," J. Opt. Soc. Am. A 64, 1-10 (1974).
[CrossRef]

N. E. Molin and K. A. Stetson, "Fringe localization in hologram interferometry of mutually independent and dependent rotations around orthogonal, nonintersecting axes," Optik 33, 399-422 (1971).

Svetlík, J.

J. Svetlík, "Speckle displacement: two related approaches," J. Mod. Opt. 39, 149-157 (1992).
[CrossRef]

Thalmann, R.

R. Thalmann, "Electronic fringe interpolation in holographic interferometry, applied to deformation measurement of solid objects," PhD thesis (Institute of Microtechnology, Neuchatel, Switzerland, 1986).

Walles, S.

S. Walles, "On the concept of homologous rays in holographic interferometry of diffusely reflecting surfaces," Opt. Act. 17, 899-913 (1970).
[CrossRef]

Yamaguchi, I.

I. Yamaguchi, "Speckle displacement and decorrelation in the diffraction and image fields for small object deformation," Opt. Act. 28, 1359-1376 (1981).
[CrossRef]

Zuercher, Z. P.

W. Schumann, Z. P. Zuercher, and D. Cuche, Holography and Deformation Analysis, Vol. 46 of Springer Series in Optical Sciences (Springer, 1985).

Appl. Opt.

J. Mod. Opt.

J. Svetlík, "Speckle displacement: two related approaches," J. Mod. Opt. 39, 149-157 (1992).
[CrossRef]

J. Opt. Soc. Am. A

K. A. Stetson, "Problem of defocusing in speckle photography, its connection to hologram interferometry, and its solutions," J. Opt. Soc. Am. A 66, 1267-1271 (1976).
[CrossRef]

K. A. Stetson, "Fringe interpretation for hologram interferometry of rigid-body motions and homogeneous deformations," J. Opt. Soc. Am. A 64, 1-10 (1974).
[CrossRef]

Opt. Act.

I. Yamaguchi, "Speckle displacement and decorrelation in the diffraction and image fields for small object deformation," Opt. Act. 28, 1359-1376 (1981).
[CrossRef]

C. Froehly, J. Monneret, J. Pasteur, and J. Ch. Viénot, "Étude des faibles déplacements d'objets opaques et de la distorsion optique dans les lasers à solide par interférométrie holographique," Opt. Act. 16, 343-362 (1969).
[CrossRef]

S. Walles, "On the concept of homologous rays in holographic interferometry of diffusely reflecting surfaces," Opt. Act. 17, 899-913 (1970).
[CrossRef]

M. Dubas and W. Schumann, "Sur la détermination holographique de l'état de déformation à la surface d'un corps non-transparent," Opt. Act. 21, 547-562 (1974).
[CrossRef]

Opt. Eng.

P. K. Rastogi and P. Jacquot, "Speckle metrology techniques: a parametric examination of the observed fringes," Opt. Eng. 21, 411-426 (1982).

H. C. Lee, "Review of image-blur models in a photographic system using the principles of optics," Opt. Eng. 29, 405-421 (1990).
[CrossRef]

B. Lutz and W. Schumann, "Approach to extend the domain of visibility of recovered modified fringes when holographic interferometry is applied to large deformations," Opt. Eng. 34, 1879-1886 (1995).
[CrossRef]

T. M. Kreis, "Frequency analysis of digital holography," Opt. Eng. 41, 771-778 (2002).
[CrossRef]

Opt. Laser Eng.

C. Forno, "Deformation measurement using high-resolution Moiré photography," Opt. Laser Eng. 8, 189-212 (1988).
[CrossRef]

Opt. Laser Technol.

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

Optik

N. E. Molin and K. A. Stetson, "Fringe localization in hologram interferometry of mutually independent and dependent rotations around orthogonal, nonintersecting axes," Optik 33, 399-422 (1971).

Proc. SPIE

W. Schumann and B. Lutz, "Holographic interferometry applied to large deformations, the role of the second derivative of the optical path difference in real-time technique," in Interferometry `94: Photomechanics, R. J. Pryputniewicz and J. Stupnicki, eds., Proc. SPIE 2342, 219-231 (1994).
[CrossRef]

Prog. Quantum Electron.

J. D. Briers, "Holographic, speckle, and Moiré techniques in optical metrology," Prog. Quantum Electron. 17, 167-233 (1993).
[CrossRef]

Other

M. Lehmann, "Statistical theory of two-wave speckle interferometry and its application to the optimization of deformation measurements," PhD thesis EPFL No. 1797 (Swiss Federal Institute of Technology Lausanne, 1998).

H. Helmers, M. Bischoff, and L. Ehlkes, "ESPI-system with active in-line digital phase phase stabilization," in Fringe '01, Fourth International Workshop on Automatic Processing of Fringe Patterns, W.Jüptner and W.Osten, eds. (Elsevier, 2001), pp. 673-679.

R. Dändliker, "The story of speckles in interferometry," in Interferometry in Speckle Light, Theory and Applications, P.Jacquot and J.M.Fournier, eds. (Springer Verlag, 2000), pp. 3-10.

A. E. Ennos, "Speckle interferometry," in Laser Speckle and Related Phenomena, J.C.Dainty, ed. (Springer Verlag, 1975), pp. 203-253.
[CrossRef]

P. Jacquot, "Photographie de speckles: exemples tirés de l'analyse de déformation de corps solides," in Lasers et Applications Industrielles, C.Bonjour and M.Matthey, eds. (Presses Polytechniques Romandes, 1982), pp. 149-204.

W. Schumann and M. Dubas, Holographic Interferometry From the Scope of Deformation Analysis of Opaque Bodies, Vol. 16 of Springer Series in Optical Sciences (Springer, 1979).

W. Schumann, Z. P. Zuercher, and D. Cuche, Holography and Deformation Analysis, Vol. 46 of Springer Series in Optical Sciences (Springer, 1985).

D.W.Robinson and G.T.Reid, eds., Interferogram Analysis, Digital Fringe Pattern Measurement Techniques (Institute of Physics Publishing, 1993).

J. W. Goodman, "Statistical properties of laser speckle patterns," in Laser Speckle and Related Phenomena, J.C.Dainty, ed. (Springer Verlag, 1975), pp. 9-75.
[CrossRef]

M. Dubas, "Sur l'Analyse Expérimentale de l'État de Déformation à la Surface d'un Corps Opaque par Interférométrie Holographique en particulier à l'Aide de la Localisation des Franges," PhD thesis ETHZ No. 5673 (Swiss Federal Institute of Technology Zurich, 1976).

P. Jacquot, "Speckle motions in three-dimensional image fields," presented at the Topical Meeting on Hologram Interferometry and Speckle Metrology, Cape Cod, Massachusetts, 2-4 June 1980.

R. Dändliker and P. Jacquot, "Holographic interferometry and speckle methods," in Optical Sensors, E.Wagner, R.Dändliker, and K.Spenner, eds. (VCH Verlagsgesellschaft, 1992), Chap. 23, pp. 589-628.

R. Thalmann, "Electronic fringe interpolation in holographic interferometry, applied to deformation measurement of solid objects," PhD thesis (Institute of Microtechnology, Neuchatel, Switzerland, 1986).

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Figures (5)

Fig. 1
Fig. 1

Formation of a fringe at K. A n and A n ′ are, respectively, the part of the undeformed surface and the part of the deformed surface, which correspond to the light rays entering the aperture A n of the imaging system. A n * is the part of the deformed surface, which corresponds to the part A n of the undeformed surface.

Fig. 2
Fig. 2

Model of CCD array [after Kreis (Ref. 37)]: the pixel pitch is Δx in the x direction and Δy in the y direction. The fill factors are, respectively, α and β. The pixel numbers are, respectively, N and M. The periodic positioning of pixels can be expressed by a convolution with comb functions, so that the CCD array as a whole can be characterized by [rect(x∕αΔx)rect(y∕βΔy)* comb(x∕Δx)comb(y∕Δy)] × rect(xNΔx)rect(yMΔy).

Fig. 3
Fig. 3

Image plane decorrelation [after Lehmann (Ref. 10)]: different regions of the same speckle field contribute to the pixel intensity before (A and B) and after (A and B′) deformation.

Fig. 4
Fig. 4

Application of the permutation tensor E onto an interior vector v (Ref. 24).

Fig. 5
Fig. 5

Curvatures of an arbitrary curve r(s) on a surface. c S , circle of curvature of r(s) with center C S and radius ρ S = 1∕|d2 r∕ds 2|. c, circle of normal curvature with center C and radius ρ = 1∕B. In the tangent plane, we have the center C g of the geodesic curvature of r(s), with radius ρ g = 1∕|b|.

Equations (311)

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100   μm
1   mm
2 π
s W
s L
s W = λ d a ,
s L = 8 λ d 2 a 2 ,
p ( I ) = 4 ( I / I av ) exp ( 2 I / I av ) .
N b
1 / N b
N b
N b
p I ( I ) = [ N b I ] N b I N b 1 Γ ( N b ) exp ( N b I I ) ,
N b
I
Γ ( N b )
N b
2 N b
S 1
S 2
K *
K *
P ,
P K * P K * = ( P K P K )
G ( r )
E [ G ( r ) ] = 0 , E [ G ( r ) G * ( r ¯ ) ] = C δ ( r r ¯ ) ,
S 1
S 2
U S 1 = C 1 n h 1 λ p 1 exp ( 2 π i λ p 1 ) ,
U S 2 = C 2 n h 2 λ p 2 exp ( 2 π i λ p 2 ) exp ( Δ φ 12 ) ,
h i ( i = 1 , 2 )
S i P , n
P , L S i = p i
S i P
Δ φ 12
Δ φ 12
Δ φ 12 = Δ φ 12 , 0 + Δ φ stepping ,
Δ φ 12 , 0
Δ φ stepping
A n A
A n ˙
U ( ρ ) = A n G ( r ) n k λ q exp ( 2 π i λ q ) [ U S 1 + U S 2 ] d A n = U 1 + U 2 ,
q = L = k ( r ρ )
U ( ρ ) = A n G ( r ) n k λ q [ C 1 n h 1 λ p 1 × exp [ 2 π i λ ( p 1 q ) ] + C 2 n h 2 λ p 2 × exp [ 2 π i λ ( p 2 q ) ] exp ( Δ φ 12 ) ] d A n = U 1 + U 2 .
G ( r ) = G ( r )
γ ˜
p 1 = p 1 - u , p 2 = p 2 - u ,
q = q - u , n = n + w ,
u p i , w 1
tr ( γ ˜ ) 1
n h i λ p i n h i λ p i , d A n [ 1 + tr ( γ ˜ ) ] d A n d A n .
u q
n k λ q n k λ q .
u q
d A n / q
A n
A n
U 1 ( ρ ) = A n C 1 G ( r ) n k λ q n h 1 λ p 1 × exp [ 2 π i λ ( p 1 q ) ] d A n ,
U 2 ( ρ ) = A n C 2 G ( r ) n k λ q n h 2 λ p 2 × exp [ 2 π i λ ( p 2 q ) ] × [ exp ( Δ φ 12 ) ] d A n .
S i
U i ( ρ ) = exp [ 2 π i λ ( u h i + D t i ) ] × U i ( ρ u L M w i d t i ) ,
w i = ( u ) g 1 L S i H i u ,
M = I n k n k .
D t i
d t i
K
ρ + u + L M w + d t ,
P K
J ( ρ )
J ( ρ )
J ( ρ ) = 1 2 ( E [ U 1 U 1 * ] + E [ U 2 U 2 * ] + E [ U 1 * U 2 ] + E [ U 1 U 2 * ] )
= I 1 + I 2 + Γ 12 + Γ 12 * ,
J ( ρ ) = 1 2 ( E [ U 1 U 1 * ] + E [ U 2 U 2 * ] + E [ U 1 * U 2 ] + E [ U 1 U 2 * ] )
= I 1 + I 2 + Γ 12 + Γ 12 * ,
I 1 = 1 2 E [ U 1 U 1 * ]
S 1
E [ U i * U j ] = E [ U i * ( ρ ) U j ( ρ ¯ ) ] = E [ A n A n C i * G * ( r ) n k λ q n h i λ p i × exp [ 2 π i λ ( p i q ) ] C j G ( r ¯ ) n ¯ k ¯ λ q ¯ n ¯ h ¯ j λ p ¯ j × exp [ 2 π i λ ( p ¯ j q ¯ ) ] exp ( Δ φ i j ) d A n d A ¯ n ] ,
Δ φ 21 = Δ φ 12
Δ φ 22 = Δ φ 11 = 0
E [ U i * U j ] = exp ( Δ φ i j ) A n C C i * C j | n ¯ k ¯ | 2 λ 4 q ¯  2 n ¯ h ¯ i p ¯  i n ¯ h ¯ j p ¯  j × exp [ 2 π i λ ( p ¯  j - p ¯  i ) ] d A n .
½ 1 2 E [ U 1 * U 2 ] = Γ 12 = | Γ 12 | exp ( Δ φ 12 + δ 12 ) ,
δ 12 = arg { A n C C 1 * C 2 | n ¯ k ¯ | 2 λ 4 q ¯  2 n ¯ h ¯ 1 p ¯  1 n ¯ h ¯ 2 p ¯  2 × exp [ 2 π i λ ( p ¯  2 p ¯  1 ) ] d A n }
( 2 π / λ ) ( p ¯ 2 - p ¯ 1 )
J ( ρ ) = I 1 + I 2 + 2 | Γ 12 | cos ( Δ φ 12 + δ 12 )
= I av [ 1 + m cos ( φ ) ] ,
I av = I 1 + I 2
m = 2 | Γ 12 | / I av
E [ U i * U j ]
E [ U i * U j ] = E [ U i * ( ρ ) U j ( ρ ¯ ) ] E ( exp { 2 π i λ [ ( u ¯ h ¯ j D ¯ t j ) ( u h i D t i ) ] } × U i * ( ρ  −  u L M w i d t i ) × U j ( ρ ¯ u ¯ L ¯ M ¯ w ¯ i d ¯ t i ) ) ,
G ( r ) = G ( r )
I i ( ρ ) I i ( ρ  −  u  −  L M w i d t i ) ,
E [ U 1 * U 2 ] exp { Δ φ 12 + 2 π i λ [ u ( h 2 h 1 ) + ( D t 2 D t 1 ) ] } × A n C C 1 * C 2 | n ¯ k ¯ | 2 λ 4 q ¯  2 n ¯ h ¯ 1 p ¯  1 n ¯ h ¯ 2 p ¯  2 × exp ( 2 π i λ { ( p ¯  2 - p ¯  1 ) + k ¯ [ L M ( w 2 w 1 ) + ( d t 2 d t 1 ) ] } ) d A n .
½ 1 2 E [ U 1 * U 2 ] = Γ 12 = | Γ 12 | exp [ Δ φ 12 + 2 π i λ [ u ( h 2 h 1 ) + ( D t 2 D t 1 ) ] + δ 12 ] ,
δ 12 = arg [ A n C C 1 * C 2 | n ¯ k ¯ | 2 λ 4 q ¯  2 n ¯ h ¯ 1 p ¯  1 n ¯ h ¯ 2 p ¯  2 × exp ( 2 π i λ { ( p ¯  2 p ¯  1 ) + k ¯ [ L M ( w 2 w 1 ) + ( d t 2 d t 1 ) ] } ) d A n ]
{ ( p ¯  2 p ¯  1 ) + k ¯ [ L M ( w 2 w 1 ) + ( d t 2 d t 1 ) ] }
J ( ρ ) = I 1 + I 2 + 2 | Γ 12 | × cos { Δ φ 12 + 2 π λ [ u ( h 2 h 1 ) + ( D t 2 D t 1 ) ] + δ 12 } = I av [ 1 + m cos ( φ ) ] ,
I av = I 1 + I 2
m = 2 | Γ 12 | / I av
D t 2 D t 1
d t 2 d t 1
D ¯ S 2 D ¯ S 1
I
φ
φ
2 π
2 π
J ( ρ )
J ( ρ )
A P
I ( K ) = A P J ( ρ ) d A P ,
I ( K ) = A P J ( ρ ) d A P .
A n
J ( ρ )
J ( ρ )
N b
I i
I ( K ) = 1 N b p = 1 N b I p ( K ) = 1 N b p = 1 N b [ I av p ( 1 + m p cos φ p ) ] ,
I ( K ) = 1 N b q = 1 N b I q ( K ) = 1 N b q = 1 N b [ I av q ( 1 + m q cos φ q ) ] ,
I i
N b N b
1 / N b
48   dB
100 120   dB
N b
I ( K )
I ( K )
I = I av [ 1 + m cos φ ] ,
I av = p = 1 N b I av p ,
m cos φ = p = 1 N b m p cos φ p ,
( L = 0 )
L S i
δ 12 δ 12 ,
I i ( ρ ) I i ( ρ u ) ,
Γ 12 | Γ 12 | exp { Δ φ 12 + δ 12 + 2 π i λ × [ u ( h 2 h 1 ) ] } ,
I av p ( ρ ) I av p ( ρ u ) ,
φ p φ p + 2 π λ [ u ( h 2 h 1 ) ] .
A P
I av I av ,
φ φ + 2 π λ [ u ( h 2 h 1 ) ] ;
[ π , π ]
A P
A n ˙
A n
A n
N b
( 1 δ )
u z
u z
s L
u z
s L
d z = u z / s L
s L
s W
G ( r ) G ( r )
L 0
u L
D u ( k h )
L L
D ¯ 1 D ¯ 2 = ( p 2 p 2 ) ( L L ) [ ( p 1 p 1 ) ( L L ) ] = D ¯ S 1 D ¯ S 2
( L 0 )
U i ( ρ )
U i ( ρ u )
L M w i
u + L M w < u
δ 12
A n
A n
A n ˙
A n
A P
A n
A P
L near
L far
L near L L far
L far L near
L near = r A P ( L ˙ + L ) r ˙ + r A P ,
L far = r A P ( L ˙ + L ) r ˙ r A P ,
r ˙
r A P
A P
( L ˙ + L )
R ˙
| L near |
| L far |
r A * P
A P *
( r A * P
r A P = r A * P ( L ˙ + L f ) / f
L ˙ + L = [ 1 + ( r ˙ / r A * P ) ] f
r A P = r ˙
L far =
L near = ( L ˙ + L ) / 2
L ˙ + L = [ 1 + ( r ˙ / r A * P ) ] f
d 2 D K = 0
( 2 π / λ ) u ( h 2 h 1 )
L S i
d 2 D K
u K
N N
K N
T = p q , T ik = p i q k .
s = T r = ( p q ) r = p ( q r ) , s i = p i q k r k ,
q r
u N
x N
u N = N N u = ( I n n ) u = u n ( n u ) ,
( u N ) i = ( N N ) i k u k = ( δ i k n i n k ) u k ,
n n = 1
w N
x N
w N = K N u = ( I k n k n ) u = u n u k n k ,
( w N ) i = ( K N ) i k u k = ( δ i k k i n k k j n j ) u k .
K N T
K N T = ( I k n k n ) T = ( I n k k n ) = N K .
( ϕ ) i = ϕ x i ,
( u ) i k = ( u ) i k = x i u k = u k x i .
x N
N N
          N ϕ = N N ( ϕ ) = ( N N ) ϕ ,
N u = N u = N N ( u ) = ( N N ) u .
π / 2
B = n n .
p q r
( p q r ) T = r q p ,
p q r ) T = p r q .
S 1
U 1 ( ρ ) = A n C 1 G ( r ) n k λ q n h 1 λ p 1 ×   exp [ 2 π i λ ( p 1 q ) ] d A n ,
U 1 ( ρ ) = A n C 1 G ( r ) n k λ q n h 1 λ p 1 ×   exp [ 2 π i λ ( p 1 q ) ] d A n ,
= A n C 1 G ( r ) n k λ q n h 1 λ p 1 ×   exp [ 2 π i λ ( p 1 q D ) ] d A n ,
D = ( q q ) ( p 1 p 1 )
A n ˙
A n
L + L ˙
K R ˙
R ˙
D ¯
P ¯
D ¯ D + d D K + 1 2 d 2 D K ,
d D K = d r n D K = d k k D K ,
= d k ( L M w + u ) ,
d 2 D K = d r ( n n D K ) d r + d s 2 b n D K ,
= d k ( k k D K ) d k + d ϕ 2 b k k D K ,
b k
d r
d k
d r = e d s
d k = m d ϕ
L R
k = L M n
k D K = L M w + K u ,
k k D K = k L M w + L ( k M ) N w + L [ k ( N w ) ] M T + ( k K ) u + ( k u ) K .
k L = L Mk ,
k K = K k K k ) T ,
k M = 1 n k [ M n ) T L M B M T ] ,
B = n n
( k k D K ) K = L 2 M [ n ( N w ) + 1 n k B ( k N w ) ] K ( k u ) + L ( M k M w + M w M k ) + L M ( n u ) K M T .
n ( N w )
n N = B n + B n ) T ,
n ( 1 L S H ) = 1 L S 2 N H ^ ,
H ^ = h H + H h + H h ) T
N u
N u = γ ˜ + Ω E + ω n ,
γ ˜
Ω E
( ω = n n )
[ κ = ( n ω ) N ]
n γ ˜ = n ( N γ ˜ N ) = n γ ˜ ) T N ] T N + B γ ˜ n + B γ ˜ n ) T ,
n ( Ω E ) = n Ω E Ω B E n + Ω B E n ) T ,
n ( ω n ) = κ n + B ω n n B ω ) T ,
( k k D K ) K = M { L 2 ( n g ) κ c + L 2 2 [ [ ( n γ ˜ ) Ng ] N + N [ ( n γ ˜ ) N g ] T + n Ω E g + Eg n Ω B g ω ω Bg ] + L [ 2 γ ˜ γ ˜ h k k γ ˜ h Ω ( E h k + k E h ) ( n h ) ( ω k + k ω ) ] - L 2 L S [ γ ˜ H + H γ ˜ + Ω ( EH - HE ) + ω Hn + Hn ω ] + L 2 L S 2 H ^ u L L S ( H u k + k H u ) ( k u ) K + L 2 n k [ k ( γ ˜ + Ω E + ω n ) g 1 L S k H u ] B } M T ,
κ c = 1 2 [ ( κ + κ T ) + Ω ( B E E B ) ( B γ ˜ + γ ˜ B ) ]
K d k = d k
d 2 D K
d k = k ¯ k
k M w = 0
d D K
d 2 D K
k ¯
d D K = k ¯ ( L M w + u ) k u ,
d 2 D K = d k ( k k D K ) d k + d ϕ 2 b k k D K = k ¯ ( k k D K ) K d k k ( k k D K ) K d k + d ϕ 2 b k ( L M w + K u ) = k ¯ ( 2 d t ) + 2 D t .
D u ( k h ) , D ¯
D ¯ D + d D K + 1 2 d 2 D K u h + D t + k ¯ ( L M w + u + d t ) .
q ¯ = k ¯ ( r ¯ ρ )
U 1 ( ρ )
U 1 ( ρ )
U 1 ( ρ ) = A n C 1 G ( r ¯ ) n ¯ k ¯ λ q ¯ n ¯ h ¯ 1 λ p ¯  1 exp [ 2 π i λ ( p ¯  1 q ¯ ) ] d A n = A n C 1 G ( r ¯ ) n ¯ k ¯ λ q ¯ n h ¯ 1 λ p ¯  1 exp [ 2 π i λ ( p ¯  1 k ¯ r ¯ ) ] × exp [ 2 π i λ ( k ¯ ρ ) ] d A n ,
U 1 ( ρ ) = A n C 1 G ( r ¯ ) n ¯ k ¯ λ q ¯ n ¯ h ¯ 1 λ p ¯  1 exp [ 2 π i λ ( p ¯ 1 q ¯ ) ] d A n = A n C 1 G ( r ¯ ) n ¯ k ¯ λ q ¯ n ¯ h ¯ 1 λ p ¯  1 × exp [ 2 π i λ ( p ¯  1 q ¯ D ¯ ) ] d A n
= exp [ 2 π i λ ( u h 1 + D t 1 ) ] × A n C 1 G ( r ¯ ) n ¯ k ¯ λ q ¯ n ¯ h ¯ 1 λ p ¯ 1 exp [ 2 π i λ ( p ¯ 1 k ¯ r ¯ ) ] × exp { 2 π i λ [ k ¯ ( ρ u L M w 1 d t 1 ) ] } d A n .
U 1 ( ρ ) = exp [ 2 π i λ ( u h 1 + D t 1 ) ] × U 1 ( ρ u L M w 1 d t 1 ) .
K
ρ + u + L M w + d t
P K
D S = L S L S
D ¯ S
D ¯
P
P ¯
P ¯
D ¯ S D S + d D S + 1 2 d 2 D S ,
D S u h ,
d D S = d k k D S = d k ( L M n D S )
= d k ( L M N w s ) = d k ( L Mw S )
= k ¯ ( L M w S ) ,
w S = ( u ) h 1 L S H u ,
    d 2 D S = d k ( k k D S ) K d k + d ϕ 2 b k k D S ,
( k k D S ) K
( k k D S ) K = L 2 M [ n ( N w S ) + 1 n k B ( k N w S ) ] + L ( M k M w S + M w S M k )
= M { L 2 ( n h ) κ c L 2 2 [ [ ( n γ ˜ ) Nh ] N + N [ ( n γ ˜ ) N h ] T + n Ω E h + E h n Ω B h ω ω B h ] L [ γ ˜ h k + k γ ˜ h + Ω ( E h k + k E h ) + ( n h ) ( ω k + k ω ) ] L 2 L S [ γ ˜ H + H γ ˜ + Ω ( E H H E ) + ω H n + H n ω ] + L 2 L S       2 H ^ u L L S ( H u k + k H u ) L 2 n k [ k ( γ ˜ + Ω E + ω n ) h + 1 L S k H u ] B } M T .
D ¯ S 1 D ¯ S 2

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