Abstract

We present an all-numeric multiplexing/demultiplexing technique for digitally recorded holograms that allow the simultaneous determination of the in-plane and the out-of-plane components of the displacement vector of an object submitted to some loading. The twin-sensitivity measurement is obtained from two different illumination directions that give two sensitivity vectors. The spatial multiplexing is achieved by an incoherent mixing of two duplets of coherent waves that produce holograms carried with orthogonal polarized reference waves. The spatial demultiplexing uses the autocorrelation function of the multiplexed holograms as a position estimator. The estimator then enables the determination of the pixel-to-pixel correspondence between the holograms for the double component determination. Results of the experiment are presented.

© 2003 Optical Society of America

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References

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  1. P. K. Rastogi, Holographic Interferometry (Springer series in Optical Sciences Vol. 68, Springer-Verlag, Berlin, 1994).
    [CrossRef]
  2. Th. Kreis, Holographic Interferometry—Principles and Methods (Akademie Verlag series in Optical Metrology Vol. 1, Akademie, Berlin, 1996).
  3. P. Smigielski, Holographie Industrielle (Edition Teknéa, Toulouse, 1994).
  4. M. A. Kronrod, N. S. Merzlyakov, L. P. Yaroslavskii, “Reconstruction of a hologram with a computer,” Sov. Phys. Tech. Phys. 17, 333–334 (1972).
  5. L. P. Yaroslavskii, N. S. Merzlyakov, Methods of Digital Holography (Consultant Bureau, New York, 1980), Chap. 1.
  6. U. Schnars, W. Jüptner, “Direct recording of holograms by a CCD target and numerical reconstruction,” Appl. Opt. 33, 179–181 (1994).
    [CrossRef] [PubMed]
  7. U. Schnars, “Direct phase determination in hologram interferometry with use of digitally recorded holograms,” J. Opt. Soc. Am. A 11, 2011–2015 (1994).
    [CrossRef]
  8. U. Schnars, Th. Kreis, W. Jüptner, “Digital recording and numerical reconstruction of holograms: reduction of the spatial frequency spectrum,” Opt. Eng. 35, 977–982 (1996).
    [CrossRef]
  9. F. Le Clerc, L. Collot, M. Gross, “Numerical heterodyne holography with two-dimensional photodetector arrays,” Opt. Lett. 25, 716–718 (2000).
    [CrossRef]
  10. M. Jacquot, P. Sandoz, G. Tribillon, “High resolution digital holography,” Opt. Commun. 190, 87–94 (2001).
    [CrossRef]
  11. Th. Kreis, M. Adams, W. Jüptner, “Methods of digital holography: a comparison,” in Optical Inspection and Micromeasurements II, C. Gorecki, ed., Proc. SPIE3098, 224–233 (1997).
    [CrossRef]
  12. I. Yamaguchi, T. Zhang, “Phase shifting digital holography,” Opt. Lett. 22, 1268–1270 (1997).
    [CrossRef] [PubMed]
  13. G. Pedrini, H. J. Tiziani, “Digital double pulse holographic interferometry using Fresnel and image plane holograms,” Measurement 18, 251–260 (1995).
    [CrossRef]
  14. G. Pedrini, Y. L. Zou, H. J. Tiziani, “Digital double pulse holographic interferometry for vibration analysis,” J. Mod. Opt. 42, 367–374 (1995).
    [CrossRef]
  15. C. Wagner, S. Seebacher, W. Osten, W. Jüptner, “Digital recording and numerical reconstruction of lensless Fourier holograms in optical metrology,” Appl. Opt. 38, 4812–4820 (1999).
    [CrossRef]
  16. E. Cuche, F. Bevilacqua, C. Depeursinge, “Digital holography for quantitative phase contrast imaging,” Opt. Lett. 24, 291–293 (1999).
    [CrossRef]
  17. I. Yamaguchi, J. Kato, S. Ohta, “Surface shape measurement by phase shifting digital holography,” Opt. Rev. 8, 85–89 (2001).
    [CrossRef]
  18. V. Linet, X. Bohineust, F. Dupuy, “Three dimensional dynamic analysis of parts of automobile body by holographic interferometry,” in Proceedings of 3rd French-German Congress on Applications of Holography, P. Smigielski, ed., 213–224 (1991).
  19. M. Takeda, H. Ina, S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. A 72, 156–160 (1982).
    [CrossRef]
  20. H. A. Aebischer, S. Waldner, “Strain distributions made visible with image-shearing speckle pattern interferometry,” Opt. Lasers Eng. 26, 407–420 (1997).
    [CrossRef]
  21. G. Pedrini, H. J. Tiziani, “Quantitative Evaluation of Two-Dimensional Dynamic Deformation Using Digital Holography,” Opt. Laser Technol. 29, 249–256 (1997).
    [CrossRef]
  22. G. Pedrini, Y. L. Zou, H. J. Tiziani, “Simultaneous quantitative evaluation of in-plane and out-of-plane deformations by use of a multidirectional spatial carrier,” Appl. Opt. 36, 786–792 (1997).
    [CrossRef] [PubMed]
  23. S. Schedin, G. Pedrini, H. J. Tiziani, F. M. Santoyo, “Simultaneous three-dimensional dynamic deformation measurements with pulsed digital holography,” Appl. Opt. 38, 7056–7062 (1999).
    [CrossRef]
  24. R. Krupka, T. Walz, A. Ettemeyer, “New Techniques and Application for 3D Brake Analysis,” Dr. Ettemeyer Application Report No. 04-00 (2000), http://www.ettemeyer.de .
  25. E. Cuche, P. Marquet, C. Depeursinge, “Spatial filtering for zero-order and twin-image elimination on digital off axis holography,” Appl. Opt. 39, 4070–4075 (2000).
    [CrossRef]
  26. J. Max, J. L. Lacoume, Méthodes et Techniques de Traitement du Signal et Applications aux Mesures Physiques, 1. Principes Généraux et Méthodes Classiques (Editions Masson, Paris, 1996).
  27. P. Picart, “Error analysis for a Mach-Zehnder-type speckle interferometer,” Opt. Lasers Eng. 35, 335–353 (2001).
    [CrossRef]
  28. S. de Nicola, P. Ferraro, A. Finizio, G. Pierattini, “Wave front reconstruction of Fresnel off-axis holograms with compensation of aberrations by means of phase shifting digital holography,” Opt. Lasers Eng. 37, 331–340 (2002).
    [CrossRef]

2002 (1)

S. de Nicola, P. Ferraro, A. Finizio, G. Pierattini, “Wave front reconstruction of Fresnel off-axis holograms with compensation of aberrations by means of phase shifting digital holography,” Opt. Lasers Eng. 37, 331–340 (2002).
[CrossRef]

2001 (3)

P. Picart, “Error analysis for a Mach-Zehnder-type speckle interferometer,” Opt. Lasers Eng. 35, 335–353 (2001).
[CrossRef]

M. Jacquot, P. Sandoz, G. Tribillon, “High resolution digital holography,” Opt. Commun. 190, 87–94 (2001).
[CrossRef]

I. Yamaguchi, J. Kato, S. Ohta, “Surface shape measurement by phase shifting digital holography,” Opt. Rev. 8, 85–89 (2001).
[CrossRef]

2000 (2)

1999 (3)

1997 (4)

H. A. Aebischer, S. Waldner, “Strain distributions made visible with image-shearing speckle pattern interferometry,” Opt. Lasers Eng. 26, 407–420 (1997).
[CrossRef]

G. Pedrini, H. J. Tiziani, “Quantitative Evaluation of Two-Dimensional Dynamic Deformation Using Digital Holography,” Opt. Laser Technol. 29, 249–256 (1997).
[CrossRef]

G. Pedrini, Y. L. Zou, H. J. Tiziani, “Simultaneous quantitative evaluation of in-plane and out-of-plane deformations by use of a multidirectional spatial carrier,” Appl. Opt. 36, 786–792 (1997).
[CrossRef] [PubMed]

I. Yamaguchi, T. Zhang, “Phase shifting digital holography,” Opt. Lett. 22, 1268–1270 (1997).
[CrossRef] [PubMed]

1996 (1)

U. Schnars, Th. Kreis, W. Jüptner, “Digital recording and numerical reconstruction of holograms: reduction of the spatial frequency spectrum,” Opt. Eng. 35, 977–982 (1996).
[CrossRef]

1995 (2)

G. Pedrini, H. J. Tiziani, “Digital double pulse holographic interferometry using Fresnel and image plane holograms,” Measurement 18, 251–260 (1995).
[CrossRef]

G. Pedrini, Y. L. Zou, H. J. Tiziani, “Digital double pulse holographic interferometry for vibration analysis,” J. Mod. Opt. 42, 367–374 (1995).
[CrossRef]

1994 (2)

1982 (1)

M. Takeda, H. Ina, S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. A 72, 156–160 (1982).
[CrossRef]

1972 (1)

M. A. Kronrod, N. S. Merzlyakov, L. P. Yaroslavskii, “Reconstruction of a hologram with a computer,” Sov. Phys. Tech. Phys. 17, 333–334 (1972).

Adams, M.

Th. Kreis, M. Adams, W. Jüptner, “Methods of digital holography: a comparison,” in Optical Inspection and Micromeasurements II, C. Gorecki, ed., Proc. SPIE3098, 224–233 (1997).
[CrossRef]

Aebischer, H. A.

H. A. Aebischer, S. Waldner, “Strain distributions made visible with image-shearing speckle pattern interferometry,” Opt. Lasers Eng. 26, 407–420 (1997).
[CrossRef]

Bevilacqua, F.

Bohineust, X.

V. Linet, X. Bohineust, F. Dupuy, “Three dimensional dynamic analysis of parts of automobile body by holographic interferometry,” in Proceedings of 3rd French-German Congress on Applications of Holography, P. Smigielski, ed., 213–224 (1991).

Collot, L.

Cuche, E.

de Nicola, S.

S. de Nicola, P. Ferraro, A. Finizio, G. Pierattini, “Wave front reconstruction of Fresnel off-axis holograms with compensation of aberrations by means of phase shifting digital holography,” Opt. Lasers Eng. 37, 331–340 (2002).
[CrossRef]

Depeursinge, C.

Dupuy, F.

V. Linet, X. Bohineust, F. Dupuy, “Three dimensional dynamic analysis of parts of automobile body by holographic interferometry,” in Proceedings of 3rd French-German Congress on Applications of Holography, P. Smigielski, ed., 213–224 (1991).

Ferraro, P.

S. de Nicola, P. Ferraro, A. Finizio, G. Pierattini, “Wave front reconstruction of Fresnel off-axis holograms with compensation of aberrations by means of phase shifting digital holography,” Opt. Lasers Eng. 37, 331–340 (2002).
[CrossRef]

Finizio, A.

S. de Nicola, P. Ferraro, A. Finizio, G. Pierattini, “Wave front reconstruction of Fresnel off-axis holograms with compensation of aberrations by means of phase shifting digital holography,” Opt. Lasers Eng. 37, 331–340 (2002).
[CrossRef]

Gross, M.

Ina, H.

M. Takeda, H. Ina, S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. A 72, 156–160 (1982).
[CrossRef]

Jacquot, M.

M. Jacquot, P. Sandoz, G. Tribillon, “High resolution digital holography,” Opt. Commun. 190, 87–94 (2001).
[CrossRef]

Jüptner, W.

C. Wagner, S. Seebacher, W. Osten, W. Jüptner, “Digital recording and numerical reconstruction of lensless Fourier holograms in optical metrology,” Appl. Opt. 38, 4812–4820 (1999).
[CrossRef]

U. Schnars, Th. Kreis, W. Jüptner, “Digital recording and numerical reconstruction of holograms: reduction of the spatial frequency spectrum,” Opt. Eng. 35, 977–982 (1996).
[CrossRef]

U. Schnars, W. Jüptner, “Direct recording of holograms by a CCD target and numerical reconstruction,” Appl. Opt. 33, 179–181 (1994).
[CrossRef] [PubMed]

Th. Kreis, M. Adams, W. Jüptner, “Methods of digital holography: a comparison,” in Optical Inspection and Micromeasurements II, C. Gorecki, ed., Proc. SPIE3098, 224–233 (1997).
[CrossRef]

Kato, J.

I. Yamaguchi, J. Kato, S. Ohta, “Surface shape measurement by phase shifting digital holography,” Opt. Rev. 8, 85–89 (2001).
[CrossRef]

Kobayashi, S.

M. Takeda, H. Ina, S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. A 72, 156–160 (1982).
[CrossRef]

Kreis, Th.

U. Schnars, Th. Kreis, W. Jüptner, “Digital recording and numerical reconstruction of holograms: reduction of the spatial frequency spectrum,” Opt. Eng. 35, 977–982 (1996).
[CrossRef]

Th. Kreis, M. Adams, W. Jüptner, “Methods of digital holography: a comparison,” in Optical Inspection and Micromeasurements II, C. Gorecki, ed., Proc. SPIE3098, 224–233 (1997).
[CrossRef]

Th. Kreis, Holographic Interferometry—Principles and Methods (Akademie Verlag series in Optical Metrology Vol. 1, Akademie, Berlin, 1996).

Kronrod, M. A.

M. A. Kronrod, N. S. Merzlyakov, L. P. Yaroslavskii, “Reconstruction of a hologram with a computer,” Sov. Phys. Tech. Phys. 17, 333–334 (1972).

Lacoume, J. L.

J. Max, J. L. Lacoume, Méthodes et Techniques de Traitement du Signal et Applications aux Mesures Physiques, 1. Principes Généraux et Méthodes Classiques (Editions Masson, Paris, 1996).

Le Clerc, F.

Linet, V.

V. Linet, X. Bohineust, F. Dupuy, “Three dimensional dynamic analysis of parts of automobile body by holographic interferometry,” in Proceedings of 3rd French-German Congress on Applications of Holography, P. Smigielski, ed., 213–224 (1991).

Marquet, P.

Max, J.

J. Max, J. L. Lacoume, Méthodes et Techniques de Traitement du Signal et Applications aux Mesures Physiques, 1. Principes Généraux et Méthodes Classiques (Editions Masson, Paris, 1996).

Merzlyakov, N. S.

M. A. Kronrod, N. S. Merzlyakov, L. P. Yaroslavskii, “Reconstruction of a hologram with a computer,” Sov. Phys. Tech. Phys. 17, 333–334 (1972).

L. P. Yaroslavskii, N. S. Merzlyakov, Methods of Digital Holography (Consultant Bureau, New York, 1980), Chap. 1.

Ohta, S.

I. Yamaguchi, J. Kato, S. Ohta, “Surface shape measurement by phase shifting digital holography,” Opt. Rev. 8, 85–89 (2001).
[CrossRef]

Osten, W.

Pedrini, G.

S. Schedin, G. Pedrini, H. J. Tiziani, F. M. Santoyo, “Simultaneous three-dimensional dynamic deformation measurements with pulsed digital holography,” Appl. Opt. 38, 7056–7062 (1999).
[CrossRef]

G. Pedrini, H. J. Tiziani, “Quantitative Evaluation of Two-Dimensional Dynamic Deformation Using Digital Holography,” Opt. Laser Technol. 29, 249–256 (1997).
[CrossRef]

G. Pedrini, Y. L. Zou, H. J. Tiziani, “Simultaneous quantitative evaluation of in-plane and out-of-plane deformations by use of a multidirectional spatial carrier,” Appl. Opt. 36, 786–792 (1997).
[CrossRef] [PubMed]

G. Pedrini, H. J. Tiziani, “Digital double pulse holographic interferometry using Fresnel and image plane holograms,” Measurement 18, 251–260 (1995).
[CrossRef]

G. Pedrini, Y. L. Zou, H. J. Tiziani, “Digital double pulse holographic interferometry for vibration analysis,” J. Mod. Opt. 42, 367–374 (1995).
[CrossRef]

Picart, P.

P. Picart, “Error analysis for a Mach-Zehnder-type speckle interferometer,” Opt. Lasers Eng. 35, 335–353 (2001).
[CrossRef]

Pierattini, G.

S. de Nicola, P. Ferraro, A. Finizio, G. Pierattini, “Wave front reconstruction of Fresnel off-axis holograms with compensation of aberrations by means of phase shifting digital holography,” Opt. Lasers Eng. 37, 331–340 (2002).
[CrossRef]

Rastogi, P. K.

P. K. Rastogi, Holographic Interferometry (Springer series in Optical Sciences Vol. 68, Springer-Verlag, Berlin, 1994).
[CrossRef]

Sandoz, P.

M. Jacquot, P. Sandoz, G. Tribillon, “High resolution digital holography,” Opt. Commun. 190, 87–94 (2001).
[CrossRef]

Santoyo, F. M.

Schedin, S.

Schnars, U.

Seebacher, S.

Smigielski, P.

P. Smigielski, Holographie Industrielle (Edition Teknéa, Toulouse, 1994).

Takeda, M.

M. Takeda, H. Ina, S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. A 72, 156–160 (1982).
[CrossRef]

Tiziani, H. J.

S. Schedin, G. Pedrini, H. J. Tiziani, F. M. Santoyo, “Simultaneous three-dimensional dynamic deformation measurements with pulsed digital holography,” Appl. Opt. 38, 7056–7062 (1999).
[CrossRef]

G. Pedrini, Y. L. Zou, H. J. Tiziani, “Simultaneous quantitative evaluation of in-plane and out-of-plane deformations by use of a multidirectional spatial carrier,” Appl. Opt. 36, 786–792 (1997).
[CrossRef] [PubMed]

G. Pedrini, H. J. Tiziani, “Quantitative Evaluation of Two-Dimensional Dynamic Deformation Using Digital Holography,” Opt. Laser Technol. 29, 249–256 (1997).
[CrossRef]

G. Pedrini, Y. L. Zou, H. J. Tiziani, “Digital double pulse holographic interferometry for vibration analysis,” J. Mod. Opt. 42, 367–374 (1995).
[CrossRef]

G. Pedrini, H. J. Tiziani, “Digital double pulse holographic interferometry using Fresnel and image plane holograms,” Measurement 18, 251–260 (1995).
[CrossRef]

Tribillon, G.

M. Jacquot, P. Sandoz, G. Tribillon, “High resolution digital holography,” Opt. Commun. 190, 87–94 (2001).
[CrossRef]

Wagner, C.

Waldner, S.

H. A. Aebischer, S. Waldner, “Strain distributions made visible with image-shearing speckle pattern interferometry,” Opt. Lasers Eng. 26, 407–420 (1997).
[CrossRef]

Yamaguchi, I.

I. Yamaguchi, J. Kato, S. Ohta, “Surface shape measurement by phase shifting digital holography,” Opt. Rev. 8, 85–89 (2001).
[CrossRef]

I. Yamaguchi, T. Zhang, “Phase shifting digital holography,” Opt. Lett. 22, 1268–1270 (1997).
[CrossRef] [PubMed]

Yaroslavskii, L. P.

M. A. Kronrod, N. S. Merzlyakov, L. P. Yaroslavskii, “Reconstruction of a hologram with a computer,” Sov. Phys. Tech. Phys. 17, 333–334 (1972).

L. P. Yaroslavskii, N. S. Merzlyakov, Methods of Digital Holography (Consultant Bureau, New York, 1980), Chap. 1.

Zhang, T.

Zou, Y. L.

Appl. Opt. (5)

J. Mod. Opt. (1)

G. Pedrini, Y. L. Zou, H. J. Tiziani, “Digital double pulse holographic interferometry for vibration analysis,” J. Mod. Opt. 42, 367–374 (1995).
[CrossRef]

J. Opt. Soc. Am. A (2)

M. Takeda, H. Ina, S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. A 72, 156–160 (1982).
[CrossRef]

U. Schnars, “Direct phase determination in hologram interferometry with use of digitally recorded holograms,” J. Opt. Soc. Am. A 11, 2011–2015 (1994).
[CrossRef]

Measurement (1)

G. Pedrini, H. J. Tiziani, “Digital double pulse holographic interferometry using Fresnel and image plane holograms,” Measurement 18, 251–260 (1995).
[CrossRef]

Opt. Commun. (1)

M. Jacquot, P. Sandoz, G. Tribillon, “High resolution digital holography,” Opt. Commun. 190, 87–94 (2001).
[CrossRef]

Opt. Eng. (1)

U. Schnars, Th. Kreis, W. Jüptner, “Digital recording and numerical reconstruction of holograms: reduction of the spatial frequency spectrum,” Opt. Eng. 35, 977–982 (1996).
[CrossRef]

Opt. Laser Technol. (1)

G. Pedrini, H. J. Tiziani, “Quantitative Evaluation of Two-Dimensional Dynamic Deformation Using Digital Holography,” Opt. Laser Technol. 29, 249–256 (1997).
[CrossRef]

Opt. Lasers Eng. (3)

H. A. Aebischer, S. Waldner, “Strain distributions made visible with image-shearing speckle pattern interferometry,” Opt. Lasers Eng. 26, 407–420 (1997).
[CrossRef]

P. Picart, “Error analysis for a Mach-Zehnder-type speckle interferometer,” Opt. Lasers Eng. 35, 335–353 (2001).
[CrossRef]

S. de Nicola, P. Ferraro, A. Finizio, G. Pierattini, “Wave front reconstruction of Fresnel off-axis holograms with compensation of aberrations by means of phase shifting digital holography,” Opt. Lasers Eng. 37, 331–340 (2002).
[CrossRef]

Opt. Lett. (3)

Opt. Rev. (1)

I. Yamaguchi, J. Kato, S. Ohta, “Surface shape measurement by phase shifting digital holography,” Opt. Rev. 8, 85–89 (2001).
[CrossRef]

Sov. Phys. Tech. Phys. (1)

M. A. Kronrod, N. S. Merzlyakov, L. P. Yaroslavskii, “Reconstruction of a hologram with a computer,” Sov. Phys. Tech. Phys. 17, 333–334 (1972).

Other (8)

L. P. Yaroslavskii, N. S. Merzlyakov, Methods of Digital Holography (Consultant Bureau, New York, 1980), Chap. 1.

P. K. Rastogi, Holographic Interferometry (Springer series in Optical Sciences Vol. 68, Springer-Verlag, Berlin, 1994).
[CrossRef]

Th. Kreis, Holographic Interferometry—Principles and Methods (Akademie Verlag series in Optical Metrology Vol. 1, Akademie, Berlin, 1996).

P. Smigielski, Holographie Industrielle (Edition Teknéa, Toulouse, 1994).

V. Linet, X. Bohineust, F. Dupuy, “Three dimensional dynamic analysis of parts of automobile body by holographic interferometry,” in Proceedings of 3rd French-German Congress on Applications of Holography, P. Smigielski, ed., 213–224 (1991).

Th. Kreis, M. Adams, W. Jüptner, “Methods of digital holography: a comparison,” in Optical Inspection and Micromeasurements II, C. Gorecki, ed., Proc. SPIE3098, 224–233 (1997).
[CrossRef]

J. Max, J. L. Lacoume, Méthodes et Techniques de Traitement du Signal et Applications aux Mesures Physiques, 1. Principes Généraux et Méthodes Classiques (Editions Masson, Paris, 1996).

R. Krupka, T. Walz, A. Ettemeyer, “New Techniques and Application for 3D Brake Analysis,” Dr. Ettemeyer Application Report No. 04-00 (2000), http://www.ettemeyer.de .

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Figures (11)

Fig. 1
Fig. 1

Geometry of the illuminating beams.

Fig. 2
Fig. 2

Experimental setup (HWP: half-wave plate, PBS: polarizing beam splitter, POL: linear polarizer).

Fig. 3
Fig. 3

Image amplitude of the spatially multiplexed digital holograms.

Fig. 4
Fig. 4

Twin spatially multiplexed modulo 2π phase maps.

Fig. 5
Fig. 5

Spatial autocorrelation function of the twin holograms.

Fig. 6
Fig. 6

Out-of-plane (top) and in-plane (bottom) displacement of the surface of the washer after z tilting.

Fig. 7
Fig. 7

Out-of-plane unwrapped displacement field.

Fig. 8
Fig. 8

Out-of-plane (top) and in-plane (bottom) displacement of the surface of the washer after the in-plane rotation.

Fig. 9
Fig. 9

In-plane unwrapped displacement field.

Fig. 10
Fig. 10

Out-of-plane (top) and in-plane (bottom) displacement of the surface of the washer after crushing along the x direction.

Fig. 11
Fig. 11

Out-of-plane (left-hand side) (a) and in-plane (right-hand side) (b) unwrapped displacement field of the surface of the washer after the crushing test.

Equations (18)

Equations on this page are rendered with MathJax. Learn more.

H=O0+R1*O1+R2*O2+R1O1*+R2O2*,
Ax, y, d=iE0λdexp- 2iπdλexp- iπλdx2+y2×-+-+ Hx, yexp-iπλdx2+y2exp2iπλdxx+yydxdy,
An, m, d=k=0k=N-1l=0l=M-1 Hk, lexp-iπλdk2Δx2+l2Δy2exp2iπknN+lmM,
ψn, m, d=arctanImAn, m, dReAn, m, d.
Δφn, m, d=ψan, m, d-ψbn, m, d.
An, m, d=IFFTO0 exp-iπλdk2Δx2+l2Δy2×n, m+IFFTR1*O1+R2*O2×exp-iπλdk2Δx2+l2Δy2n, m+IFFTR1O1*+R2O2*×exp-iπλdk2Δx2+l2Δy2n, m.
ArRn, m, d=IFFTOrk, lexp-iπλdk2Δx2+l2Δy2n, m*IFFT×Rr*k, ln, m,
-+-+ ar exp2iπurx+vryexp2iπλdxx+yydxdy=arδxλd-ur, yλd-vr,
ArRn, m, d=arArn, m*δn-NurΔx, m-MvrΔy=arArn-NurΔx, m-MvrΔy,
In, m, d|a1A1n-n1, m-m1|2+|a2A2n-n2, m-n2|2,
In, m, d=α1I1n-n1, m-m1+α2I2×n-n2, m-m2,
RIIp, q, d=EIn+p, m+q, dI*n, m, d,
RIIp, q, d=α12RI1I1p, q, d+α22RI2I2p, q, d+α1α2RI1I2p-Δn, q-Δm, d+α1α2RI2I1p+Δn, q+Δm, d,
A1n+Δn, m+Δm=IFFTFFTA1n, mk, l×exp2iπkΔn+lΔm×n, m.
A2n-Δn, m-Δm=IFFTFFTA2n, mk, l×exp-2iπkΔn+lΔm×n, m.
Δφs=Δφ2+Δφ1=2πλL2k0-k1-k2=4πλ1+cos θdz,
Δφd=Δφ2-Δφ1=2πλLk1-k2=4πλsin θdx,
u1, v1=δz/f1λ1+δz2/f12, 0,u2, v2=0, δy/f3λ1+δy2/f32,

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