Abstract

An electronic speckle interferometer, arranged for out-of-plane sensitivity and with an off-axis reference beam to produce spatial phase bias, is used for three-dimensional deformation field measurements. The complex amplitude of the object wave is calculated by application of a Fourier-transform method to a single interferogram. The change in phase after object deformation yields the out-of-plane component of the displacement field. The two in-plane components are obtained by cross correlation of subimages of the reconstructed object wave’s intensity, a method that is also referred to as digital speckle photography. The Fourier-transform algorithm is extended and modified, leading to random measurement errors that are below widely accepted theoretical limits and also to an extended measuring range. These properties and the mutually combined information improve the accuracy of both methods compared with their usual single implementation. The performance is evaluated in experiments with pure out-of-plane, pure in-plane, and combined deformations and compared with theoretical values. An example of a practical application is given.

© 2001 Optical Society of America

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References

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  1. P. Meinlschmidt, K. D. Hinsch, R. S. Sirohi, eds., Electronic Speckle Pattern Interferometry, Vol. MS 132 of SPIE Milestone Series (SPIE Optical Engineering Press, Bellingham, Wash., 1996).
  2. H. Bruning, D. Herriott, J. Gallagher, D. Rosenfeld, A. White, D. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. 13, 2693–2703 (1974).
    [CrossRef] [PubMed]
  3. D. Robinson, D. Williams, “Digital phase stepping interferometry,” Opt. Commun. 57, 26–30 (1986).
    [CrossRef]
  4. T. Bothe, J. Burke, H. Helmers, “Spatial phase shifting in electronic speckle pattern interferometry: minimization of phase reconstruction errors,” Appl. Opt. 36, 5310–5316 (1997).
    [CrossRef] [PubMed]
  5. J. Burke, “Application and optimisation of the spatial phase shifting technique in digital speckle interferometry,” Ph.D. dissertation (Carl von Ossietzky Universität Oldenburg, Oldenburg, Germany).
  6. M. Takeda, H. Ina, S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72, 156–160 (1982).
    [CrossRef]
  7. H. Saldner, N. Molin, K. Stetson, “Fourier-transform evaluation of phase data in spatially phase-biased TV holograms,” Appl. Opt. 35, 332–336 (1996).
    [CrossRef] [PubMed]
  8. R. Sirohi, J. Burke, H. Helmers, K. Hinsch, “Spatial phase shifting for pure in-plane displacement and displacement-derivative measurements in ESPI,” Appl. Opt. 36, 5787–5791 (1997).
    [CrossRef] [PubMed]
  9. M. Sjödahl, L. R. Benckert, “Electronic speckle photography: analysis of an algorithm giving the displacement with subpixel accuracy,” Appl. Opt. 32, 2278–2284 (1993).
    [CrossRef] [PubMed]
  10. M. Sjödahl, “Some recent advances in electronic speckle photography,” Opt. Lasers Eng. 29, 125–144 (1998).
    [CrossRef]
  11. P. Synnergren, M. Sjödahl, “A stereoscopic digital speckle photography system for 3-D displacement field measurements,” Opt. Lasers Eng. 31, 425–443 (1999).
    [CrossRef]
  12. F. D. Adams, G. E. Maddux, “Synthesis of holography and speckle photography to measure 3-D displacements,” Appl. Opt. 13, 219 (1974).
    [CrossRef] [PubMed]
  13. M. Sjödahl, H. O. Saldner, “Three-dimensional deformation field measurements with simultaneous TV holography and electronic speckle photography,” Appl. Opt. 36, 3645–3648 (1997).
    [CrossRef] [PubMed]
  14. A. Andersson, A. Runnemalm, M. Sjödahl, “Digital speckle-pattern interferometry: fringe retrieval for large in-plane deformation with digital speckle photography,” Appl. Opt. 38, 5408–5412 (1999).
    [CrossRef]
  15. M. Sjödahl, “Accuracy in electronic speckle photography,” Appl. Opt. 36, 2875–2885 (1997).
    [CrossRef] [PubMed]
  16. J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1975), pp. 9–75.
  17. J. Burke, H. Helmers, “Spatial vs. temporal phase shifting in electronic speckle-pattern interferometry: noise comparison in phase maps,” Appl. Opt. 39, 4598–4606 (2000).
    [CrossRef]
  18. J. Schwider, O. Falkenstörfer, H. Schreiber, A. Zöllner, N. Streibl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32, 1883–1885 (1993).
    [CrossRef]
  19. K. Freischlad, C. Koliopoulos, “Fourier description of digital phase-measuring interferometry,” J. Opt. Soc. Am. A 7, 542–551 (1990).
    [CrossRef]
  20. S. Donati, G. Martini, “Speckle-pattern intensity and phase: second-order conditional statistics,” J. Opt. Soc. Am. 69, 1690–1694 (1979).
    [CrossRef]
  21. K. Creath, “Phase-shifting interferometry,” in Speckle, H. H. Arsenault, ed., Proc. SPIE556, 337–346 (1985).
  22. J. M. Huntley, “Random phase measurement errors in digital speckle interferometry,” Opt. Lasers Eng. 26, 131–150 (1997).
    [CrossRef]
  23. M. Sjödahl, “Electronic speckle photography: increased accuracy by nonintegral pixel shifting,” Appl. Opt. 33, 6667–6673 (1994).
    [CrossRef] [PubMed]
  24. R. Feiel, P. Wilksch, “High-resolution laser speckle correlation for displacement and strain measurement,” Appl. Opt. 39, 54–60 (2000).
    [CrossRef]

2000 (2)

1999 (2)

P. Synnergren, M. Sjödahl, “A stereoscopic digital speckle photography system for 3-D displacement field measurements,” Opt. Lasers Eng. 31, 425–443 (1999).
[CrossRef]

A. Andersson, A. Runnemalm, M. Sjödahl, “Digital speckle-pattern interferometry: fringe retrieval for large in-plane deformation with digital speckle photography,” Appl. Opt. 38, 5408–5412 (1999).
[CrossRef]

1998 (1)

M. Sjödahl, “Some recent advances in electronic speckle photography,” Opt. Lasers Eng. 29, 125–144 (1998).
[CrossRef]

1997 (5)

1996 (1)

1994 (1)

1993 (2)

M. Sjödahl, L. R. Benckert, “Electronic speckle photography: analysis of an algorithm giving the displacement with subpixel accuracy,” Appl. Opt. 32, 2278–2284 (1993).
[CrossRef] [PubMed]

J. Schwider, O. Falkenstörfer, H. Schreiber, A. Zöllner, N. Streibl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32, 1883–1885 (1993).
[CrossRef]

1990 (1)

1986 (1)

D. Robinson, D. Williams, “Digital phase stepping interferometry,” Opt. Commun. 57, 26–30 (1986).
[CrossRef]

1982 (1)

1979 (1)

1974 (2)

Adams, F. D.

Andersson, A.

Benckert, L. R.

Bothe, T.

Brangaccio, D.

Bruning, H.

Burke, J.

Creath, K.

K. Creath, “Phase-shifting interferometry,” in Speckle, H. H. Arsenault, ed., Proc. SPIE556, 337–346 (1985).

Donati, S.

Falkenstörfer, O.

J. Schwider, O. Falkenstörfer, H. Schreiber, A. Zöllner, N. Streibl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32, 1883–1885 (1993).
[CrossRef]

Feiel, R.

Freischlad, K.

Gallagher, J.

Goodman, J. W.

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1975), pp. 9–75.

Helmers, H.

Herriott, D.

Hinsch, K.

Huntley, J. M.

J. M. Huntley, “Random phase measurement errors in digital speckle interferometry,” Opt. Lasers Eng. 26, 131–150 (1997).
[CrossRef]

Ina, H.

Kobayashi, S.

Koliopoulos, C.

Maddux, G. E.

Martini, G.

Molin, N.

Robinson, D.

D. Robinson, D. Williams, “Digital phase stepping interferometry,” Opt. Commun. 57, 26–30 (1986).
[CrossRef]

Rosenfeld, D.

Runnemalm, A.

Saldner, H.

Saldner, H. O.

Schreiber, H.

J. Schwider, O. Falkenstörfer, H. Schreiber, A. Zöllner, N. Streibl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32, 1883–1885 (1993).
[CrossRef]

Schwider, J.

J. Schwider, O. Falkenstörfer, H. Schreiber, A. Zöllner, N. Streibl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32, 1883–1885 (1993).
[CrossRef]

Sirohi, R.

Sjödahl, M.

Stetson, K.

Streibl, N.

J. Schwider, O. Falkenstörfer, H. Schreiber, A. Zöllner, N. Streibl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32, 1883–1885 (1993).
[CrossRef]

Synnergren, P.

P. Synnergren, M. Sjödahl, “A stereoscopic digital speckle photography system for 3-D displacement field measurements,” Opt. Lasers Eng. 31, 425–443 (1999).
[CrossRef]

Takeda, M.

White, A.

Wilksch, P.

Williams, D.

D. Robinson, D. Williams, “Digital phase stepping interferometry,” Opt. Commun. 57, 26–30 (1986).
[CrossRef]

Zöllner, A.

J. Schwider, O. Falkenstörfer, H. Schreiber, A. Zöllner, N. Streibl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32, 1883–1885 (1993).
[CrossRef]

Appl. Opt. (12)

H. Bruning, D. Herriott, J. Gallagher, D. Rosenfeld, A. White, D. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. 13, 2693–2703 (1974).
[CrossRef] [PubMed]

T. Bothe, J. Burke, H. Helmers, “Spatial phase shifting in electronic speckle pattern interferometry: minimization of phase reconstruction errors,” Appl. Opt. 36, 5310–5316 (1997).
[CrossRef] [PubMed]

H. Saldner, N. Molin, K. Stetson, “Fourier-transform evaluation of phase data in spatially phase-biased TV holograms,” Appl. Opt. 35, 332–336 (1996).
[CrossRef] [PubMed]

R. Sirohi, J. Burke, H. Helmers, K. Hinsch, “Spatial phase shifting for pure in-plane displacement and displacement-derivative measurements in ESPI,” Appl. Opt. 36, 5787–5791 (1997).
[CrossRef] [PubMed]

M. Sjödahl, L. R. Benckert, “Electronic speckle photography: analysis of an algorithm giving the displacement with subpixel accuracy,” Appl. Opt. 32, 2278–2284 (1993).
[CrossRef] [PubMed]

F. D. Adams, G. E. Maddux, “Synthesis of holography and speckle photography to measure 3-D displacements,” Appl. Opt. 13, 219 (1974).
[CrossRef] [PubMed]

M. Sjödahl, H. O. Saldner, “Three-dimensional deformation field measurements with simultaneous TV holography and electronic speckle photography,” Appl. Opt. 36, 3645–3648 (1997).
[CrossRef] [PubMed]

A. Andersson, A. Runnemalm, M. Sjödahl, “Digital speckle-pattern interferometry: fringe retrieval for large in-plane deformation with digital speckle photography,” Appl. Opt. 38, 5408–5412 (1999).
[CrossRef]

M. Sjödahl, “Accuracy in electronic speckle photography,” Appl. Opt. 36, 2875–2885 (1997).
[CrossRef] [PubMed]

J. Burke, H. Helmers, “Spatial vs. temporal phase shifting in electronic speckle-pattern interferometry: noise comparison in phase maps,” Appl. Opt. 39, 4598–4606 (2000).
[CrossRef]

M. Sjödahl, “Electronic speckle photography: increased accuracy by nonintegral pixel shifting,” Appl. Opt. 33, 6667–6673 (1994).
[CrossRef] [PubMed]

R. Feiel, P. Wilksch, “High-resolution laser speckle correlation for displacement and strain measurement,” Appl. Opt. 39, 54–60 (2000).
[CrossRef]

J. Opt. Soc. Am. (2)

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

Opt. Commun. (1)

D. Robinson, D. Williams, “Digital phase stepping interferometry,” Opt. Commun. 57, 26–30 (1986).
[CrossRef]

Opt. Eng. (1)

J. Schwider, O. Falkenstörfer, H. Schreiber, A. Zöllner, N. Streibl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32, 1883–1885 (1993).
[CrossRef]

Opt. Lasers Eng. (3)

M. Sjödahl, “Some recent advances in electronic speckle photography,” Opt. Lasers Eng. 29, 125–144 (1998).
[CrossRef]

P. Synnergren, M. Sjödahl, “A stereoscopic digital speckle photography system for 3-D displacement field measurements,” Opt. Lasers Eng. 31, 425–443 (1999).
[CrossRef]

J. M. Huntley, “Random phase measurement errors in digital speckle interferometry,” Opt. Lasers Eng. 26, 131–150 (1997).
[CrossRef]

Other (4)

K. Creath, “Phase-shifting interferometry,” in Speckle, H. H. Arsenault, ed., Proc. SPIE556, 337–346 (1985).

J. Burke, “Application and optimisation of the spatial phase shifting technique in digital speckle interferometry,” Ph.D. dissertation (Carl von Ossietzky Universität Oldenburg, Oldenburg, Germany).

P. Meinlschmidt, K. D. Hinsch, R. S. Sirohi, eds., Electronic Speckle Pattern Interferometry, Vol. MS 132 of SPIE Milestone Series (SPIE Optical Engineering Press, Bellingham, Wash., 1996).

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1975), pp. 9–75.

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Figures (16)

Fig. 1
Fig. 1

Optical configuration of the speckle interferometer.

Fig. 2
Fig. 2

Power spectrum of a spatially phase-biased interferogram.

Fig. 3
Fig. 3

Schematic profile along the diagonal of the power spectrum in Fig. 2.

Fig. 4
Fig. 4

Transmission profile of filter function F BW compared with the envelope of the sideband and with the filter characteristics of a local phase-reconstruction algorithm explained in Section 4.

Fig. 5
Fig. 5

Out-of-plane displacement error versus fringe density for various phase-reconstruction algorithms and comparison with the expected theoretical limit.

Fig. 6
Fig. 6

Out-of-plane displacement error versus fringe density determined from the same set of interferograms as in Fig. 5 for more variations of the FTM. Some curves are repeated from Fig. 5 for comparison.

Fig. 7
Fig. 7

Shift of the filter function in frequency space: sideband spectrum left, before deformation and right, after deformation. Dashed circles represent the cutoff frequency of F BW.

Fig. 8
Fig. 8

Mod 2π sawtooth fringes obtained from various phase-reconstruction algorithms as indicated. Displacement error σΔz is given in units of 10-3 λ. The fringe density is N x = 20.

Fig. 9
Fig. 9

Same as Fig. 8, except that the sine and the cosine of the phase difference have been low-pass filtered with a 3 × 3 unit convolution kernel.

Fig. 10
Fig. 10

In-plane displacement error versus displacement for various intensity reconstruction algorithms and comparison with ordinary DSP and the theoretically expected error.

Fig. 11
Fig. 11

Horizontal in-plane displacement error versus fringe density for various intensity reconstruction algorithms and comparison with the standard DSP method.

Fig. 12
Fig. 12

Vertical in-plane displacement error versus fringe density calculated from the same images and for the same algorithms as in Fig. 11.

Fig. 13
Fig. 13

Out-of-plane displacement error versus in-plane translation for various phase-reconstruction algorithms. The fringe density is N x ≈ 10.

Fig. 14
Fig. 14

Mod 2π phase maps obtained for in-plane rotation and out-of-plane tilt. Left, local four-step algorithm, σΔz = 0.100 λ; right, extended FTM with amplitude backshift, σΔz = 0.035 λ.

Fig. 15
Fig. 15

Demonstration object to induce a 3-D deformation field. The observed region is indicated by the dashed rectangle.

Fig. 16
Fig. 16

3-D deformation measurement of the object in Fig. 15. The phase map of the out-of-plane component is superimposed with arrows that indicate the in-plane displacement.

Equations (16)

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ix, y=|ax, y+rx, yexp-2πiν0xx+ν0yy|2,
Iνx, νy=|a|2+|r0|2δνx, νy+r0A*-νx-ν0x, -νy-ν0y+r0*Aνx-ν0x, νy-ν0y,
-1r0*Aνx, νy=r0*ax, y=r0*|ax, y|expiφx, y,
ãx, yax, y=ãx, yax, yexpiΔφx, y
Δzx, y=λ2π1+cos Θ Δφx, y.
iSx, y=|ax, y|2.
Δx=dpM Δx,
ix, y=|a|2+|r|2+γa*r exp-2πiν0xx+ν0yy+ar* exp2πiν0xx+ν0yy,
Iνx, νy=|a|2+|r|2+γA*-νx, -νy*Rνx+ν0x, νy+ν0y+γAνx, νy*R*-νx+ν0x, -νy+ν0yHνx, νy,
ISBνx, νy=|a|2νx+ν0x, νy+ν0y+γAνx, νy*R*-νx, -νyHνx+ν0x, νy+ν0yFνx, νy.
aγr0=-1ISB=|a|2*f+γar**f,
|a0|2=|aγr0|2γ2|r|2.
φx2=arctan2ix3-ix2ix1-ix2-ix3+ix4.
σΔφ2=π23-π arcsin cI+arcsin2cI-0.5 n=0cInn2,
cI=2πarccosβ/βmax-β/βmax1-β/βmax21/2,
σΔx=k dsp2N1-cIcI1/2,

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