Abstract

Recently, a new method to measure object shape and deformation with temporal evolution of speckles in speckle interferometry was reported. In this method, certain parameters, sensitive to shape or deformation are changed continuously, and the fluctuations in the irradiance of each speckle is recorded. The information over the whole object deformation is retrieved by Fourier-transformation techniques. We present a detailed theory and analyze the influence of decorrelation due to longitudinal and lateral size of the speckles. It is also shown that the method can be used to measure small deformations (less than 5 µm) with higher resolution. Further, the nonlinearity of the camera is shown to enhance the sensitivity.

© 1999 Optical Society of America

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References

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  1. K. Erf, Speckle Metrology (Academic, New York, 1978).
  2. R. Jones, C. Wykes, Holographic and Speckle Interferometry (Cambridge University Press, London, 1983).
  3. R. S. Sirohi, Speckle Metrology, (Marcel Dekker, New York, 1993).
  4. C. Joenathan, “Speckle photography, shearography, and ESPI,” in Optical Measurement Techniques and Applications, P. K. Rastogi, ed. (Artech House, Boston, London, 1997), Chap. 6.
  5. C. Joenathan, B. Franze, P. Haible, H. J. Tiziani, “Speckle interferometry with temporal phase evaluation for measuring large-object deformation,” Appl. Opt. 37, 2608–2614 (1998).
    [CrossRef]
  6. C. Joenathan, B. Franze, P. Haible, H. J. Tiziani, “Large in-plane displacement measurement in dual beam speckle interferometry using temporal phase measurement,” J. Mod. Opt. 45, 1975–1984 (1998).
    [CrossRef]
  7. C. Joenathan, B. Franze, P. Haible, H. J. Tiziani, “Novel temporal Fourier transform speckle pattern shearing interferometer,” Opt. Eng. 37, 1790–1795 (1998).
    [CrossRef]
  8. C. Joenathan, B. Franze, P. Haible, H. J. Tiziani, “Shape measurement by use of temporal Fourier transformation in dual-beam illumination speckle interferometry,” Appl. Opt. 37, 3385–3390 (1998).
    [CrossRef]
  9. H. J. Tiziani, B. Franze, P. Haible, “Wavelength-shift speckle interferometry for absolute profilometry using a mode-hop free external cavity diode laser,” J. Mod. Opt. 44, 1485–1496 (1997).
    [CrossRef]
  10. I. Yamaguchi, “Speckle displacement and decorrelation in the diffraction and image fields for small object deformation,” Opt. Acta (J. Mod. Opt.) 28, 1359–1376 (1981).
    [CrossRef]
  11. T. Yoshimura, M. Zhou, K. Yamahai, Z. Liyan, “Optimum determination of speckle size to be used in electronic speckle pattern interferometry,” Appl. Opt. 34, 87–91 (1995).
    [CrossRef] [PubMed]
  12. M. O. Petersen, “Decorrelation and fringe visibility: on the limiting behavior of various electronic speckle-pattern correlation interferometer,” J. Opt. Soc. Am. A 8, 1082–1089 (1991).
    [CrossRef]
  13. T. W. Ng, F. S. Chau, “Effect of camera resolution on fringe contrast in digital speckle correlation interferometry,” Optik 97, 183–185 (1994).
  14. L. Leushacke, M. Kirchner, “Three-dimensional correlation coefficient of speckle intensity for rectangular and circular apertures,” J. Opt. Soc. Am. A 7, 827–832 (1992).
    [CrossRef]

1998 (4)

C. Joenathan, B. Franze, P. Haible, H. J. Tiziani, “Speckle interferometry with temporal phase evaluation for measuring large-object deformation,” Appl. Opt. 37, 2608–2614 (1998).
[CrossRef]

C. Joenathan, B. Franze, P. Haible, H. J. Tiziani, “Large in-plane displacement measurement in dual beam speckle interferometry using temporal phase measurement,” J. Mod. Opt. 45, 1975–1984 (1998).
[CrossRef]

C. Joenathan, B. Franze, P. Haible, H. J. Tiziani, “Novel temporal Fourier transform speckle pattern shearing interferometer,” Opt. Eng. 37, 1790–1795 (1998).
[CrossRef]

C. Joenathan, B. Franze, P. Haible, H. J. Tiziani, “Shape measurement by use of temporal Fourier transformation in dual-beam illumination speckle interferometry,” Appl. Opt. 37, 3385–3390 (1998).
[CrossRef]

1997 (1)

H. J. Tiziani, B. Franze, P. Haible, “Wavelength-shift speckle interferometry for absolute profilometry using a mode-hop free external cavity diode laser,” J. Mod. Opt. 44, 1485–1496 (1997).
[CrossRef]

1995 (1)

1994 (1)

T. W. Ng, F. S. Chau, “Effect of camera resolution on fringe contrast in digital speckle correlation interferometry,” Optik 97, 183–185 (1994).

1992 (1)

1991 (1)

1981 (1)

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

Chau, F. S.

T. W. Ng, F. S. Chau, “Effect of camera resolution on fringe contrast in digital speckle correlation interferometry,” Optik 97, 183–185 (1994).

Erf, K.

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

Franze, B.

C. Joenathan, B. Franze, P. Haible, H. J. Tiziani, “Speckle interferometry with temporal phase evaluation for measuring large-object deformation,” Appl. Opt. 37, 2608–2614 (1998).
[CrossRef]

C. Joenathan, B. Franze, P. Haible, H. J. Tiziani, “Large in-plane displacement measurement in dual beam speckle interferometry using temporal phase measurement,” J. Mod. Opt. 45, 1975–1984 (1998).
[CrossRef]

C. Joenathan, B. Franze, P. Haible, H. J. Tiziani, “Novel temporal Fourier transform speckle pattern shearing interferometer,” Opt. Eng. 37, 1790–1795 (1998).
[CrossRef]

C. Joenathan, B. Franze, P. Haible, H. J. Tiziani, “Shape measurement by use of temporal Fourier transformation in dual-beam illumination speckle interferometry,” Appl. Opt. 37, 3385–3390 (1998).
[CrossRef]

H. J. Tiziani, B. Franze, P. Haible, “Wavelength-shift speckle interferometry for absolute profilometry using a mode-hop free external cavity diode laser,” J. Mod. Opt. 44, 1485–1496 (1997).
[CrossRef]

Haible, P.

C. Joenathan, B. Franze, P. Haible, H. J. Tiziani, “Shape measurement by use of temporal Fourier transformation in dual-beam illumination speckle interferometry,” Appl. Opt. 37, 3385–3390 (1998).
[CrossRef]

C. Joenathan, B. Franze, P. Haible, H. J. Tiziani, “Novel temporal Fourier transform speckle pattern shearing interferometer,” Opt. Eng. 37, 1790–1795 (1998).
[CrossRef]

C. Joenathan, B. Franze, P. Haible, H. J. Tiziani, “Large in-plane displacement measurement in dual beam speckle interferometry using temporal phase measurement,” J. Mod. Opt. 45, 1975–1984 (1998).
[CrossRef]

C. Joenathan, B. Franze, P. Haible, H. J. Tiziani, “Speckle interferometry with temporal phase evaluation for measuring large-object deformation,” Appl. Opt. 37, 2608–2614 (1998).
[CrossRef]

H. J. Tiziani, B. Franze, P. Haible, “Wavelength-shift speckle interferometry for absolute profilometry using a mode-hop free external cavity diode laser,” J. Mod. Opt. 44, 1485–1496 (1997).
[CrossRef]

Joenathan, C.

C. Joenathan, B. Franze, P. Haible, H. J. Tiziani, “Novel temporal Fourier transform speckle pattern shearing interferometer,” Opt. Eng. 37, 1790–1795 (1998).
[CrossRef]

C. Joenathan, B. Franze, P. Haible, H. J. Tiziani, “Shape measurement by use of temporal Fourier transformation in dual-beam illumination speckle interferometry,” Appl. Opt. 37, 3385–3390 (1998).
[CrossRef]

C. Joenathan, B. Franze, P. Haible, H. J. Tiziani, “Speckle interferometry with temporal phase evaluation for measuring large-object deformation,” Appl. Opt. 37, 2608–2614 (1998).
[CrossRef]

C. Joenathan, B. Franze, P. Haible, H. J. Tiziani, “Large in-plane displacement measurement in dual beam speckle interferometry using temporal phase measurement,” J. Mod. Opt. 45, 1975–1984 (1998).
[CrossRef]

C. Joenathan, “Speckle photography, shearography, and ESPI,” in Optical Measurement Techniques and Applications, P. K. Rastogi, ed. (Artech House, Boston, London, 1997), Chap. 6.

Jones, R.

R. Jones, C. Wykes, Holographic and Speckle Interferometry (Cambridge University Press, London, 1983).

Kirchner, M.

Leushacke, L.

Liyan, Z.

Ng, T. W.

T. W. Ng, F. S. Chau, “Effect of camera resolution on fringe contrast in digital speckle correlation interferometry,” Optik 97, 183–185 (1994).

Petersen, M. O.

Sirohi, R. S.

R. S. Sirohi, Speckle Metrology, (Marcel Dekker, New York, 1993).

Tiziani, H. J.

C. Joenathan, B. Franze, P. Haible, H. J. Tiziani, “Speckle interferometry with temporal phase evaluation for measuring large-object deformation,” Appl. Opt. 37, 2608–2614 (1998).
[CrossRef]

C. Joenathan, B. Franze, P. Haible, H. J. Tiziani, “Large in-plane displacement measurement in dual beam speckle interferometry using temporal phase measurement,” J. Mod. Opt. 45, 1975–1984 (1998).
[CrossRef]

C. Joenathan, B. Franze, P. Haible, H. J. Tiziani, “Novel temporal Fourier transform speckle pattern shearing interferometer,” Opt. Eng. 37, 1790–1795 (1998).
[CrossRef]

C. Joenathan, B. Franze, P. Haible, H. J. Tiziani, “Shape measurement by use of temporal Fourier transformation in dual-beam illumination speckle interferometry,” Appl. Opt. 37, 3385–3390 (1998).
[CrossRef]

H. J. Tiziani, B. Franze, P. Haible, “Wavelength-shift speckle interferometry for absolute profilometry using a mode-hop free external cavity diode laser,” J. Mod. Opt. 44, 1485–1496 (1997).
[CrossRef]

Wykes, C.

R. Jones, C. Wykes, Holographic and Speckle Interferometry (Cambridge University Press, London, 1983).

Yamaguchi, I.

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

Yamahai, K.

Yoshimura, T.

Zhou, M.

Appl. Opt. (3)

J. Mod. Opt. (2)

H. J. Tiziani, B. Franze, P. Haible, “Wavelength-shift speckle interferometry for absolute profilometry using a mode-hop free external cavity diode laser,” J. Mod. Opt. 44, 1485–1496 (1997).
[CrossRef]

C. Joenathan, B. Franze, P. Haible, H. J. Tiziani, “Large in-plane displacement measurement in dual beam speckle interferometry using temporal phase measurement,” J. Mod. Opt. 45, 1975–1984 (1998).
[CrossRef]

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

Opt. Acta (J. Mod. Opt.) (1)

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

Opt. Eng. (1)

C. Joenathan, B. Franze, P. Haible, H. J. Tiziani, “Novel temporal Fourier transform speckle pattern shearing interferometer,” Opt. Eng. 37, 1790–1795 (1998).
[CrossRef]

Optik (1)

T. W. Ng, F. S. Chau, “Effect of camera resolution on fringe contrast in digital speckle correlation interferometry,” Optik 97, 183–185 (1994).

Other (4)

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

R. Jones, C. Wykes, Holographic and Speckle Interferometry (Cambridge University Press, London, 1983).

R. S. Sirohi, Speckle Metrology, (Marcel Dekker, New York, 1993).

C. Joenathan, “Speckle photography, shearography, and ESPI,” in Optical Measurement Techniques and Applications, P. K. Rastogi, ed. (Artech House, Boston, London, 1997), Chap. 6.

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Figures (7)

Fig. 1
Fig. 1

Schematic of the experimental arrangement of the temporal Fourier-transform speckle interferometric method.

Fig. 2
Fig. 2

Experimental results for an object tilted continuously. (a) Intensity variation at one pixel as a function of time. (b) Fourier transformation of the temporal signal shown in Fig. 2(a). (c) Inverse Fourier transform of the filtered spectrum. The phase varies between -π and +π and only 100 frames are shown for clarity.

Fig. 3
Fig. 3

Signal modulation at one pixel against frame number. Only 200 of the 1024 frames are shown. (a) Experimental modulations near the transition region from one speckle to another. (b) Theoretical plot of the intensity variation of two cigar-shaped structure for the speckles. The object to reference-beam ratio was 1:9. The phase of the speckles and the intensity of the beams were set to match the experimental results. (c) Phase plot obtained with the Fourier-transform method of analysis, which exhibits no anomalous phase jumps in the transition zone. Anomalous jumps can occur in some transition zone if no signal is present.

Fig. 4
Fig. 4

Signal modulation in one speckle; 400 of the 1024 frames are shown. (a) Experimental modulation taken from a data for an object deformed during the exposure. (b) Theoretical plot for which the object to reference-beam ratio was taken to be 1:4. (c) Raw-phase plot obtained with the speckle intensity modulation of Fig. 4(a) and matches the results of Fig. 2(c).

Fig. 5
Fig. 5

Effect of the width of the bandpass filter in retrieving data in the transition zone when no signal is present. (a) Signal modulation is reduced considerably at the beginning and then is good after 200 frames. (b) Raw-phase plot obtained with a broad bandpass filter. Anomalous phase jumps occur in the region of poor signal. (c) Raw-phase map with a narrow bandpass filter centered close to the peak in the side band.

Fig. 6
Fig. 6

Effect of the number of speckles in one pixel. Here two speckles were taken to fill one pixel. Only 200 of the 1024 frames are displayed. (a) Signal modulation obtained in one pixel is good and well above the quantization level of the camera. (b) However, the signal modulation in the adjacent pixel is reduced considerably in spite of the high average intensity. This reveals that the phase difference in the two speckles is π.

Fig. 7
Fig. 7

Spectra obtained in the Fourier-transform plane. The second peak is due to the nonlinear response of the camera and is the second-order term in the nonlinearity.

Equations (24)

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Ix, y, t=Iox, y1+m cosΦ0x, y+Φx, y, t,
Φx, y, t=4πhx, yΔλtλ2=2πhx, yλeff  shape,
λeff=λ22Δλ,Φx, y, t=2πWx, y, t1+cos θλ=2πWx, y, tλeff  out of plane,
λeff=λ1+cos θ,Φx, y, t=4πUx, y, tsin θλ=2πUx, y, tλeff  in plane,
λeff=λ2 sin θ,Φx, y, t=2πWx, y, t/xΔx1+cos θλ=2πΔxWx, y, t/xλeff  slope,
λeff=λ1+cos θΦx, y, t=4πhx, ysin θΔθtλ=2πhx, yλeff  shape,
λeff=λ2 sin θΔθ.
finsx, y, t=12πλΦx, y, tt.
FTIx, y, t=Ã+B˜f-fmed+C˜-f-fmed.
Ã=FTI0x, y,B˜f-fmed=FTI0x, y2 m expiΦox, y+Φx, y, t,  C˜-f-fmed=FTI0x, y2 m exp-iΦox, y+Φx, y, t.
FT-1B˜f-fmed=C1I0x, y2 m expiΦox, y+Φx, y, t,
Ix, y, t=a12+a22 sinc2ΦM+2a1a2 sincΦMcosΦox, y+Φx, y, t.
Ix, y, t=a12+a22 sinc2ΦM+2a1a2sinp+qΦ2pΦ+sinp-qΦ2pΦ.
Ix, y, t=Iox, y1+m1 cosΦ0x, y+Φx, y, t+m2 cosΦ2x, y,
ITx, y, t=i=1nAi+Bi cosΦ0ix, y, t,
Vt=C+αIt+βI2t+γI3t+,
Vx, y, t=Ko+K1 cosΦo+Φx, y, t+K2 cos 2Φo+Φx, y, t+K3 cos 3Φo+Φx, y, t+,
Φx, y=4NπΔZx, yλ,
Ix, y, t=Iox, y1+m cosΦox, y+Φ1t+Φx, y, t,
hx, y=Nnqλ2Δλ  shape by wavelength shift,
Zx, y=Nnqλ1+cos θ  out-of-plane displacement,
Ux, y, Vx, y=Nnqλ2 sin θ  in-plane displacement,
Zx, yx Δx=Nnqλ2  shear interferometry,
hx, y=Nnq λ sin θΔθ2  shape by dual-beam illumination,

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