Abstract

Interferometric measurement techniques such as holographic interferometry and electronic speckle-pattern interferometry are valuable for measuring the deformation of objects. Conventional theoretical models of deformation measurement assume collimated illumination and telecentric imaging, which are usually only practical for small objects. Large objects often require divergent illumination, for which the models are valid only when the object is planar, and then only in the paraxial region. We present an analysis and discussion of the three-dimensional systematic sensitivity errors for both in-plane and out-of-plane interferometer configurations, where it is shown that the errors can be significant. A dimensionless approach is adopted to make the analysis generic and hence scalable to a system of any size.

© 2003 Optical Society of America

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  1. Y. Surrel, “Phase stepping: a new self-calibrating algorithm,” Appl. Opt. 32, 3598–3600 (1993).
    [CrossRef] [PubMed]
  2. K. Hibino, B. F. Oreb, D. I. Farrant, K. G. Larkin, “Phase-shifting algorithms for nonlinear and spatially nonuniform phase shifts,” J. Opt. Soc. Am. 14, 918–930 (1997).
    [CrossRef]
  3. P. Picart, J. C. Pascal, J. M. Breteau, “Systematic errors of phase-shifting speckle interferometry,” Appl. Opt. 40, 2107–2116 (2001).
    [CrossRef]
  4. M. Kujawinska, J. Wojciak, “High accuracy Fourier transform fringe pattern analysis,” Opt. Lasers Eng. 14, 325–339 (1991).
    [CrossRef]
  5. J. H. Massig, J. Heppner, “Fringe-pattern analysis with high accuracy by use of the Fourier-transform method: theory and experimental tests,” Appl. Opt. 40, 2081–2088 (2001).
    [CrossRef]
  6. A. Fernandez, G. H. Kaufmann, A. F. Doval, J. Blanco-Garcia, J. L. Fernandez, “Comparison of carrier removal methods in the analysis of TV holography fringes by the Fourier transform method,” Opt. Eng. 37, 2899–2905 (1998).
    [CrossRef]
  7. T. Takatsuji, B. F. Oreb, D. I. Farrant, J. R. Tyrer, “Simultaneous measurement of three orthogonal components of displacement by electronic speckle-pattern interferometry and the Fourier transform method,” Appl. Opt. 36, 1438–1445 (1997).
    [CrossRef] [PubMed]
  8. B. Kemper, D. Dirksen, J. Kandulla, G. von Bally, “Quantitative determination of out-of-plane displacements by endoscopic electronic-speckle-pattern interferometry,” Opt. Commun. 194, 75–82 (2001).
    [CrossRef]
  9. D. I. Farrant, J. N. Petzing, J. R. Tyrer, “Geometrically qualified ESPI vibration analysis,” Opt. Lasers Eng., to be published.
  10. C. Joenathan, “Speckle photography, shearography, and ESPI,” in Optical Measurement Techniques and Applications, P. K. Rastogi, ed. (Artech, Boston, Mass., 1997).
  11. W. Osten, “Application of optical shape measurement for the nondestructive evaluation of complex objects,” Opt. Eng. 39, 232–243 (2000).
    [CrossRef]
  12. Ch. De Veuster, P. Slangen, Y. Renotte, L. Berwart, Y. Lion, “Influence of the geometry of illumination and viewing beams on displacement measurement errors in interferometric metrology,” Opt. Commun. 143, 95–101 (1997).
    [CrossRef]
  13. D. Albrecht, “Estimation of the 2d measurement error introduced by in-plane and out-of-plane electronic speckle pattern interferometry instruments,” Opt. Lasers Eng. 31, 63–81 (1999).
    [CrossRef]
  14. W. S. W. Abdullah, J. N. Petzing, J. R. Tyrer, “Wavefront divergence: a source of error in quantified speckle shearing data,” J. Mod. Opt. 48, 757–772 (2001).
  15. S. Schedin, G. Pedrini, H. J. Tiziani, F. Mendoza Santoyo, “Simultaneous three-dimensional dynamic deformation measurements with pulsed digital holography,” Appl. Opt. 38, 7056–7062 (1999).
    [CrossRef]

2001 (4)

B. Kemper, D. Dirksen, J. Kandulla, G. von Bally, “Quantitative determination of out-of-plane displacements by endoscopic electronic-speckle-pattern interferometry,” Opt. Commun. 194, 75–82 (2001).
[CrossRef]

W. S. W. Abdullah, J. N. Petzing, J. R. Tyrer, “Wavefront divergence: a source of error in quantified speckle shearing data,” J. Mod. Opt. 48, 757–772 (2001).

J. H. Massig, J. Heppner, “Fringe-pattern analysis with high accuracy by use of the Fourier-transform method: theory and experimental tests,” Appl. Opt. 40, 2081–2088 (2001).
[CrossRef]

P. Picart, J. C. Pascal, J. M. Breteau, “Systematic errors of phase-shifting speckle interferometry,” Appl. Opt. 40, 2107–2116 (2001).
[CrossRef]

2000 (1)

W. Osten, “Application of optical shape measurement for the nondestructive evaluation of complex objects,” Opt. Eng. 39, 232–243 (2000).
[CrossRef]

1999 (2)

D. Albrecht, “Estimation of the 2d measurement error introduced by in-plane and out-of-plane electronic speckle pattern interferometry instruments,” Opt. Lasers Eng. 31, 63–81 (1999).
[CrossRef]

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

1998 (1)

A. Fernandez, G. H. Kaufmann, A. F. Doval, J. Blanco-Garcia, J. L. Fernandez, “Comparison of carrier removal methods in the analysis of TV holography fringes by the Fourier transform method,” Opt. Eng. 37, 2899–2905 (1998).
[CrossRef]

1997 (3)

T. Takatsuji, B. F. Oreb, D. I. Farrant, J. R. Tyrer, “Simultaneous measurement of three orthogonal components of displacement by electronic speckle-pattern interferometry and the Fourier transform method,” Appl. Opt. 36, 1438–1445 (1997).
[CrossRef] [PubMed]

Ch. De Veuster, P. Slangen, Y. Renotte, L. Berwart, Y. Lion, “Influence of the geometry of illumination and viewing beams on displacement measurement errors in interferometric metrology,” Opt. Commun. 143, 95–101 (1997).
[CrossRef]

K. Hibino, B. F. Oreb, D. I. Farrant, K. G. Larkin, “Phase-shifting algorithms for nonlinear and spatially nonuniform phase shifts,” J. Opt. Soc. Am. 14, 918–930 (1997).
[CrossRef]

1993 (1)

1991 (1)

M. Kujawinska, J. Wojciak, “High accuracy Fourier transform fringe pattern analysis,” Opt. Lasers Eng. 14, 325–339 (1991).
[CrossRef]

Abdullah, W. S. W.

W. S. W. Abdullah, J. N. Petzing, J. R. Tyrer, “Wavefront divergence: a source of error in quantified speckle shearing data,” J. Mod. Opt. 48, 757–772 (2001).

Albrecht, D.

D. Albrecht, “Estimation of the 2d measurement error introduced by in-plane and out-of-plane electronic speckle pattern interferometry instruments,” Opt. Lasers Eng. 31, 63–81 (1999).
[CrossRef]

Berwart, L.

Ch. De Veuster, P. Slangen, Y. Renotte, L. Berwart, Y. Lion, “Influence of the geometry of illumination and viewing beams on displacement measurement errors in interferometric metrology,” Opt. Commun. 143, 95–101 (1997).
[CrossRef]

Blanco-Garcia, J.

A. Fernandez, G. H. Kaufmann, A. F. Doval, J. Blanco-Garcia, J. L. Fernandez, “Comparison of carrier removal methods in the analysis of TV holography fringes by the Fourier transform method,” Opt. Eng. 37, 2899–2905 (1998).
[CrossRef]

Breteau, J. M.

De Veuster, Ch.

Ch. De Veuster, P. Slangen, Y. Renotte, L. Berwart, Y. Lion, “Influence of the geometry of illumination and viewing beams on displacement measurement errors in interferometric metrology,” Opt. Commun. 143, 95–101 (1997).
[CrossRef]

Dirksen, D.

B. Kemper, D. Dirksen, J. Kandulla, G. von Bally, “Quantitative determination of out-of-plane displacements by endoscopic electronic-speckle-pattern interferometry,” Opt. Commun. 194, 75–82 (2001).
[CrossRef]

Doval, A. F.

A. Fernandez, G. H. Kaufmann, A. F. Doval, J. Blanco-Garcia, J. L. Fernandez, “Comparison of carrier removal methods in the analysis of TV holography fringes by the Fourier transform method,” Opt. Eng. 37, 2899–2905 (1998).
[CrossRef]

Farrant, D. I.

K. Hibino, B. F. Oreb, D. I. Farrant, K. G. Larkin, “Phase-shifting algorithms for nonlinear and spatially nonuniform phase shifts,” J. Opt. Soc. Am. 14, 918–930 (1997).
[CrossRef]

T. Takatsuji, B. F. Oreb, D. I. Farrant, J. R. Tyrer, “Simultaneous measurement of three orthogonal components of displacement by electronic speckle-pattern interferometry and the Fourier transform method,” Appl. Opt. 36, 1438–1445 (1997).
[CrossRef] [PubMed]

D. I. Farrant, J. N. Petzing, J. R. Tyrer, “Geometrically qualified ESPI vibration analysis,” Opt. Lasers Eng., to be published.

Fernandez, A.

A. Fernandez, G. H. Kaufmann, A. F. Doval, J. Blanco-Garcia, J. L. Fernandez, “Comparison of carrier removal methods in the analysis of TV holography fringes by the Fourier transform method,” Opt. Eng. 37, 2899–2905 (1998).
[CrossRef]

Fernandez, J. L.

A. Fernandez, G. H. Kaufmann, A. F. Doval, J. Blanco-Garcia, J. L. Fernandez, “Comparison of carrier removal methods in the analysis of TV holography fringes by the Fourier transform method,” Opt. Eng. 37, 2899–2905 (1998).
[CrossRef]

Heppner, J.

Hibino, K.

K. Hibino, B. F. Oreb, D. I. Farrant, K. G. Larkin, “Phase-shifting algorithms for nonlinear and spatially nonuniform phase shifts,” J. Opt. Soc. Am. 14, 918–930 (1997).
[CrossRef]

Joenathan, C.

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

Kandulla, J.

B. Kemper, D. Dirksen, J. Kandulla, G. von Bally, “Quantitative determination of out-of-plane displacements by endoscopic electronic-speckle-pattern interferometry,” Opt. Commun. 194, 75–82 (2001).
[CrossRef]

Kaufmann, G. H.

A. Fernandez, G. H. Kaufmann, A. F. Doval, J. Blanco-Garcia, J. L. Fernandez, “Comparison of carrier removal methods in the analysis of TV holography fringes by the Fourier transform method,” Opt. Eng. 37, 2899–2905 (1998).
[CrossRef]

Kemper, B.

B. Kemper, D. Dirksen, J. Kandulla, G. von Bally, “Quantitative determination of out-of-plane displacements by endoscopic electronic-speckle-pattern interferometry,” Opt. Commun. 194, 75–82 (2001).
[CrossRef]

Kujawinska, M.

M. Kujawinska, J. Wojciak, “High accuracy Fourier transform fringe pattern analysis,” Opt. Lasers Eng. 14, 325–339 (1991).
[CrossRef]

Larkin, K. G.

K. Hibino, B. F. Oreb, D. I. Farrant, K. G. Larkin, “Phase-shifting algorithms for nonlinear and spatially nonuniform phase shifts,” J. Opt. Soc. Am. 14, 918–930 (1997).
[CrossRef]

Lion, Y.

Ch. De Veuster, P. Slangen, Y. Renotte, L. Berwart, Y. Lion, “Influence of the geometry of illumination and viewing beams on displacement measurement errors in interferometric metrology,” Opt. Commun. 143, 95–101 (1997).
[CrossRef]

Massig, J. H.

Mendoza Santoyo, F.

Oreb, B. F.

Osten, W.

W. Osten, “Application of optical shape measurement for the nondestructive evaluation of complex objects,” Opt. Eng. 39, 232–243 (2000).
[CrossRef]

Pascal, J. C.

Pedrini, G.

Petzing, J. N.

W. S. W. Abdullah, J. N. Petzing, J. R. Tyrer, “Wavefront divergence: a source of error in quantified speckle shearing data,” J. Mod. Opt. 48, 757–772 (2001).

D. I. Farrant, J. N. Petzing, J. R. Tyrer, “Geometrically qualified ESPI vibration analysis,” Opt. Lasers Eng., to be published.

Picart, P.

Renotte, Y.

Ch. De Veuster, P. Slangen, Y. Renotte, L. Berwart, Y. Lion, “Influence of the geometry of illumination and viewing beams on displacement measurement errors in interferometric metrology,” Opt. Commun. 143, 95–101 (1997).
[CrossRef]

Schedin, S.

Slangen, P.

Ch. De Veuster, P. Slangen, Y. Renotte, L. Berwart, Y. Lion, “Influence of the geometry of illumination and viewing beams on displacement measurement errors in interferometric metrology,” Opt. Commun. 143, 95–101 (1997).
[CrossRef]

Surrel, Y.

Takatsuji, T.

Tiziani, H. J.

Tyrer, J. R.

W. S. W. Abdullah, J. N. Petzing, J. R. Tyrer, “Wavefront divergence: a source of error in quantified speckle shearing data,” J. Mod. Opt. 48, 757–772 (2001).

T. Takatsuji, B. F. Oreb, D. I. Farrant, J. R. Tyrer, “Simultaneous measurement of three orthogonal components of displacement by electronic speckle-pattern interferometry and the Fourier transform method,” Appl. Opt. 36, 1438–1445 (1997).
[CrossRef] [PubMed]

D. I. Farrant, J. N. Petzing, J. R. Tyrer, “Geometrically qualified ESPI vibration analysis,” Opt. Lasers Eng., to be published.

von Bally, G.

B. Kemper, D. Dirksen, J. Kandulla, G. von Bally, “Quantitative determination of out-of-plane displacements by endoscopic electronic-speckle-pattern interferometry,” Opt. Commun. 194, 75–82 (2001).
[CrossRef]

Wojciak, J.

M. Kujawinska, J. Wojciak, “High accuracy Fourier transform fringe pattern analysis,” Opt. Lasers Eng. 14, 325–339 (1991).
[CrossRef]

Appl. Opt. (5)

J. Mod. Opt. (1)

W. S. W. Abdullah, J. N. Petzing, J. R. Tyrer, “Wavefront divergence: a source of error in quantified speckle shearing data,” J. Mod. Opt. 48, 757–772 (2001).

J. Opt. Soc. Am. (1)

K. Hibino, B. F. Oreb, D. I. Farrant, K. G. Larkin, “Phase-shifting algorithms for nonlinear and spatially nonuniform phase shifts,” J. Opt. Soc. Am. 14, 918–930 (1997).
[CrossRef]

Opt. Commun. (2)

B. Kemper, D. Dirksen, J. Kandulla, G. von Bally, “Quantitative determination of out-of-plane displacements by endoscopic electronic-speckle-pattern interferometry,” Opt. Commun. 194, 75–82 (2001).
[CrossRef]

Ch. De Veuster, P. Slangen, Y. Renotte, L. Berwart, Y. Lion, “Influence of the geometry of illumination and viewing beams on displacement measurement errors in interferometric metrology,” Opt. Commun. 143, 95–101 (1997).
[CrossRef]

Opt. Eng. (2)

W. Osten, “Application of optical shape measurement for the nondestructive evaluation of complex objects,” Opt. Eng. 39, 232–243 (2000).
[CrossRef]

A. Fernandez, G. H. Kaufmann, A. F. Doval, J. Blanco-Garcia, J. L. Fernandez, “Comparison of carrier removal methods in the analysis of TV holography fringes by the Fourier transform method,” Opt. Eng. 37, 2899–2905 (1998).
[CrossRef]

Opt. Lasers Eng. (2)

M. Kujawinska, J. Wojciak, “High accuracy Fourier transform fringe pattern analysis,” Opt. Lasers Eng. 14, 325–339 (1991).
[CrossRef]

D. Albrecht, “Estimation of the 2d measurement error introduced by in-plane and out-of-plane electronic speckle pattern interferometry instruments,” Opt. Lasers Eng. 31, 63–81 (1999).
[CrossRef]

Other (2)

D. I. Farrant, J. N. Petzing, J. R. Tyrer, “Geometrically qualified ESPI vibration analysis,” Opt. Lasers Eng., to be published.

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

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Figures (7)

Fig. 1
Fig. 1

OOP deformation interferometer configuration.

Fig. 2
Fig. 2

Detector array configuration.

Fig. 3
Fig. 3

IP deformation interferometer configuration.

Fig. 4
Fig. 4

Relative sensitivity error for the OOP system with d 3 displacement plotted along the diagonal of the object.

Fig. 5
Fig. 5

Relative sensitivity error difference between convex-cylindrical and planar objects for an OOP system with d 3 displacement.

Fig. 6
Fig. 6

Maximum relative sensitivity errors for an OOP system: (a) d 1 displacement, (b) d 2 displacement, (c) d 3 displacement.

Fig. 7
Fig. 7

Maximum relative sensitivity errors for an IP system: (a) d 1 displacement, (b) d 2 displacement, (c) d 3 displacement.

Equations (36)

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Δϕ=K · d=2πλrˆ2-rˆ1 · d,
Δϕ=2πλcos θ1+cos θ2d3+sin θ1-sin θ2d1.
R1=(x-s)û+yvˆ+zŵ.
|R1|=x-s2+y2+z21/2.
rˆ1=x-s|R1|û+y|R1|vˆ+z|R1|ŵ.
rˆ2=t-x|R2|û-y|R2|vˆ-z|R2|ŵ.
Δϕx, y, z=2πλt-xt-x2+y2+z21/2 -x-sx-s2+y2+z21/2d1 +-yt-x2+y2+z21/2-yx-s2+y2+z21/2d2 +-zt-x2+y2+z21/2-zx-s2+y2+z21/2d3.
Δϕx, y, z=2πλK1d1+K2d2+K3d3,
K1c=K10, 0, z
K1c=ss2+z21/2,
K2c=0,
K3c=-1+zs2+z21/2=-1+cos θ1,
K1c/K3c=-sz+s2+z21/2.
ρ31=K1/K3c,
ρ32=K2/K3c,
ρ33=K3-K3c/K3c,
|R2d|c1+c2±d2+Xm21/2,
|R2d|c1+c2±d2+Xm+Xtm21/2,
ρt=δR2-δR2/δR2.
Δϕ=2πλrˆ1-rˆ2 · d.
Δϕ=2πλsin θ1+sin θ2d1+cos θ1-cos θ2d3.
R1=x-s1û+yvˆ+zŵ,
R2=x-s2û+yvˆ+zŵ.
rˆ1=x-s1|R1|û+y|R1|vˆ+z|R1|ŵ,
rˆ2=x-s2|R2|û+y|R2|vˆ+z|R2|ŵ.
Δϕx, y, z=2πλx-s1x-s12+y2+z21/2-x-s2x-s22+y2+z21/2d1 +yx-s12+y2+z21/2 -yx-s22+y2+z21/2d2 +zx-s12+y2+z21/2 -zx-s22+y2+z21/2d3.
K1c=2ss2+z21/2=2 sin θ,
K2c=0,
K3c=0.
ρ11=K1-K1c/K1c,
ρ12=K2/K1c,
ρ13=K3/K1c.
aspect ratio, RA=|L/2Xm|,
profile relief ratio, RP=|L/Zm|,
source offset ratio, RS=|s/L|,
object ratio,   RO=|Xm/Zm|=RP/2RA.

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