Abstract

A theoretical and numerical investigation of the systematic phase errors in phase-shifting speckle interferometry is presented. The theoretical investigation analyzes the behavior of some systematic error induced by intensity variations in two cases of data-computing techniques. The first case deals with the technique in which the phase change is computed, unwrapped, and then linearly filtered; the second case deals with the technique in which the data are linearly filtered before the arctangent calculation and then unwrapped. With the first filtering technique it is shown that it is preferable when the phase change is of relatively low spatial frequency, leading to a particularly accurate method. With the second case it is demonstrated that an important parameter of speckle interferometry is the modulation factor of the interference frame that induces phase errors when the data are filtered before the arctangent calculation. We show that this technique is better than the first when the phase change is composed of high-spatial-frequency variations. The theoretical investigation of the two techniques is compared with numerical simulations, considering the frequency characteristics of the phase change, and this shows a good match between theory and simulations.

© 2001 Optical Society of America

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References

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  1. H. Steinbichler, G. Gehring, “TV holography and holographic interferometry: industrial applications,” Opt. Lasers Eng. 24, 111–127 (1996).
    [CrossRef]
  2. O. J. Lokberg, J. T. Malmo, “Detection of defects in composite materials by TV-holography,” NDT & E Int. 21, 223–228 (1988).
  3. M. Facchini, P. Zanetta, L. Binda, G. Mirabella, C. Tiraboschi, “Estimation of masonry mechanical characteristics by ESPI fringe interpretation,” Opt. Lasers Eng. 23, 277–290 (1995).
    [CrossRef]
  4. A. A. M. Maas, P. A. A. M. Somers, “Two-dimensional deconvolution applied to phase stepped shearography,” Opt. Lasers Eng. 26, 251–360 (1997).
  5. P. K. Rastogi, Holographic Interferometry (Springer-Verlag, Berlin, 1994).
    [CrossRef]
  6. P. Picart, “Evaluation of phase shifting speckle interferometry accuracy,” in Proceedings of the International Conference on Interferometry in Speckle Light, P. Jacquot, J.-M. Fournier, eds. (Springer-Verlag, Berlin, 2000), pp. 431–438.
    [CrossRef]
  7. K. A. Stetson, “Theory and application of electronic holography,” in Proceedings of the International Conference on Hologram Interferometry and Speckle Metrology, K. A. Stetson, R. J. Pryputniewicz, eds. (Society for Experimental Mechanics, Bethel, Conn., 1990), pp. 294–300.
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2001 (1)

P. Picart, E. Lolive, J.-M. Berthelot, “Characterization of composite materials using a polarizing speckle interferometer,” Opt. Eng. 40, 81–89 (2001).
[CrossRef]

1999 (2)

B. V. Dorrio, J. L. Fernandez, “Phase evaluation methods in whole-field optical measurement techniques,” Meas. Sci. Tech. 10, 33–55 (1999).
[CrossRef]

H. A. Aebischer, S. Waldner, “A simple and effective method for filtering speckle-interferometric phase fringe patterns,” Opt. Commun. 162, 205–230 (1999).
[CrossRef]

1997 (1)

A. A. M. Maas, P. A. A. M. Somers, “Two-dimensional deconvolution applied to phase stepped shearography,” Opt. Lasers Eng. 26, 251–360 (1997).

1996 (1)

H. Steinbichler, G. Gehring, “TV holography and holographic interferometry: industrial applications,” Opt. Lasers Eng. 24, 111–127 (1996).
[CrossRef]

1995 (2)

M. Facchini, P. Zanetta, L. Binda, G. Mirabella, C. Tiraboschi, “Estimation of masonry mechanical characteristics by ESPI fringe interpretation,” Opt. Lasers Eng. 23, 277–290 (1995).
[CrossRef]

B. Zhao, Y. Surrel, “Phase shifting: six sample self calibrating algorithm insensitive to the second harmonic in the fringe signal,” Opt. Eng. 34, 2821–2822 (1995).
[CrossRef]

1993 (1)

J. M. Huntley, H. Saldner, “Temporal phase unwrapping algorithm for automated interferogram analysis,” Appl. Opt. 26, 3047–3052 (1993).
[CrossRef]

1988 (1)

O. J. Lokberg, J. T. Malmo, “Detection of defects in composite materials by TV-holography,” NDT & E Int. 21, 223–228 (1988).

1987 (1)

1985 (1)

1983 (1)

1979 (1)

1966 (1)

P. Carré, “Installation et Utilisation du Comparateur Photoélectrique et Interférentiel du Bureau International des Poids et Mesures,” Metrologia 2, 13–23 (1966).
[CrossRef]

Aebischer, H. A.

H. A. Aebischer, S. Waldner, “A simple and effective method for filtering speckle-interferometric phase fringe patterns,” Opt. Commun. 162, 205–230 (1999).
[CrossRef]

Ai, C.

Berthelot, J.-M.

P. Picart, E. Lolive, J.-M. Berthelot, “Characterization of composite materials using a polarizing speckle interferometer,” Opt. Eng. 40, 81–89 (2001).
[CrossRef]

Binda, L.

M. Facchini, P. Zanetta, L. Binda, G. Mirabella, C. Tiraboschi, “Estimation of masonry mechanical characteristics by ESPI fringe interpretation,” Opt. Lasers Eng. 23, 277–290 (1995).
[CrossRef]

Burow, R.

Carré, P.

P. Carré, “Installation et Utilisation du Comparateur Photoélectrique et Interférentiel du Bureau International des Poids et Mesures,” Metrologia 2, 13–23 (1966).
[CrossRef]

Cheng, Y. Y.

Creath, K.

K. Creath, “Phase measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1988), Vol. XXVI, pp. 349–393.
[CrossRef]

Dorrio, B. V.

B. V. Dorrio, J. L. Fernandez, “Phase evaluation methods in whole-field optical measurement techniques,” Meas. Sci. Tech. 10, 33–55 (1999).
[CrossRef]

Elssner, K. E.

Facchini, M.

M. Facchini, P. Zanetta, L. Binda, G. Mirabella, C. Tiraboschi, “Estimation of masonry mechanical characteristics by ESPI fringe interpretation,” Opt. Lasers Eng. 23, 277–290 (1995).
[CrossRef]

Fernandez, J. L.

B. V. Dorrio, J. L. Fernandez, “Phase evaluation methods in whole-field optical measurement techniques,” Meas. Sci. Tech. 10, 33–55 (1999).
[CrossRef]

Frantz, L. M.

Gehring, G.

H. Steinbichler, G. Gehring, “TV holography and holographic interferometry: industrial applications,” Opt. Lasers Eng. 24, 111–127 (1996).
[CrossRef]

Grzanna, J.

Huntley, J. M.

J. M. Huntley, H. Saldner, “Temporal phase unwrapping algorithm for automated interferogram analysis,” Appl. Opt. 26, 3047–3052 (1993).
[CrossRef]

Lokberg, O. J.

O. J. Lokberg, J. T. Malmo, “Detection of defects in composite materials by TV-holography,” NDT & E Int. 21, 223–228 (1988).

Lolive, E.

P. Picart, E. Lolive, J.-M. Berthelot, “Characterization of composite materials using a polarizing speckle interferometer,” Opt. Eng. 40, 81–89 (2001).
[CrossRef]

Maas, A. A. M.

A. A. M. Maas, P. A. A. M. Somers, “Two-dimensional deconvolution applied to phase stepped shearography,” Opt. Lasers Eng. 26, 251–360 (1997).

Malmo, J. T.

O. J. Lokberg, J. T. Malmo, “Detection of defects in composite materials by TV-holography,” NDT & E Int. 21, 223–228 (1988).

Merkel, K.

Mirabella, G.

M. Facchini, P. Zanetta, L. Binda, G. Mirabella, C. Tiraboschi, “Estimation of masonry mechanical characteristics by ESPI fringe interpretation,” Opt. Lasers Eng. 23, 277–290 (1995).
[CrossRef]

Ohe, W.

Picart, P.

P. Picart, E. Lolive, J.-M. Berthelot, “Characterization of composite materials using a polarizing speckle interferometer,” Opt. Eng. 40, 81–89 (2001).
[CrossRef]

P. Picart, “Evaluation of phase shifting speckle interferometry accuracy,” in Proceedings of the International Conference on Interferometry in Speckle Light, P. Jacquot, J.-M. Fournier, eds. (Springer-Verlag, Berlin, 2000), pp. 431–438.
[CrossRef]

Rastogi, P. K.

P. K. Rastogi, Holographic Interferometry (Springer-Verlag, Berlin, 1994).
[CrossRef]

Saldner, H.

J. M. Huntley, H. Saldner, “Temporal phase unwrapping algorithm for automated interferogram analysis,” Appl. Opt. 26, 3047–3052 (1993).
[CrossRef]

Sawchuk, A. A.

Schwider, J.

Somers, P. A.

H. H. van Brug, P. A. Somers, “Temporal phase unwrapping with two or four images per time frame: a comparison,” in Interferometry ’99: Techniques and Technologies, M. Kujawinska, M. Takeda, eds., Proc. SPIE3744, 358–365 (1999).

Somers, P. A. A. M.

A. A. M. Maas, P. A. A. M. Somers, “Two-dimensional deconvolution applied to phase stepped shearography,” Opt. Lasers Eng. 26, 251–360 (1997).

Spolaczyk, R.

Steinbichler, H.

H. Steinbichler, G. Gehring, “TV holography and holographic interferometry: industrial applications,” Opt. Lasers Eng. 24, 111–127 (1996).
[CrossRef]

Stetson, K. A.

K. A. Stetson, “Theory and application of electronic holography,” in Proceedings of the International Conference on Hologram Interferometry and Speckle Metrology, K. A. Stetson, R. J. Pryputniewicz, eds. (Society for Experimental Mechanics, Bethel, Conn., 1990), pp. 294–300.

Surrel, Y.

B. Zhao, Y. Surrel, “Phase shifting: six sample self calibrating algorithm insensitive to the second harmonic in the fringe signal,” Opt. Eng. 34, 2821–2822 (1995).
[CrossRef]

Tiraboschi, C.

M. Facchini, P. Zanetta, L. Binda, G. Mirabella, C. Tiraboschi, “Estimation of masonry mechanical characteristics by ESPI fringe interpretation,” Opt. Lasers Eng. 23, 277–290 (1995).
[CrossRef]

van Brug, H. H.

H. H. van Brug, P. A. Somers, “Temporal phase unwrapping with two or four images per time frame: a comparison,” in Interferometry ’99: Techniques and Technologies, M. Kujawinska, M. Takeda, eds., Proc. SPIE3744, 358–365 (1999).

Waldner, S.

H. A. Aebischer, S. Waldner, “A simple and effective method for filtering speckle-interferometric phase fringe patterns,” Opt. Commun. 162, 205–230 (1999).
[CrossRef]

Wyant, J. C.

Zanetta, P.

M. Facchini, P. Zanetta, L. Binda, G. Mirabella, C. Tiraboschi, “Estimation of masonry mechanical characteristics by ESPI fringe interpretation,” Opt. Lasers Eng. 23, 277–290 (1995).
[CrossRef]

Zhao, B.

B. Zhao, Y. Surrel, “Phase shifting: six sample self calibrating algorithm insensitive to the second harmonic in the fringe signal,” Opt. Eng. 34, 2821–2822 (1995).
[CrossRef]

Appl. Opt. (5)

Meas. Sci. Tech. (1)

B. V. Dorrio, J. L. Fernandez, “Phase evaluation methods in whole-field optical measurement techniques,” Meas. Sci. Tech. 10, 33–55 (1999).
[CrossRef]

Metrologia (1)

P. Carré, “Installation et Utilisation du Comparateur Photoélectrique et Interférentiel du Bureau International des Poids et Mesures,” Metrologia 2, 13–23 (1966).
[CrossRef]

NDT & E Int. (1)

O. J. Lokberg, J. T. Malmo, “Detection of defects in composite materials by TV-holography,” NDT & E Int. 21, 223–228 (1988).

Opt. Commun. (1)

H. A. Aebischer, S. Waldner, “A simple and effective method for filtering speckle-interferometric phase fringe patterns,” Opt. Commun. 162, 205–230 (1999).
[CrossRef]

Opt. Eng. (2)

P. Picart, E. Lolive, J.-M. Berthelot, “Characterization of composite materials using a polarizing speckle interferometer,” Opt. Eng. 40, 81–89 (2001).
[CrossRef]

B. Zhao, Y. Surrel, “Phase shifting: six sample self calibrating algorithm insensitive to the second harmonic in the fringe signal,” Opt. Eng. 34, 2821–2822 (1995).
[CrossRef]

Opt. Lasers Eng. (3)

M. Facchini, P. Zanetta, L. Binda, G. Mirabella, C. Tiraboschi, “Estimation of masonry mechanical characteristics by ESPI fringe interpretation,” Opt. Lasers Eng. 23, 277–290 (1995).
[CrossRef]

A. A. M. Maas, P. A. A. M. Somers, “Two-dimensional deconvolution applied to phase stepped shearography,” Opt. Lasers Eng. 26, 251–360 (1997).

H. Steinbichler, G. Gehring, “TV holography and holographic interferometry: industrial applications,” Opt. Lasers Eng. 24, 111–127 (1996).
[CrossRef]

Other (6)

P. K. Rastogi, Holographic Interferometry (Springer-Verlag, Berlin, 1994).
[CrossRef]

P. Picart, “Evaluation of phase shifting speckle interferometry accuracy,” in Proceedings of the International Conference on Interferometry in Speckle Light, P. Jacquot, J.-M. Fournier, eds. (Springer-Verlag, Berlin, 2000), pp. 431–438.
[CrossRef]

K. A. Stetson, “Theory and application of electronic holography,” in Proceedings of the International Conference on Hologram Interferometry and Speckle Metrology, K. A. Stetson, R. J. Pryputniewicz, eds. (Society for Experimental Mechanics, Bethel, Conn., 1990), pp. 294–300.

K. Creath, “Phase measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1988), Vol. XXVI, pp. 349–393.
[CrossRef]

H. H. van Brug, P. A. Somers, “Temporal phase unwrapping with two or four images per time frame: a comparison,” in Interferometry ’99: Techniques and Technologies, M. Kujawinska, M. Takeda, eds., Proc. SPIE3744, 358–365 (1999).

Guide to the Expression of Uncertainty in Measurement, ICS 03.120.30;17.020 (International Organization for Standardization, Geneva, 1996).

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Figures (12)

Fig. 1
Fig. 1

Simulated low-frequency phase change.

Fig. 2
Fig. 2

Influence of the calibration parameter, 0° profile of raw phase error (solid curve, numerical; circles, analytical).

Fig. 3
Fig. 3

0° profile of phase error of iteration 30, influence of the calibration parameter (solid curve, numerical; circles, analytical).

Fig. 4
Fig. 4

0° profile of phase error of iteration 30, case b, influence of the filtering process (solid curve, numerical; circles, analytical).

Fig. 5
Fig. 5

Cases A and B: rms phase error due to the calibration parameter (case A: solid curve, numerical; circles, analytical; asterisks, σΔφ/g h n . Case B: dotted–dashed curve, numerical; crosses, analytical).

Fig. 6
Fig. 6

Simulated high-frequency phase change.

Fig. 7
Fig. 7

Influence of the calibration parameter, 0° profile of raw phase error (solid curve, numerical; circles, analytical).

Fig. 8
Fig. 8

Cases A and B: rms phase error due to the calibration parameter (Case A: solid curves, numerical; circles, analytical; asterisks, σΔφ/g h n . Case B: dotted–dashed curve, numerical; crosses, analytical).

Fig. 9
Fig. 9

0° profile of phase error of iteration 1, influence of the calibration parameter (solid curve, numerical; circles, analytical).

Fig. 10
Fig. 10

0° profile of phase error of iteration 1, case B, influence of the filtering process (solid curves, numerical; circles, analytical).

Fig. 11
Fig. 11

0° profile of phase error of iteration 30, case B, influence of the calibration parameter (solid curves, numerical; circles, analytical).

Fig. 12
Fig. 12

0° profile of phase error of iteration 30, case B, influence of the filtering process (solid curve, numerical; circles, analytical).

Equations (28)

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Ei=a+b cosψ+i-1ϕ, i=1, 2, 3,, N,
Fi=a+b cosψ+Δφ+i-1ϕ, i=1, 2, 3,, N,
Δˆφ=atanSEi, FiCEi, Fi,
Δˆφ=atanSfEi, FiCfEi, Fi,
h=h-M, -Nh-M, N|||h0, -Nh0, 0h0, N|||hM, NhM, N with m=-Mm=+Ml=-Nl=+N hm, l=1.
dΔφ=Δφpdp+122Δφp2dp2++1k!kΔφpkdpk+ .
dΔφ=ΔφSdSEi, Fi+ΔφCdCEi, Fi+122ΔφS2dSEi, Fi2+122ΔφC2dCEi, Fi2++1k!kΔφSkdSEi, Fik+1k!kΔφCkdCEi, Fik+ .
dΔφ=1κβdSEi, Ficos Δφ-1κβdCEi, Fisin Δφ.
dΔφf=fdΔφ=f1κβdSEi, Ficos Δφ-f1κβdCEi, Fisin Δφ.
PdΔφf=PdΔφm=-Mm=+Ml=-Nl=+N |hm, l|2,
PdΔφ=dΔφ2ψ, Δφ, ϕ, a, b,
dΔφnψ, Δφ, ϕ, a, b=-+-+dΔφnu, v, ϕ, a, b×Pψ,Δφu, vdudv,
PdΔφ=14π2-π+π02πdΔφ2u, v, ϕ, a, bdudv.
gh=1m=-Mm=Ml=-Nl=+N|hm, l|21/2.
hni, j=hi, j * hi, j *  * hi, jn times,
ghn=1m=-nMm=+nMl=-nNl=+nN|hnm, l|21/2.
dΔφR=atanfβ sin Δφcos Δφ-fβ cos Δφsin Δφfβ cos Δφcos Δφ+fβ sin Δφsin Δφ.
tan ΔφE=Sf+dSfCf+dCf,
dΔφ=ΔφE-ΔφR=atanCfdSf-SfdCfκCf2+κSf2+CfdCf+SfdSf.
dΔφ1κCfdSfCf2+Sf2-1κSfdCfCf2+Sf2+1κ2Sf2-Cf2dSfdCfCf2+Sf22+1κ2SfCfdCf2-dSf2Cf2+Sf22.
dΔφ=1κfβ cos Δφfβ cos Δφ2+fβ sin Δφ2×fdSEi, Fi-1κfβ sin Δφfβ cos Δφ2+fβ sin Δφ2×fdCEi, Fi.
sin Φk=fsin Φk-1, cos Φk=fcos Φk-1, k1, dΔφRk=atansin Φk cos Δφ-cos Φk sin Δφcos Φk cos Δφ+sin Φk sin Δφ, Ψk=atansin Φkcos Φk, sin Φk=sin Ψk, cos Φk=cos Ψk, with sin Φ0=β sin Δφ, cos Φ0=β cos Δφ.
sin Φk=fsin Φk-1, cos Φk=fcos Φk-1, sin Θk=fsin Θk-1, cos Θk=fcos Θk-1, k1, dΔφk=atansin Θk cos Φk-cos Θk sin Φkcos Θk cos Φk+sin Θk sin Φk, Ψk=atansin Φkcos Φk,  Ωk=atansin Θkcos Θk, sin Φk=sin Ψk,  sin Θk=sin Ωk, cos Φk=cos Ψk,  cos Θk=cos Ωk, with sin Φ0=β sin Δφ, sin Θ0=β sin Δφ+dSEi, Fi/κ, cos Φ0=β cos Δφ, cos Θ0=β cos Δφ+dCEi, Fi/κ.
Δφ=atanF4-F2E1-E3-F1-F3E4-E2F1-F3E1-E3+F4-F2E4-E2,
dSEi, Fi=0, dCEi, Fi=-4πσc/cb2 sin2ψ+Δφ.
dΔφψ, Δφ, c=π2σcccos2ψ+2Δφ-cos 2ψ,
Δφm, n=αm2+n2.
Δφm, n=αm2+n2+α exp-m-m02+n22σn2.

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