Abstract

The influence of random vibrations on the performance of a dynamic phase-shifting speckle pattern interferometer is investigated by means of experiments and numerical simulations. Two aspects are evaluated: first, temporal unwrapping reliability, second, vibration-induced phase noise. The former is found to be a significant constraint, even for peak velocities well below the Nyquist velocity limit of the interferometer. Shorter sampling windows and higher framing rates are shown to increase the unwrapping success rate, but longer windows reduce the phase error. Three analytical criteria for determining the expected unwrapping success rate are proposed and compared.

© 2002 Optical Society of America

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References

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  1. X. C. de Lega, “Processing of nonstationary interference patterns: adapted phase-shifting algorithms and wavelet analysis. Application to dynamic deformation measurements by holographic and speckle interferometry,” Ph.D. dissertation (Ecole Polytechnique Fédérale de Lausanne, Lausanne, France, 1997).
  2. J. M. Huntley, G. H. Kaufmann, D. Kerr, “Phase-shifted dynamic speckle pattern interferometry at 1 kHz,” Appl. Opt. 38, 6556–6563 (1999).
    [CrossRef]
  3. P. Haible, M. P. Kothiyal, H. J. Tiziani, “Heterodyne temporal speckle-pattern interferometry,” Appl. Opt. 39, 114–117 (2000).
    [CrossRef]
  4. J. M. Kilpatrick, A. J. Moore, J. S. Barton, J. D. C. Jones, M. Reeves, C. Buckberry, “Measurement of complex surface deformation at audio acoustic frequencies by high-speed dynamic phase-stepped digital speckle pattern interferometry,” Opt. Lett. 25, 1068–1070 (2000).
    [CrossRef]
  5. J. M. Huntley, H. Saldner, “Temporal phase-unwrapping algorithm for automated interferogram analysis,” Appl. Opt. 32, 3047–3052 (1993).
    [CrossRef] [PubMed]
  6. See, for example,D. W. Robinson, G. T. Reid, eds., Interferogram Analysis (Institute of Physics, Bristol, UK, 1993).
  7. J. Schwider, R. Burow, K. E. Elssner, J. Grzanna, R. Spolaczyk, K. Merkel, “Digital wave-front measuring interferometry—some systematic error sources,” Appl. Opt. 22, 3421–3432 (1983).
    [CrossRef]
  8. P. de Groot, “Vibration in phase-shifting interferometry,” J. Opt. Soc. Am. A 12, 354–365 (1995).
    [CrossRef]
  9. P. de Groot, L. L. Deck, “Numerical simulations of vibration in phase-shifting interferometry,” Appl. Opt. 35, 2172–2178 (1996).
    [CrossRef] [PubMed]
  10. P. D. Ruiz, J. M. Huntley, J. Shen, C. R. Coggrave, G. H. Kaufmann, “Vibration-induced phase errors in high-speed phase-shifting speckle pattern interferometry,” Appl. Opt. 40, 2117–2125 (2001).
    [CrossRef]
  11. B. Bessason, C. Madshus, H. A. Frøystein, H. Kolbjørnsen, “Vibration criteria for metrology laboratories,” Meas. Sci. Technol. 10, 1009–1014 (1999).
    [CrossRef]
  12. E. E. Ungar, D. H. Sturz, C. H. Amick, “Vibration control design of high technology facilities,” Sound Vib. 7, 20–27 (1990).
  13. Y. Surrel, “Additive noise effect in digital phase detection,” Appl. Opt. 36, 271–276 (1997).
    [CrossRef] [PubMed]
  14. C. P. Brophy, “Effect of intensity error correlation on the computed phase of phase-shifting interferometry,” J. Opt. Soc. Am. A 7, 537–541 (1990).
    [CrossRef]
  15. P. de Groot, “Derivation of algorithms for phase-shifting interferometry using the concept of a data-sampling window,” Appl. Opt. 34, 4723–4730 (1995).
    [CrossRef]
  16. Y. Surrel, “Design of algorithms for phase measurements by the use of phase stepping,” Appl. Opt. 35, 51–60 (1996).
    [CrossRef] [PubMed]
  17. J. M. Huntley, H. O. Saldner, “Error-reduction methods for shape measurement by temporal phase unwrapping,” J. Opt. Soc. Am. A 14, 3188–3196 (1997).
    [CrossRef]
  18. J. D. Robson, An Introduction to Random Vibration (Edinburgh University Press, Edinburgh, Scotland, 1963).

2001 (1)

2000 (2)

1999 (2)

B. Bessason, C. Madshus, H. A. Frøystein, H. Kolbjørnsen, “Vibration criteria for metrology laboratories,” Meas. Sci. Technol. 10, 1009–1014 (1999).
[CrossRef]

J. M. Huntley, G. H. Kaufmann, D. Kerr, “Phase-shifted dynamic speckle pattern interferometry at 1 kHz,” Appl. Opt. 38, 6556–6563 (1999).
[CrossRef]

1997 (2)

1996 (2)

1995 (2)

1993 (1)

1990 (2)

E. E. Ungar, D. H. Sturz, C. H. Amick, “Vibration control design of high technology facilities,” Sound Vib. 7, 20–27 (1990).

C. P. Brophy, “Effect of intensity error correlation on the computed phase of phase-shifting interferometry,” J. Opt. Soc. Am. A 7, 537–541 (1990).
[CrossRef]

1983 (1)

Amick, C. H.

E. E. Ungar, D. H. Sturz, C. H. Amick, “Vibration control design of high technology facilities,” Sound Vib. 7, 20–27 (1990).

Barton, J. S.

Bessason, B.

B. Bessason, C. Madshus, H. A. Frøystein, H. Kolbjørnsen, “Vibration criteria for metrology laboratories,” Meas. Sci. Technol. 10, 1009–1014 (1999).
[CrossRef]

Brophy, C. P.

Buckberry, C.

Burow, R.

Coggrave, C. R.

de Groot, P.

de Lega, X. C.

X. C. de Lega, “Processing of nonstationary interference patterns: adapted phase-shifting algorithms and wavelet analysis. Application to dynamic deformation measurements by holographic and speckle interferometry,” Ph.D. dissertation (Ecole Polytechnique Fédérale de Lausanne, Lausanne, France, 1997).

Deck, L. L.

Elssner, K. E.

Frøystein, H. A.

B. Bessason, C. Madshus, H. A. Frøystein, H. Kolbjørnsen, “Vibration criteria for metrology laboratories,” Meas. Sci. Technol. 10, 1009–1014 (1999).
[CrossRef]

Grzanna, J.

Haible, P.

Huntley, J. M.

Jones, J. D. C.

Kaufmann, G. H.

Kerr, D.

Kilpatrick, J. M.

Kolbjørnsen, H.

B. Bessason, C. Madshus, H. A. Frøystein, H. Kolbjørnsen, “Vibration criteria for metrology laboratories,” Meas. Sci. Technol. 10, 1009–1014 (1999).
[CrossRef]

Kothiyal, M. P.

Madshus, C.

B. Bessason, C. Madshus, H. A. Frøystein, H. Kolbjørnsen, “Vibration criteria for metrology laboratories,” Meas. Sci. Technol. 10, 1009–1014 (1999).
[CrossRef]

Merkel, K.

Moore, A. J.

Reeves, M.

Robson, J. D.

J. D. Robson, An Introduction to Random Vibration (Edinburgh University Press, Edinburgh, Scotland, 1963).

Ruiz, P. D.

Saldner, H.

Saldner, H. O.

Schwider, J.

Shen, J.

Spolaczyk, R.

Sturz, D. H.

E. E. Ungar, D. H. Sturz, C. H. Amick, “Vibration control design of high technology facilities,” Sound Vib. 7, 20–27 (1990).

Surrel, Y.

Tiziani, H. J.

Ungar, E. E.

E. E. Ungar, D. H. Sturz, C. H. Amick, “Vibration control design of high technology facilities,” Sound Vib. 7, 20–27 (1990).

Appl. Opt. (9)

J. Opt. Soc. Am. A (3)

Meas. Sci. Technol. (1)

B. Bessason, C. Madshus, H. A. Frøystein, H. Kolbjørnsen, “Vibration criteria for metrology laboratories,” Meas. Sci. Technol. 10, 1009–1014 (1999).
[CrossRef]

Opt. Lett. (1)

Sound Vib. (1)

E. E. Ungar, D. H. Sturz, C. H. Amick, “Vibration control design of high technology facilities,” Sound Vib. 7, 20–27 (1990).

Other (3)

See, for example,D. W. Robinson, G. T. Reid, eds., Interferogram Analysis (Institute of Physics, Bristol, UK, 1993).

J. D. Robson, An Introduction to Random Vibration (Edinburgh University Press, Edinburgh, Scotland, 1963).

X. C. de Lega, “Processing of nonstationary interference patterns: adapted phase-shifting algorithms and wavelet analysis. Application to dynamic deformation measurements by holographic and speckle interferometry,” Ph.D. dissertation (Ecole Polytechnique Fédérale de Lausanne, Lausanne, France, 1997).

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Figures (5)

Fig. 1
Fig. 1

Dynamic high-speed phase-shifting speckle interferometer: F, frame store; P, Pockels cell; D, high-voltage driver; G, function generator; DA, digital-to-analog converter; SC, signal conditioner; HA, high-voltage amplifier; O, test object; BS, 90:10 beam splitters; M, mirrors; L, lenses.

Fig. 2
Fig. 2

Velocity spectrum of the random vibration used to excite the PZT transducer for a λ/2 rms amplitude of the disk displacement.

Fig. 3
Fig. 3

Detail of the unwrapped phase change measured at different locations of a disk submitted to piston random vibration for a rms displacement amplitude of σz = λ/4. Each curve corresponds to a phase of pixel clusters with a different starting phase. The mean phase over all the clusters is also shown in bold.

Fig. 4
Fig. 4

(a) Velocity of a surface excited with a random vibration with a rms phase amplitude of π/2 rad. Modulus, in arbitrary units, calculated from the windowed Fourier transform of the intensity signal by use of a Hanning window (b) with M = 32; (c) with M = 64. The framing rate of the camera corresponds to 1 kHz.

Fig. 5
Fig. 5

(a) Velocity of a surface excited with a random vibration with an rms phase amplitude of 2π rad. Modulus, in arbitrary units, calculated from the windowed Fourier transform of the intensity signal by using a Hanning window (b) with M = 32; (c) with M = 64. The framing rate of the camera corresponds to 1 kHz.

Tables (4)

Tables Icon

Table 1 Unwrapping Success Rate for Different Vibration-Displacement rms Amplitudes σz, Camera Framing Rates, and PS Algorithms

Tables Icon

Table 2 Comparison of Simulation and Analytical (by Using Criterion 3) Unwrapping Success Rates for a Range of Vibration Amplitudes σϕ and Hanning-Window Durations M

Tables Icon

Table 3 Average rms Phase Error (in radians) Obtained for Different PS Algorithms by Use of Simulated Data at Different Framing Rates

Tables Icon

Table 4 Root-Mean-Square Phase Error Averaged over Different Pixel Clusters and 500 Starting Points (Reference Frames) Evaluated from Experimental Data for Different PS Algorithms and Vibration Amplitudes

Equations (19)

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Sf=S00<ff0S0f0ff0<f,
Iτj=I01+VcosΦτj+Φvτj+ϕτj,
σϕ=4πλ σz.
Φˆwt= tan-1NtDt,
Nt=Imzt, Dt=Rezt,
zt=t=0M-1atI¯t+t+it=0M-1btI¯t+texp-iΔϕt,
at+ibt=wtexp-iΔϕt, t=0, 1, 2,  , M-1,
Ĩkt, t=t=0M-1I¯t+twtexp-2πiktt/N.
Φˆwt+3/2=tan-1S(I¯t-I¯t+3+I¯t+1-I¯t+23I¯t+1-I¯t+2-I¯t+I¯t+3)1/2I¯t+1+I¯t+2-I¯t-I¯t+3,
wt=12+12cos2πMt-M-12, t=0, 1,  , M-2.
dt=NINTΦˆwt-Φˆwt-1/2π, t=1,2,  , s,
νt=t=1tdt, t=1, 2,  , s,
ΔΦˆut, 0=Φˆwt-Φˆw0-2πνt, t=1, 2,  , s 
σt, tj=1Ll=0L-1ΔΦˆut, tj, l-ΔΦˆut, tj21/2,
ΔΦˆut, tj=1Ll=0L-1ΔΦˆut, tj, l.
σ¯=1s×sjt=1stj=1sjσt, tj,
μ=2f1f2f2Sfdff1f2Sfdf1/2 exp- νn22f1f2Sfdf,
P0=exp-μτ.
δkt=8Tfνz/λ,

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