Abstract

We present results from numerical simulations of a dynamic phase-shifting speckle interferometer used in the presence of mechanical vibrations. The simulation is based on a detailed mathematical model of the system, which is used to predict the expected frequency response of the rms measurement error, in the time-varying phase difference maps, as a result of vibration. The performance of different phase-shifting algorithms is studied over a range of vibrational frequencies. Phase-difference evaluation is performed by means of temporal phase shifting and temporal phase unwrapping. It is demonstrated that longer sampling windows and higher framing rates are preferred in order to reduce the phase-change error that is due to vibration. A numerical criterion for an upper limit on the length of time window for the phase-shifting algorithm is also proposed. The numerical results are finally compared with experimental data, acquired with a phase-shifting speckle interferometer of 1000 frames/s.

© 2001 Optical Society of America

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References

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  1. See, for example, D. W. Robinson, G. T. Reid, eds., Interferogram Analysis (Institute of Physics, Bristol, UK, 1993).
  2. I. Yamaguchi, J. Y. Liu, J. Kato, “Active phase-shifting interferometers for shape and deformation measurements,” Opt. Eng. 35, 2930–2937 (1996).
    [CrossRef]
  3. X. C. de Lega, “Processing of non-stationary interference patterns: adapted phase-shifting algorithms and wavelet analysis. Application to dynamic deformation measurements by holographic and speckle interferometry,” Ph.D. dissertation (l’Ecole Polytechnique Fédérale de Lausanne, Lausanne, Switzerland, 1997).
  4. J. M. Huntley, G. H. Kaufmann, D. Kerr, “Phase-shifted dynamic speckle pattern interferometry at 1 kHz,” Appl. Opt. 38, 6556–6563 (1999).
    [CrossRef]
  5. P. Haible, M. P. Kothiyal, H. J. Tiziani, “Heterodyne temporal speckle-pattern interferometry,” Appl. Opt. 39, 114–117 (2000).
    [CrossRef]
  6. J. M. Kilpatrick, A. J. Moore, J. S. Barton, J. D. C. Jones, M. Reeves, C. Buckberry, “Measurement of complex surface deformation by high-speed dynamic phase-stepped digital speckle pattern interferometry,” Opt. Lett. 25, 1068–1070 (2000).
    [CrossRef]
  7. P. de Groot, “Vibration in phase-shifting interferometry,” J. Opt. Soc. Am. A 12, 354–365 (1995).
    [CrossRef]
  8. P. de Groot, L. L. Deck, “Numerical simulations of vibration in phase-shifting interferometry,” Appl. Opt. 35, 2172–2178 (1996).
    [CrossRef] [PubMed]
  9. J. M. Huntley, H. Saldner, “Temporal phase-unwrapping algorithm for automated interferogram analysis,” Appl. Opt. 32, 3047–3052 (1993).
    [CrossRef] [PubMed]
  10. 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]
  11. Y. Surrel, “Design of algorithms for phase measurements by the use of phase stepping,” Appl. Opt. 35, 51–60 (1996).
    [CrossRef] [PubMed]
  12. 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]
  13. P. Hariharan, B. F. Oreb, T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. 26, 2504–2506 (1987).
    [CrossRef] [PubMed]
  14. J. M. Huntley, “Suppression of phase errors from vibration in phase-shifting interferometry,” J. Opt. Soc. Am. A 15, 2233–2241 (1998).
    [CrossRef]

2000 (2)

1999 (1)

1998 (1)

1996 (3)

1995 (2)

1993 (1)

1987 (1)

1983 (1)

Barton, J. S.

Buckberry, C.

Burow, R.

de Groot, P.

de Lega, X. C.

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

Deck, L. L.

Eiju, T.

Elssner, K. E.

Grzanna, J.

Haible, P.

Hariharan, P.

Huntley, J. M.

Jones, J. D. C.

Kato, J.

I. Yamaguchi, J. Y. Liu, J. Kato, “Active phase-shifting interferometers for shape and deformation measurements,” Opt. Eng. 35, 2930–2937 (1996).
[CrossRef]

Kaufmann, G. H.

Kerr, D.

Kilpatrick, J. M.

Kothiyal, M. P.

Liu, J. Y.

I. Yamaguchi, J. Y. Liu, J. Kato, “Active phase-shifting interferometers for shape and deformation measurements,” Opt. Eng. 35, 2930–2937 (1996).
[CrossRef]

Merkel, K.

Moore, A. J.

Oreb, B. F.

Reeves, M.

Saldner, H.

Schwider, J.

Spolaczyk, R.

Surrel, Y.

Tiziani, H. J.

Yamaguchi, I.

I. Yamaguchi, J. Y. Liu, J. Kato, “Active phase-shifting interferometers for shape and deformation measurements,” Opt. Eng. 35, 2930–2937 (1996).
[CrossRef]

Appl. Opt. (8)

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

Opt. Eng. (1)

I. Yamaguchi, J. Y. Liu, J. Kato, “Active phase-shifting interferometers for shape and deformation measurements,” Opt. Eng. 35, 2930–2937 (1996).
[CrossRef]

Opt. Lett. (1)

Other (2)

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

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

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Figures (9)

Fig. 1
Fig. 1

Expected rms phase-change error versus normalized vibration frequency for the five-frame Schwider–Hariharan formula.

Fig. 2
Fig. 2

Predicted rms phase-change error versus normalized vibration frequency for a PS formula with a rectangular sampling window: (a) 8-frame formula, (b) 16-frame formula (c), 32-frame formula.

Fig. 3
Fig. 3

Predicted rms phase-change error versus normalized vibration frequency for a PS algorithm based on a Hanning window and for different window widths: (a) 8-frame formula, (b) 16-frame formula, (c) 32-frame formula.

Fig. 4
Fig. 4

Dynamic PS speckle interferometer showing frame store (F), Pockels cell (P), high-voltage driver (D), function generator (G), test object (O), 90:10 beam splitters (BS), mirrors (M), and lenses (L).

Fig. 5
Fig. 5

Comparison of the phase change measured by means of the high-speed PS speckle interferometer and the four-frame PS algorithm (solid curve) and also by means of a LV (dashed curve), for a vibration frequency of 60 Hz and a vibration amplitude of ∼λ/10.

Fig. 6
Fig. 6

Experimental (circles) and simulated (curve) rms phase-change error obtained with a PS formula based on a 32-frame Hanning window, for vibration frequencies in the 8 Hz–1 kHz range and a vibration amplitude of ∼λ/10.

Fig. 7
Fig. 7

Modulus calculated from the windowed Fourier transform of the intensity signal for a vibration with an amplitude of π rad (horizontal axis shows the frame number). A PS formula based on a 32-frame Hanning window is used, together with a framing rate of 1000 s-1 and vibration frequencies of (a) 12.5 Hz and (b) 175 Hz.

Fig. 8
Fig. 8

Schematic illustration of large-M-value rms phase-change error plots for a rectangular window, showing the transition between cases T w T v and T w T v .

Fig. 9
Fig. 9

Velocity spectrum of vibrations measured on the laboratory floor (out of the optical bench) by means of a LV.

Equations (21)

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Iτj=I01+V cosΦτj+ϕτj+Φvτj,
Φvτj=ϕ0 cos2πfτj+α,
ϕτj=Δϕ REMNINTj+p+1/2p+N-1, N,
Īt=1pj=tptp+p-1 Iτj,  t=0, 1, 2,, Nt-1,
Φˆwt=tan-1NtDt,
Nt=Imzt,  Dt=Rezt,
zt=t=0M-1 atĪt+t+i t=0M-1 btĪt+t×exp-iΔϕt,
at+ibt=wtexp-iΔϕt, t=0, 1, 2,, M-1,
Ĩkt, t=t=0M-1 Īt+twtexp-2πiktt/N.
dt=NINTΦˆwt-Φˆwt-1/2π, t=1, 2,, s,
νt=t=1t dt,  t=1, 2,, s,
ΔΦˆut, 0=Φˆwt-Φˆw0-2πνt, t=1, 2,, s.
δΦl, t, 0=ΔΦˆul, t, 0-ΔΦl, t, 0.
δΦl, tj, ti=ΔΦˆul, tj, ti=ΔΦˆul, tj, 0-ΔΦˆul, ti, 0.
Ef=1LP2l=0L-1ti=0P-1tj=ti+1ti+PΔΦˆul, tj, ti-ΔΦˆu21/2,
ΔΦˆu=1LP2l=0L-1ti=0P-1tj=ti+1ti+P ΔΦˆul, ti, tj.
wt=12+12cos2πMt-M-12, t=0, 1,, M-2.
δkt=4ϕ0Tf/Tν.
Tf<Tν/4ϕ0.
Tf<Tν/Mϕ0.
Tf<Tν/4.

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