Abstract

Incorporation of two phase-shifting devices in a holographic moiré configuration not only renders the interferometer compatible with automated measurements but also allows for simultaneous measurement of multiple phase information in the interferometer. However, simultaneous handling of multiple phase steps and subsequent simultaneous determination of multiple phase distributions requires the introduction of novel tools in phase-shifting interferometry. In this context, the aim of this paper is to propose a subspace invariance approach to address these issues. This approach takes advantage of the rotational invariance of signal subspaces spanned by two temporally displaced data sets formed from the intensity fringes recorded temporally on pixels of the CCD camera. The method first identifies the arbitrary phase steps imparted to the piezoactuator devices. The estimated phase steps are subsequently applied in the linear Vandermonde system of equations to determine the phase distributions. The method also allows for handling nonsinusoidal wavefronts. Since the phase steps are extracted at every point on the interferogram, the method is applicable to configurations that use spherical beams. The robustness of the method is investigated by adding white Gaussian noise during the simulations.

© 2005 Optical Society of America

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References

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  1. P. Carré, “Installation et utilisation du comparateur photoélectrique et interférentiel du Bureau International des Poids et Mesures,” Metrologia 2, 13–23 (1966).
    [CrossRef]
  2. J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, D. J. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. 13, 2693–2703 (1974).
    [CrossRef] [PubMed]
  3. K. Creath, “Phase-shifting holographic interferometry,” Holographic Interferometry, P. K. Rastogi, ed. (Springer Series in Optical Sciences, 1994), Vol. 68, pp. 109–150.
    [CrossRef]
  4. T. Kreis, Holographic Interferometry Principles and Methods (Akademie Verlag, 1996) pp. 101–170.
  5. J. E. Greivenkamp, J. H. Bruning, Phase shifting interferometryOptical Shop Testing, D. Malacara, ed. (Wiley, 1992) pp. 501-598.
  6. 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] [PubMed]
  7. Y. Zhu, T. Gemma, “Method for designing error-compensating phase-calculation algorithms for phase-shifting interferometry,” Appl. Opt. 40, 4540–4546 (2001).
    [CrossRef]
  8. 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]
  9. J. Schwider, O. Falkenstorfer, H. Schreiber, A. Zoller, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. (Bellingham) 32, 1883–1885 (1993).
    [CrossRef]
  10. Y. Surrel, “Phase stepping: a new self-calibrating algorithm,” Appl. Opt. 32, 3598–3600 (1993).
    [CrossRef] [PubMed]
  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. van Wingerden, H. J. Frankena, C. Smorenburg, “Linear approximation for measurement errors in phase shifting interferometry,” Appl. Opt. 30, 2718–2729 (1991).
    [CrossRef] [PubMed]
  13. K. G. Larkin, B. F. Oreb, “Design and assessment of symmetrical phase-shifting algorithms,” J. Opt. Soc. Am. A 9, 1740–1748 (1992).
    [CrossRef]
  14. K. Hibino, B. F. Oreb, D. I. Farrant, K. G. Larkin, “Phase shifting for nonsinusoidal waveforms with phase-shift errors,” J. Opt. Soc. Am. A 12, 761–768 (1995).
    [CrossRef]
  15. Y.-Y. Cheng, J. C. Wyant, “Phase-shifter calibration in phase-shifting interferometry,” Appl. Opt. 24, 3049–3052 (1985).
    [CrossRef]
  16. B. Zhao, “A statistical method for fringe intensity-correlated error in phase-shifting measurement: The effect of quantization error on the N-bucket algorithm,” Meas. Sci. Technol. 8, 147–153 (1997).
    [CrossRef]
  17. B. Zhao, Y. Surrel, “Effect of quantization error on the computed phase of phase-shifting measurements,” Appl. Opt. 36, 2070–2075 (1997).
    [CrossRef] [PubMed]
  18. R. Józwicki, M. Kujawinska, M. Salbut, “New contra old wavefront measurement concepts for interferometric optical testing,” Opt. Eng. (Bellingham) 31, 422–433 (1992).
    [CrossRef]
  19. P. de Groot, “Vibration in phase-shifting interferometry,” J. Opt. Soc. Am. A 12, 354–365 (1995).
    [CrossRef]
  20. P. de Groot, L. L. Deck, “Numerical simulations of vibration in phase-shifting interferometry,” Appl. Opt. 35, 2172–2178 (1996).
    [CrossRef] [PubMed]
  21. C. Rathjen, “Statistical properties of phase-shift algorithms,” J. Opt. Soc. Am. A 12, 1997–2008 (1995).
    [CrossRef]
  22. P. L. Wizinowich, “Phase-shifting interferometry in the presence of vibration: A new algorithm and system,” Appl. Opt. 29, 3271–3279 (1990).
    [CrossRef] [PubMed]
  23. C. Joenathan, B. M. Khorana, “Phase measurement by differentiating interferometric fringes,” J. Mod. Opt. 39, 2075–2087 (1992).
    [CrossRef]
  24. K. Kinnnstaetter, A. W. Lohmann, J Schwider, N. Streibl, “Accuracy of phase shifting interferometry,” Appl. Opt. 27, 5082–5089 (1988).
    [CrossRef]
  25. C. J. Morgan, “Least squares estimation in phase-measurement interferometry,” Opt. Lett. 7, 368–370 (1982).
    [CrossRef] [PubMed]
  26. J. E. Grievenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. (Bellingham) 23, 350–352 (1984).
  27. P. K. Rastogi, “Phase shifting applied to four-wave holographic interferometers,” Appl. Opt. 31, 1680–1681 (1992).
    [CrossRef] [PubMed]
  28. P. K. Rastogi, “Phase-shifting holographic moiré: Phase-shifter error-insensitive algorithms for the extraction of the difference and sum of phases in holographic moiré,” Appl. Opt. 32, 3669–3675 (1993).
    [CrossRef] [PubMed]
  29. R. Roy, T. Kailath, “ESPRIT-estimation of signal parameters via rotational invariance techniques,” IEEE Trans. Acoust., Speech, Signal Process. 37, 984–995 (1989).
    [CrossRef]
  30. J. J. Fuchs, “Estimating the number of sinusoids in additive white noise,” IEEE Trans. Acoust., Speech, Signal Process. 36, 1846–1853 (1988).
    [CrossRef]
  31. P. K. Rastogi, M. Spajer, J. Monneret, “In-plane deformation measurement using holographic moiré,” Opt. Lasers Eng. 2, 79–103 (1981).
    [CrossRef]
  32. T. Söderström, P. Stoica, “Accuracy of high-order Yule–Walker methods for frequency estimation of complex sine waves,” IEE Proc. F, Radar Signal Process. 140, 71–80 (1993).
    [CrossRef]
  33. R. Kumaresan, D. W. Tufts, “Estimating the angles of arrival of multiple plane waves,” IEEE Trans. Aerosp. Electron. Syst. AES-19, 134–139 (1983).
    [CrossRef]
  34. B. D. Rao, K. V.S. Hari, “Weighted subspace methods and spatial smoothing: analysis and comparison,” IEEE Trans. Signal Process. 41, 788–803 (1993).
    [CrossRef]

2001 (1)

1997 (2)

B. Zhao, “A statistical method for fringe intensity-correlated error in phase-shifting measurement: The effect of quantization error on the N-bucket algorithm,” Meas. Sci. Technol. 8, 147–153 (1997).
[CrossRef]

B. Zhao, Y. Surrel, “Effect of quantization error on the computed phase of phase-shifting measurements,” Appl. Opt. 36, 2070–2075 (1997).
[CrossRef] [PubMed]

1996 (2)

1995 (3)

1993 (5)

P. K. Rastogi, “Phase-shifting holographic moiré: Phase-shifter error-insensitive algorithms for the extraction of the difference and sum of phases in holographic moiré,” Appl. Opt. 32, 3669–3675 (1993).
[CrossRef] [PubMed]

T. Söderström, P. Stoica, “Accuracy of high-order Yule–Walker methods for frequency estimation of complex sine waves,” IEE Proc. F, Radar Signal Process. 140, 71–80 (1993).
[CrossRef]

B. D. Rao, K. V.S. Hari, “Weighted subspace methods and spatial smoothing: analysis and comparison,” IEEE Trans. Signal Process. 41, 788–803 (1993).
[CrossRef]

J. Schwider, O. Falkenstorfer, H. Schreiber, A. Zoller, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. (Bellingham) 32, 1883–1885 (1993).
[CrossRef]

Y. Surrel, “Phase stepping: a new self-calibrating algorithm,” Appl. Opt. 32, 3598–3600 (1993).
[CrossRef] [PubMed]

1992 (4)

K. G. Larkin, B. F. Oreb, “Design and assessment of symmetrical phase-shifting algorithms,” J. Opt. Soc. Am. A 9, 1740–1748 (1992).
[CrossRef]

R. Józwicki, M. Kujawinska, M. Salbut, “New contra old wavefront measurement concepts for interferometric optical testing,” Opt. Eng. (Bellingham) 31, 422–433 (1992).
[CrossRef]

P. K. Rastogi, “Phase shifting applied to four-wave holographic interferometers,” Appl. Opt. 31, 1680–1681 (1992).
[CrossRef] [PubMed]

C. Joenathan, B. M. Khorana, “Phase measurement by differentiating interferometric fringes,” J. Mod. Opt. 39, 2075–2087 (1992).
[CrossRef]

1991 (1)

1990 (1)

1989 (1)

R. Roy, T. Kailath, “ESPRIT-estimation of signal parameters via rotational invariance techniques,” IEEE Trans. Acoust., Speech, Signal Process. 37, 984–995 (1989).
[CrossRef]

1988 (2)

J. J. Fuchs, “Estimating the number of sinusoids in additive white noise,” IEEE Trans. Acoust., Speech, Signal Process. 36, 1846–1853 (1988).
[CrossRef]

K. Kinnnstaetter, A. W. Lohmann, J Schwider, N. Streibl, “Accuracy of phase shifting interferometry,” Appl. Opt. 27, 5082–5089 (1988).
[CrossRef]

1987 (1)

1985 (1)

1984 (1)

J. E. Grievenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. (Bellingham) 23, 350–352 (1984).

1983 (2)

R. Kumaresan, D. W. Tufts, “Estimating the angles of arrival of multiple plane waves,” IEEE Trans. Aerosp. Electron. Syst. AES-19, 134–139 (1983).
[CrossRef]

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] [PubMed]

1982 (1)

1981 (1)

P. K. Rastogi, M. Spajer, J. Monneret, “In-plane deformation measurement using holographic moiré,” Opt. Lasers Eng. 2, 79–103 (1981).
[CrossRef]

1974 (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]

Brangaccio, D. J.

Bruning, J. H.

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-shifting holographic interferometry,” Holographic Interferometry, P. K. Rastogi, ed. (Springer Series in Optical Sciences, 1994), Vol. 68, pp. 109–150.
[CrossRef]

de Groot, P.

Deck, L. L.

Eiju, T.

Elssner, K. E.

Falkenstorfer, O.

J. Schwider, O. Falkenstorfer, H. Schreiber, A. Zoller, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. (Bellingham) 32, 1883–1885 (1993).
[CrossRef]

Farrant, D. I.

Frankena, H. J.

Fuchs, J. J.

J. J. Fuchs, “Estimating the number of sinusoids in additive white noise,” IEEE Trans. Acoust., Speech, Signal Process. 36, 1846–1853 (1988).
[CrossRef]

Gallagher, J. E.

Gemma, T.

Greivenkamp, J. E.

J. E. Greivenkamp, J. H. Bruning, Phase shifting interferometryOptical Shop Testing, D. Malacara, ed. (Wiley, 1992) pp. 501-598.

Grievenkamp, J. E.

J. E. Grievenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. (Bellingham) 23, 350–352 (1984).

Grzanna, J.

Hari, K. V.S.

B. D. Rao, K. V.S. Hari, “Weighted subspace methods and spatial smoothing: analysis and comparison,” IEEE Trans. Signal Process. 41, 788–803 (1993).
[CrossRef]

Hariharan, P.

Herriott, D. R.

Hibino, K.

Joenathan, C.

C. Joenathan, B. M. Khorana, “Phase measurement by differentiating interferometric fringes,” J. Mod. Opt. 39, 2075–2087 (1992).
[CrossRef]

Józwicki, R.

R. Józwicki, M. Kujawinska, M. Salbut, “New contra old wavefront measurement concepts for interferometric optical testing,” Opt. Eng. (Bellingham) 31, 422–433 (1992).
[CrossRef]

Kailath, T.

R. Roy, T. Kailath, “ESPRIT-estimation of signal parameters via rotational invariance techniques,” IEEE Trans. Acoust., Speech, Signal Process. 37, 984–995 (1989).
[CrossRef]

Khorana, B. M.

C. Joenathan, B. M. Khorana, “Phase measurement by differentiating interferometric fringes,” J. Mod. Opt. 39, 2075–2087 (1992).
[CrossRef]

Kinnnstaetter, K.

Kreis, T.

T. Kreis, Holographic Interferometry Principles and Methods (Akademie Verlag, 1996) pp. 101–170.

Kujawinska, M.

R. Józwicki, M. Kujawinska, M. Salbut, “New contra old wavefront measurement concepts for interferometric optical testing,” Opt. Eng. (Bellingham) 31, 422–433 (1992).
[CrossRef]

Kumaresan, R.

R. Kumaresan, D. W. Tufts, “Estimating the angles of arrival of multiple plane waves,” IEEE Trans. Aerosp. Electron. Syst. AES-19, 134–139 (1983).
[CrossRef]

Larkin, K. G.

Lohmann, A. W.

Merkel, K.

Monneret, J.

P. K. Rastogi, M. Spajer, J. Monneret, “In-plane deformation measurement using holographic moiré,” Opt. Lasers Eng. 2, 79–103 (1981).
[CrossRef]

Morgan, C. J.

Oreb, B. F.

Rao, B. D.

B. D. Rao, K. V.S. Hari, “Weighted subspace methods and spatial smoothing: analysis and comparison,” IEEE Trans. Signal Process. 41, 788–803 (1993).
[CrossRef]

Rastogi, P. K.

Rathjen, C.

Rosenfeld, D. P.

Roy, R.

R. Roy, T. Kailath, “ESPRIT-estimation of signal parameters via rotational invariance techniques,” IEEE Trans. Acoust., Speech, Signal Process. 37, 984–995 (1989).
[CrossRef]

Salbut, M.

R. Józwicki, M. Kujawinska, M. Salbut, “New contra old wavefront measurement concepts for interferometric optical testing,” Opt. Eng. (Bellingham) 31, 422–433 (1992).
[CrossRef]

Schreiber, H.

J. Schwider, O. Falkenstorfer, H. Schreiber, A. Zoller, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. (Bellingham) 32, 1883–1885 (1993).
[CrossRef]

Schwider, J

Schwider, J.

J. Schwider, O. Falkenstorfer, H. Schreiber, A. Zoller, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. (Bellingham) 32, 1883–1885 (1993).
[CrossRef]

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] [PubMed]

Smorenburg, C.

Söderström, T.

T. Söderström, P. Stoica, “Accuracy of high-order Yule–Walker methods for frequency estimation of complex sine waves,” IEE Proc. F, Radar Signal Process. 140, 71–80 (1993).
[CrossRef]

Spajer, M.

P. K. Rastogi, M. Spajer, J. Monneret, “In-plane deformation measurement using holographic moiré,” Opt. Lasers Eng. 2, 79–103 (1981).
[CrossRef]

Spolaczyk, R.

Stoica, P.

T. Söderström, P. Stoica, “Accuracy of high-order Yule–Walker methods for frequency estimation of complex sine waves,” IEE Proc. F, Radar Signal Process. 140, 71–80 (1993).
[CrossRef]

Streibl, N.

Surrel, Y.

Tufts, D. W.

R. Kumaresan, D. W. Tufts, “Estimating the angles of arrival of multiple plane waves,” IEEE Trans. Aerosp. Electron. Syst. AES-19, 134–139 (1983).
[CrossRef]

van Wingerden, J.

White, A. D.

Wizinowich, P. L.

Wyant, J. C.

Zhao, B.

B. Zhao, “A statistical method for fringe intensity-correlated error in phase-shifting measurement: The effect of quantization error on the N-bucket algorithm,” Meas. Sci. Technol. 8, 147–153 (1997).
[CrossRef]

B. Zhao, Y. Surrel, “Effect of quantization error on the computed phase of phase-shifting measurements,” Appl. Opt. 36, 2070–2075 (1997).
[CrossRef] [PubMed]

Zhu, Y.

Zoller, A.

J. Schwider, O. Falkenstorfer, H. Schreiber, A. Zoller, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. (Bellingham) 32, 1883–1885 (1993).
[CrossRef]

Appl. Opt. (14)

J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, D. J. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. 13, 2693–2703 (1974).
[CrossRef] [PubMed]

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] [PubMed]

Y. Zhu, T. Gemma, “Method for designing error-compensating phase-calculation algorithms for phase-shifting interferometry,” Appl. Opt. 40, 4540–4546 (2001).
[CrossRef]

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]

Y. Surrel, “Phase stepping: a new self-calibrating algorithm,” Appl. Opt. 32, 3598–3600 (1993).
[CrossRef] [PubMed]

Y. Surrel, “Design of algorithms for phase measurements by the use of phase stepping,” Appl. Opt. 35, 51–60 (1996).
[CrossRef] [PubMed]

J. van Wingerden, H. J. Frankena, C. Smorenburg, “Linear approximation for measurement errors in phase shifting interferometry,” Appl. Opt. 30, 2718–2729 (1991).
[CrossRef] [PubMed]

Y.-Y. Cheng, J. C. Wyant, “Phase-shifter calibration in phase-shifting interferometry,” Appl. Opt. 24, 3049–3052 (1985).
[CrossRef]

B. Zhao, Y. Surrel, “Effect of quantization error on the computed phase of phase-shifting measurements,” Appl. Opt. 36, 2070–2075 (1997).
[CrossRef] [PubMed]

P. de Groot, L. L. Deck, “Numerical simulations of vibration in phase-shifting interferometry,” Appl. Opt. 35, 2172–2178 (1996).
[CrossRef] [PubMed]

P. L. Wizinowich, “Phase-shifting interferometry in the presence of vibration: A new algorithm and system,” Appl. Opt. 29, 3271–3279 (1990).
[CrossRef] [PubMed]

K. Kinnnstaetter, A. W. Lohmann, J Schwider, N. Streibl, “Accuracy of phase shifting interferometry,” Appl. Opt. 27, 5082–5089 (1988).
[CrossRef]

P. K. Rastogi, “Phase shifting applied to four-wave holographic interferometers,” Appl. Opt. 31, 1680–1681 (1992).
[CrossRef] [PubMed]

P. K. Rastogi, “Phase-shifting holographic moiré: Phase-shifter error-insensitive algorithms for the extraction of the difference and sum of phases in holographic moiré,” Appl. Opt. 32, 3669–3675 (1993).
[CrossRef] [PubMed]

IEE Proc. F, Radar Signal Process. (1)

T. Söderström, P. Stoica, “Accuracy of high-order Yule–Walker methods for frequency estimation of complex sine waves,” IEE Proc. F, Radar Signal Process. 140, 71–80 (1993).
[CrossRef]

IEEE Trans. Acoust., Speech, Signal Process. (2)

R. Roy, T. Kailath, “ESPRIT-estimation of signal parameters via rotational invariance techniques,” IEEE Trans. Acoust., Speech, Signal Process. 37, 984–995 (1989).
[CrossRef]

J. J. Fuchs, “Estimating the number of sinusoids in additive white noise,” IEEE Trans. Acoust., Speech, Signal Process. 36, 1846–1853 (1988).
[CrossRef]

IEEE Trans. Aerosp. Electron. Syst. (1)

R. Kumaresan, D. W. Tufts, “Estimating the angles of arrival of multiple plane waves,” IEEE Trans. Aerosp. Electron. Syst. AES-19, 134–139 (1983).
[CrossRef]

IEEE Trans. Signal Process. (1)

B. D. Rao, K. V.S. Hari, “Weighted subspace methods and spatial smoothing: analysis and comparison,” IEEE Trans. Signal Process. 41, 788–803 (1993).
[CrossRef]

J. Mod. Opt. (1)

C. Joenathan, B. M. Khorana, “Phase measurement by differentiating interferometric fringes,” J. Mod. Opt. 39, 2075–2087 (1992).
[CrossRef]

J. Opt. Soc. Am. A (4)

Meas. Sci. Technol. (1)

B. Zhao, “A statistical method for fringe intensity-correlated error in phase-shifting measurement: The effect of quantization error on the N-bucket algorithm,” Meas. Sci. Technol. 8, 147–153 (1997).
[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]

Opt. Eng. (Bellingham) (3)

J. E. Grievenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. (Bellingham) 23, 350–352 (1984).

R. Józwicki, M. Kujawinska, M. Salbut, “New contra old wavefront measurement concepts for interferometric optical testing,” Opt. Eng. (Bellingham) 31, 422–433 (1992).
[CrossRef]

J. Schwider, O. Falkenstorfer, H. Schreiber, A. Zoller, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. (Bellingham) 32, 1883–1885 (1993).
[CrossRef]

Opt. Lasers Eng. (1)

P. K. Rastogi, M. Spajer, J. Monneret, “In-plane deformation measurement using holographic moiré,” Opt. Lasers Eng. 2, 79–103 (1981).
[CrossRef]

Opt. Lett. (1)

Other (3)

K. Creath, “Phase-shifting holographic interferometry,” Holographic Interferometry, P. K. Rastogi, ed. (Springer Series in Optical Sciences, 1994), Vol. 68, pp. 109–150.
[CrossRef]

T. Kreis, Holographic Interferometry Principles and Methods (Akademie Verlag, 1996) pp. 101–170.

J. E. Greivenkamp, J. H. Bruning, Phase shifting interferometryOptical Shop Testing, D. Malacara, ed. (Wiley, 1992) pp. 501-598.

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Figures (7)

Fig. 1
Fig. 1

Schematic of the optical setup in holographic moiré.

Fig. 2
Fig. 2

Fringe contours corresponding to Eq. (20) for κ = 2 and SNRs (a) 10 dB and (b) 60 dB.

Fig. 3
Fig. 3

Plots for estimating the number of harmonics present in a signal with SNR = 10 dB . (a) Typical plot when Fourier transform is applied temporally at pixel ( x , y ) for N = 100 sample points. (b) Plot for one-dimensional FFT applied spatially across any row on the data frame. To separate the frequencies, the carrier has been added to the signal. (c) Diagonal entries for the matrix M obtained from singular value decomposition of R I in Eq. (4). The value of κ can be reliably estimated from this plot.

Fig. 4
Fig. 4

Plots for phase steps α and β (in degrees) obtained by using the forward approach at an arbitrary pixel location on a data frame for different values of N and m.

Fig. 5
Fig. 5

Plots of phase steps α and β (in degrees) obtained by using the forward–backward approach at an arbitrary pixel location on a data frame for different values of N and m.

Fig. 6
Fig. 6

Typical errors in computation of phase distributions (a) φ 1 and (b) φ 2 , in radians, for phase steps obtained from Fig. 5(e) for 30 dB noise.

Fig. 7
Fig. 7

Typical wrapped phase φ 1 (solid curve) and φ 2 (dashed curve) in radians, for phase steps obtained from Fig. 5(e) for 30 dB noise.

Equations (41)

Equations on this page are rendered with MathJax. Learn more.

I ( t ) = I d c + k = 1 κ a k exp [ i k ( φ 1 + t α ) ] + k = 1 κ a k exp [ i k ( φ 1 + t α ) ] + k = 1 κ b k exp [ i k ( φ 2 + t β ) ] + k = 1 κ b k exp [ i k ( φ 2 + t β ) ]
t = 0 , 1 , 2 , , m , , N 1 ,
I ( t ) = I d c + k = 1 κ k u k t + k = 1 κ k * ( u k * ) t + k = 1 κ k v k t + k = 1 κ k * ( v k * ) t + η ( t ) for t = 0 , 1 , , m , , N 1 ,
r ( p ) = E [ I ( t ) I * ( t p ) ] = n = 0 4 κ A n 2 exp ( i ω n p ) + σ 2 δ p , 0 ,
R I = E { [ I * ( t 1 ) I * ( t 2 ) . . I * ( t m ) ] [ I ( t 1 ) I ( t 2 ) . . I ( t m ) ] } = [ r ( 0 ) r * ( 1 ) . . r * ( m 1 ) r ( 1 ) . . . . . . . . . . . . . r * ( 1 ) r ( m 1 ) r ( m 2 ) . . r ( 0 ) ] ,
A m × n = [ a ( ω 0 ) a ( ω 1 ) . a ( ω 4 κ ) ] ,
P = [ A 0 2 0 . . 0 0 A 1 2 . . . . . . . . . . . . 0 0 0 . . A 4 κ 2 ] .
λ 1 λ 2 λ n σ 2 ,
λ n + 1 = λ n + 2 = λ m = σ 2 .
R I G = G [ λ n + 1 0 . . 0 0 λ n + 1 . . . . . . . . . . . . 0 0 0 . . λ m ] = σ 2 G = APA c G + σ 2 G .
A c G = 0
Λ = [ λ 1 σ 2 0 . . 0 0 λ 2 σ 2 . . . . . . . . . . . . 0 0 0 . . λ n σ 2 ] .
R I S = S [ λ 1 0 . . 0 0 λ 2 . . . . . . . . . . . . 0 0 0 . . λ n ] = APA c S + σ 2 S .
A 1 ( m 1 ) × n = [ I ( m 1 ) × ( m 1 ) 0 ( m 1 ) × 1 ] A ,
A 2 ( m 1 ) × n = [ 0 ( m 1 ) × 1 I ( m 1 ) × ( m 1 ) ] A ,
S 1 ( m 1 ) × n = [ I ( m 1 ) × ( m 1 ) 0 ( m 1 ) × 1 ] S ,
S 2 ( m 1 ) × n = [ 0 ( m 1 ) × 1 I ( m 1 ) × ( m 1 ) ] S .
S 2 = A 2 Γ = A 1 D Γ = S 1 Γ 1 D Γ = S 1 Υ ,
Υ = ( S 1 c S 1 ) 1 S 1 c S 2 .
Υ ̂ = ( S ̂ 1 c S ̂ 1 ) 1 S ̂ 1 c S ̂ 2 .
I ( x , y ; t ) = I d c + a 1 exp [ i ( φ 1 + t α ) ] + a 1 exp [ i ( φ 1 + t α ) ] + a 2 exp [ 2 i ( φ 1 + t α ) ] + a 2 exp [ 2 i ( φ 1 + t α ) ] + b 1 exp [ i ( φ 2 + t β ) ] + b 1 exp [ i ( φ 2 + t β ) ] + b 2 exp [ 2 i ( φ 2 + t β ) ] + b 2 exp [ 2 i ( φ 2 + t β ) ] + η ( t ) for t = 0 , 1 , , N 1 .
φ 1 ( x , y ) = 2 π λ ( p x ) 2 + ( p y ) 2 ,
φ 2 ( x , y ) = 2 π λ ( p x ) 2 + ( p y ) 2 ,
R ̂ I = 1 N t = m N [ I * ( t 1 ) I * ( t 2 ) . . I * ( t m ) ] [ I ( t 1 ) I ( t 2 ) . . I ( t m ) ] ,
R ̂ I = 1 2 N t = m N { [ I * ( t 1 ) I * ( t 2 ) . . I * ( t m ) ] [ I ( t 1 ) I ( t 2 ) . . I ( t m ) ] + [ I * ( t m ) . . I * ( t 2 ) I * ( t 1 ) ] [ I ( t m ) . . I ( t 2 ) I ( t 1 ) ] } .
[ exp ( i κ α 0 ) exp ( i κ α 0 ) exp ( i κ β 0 ) . . . . 1 exp ( i κ α 1 ) exp ( i κ α 1 ) exp ( i κ β 1 ) . . . . 1 . . . . . . . . . . . . . . . . exp ( i κ α N 1 ) exp ( i κ α N 1 ) exp ( i κ β N 1 ) . . . . 1 ] [ κ κ * κ . I d c ] = [ I 0 I 1 I 2 . I N 1 ] ,
I ( t ) = I d c + k = 1 κ a k exp ( i k φ 1 ) exp ( i α k t ) + k = 1 κ a k exp ( i k φ 1 ) exp ( i α k t ) + k = 1 κ b k exp ( i k φ 2 ) exp ( i β k t ) + k = 1 κ b k exp ( i k φ 2 ) exp ( i β k t ) + η ( t ) for t = 0 , 1 , 2 , m , , N 1 .
r ( p ) = E [ I ( t ) I * ( t p ) ] .
I ( t ) = I d c + a 1 exp ( i φ 1 ) exp ( i α t ) + a 1 exp ( i φ 1 ) exp ( i α t ) + b 1 exp ( i φ 2 ) exp ( i β t ) + b 1 exp ( i φ 2 ) exp ( i β t ) + η ( t ) .
I * ( t p ) = I d c + a 1 exp ( i φ 1 ) exp [ i α ( t p ) ] + a 1 exp ( i φ 1 ) exp [ i α ( t p ) ] + b 1 exp ( i φ 2 ) exp [ i β ( t p ) ] + b 1 exp ( i φ 2 ) exp [ i β ( t p ) ] + η * ( t p ) .
r ( p ) = E [ I d c 2 + c 1 + exp ( i α p ) ( a 1 2 + c 2 ) + exp ( i α p ) ( a 1 2 + c 3 ) + exp ( i β p ) ( b 1 2 + c 4 ) + exp ( i β p ) ( b 1 2 + c 5 ) + η ( t ) η * ( t p ) ] ,
c 1 = I d c a 1 exp [ i ( φ 1 + α t ) ] + I d c a 1 exp [ i ( φ 1 + α t ) ] + I d c b 1 exp [ i ( φ 2 + β t ) ] + I d c b 1 exp [ i ( φ 2 + β t ) ]
c 2 = I d c a 1 exp [ i ( φ 1 + α t ) ] + a 1 2 exp [ 2 i ( φ 1 + α t ) ] + a 1 b 1 exp { i [ φ 1 φ 2 + t ( α β ) ] } + a 1 b 1 exp { i [ ( φ 1 + φ 2 + t ( α + β ) ) ] } ,
c 3 = I d c a 1 exp [ i ( φ 1 + α t ) ] + a 1 2 exp [ 2 i ( φ 1 + α t ) ] + a 1 b 1 exp { i [ φ 1 + φ 2 + t ( α + β ) ] } + a 1 b 1 exp { i [ φ 1 φ 2 + t ( α β ) ] } ,
c 4 = I d c b 1 exp [ i ( φ 2 + β t ) ] + b 1 2 exp [ 2 i ( φ 1 + α t ) ] + a 1 b 1 exp { i [ φ 1 φ 2 + t ( α β ) ] } + a 1 b 1 exp { i [ φ 1 + φ 2 + t ( α + β ) ] } ,
c 5 = I d c b 1 exp [ i ( φ 2 + β t ) ] + b 1 2 exp [ 2 i ( φ 2 + β t ) ] + a 1 b 1 exp { i [ φ 1 + φ 2 + t ( α + β ) ] } + a 1 b 1 exp { i [ φ 1 φ 2 + t ( α β ) ] } .
E ( I d c 2 + c 1 ) = A 0 2 , E ( a 1 2 + c 2 ) = A 1 2 , E ( a 1 2 + c 3 ) = A 2 2 , E ( b 1 2 + c 4 ) = A 3 2 , E ( b 1 2 + c 5 ) = A 4 2 .
r ( p ) = A 0 2 + A 1 2 exp ( i α p ) + A 2 2 exp ( i α p ) + A 3 2 exp ( i β p ) + A 4 2 exp ( i β p ) + σ 2 δ p , 0 .
E [ η ( k ) η * ( j ) ] = σ 2 δ k , j ,
0 2 π exp ( i ψ ) d ψ = 0 .
r ( p ) = E [ I ( t ) I * ( t p ) ] = n = 0 4 κ A n 2 exp ( i ω n p ) + σ 2 δ p , 0 .

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