Abstract

The advantages of spatial phase shifting (SPS) compared with temporal phase shifting in the field of electronic speckle pattern interferometry are described. Some periodic phase reconstruction errors occurring in SPS are discussed. It is shown that these errors become minimal for a spatial phase-shift angle of 2π/3. Furthermore, a modified phase reconstruction formula is presented by which the noise in the reconstructed phase map is reduced.

© 1997 Optical Society of America

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References

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  1. J. E. Greivenkamp, J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing, D. Malacara ed. (Wiles, New York, 1992), pp. 501–598.
  2. K. Creath, “Temporal phase measurement methods,” in Interferogram Analysis, D. W. Robinson, G. T. Reid, eds. (Institute of Physics Publishing, Bristol, 1993), pp. 94–140.
  3. S. Nakadate, H. Saito, “Fringe scanning speckle-pattern interferometry,” Appl. Opt. 24, 2172–2180 (1985).
    [CrossRef] [PubMed]
  4. K. Creath, “Phase-shifting speckle interferometry,” Appl. Opt. 24, 3053–3058 (1985).
    [CrossRef] [PubMed]
  5. R. Smythe, R. Moore, “Instantaneous phase measuring interferometry,” Opt. Eng. 23, 361–364 (1984).
    [CrossRef]
  6. O. Y. Kwon, “Multichannel phase-shifted interferometer,” Opt. Lett. 9, 59–61 (1984).
    [CrossRef] [PubMed]
  7. M. Kujawinska, “Spatial phase measurement methods,” in Interferogram Analysis, D. W. Robinson, G. T. Reid, eds. (Institute of Physics Publishing, Bristol, 1993), pp. 141–193.
  8. Y. Ichioka, M. Inuiya, “Direct phase detecting system,” Appl. Opt. 11, 1507–1514 (1972).
    [CrossRef] [PubMed]
  9. L. Mertz, “Real-time fringe-pattern analysis,” Appl. Opt. 22, 1535–1539 (1983).
    [CrossRef] [PubMed]
  10. D. M. Shough, O. Y. Kwon, D. F. Leary, “High-speed interferometric measurements of aerodynamic phenomena,” in Propagation of High-Energy Laser Beams Through the Earth’s Atmosphere, P. B. Ulrich, L. E. Wilson, eds., Proc. SPIE Vol. 1221, 394–403 (1990).
    [CrossRef]
  11. M. Küchel, “Verfahren zur Messung eines phasenmodulierten Signals,” Offenlegungsschrift Deutsches PatentamtDE 4014019 (7Nov.1991).
  12. D. C. Williams, N. S. Nassar, J. E. Banyard, M. S. Virdee, “Digital phase-step interferometry: a simplified approach,” Opt. Laser Technol. 23, 147–150 (1991).
    [CrossRef]
  13. P. H. Chan, P. J. Bryanston–Cross, S. C. Parker, “Fringe-pattern analysis using a spatial phase-stepping method with automatic phase unwrapping,” Meas. Sci. Technol. 6, 1250–1259 (1995).
    [CrossRef]
  14. M. Servin, F. J. Cuevas, “A novel technique for spatial phase-shifting interferometry,” J. Mod. Opt. 42, 1853–1862 (1995).
    [CrossRef]
  15. H. Steinbichler, J. Gutjahr, “Verfahren zur direkten Phasenmessung von Strahlung, insbesondere Lichtstrahlung, und Vorrichtung zur Durchführung dieses Verfahrens,” Offenlegungsschrift Deutsches PatentamtDE 3930632 A1 (14Mar.1991).
  16. S. Leidenbach, “Die direkte Phasenmessung—ein neues Verfahren zur Berechnung von Phasenbildern aus nur einem Intensitätsbild,” Proc. Laser 1, 68–72 (1991).
  17. G. Pedrini, B. Pfister, H. Tiziani, “Double pulse-electronic speckle interferometry,” J. Mod. Opt. 40, 89–96 (1993).
    [CrossRef]
  18. 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]
  19. J. Schwider, “Advanced evaluation techniques in interferometry,” in Progress in Optics XXVIII, E. Wolff, ed. (Elsevier, Amsterdam, 1990), pp. 271–359.
    [CrossRef]
  20. J. Schmit-Wojciak, K. Creath, “Spatial and temporal phase-measurement techniques: a comparison of major error sources in one dimension,” in Interferometry: Techniques and Analysis, G. M. Brown, M. Kujawinska, O. Y. Kwon, G. T. Reid, eds., Proc. SPIE1755, 202–211 (1992).
  21. 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]
  22. T. Maack, R. Kowarschik, “Camera influence on the phase measurement accuracy of a phase shifting speckle interferometer,” Appl. Opt. 35, 3514–3524 (1996).
    [CrossRef] [PubMed]
  23. H. Kadono, S. Toyooka, “Statistical interferometry based on the statistics of speckle phase,” Opt. Lett. 16, 883–885 (1991).
    [CrossRef] [PubMed]

1996 (1)

1995 (2)

P. H. Chan, P. J. Bryanston–Cross, S. C. Parker, “Fringe-pattern analysis using a spatial phase-stepping method with automatic phase unwrapping,” Meas. Sci. Technol. 6, 1250–1259 (1995).
[CrossRef]

M. Servin, F. J. Cuevas, “A novel technique for spatial phase-shifting interferometry,” J. Mod. Opt. 42, 1853–1862 (1995).
[CrossRef]

1993 (1)

G. Pedrini, B. Pfister, H. Tiziani, “Double pulse-electronic speckle interferometry,” J. Mod. Opt. 40, 89–96 (1993).
[CrossRef]

1991 (3)

H. Kadono, S. Toyooka, “Statistical interferometry based on the statistics of speckle phase,” Opt. Lett. 16, 883–885 (1991).
[CrossRef] [PubMed]

S. Leidenbach, “Die direkte Phasenmessung—ein neues Verfahren zur Berechnung von Phasenbildern aus nur einem Intensitätsbild,” Proc. Laser 1, 68–72 (1991).

D. C. Williams, N. S. Nassar, J. E. Banyard, M. S. Virdee, “Digital phase-step interferometry: a simplified approach,” Opt. Laser Technol. 23, 147–150 (1991).
[CrossRef]

1990 (1)

1985 (2)

1984 (2)

R. Smythe, R. Moore, “Instantaneous phase measuring interferometry,” Opt. Eng. 23, 361–364 (1984).
[CrossRef]

O. Y. Kwon, “Multichannel phase-shifted interferometer,” Opt. Lett. 9, 59–61 (1984).
[CrossRef] [PubMed]

1983 (2)

1972 (1)

Banyard, J. E.

D. C. Williams, N. S. Nassar, J. E. Banyard, M. S. Virdee, “Digital phase-step interferometry: a simplified approach,” Opt. Laser Technol. 23, 147–150 (1991).
[CrossRef]

Brophy, C. P.

Bruning, J. H.

J. E. Greivenkamp, J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing, D. Malacara ed. (Wiles, New York, 1992), pp. 501–598.

Bryanston–Cross, P. J.

P. H. Chan, P. J. Bryanston–Cross, S. C. Parker, “Fringe-pattern analysis using a spatial phase-stepping method with automatic phase unwrapping,” Meas. Sci. Technol. 6, 1250–1259 (1995).
[CrossRef]

Burow, R.

Chan, P. H.

P. H. Chan, P. J. Bryanston–Cross, S. C. Parker, “Fringe-pattern analysis using a spatial phase-stepping method with automatic phase unwrapping,” Meas. Sci. Technol. 6, 1250–1259 (1995).
[CrossRef]

Creath, K.

K. Creath, “Phase-shifting speckle interferometry,” Appl. Opt. 24, 3053–3058 (1985).
[CrossRef] [PubMed]

K. Creath, “Temporal phase measurement methods,” in Interferogram Analysis, D. W. Robinson, G. T. Reid, eds. (Institute of Physics Publishing, Bristol, 1993), pp. 94–140.

J. Schmit-Wojciak, K. Creath, “Spatial and temporal phase-measurement techniques: a comparison of major error sources in one dimension,” in Interferometry: Techniques and Analysis, G. M. Brown, M. Kujawinska, O. Y. Kwon, G. T. Reid, eds., Proc. SPIE1755, 202–211 (1992).

Cuevas, F. J.

M. Servin, F. J. Cuevas, “A novel technique for spatial phase-shifting interferometry,” J. Mod. Opt. 42, 1853–1862 (1995).
[CrossRef]

Elssner, K. E.

Greivenkamp, J. E.

J. E. Greivenkamp, J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing, D. Malacara ed. (Wiles, New York, 1992), pp. 501–598.

Grzanna, J.

Gutjahr, J.

H. Steinbichler, J. Gutjahr, “Verfahren zur direkten Phasenmessung von Strahlung, insbesondere Lichtstrahlung, und Vorrichtung zur Durchführung dieses Verfahrens,” Offenlegungsschrift Deutsches PatentamtDE 3930632 A1 (14Mar.1991).

Ichioka, Y.

Inuiya, M.

Kadono, H.

Kowarschik, R.

Küchel, M.

M. Küchel, “Verfahren zur Messung eines phasenmodulierten Signals,” Offenlegungsschrift Deutsches PatentamtDE 4014019 (7Nov.1991).

Kujawinska, M.

M. Kujawinska, “Spatial phase measurement methods,” in Interferogram Analysis, D. W. Robinson, G. T. Reid, eds. (Institute of Physics Publishing, Bristol, 1993), pp. 141–193.

Kwon, O. Y.

O. Y. Kwon, “Multichannel phase-shifted interferometer,” Opt. Lett. 9, 59–61 (1984).
[CrossRef] [PubMed]

D. M. Shough, O. Y. Kwon, D. F. Leary, “High-speed interferometric measurements of aerodynamic phenomena,” in Propagation of High-Energy Laser Beams Through the Earth’s Atmosphere, P. B. Ulrich, L. E. Wilson, eds., Proc. SPIE Vol. 1221, 394–403 (1990).
[CrossRef]

Leary, D. F.

D. M. Shough, O. Y. Kwon, D. F. Leary, “High-speed interferometric measurements of aerodynamic phenomena,” in Propagation of High-Energy Laser Beams Through the Earth’s Atmosphere, P. B. Ulrich, L. E. Wilson, eds., Proc. SPIE Vol. 1221, 394–403 (1990).
[CrossRef]

Leidenbach, S.

S. Leidenbach, “Die direkte Phasenmessung—ein neues Verfahren zur Berechnung von Phasenbildern aus nur einem Intensitätsbild,” Proc. Laser 1, 68–72 (1991).

Maack, T.

Merkel, K.

Mertz, L.

Moore, R.

R. Smythe, R. Moore, “Instantaneous phase measuring interferometry,” Opt. Eng. 23, 361–364 (1984).
[CrossRef]

Nakadate, S.

Nassar, N. S.

D. C. Williams, N. S. Nassar, J. E. Banyard, M. S. Virdee, “Digital phase-step interferometry: a simplified approach,” Opt. Laser Technol. 23, 147–150 (1991).
[CrossRef]

Parker, S. C.

P. H. Chan, P. J. Bryanston–Cross, S. C. Parker, “Fringe-pattern analysis using a spatial phase-stepping method with automatic phase unwrapping,” Meas. Sci. Technol. 6, 1250–1259 (1995).
[CrossRef]

Pedrini, G.

G. Pedrini, B. Pfister, H. Tiziani, “Double pulse-electronic speckle interferometry,” J. Mod. Opt. 40, 89–96 (1993).
[CrossRef]

Pfister, B.

G. Pedrini, B. Pfister, H. Tiziani, “Double pulse-electronic speckle interferometry,” J. Mod. Opt. 40, 89–96 (1993).
[CrossRef]

Saito, H.

Schmit-Wojciak, J.

J. Schmit-Wojciak, K. Creath, “Spatial and temporal phase-measurement techniques: a comparison of major error sources in one dimension,” in Interferometry: Techniques and Analysis, G. M. Brown, M. Kujawinska, O. Y. Kwon, G. T. Reid, eds., Proc. SPIE1755, 202–211 (1992).

Schwider, J.

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]

J. Schwider, “Advanced evaluation techniques in interferometry,” in Progress in Optics XXVIII, E. Wolff, ed. (Elsevier, Amsterdam, 1990), pp. 271–359.
[CrossRef]

Servin, M.

M. Servin, F. J. Cuevas, “A novel technique for spatial phase-shifting interferometry,” J. Mod. Opt. 42, 1853–1862 (1995).
[CrossRef]

Shough, D. M.

D. M. Shough, O. Y. Kwon, D. F. Leary, “High-speed interferometric measurements of aerodynamic phenomena,” in Propagation of High-Energy Laser Beams Through the Earth’s Atmosphere, P. B. Ulrich, L. E. Wilson, eds., Proc. SPIE Vol. 1221, 394–403 (1990).
[CrossRef]

Smythe, R.

R. Smythe, R. Moore, “Instantaneous phase measuring interferometry,” Opt. Eng. 23, 361–364 (1984).
[CrossRef]

Spolaczyk, R.

Steinbichler, H.

H. Steinbichler, J. Gutjahr, “Verfahren zur direkten Phasenmessung von Strahlung, insbesondere Lichtstrahlung, und Vorrichtung zur Durchführung dieses Verfahrens,” Offenlegungsschrift Deutsches PatentamtDE 3930632 A1 (14Mar.1991).

Tiziani, H.

G. Pedrini, B. Pfister, H. Tiziani, “Double pulse-electronic speckle interferometry,” J. Mod. Opt. 40, 89–96 (1993).
[CrossRef]

Toyooka, S.

Virdee, M. S.

D. C. Williams, N. S. Nassar, J. E. Banyard, M. S. Virdee, “Digital phase-step interferometry: a simplified approach,” Opt. Laser Technol. 23, 147–150 (1991).
[CrossRef]

Williams, D. C.

D. C. Williams, N. S. Nassar, J. E. Banyard, M. S. Virdee, “Digital phase-step interferometry: a simplified approach,” Opt. Laser Technol. 23, 147–150 (1991).
[CrossRef]

Appl. Opt. (6)

J. Mod. Opt. (2)

M. Servin, F. J. Cuevas, “A novel technique for spatial phase-shifting interferometry,” J. Mod. Opt. 42, 1853–1862 (1995).
[CrossRef]

G. Pedrini, B. Pfister, H. Tiziani, “Double pulse-electronic speckle interferometry,” J. Mod. Opt. 40, 89–96 (1993).
[CrossRef]

J. Opt. Soc. Am. A (1)

Meas. Sci. Technol. (1)

P. H. Chan, P. J. Bryanston–Cross, S. C. Parker, “Fringe-pattern analysis using a spatial phase-stepping method with automatic phase unwrapping,” Meas. Sci. Technol. 6, 1250–1259 (1995).
[CrossRef]

Opt. Eng. (1)

R. Smythe, R. Moore, “Instantaneous phase measuring interferometry,” Opt. Eng. 23, 361–364 (1984).
[CrossRef]

Opt. Laser Technol. (1)

D. C. Williams, N. S. Nassar, J. E. Banyard, M. S. Virdee, “Digital phase-step interferometry: a simplified approach,” Opt. Laser Technol. 23, 147–150 (1991).
[CrossRef]

Opt. Lett. (2)

Proc. Laser (1)

S. Leidenbach, “Die direkte Phasenmessung—ein neues Verfahren zur Berechnung von Phasenbildern aus nur einem Intensitätsbild,” Proc. Laser 1, 68–72 (1991).

Other (8)

H. Steinbichler, J. Gutjahr, “Verfahren zur direkten Phasenmessung von Strahlung, insbesondere Lichtstrahlung, und Vorrichtung zur Durchführung dieses Verfahrens,” Offenlegungsschrift Deutsches PatentamtDE 3930632 A1 (14Mar.1991).

J. E. Greivenkamp, J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing, D. Malacara ed. (Wiles, New York, 1992), pp. 501–598.

K. Creath, “Temporal phase measurement methods,” in Interferogram Analysis, D. W. Robinson, G. T. Reid, eds. (Institute of Physics Publishing, Bristol, 1993), pp. 94–140.

M. Kujawinska, “Spatial phase measurement methods,” in Interferogram Analysis, D. W. Robinson, G. T. Reid, eds. (Institute of Physics Publishing, Bristol, 1993), pp. 141–193.

D. M. Shough, O. Y. Kwon, D. F. Leary, “High-speed interferometric measurements of aerodynamic phenomena,” in Propagation of High-Energy Laser Beams Through the Earth’s Atmosphere, P. B. Ulrich, L. E. Wilson, eds., Proc. SPIE Vol. 1221, 394–403 (1990).
[CrossRef]

M. Küchel, “Verfahren zur Messung eines phasenmodulierten Signals,” Offenlegungsschrift Deutsches PatentamtDE 4014019 (7Nov.1991).

J. Schwider, “Advanced evaluation techniques in interferometry,” in Progress in Optics XXVIII, E. Wolff, ed. (Elsevier, Amsterdam, 1990), pp. 271–359.
[CrossRef]

J. Schmit-Wojciak, K. Creath, “Spatial and temporal phase-measurement techniques: a comparison of major error sources in one dimension,” in Interferometry: Techniques and Analysis, G. M. Brown, M. Kujawinska, O. Y. Kwon, G. T. Reid, eds., Proc. SPIE1755, 202–211 (1992).

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Figures (5)

Fig. 1
Fig. 1

ESPI setup modified for SPS: S, sensor; O, objective; BS, beam splitter.

Fig. 2
Fig. 2

Statistical phase error σψ due to statistical intensity fluctuations σ I .

Fig. 3
Fig. 3

Phase reconstruction errors by intensity saturation for different phase shift angles β.

Fig. 4
Fig. 4

Deviations from the expected uniform phase distribution in a speckle field by wrong phase-shift angles. Left, phase shift too small; center, correct phase shift; right, phase shift too great.

Fig. 5
Fig. 5

Improved results of a deformation measurement by use of a modified phase reconstruction formula (see text).

Equations (14)

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αx, y=x+Δx2+y2+Δz21/2-x2+y2+Δz21/22πλ,
αx, yαx=x+Δx2+Δz21/2-x2+Δz21/22πλ.
Ixn, y=I0xn, y1+γxn, ysincΦ2×cosφxn, y+nβ+C,  n=1, , M.
αxx=x+Δxx+Δx2+Δz21/2-xx2+Δz21/22πλ
φn=tan-11-cos βsin βIn-1-In+12In-In-1-In+1mod π,
In+m=I01+γ sincΦ2cosφ+nβ+mβ,  n=2, , M-1, m=-1, 0, +1.
σψ=m=-11ψIn+m2σI21/2.
ψIn+m=fIn-1, In, In+1In+m11+f2In-1, In, In+1.
σψ=σI cos2 ψ1-cos βsin βm=-11fIn-1, In, In+1In+m21/2.
In-1-In+1sin β=2γI0 sin ψ,  2In-In-1-In+11-cos β=2γI0 cos ψ,
σψ=K2 cos2 ψsin2 β+6 sin2 ψ1-cos β21/2,  with K=σI2γI0.
σ¯ψ=K1sin2 β+31-cos β21/2,
I0xn, y=Oxn, y+Rxn, y, γxn, y=2Oxn, y1/2Rxn, y1/2Oxn, y+Rxn, y,
φn=tan-1On1/2In-1-In+1-cos βOn-11/2In-In+1+On+11/2In-1-Insin βOn-11/2In-In+1+On+11/2In-In-1mod π.

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