Abstract

As a phase shifter usually suffers from both translational and tilt-shift errors during shifting, so every pixel in the same interferogram will have a different phase-shift value. Thus nonlinear phase-measurement errors cannot be avoided, but even translational-shift error has been corrected effectively. However, based on the fact that the shifted phases of all the pixels in the same interferogram remain on the phase-shift plane, by defining this plane one can eliminate a significant number of phase errors. A new algorithm that is immune to both translational- and tilt-shift errors in a phase shifter for phase-stepping interferometers is presented. A first-order Taylor series expansion replaces the nonlinear equations for defining the phase-shift plane, and iteration of the algorithm guarantees its accuracy. Results of a computer simulation show that phase-measurement errors caused by both translation- and tilt-shift error can be compensated for completely, even when the tilt-shift error is not more than ±1%.

© 2000 Optical Society of America

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References

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  1. K. Creath, “Temporal phase measurement method,” in Interferogram Analysis: Digital Fringe Pattern Measurement, D. W. Robinson, G. Reid, eds. (IOP, Bristol, UK, 1993), pp. 94–140.
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    [CrossRef]
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    [CrossRef] [PubMed]
  4. C. J. Morgan, “Least-squares estimation in phase-measurement interferometry,” Opt. Lett. 7, 368–370 (1982).
    [CrossRef] [PubMed]
  5. K. Okada, A. Sato, J. Tsujiuchi, “Simultaneous calculation of phase distribution and scanning phase shifting interferometry,” Opt. Commun. 84, 118–124 (1991).
    [CrossRef]
  6. I.-B. Kong, S.-W. Kim, “General algorithm of phase-shifting interferometry by iterative least-squares fitting,” Opt. Eng. 34, 183–187 (1995).
    [CrossRef]
  7. I.-B. Kong, S.-W. Kim, “Portable inspection of precision surfaces by phase-shifting interferometry with automatic suppression of phase-shift errors,” Opt. Eng. 34, 1400–1404 (1995).
    [CrossRef]
  8. C. Wei, M. Chen, H. Guo, Z. Wang, “General phase-stepping algorithm using Lissajous figures technique,” in Laser Interferometry IX: Techniques and Analysis, M. Kujawińsk, G. M. Brown, M. Takeda, eds., Proc. SPIE3478, 411–416 (1998).
    [CrossRef]
  9. C. Wei, M. Chen, Z. Wang, “General phase-stepping algorithm with automatic calibration of phase steps,” Opt. Eng. 38, 1357–1360 (1999).
    [CrossRef]

1999 (1)

C. Wei, M. Chen, Z. Wang, “General phase-stepping algorithm with automatic calibration of phase steps,” Opt. Eng. 38, 1357–1360 (1999).
[CrossRef]

1995 (2)

I.-B. Kong, S.-W. Kim, “General algorithm of phase-shifting interferometry by iterative least-squares fitting,” Opt. Eng. 34, 183–187 (1995).
[CrossRef]

I.-B. Kong, S.-W. Kim, “Portable inspection of precision surfaces by phase-shifting interferometry with automatic suppression of phase-shift errors,” Opt. Eng. 34, 1400–1404 (1995).
[CrossRef]

1991 (1)

K. Okada, A. Sato, J. Tsujiuchi, “Simultaneous calculation of phase distribution and scanning phase shifting interferometry,” Opt. Commun. 84, 118–124 (1991).
[CrossRef]

1982 (1)

1974 (1)

1966 (1)

P. Carré, “Installation et utilisation du comparateur photoélectrique et intertérentiel du Bureau International des Poids et Mesures,” Metrologia 2, 13–23 (1966).
[CrossRef]

Brangaccio, D. J.

Bruning, J. H.

Carré, P.

P. Carré, “Installation et utilisation du comparateur photoélectrique et intertérentiel du Bureau International des Poids et Mesures,” Metrologia 2, 13–23 (1966).
[CrossRef]

Chen, M.

C. Wei, M. Chen, Z. Wang, “General phase-stepping algorithm with automatic calibration of phase steps,” Opt. Eng. 38, 1357–1360 (1999).
[CrossRef]

C. Wei, M. Chen, H. Guo, Z. Wang, “General phase-stepping algorithm using Lissajous figures technique,” in Laser Interferometry IX: Techniques and Analysis, M. Kujawińsk, G. M. Brown, M. Takeda, eds., Proc. SPIE3478, 411–416 (1998).
[CrossRef]

Creath, K.

K. Creath, “Temporal phase measurement method,” in Interferogram Analysis: Digital Fringe Pattern Measurement, D. W. Robinson, G. Reid, eds. (IOP, Bristol, UK, 1993), pp. 94–140.

Gallagher, J. E.

Guo, H.

C. Wei, M. Chen, H. Guo, Z. Wang, “General phase-stepping algorithm using Lissajous figures technique,” in Laser Interferometry IX: Techniques and Analysis, M. Kujawińsk, G. M. Brown, M. Takeda, eds., Proc. SPIE3478, 411–416 (1998).
[CrossRef]

Herriott, D. R.

Kim, S.-W.

I.-B. Kong, S.-W. Kim, “General algorithm of phase-shifting interferometry by iterative least-squares fitting,” Opt. Eng. 34, 183–187 (1995).
[CrossRef]

I.-B. Kong, S.-W. Kim, “Portable inspection of precision surfaces by phase-shifting interferometry with automatic suppression of phase-shift errors,” Opt. Eng. 34, 1400–1404 (1995).
[CrossRef]

Kong, I.-B.

I.-B. Kong, S.-W. Kim, “General algorithm of phase-shifting interferometry by iterative least-squares fitting,” Opt. Eng. 34, 183–187 (1995).
[CrossRef]

I.-B. Kong, S.-W. Kim, “Portable inspection of precision surfaces by phase-shifting interferometry with automatic suppression of phase-shift errors,” Opt. Eng. 34, 1400–1404 (1995).
[CrossRef]

Morgan, C. J.

Okada, K.

K. Okada, A. Sato, J. Tsujiuchi, “Simultaneous calculation of phase distribution and scanning phase shifting interferometry,” Opt. Commun. 84, 118–124 (1991).
[CrossRef]

Rosenfeld, D. P.

Sato, A.

K. Okada, A. Sato, J. Tsujiuchi, “Simultaneous calculation of phase distribution and scanning phase shifting interferometry,” Opt. Commun. 84, 118–124 (1991).
[CrossRef]

Tsujiuchi, J.

K. Okada, A. Sato, J. Tsujiuchi, “Simultaneous calculation of phase distribution and scanning phase shifting interferometry,” Opt. Commun. 84, 118–124 (1991).
[CrossRef]

Wang, Z.

C. Wei, M. Chen, Z. Wang, “General phase-stepping algorithm with automatic calibration of phase steps,” Opt. Eng. 38, 1357–1360 (1999).
[CrossRef]

C. Wei, M. Chen, H. Guo, Z. Wang, “General phase-stepping algorithm using Lissajous figures technique,” in Laser Interferometry IX: Techniques and Analysis, M. Kujawińsk, G. M. Brown, M. Takeda, eds., Proc. SPIE3478, 411–416 (1998).
[CrossRef]

Wei, C.

C. Wei, M. Chen, Z. Wang, “General phase-stepping algorithm with automatic calibration of phase steps,” Opt. Eng. 38, 1357–1360 (1999).
[CrossRef]

C. Wei, M. Chen, H. Guo, Z. Wang, “General phase-stepping algorithm using Lissajous figures technique,” in Laser Interferometry IX: Techniques and Analysis, M. Kujawińsk, G. M. Brown, M. Takeda, eds., Proc. SPIE3478, 411–416 (1998).
[CrossRef]

White, A. D.

Appl. Opt. (1)

Metrologia (1)

P. Carré, “Installation et utilisation du comparateur photoélectrique et intertérentiel du Bureau International des Poids et Mesures,” Metrologia 2, 13–23 (1966).
[CrossRef]

Opt. Commun. (1)

K. Okada, A. Sato, J. Tsujiuchi, “Simultaneous calculation of phase distribution and scanning phase shifting interferometry,” Opt. Commun. 84, 118–124 (1991).
[CrossRef]

Opt. Eng. (3)

I.-B. Kong, S.-W. Kim, “General algorithm of phase-shifting interferometry by iterative least-squares fitting,” Opt. Eng. 34, 183–187 (1995).
[CrossRef]

I.-B. Kong, S.-W. Kim, “Portable inspection of precision surfaces by phase-shifting interferometry with automatic suppression of phase-shift errors,” Opt. Eng. 34, 1400–1404 (1995).
[CrossRef]

C. Wei, M. Chen, Z. Wang, “General phase-stepping algorithm with automatic calibration of phase steps,” Opt. Eng. 38, 1357–1360 (1999).
[CrossRef]

Opt. Lett. (1)

Other (2)

K. Creath, “Temporal phase measurement method,” in Interferogram Analysis: Digital Fringe Pattern Measurement, D. W. Robinson, G. Reid, eds. (IOP, Bristol, UK, 1993), pp. 94–140.

C. Wei, M. Chen, H. Guo, Z. Wang, “General phase-stepping algorithm using Lissajous figures technique,” in Laser Interferometry IX: Techniques and Analysis, M. Kujawińsk, G. M. Brown, M. Takeda, eds., Proc. SPIE3478, 411–416 (1998).
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

Twyman–Green interferometer with tilt phase-shift error.

Fig. 2
Fig. 2

Phase distribution of a wave front.

Fig. 3
Fig. 3

Computer simulation results: (a) w = 0.025, (b) w = 0.0125, (c) w = 0.0025, (d) w = 0.00125. I, Bruning algorithm; II, Kong algorithm; III, suggested algorithm.

Fig. 4
Fig. 4

Relationship between residual PV errors and number of iterations (w = 0.00125).

Tables (1)

Tables Icon

Table 1 Simulation Results: Residual PV Errors (rad)

Equations (13)

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Ini, j=I0i, j1+ri, jcosϕi, j-δni, j,
δni, j=ani+bnj+dn.
Jni, j=Ini, j-I1i, j.
Jni, j=Ci, jcosani+bnj+dn-1+Si, jsinani+bnj+dn,
Ci, j=I0i, jri, jcosϕi, j
Si, j=I0i, jri, jsinϕi, j.
ϕi, j=tan-1Si, jCi, j.
an=an*+Δan,
bn=bn*+Δbn,
dn=dn*+Δdn.
Jni, j=Ci, jcosan*i+bn*j+dn*-1+Si, jsinan*i+bn*j+dn*+Si, jcosan*i+bn*j+dn*-Ci, jsinan*i+bn*j+dn*×Δani+Δbnj+Δdn.
Si, jcosan*i+bn*j+dn*-Ci, jsinan*i+bn*j+dn*Δani+Δbnj+Δdn=Jni, j-Ci, jcosan*i+bn*j+dn*-1-Si, jsinan*i+bn*j+dn*.
δni, j=0.25n-1π+0.025pnπ+wsni+tnjπ/127,

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