Abstract

A computationally efficient algorithm for phase-shifting interferometry with imprecise phase shifts is developed. It permits the use of an uncalibrated phase shifter and is also insensitive to spatial intensity variations. The measurement has both spatial and temporal aspects. Comparisons are made between pixels within the same interferogram, and these comparisons are extended across a set of interferograms by a maximum–minimum procedure. A test experiment is performed and confirms the theoretical results. An additional advantage of the algorithm is that an error measure can be developed. This error measure is used to implement an error correction scheme.

© 2000 Optical Society of America

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References

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  1. P. K. Rastogi, Holographic Interferometry (Springer-Verlag, Berlin, 1994).
    [CrossRef]
  2. D. Malacara, M. Servin, Zacarias Malacara, Interferogram Analysis for Optical Testing (Marcel Dekker, New York, 1998).
  3. N. A. Ochoa, J. M. Huntley, “Convenient method for calibrating nonlinear phase modulators for use in phase-shifting interferometry,” Opt. Eng. 37, 2501–2505 (1998).
    [CrossRef]
  4. T. Kreis, “Digital holographic interference-phase measurement using the Fourier-transform method,” J. Opt. Soc. Am. A 3, 847–855 (1986).
    [CrossRef]
  5. R. Windecker, H. J. Tiziani, “Semispatial, robust, and accurate phase evaluation algorithm,” Appl. Opt. 34, 7321–7326 (1995).
    [CrossRef] [PubMed]
  6. J. E. Greivenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23, 350–352 (1984).
    [CrossRef]
  7. G. D. Lassahn, J. K. Lassahn, P. L. Taylor, V. A. Deason, “Multiphase fringe analysis with unknown phase shifts,” Opt. Eng. 33, 2039–2044 (1994).
    [CrossRef]
  8. C. T. Farrell, M. A. Player, “Phase-step insensitive algorithms for phase-shifting interferometry,” Meas. Sci. Technol. 5, 648–652 (1994).
    [CrossRef]
  9. F. Bookstein, “Fitting conic sections to scattered data,” Comput. Graph. Image Process. 9, 56–71 (1979).
    [CrossRef]
  10. C. T. Farrell, M. A. Player, “Phase step measurement and variable step algorithms in phase-shifting interferometry,” Meas. Sci. Technol. 3, 953–958 (1992).
    [CrossRef]
  11. G.-S. Han, S.-W. Kim, “Numerical correction of reference phases in phase-shifting interferometry by iterative least-squares fitting,” Appl. Opt. 33, 7321–732 (1994).
    [CrossRef] [PubMed]
  12. A. Dobroiu, D. Apostol, V. Nascov, V. Damian, “Statistical self-calibrating algorithm for phase-shift interferometry based on a smoothness assessment of the intensity offset map,” Meas. Sci. Technol. 9, 1451–1455 (1998).
    [CrossRef]
  13. G. Lai, T. Yatagai, “Generalized phase-shifting interferometry,” J. Opt. Soc. Am. A 8, 822–827 (1991).
    [CrossRef]
  14. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, 1995).
    [CrossRef]
  15. A more comprehensive description of the use of correlations to find the phase difference between two pixels i and j is given by cos(δi - δj) = 〈IiIj〉 - 〈Ii〉〈Ij〉/[〈Ii2〉 - 〈Ii〉2 〈Ii2〉 - 〈Ii〉2]1/2, where Ii and Ij are the intensities at the pixels i and j, respectively. The angle brackets denote an ensemble average over the randomly varying phase ϕ.

1998

N. A. Ochoa, J. M. Huntley, “Convenient method for calibrating nonlinear phase modulators for use in phase-shifting interferometry,” Opt. Eng. 37, 2501–2505 (1998).
[CrossRef]

A. Dobroiu, D. Apostol, V. Nascov, V. Damian, “Statistical self-calibrating algorithm for phase-shift interferometry based on a smoothness assessment of the intensity offset map,” Meas. Sci. Technol. 9, 1451–1455 (1998).
[CrossRef]

1995

1994

G.-S. Han, S.-W. Kim, “Numerical correction of reference phases in phase-shifting interferometry by iterative least-squares fitting,” Appl. Opt. 33, 7321–732 (1994).
[CrossRef] [PubMed]

G. D. Lassahn, J. K. Lassahn, P. L. Taylor, V. A. Deason, “Multiphase fringe analysis with unknown phase shifts,” Opt. Eng. 33, 2039–2044 (1994).
[CrossRef]

C. T. Farrell, M. A. Player, “Phase-step insensitive algorithms for phase-shifting interferometry,” Meas. Sci. Technol. 5, 648–652 (1994).
[CrossRef]

1992

C. T. Farrell, M. A. Player, “Phase step measurement and variable step algorithms in phase-shifting interferometry,” Meas. Sci. Technol. 3, 953–958 (1992).
[CrossRef]

1991

1986

1984

J. E. Greivenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23, 350–352 (1984).
[CrossRef]

1979

F. Bookstein, “Fitting conic sections to scattered data,” Comput. Graph. Image Process. 9, 56–71 (1979).
[CrossRef]

Apostol, D.

A. Dobroiu, D. Apostol, V. Nascov, V. Damian, “Statistical self-calibrating algorithm for phase-shift interferometry based on a smoothness assessment of the intensity offset map,” Meas. Sci. Technol. 9, 1451–1455 (1998).
[CrossRef]

Bookstein, F.

F. Bookstein, “Fitting conic sections to scattered data,” Comput. Graph. Image Process. 9, 56–71 (1979).
[CrossRef]

Damian, V.

A. Dobroiu, D. Apostol, V. Nascov, V. Damian, “Statistical self-calibrating algorithm for phase-shift interferometry based on a smoothness assessment of the intensity offset map,” Meas. Sci. Technol. 9, 1451–1455 (1998).
[CrossRef]

Deason, V. A.

G. D. Lassahn, J. K. Lassahn, P. L. Taylor, V. A. Deason, “Multiphase fringe analysis with unknown phase shifts,” Opt. Eng. 33, 2039–2044 (1994).
[CrossRef]

Dobroiu, A.

A. Dobroiu, D. Apostol, V. Nascov, V. Damian, “Statistical self-calibrating algorithm for phase-shift interferometry based on a smoothness assessment of the intensity offset map,” Meas. Sci. Technol. 9, 1451–1455 (1998).
[CrossRef]

Farrell, C. T.

C. T. Farrell, M. A. Player, “Phase-step insensitive algorithms for phase-shifting interferometry,” Meas. Sci. Technol. 5, 648–652 (1994).
[CrossRef]

C. T. Farrell, M. A. Player, “Phase step measurement and variable step algorithms in phase-shifting interferometry,” Meas. Sci. Technol. 3, 953–958 (1992).
[CrossRef]

Greivenkamp, J. E.

J. E. Greivenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23, 350–352 (1984).
[CrossRef]

Han, G.-S.

Huntley, J. M.

N. A. Ochoa, J. M. Huntley, “Convenient method for calibrating nonlinear phase modulators for use in phase-shifting interferometry,” Opt. Eng. 37, 2501–2505 (1998).
[CrossRef]

Kim, S.-W.

Kreis, T.

Lai, G.

Lassahn, G. D.

G. D. Lassahn, J. K. Lassahn, P. L. Taylor, V. A. Deason, “Multiphase fringe analysis with unknown phase shifts,” Opt. Eng. 33, 2039–2044 (1994).
[CrossRef]

Lassahn, J. K.

G. D. Lassahn, J. K. Lassahn, P. L. Taylor, V. A. Deason, “Multiphase fringe analysis with unknown phase shifts,” Opt. Eng. 33, 2039–2044 (1994).
[CrossRef]

Malacara, D.

D. Malacara, M. Servin, Zacarias Malacara, Interferogram Analysis for Optical Testing (Marcel Dekker, New York, 1998).

Malacara, Zacarias

D. Malacara, M. Servin, Zacarias Malacara, Interferogram Analysis for Optical Testing (Marcel Dekker, New York, 1998).

Mandel, L.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, 1995).
[CrossRef]

Nascov, V.

A. Dobroiu, D. Apostol, V. Nascov, V. Damian, “Statistical self-calibrating algorithm for phase-shift interferometry based on a smoothness assessment of the intensity offset map,” Meas. Sci. Technol. 9, 1451–1455 (1998).
[CrossRef]

Ochoa, N. A.

N. A. Ochoa, J. M. Huntley, “Convenient method for calibrating nonlinear phase modulators for use in phase-shifting interferometry,” Opt. Eng. 37, 2501–2505 (1998).
[CrossRef]

Player, M. A.

C. T. Farrell, M. A. Player, “Phase-step insensitive algorithms for phase-shifting interferometry,” Meas. Sci. Technol. 5, 648–652 (1994).
[CrossRef]

C. T. Farrell, M. A. Player, “Phase step measurement and variable step algorithms in phase-shifting interferometry,” Meas. Sci. Technol. 3, 953–958 (1992).
[CrossRef]

Rastogi, P. K.

P. K. Rastogi, Holographic Interferometry (Springer-Verlag, Berlin, 1994).
[CrossRef]

Servin, M.

D. Malacara, M. Servin, Zacarias Malacara, Interferogram Analysis for Optical Testing (Marcel Dekker, New York, 1998).

Taylor, P. L.

G. D. Lassahn, J. K. Lassahn, P. L. Taylor, V. A. Deason, “Multiphase fringe analysis with unknown phase shifts,” Opt. Eng. 33, 2039–2044 (1994).
[CrossRef]

Tiziani, H. J.

Windecker, R.

Wolf, E.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, 1995).
[CrossRef]

Yatagai, T.

Appl. Opt.

Comput. Graph. Image Process.

F. Bookstein, “Fitting conic sections to scattered data,” Comput. Graph. Image Process. 9, 56–71 (1979).
[CrossRef]

J. Opt. Soc. Am. A

Meas. Sci. Technol.

C. T. Farrell, M. A. Player, “Phase-step insensitive algorithms for phase-shifting interferometry,” Meas. Sci. Technol. 5, 648–652 (1994).
[CrossRef]

C. T. Farrell, M. A. Player, “Phase step measurement and variable step algorithms in phase-shifting interferometry,” Meas. Sci. Technol. 3, 953–958 (1992).
[CrossRef]

A. Dobroiu, D. Apostol, V. Nascov, V. Damian, “Statistical self-calibrating algorithm for phase-shift interferometry based on a smoothness assessment of the intensity offset map,” Meas. Sci. Technol. 9, 1451–1455 (1998).
[CrossRef]

Opt. Eng.

N. A. Ochoa, J. M. Huntley, “Convenient method for calibrating nonlinear phase modulators for use in phase-shifting interferometry,” Opt. Eng. 37, 2501–2505 (1998).
[CrossRef]

J. E. Greivenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23, 350–352 (1984).
[CrossRef]

G. D. Lassahn, J. K. Lassahn, P. L. Taylor, V. A. Deason, “Multiphase fringe analysis with unknown phase shifts,” Opt. Eng. 33, 2039–2044 (1994).
[CrossRef]

Other

P. K. Rastogi, Holographic Interferometry (Springer-Verlag, Berlin, 1994).
[CrossRef]

D. Malacara, M. Servin, Zacarias Malacara, Interferogram Analysis for Optical Testing (Marcel Dekker, New York, 1998).

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, 1995).
[CrossRef]

A more comprehensive description of the use of correlations to find the phase difference between two pixels i and j is given by cos(δi - δj) = 〈IiIj〉 - 〈Ii〉〈Ij〉/[〈Ii2〉 - 〈Ii〉2 〈Ii2〉 - 〈Ii〉2]1/2, where Ii and Ij are the intensities at the pixels i and j, respectively. The angle brackets denote an ensemble average over the randomly varying phase ϕ.

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Figures (2)

Fig. 1
Fig. 1

Flow chart of the max–min algorithm. The reference pixels are denoted 1 and 2, and the test pixel is denoted X = (x, y).

Fig. 2
Fig. 2

Experimentally measured phase distribution of the object beam. The object beam is created by introduction of a tilt along the x axis (note that the construction of the mirror mount also introduces a slight tilt along the y axis).

Tables (3)

Tables Icon

Table 1 Variation of Maximum Error (in radians) in the Phase Distribution with the Phase Difference (ϕ12) between the Reference Pixelsa

Tables Icon

Table 2 Variation of Maximum Error (in radians) in the Phase Distribution with the Phase Difference (ϕ12) between the Reference Pixelsa

Tables Icon

Table 3 Variation of Maximum Error (in radians) in the Phase Distribution with Added Intensity Noisea

Equations (25)

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

Ix, y=Ibx, y+Imx, ycosδx, y+ϕ,
Inx, y=cosδx, y+ϕ.
cosδx, y-δx, y=2cosδx, y+ϕcosδx, y+ϕ,
QXX±=cos δx, y±cos δx, ycos ϕ-sin δx, y±sin δx, ysin ϕ.
RXX±=maxQXX±-minQXX±=22±2 cosδx, y-δx, y1/2.
cosδx, y-δx, y=RXX+2-RXX-216.
RXX+2+RXX-2=16.
cosδx, y=a,
cosδx, y-δx2, y2=b,
cosδx2, y2=c.
a=RX1+2-RX1-216,
b=RX2+2-RX2-216,
c=R12+2-R12-216,
sin δx, y=±ac-b1-c2.
δx, y=arctan±ac-ba1-c2.
|cosδx, y-δx, y-cosδ0x, y-δ0x, y|=|ε-ε+cosδ0x, y-δ0x, yε+ε|2 maxε, ε1/4δα2.
tanδx, y=± ac-ba1-c2.
1cos2ΦδΦ=δa a+δb b+δc c±ac-ba1-c2.
δΦ=b1-c2 δa-a1-c2 δb+a2a-bc1-c23/2 δc.
δΦ=|bδa-aδb+a2a-bcδc|1/2δα2.
η=16-RXX+2+RXX-2.
R++ΔR+2+R-+ΔR-2=16,
2R+ΔR++2R-ΔR-=16-R+2+R-2=η.
ΔR+=ηR+2R+2+R-2,
ΔR-=ηR-2R+2+R-2.

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