Abstract

We present five different eight-point phase-shifting algorithms, each with a different window function. The window function plays a crucial role in determining the phase (wavefront) because it significantly influences phase error. We begin with a simple eight-point algorithm that uses a rectangular window function. We then present alternative algorithms with triangular and bell-shaped window functions that were derived from a new error-reducing multiple-averaging technique. The algorithms with simple (rectangular and triangular) window functions show a large phase error, whereas the algorithms with bell-shaped window functions are considerably less sensitive to different phase-error sources. We demonstrate that the shape of the window function significantly influences phase error.

© 1996 Optical Society of America

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References

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  1. K. Creath, “Temporal phase measurement methods,” in Interferogram Analysis: Digital Fringe Pattern Measurement Technique, D. W. Robinson, G. T. Reid, eds. (Institute of Physics, Bristol, U.K., 1993), Chap. 4, pp. 94–140.
  2. J. E. Greivenkamp, J. H. Bruning, “Phase shifting interferometers,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, New York, 1992), Chap. 14.
  3. See, for example, M. Kujawinska, “Spatial phase measurement methods,” in Interferogram Analysis: Digital Fringe Pattern Measurement Technique, D. W. Robinson, G. T. Reid, eds. (Institute of Physics, Bristol, U.K., 1993), Chap. 5, pp. 141–193.
  4. D. Malacara, S. L. DeVore, “Direct measuring interferometry,” in Interferogram Analysis: Digital Fringe Pattern Measurement Technique, D. W. Robinson, G. T. Reid, eds. (Institute of Physics, Bristol, U.K., 1993), Chap. 13.6, pp. 494–500, and references within.
  5. J. Schmit, K. Creath, M. Kujawinska, “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).
  6. K. Creath, J. Schmit, “N-point spatial phase measurement techniques for nondestructive testing,” Opt. Lasers Eng. 24, 365–379 (1996).
    [Crossref]
  7. C. J. Morgan, “Least-squares estimation in phase-measurement interferometry,” Opt. Lett. 7, 368–373 (1982).
    [Crossref] [PubMed]
  8. J. E. Greivenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23, 350–352 (1984).
  9. K. H. Womack, “Interferometric phase measurement using spatial synchronous detection,” Opt. Eng. 23, 391–395 (1984).
  10. K. G. Larkin, B. F. Oreb, “Design and assessment of symmetrical phase-shifting algorithms,” J. Opt. Soc. Am. A 9, 1740–1748 (1992).
    [Crossref]
  11. K. Freischald, C. Koliopoulos, “Fourier description of digital phase-measuring interferometry,” J. Opt. Soc. Am. A 7, 542–551 (1990).
    [Crossref]
  12. Y. Surrel, “Phase stepping: a new self-calibrating algorithm,” Appl. Opt. 32, 3598–3600 (1993).
    [Crossref] [PubMed]
  13. J. Schwider, “Advanced evaluation techniques in interferometry,” in Progress in Optics XXVIII, E. Wolf, ed. (Elsevier, New York, 1990), Chap. 4, pp. 271–359.
    [Crossref]
  14. J. Schmit, K. Creath, “Extended averaging technique for derivation of error-compensating algorithms in phase shifting interferometry,” Appl. Opt. 34, 3610–3619 (1995).
    [Crossref] [PubMed]
  15. 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]
  16. J. Bruning, D. H. Herriot, J. E. Gallagher, D. P. Rosenfeld, A. D. White, D. J. Brangccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. 13, 2693–2703 (1974).
    [Crossref] [PubMed]
  17. J. Schwider, R. Burow, K. E. Elsner, J. Grzanna, R. Spolaczyk, K. Merkel, “Digital wave-front measuring interferometry: some systematic error sources,” Appl. Opt. 22, 3421–3432 (1983).
    [Crossref] [PubMed]
  18. P. Hariharan, B. F. Oreb, T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. 26, 2504–2505 (1987).
    [Crossref] [PubMed]
  19. J. Schwider, O. Falkenstorfer, H. Screiber, A. Zoller, N. Streibl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32, 1883–1885 (1993).
    [Crossref]
  20. 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]
  21. P. de Groot, L. Deck, “Long-wavelength laser diode interferometer for surface flatness measurement,” in Optical Measurements and Sensors for the Process Industries, C. Gorecki, R. W. Preater, eds., Proc. SPIE2248, 136–140 (1994).
  22. M. Takeda, “Spatial carrier fringe pattern analysis and its application to precision interferometry: an overview,” Ind. Met. 1, 79–99 (1990).
  23. M. Kujawinska, J. Wojciak, “High accuracy fourier transform fringe pattern analysis,” Opt. Lasers Eng. 14, 325–339 (1991).
    [Crossref]
  24. R. W. Ramirez, The FFT: Fundamentals and Concepts (Prentice-Hall, Englewood Cliffs, N.J., 1985).
  25. A. A. Malcolm, D. R. Burton, M. J. A. Lalor, “A study of the effect of windowing on the accuracy of surface measurements obtained from the Fourier analysis of fringe pattern,” in Proceedings of FASIG Fringe Analysis ’89, Loughborough University of Technology, 4–5 April 1989 (The Fringe Analysis Special Interest Group, 1989).
  26. C. Joenathan, “Phase-measuring interferometry: new methods and error analysis,” Appl. Opt. 33, 4147–4155 (1994).
    [Crossref] [PubMed]
  27. P. de Groot, “Phase-shift calibration errors in interferometers with spherical Fizeau cavities,” Appl. Opt. 34, 2856–2863 (1995).
    [Crossref]
  28. R. C. Moore, F. H. Slaymaker, “Direct measurement of phase in a spherical-wave Fizeau interferometer,” Appl. Opt. 19, 2196–2200 (1980).
    [Crossref] [PubMed]
  29. K. Creath, P. Hariharan, “Phase-shifting errors in interferometric tests with high-numerical-aperture reference surfaces,” Appl. Opt. 33, 24–25 (1994).
    [Crossref] [PubMed]

1996 (1)

K. Creath, J. Schmit, “N-point spatial phase measurement techniques for nondestructive testing,” Opt. Lasers Eng. 24, 365–379 (1996).
[Crossref]

1995 (4)

1994 (2)

1993 (2)

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

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

1992 (1)

1991 (1)

M. Kujawinska, J. Wojciak, “High accuracy fourier transform fringe pattern analysis,” Opt. Lasers Eng. 14, 325–339 (1991).
[Crossref]

1990 (2)

M. Takeda, “Spatial carrier fringe pattern analysis and its application to precision interferometry: an overview,” Ind. Met. 1, 79–99 (1990).

K. Freischald, C. Koliopoulos, “Fourier description of digital phase-measuring interferometry,” J. Opt. Soc. Am. A 7, 542–551 (1990).
[Crossref]

1987 (1)

1984 (2)

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

K. H. Womack, “Interferometric phase measurement using spatial synchronous detection,” Opt. Eng. 23, 391–395 (1984).

1983 (1)

1982 (1)

1980 (1)

1974 (1)

Brangccio, D. J.

Bruning, J.

Bruning, J. H.

J. E. Greivenkamp, J. H. Bruning, “Phase shifting interferometers,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, New York, 1992), Chap. 14.

Burow, R.

Burton, D. R.

A. A. Malcolm, D. R. Burton, M. J. A. Lalor, “A study of the effect of windowing on the accuracy of surface measurements obtained from the Fourier analysis of fringe pattern,” in Proceedings of FASIG Fringe Analysis ’89, Loughborough University of Technology, 4–5 April 1989 (The Fringe Analysis Special Interest Group, 1989).

Creath, K.

K. Creath, J. Schmit, “N-point spatial phase measurement techniques for nondestructive testing,” Opt. Lasers Eng. 24, 365–379 (1996).
[Crossref]

J. Schmit, K. Creath, “Extended averaging technique for derivation of error-compensating algorithms in phase shifting interferometry,” Appl. Opt. 34, 3610–3619 (1995).
[Crossref] [PubMed]

K. Creath, P. Hariharan, “Phase-shifting errors in interferometric tests with high-numerical-aperture reference surfaces,” Appl. Opt. 33, 24–25 (1994).
[Crossref] [PubMed]

K. Creath, “Temporal phase measurement methods,” in Interferogram Analysis: Digital Fringe Pattern Measurement Technique, D. W. Robinson, G. T. Reid, eds. (Institute of Physics, Bristol, U.K., 1993), Chap. 4, pp. 94–140.

J. Schmit, K. Creath, M. Kujawinska, “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).

de Groot, P.

P. de Groot, “Phase-shift calibration errors in interferometers with spherical Fizeau cavities,” Appl. Opt. 34, 2856–2863 (1995).
[Crossref]

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]

P. de Groot, L. Deck, “Long-wavelength laser diode interferometer for surface flatness measurement,” in Optical Measurements and Sensors for the Process Industries, C. Gorecki, R. W. Preater, eds., Proc. SPIE2248, 136–140 (1994).

Deck, L.

P. de Groot, L. Deck, “Long-wavelength laser diode interferometer for surface flatness measurement,” in Optical Measurements and Sensors for the Process Industries, C. Gorecki, R. W. Preater, eds., Proc. SPIE2248, 136–140 (1994).

DeVore, S. L.

D. Malacara, S. L. DeVore, “Direct measuring interferometry,” in Interferogram Analysis: Digital Fringe Pattern Measurement Technique, D. W. Robinson, G. T. Reid, eds. (Institute of Physics, Bristol, U.K., 1993), Chap. 13.6, pp. 494–500, and references within.

Eiju, T.

Elsner, K. E.

Falkenstorfer, O.

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

Farrant, D. I.

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]

Freischald, K.

Gallagher, J. E.

Greivenkamp, J. E.

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

J. E. Greivenkamp, J. H. Bruning, “Phase shifting interferometers,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, New York, 1992), Chap. 14.

Grzanna, J.

Hariharan, P.

Herriot, D. H.

Hibino, K.

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]

Joenathan, C.

Koliopoulos, C.

Kujawinska, M.

M. Kujawinska, J. Wojciak, “High accuracy fourier transform fringe pattern analysis,” Opt. Lasers Eng. 14, 325–339 (1991).
[Crossref]

See, for example, M. Kujawinska, “Spatial phase measurement methods,” in Interferogram Analysis: Digital Fringe Pattern Measurement Technique, D. W. Robinson, G. T. Reid, eds. (Institute of Physics, Bristol, U.K., 1993), Chap. 5, pp. 141–193.

J. Schmit, K. Creath, M. Kujawinska, “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).

Lalor, M. J. A.

A. A. Malcolm, D. R. Burton, M. J. A. Lalor, “A study of the effect of windowing on the accuracy of surface measurements obtained from the Fourier analysis of fringe pattern,” in Proceedings of FASIG Fringe Analysis ’89, Loughborough University of Technology, 4–5 April 1989 (The Fringe Analysis Special Interest Group, 1989).

Larkin, K. G.

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]

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

Malacara, D.

D. Malacara, S. L. DeVore, “Direct measuring interferometry,” in Interferogram Analysis: Digital Fringe Pattern Measurement Technique, D. W. Robinson, G. T. Reid, eds. (Institute of Physics, Bristol, U.K., 1993), Chap. 13.6, pp. 494–500, and references within.

Malcolm, A. A.

A. A. Malcolm, D. R. Burton, M. J. A. Lalor, “A study of the effect of windowing on the accuracy of surface measurements obtained from the Fourier analysis of fringe pattern,” in Proceedings of FASIG Fringe Analysis ’89, Loughborough University of Technology, 4–5 April 1989 (The Fringe Analysis Special Interest Group, 1989).

Merkel, K.

Moore, R. C.

Morgan, C. J.

Oreb, B. F.

Ramirez, R. W.

R. W. Ramirez, The FFT: Fundamentals and Concepts (Prentice-Hall, Englewood Cliffs, N.J., 1985).

Rosenfeld, D. P.

Schmit, J.

K. Creath, J. Schmit, “N-point spatial phase measurement techniques for nondestructive testing,” Opt. Lasers Eng. 24, 365–379 (1996).
[Crossref]

J. Schmit, K. Creath, “Extended averaging technique for derivation of error-compensating algorithms in phase shifting interferometry,” Appl. Opt. 34, 3610–3619 (1995).
[Crossref] [PubMed]

J. Schmit, K. Creath, M. Kujawinska, “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, O. Falkenstorfer, H. Screiber, A. Zoller, N. Streibl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32, 1883–1885 (1993).
[Crossref]

J. Schwider, R. Burow, K. E. Elsner, 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. Wolf, ed. (Elsevier, New York, 1990), Chap. 4, pp. 271–359.
[Crossref]

Screiber, H.

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

Slaymaker, F. H.

Spolaczyk, R.

Streibl, N.

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

Surrel, Y.

Takeda, M.

M. Takeda, “Spatial carrier fringe pattern analysis and its application to precision interferometry: an overview,” Ind. Met. 1, 79–99 (1990).

White, A. D.

Wojciak, J.

M. Kujawinska, J. Wojciak, “High accuracy fourier transform fringe pattern analysis,” Opt. Lasers Eng. 14, 325–339 (1991).
[Crossref]

Womack, K. H.

K. H. Womack, “Interferometric phase measurement using spatial synchronous detection,” Opt. Eng. 23, 391–395 (1984).

Zoller, A.

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

Appl. Opt. (10)

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

J. Schmit, K. Creath, “Extended averaging technique for derivation of error-compensating algorithms in phase shifting interferometry,” Appl. Opt. 34, 3610–3619 (1995).
[Crossref] [PubMed]

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]

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

J. Schwider, R. Burow, K. E. Elsner, J. Grzanna, R. Spolaczyk, K. Merkel, “Digital wave-front measuring interferometry: some systematic error sources,” Appl. Opt. 22, 3421–3432 (1983).
[Crossref] [PubMed]

P. Hariharan, B. F. Oreb, T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. 26, 2504–2505 (1987).
[Crossref] [PubMed]

C. Joenathan, “Phase-measuring interferometry: new methods and error analysis,” Appl. Opt. 33, 4147–4155 (1994).
[Crossref] [PubMed]

P. de Groot, “Phase-shift calibration errors in interferometers with spherical Fizeau cavities,” Appl. Opt. 34, 2856–2863 (1995).
[Crossref]

R. C. Moore, F. H. Slaymaker, “Direct measurement of phase in a spherical-wave Fizeau interferometer,” Appl. Opt. 19, 2196–2200 (1980).
[Crossref] [PubMed]

K. Creath, P. Hariharan, “Phase-shifting errors in interferometric tests with high-numerical-aperture reference surfaces,” Appl. Opt. 33, 24–25 (1994).
[Crossref] [PubMed]

Ind. Met. (1)

M. Takeda, “Spatial carrier fringe pattern analysis and its application to precision interferometry: an overview,” Ind. Met. 1, 79–99 (1990).

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

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

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]

Opt. Eng. (3)

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

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

K. H. Womack, “Interferometric phase measurement using spatial synchronous detection,” Opt. Eng. 23, 391–395 (1984).

Opt. Lasers Eng. (2)

K. Creath, J. Schmit, “N-point spatial phase measurement techniques for nondestructive testing,” Opt. Lasers Eng. 24, 365–379 (1996).
[Crossref]

M. Kujawinska, J. Wojciak, “High accuracy fourier transform fringe pattern analysis,” Opt. Lasers Eng. 14, 325–339 (1991).
[Crossref]

Opt. Lett. (1)

Other (9)

K. Creath, “Temporal phase measurement methods,” in Interferogram Analysis: Digital Fringe Pattern Measurement Technique, D. W. Robinson, G. T. Reid, eds. (Institute of Physics, Bristol, U.K., 1993), Chap. 4, pp. 94–140.

J. E. Greivenkamp, J. H. Bruning, “Phase shifting interferometers,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, New York, 1992), Chap. 14.

See, for example, M. Kujawinska, “Spatial phase measurement methods,” in Interferogram Analysis: Digital Fringe Pattern Measurement Technique, D. W. Robinson, G. T. Reid, eds. (Institute of Physics, Bristol, U.K., 1993), Chap. 5, pp. 141–193.

D. Malacara, S. L. DeVore, “Direct measuring interferometry,” in Interferogram Analysis: Digital Fringe Pattern Measurement Technique, D. W. Robinson, G. T. Reid, eds. (Institute of Physics, Bristol, U.K., 1993), Chap. 13.6, pp. 494–500, and references within.

J. Schmit, K. Creath, M. Kujawinska, “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).

J. Schwider, “Advanced evaluation techniques in interferometry,” in Progress in Optics XXVIII, E. Wolf, ed. (Elsevier, New York, 1990), Chap. 4, pp. 271–359.
[Crossref]

R. W. Ramirez, The FFT: Fundamentals and Concepts (Prentice-Hall, Englewood Cliffs, N.J., 1985).

A. A. Malcolm, D. R. Burton, M. J. A. Lalor, “A study of the effect of windowing on the accuracy of surface measurements obtained from the Fourier analysis of fringe pattern,” in Proceedings of FASIG Fringe Analysis ’89, Loughborough University of Technology, 4–5 April 1989 (The Fringe Analysis Special Interest Group, 1989).

P. de Groot, L. Deck, “Long-wavelength laser diode interferometer for surface flatness measurement,” in Optical Measurements and Sensors for the Process Industries, C. Gorecki, R. W. Preater, eds., Proc. SPIE2248, 136–140 (1994).

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Figures (7)

Fig. 1
Fig. 1

Window function shapes.

Fig. 2
Fig. 2

Values of the reference signals (designated by circles) at the sampling points for reference signals with initial phases equal to 0 and π/4. The corresponding sample numbers are given below each reference signal.

Fig. 3
Fig. 3

Intensity signal truncated by the rectangular window functions in FFT and n-point techniques.

Fig. 4
Fig. 4

Fourier spectra of signals multiplied by the 8-RECT and 8-BELL7 window functions, shown only for positive frequencies up to the Nyquist frequency. Signals are sampled with spacing s. The period of the measured sinusoidal signal equals 4s, and the width of the window equals 8s. Dots represent sampling points in each spectrum.

Fig. 5
Fig. 5

P–V phase error versus percent of phase-shift miscalibration.

Fig. 6
Fig. 6

Phase errors for three eight-point algorithms, due to 6.25% phase-shift miscalibration (2-fringe tilt miscalibration).

Fig. 7
Fig. 7

Phase retrieved from real data with 8-RECT and 8-BELL7 algorithms.

Equations (22)

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

tan [ φ ( x i ) ] = i = 1 M I i ( x i ) sin ( 2 π K x i + θ ) h ( x i ) i = 1 M I i ( x i ) cos ( 2 π K x i + θ ) h ( x i ) .
tan φ = N / D ,
( M + 1 ) p θ = M p θ + M p θ + π / 2 = N 1 + N 2 D 1 + D 2 = N D ,
( M + 2 ) p θ = ( M + 1 ) p θ + ( M + 1 ) p θ + π / 2 = N + N D + D = N 1 + 2 N 2 + N 3 D 1 + 2 D 2 + D 3 .
4 p π / 4 = I 1 + I 2 - I 3 - I 4 I 1 - I 2 - I 3 + I 4 ,
5 p π / 4 = I 1 + 2 I 2 - 2 I 3 - 2 I 4 + I 5 I 1 - 2 I 2 - 2 I 3 + 2 I 4 + I 5 = 4 p π / 4 + 4 p 3 π / 4 ,
6 p π / 4 = I 1 + 3 I 2 - 4 I 3 - 4 I 4 + 3 I 5 + I 6 I 1 - 3 I 2 - 4 I 3 + 4 I 4 + 3 I 5 - I 6 = 5 p π / 4 + 5 p 3 π / 4 ,
7 p π / 4 = I 1 + 4 I 2 - 7 I 3 - 8 I 4 + 7 I 5 + 4 I 6 - I 7 I 1 - 4 I 2 - 7 I 3 + 8 I 4 + 7 I 5 - 4 I 6 - I 7 = 6 p π / 4 + 6 p 3 π / 4 .
( M + 2 ) p θ = M p θ + M p θ + π / 2 + M p θ + π = N 1 + N 2 + N 3 D 1 + D 2 + D 3 = N D ,
( M + 3 ) p θ = M p θ + M p θ + π / 2 + M p θ + π + M p θ + 3 π / 2 = N 1 + N 2 + N 3 + N 4 D 1 + D 2 + D 3 + D 4 = N D .
8 - RECT = I 1 + I 2 - I 3 - I 4 + I 5 + I 6 - I 7 - I 8 I 1 - I 2 - I 3 + I 4 + I 5 - I 6 - I 7 + I 8 .
8 - TRI 4 = I 1 + 2 I 2 - 3 I 3 - 4 I 4 + 4 I 5 + 3 I 6 - 2 I 7 - I 8 I 1 - 2 I 2 - 3 I 3 + 4 I 4 + 4 I 5 - 3 I 6 - 2 I 7 + I 8 = 4 p π / 4 + 4 p 3 π / 4 + 4 p 5 π / 4 + 4 p 7 π / 4 + 4 p 9 π / 4 ,
8 - TRI 5 = I 1 + 3 I 2 - 5 I 3 - 7 I 4 + 7 I 5 + 5 I 6 - 3 I 7 - I 8 I 1 - 3 I 2 - 5 I 3 + 7 I 4 + 7 I 5 - 5 I 6 - 3 I 7 + I 8 = 5 p π / 4 + 5 p 3 π / 4 + 5 p 5 π / 4 + 5 p 7 π / 4 .
8 - BELL 6 = I 1 + 4 I 2 - 8 I 3 - 11 I 4 + 11 I 5 + 8 I 6 - 4 I 7 - I 8 I 1 - 4 I 2 - 8 I 3 + 11 I 4 + 11 I 5 - 8 I 6 - 4 I 7 + I 8 = 6 p π / 4 + 6 p 3 π / 4 + 6 p 5 π / 4 ,
8 - BELL 7 = I 1 + 5 I 2 - 11 I 3 - 15 I 4 + 15 I 5 + 11 I 6 - 5 I 7 - I 8 I 1 - 5 I 2 - 11 I 3 + 15 I 4 + 15 I 5 - 11 I 6 - 5 I 7 + I 8 = 7 p π / 4 + 7 p 3 π / 4 .
4 p 0 = 2 ( I 2 - I 3 ) I 1 - I 2 - I 3 + I 4 .
4 p 0 = I 2 - I 4 I 1 - I 3 .
8 - REC T = I 2 - I 4 + I 6 - I 8 I 1 - I 3 + I 5 - I 7 ,
8 - TRI 4 = 2 I 2 - 4 I 4 + 3 I 6 - I 8 I 1 - 3 I 3 + 4 I 5 - 2 I 7 = 4 p 0 + 4 p π / 2 + 4 p π + 4 p 3 π / 2 + 4 p 0 ,
8 - TRI 5 = 3 I 2 - 7 I 4 + 5 I 6 - I 8 I 1 - 7 I 3 + 5 I 5 - 3 I 7 = 5 p 0 + 5 p π / 2 + 5 p π + 5 p 3 π / 2 ,
8 - BELL 6 = 4 I 2 - 11 I 4 + 8 I 6 - I 8 I 1 - 8 I 3 + 11 I 5 - 4 I 7 = 6 p 0 + 6 p π / 2 + 6 p π ,
8 - BELL 7 = 5 I 2 - 15 I 4 + 11 I 6 - I 8 I 1 - 11 I 3 + 15 I 5 - 5 I 7 = 7 p 0 + 7 p π / 2 .

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