Abstract

A novel means of quantitatively assessing the performance of a phase-shifting interferometer is further investigated. We show how maximum-likelihood estimation theory can be used to estimate the surface profile from the general case of M noisy, phase-shifted measurements. Monte Carlo experiments show that the maximum-likelihood estimator is unbiased and efficient, achieving the theoretical Cramér–Rao lower bound on the variance of the error. We then use Monte Carlo experiments to compare the performance of the maximum-likelihood estimator with that of two conventional algorithms.

© 1998 Optical Society of America

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References

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  1. E. W. Rogala, H. H. Barrett, “Phase-shifting interferometry and maximum-likelihood estimation theory,” Appl. Opt. 36, 8871–8876 (1997).
    [CrossRef]
  2. C. Rathjen, “Statistical properties of phase-shift algorithms,” J. Opt. Soc. Am. A 12, 1997–2008 (1995).
    [CrossRef]
  3. H. L. Van Trees, Detection, Estimation, and Linear Modulation Theory (Wiley, New York, 1968), Part 1.
  4. M. G. Kendall, A. Stuart, The Advanced Theory of Statistics, 3rd ed. (Hafner, New York, 1973), Vol. 2.
  5. B. R. Frieden, Probability, Statistical Optics, and Data Testing, 2nd ed. (Springer-Verlag, New York, 1991).
    [CrossRef]
  6. D. L. Cohn, J. L. Melsa, Decision and Estimation Theory (McGraw-Hill, New York, 1978).
  7. R. A. Fisher, “Theory of statistical estimation,” Proc. Camb. Phil. Soc. XXII Part 5, 700–725 (1925).
    [CrossRef]
  8. I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965).
  9. J. H. Bruning, D. R. Herriot, J. E. Gallagher, D. P. Resenfeld, A. D. White, D. J. Bangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. 13, 2693–2703 (1974).
    [CrossRef] [PubMed]
  10. C. J. Morgan, “Least-squares estimation in phase-measurement interferometry,” Opt. Lett. 7, 368–370 (1982).
    [CrossRef] [PubMed]
  11. J. E. Greivenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23, 350–352 (1984).
    [CrossRef]
  12. Ref. 5, pp. 378–382.
  13. Ref. 4, pp. 8–18.
  14. Ref. 3, pp. 65–73.
  15. W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, New York, 1992), pp. 274–328.
  16. P. L’Ecuyer, “Efficient and portable combined random number generators,” Commun. ACM 31, 742–774 (1988).
    [CrossRef]
  17. P. Carré, “Installation et utilisation du comparateur photoelectrique et interferentiel du bureau international des poids de mesures,” Metrologia 2, 13–23 (1966).
    [CrossRef]
  18. 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]

1997 (1)

1995 (1)

1988 (1)

P. L’Ecuyer, “Efficient and portable combined random number generators,” Commun. ACM 31, 742–774 (1988).
[CrossRef]

1987 (1)

1984 (1)

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

1982 (1)

1974 (1)

1966 (1)

P. Carré, “Installation et utilisation du comparateur photoelectrique et interferentiel du bureau international des poids de mesures,” Metrologia 2, 13–23 (1966).
[CrossRef]

1925 (1)

R. A. Fisher, “Theory of statistical estimation,” Proc. Camb. Phil. Soc. XXII Part 5, 700–725 (1925).
[CrossRef]

Bangaccio, D. J.

Barrett, H. H.

Bruning, J. H.

Carré, P.

P. Carré, “Installation et utilisation du comparateur photoelectrique et interferentiel du bureau international des poids de mesures,” Metrologia 2, 13–23 (1966).
[CrossRef]

Cohn, D. L.

D. L. Cohn, J. L. Melsa, Decision and Estimation Theory (McGraw-Hill, New York, 1978).

Eiju, T.

Fisher, R. A.

R. A. Fisher, “Theory of statistical estimation,” Proc. Camb. Phil. Soc. XXII Part 5, 700–725 (1925).
[CrossRef]

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, New York, 1992), pp. 274–328.

Frieden, B. R.

B. R. Frieden, Probability, Statistical Optics, and Data Testing, 2nd ed. (Springer-Verlag, New York, 1991).
[CrossRef]

Gallagher, J. E.

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965).

Greivenkamp, J. E.

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

Hariharan, P.

Herriot, D. R.

Kendall, M. G.

M. G. Kendall, A. Stuart, The Advanced Theory of Statistics, 3rd ed. (Hafner, New York, 1973), Vol. 2.

L’Ecuyer, P.

P. L’Ecuyer, “Efficient and portable combined random number generators,” Commun. ACM 31, 742–774 (1988).
[CrossRef]

Melsa, J. L.

D. L. Cohn, J. L. Melsa, Decision and Estimation Theory (McGraw-Hill, New York, 1978).

Morgan, C. J.

Oreb, B. F.

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, New York, 1992), pp. 274–328.

Rathjen, C.

Resenfeld, D. P.

Rogala, E. W.

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965).

Stuart, A.

M. G. Kendall, A. Stuart, The Advanced Theory of Statistics, 3rd ed. (Hafner, New York, 1973), Vol. 2.

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, New York, 1992), pp. 274–328.

Van Trees, H. L.

H. L. Van Trees, Detection, Estimation, and Linear Modulation Theory (Wiley, New York, 1968), Part 1.

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, New York, 1992), pp. 274–328.

White, A. D.

Appl. Opt. (3)

Commun. ACM (1)

P. L’Ecuyer, “Efficient and portable combined random number generators,” Commun. ACM 31, 742–774 (1988).
[CrossRef]

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

Metrologia (1)

P. Carré, “Installation et utilisation du comparateur photoelectrique et interferentiel du bureau international des poids de mesures,” Metrologia 2, 13–23 (1966).
[CrossRef]

Opt. Eng. (1)

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

Opt. Lett. (1)

Proc. Camb. Phil. Soc. (1)

R. A. Fisher, “Theory of statistical estimation,” Proc. Camb. Phil. Soc. XXII Part 5, 700–725 (1925).
[CrossRef]

Other (9)

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965).

H. L. Van Trees, Detection, Estimation, and Linear Modulation Theory (Wiley, New York, 1968), Part 1.

M. G. Kendall, A. Stuart, The Advanced Theory of Statistics, 3rd ed. (Hafner, New York, 1973), Vol. 2.

B. R. Frieden, Probability, Statistical Optics, and Data Testing, 2nd ed. (Springer-Verlag, New York, 1991).
[CrossRef]

D. L. Cohn, J. L. Melsa, Decision and Estimation Theory (McGraw-Hill, New York, 1978).

Ref. 5, pp. 378–382.

Ref. 4, pp. 8–18.

Ref. 3, pp. 65–73.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, New York, 1992), pp. 274–328.

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Figures (4)

Fig. 1
Fig. 1

Monte Carlo results for the four-step Carré algorithm (open circles) and the numerical ML estimator (asterisks). σ ĥ is plotted as a function of phase shift. Solid curve, the theoretical Cramér–Rao lower bound.

Fig. 2
Fig. 2

Monte Carlo results for the five-step Hariharan algorithm (open circles) and the ML estimator (asterisks). σ ĥ is plotted as a function of phase shift. Solid curve, the theoretical Cramér–Rao lower bound.

Fig. 3
Fig. 3

Monte Carlo results comparing the five-step Hariharan algorithm (open circles) and the numerical ML estimator (asterisks) at 90° phase shifts. σ ĥ is plotted as a function of the true value of h. Solid curve, the theoretical Cramér–Rao lower bound.

Fig. 4
Fig. 4

Monte Carlo results comparing the five-step Hariharan algorithm (open circles) and the analytical ML estimator (asterisks) at 72° phase shifts. σ ĥ is plotted as a function of the true value of h. Solid curve, the theoretical Cramér–Rao lower bound.

Tables (1)

Tables Icon

Table 1 Monte Carlo Results for Analytic ML Estimators

Equations (20)

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I m = I ¯ m h + η m ,     m = 0 ,   1 ,   2 , ,   M - 1 ,
I ¯ m h = I a + I b   cos m α - 4 π h λ + Δ Φ ,
ln p I | h = - m = 0 M - 1 ln 2 π σ - 1 2 σ 2 m = 0 M - 1 I m - I ¯ m h 2 .
d d h   { ln [ p I | h ] } h = h ˆ ML I = 1 σ 2 m = 0 M - 1 { [ I m - I ¯ m h ] × d d h   [ I ¯ m h ] } h = h ˆ ML ( I ) = 0 .
4 π I b λ σ 2 m = 0 M - 1 I m - I a - I b cos m α + β sin m α + β = 0 ,
m = 0 M - 1 I m   sin m α + β - I a   sin m α + β - I b   sin m α + β cos m α + β = 0 .
sin β 1 + cos α + cos 2 α +     + cos M - 1 α + cos β 0 + sin α + sin 2 α +   + sin M - 1 α = 0 .
sin β 1 + cos 2 α + cos 4 α +   + cos 2 M - 1 α + cos β 0 + sin 2 α + sin 4 α +   + sin 2 M - 1 α = 0 .
m = 0 M - 1 sin m α = m = 0 M - 1 cos m α = m = 0 M - 1 sin 2 m α = m = 0 M - 1 cos 2 m α = 0 .
I 0   sin β + I 1   sin α + β + I 2   sin 2 α + β + + I M - 1   sin M - 1 α + β = 0 .
I 0 + I 1   cos α + I 2   cos 2 α + + I M - 1   cos M - 1 α sin β = - I 1   sin α + I 2   sin 2 α + + I M - 1   sin M - 1 α cos β .
h ˆ ML I = λ 4 π Δ Φ - tan - 1 - m = 0 M - 1   I m   sin m α m = 0 M   I m   cos m α ,
σ h ˆ 2 = - d M I h ˆ I - h 2 p I | h - d M I d d h ln p I | h 2 p I | h - 1 ,
σ h ˆ 2 σ 2 m = 0 M - 1 d I ¯ m h d h 2 .
σ h ˆ 2 σ 2 4 π λ 2 I b 2 m = 0 M - 1 sin 2 m α - 4 π h λ + .
m = 0 M - 1 sin 2 m α + β = 1 2 M - cos 2 β m = 0 M - 1 cos 2 m α + sin 2 β m = 0 M - 1 sin 2 m α .
m = 0 M - 1 sin 2 m α + β = M 2 .
σ h ˆ 2 λ 2 8 π 2 I b 2 M   σ 2 .
σ h ˆ 2 0.0001 λ 2 8 π 2 M ,
β = tan - 1 2   sin   α   I 1 - I 3 2 I 2 - I 4 - I 0 .

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