Abstract

A novel means of quantitatively assessing the performance of a phase-shifting interferometer is presented. We show how maximum-likelihood estimation theory can be used to estimate the surface-height profile from four noisy phase-shifted measurements. Remarkably, the analytical expression for the maximum-likelihood estimator is identical to the classical four-step algorithm, thereby rooting the traditional method on a statistically sound foundation. Furthermore, a Monte Carlo experiment shows the maximum-likelihood estimator is unbiased and efficient, achieving the theoretical Cramer–Rao lower bound on the variance of the error. This technique is then used to show that the performance is a function of the ratio of the irradiances from each arm, with the optimal performance occurring, not surprisingly, when the irradiances from the two arms are equal.

© 1997 Optical Society of America

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References

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  1. J. E. Greivenkamp, J. H. Bruning, “Phase-shifting interferometers,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, New York, 1992), pp. 501–598.
  2. K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1988), Vol. 26, pp. 349–393.
    [CrossRef]
  3. K. Creath, “Comparison of phase-measurement algorithms,” in Surface Characterization and Testing, K. Creath, ed., Proc. SPIE, 680, 19–29 (1986).
    [CrossRef]
  4. H. L. Van Trees, Detection, Estimation, and Linear Modulation Theory (Wiley, New York, 1968), Part 1.
  5. M. G. Kendall, A. Stuart, The Advanced Theory of Statistics, 3rd ed. (Hafner, New York, 1973), Vol. 2.
  6. B. R. Frieden, Probability, Statistical Optics, and Data Testing, 2nd ed. (Springer-Verlag, New York, 1991).
    [CrossRef]
  7. D. L. Cohn, J. L. Melsa, Decision and Estimation Theory (McGraw-Hill, New York, 1978).
  8. Ref. 4, p. 66.
  9. R. A. Fisher, “Theory of statistical estimation,” Proc. Cambridge Philos. Soc. 22, 700–725 (1925).
    [CrossRef]
  10. Ref. 4, p. 68.
  11. Ref. 5, p. 38.
  12. 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.
  13. P. L’Ecuyer, “Efficient and portable combined random number generators,” Commun. ACM 31, 742–749 (1988).
    [CrossRef]
  14. M. Quenouille, “Approximate tests of correlation in time series,” J. R. Stat. Soc. Ser. B. 11, 18–84 (1949).
  15. J. Tukey, “Bias and confidence in not quite large samples,” Ann. Math. Stat. Soc. Ser. B. 29, 614 (1958).
  16. B. Efron, The Jackknife, the Bootstrap and Other Resampling Plans (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1982).
    [CrossRef]
  17. K. M. Wolter, Introduction to Variance Estimation (Springer-Verlag, New York, 1985), pp. 153–200.
  18. F. Mostellar, J. W. Tukey, Data Analysis and Regression: A Second Course in Statistics (Addison-Wesley, Reading, Mass., 1977), pp. 119–162.

1988 (1)

P. L’Ecuyer, “Efficient and portable combined random number generators,” Commun. ACM 31, 742–749 (1988).
[CrossRef]

1958 (1)

J. Tukey, “Bias and confidence in not quite large samples,” Ann. Math. Stat. Soc. Ser. B. 29, 614 (1958).

1949 (1)

M. Quenouille, “Approximate tests of correlation in time series,” J. R. Stat. Soc. Ser. B. 11, 18–84 (1949).

1925 (1)

R. A. Fisher, “Theory of statistical estimation,” Proc. Cambridge Philos. Soc. 22, 700–725 (1925).
[CrossRef]

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), pp. 501–598.

Cohn, D. L.

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

Creath, K.

K. Creath, “Comparison of phase-measurement algorithms,” in Surface Characterization and Testing, K. Creath, ed., Proc. SPIE, 680, 19–29 (1986).
[CrossRef]

K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1988), Vol. 26, pp. 349–393.
[CrossRef]

Efron, B.

B. Efron, The Jackknife, the Bootstrap and Other Resampling Plans (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1982).
[CrossRef]

Fisher, R. A.

R. A. Fisher, “Theory of statistical estimation,” Proc. Cambridge Philos. Soc. 22, 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]

Greivenkamp, J. E.

J. E. Greivenkamp, J. H. Bruning, “Phase-shifting interferometers,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, New York, 1992), pp. 501–598.

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, 742–749 (1988).
[CrossRef]

Melsa, J. L.

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

Mostellar, F.

F. Mostellar, J. W. Tukey, Data Analysis and Regression: A Second Course in Statistics (Addison-Wesley, Reading, Mass., 1977), pp. 119–162.

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.

Quenouille, M.

M. Quenouille, “Approximate tests of correlation in time series,” J. R. Stat. Soc. Ser. B. 11, 18–84 (1949).

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.

Tukey, J.

J. Tukey, “Bias and confidence in not quite large samples,” Ann. Math. Stat. Soc. Ser. B. 29, 614 (1958).

Tukey, J. W.

F. Mostellar, J. W. Tukey, Data Analysis and Regression: A Second Course in Statistics (Addison-Wesley, Reading, Mass., 1977), pp. 119–162.

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.

Wolter, K. M.

K. M. Wolter, Introduction to Variance Estimation (Springer-Verlag, New York, 1985), pp. 153–200.

Ann. Math. Stat. Soc. Ser. B. (1)

J. Tukey, “Bias and confidence in not quite large samples,” Ann. Math. Stat. Soc. Ser. B. 29, 614 (1958).

Commun. ACM (1)

P. L’Ecuyer, “Efficient and portable combined random number generators,” Commun. ACM 31, 742–749 (1988).
[CrossRef]

J. R. Stat. Soc. Ser. B. (1)

M. Quenouille, “Approximate tests of correlation in time series,” J. R. Stat. Soc. Ser. B. 11, 18–84 (1949).

Proc. Cambridge Philos. Soc. (1)

R. A. Fisher, “Theory of statistical estimation,” Proc. Cambridge Philos. Soc. 22, 700–725 (1925).
[CrossRef]

Other (14)

Ref. 4, p. 68.

Ref. 5, p. 38.

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.

B. Efron, The Jackknife, the Bootstrap and Other Resampling Plans (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1982).
[CrossRef]

K. M. Wolter, Introduction to Variance Estimation (Springer-Verlag, New York, 1985), pp. 153–200.

F. Mostellar, J. W. Tukey, Data Analysis and Regression: A Second Course in Statistics (Addison-Wesley, Reading, Mass., 1977), pp. 119–162.

J. E. Greivenkamp, J. H. Bruning, “Phase-shifting interferometers,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, New York, 1992), pp. 501–598.

K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1988), Vol. 26, pp. 349–393.
[CrossRef]

K. Creath, “Comparison of phase-measurement algorithms,” in Surface Characterization and Testing, K. Creath, ed., Proc. SPIE, 680, 19–29 (1986).
[CrossRef]

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. 4, p. 66.

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Figures (3)

Fig. 1
Fig. 1

Estimate of the mean value, 〈ĥ〉, for five independent Monte Carlo experiments, each consisting of 106 trials. The noise level was σ = 1% of (Iref + Itest).

Fig. 2
Fig. 2

Estimate of the standard deviation on ĥ, σĥ, for five independent Monte Carlo experiments, each consisting of 106 trials. The noise level was σ = 1% of (Iref + Itest).

Fig. 3
Fig. 3

Lower bound as a function of irradiance ratio. X denotes Monte Carlo results, while the solid curve denotes theoretically calculated Cramer–Rao lower bounds. The errors on the Monte Carlo results are too small for the scale shown here, and error bars are not shown.

Tables (3)

Tables Icon

Table 1 Results from Five Independent Monte Carlo Experiments, Each Consisting of 1 × 106 Trialsa

Tables Icon

Table 2 Results from Monte Carlo Experiment and Theoretically Calculated Cramer–Rao Lower Boundsa

Tables Icon

Table 3 σĥ As a Function of Irradiance Ratio for Noise σ = 0.01 (Iref + Itest)

Equations (19)

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

Im=Īmh+nm,  m=0, 1, 2, 3,
pIm|h=12πσ exp-Im-Īmh22σ2.
pI|h=m=0312πσ exp-Im-I¯mh22σ2.
lnpI|h=m=03 ln12πσ+m=03-Im-Īmh22σ2.
ddhlnpI|hh=ĥMLI=m=031σ2Im-Īmh×ddhĪmhh=ĥMLI=0.
σĥ2=-d4IĥI-h2pI|h1-d4IddhlnpI|h2pI|h,
Īmh=Iref+Itest+2IrefItest cosmπ2-4πhλ+ΔΦ.
1σ2m=03Im-Iref+Itest+2IrefItest×cosmπ2-4πhλ+ΔΦddhIref+Itest+2IrefItest×cosmπ2-4πhλ+ΔΦh=ĥMLI=0.
m=03Im-Iref-Itest-2IrefItest cosmπ2-4πĥMLλ+ΔΦsinmπ2-4πĥMLλ+ΔΦ=0.
I0-Iref-Itest-2IrefItest cos4πĥMLλ-ΔΦ×-sin4πĥMLλ-ΔΦ+I1-Iref-Itest-2IrefItest sin4πĥMLλ-ΔΦcos4πĥMLλ-ΔΦ+I2-Iref-Itest+2IrefItest cos4πĥMLλ-ΔΦ×sin4πĥMLλ-ΔΦ+I3-Iref-Itest+2IrefItest sin4πĥMLλ-ΔΦ×-cos4πĥMLλ-ΔΦ=0.
sin4πĥMLλ-ΔΦI2-I0+cos4πĥMLλ-ΔΦ×I1-I3=0.
ĥML=λ4πtan-1I3-I1I2-I0+ΔΦ.
f=ddhfh,  f=d2dh2fh,
f+Δhf=0.
hminj=hminj-1+Δh,  hmin1=hb from the bracketing scheme above.
fj-fj-110-12.
θˆα=kθˆ-k-1θˆα.
θˆJK=α=1kθˆαk,
σθˆJK2=1kk-1α=1kθˆα-θˆ2.

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