OSA's Digital Library

Applied Optics

Applied Optics


  • Vol. 36, Iss. 34 — Dec. 1, 1997
  • pp: 8871–8876

Phase-shifting interferometry and maximum-likelihood estimation theory

Eric W. Rogala and Harrison H. Barrett  »View Author Affiliations

Applied Optics, Vol. 36, Issue 34, pp. 8871-8876 (1997)

View Full Text Article

Enhanced HTML    Acrobat PDF (249 KB)

Browse Journals / Lookup Meetings

Browse by Journal and Year


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools



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

Original Manuscript: April 28, 1997
Revised Manuscript: August 1, 1997
Published: December 1, 1997

Eric W. Rogala and Harrison H. Barrett, "Phase-shifting interferometry and maximum-likelihood estimation theory," Appl. Opt. 36, 8871-8876 (1997)

Sort:  Author  |  Year  |  Journal  |  Reset  


  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.

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.


Fig. 1 Fig. 2 Fig. 3

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited