## Maximum-likelihood curve-fitting scheme for experiments with pulsed lasers subject to intensity fluctuations

Applied Optics, Vol. 42, Issue 9, pp. 1551-1563 (2003)

http://dx.doi.org/10.1364/AO.42.001551

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### Abstract

Evaluation schemes, e.g., least-squares fitting, are not generally applicable to any types of experiments. If the evaluation schemes were not derived from a measurement model that properly described the experiment to be evaluated, poorer precision or accuracy than attainable from the measured data could result. We outline ways in which statistical data evaluation schemes should be derived for all types of experiment, and we demonstrate them for laser-spectroscopic experiments, in which pulse-to-pulse fluctuations of the laser power cause correlated variations of laser intensity and generated signal intensity. The method of maximum likelihood is demonstrated in the derivation of an appropriate fitting scheme for this type of experiment. Statistical data evaluation contains the following steps. First, one has to provide a measurement model that considers statistical variation of all enclosed variables. Second, an evaluation scheme applicable to this particular model has to be derived or provided. Third, the scheme has to be characterized in terms of accuracy and precision. A criterion for accepting an evaluation scheme is that it have accuracy and precision as close as possible to the theoretical limit. The fitting scheme derived for experiments with pulsed lasers is compared to well-established schemes in terms of fitting power and rational functions. The precision is found to be as much as three times better than for simple least-squares fitting. Our scheme also suppresses the bias on the estimated model parameters that other methods may exhibit if they are applied in an uncritical fashion. We focus on experiments in nonlinear spectroscopy, but the fitting scheme derived is applicable in many scientific disciplines.

© 2003 Optical Society of America

**OCIS Codes**

(000.4430) General : Numerical approximation and analysis

(000.5490) General : Probability theory, stochastic processes, and statistics

(120.3940) Instrumentation, measurement, and metrology : Metrology

(300.6360) Spectroscopy : Spectroscopy, laser

**History**

Original Manuscript: December 10, 2001

Revised Manuscript: November 3, 2002

Published: March 20, 2003

**Citation**

Thomas Metz, Joachim Walewski, and Clemens F. Kaminski, "Maximum-likelihood curve-fitting scheme for experiments with pulsed lasers subject to intensity fluctuations," Appl. Opt. **42**, 1551-1563 (2003)

http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-42-9-1551

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### References

- P. H. Garthwaite, I. T. Jolliffe, B. Jones, Statistical Inference (Prentice-Hall, London, 1995).
- M. G. Kendall, A. Stuart, The Advanced Theory of Statistics, 3rd ed. (Griffin, London, 1973), Vol. 2.
- J. F. Kenney, E. S. Keeping, Mathematics of Statistics, 2nd ed. (Van Nostrand, Princeton, N.J., 1951), Part II.
- R. A. Fisher, “On an absolute criterion for fitting frequency curves,” Mess. Math. 41, 155–160 (1912).
- P. J. Mohr, B. N. Taylor, “CODATA recommended values of the fundamental physical constants: 1998,” Rev. Mod. Phys. 72, 351–495 (2000). [CrossRef]
- S. Brandt, Statistical and Computational Methods in Data Analysis (North-Holland, Amsterdam, 1970).
- H. I. Britt, R. H. Luecke, “The estimation of parameters in nonlinear implicit models,” Technometrics 15, 233–247 (1973). [CrossRef]
- A. Celmiņš, “Least squares adjustment with finite residuals for non-linear constraints and partially correlated data,” in Nineteenth Conference of Army Mathematicians (U.S. Army Research Office, Washington, D.C., 1973), AROD-73-3-PT-2, pp. 809–858.
- W. H. Jefferys, “On the method of least squares,” Astron. J. 85, 177–181 (1980). [CrossRef]
- W. H. Jefferys, “On the method of least squares. II,” Astron. J. 86, 149–155 (1981). [CrossRef]
- MATLAB Optimization Toolbox, V. 2.2. Mathworks: http://www.mathworks.com/access/helpdesk/help/toolbox/optim/optim.shtml .
- D. Taupin, Probabilities, Data Reduction and Error Analysis in the Physical Sciences (Éditions de Physique, Les Ulis, France, 1988).
- W. Feller, An Introduction to Probability Theory and Its Applications (Wiley, New York, 1971), Vol. II.
- P. R. Bevington, D. K. Robinson, Data Reduction and Error Analysis for the Physical Sciences, 2nd ed. (McGraw-Hill, Boston, Mass., 1992).
- International Organization for Standardization, Guide to Expression of Uncertainty in Measurement, 2nd ed. (ISO, Geneva, 1995).
- W. A. Fuller, Measurement Error Models (Wiley, New York, 1987). [CrossRef]
- R. J. Caroll, D. Ruppert, L. A. Stefanski, Measurement Error in Nonlinear Models, Vol. 63 of Monographs on Statistics and Applied Probability (Chapman Hall, London, 1995).
- J. Walewski, C. F. Kaminski, S. F. Hanna, R. P. Lucht, “Dependence of partially saturated polarization spectroscopy signals on pump intensity and collision rate,” Phys. Rev. A 64, 063816 (2001). [CrossRef]
- B. N. Taylor, C. E. Kuyatt, “Guidelines for evaluating and expressing the uncertainty of NIST measurement results,” Tech. Rep. (National Institute of Standards and Technology, Gaithersburg, Md., 1994); http://physics.nist.gov/pubs/guidelines/contents.html .
- W. Demtröder, Laser Spectroscopy (Springer-Verlag, Berlin, 1996). [CrossRef]
- A. C. Eckbreth, Laser Diagnostics for Combustion Temperature and Species (Overseas Publishers, Amsterdam, 1996).
- E. L. Dereniak, D. G. Growe, Optical Radiation Detectors (Wiley, New York, 1984).
- W. J. Thompson, J. R. Macdonald, “Correcting parameter bias caused by taking logs of exponential data,” Am. J. Phys. 59, 854–856 (1991). [CrossRef]
- J. W. Tukey, Exploratory Data Analysis (Addison-Wesley, Reading, Mass., 1977), pp. 39–41, 44.

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