## 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

**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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