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Journal of Physics A Mathematical and General

Normalization and statistical noise level in the normalized autocorrelation function. Compensated normalization

Zbigniew Kojro

Published Mar 7, 1991• 7 citations
Publisher

Abstract

The use of large delay times in modern correlation experiments must be supported by an appropriate normalization scheme. The already available symmetric normalization does not eliminate all those statistical errors from a measured intensity autocorrelation function which can be eliminated with the help of a normalization procedure. A new kind of compensated normalization is presented.

DOI
10.1088/0305-4470/24/5/003

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The use of large delay times in modern correlation experiments must be supported by an appropriate normalization scheme. The already available symmetric normalization does not eliminate all those statistical errors from a measured intensity autocorrelation function which can be eliminated with the help of a normalization procedure. A new kind of compensated normalization is presented.
An adequate analysis of photon correlation data requires knowledge about the statistical accuracy of the measured data. For the model of gamma-distributed intensities, that is including the effect of a finite intercept, the full covariance matrix is calculated for all the channels of the photon autocorrelation functions. A thorough discussion of multiple sample time correlation illuminates the importance of temporal averaging effects at large lag times. A practical estimation scheme is given for the noise in photon correlation data from a multiple sample time measurement.
The results of a theory of noise on photon correlation data is presented for the rather general case of a detector signal which can be modeled as a Gamma process. This model includes finite intercept effects and may even be used for heterodyne measurements. Multiple sample time measurements are considered with particular care and significant decreases of noise with increasing sample time are typically found at small as well as at large lag (and sample) times, if the standard normalization procedure is replaced by the symmetric or compensated normalization schemes. Large estimator covariances are observed across lag time ranges up to one coherence time. This type of correlated noise is typical for correlation measurements and may easily lead to overfitting problems. Experimental verifications are included for the noise calculations.
The authors investigate the statistical errors which arise in digital autocorrelation of photon counting fluctuations due to the finite duration of experiments. An expression for the accuracy with which individual points of the autocorrelation function of intensities can be measured is derived analytically and evaluated for Gaussian- Lorentzian light. The appropriate weighting for a 'least-squares' fitting procedure is determined and the error in spectral linewidth, obtained by this procedure from experiment, is calculated.
Experimental determinations of correlation functions typically involve events of differing particle number. In order to interpret such data, it is important to relate these measurements to fixed-number correlations. We exhibit the total correlation function in terms of fixed n correlations and fluctuations with respect to the average. We note that moments, constructed as integrals of the appropriate correlation functions, can be dominated in the case of broad distributions by cumulant moments constructed from single-particle density fluctuations.