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Journal of the Optical Society of America B

Problems of channel correlation and statistical bias in photon-correlation spectroscopy

Richard C. Haskell, Gary L. Pisciotta

Published May 1, 1985• 5 citations
Publisher

Abstract

Correlation between channels of the normalized photocount-rate correlation function g(2)(τ) becomes significant at high count rates and leads to a number of data-analysis problems. We derive an expression for channel correlation that is valid for a detector area of arbitrary extent and compare the theoretical predictions with measured values. A data-analysis procedure is demonstrated that employs the theoretical expression for channel correlation and provides a rigorous test of an assumed fitting function. The procedure facilitates the use of the cumulant method in determining the polydispersity of scatterers. An expression for the statistical bias of g(2)(τ) is also derived and compared with measured values.

DOI
10.1364/josab.2.000714

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Correlation between channels of the normalized photocount-rate correlation function g(2)(τ) becomes significant at high count rates and leads to a number of data-analysis problems. We derive an expression for channel correlation that is valid for a detector area of arbitrary extent and compare the theoretical predictions with measured values. A data-analysis procedure is demonstrated that employs the theoretical expression for channel correlation and provides a rigorous test of an assumed fitting function. The procedure facilitates the use of the cumulant method in determining the polydispersity of scatterers. An expression for the statistical bias of g(2)(τ) is also derived and compared with measured values.
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In recent years the utilization of "optical mixing" spectroscopic techniques in laser light scattering experiments has proved to be a successful method for the investigation of the dynamical properties of physical systems. By illuminating a photosensitive detector with the scattered light and measuring the spectrum or autocorrelation function of the resulting photocurrent, one can, in general, obtain the ensemble-average time dependence of specific collective excitations within the scattering medium. We present here a detailed quantitative analysis of the statistical errors inherent in such measurements due to the stochastic nature of both the scattering and photoemission processes. We determine the statistical errors on the optical-intensity correlation function as measured by two photocounting digital correlator models and on the intensity spectrum as measured by a "self-beat" optical mixing spectrometer. From these errors and a generalized least-mean-squares fitting procedure we calculate the uncertainty on the measured correlation time (linewidth) for the case of a Gaussian optical field with an exponential intensity correlation function (a Lorentzian spectrum). Scaling relationships are given which permit our numerical results to be applied to an arbitrary set of experimental parameters.
The corrections to the photocount distributions, due to dead-time effects, are evaluated by use of the time-interval probability density of the counts. The normalized cumulative distribution function J k ( T , τ), associated with the detection of fluctuating light beams during a counting time interval T , with a counter having a constant dead-time τ, appear naturally as a convenient quantity to be used in comparing experimental results with theoretical predictions. The quantities J k ( T , τ) obey a simple recurrence relation and are readily derived from the photocount distribution p ( k, T , τ). A general expression is derived for J k ( T , τ) and p ( k, T , τ) associated with the detection of fluctuating light beams. Typical cumulative distribution functions, relating to the detection of thermal light, are evaluated as functions of the degree of polarization of the light, of the average number of photocounts < n 0 > in the absence of dead-time effects and of the ratio τ/ T . The distributions J k ( T , τ) and p ( k, T , τ) relating to detection of laser light are also calculated in a typical case.
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 results for the noise of photon-correlation and photon-structure functions are presented. The noise dependences on the mean count rate and on the sampling time are investigated by means of a specifically built linear fluctuation analyzer, and the results are compared with theoretical predictions. The predicted mean count rate dependence and sampling time dependence of the noise are confirmed by the experimental analysis. Some discrepancies between analytical expressions and experimental results are observed in the crossover predictions. The results show that the structure function has better noise performance for relatively high mean count rates and for long sampling times, which strongly suggests a wider use of this statistical estimator in measurements requiring good accuracy, as in the case of uniform sampling.
The statistics of the sum of photocounts in partially coherent Gaussian light is investigated. Both the case of detection at LLL discrete points and the case of an extended surface of detection are considered. The combinatorial-analysis approach is applied to derive the moments and cumulants of the distribution. They all can be expressed in terms of the second-order correlation function. The results are generally valid independent of the ratio of the observation time to the coherence time of light. A comparison between the theoretical moments and the moments computed from experimental distributions as a function of the detector surface is carried out with the help of a pseudothermal source.