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Optica Acta International Journal of Optics

Statistical Accuracy in the Photon Counting Structure Function of Fluctuating Light Fields

Christopher J. Oliver, Edward R. Pike

Published Oct 1, 1981• 14 citations
Publisher

Abstract

An analysis of the variance of the structure function coefficients of a train of digital photodetections, both unnormalized and with various normalizations, is made and compared with known corresponding results for the photon correlation function. A gaussian-lorentzian model light field is used in both direct and homodyne detection. Numerical simulations confirm and extend the analytical results, which show certain circumstances when the structure function may be a useful alternative measure of spectral content.

DOI
10.1080/713820449

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An analysis of the variance of the structure function coefficients of a train of digital photodetections, both unnormalized and with various normalizations, is made and compared with known corresponding results for the photon correlation function. A gaussian-lorentzian model light field is used in both direct and homodyne detection. Numerical simulations confirm and extend the analytical results, which show certain circumstances when the structure function may be a useful alternative measure of spectral content.
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.
We discuss the accuracy of a measurement of a spectral linewidth using the method of digital correlation of single-clipped photon counting fluctuations with a given light flux and a given duration of experiment. Statistical effects due to the random nature of the light field and the photoelectric process limit this accuracy, so that the longer the duration of the experiment the more accurate the result, if all other factors are equal. Theoretical and experimental results are presented which show the accuracies obtainable as functions of the sample time, detector area and clipping level.
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.
It is well known that the photocount distribution associated with the photoelectric detection of fluctuating light beams carries information about the probability density of the light intensity. In this paper, we present an exact solution to a problem of particular interest which arises in connection with the photoelectric detection of narrow-band Gaussian (thermal or pseudothermal) light, namely, the problem of determining the photocount distribution when the counting-time interval is not necessarily short compared to the coherence time of the light. The method developed in this paper is applied to the case where the spectral profile of the light is Lorentzian; it leads to an exact expression for the photocount generating function and it provides simple recurrence relations for the photocount distribution and for its factorial moments.
This paper studies the limiting accuracy with which power spectra and the values of spectral parameters for random signals can be achieved from short batches of data. Initially, general expressions for arbitrary data are derived; these are then restricted for simplicity to wide-band signals. In addition, comparison is made with computer simulation for two well defined models, namely, photodetection of light of constant intensity and heterodyne photodetection of narrow-band Gaussian-Lorentzian light. It is shown that there is no analytical difference between operation in time or frequency space for batch data, and also that long data sets can be analysed at least as well, in terms of the accuracy with which spectral parameters can be determined by fitting, by averaging over many short batches as by processing the set as a whole.