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Physical Review A

N -Fold Photoelectric Counting Statistics of Gaussian Light

Cyrus D. Cantrell

Published Mar 1, 1970• 38 citations
Publisher

Abstract

This paper deals with the theory of the statistics of photoelectric counting experiments which involve $N$ separate counting intervals. The main new results reported here are: (i) a simple formula for the $N$-fold cumulants of the intensity of Gaussian light; (ii) a detailed study of the relations between the $N$-fold cumulants of the integrated intensity and the cumulants of the $N$-fold photoelectric counts, for light of arbitrary coherence properties. These results are derived and discussed in the context of recent work by other authors.

DOI
10.1103/physreva.1.672

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The one-detector photoelectron counting statistics of a harmonic signal mixed with thermal light was considered in paper I. In the present paper, we consider the extension to the counting statistics with several detectors. An expression for the factorial moment generating function of the number of counts is obtained in terms of the eigenvalues of an associated matrix integral equation. This expression is used to evaluate the factorial cumulants of the number of counts.

Physical Review A•Dec 1, 1970•6 citations
This paper deals with the theory of the statistics of photoelectric counting experiments which involve NNN separate counting intervals. The main new results reported here are: (i) a simple formula for the NNN-fold cumulants of the intensity of Gaussian light; (ii) a detailed study of the relations between the NNN-fold cumulants of the integrated intensity and the cumulants of the NNN-fold photoelectric counts, for light of arbitrary coherence properties. These results are derived and discussed in the context of recent work by other authors.
General formulas for the factorial cumulants of the photocounting distribution and the cumulants of the integrated intensity are obtained for chaotic light superposed with coherent light, for experiments involving NNN different counting intervals. The chaotic light may be partially polarized, need not be time stationary or single mode, may differ in frequency from the coherent light, and may have any spectral distribution.
Exact formulas are derived for the factorial cumulants of the photoelectric counting distribution of partially polarized Gaussian and Gaussian-plus-coherent light for arbitrary detector areas. The results make it possible to follow the transition from Bose-Einstein statistics in the small-area limit to Poisson statistics in the large-area limit. Numerical results are presented for a circular geometry. These methods, which enable experimentally measured photocount statistics to be extrapolated to zero detector area, are expected to be useful in measurements of small departures from Gaussian statistics. Other theoretical results make it possible to express the statistics of the sum of photocounts from different photocathodes, or the multiaperture, single-cathode (MASC) photocount statistics of light of arbitrary coherence properties, in terms of the multicathode (MC) counting statistics. Explicit expressions are derived for the MASC photocount cumulants for the special case of partially polarized Gaussian light.
A general formula is derived relating the generating function of the N-fold photoelectron counting distribution to the characteristic functional of the optical field. For a stationary Gaussian radiation field the generating function is expressed as an infinite product involving eigenvalues of a set of integral equations. The results of Bédard, Dialetis, and Cantrell are shown to hold as special cases. Addition of photoelectric counts from correlated Gaussian light is analyzed and the appropriate generating function is given.
An exact expression is obtained for the generating function of the N-fold (i.e., N-detector) photoelectron counting distribution for cross-spectrally pure Gaussian light, in terms of the onefold generating function. It is assumed that the counting times at the different detectors are all equal, but not necessarily short compared to the coherence time of the light. In cases where the onefold generating function can be calculated, a closed formula for the N-fold generating function can be obtained.
Results have been presented by Bedard (see abstr. A33736 of 1967) on the derivation of the N-fold joint photocount distribution of a Gaussian (thermal) radiation field of arbitrary spectral profile, when the counting-time intervals are short compared with the coherence time of the light. In this paper these results are extended for arbitrary counting-time intervals. The authors assume that the fluctuating radiation field has a fixed direction of propagation and that all the photodetector surfaces lie within one coherence area on a plane normal to this direction of propagation. The evaluation of the -fold joint photocount distribution and the photocount moments can be reduced to the solution of either a homogeneous or an inhomogeneous (Fredholm) integral equation of the second kind.