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Physical Review

N -Fold Joint Photon Counting Distributions Associated with the Photodetection of Gaussian Light

Gabriel Bédard

Published Sep 25, 1967• 55 citations
Publisher

Abstract

Recently, Arecchi, Bern\'e, and Sona reported results of theoretical and experimental investigations on the twofold joint photocount distributions of a stationary Gaussian-Markovian radiation field. In this paper, we generalize their results by deriving the $N$-fold joint photocount distribution of a Gaussian (thermal) radiation field of arbitrary spectral profile, when the counting-time intervals are short compared to the coherence time of the light. The present analysis provides simple recurrence relations for the $N$-fold joint photocount distributions and for their generalized factorial moments. These relations are derived with the help of an $N\ifmmode\times\else\texttimes\fi{}N$ generating matrix which is introduced here. The present analysis also indicates to what extent the $N$-fold joint photocount distributions contain useful information about the higher-order coherence properties of the radiation field.

DOI
10.1103/physrev.161.1304

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Recently, Arecchi, Bern'e, and Sona reported results of theoretical and experimental investigations on the twofold joint photocount distributions of a stationary Gaussian-Markovian radiation field. In this paper, we generalize their results by deriving the NNN-fold joint photocount distribution of a Gaussian (thermal) radiation field of arbitrary spectral profile, when the counting-time intervals are short compared to the coherence time of the light. The present analysis provides simple recurrence relations for the NNN-fold joint photocount distributions and for their generalized factorial moments. These relations are derived with the help of an N\ifmmode×\else\texttimes\fiNN\ifmmode\times\else\texttimes\fi{}NN\ifmmode×\else\texttimes\fiN generating matrix which is introduced here. The present analysis also indicates to what extent the NNN-fold joint photocount distributions contain useful information about the higher-order coherence properties of the radiation field.
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.
The conventional model of the photodetection process and the short-time approximation are used to obtain the factorial moment generating function for the NNN-fold detection of mixed thermal and coherent radiation. This generating function is then used to obtain the photocount distribution for twofold joint photodetection. The photocount distribution is expressed as a threefold summation over Laguerre polynomials. The distribution is examined for the limiting cases of purely thermal radiation and uncorrelated fields.
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.
We study the photocounting statistics and the integrated intensity distribution of partially polarized Gaussian light. Expressions for the characteristic functions of the two distributions are obtained in terms of the eigenvalues of an associated integral equation. Closed-form expressions for these characteristic functions are derived for the special case when the spectral profile of the light is Lorentzian. Results from a numerical evaluation of the factorial moments of the number of photoelectrons in a given time interval and of the integrated intensity distribution are also presented.
In recent years there has been considerable utilization of photocount statistics for studying theoretical and experimental properties of a quantized electromagnetic field. The electromagnetic field, which has been of greatest interest in these areas, is a superposition of coherent and chaotic radiation. In this paper closed-form recurrence equations are derived for the photocount distribution and the factorial moments for this case. The photocount distribution is then obtained by using a well-known solution for p(0, T), and the factorial moments are generated by using the fact that the zeroth factorial moment is unity.
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.