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Physical Review

Photon Counting Statistics of Gaussian Light

Gabriel Bédard

Published Nov 25, 1966• 97 citations
Publisher

Abstract

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.

DOI
10.1103/physrev.151.1038

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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.
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.
It is well known that photoelectric measurements yield information about the statistical behavior of fluctuating light beams. The probability of n photoelectrons being ejected, in a fixed time interval, from the photosensitive surface of the photoelectric detector, upon which the light beam is normally incident, is a linear (Poisson) transform of the probability density for the intensity of the beam. With the help of some plausible assumptions, the present analysis provides solutions to the problem of inverting the Poisson transform, thus determining the probability density for the intensity from experimentally obtained photocounting distributions. The effectiveness of the method is demonstrated by an actual inversion of a typical experimental counting distribution. The technique is of particular interest in connection with efforts to understand the statistical behavior of optical fields, especially of laser fields.
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 $N\ifmmode\times\else\texttimes\fi{}N$ 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.
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.
The practical treatment of statistics for Gaussian-Lorentzian light is studied theoretically and experimentally by using the time interval distribution of photoelectric pulses, which is derived in two approaches that are based on Markovian processes and the generating function of photon counting statistics, taking the experimental conditions into account. The limits of the validity of the Markovian assumption are made clear. In arbitrary light intensity it is valid experimentally to employ the practical generating function of photon counting distribution expressed as the product of an ideal generating function and a Poissonian one.