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Optica Acta International Journal of Optics

Photon Counting Statistics of a Stationary Gaussian Field with Brillouin Spectrum

Manmohan Singh, S. K. Srinivasan

Published Dec 1, 1981• 4 citations
Publisher

Abstract

The statistical characteristics of a stationary gaussian field endowed with Brillouin spectrum are studied. A closed-form expression for the exact one-fold generating function (g.f.) corresponding to the arbitrary count rates and the sample times is derived. The time-interval distribution of the photon counting statistics is extracted from the g.f. without resort to any approximate procedures. Comparison with the earlier results obtained by use of the approximate methods shows that the error in such approximations is considerable.

DOI
10.1080/713820504

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The statistical characteristics of a stationary gaussian field endowed with Brillouin spectrum are studied. A closed-form expression for the exact one-fold generating function (g.f.) corresponding to the arbitrary count rates and the sample times is derived. The time-interval distribution of the photon counting statistics is extracted from the g.f. without resort to any approximate procedures. Comparison with the earlier results obtained by use of the approximate methods shows that the error in such approximations is considerable.
Based on the unique analytic form of the one-fold generating-function very recently given by the present author in [3], we provide in this paper a complete and an exact anaysis of the time-interval-statistics of the Brillouin spectrum with its central Rayleigh line. We also distinguish the time-interval-distributions for the present Brillouin triplet from the time-interval-distributions for the pure Brillouin spectrum (a doublet) given in Ref. [4]. As in Ref. [4], we establish here too the invalidity of the earlier numerical-results due to Blake and Barakat [5] for the present Brillouin triplet, as they violate even the well known equation (32) for the time-interval-distribution for the conditional photocount
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.
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.
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 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.