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Canadian Journal of Physics

Distribution de la Somme des Photoélectrons Détectés en L Points sous Eclairement Gaussien Partiellement Cohérent

Jacques Bures, Claude A. Delisle, Andrzej Zardecki

Published Dec 15, 1971• 4 citations
Publisher

Abstract

The analysis of the photocount distribution in incoherent Gaussian light detected in L space points of a photodetector is extended to the case of partially coherent light. The second moment of the derived distribution is exact. A function η is introduced which accounts for the spatial coherence aspect of light as does the function ξ, (Mandel) for the temporal coherence aspect. A more general function ζ, which for cross-spectrally pure light reduces to the product ζ = ηξ, depends on both spatial and temporal characteristics of coherence. It is shown both theoretically and experimentally that many different geometrical configurations yield the same normalized second moment. A pseudo-thermal source [Formula: see text] where T is the detection time and Tc the coherence time) is used for the experimental verification.

DOI
10.1139/p71-366

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We consider the photocount statistics of a field formed by superposition of a coherent signal of an arbitrary form, and the Brownian noise (Ornstein--Uhlenbeck process). We obtain an analytical expression for the generating function of photocounts. In the case of a harmonic coherent signal, we analyze the count statistics of the superposed field as a function of the shift of the carrier frequency of the harmonic signal relative to the maximum of the Lorentz spectral contour, as a function of the partial intensity components of the superposed radiation, and also as a function of the product of the duration of photodetection and the spectral line width of the noise. I. The aim of the present work is to derive and study a formula for the probability of counts under the assumption that the photodetector registers an optical field formed by the superposition of a coherent signal and Brownian noise. This problem is of interest in the study of the count distribution in a detector which absorbs a mixture of coherent and incoherent radiation [i]. In the derivation of the formula for the probability density of the number of counts, one encounters specific mathematical difficulties associated with the fact that the photodetection process is extended in time. The probability P(m, T) that, during a time interval T, m photoelectrons are detected when the superposition of a coherent signal B(t) and noise ~(t) falls on the detector, is given by the Mandel formula [2]

•Jan 1, 1985•0 citations
The analysis of the photocount distribution in incoherent Gaussian light detected in L space points of a photodetector is extended to the case of partially coherent light. The second moment of the derived distribution is exact. A function η is introduced which accounts for the spatial coherence aspect of light as does the function ξ, (Mandel) for the temporal coherence aspect. A more general function ζ, which for cross-spectrally pure light reduces to the product ζ = ηξ, depends on both spatial and temporal characteristics of coherence. It is shown both theoretically and experimentally that many different geometrical configurations yield the same normalized second moment. A pseudo-thermal source [Formula: see text] where T is the detection time and Tc the coherence time) is used for the experimental verification.
The statistical behavior of photoelectrons emitted at L points of a photocathode in partially coherent Gaussian light is considered. The exact formula for the cumulants and the factorial moment generating function is derived. An exact expression for the photocount distribution and for the factorial moments is obtained in the case when the detection time is much smaller than the coherence time. Two cases of interest which were treated earlier by the same authors are also considered. In particular, the case of two points of detection is of special interest because a physical meaning can be given to the difference of the normalized eigenvalues. Finally, both the exact and approximate theories are compared with the experimental photocount distribution for four different geometrical arrangements.
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 probability distribution of photoelectrons detected simultaneously at L different points of a photocathode is studied under the assumptions that the incident light is Gaussian with arbitrary state of coherence and polarization. The generating function is expressed as an infinite product of eigenvalues of a matrix integral equation, the kernel being the correlation matrix. The case of spectrally pure light beams is discussed in detail and the relation to previous work is given. The notion of polarizationally pure beams is introduced, for which the polarization state of light is independent of the spatial and temporal coherence. The generating function is then expressed in terms of eigenvalues of a simpler integral equation. An approximate distribution valid up to the second moment is derived.
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.
The experimental photocount distribution from a photocathode illuminated by a pseudothermal source is compared with theory. The detection time is much smaller than the coherence time [Formula: see text]. The agreement is excellent for an illuminated area composed of holes small enough to be considered as points and far enough to guarantee incoherence among them. But, for an extended circular area, the agreement is good only in the limits when that area is very small or very large with respect to the coherence area. Finally, we consider on the one hand, the evaluation of an extended detection area as a function of the number of statistically independent points, L, and on the other hand, the relative size of a physical point.
We consider the photocount statistics of a field formed by superposition of a coherent signal of an arbitrary form, and the Brownian noise (Ornstein--Uhlenbeck process). We obtain an analytical expression for the generating function of photocounts. In the case of a harmonic coherent signal, we analyze the count statistics of the superposed field as a function of the shift of the carrier frequency of the harmonic signal relative to the maximum of the Lorentz spectral contour, as a function of the partial intensity components of the superposed radiation, and also as a function of the product of the duration of photodetection and the spectral line width of the noise. I. The aim of the present work is to derive and study a formula for the probability of counts under the assumption that the photodetector registers an optical field formed by the superposition of a coherent signal and Brownian noise. This problem is of interest in the study of the count distribution in a detector which absorbs a mixture of coherent and incoherent radiation [i]. In the derivation of the formula for the probability density of the number of counts, one encounters specific mathematical difficulties associated with the fact that the photodetection process is extended in time. The probability P(m, T) that, during a time interval T, m photoelectrons are detected when the superposition of a coherent signal B(t) and noise ~(t) falls on the detector, is given by the Mandel formula [2]