Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

IEEE Transactions on Information Theory

Probability distribution of time of arrival of photoevents for a stationary optical field (Corresp.)

Bahaa E. A. Saleh

Published Mar 1, 1974• 3 citations
Publisher

Abstract

A simple relation is found between the probability distribution of time of arrival of photoevents and the moment-generating function of the detected optical energy of a general stationary optical field. This is applied to the case of a Gaussian-Lorentzian field and the case of a mixture of coherent and thermal light.
DOI
10.1109/tit.1974.1055188

Related papers

Photon Counting Statistics of Gaussian Light

Gabriel Bédard

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.

Physical Review•Nov 25, 1966•97 citations

Joint probability of instants of occurrence of photon events and estimation of optical parameters

Bahaa E. A. Saleh

Using a Karhunen-Loeve modal expansion of a stationary and gaussian optical field an expression is derived for the joint probability of instants of occurrence of photon events. When the observation time is much longer than the coherence time, this expression takes a simple form. In this case, and for lorentzian light, it is shown that measurement of the total number of counts is sufficient to determine the maximum likelihood estimate (MLE) of the average count rate. A nonlinear equation is found whose solution gives the MLE of the bandwidth.

Journal of Physics A Mathematical Nuclear and General•Jul 21, 1974•2 citations

Photon time of arrival, time between consecutive photons and the moment generating function

Bahaa E. A. Saleh

The probability densities of the time of arrival of photoelectrons and the time interval between consecutive photoelectrons are respectively proportional to the first and second time derivatives of the moment generating function. Expressions for these densities are given for several classes of optical fields.

Journal of Physics A Mathematical Nuclear and General•Dec 1, 1973•8 citations

Relation between the Statistical Representations of Real and Associated Complex Fields in Optical Coherence Theory

G. S. Agarwal, E. Wolf

In the general theory of optical coherence, the following problem discussed in the present paper, arises: to determine the statistical properties of a field represented by an analytic signal from the knowledge of the statistical properties of the corresponding real field. It is shown by the use of the characteristic functionals that in order to determine the joint probability distributions of the complex field at N space-time points, the knowledge of the complete statistical description of the real field is required; on the other hand, the moments of the complex field up to that order can be determined from the knowledge of the moments of the real field up to the same order. The results are illustrated by explicit calculations relating to the Gaussian random process, which, as is well known, characterizes the fluctuations of thermal light. A converse of a well-known theorem of Kac and Siegert relating to a Gaussian random process is derived as an immediate consequence of our analysis.

Journal of Mathematical Physics•Nov 1, 1972•4 citations

Moments of the Sum of Photocounts in Gaussian Light

Jacques Bures, Claude A. Delisle, Andrzej Zardecki

The statistics of the sum of photocounts in partially coherent Gaussian light is investigated. Both the case of detection at $L$ 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.

Physical Review A•Dec 1, 1972•6 citations

Application of the Functional Formalism to Investigation of the Photoelectric Counting Distribution

Andrzej Zardecki

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.

Canadian Journal of Physics•Jul 1, 1971•7 citations
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.
Using a Karhunen-Loeve modal expansion of a stationary and gaussian optical field an expression is derived for the joint probability of instants of occurrence of photon events. When the observation time is much longer than the coherence time, this expression takes a simple form. In this case, and for lorentzian light, it is shown that measurement of the total number of counts is sufficient to determine the maximum likelihood estimate (MLE) of the average count rate. A nonlinear equation is found whose solution gives the MLE of the bandwidth.
In the general theory of optical coherence, the following problem discussed in the present paper, arises: to determine the statistical properties of a field represented by an analytic signal from the knowledge of the statistical properties of the corresponding real field. It is shown by the use of the characteristic functionals that in order to determine the joint probability distributions of the complex field at N space-time points, the knowledge of the complete statistical description of the real field is required; on the other hand, the moments of the complex field up to that order can be determined from the knowledge of the moments of the real field up to the same order. The results are illustrated by explicit calculations relating to the Gaussian random process, which, as is well known, characterizes the fluctuations of thermal light. A converse of a well-known theorem of Kac and Siegert relating to a Gaussian random process is derived as an immediate consequence of our analysis.
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.
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.