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

Physical Review

Approximate Formulas for Photoelectric Counting Distributions

Gabriel Bédard, Jane C. Chang, L. Mandeļ

Published Aug 25, 1967• 71 citations
Publisher

Abstract

The validity of a simple approximate formula for the photoelectric counting probability in a thermal optical field, which was proposed by one of us (L.M.) in 1959, is investigated. The formula is based on a generalization of the Bose-Einstein distribution and should hold for light of arbitrary spectral density. It is shown by explicit calculation for three different spectral distributions that the formula holds with good accuracy over a very wide range of conditions. It should therefore prove useful when the spectral distribution of the light being studied is not known.

DOI
10.1103/physrev.160.1496

Related papers

Approximate Photocount Statistics

L. Mandeļ

Attention is drawn to the fact that the approximation formula of Bédard, Chang, and Mandel for photoelectric counting yields different probabilities for different spectral distributions, and not just one function as implied in a recent paper of Lachs.

Journal of Applied Physics•Jan 1, 1972•3 citations

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

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

Generating Function of the N-fold Photoelectric Counting Distribution for Gaussian Light

Cyrus D. Cantrell

An exact expression is obtained for the generating function of the N-fold (i.e., N-detector) photoelectron counting distribution for cross-spectrally pure Gaussian light, in terms of the onefold generating function. It is assumed that the counting times at the different detectors are all equal, but not necessarily short compared to the coherence time of the light. In cases where the onefold generating function can be calculated, a closed formula for the N-fold generating function can be obtained.

Journal of Mathematical Physics•Jun 1, 1971•8 citations

Approximate formulae for the superposition of coherent and chaotic fields

R. Horák, Ladislav Mišta, Jan Peřina

Recently derived formulae for the integrated intensity distribution, the photon-counting distribution and its factorial moments in the statistics of the superposition of coherent and chaotic multimode fields are proposed as approximate formulae for light of arbitrary spectrum. It is shown by explicit calculations of the third factorial moment for the superposition of a one-mode coherent field with a Gaussian-Lorentzian field that the proposed formulae hold with good accuracy over a wide range of conditions. An application to the determination of spectral parameters of light is given.

Journal of physics. A, Proceedings of the Physical Society. General•Mar 1, 1971•14 citations

Analysis of Light Fluctuations from Photon Counting Statistics*

Gabriel Bédard

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.

Journal of the Optical Society of America•Oct 1, 1967•32 citations
The validity of a simple approximate formula for the photoelectric counting probability in a thermal optical field, which was proposed by one of us (L.M.) in 1959, is investigated. The formula is based on a generalization of the Bose-Einstein distribution and should hold for light of arbitrary spectral density. It is shown by explicit calculation for three different spectral distributions that the formula holds with good accuracy over a very wide range of conditions. It should therefore prove useful when the spectral distribution of the light being studied is not known.
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.
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.
An exact expression is obtained for the generating function of the N-fold (i.e., N-detector) photoelectron counting distribution for cross-spectrally pure Gaussian light, in terms of the onefold generating function. It is assumed that the counting times at the different detectors are all equal, but not necessarily short compared to the coherence time of the light. In cases where the onefold generating function can be calculated, a closed formula for the N-fold generating function can be obtained.
Recently derived formulae for the integrated intensity distribution, the photon-counting distribution and its factorial moments in the statistics of the superposition of coherent and chaotic multimode fields are proposed as approximate formulae for light of arbitrary spectrum. It is shown by explicit calculations of the third factorial moment for the superposition of a one-mode coherent field with a Gaussian-Lorentzian field that the proposed formulae hold with good accuracy over a wide range of conditions. An application to the determination of spectral parameters of light is given.
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.