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Optica Acta International Journal of Optics

Statistics of Integrated Gamma-lorentzian Intensity Fluctuations

Eric Jakeman

Published Jun 1, 1980• 30 citations
Publisher

Abstract

Intensity fluctuations arising from a random walk in 2α dimensions are governed by the gamma-distribution of parameter α. When 2α is fractional, conventional methods of calculating statistical properties of the integrated intensity, based on a gaussian noise theory approach, can no longer be used. In this paper the single interval statistics of an integrated gamma-distributed variable with lorentzian spectrum are derived directly from a rate equation for the process. The results include, as a special case, the integrated photon statistics predicted by a model for single-mode thermal radiation in a cavity.

DOI
10.1080/713820299

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Intensity fluctuations arising from a random walk in 2α dimensions are governed by the gamma-distribution of parameter α. When 2α is fractional, conventional methods of calculating statistical properties of the integrated intensity, based on a gaussian noise theory approach, can no longer be used. In this paper the single interval statistics of an integrated gamma-distributed variable with lorentzian spectrum are derived directly from a rate equation for the process. The results include, as a special case, the integrated photon statistics predicted by a model for single-mode thermal radiation in a cavity.
Discusses the statistical properties of the intensity fluctuations of light fields consisting of an incoherent Gaussian component heterodyned with a single-frequency coherent beam. The second moment of the intensity- fluctuation distribution is obtained in an analytic form for a Lorentzian incoherent spectrum for arbitrary values of the bandwidth and frequency displacement from the coherent mode. The analysis includes as special cases many of the exact results obtained in statistical optics over the last decade.
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.
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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.