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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Inverse problem in 2D photocounting statistics

Vitalij N. Kurashov, Andrey V. Kurashov

Published Dec 1, 1997• 0 citations
Publisher

Abstract

An analytical method to recover the 2D probability density function (PDF) of a random light field from the PDF of photon counts is presented. Some illustrations of the inversion procedure are given with application to the problem of light intensity interferometry.
DOI
10.1117/12.295703

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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.
An analytical recursive method for solving the inverse problem of photocount statistics, that is, for reconstructing the photon-number distribution from the photocount distribution, is proposed for few-photon light. It is shown that if the inverse problem is represented as a system of linear equations, then for finite distributions one can obtain a recurrence formula relating the photocount distribution and the photon-number distribution without restrictions on the quantum efficiency of photodetection.
The multidimensional photon statistics has been determined for the first time from the measurement of the statistics of electrons generated by the detected radiation. The two-dimensional case is described in detail. The used method is based on the application of the characteristic function, which makes it possible to obtain the relation between the two-dimensional photon distribution function and the two-dimensional photocount distribution function. The domain of application of the obtained results are few-photon and nonclassical light fields.
The problem of obtaining the probability density of the fluctuating light intensity from the knowledge of photocounting statistics, when the detectors have a finite dead time, is considered. The integral relation for the probability of registering nnn photocounts in the time interval TTT in the presence of a dead time $\ensuremath{\tau}$ is inverted, and an expression is obtained for the probability density of the fluctuating intensity of the incident light radiation. The method is illustrated with a few specific examples.