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Journal of the Optical Society of America A

Computation of photoelectron counting distributions by numerical contour integration II Light with rational spectral density

Carl W. Helstrom

Published Jul 1, 1987• 3 citations
Publisher

Abstract

For a mixture of an arbitrary number of sinusoidal coherent components and chaotic light with a rational spectral density having simple poles, the probability-generating function h(z) of the number of photoelectrons ejected by the light in an interval is expressed in terms of certain finite determinants. It figures in the integrands of contour integrals in the complex plane for the cumulative and complementary cumulative distributions of the number of photoelectron counts. When these are evaluated by numerical integration, h(z) can be computed at each point on the contour by standard computer routines for solving algebraic equations and evaluating determinants, permitting precise and efficient computation of the photocount distributions.

DOI
10.1364/josaa.4.001245

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For a mixture of an arbitrary number of sinusoidal coherent components and chaotic light with a rational spectral density having simple poles, the probability-generating function h(z) of the number of photoelectrons ejected by the light in an interval is expressed in terms of certain finite determinants. It figures in the integrands of contour integrals in the complex plane for the cumulative and complementary cumulative distributions of the number of photoelectron counts. When these are evaluated by numerical integration, h(z) can be computed at each point on the contour by standard computer routines for solving algebraic equations and evaluating determinants, permitting precise and efficient computation of the photocount distributions.
We consider onefold photoelectron-counting statistics, for arbitrary counting times, when the wave field on the detector is a mixture of thermal light of a specified spectral profile (incoherent component) and a harmonic signal (coherent component). The probability density function of the time-integrated intensity W(Ω) is obtained from the inverse Laplace transform of the moment-generating function by the Dubner–Abate algorithm. The corresponding probability distribution of the detected photoelectrons, P(m), is then evaluated directly by numerical integration of W(Ω). Numerical results for W(Ω) and P(m) have been obtained when the thermal line shape is Lorentzian for both the homodyne situation (coherent line coincides with the mean frequency of the thermal spectral profile) and the heterodyne situation (coherent line is now off center with respect to the mean frequency of the thermal spectral profile).
In this paper, we consider the photoelectron counting distribution when the wave field incident on the detector is a mixture of harmonic signal and thermal noise. An expression for the factorial moment-generating function of the number of counts is obtained in terms of the eigenvalues of an associated integral equation. This expression is then used to evaluate the factorial cumulants of the number of counts. The results are valid for an arbitrary counting interval. The extension to photoelectron counting with several detectors is considered in paper II of the present investigation.
The corrections to the photocount distributions, due to dead-time effects, are evaluated by use of the time-interval probability density of the counts. The normalized cumulative distribution function J k ( T , τ), associated with the detection of fluctuating light beams during a counting time interval T , with a counter having a constant dead-time τ, appear naturally as a convenient quantity to be used in comparing experimental results with theoretical predictions. The quantities J k ( T , τ) obey a simple recurrence relation and are readily derived from the photocount distribution p ( k, T , τ). A general expression is derived for J k ( T , τ) and p ( k, T , τ) associated with the detection of fluctuating light beams. Typical cumulative distribution functions, relating to the detection of thermal light, are evaluated as functions of the degree of polarization of the light, of the average number of photocounts < n 0 > in the absence of dead-time effects and of the ratio τ/ T . The distributions J k ( T , τ) and p ( k, T , τ) relating to detection of laser light are also calculated in a typical case.
Onefold photoelectron counting distributions are evaluated for coherent light scattered by an ensemble of particles. The ensemble contains, in general, a random number of nonidentical particles. Expressions are obtained for the probability density function and the moments of the scattering intensity. These results reduce to those of Pusey, Schaefer, and Koppel when the particles are identical. Photocount probabilities are also evaluated. The variance, coefficient of skewness, and coefficient of excess are evaluated when the probability density of the intensity contributed by each scatterer is specified by a narrow Gaussian distribution centered about the deterministic mean.
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.