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Proceedings of the Physical Society

The distribution of photoelectric counts from partially polarized Gaussian light

Carl W. Helstrom

Published May 1, 1964• 22 citations
Publisher

Abstract

The distribution of the number of photoelectrons counted when partially polarized Gaussian light falls on a photosensitive surface is derived by the methods of Kac and Siegert. When the counting time is much longer than the coherence time of the light, the distribution is approximately the same as that of n bosons assigned to two sets of s states each, corresponding to beams of relative intensities ½(1 + P ) and ½(1 - P ), where P , 0 <or= P <or= 1, describes the state of polarization of the light, and s is the ratio of the counting time to the coherence time of the light.

DOI
10.1088/0370-1328/83/5/310

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The distribution of the number of photoelectrons counted when partially polarized Gaussian light falls on a photosensitive surface is derived by the methods of Kac and Siegert. When the counting time is much longer than the coherence time of the light, the distribution is approximately the same as that of n bosons assigned to two sets of s states each, corresponding to beams of relative intensities ½(1 + P ) and ½(1 - P ), where P , 0 <or= P <or= 1, describes the state of polarization of the light, and s is the ratio of the counting time to the coherence time of the light.
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.
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.
Exact formulas are derived for the factorial cumulants of the photoelectric counting distribution of partially polarized Gaussian and Gaussian-plus-coherent light for arbitrary detector areas. The results make it possible to follow the transition from Bose-Einstein statistics in the small-area limit to Poisson statistics in the large-area limit. Numerical results are presented for a circular geometry. These methods, which enable experimentally measured photocount statistics to be extrapolated to zero detector area, are expected to be useful in measurements of small departures from Gaussian statistics. Other theoretical results make it possible to express the statistics of the sum of photocounts from different photocathodes, or the multiaperture, single-cathode (MASC) photocount statistics of light of arbitrary coherence properties, in terms of the multicathode (MC) counting statistics. Explicit expressions are derived for the MASC photocount cumulants for the special case of partially polarized Gaussian light.
Statistical properties of partially polarized Gaussian light were studied experimentally for degrees of polarization PPP from 0.03 to 1 by a standard technique of photon counting. The Gaussian light with an arbitrary degree of polarization was obtained from Rayleigh scattering of orthogonally polarized laser beams. The factorial moments up to sixth order of the photoelectron distribution were measured with an arbitrary temporal and spatial coherence. After corrections were taken for the experimental conditions, the theoretical values of these factorial moments were compared with the experimental results. It is shown that the theoretical predictions were quantitatively well verified by the present measurements. The microscopic process is discussed from the standpoint of the light scattering system used in the present measurements, and it is shown that the orthogonally polarized components of the scattered light observed are statistically independent.
The probability distribution of photoelectrons detected simultaneously at L different points of a photocathode is studied under the assumptions that the incident light is Gaussian with arbitrary state of coherence and polarization. The generating function is expressed as an infinite product of eigenvalues of a matrix integral equation, the kernel being the correlation matrix. The case of spectrally pure light beams is discussed in detail and the relation to previous work is given. The notion of polarizationally pure beams is introduced, for which the polarization state of light is independent of the spatial and temporal coherence. The generating function is then expressed in terms of eigenvalues of a simpler integral equation. An approximate distribution valid up to the second moment is derived.
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.