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Physical Review A

Effect of Spatial Coherence on the Photoelectric Counting Statistics of Gaussian Light

Cyrus D. Cantrell, John R. Fields

Published Jun 1, 1973• 18 citations

Abstract

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.

DOI: 10.1103/physreva.7.2063

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Physical Review A•Oct 1, 1979•5 citations

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.

The statistical behavior of photoelectrons emitted at L points of a photocathode in partially coherent Gaussian light is considered. The exact formula for the cumulants and the factorial moment generating function is derived. An exact expression for the photocount distribution and for the factorial moments is obtained in the case when the detection time is much smaller than the coherence time. Two cases of interest which were treated earlier by the same authors are also considered. In particular, the case of two points of detection is of special interest because a physical meaning can be given to the difference of the normalized eigenvalues. Finally, both the exact and approximate theories are compared with the experimental photocount distribution for four different geometrical arrangements.

General formulas for the factorial cumulants of the photocounting distribution and the cumulants of the integrated intensity are obtained for chaotic light superposed with coherent light, for experiments involving

N

different counting intervals. The chaotic light may be partially polarized, need not be time stationary or single mode, may differ in frequency from the coherent light, and may have any spectral distribution.

This paper deals with the theory of the statistics of photoelectric counting experiments which involve

N

separate counting intervals. The main new results reported here are: (i) a simple formula for the

N

-fold cumulants of the intensity of Gaussian light; (ii) a detailed study of the relations between the

N

-fold cumulants of the integrated intensity and the cumulants of the

N

-fold photoelectric counts, for light of arbitrary coherence properties. These results are derived and discussed in the context of recent work by other authors.

The statistics of the sum of photocounts in partially coherent Gaussian light is investigated. Both the case of detection at

L

discrete points and the case of an extended surface of detection are considered. The combinatorial-analysis approach is applied to derive the moments and cumulants of the distribution. They all can be expressed in terms of the second-order correlation function. The results are generally valid independent of the ratio of the observation time to the coherence time of light. A comparison between the theoretical moments and the moments computed from experimental distributions as a function of the detector surface is carried out with the help of a pseudothermal source.

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.

Statistical properties of partially polarized Gaussian light were studied experimentally for degrees of polarization

P

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.