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Physical Review A

Finite-aperture corrections to photocount statistics of Gaussian light

A P Bark, Steven Rathgeb Smith

Published Jan 1, 1977• 6 citations
Publisher

Abstract

Experimental results on the effects of finite-aperture detectors on the statistics of Gaussian light are presented. A laser illuminating a rotating diffusing screen was used to provide a source with a long coherence time, and the photoelectron statistics of a photomultiplier detector were recorded electronically. The data, which were taken with circular apertures and a linear polarizer in front of the detector, agree with calculations of factorial cumulant correction factors published by Cantrell and Fields.

DOI
10.1103/physreva.15.269

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Physical Review A•Sep 1, 1972•2 citations
Experimental results on the effects of finite-aperture detectors on the statistics of Gaussian light are presented. A laser illuminating a rotating diffusing screen was used to provide a source with a long coherence time, and the photoelectron statistics of a photomultiplier detector were recorded electronically. The data, which were taken with circular apertures and a linear polarizer in front of the detector, agree with calculations of factorial cumulant correction factors published by Cantrell and 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 statistics of the sum of photocounts in partially coherent Gaussian light is investigated. Both the case of detection at LLL 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.
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.
The transient phase-independent solutions of the laser Fokker-Planck equation are analyzed for a short range of time following the moment of the switching on of the oscillations. The evolution of the laser light in time is then represented as a nonstationary Gaussian random process. On the basis of this fact, the factorial moment generating function of the photocount probability distribution is derived, which accounts automatically for all the finite-counting-time corrections. The time development of the first two moments is compared with the recent results of Arecchi and Degiorgio.