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Soviet Journal of Quantum Electronics

Influence of coherent properties of an optical field on the photocount statistics

В. Г. Астафуров, G. N. Glazov

Published Jan 31, 1985• 0 citations
Publisher

Abstract

The asymmetry of photocount statistics in respect of the spatial and temporal coherence factors is demonstrated and explained. This asymmetry appears because of the of a counter receiving a non-Poisson photoelectron flux. A statistical model of an optical field, photoelectron flux, and counting flux is used in a computer study of the dependences of the distributions and moments of the number of counts in a given time interval on the ratio of the coherence factors. An explanation of the features of these dependences is provided. A determination is made of the range of conditions in which the asymmetry is significant. In particular, the asymmetry disappears in the case of a Poisson photoelectron flux and for a very short dead time.

DOI
10.1070/qe1985v015n01abeh005859

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The asymmetry of photocount statistics in respect of the spatial and temporal coherence factors is demonstrated and explained. This asymmetry appears because of the of a counter receiving a non-Poisson photoelectron flux. A statistical model of an optical field, photoelectron flux, and counting flux is used in a computer study of the dependences of the distributions and moments of the number of counts in a given time interval on the ratio of the coherence factors. An explanation of the features of these dependences is provided. A determination is made of the range of conditions in which the asymmetry is significant. In particular, the asymmetry disappears in the case of a Poisson photoelectron flux and for a very short dead time.
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.
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.
We consider the photocount statistics of a field formed by superposition of a coherent signal of an arbitrary form, and the Brownian noise (Ornstein--Uhlenbeck process). We obtain an analytical expression for the generating function of photocounts. In the case of a harmonic coherent signal, we analyze the count statistics of the superposed field as a function of the shift of the carrier frequency of the harmonic signal relative to the maximum of the Lorentz spectral contour, as a function of the partial intensity components of the superposed radiation, and also as a function of the product of the duration of photodetection and the spectral line width of the noise. I. The aim of the present work is to derive and study a formula for the probability of counts under the assumption that the photodetector registers an optical field formed by the superposition of a coherent signal and Brownian noise. This problem is of interest in the study of the count distribution in a detector which absorbs a mixture of coherent and incoherent radiation [i]. In the derivation of the formula for the probability density of the number of counts, one encounters specific mathematical difficulties associated with the fact that the photodetection process is extended in time. The probability P(m, T) that, during a time interval T, m photoelectrons are detected when the superposition of a coherent signal B(t) and noise ~(t) falls on the detector, is given by the Mandel formula [2]
We consider the probability P(n)(t0,t){P}^{(n)}({t}_{0}, t)P(n)(t0​,t) of counting nnn photoelectrons during any time interval [t0,t{t}_{0}, tt0​,t] in a photodetection model built in the general framework of quantum field theory. Owing to an equilibrium assumption on the detector state, P(n)(t0,t){P}^{(n)}({t}_{0}, t)P(n)(t0​,t) is evaluated from the incident field and some electron-correlation functions. The conditions for P(n)(t0,t){P}^{(n)}({t}_{0}, t)P(n)(t0​,t) to be a compound Poisson distribution as expected are discussed. This leads to the definition of coherence properties for the electrons of the detector in connection with the reliability of the detection operation in counting experiments.
This paper is concerned with the study of statistical properties of various optical fields obtained by several types of laser intensity modulation from the stationary random point process point of view. We essentially consider intensity correlations and time intervals distributions (TID). A generalization of the already known TID is given: It is the probability distribution (MTID) of the time interval between one photoelectron and the n th following photoelectron. Some theoretical and experimental results are presented in the general case (the time intervals considered are not necessarily small compared to the coherence time of the field). The modulation effect can be characterized by the photoelectron bunching effect. The TID and the MTID are significantly dependent upon the spectral profile of the field, if the used time scale is of the order of the coherence time. On the other hand, the MTID can be interpreted as the probability of the sum over time intervals between two successive photoelectrons. We treat two cases: If the variables of this sum are uncorrelated, we verify the central-limit theorem; if the variables are correlated we study the influence of the correlations when the number of summed variables becomes large.