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IEEE Journal of Quantum Electronics

Determination of intensity correlation functions from photoelectric counting distributions

David W. Meltzer, L. Mandeļ

Published Nov 1, 1970• 21 citations
Publisher

Abstract

Two techniques are described for obtaining information about the intensity correlation function of an optical field, from measurements of photoelectric counting distributions. The first method is based on double differentiation of the second factorial moment with respect to the counting time. It requires very accurate data, but is shown to be quite feasible, and yields the intensity-correlation function explicitly. The second procedure is less sensitive to inaccuracies and allows the correlation time and certain other parameters to be determined from the factorial moments. In addition, it yields some information about the general form of the correlation function, but no explicit function. The procedures are illustrated and compared by application to photoelectric counting measurements of a laser far below the threshold of oscillation.

DOI
10.1109/jqe.1970.1076341

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Two techniques are described for obtaining information about the intensity correlation function of an optical field, from measurements of photoelectric counting distributions. The first method is based on double differentiation of the second factorial moment with respect to the counting time. It requires very accurate data, but is shown to be quite feasible, and yields the intensity-correlation function explicitly. The second procedure is less sensitive to inaccuracies and allows the correlation time and certain other parameters to be determined from the factorial moments. In addition, it yields some information about the general form of the correlation function, but no explicit function. The procedures are illustrated and compared by application to photoelectric counting measurements of a laser far below the threshold of oscillation.
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.
A relation is derived between the probability distribution of the integrated intensity of light incident on a photoelectric detector and that of the emission of n photoelectrons in a time interval (t, t + T) from the properties of a quantized electromagnetic field. Both the direct and the inverse relations are derived, thereby enabling the statistical properties of light to be determined from photoelectric measurements. An explicit expression is given for the coefficients for calculating the moments ( W k) by means of the moments (n J) (] = 1, 2 ..... k). The correctness of the operations in which generalized functions occur is confirmed by some direct calculations. The significance of photoelectric detectors has been increasing of late. Since their response is proportional to the intensity of incident light (to the square of the amplitude of light) (more exactly, to the local mean time value of the intensity), one speaks of quadratic detectors. These detectors are of basic importance in the theory of the coherence of the electromagnetic field, since if we place N such detectors into N different points in space and if we lead the outputs of these detectors to the correlator we can measure the correlation (coherence) effects of the 2N-th order in the given field [1]. Such correlation measurements provide information on the energy distribution in the radiation spectrum [2], the angular dimensions of stars [3], and the state of polarization [4] and also enable the phase of physically important functions (e.g. the scattering amplitudes) [5] to be determined. Although in such experiments the properties of photoelectrons are studied rather than of photons, they reflect the basic properties of photons. Let the probability distribution Of the emission of n photoelectrons in a time interval (t, t + T) be p(n, T, t) - p(n) and let the probability distribution of the quantity W(T, t) = W = J'tt+r I(t') dr', where I(t) is the intensity of linearly polarized light incident on the photodetector (for simplicity we assume that the sensitivity of the photocathode is a = 1), be P(W). The quantity W can be regarded as a classical characteristic of the field, the quantity n represents the quantum characteristic of the field. Then, on the basis of such a semi-classical description, in which the field is described classically, but its interaction is described quantum-wise, one can derive the following relation [6]
A procedure for the measurement of intensity correlations of the type $〈:\stackrel{^}{I}(\mathrm{x}, t)\stackrel{^}{I}(\mathrm{x}, t+\ensuremath{\tau}):〉$ and $〈:\stackrel{^}{I}(\mathrm{x}, t)\ifmmode\times\else\texttimes\fi{}\stackrel{^}{I}(\mathrm{x}, t+{\ensuremath{\tau}}_{1})\stackrel{^}{I}(\mathrm{x}, t+{\ensuremath{\tau}}_{2}):〉$ with the use of a time-to-amplitude converter is developed. The procedure is used to measure second- and third-order intensity fluctuation correlation functions of the optical field produced by a gas laser operated at various levels near its threshold of oscillation. The results are shown to be in agreement with the predictions of laser theories.
It is well known that the photocount distribution associated with the photoelectric detection of fluctuating light beams carries information about the probability density of the light intensity. In this paper, we present an exact solution to a problem of particular interest which arises in connection with the photoelectric detection of narrow-band Gaussian (thermal or pseudothermal) light, namely, the problem of determining the photocount distribution when the counting-time interval is not necessarily short compared to the coherence time of the light. The method developed in this paper is applied to the case where the spectral profile of the light is Lorentzian; it leads to an exact expression for the photocount generating function and it provides simple recurrence relations for the photocount distribution and for its factorial moments.
The theoretical expression of the first-order factorial moment of a triggered delayed photocount distribution and a description of its SNR have been developed. We explain the adventages of this thecnique, in comparison with the second-order correlation function, when we measured a temporal signal under low light level. We derived the theoretical errors involved in determining a parameter by measuring both functions for a signal with a periodically modulated intensity, since this kind of signals usually appear in spectroscopy. We conclude that the first-order factorial moment of a triggered delayed photocount distribution is a useful tool for characterizing a signal at low; light level conditions.