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Applied Physics Letters

Scaled photocount correlation of non-Gaussian scattered light

Dennis E. Koppel, Dale W. Schaefer

Published Jan 1, 1973• 14 citations
Publisher

Abstract

The photocount autocorrelation of non-Gaussian scattered laser light is discussed as a tool for the study of both very dilute or highly correlated scattering systems. A scaled photocount correlation technique is introduced and shown to produce a function directly proportional to the correlation function of the integrated scattered intensity, independent of the field statistics. This technique overcomes the serious problems of interpretation that can arise when the commonly used clipped correlation technique is applied to non-Gaussian fields. The scaled correlation technique is demonstrated experimentally with a simple scattering system, a small number of particles under uniform translational motion.

DOI
10.1063/1.1654464

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The photocount autocorrelation of non-Gaussian scattered laser light is discussed as a tool for the study of both very dilute or highly correlated scattering systems. A scaled photocount correlation technique is introduced and shown to produce a function directly proportional to the correlation function of the integrated scattered intensity, independent of the field statistics. This technique overcomes the serious problems of interpretation that can arise when the commonly used clipped correlation technique is applied to non-Gaussian fields. The scaled correlation technique is demonstrated experimentally with a simple scattering system, a small number of particles under uniform translational motion.
The theoretical treatment of clipped photocount autocorrelation of Gaussian light, introduced by Jakeman and Pike, has been extended to include the effects of finite-duration sampling intervals and incomplete spatial coherence. A system of ``interval partitioning'' is used to derive theoretical expressions for 〈Nq(0)N(τ)〉 and 〈N0(0) N0(τ)〉, the single- and double-clipped photocount autocorrelation functions. Here N(t) is the number of photons detected in a sampling interval centered at t and Nq(t) is the clipped count at level q. Each original interval is systematically partitioned in space and time. Multidimensional generating functions are applied to the sets of subintervals with the object of expressing g(1) (t), the first-order field correlation function, in terms of measurable quantities. It is shown that under certain well-defined conditions, 〈Nq(0) N(τ)〉−〈Nq〉〈N〉=γq|g(1)(τ)|2 and 〈1−N0〉−2−[1−2〈N0〉+〈N0(0)N0(τ)〉]−1=φ|g(1)(τ)|2, where, for a given set of experimental parameters, γq and φ can be taken as simple multiplicative factors independent of the delay time τ. For comparative purposes, a similar analysis is performed for the unclipped correlation function 〈N(0)N(τ)〉. Finally, computer calculated values of various quantities of interest are presented for several representative cases.
In recent years the utilization of "optical mixing" spectroscopic techniques in laser light scattering experiments has proved to be a successful method for the investigation of the dynamical properties of physical systems. By illuminating a photosensitive detector with the scattered light and measuring the spectrum or autocorrelation function of the resulting photocurrent, one can, in general, obtain the ensemble-average time dependence of specific collective excitations within the scattering medium. We present here a detailed quantitative analysis of the statistical errors inherent in such measurements due to the stochastic nature of both the scattering and photoemission processes. We determine the statistical errors on the optical-intensity correlation function as measured by two photocounting digital correlator models and on the intensity spectrum as measured by a "self-beat" optical mixing spectrometer. From these errors and a generalized least-mean-squares fitting procedure we calculate the uncertainty on the measured correlation time (linewidth) for the case of a Gaussian optical field with an exponential intensity correlation function (a Lorentzian spectrum). Scaling relationships are given which permit our numerical results to be applied to an arbitrary set of experimental parameters.
Onefold photoelectron counting distributions are evaluated for coherent light scattered by an ensemble of particles. The ensemble contains, in general, a random number of nonidentical particles. Expressions are obtained for the probability density function and the moments of the scattering intensity. These results reduce to those of Pusey, Schaefer, and Koppel when the particles are identical. Photocount probabilities are also evaluated. The variance, coefficient of skewness, and coefficient of excess are evaluated when the probability density of the intensity contributed by each scatterer is specified by a narrow Gaussian distribution centered about the deterministic mean.