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Physical Review A

Intensity-Correlation Spectroscopy

Vittorio Degiorgio, J. B. Lastovka

Published Nov 1, 1971• 103 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.4.2033

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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.
The authors investigate the statistical errors which arise in digital autocorrelation of photon counting fluctuations due to the finite duration of experiments. An expression for the accuracy with which individual points of the autocorrelation function of intensities can be measured is derived analytically and evaluated for Gaussian- Lorentzian light. The appropriate weighting for a 'least-squares' fitting procedure is determined and the error in spectral linewidth, obtained by this procedure from experiment, is calculated.
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.
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]
The generation of a broadband laser field with well-defined and controllable statistical properties, known as the real Gaussian laser field, has been achieved through the random modulation of the amplitude of a stabilized laser beam. The verification of the field-generation technique is provided by measurements of the laser power spectrum (Lorentzian) and by measurements of the intensity autocorrelation function. The latter is shown to decrease exponentially from an initial value of nearly 3 to a final value of 1 with a decay time related to the inverse bandwidth of the field. The techniques for generating this field and for its characterization are discussed in this paper.
We discuss general polarization-sensitive correlation measurements of light fluctuations based on the Hanbury Brown and Twiss interference detection scheme. Such measurements are considered as a basis for correlation spectroscopy, which allows us to observe the correlations of the fluctuations of all polarization components (Stokes parameters) of the probe light. The interferometer scheme is discussed in application to the study of the polarization-dependent interaction of light with an atomic ensemble with polarized angular momentum and put into an external magnetic field. We describe the general procedure for evaluating the photocurrent correlation function in the case of an optically thin atomic medium. Our calculations are based on the Keldysh diagram technique which is usually used for the study of non-equilibrium systems by means of the second quantized formalism. It allows us to express the Raman field-correlation functions and parametric scattering processes describing the output light fluctuations in terms of a different type of time-ordered electric field and atomic Green functions. We show that with definite conditions the photocurrent correlation function can be written as the sum of two terms proportional to the correlation functions of atomic density matrix fluctuations in symmetric and antisymmetric forms. This expression of the photocurrent correlation function can be used in various different applications.