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Physical Review

Correlation Properties of Light Scattered From Fluids

L. Mandeļ

Published May 5, 1969• 31 citations
Publisher

Abstract

The fluctuation and correlation properties of light scattered by a fluid are studied, without neglect of the spectral linewidth of the incident field. When the source is a single-mode laser, it is shown that, although the instantaneous amplitude of the scattered field obeys a Gaussian probability density, the scattered field is not a Gaussian field. The linewidth of the laser beam is reflected in the amplitude correlations of the scattered light, but not in the intensity correlations. On the other hand, when the laser is oscillating in more than one mode simultaneously, the spectral profile of the laser beam makes a contribution to the spectral density of the scattered intensity fluctuations, and cannot be neglected.

DOI
10.1103/physrev.181.75

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The fluctuation and correlation properties of light scattered by a fluid are studied, without neglect of the spectral linewidth of the incident field. When the source is a single-mode laser, it is shown that, although the instantaneous amplitude of the scattered field obeys a Gaussian probability density, the scattered field is not a Gaussian field. The linewidth of the laser beam is reflected in the amplitude correlations of the scattered light, but not in the intensity correlations. On the other hand, when the laser is oscillating in more than one mode simultaneously, the spectral profile of the laser beam makes a contribution to the spectral density of the scattered intensity fluctuations, and cannot be neglected.
Microscopic scattering processes are discussed for two structures of light incident on thermal particles. One is composed of two collinear light beams originating from different oscillating modes of a laser, and another is composed of two counter propagating beams originating from the same oscillating mode. A comparison of the cumulant expansions for the above two cases has allowed discerning which terms in the scattering structures led to the same statistical properties of the scattered photons. It is shown that, for the scattering of multimode-laser light, the statistical properties of scattered photons are not affected by the noncorrelating random phases of the incident laser fields but are the same as those in the case of single-mode laser light under practical scattering experiments. The factorial moments ${F}_{k}[=〈n(n\ensuremath{-}1)\ensuremath{\cdots}(n\ensuremath{-}k+1)〉]$ up to sixth order of the scattered photons were measured for various scattering situations including the above two cases, and they were found to support the theoretical results.
The paper contains a quantum theoretical analysis of laser beam fluctuations and of the light beat experiments with two lasers. With the help of experimental results on photon counting fluctuations in a single-mode laser field, some correlation properties of the field are derived. It is shown that the correlation equations are satisfied by states of the field which are much more general than "coherent" states. The equations lead directly to the spectral density of the intensity operator in the light beat experiments, which can be obtained from photoelectric measurements. The resulting expression is practically identical to that found by Forrester for light having thermal statistical properties. The reasons for this are discussed by a comparison of the corresponding probability distributions of photon counts and of the classical wave amplitude.
We report on intensity fluctuations of a coherent random laser based on incoherent feedback via nonresonant multiple scattering. We quantify the spectral line shape fluctuations in terms of correlations of an individual spectrum with the ensemble-averaged spectrum, which infers the signature of the gain profile of the medium. These correlations are studied in relation to the intensity of the highest coherent modes. We evaluate the distribution of the ratio of the coherent and incoherent fractions in the emission, after independently assessing their statistics. Finally, these intensity fluctuations are graphically represented in a single scatter plot, the centroid of which can be used as a characterization parameter for the laser.
In this study, we measured the temporal correlations in the electromagnetic field radiated by a laser in the threshold region of oscillation, from 110\frac{1}{10}101​ to 10 times threshold intensity. The experimental results are compared with theoretical predictions based on solutions of a Fokker-Planck equation. We stabilized the intensity of a He-Ne cw gas laser by means of a long-time-constant servo system which controlled the cavity length. Using a fast photomultiplier as detector, we recorded the photoelectron count distribution within a short counting time (3 \ensuremath{\mu}sec) while the photomultiplier was exposed to the laser light. From the photoelectron count distribution measurement, we calculated second, third, and fourth normalized cumulants of the intensity probability density function of the light field. The normalized cumulant is a measure of "pure" correlations among photons because the contributions from lower-order correlations are removed. The statistics of the photoelectron count distribution shows that the intensity fluctuations at about 110\frac{1}{10}101​ threshold are nearly those of a Gaussian field, and continuously approach those of a constant-amplitude field as the intensity is raised to about 10 times threshold. The normalized second, third, and fourth cumulants of the intensity probability density function of the laser light were also measured at 17 and 42% of threshold intensity as the counting time was increased from 3 to 1000 \ensuremath{\mu}sec. The results agree with predictions computed under the assumption that the dependence of the correlation functions on the time variables is the same as for Gaussian light.
We have studied the amplitude and temporal correlations of intensity fluctuations of multiply scattered light transmitted through a colloidal suspension. When the coherence length lc{\mathit{l}}_{\mathit{c}}lc​ of the incident light becomes comparable to the width of the probability distribution of photon-path lengths, both the amplitude and dynamics of intensity fluctuations become functions of lc{\mathit{l}}_{\mathit{c}}lc​. In this article we present a simple theory that relates these coherence effects to the photon path-length distribution P(L). We have tested this theory using P(L) calculated from computer simulations and from the diffusion approximation, and find that our theory qualitatively accounts for the observed dependences on coherence length. Our results indicate when the effects of finite laser coherence must be taken into account in quasielastic multiple scattering. In the few-scattering regime of quasielastic multiple scattering, we find that the intensity autocorrelation function generally has a stretched-exponential form, and we discuss the origin of this behavior.
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.