Scattered field, autocorrelation function

For the purpose of discussing the differences between heterodyne and homodyne scattering we define the two scattered field autocorrelation functions... 

Photon Correlation. Particles suspended in a fluid undergo Brownian motion due to collisions with the liquid molecules. This random motion results in scattering and Doppler broadening of the frequency of the scattered light. Experimentally, it is more accurate to measure the autocorrelation function in the time domain than measuring the power spectrum in the frequency domain. The normalized electric field autocorrelation function g(t) for a suspension of monodisperse particles or droplets is given by ... 

Equation (8.159) is strictly valid for a Gaussian distribution of electric fields. The electric field autocorrelation function is related to the dynamic structure factor S q, t) [compare it with the static scattering function S q) in Eq. (3.121)] ... 

In light-scattering experiments one measures the spectral density of the electric field autocorrelation function of the scattered light wave, given as... 

Electric-Field Autocorrelation Function We consider the autocorrelation function of the electric field E,(t) of the light scattered by solutes. As we have seen in Section 2.4, is a complex quantity. We introduce another normalized autocorrelation function gi(r), which is defined as... 

The electric field autocorrelation function can be obtained in a heterodyne system, in which the scattered light is mixed with unscattered light from the laser source, thus obtaining a beat frequency. The characteristic exponential decay rate for the heterodyne correlation function is q DI2 that is, one-half the decay rate of the homodyne autocorrelation function. Sometimes there will be a partial heterodyne character to the autocorrelation function if unwanted stray light from the incident laser mixes with the scattered light, termed accidental heterodyning [16]. 

In most PCS experiments the intensity autocorrelation function G2(0,t) is measured at one or several scattering angles 0 as a function of time delay x. In the first step of the interpretation procedure G2(0,x) is related to the modulus of the normalized field autocorrelation function gi(0,x) by a Siegert relation ... 

In eq. (1) A is a, in principle constant, background signal and B is an instrumental factor. Note that eq. (1) applies only to scattered fields with Gaussian statistics an hypothesis which is not always fulfilled experimentally. Especially for particles larger than roughly 0.5 to 1 im additional time delay dependent factors can be distinguished in eq. (1) In a second step the time decay of the field autocorrelation function is related to the particles Brownian motion. Thereby it is assumed that the particles scatter independently. In particular for monodisperse samples gi(0,x) is an exponentially decaying function ... 

The rate of decay of g x) is indicative of the typical fluctuation time of the scattering signal and of the rate of diffusion of the scatterers. Quantitatively, g( )(r) can be related to the electric field autocorrelation function through the relation... 

For polydisperse scatterers, the field autocorrelation function may be expressed as the Laplace transform of a continuous distribution F of decay rates ... 

This effective Q,t-range overlaps with that of DLS. DLS measures the dynamics of density or concentration fluctuations by autocorrelation of the scattered laser light intensity in time. The intensity fluctuations result from a change of the random interference pattern (speckle) from a small observation volume. The size of the observation volume and the width of the detector opening determine the contrast factor C of the fluctuations (coherence factor). The normalized intensity autocorrelation function g Q,t) relates to the field amplitude correlation function g (Q,t) in a simple way g t)=l+C g t) if Gaussian statistics holds [30]. g Q,t) represents the correlation function of the fluctuat-... 

Regarding the relationship between CARS and conventional Raman spectroscopy, as is evident from the equations above, the scattered anti-Stokes field amplitude (proportional to P) depends linearly on the autocorrelation functions. With respect to molecular dynamics and disregarding the minor point that the field amplitude is not directly measured, CARS is a... 

The intensity I of the light scattered from a dilute macromolecular or supra-molecular solution is a fluctuating quantity due to the Brownian motion of the scattering particles. These fluctuations can be analysed in terms of the normalised autocorrelation function y1 ( t ) of the scattered electrical field Es, which contains information about the structure and the dynamics of the scattering particles [80]. 

The correlation functions for the amplitude fluctuations can now be connected with the autocorrelation function for the scattered field A (see Section IV.A). This function g (A, t) is defined as ... 

It is clear from Eq. (2.4.15) and Section 1.2 that the light-scattering spectrum is determined from autocorrelation functions of the electric field at the detector. Thus the goal of any theory of light scattering is to show how important physical properties of the scattering medium can be extracted from the measured time-correlation functions. 

In a QELS experiment, the autocorrelation function of the polarized scattered field is measured with K as the scattering vector. The scattered field is proportional to the amplitude of the fluctuations of the local polymer concentration in the gel. These concentration fluctuations are related to the local deformation in the gel. 

When monochromatic light is scattered by moving particles that show thermal motion, the field amplitudes E oo) show a Gaussian distribution. The experimental arrangement for measuring the homodyne spectrum is shown in Fig. 7.31. The power spectrum P (jo) of the photocurrent (7.68), which is related to the spectral distribution I oo), is measured either directly by an electronic spectrum analyzer, or with a correlator, which determines the Fourier transform of the autocorrelation function C(r) a i t)) i t -f- r)). According to (7.63), C(r) is related to the intensity correlation function G (r), which yields (7.64), and I co). 

If the light fluctuation is caused by Brownian motion, one can relate the normalised autocorrelation function of the detected light intensity to that of the scattered field by the following equations (Siegert 1943 Xu 2000, pp. 86-89) ... 

---