Autocorrelation functions homodyne

It is important to note that the homodyne autocorrelation function is independent of the mean velocity within the scattering volume and is influenced only by the diffusion and velocity gradient induced motions of the particles. Furthermore, if there is no velocity gradient, the result is that the correlation function is the following, familiar result,... 

In the homodyne mode, G2(t) can be related to the normalised field autocorrelation function gj (t) by. 

Equations 5.446 and 5.450 are applicable in the so-called homodyne method (or self-beating method), where only scattered light is received by the detector. In some cases, it is also desirable to capture by the detector a part of the incident beam that has not undergone the scattering process. This method is called heterodyne (or method of the local oscillator) and sometimes provides information that is not accessible by the homodyne method. It can be shown that if the intensity of the scattered beam is much lower than that of the detected nonscattered (incident) beam, the detector measures the autocorrelation function of the electrical held of the scattered light, dehned as... 

If a homodyne method is used, the measured autocorrelation function (g,x) can be interpreted by using the Siegert relation. Equation 5.454. The translational and rotational diffusion coefficients for several specific shapes of the particles are given in Table 5.9. The respective power spectrum functions can be calculated by using the Fourier transform. Equation 5.449b. 

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

The data from a population of identical particles can be easily analyzed. If the homodyne autocorrelation function of monodisperse spheres (Equation 12) is measured, the data should conform to the expansion ... 

The autocorrelation function C(t) is the product of the intensity at a given time and a delayed value of the intensity averaged as a function of delay time (r). For monodisperse particle systems in homodyne detection mode (heterodyne spectroscopy allows detection of light at much lower intensities than homodyne), C(t) is a single decaying exponential defined as ... 

Fig. 7.31 Schematic experimental setup for measuring the autocorrelation function of scattered light (homodyne spectroscopy), with a correlator as an alternative to an electronic spectrum analyzer...

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). 

It can be shown in a rather lengthy derivation [25] that Pp(Q>t) is proportional to the scattered electric field heterodyne autocorrelation function, which, in turn is directly related to the scattered electric field homodyne autocorrelation function. It is this latter that is most frequently measured in DLS, where the scattered intensity is directly autocorrelated, rather than the scattered electric field. 

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]. 

Thns, in either homodyne or heterodyne mode, the DLS autocorrelation function can yield D at any given q from an exponential fit of the form in Eqnation 5.63. For polydis-perse systems, DLS provides an average over the diffnsion coefficient distribution of the particles. A nnmber of approaches have been developed for analyzing polydisperse systems, including the robust cumulant method (see Section 8.2.2), histograms, and the inverse Laplace transform method (see Section 8.2.3). To turn the coefficient distribntion into the MWD requires a known relationship between D and M, often of the form D=AM y. 

Re F stands for the real part of the function F. The function fcs(r) models a slowly varying background, which is usually present in all of the measurements. The constant background term B is measured by the autocorrelator using special time bins with extra delay. / () is the intensity of the local oscillator (may represent scattering due to the interface itself) the term 2/S0// 0 indicates the relative amount of particle-scattered light and reference scattered photons and should not exceed 0.1 for heterodyne detection. The quantity / is an instrumental constant, a value around 0.5 indicating a reasonably optimized system for homodyne detection. 

If an autocorrelator is used to analyze the PM output the correlation function in Eq. (4.3.5) is directly measured. Sometimes a spectrum analyzer4 is. used instead of an autocorrelator. This device determines the spectral density of the photocurrent from the PM and thus in a homodyne experiment it determines the time Fourier transform of h t), that is,... 

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