Intensity autocorrelation function

I will present here the properties of various sources when the random variable considered is the field intensity. In this case, one has access to the mean and variance via a simple photodetector. The autocorrelation function can be interpreted as the probability of detecting one photon at time t + t when one photon has been detected at time t. The measurement is done using a pair of photodetectors in a start stop arrangement (Kimble et al., 1977). The system is usually considered stationary so that the autocorrelation function, which is denoted depends only on r and is defined by ... 

A Fock state is a state containing a fixed number of photons, N. These states are very hard to produce experimentally for A > 2. Their photon number probability density distribution P (m) is zero everywhere except for m = N, their variance is equal to zero since the intensity is perfectly determined. Finally, the field autocorrelation function is constant... 

We have shown in this chapter how some experiments made it necessary in some cases to use a quantum description of light instead of the standard semi-classical theory where only the atomic part is quantized. A brief description of different helds in terms of their statistical properties was also given. This description makes it possible to discriminate between the different sources using the intensity autocorrelation function (r). 

Raman intensities of the molecular vibrations as well as of their crystal components have been calculated by means of a bond polarizibility model based on two different intramolecular force fields ([87], the UBFF after Scott et al. [78] and the GVFF after Eysel [83]). Vibrational spectra have also been calculated using velocity autocorrelation functions in MD simulations with respect to the symmetry of intramolecular vibrations [82]. 

Fluorescence intensity detected with a confocal microscope for the small area of diluted solution temporally fluctuates in sync with (i) motions of solute molecules going in/out of the confocal volume, (ii) intersystem crossing in the solute, and (hi) quenching by molecular interactions. The degree of fluctuation is also dependent on the number of dye molecules in the confocal area (concentration) with an increase in the concentration of the dye, the degree of fluctuation decreases. The autocorrelation function (ACF) of the time profile of the fluorescence fluctuation provides quantitative information on the dynamics of molecules. This method of measurement is well known as fluorescence correlation spectroscopy (FCS) [8, 9]. 

The autocorrelation function, G(x), of the temporal fluctuation of the fluorescence intensity at the confocal volume is analytically represented by the following equation [8, 9] ... 

The temporal evolution of P(r,t 0,0) is determined by the diffusion coefficient D. Owing to the movement of the particles the phase of the scattered light shifts and this leads to intensity fluctuations by interference of the scattered light on the detector, as illustrated in Figure 9. Depending on the size of the polymers and the viscosity of the solvent the polymer molecules diffuse more or less rapidly. From the intensity fluctuations the intensity autocorrelation function... 

After normalization to the asymptotic baseline, g2(r) decays from two to unity if measured with a perfect instrument. A real instrument always suffers from some loss of coherence, and for a monodisperse solution of ideal, non-interacting solute molecules the intensity autocorrelation function g2(r) takes the form... 

This function is a continuous analogue of the frequencies derived from the quasiharmonic approximation. Information about the intensities can be obtained by using the dipole moment autocorrelation function in place of the velocity. The advantage of using MD to build up information about the vibrational modes of the polymer is that the approach incorporates an averaging over many vibrational states of a complex molecule, which may be changing conformation... 

Fig. 11.10. Schematic illustration of fluorescence correlation spectroscopy. The autocorrelation function characterises the fluctuations of the fluorescence intensity its decay time expresses the average duration of a...

For a single fluorescent species undergoing Brownian motion with a translational diffusion coefficient Dt (see Chapter 8, Section 8.1), the autocorrelation function, in the case of Gaussian intensity distribution in the x, y plane and infinite dimension in the z-direction, is given by... 

Triplet state kinetics can also be studied by FCS (Widengren et al., 1995). In fact, with dyes such as fluoresceins and rhodamines, additional fluctuations in fluorescence are observed when increasing excitation intensities as the molecules enter and leave their triplet states. The time-dependent part of the autocorrelation function is given by... 

When the excitation light is polarized and/or if the emitted fluorescence is detected through a polarizer, rotational motion of a fluorophore causes fluctuations in fluorescence intensity. We will consider only the case where the fluorescence decay, the rotational motion and the translational diffusion are well separated in time. In other words, the relevant parameters are such that tc rp, where is the lifetime of the singlet excited state, zc is the rotational correlation time (defined as l/6Dr where Dr is the rotational diffusion coefficient see Chapter 5, Section 5.6.1), and td is the diffusion time defined above. Then, the normalized autocorrelation function can be written as (Rigler et al., 1993)... 

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-... 

As the salt concentration continues to decrease, however, matters change dramatically Q). The total scattering intensity decreases more abruptly, and the QLS autocorrelation function, which has been a simple single-exponential decay, becomes markedly two-exponential. The two decay rates differ by as much as two orders of magnitude. The faster continues the upward trend of D pp from higher salt, and is thus assigned the term "ordinary . The slower, which is about 1/10 of Dapp high salt, and appears to reflect a new mode of solution dynamics, is termed "extraordinary . 

Analysis of the relative contributions of the slow and fast decays to the total intensity, by "peeling off the slower from the linear region of a semi-log plot of the autocorrelation function, indicates that it is the ordinary scattering that decreases in intensity. The extraordinary contribution remains roughly constant once it appears. 

In contrast, in dynamic light scattering (DLS) the temporal variation of the intensity is measured and is represented usually through what is known as the intensity autocorrelation function. The diffusion coefficients of the particles, particle size, and size distribution can be deduced from such measurements. There are many variations of dynamic light scattering, and... 

FIG. 5.16 Schematic illustration of intensity measurement and the corresponding autocorrelation function in dynamic light scattering (a) variation of the intensity of the scattered light with time (b) the variation of the autocorrelation function C(s,td) with the delay time td. 

In order to be able to use the fluctuation of the intensity around the average value, we need to find a way to represent the fluctuations in a convenient manner. In Section 5.3b in our discussion of Rayleigh scattering applied to solutions, we came across the concept of fluctuations of polarizabilities and concentration of scatterers and the role they play in light scattering experiments. In the present section, what we are interested in is the time dependence of such fluctuations. In general, it is not convenient to deal with detailed records of the fluctuations of a measured quantity as a function of time. Instead, one reduces the details of the fluctuations to what is known as the autocorrelation function C(s,td), as defined below ... 

The Siegert relation is valid except in the case of scattering volumes with a very small number of scatterers or when the motion of the scatterers is limited. We ignore the exceptions, which are rare in common uses of DLS, and consider only autocorrelations of the type shown in Equation (104). As mentioned above, modern DLS instruments use computer-controlled correlators to calculate the intensity autocorrelation function automatically and to obtain the results in terms of the function gi(s,/rf) therefore we only need to concern ourselves here with the interpretation of gi(s,td). 

Solution The given DLS data can be used to obtain the intensity autocorrelation function g,(s,td) by rewriting the Siegert relation as follows ... 

Measurements at low Qs At low Qs, because of the large magnitudes of s , the measured intensity and its autocorrelation function are dominated by the cumulative diffusion of the particles. The measured decay rate thus represents the cumulative or mutual diffusion coefficient Dm given by... 

What is an autocorrelation function Sketch qualitatively the autocorrelation function of the intensity of scattered light from a dispersion for a number of angles. 

For a theoretical calculation of relaxation times one. must write the temporal autocorrelation functions of several functions Fn of the interparticle coordinates riS(t), 0y(O, and interparticle distance and where 0,/O and external magnetic field Ho (here particle refers to magnetic nuclei and atoms). The relaxation rates are proportional to the Fourier intensities of these autocorrelation functions at selected frequencies. For example, Torrey (16) has written for this autocorrelation function the equivalent ensemble average... 

Up to now, we have given a general theoretical development of the self-beat technique. As a practical illustration of the experimental apparatus used to detect autocorrelation functions in scattering experiments, the equipment currently used in our laboratory will now be described. While our treatment of the autocorrelation function has been in terms of an analog signal, the computer that measures this function is actually a digital device. This is based on the fact that it is also valid to count the scattered photons in order to calculate Ci(r) as the optical intensity signal is essentially determined by the number of photons that strike the photocathode per unit time. We have then... 

Light scattering is due to fluctuations in the local dielectric tensor e of the medium. In fluids these fluctuations are dynamic and the scattered intensity will be a function of time and the frequency spectrum of the scattered light will differ from that of the incident light. The time dependence of the total scattered intensity is analyzed by measuring the intensity autocorrelation function... 

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