Second-order autocorrelation function

Although the preparation of the excited state has been described in terms of a delta function excitation, the same results should be obtained for the case of excitation by a broad-band, random, conventional light source. We have pointed out, in Section VI, that in the case of the non-radiative decay of an excited state, the same behavior is predicted to follow excitation by a light source characterized by a second-order autocorrelation function which describes random phases and excitation by a delta function pulse. A similar situation prevails when the radiative decay channel is also taken into account. 

The fluctuations in scattered Hght are detected by a photomultiplier and recorded the data containing information on particle motion are then processed by a digital correlator. The latter compares the intensity of scattered light at time t, I(t), to the intensity at a very small time interval r later, /(t + r), and constructs the second-order autocorrelation function G2(r) of the scattered intensity. 

The most widespread technique employed to measure the duration of ultrafast pulses is based on the determination of the second-order autocorrelation function of the intensity. Figure 3 illustrates both the principles and the experimental configuration employed for CW autocorrelation measurements of ps and fs pulses in real time. There are other optically-based methods utilised to measure optical responses over times orders of magnitude faster than electronic detectors, e.g. photodiodes which are currently limited to 60ps rise time, and even faster than streak camera determinations, which are capable of resolving 1 ps.. However, the discussions below focuses on the background free autocorr-... 

This overlap integral is the fact the second-order autocorrelation function (t) as equation [3] shows. The additional term r(t) is the rapidly varying, phase interference factor which is included for completeness but actually as it is optically averaged to zero. Finally, the output from the PMT is recorded in real time on an XY oscilloscope, and the profile of G is captured from which we may deconvolve to obtain the orginal function I(t) and thus determine the pulse width. Three very important points must be made here. First, a second-order autocorrelation... 

The time dependence of the scattered photons are most efficiently analyzed by measuring the second-order autocorrelation function (ACF) of the photon pulse train. 

This second-order autocorrelation function is related to the normalized electric field correlation function = ( p) by the Siegert relation... 

This shows that the autocorrelation function C(t) of the photoelectron current is directly related to the second-order correlation function G r) of the light field. 

NMR 13C spin-lattice relaxation times are sensitive to the reorientational dynamics of 13C-1H vectors. The motion of the attached proton(s) causes fluctuations in the magnetic field at the 13C nuclei, which results in decay of their magnetization. Although the time scale for the experimentally measured decay of the magnetization of a 13C nucleus in a polymer melt is typically on the order of seconds, the corresponding decay of the 13C-1H vector autocorrelation function is on the order of nanoseconds, and, hence, is amenable to simulation. 

The values of these autocorrelation functions at times t = 0 and t = 00 are related to the two order parameters orientational averages of the second- and fourth-rank Legendre polynomial P2(cos/ ) and P4 (cos p). respectively, relative to the orientation p of the probe axis with respect to the normal to the local bilayer surface or with respect to the liquid crystal direction. The order parameters are defined as... 

If F is an operator working on the system S alone this autocorrelation function describes the fluctuations in S under influence of the bath. We compute it to second order in the coupling constant a. 

This formula resembles Eq. (2.6) for the autocorrelation signal. We can further expand 5(l)( ,r) to second order in the pump field and express the result in terms of the four-point correlation function (2.8) (see Appendix E). 

The partial correlation function overcomes the correlation transfer effect as described above and shows, in contrast to the autocorrelation function, only one spike at t — 1 for first order autoregressive processes and spikes at t = 1 and x = 2 for second order autoregressive processes, and so on. 

Three of the experiments are completely new, and all make use of optical measurements. One involves a temperature study of the birefringence in a liquid crystal to determine the evolution of nematic order as one approaches the transition to an isotropic phase. The second uses dynamic laser light scattering from an aqueous dispersion of polystyrene spheres to determine the autocorrelation function that characterizes the size of these particles. The third is a study of the absorption and fluorescence spectra of CdSe nanocrystals (quantum dots) and involves modeling of these in terms of simple quantum mechanical concepts. 

The same result follows when the average of the exponential function given in Eq. (51) is transformed using the cumulant expansion theorem and, assuming a Gaussian process, all correlations higher than second order (Stepisnik, 1981, 1985) are neglected. The particle velocity autocorrelations form a tensor... 

Theoretical Models for Local Segmental Motions. The second order orientation autocorrelation function measured in this experiment is defined by ... 

The strength of the bath coupling to each system variable is described by the coupling constants / and, because they enter at second order, the rate constant for the dissipation process arising from each term in Eq. (38) will be proportional to f I- The only important properties of the F t) are their autocorrelation and cross-correlation functions, (FJfi)F t)) and F (0)Fi,(t)), which enter the definition of the Redfield tensor in Eq. (18). These, like the classical correlation functions discussed earlier, do not satisfy the detailed-balance relation and must be corrected in the same way. It is convenient, but not necessary, that the variables be chosen to be independent, so that the cross-correlation functions vanish. 

Here 6(t) is the angle between the C-H bond vector at time zero and time t. This particular correlation function is referred to as a second order orientation autocorrelation function. 

---