The Autocorrelator

1 A light beam of 21 mW reaches a photoconduchon detector with a 1 mm thick active area. The absorption coefficient at 965 nm (the incident wavelength) is 23 cm Calcnlate the nnmber of carriers created per unit hme if the quantum efficiency of the process is 0.13. 

2 A photodiode is illuminated with a green beam (532 nm) whose power is unknown. The photodiode is operating in the photovoltaic regime at room temperature. After illumination, the voltage induced in the photodiode is 34 mV. Calculate the incident power if the quantum efficiency is 0.65 and if the electrical current generated in the photodiode in the absence of illumination is 1 mA. 

3 Calculate the current induced in a photodiode with an inhinsic quantum efficiency of 0.90 when it is illuminated at room temperature with a 0.35 mW light beam whose wavelength is 1140 nm. The photodiode is working in the photoconductor regime, and in the absence of illumination no electrical current is generated by this photodiode. What happens if the photodiode is cooled down to 5 °C  

5 Calculate the dark current intensity at room temperature (T = 300 K) of a metallic photocathode with the following characteristics area = 3 cm, eip = 1.05 eV. If the quantum efficiency of this photocathode is 0.75, calculate the minimum power that can be detected (the wavelength of the incident beam is 808 nm). Discuss the improvements if the photocathode is cooled down to 260 K. 

6 Now calculate the minimum light power that can be measured with a photomultiplier using the photocathode of Exercise 3.5 and with 10 dynodes, each of which has a secondary emission coefficient of 5 = 6. Estimate these minimum powers if the photocathode is cooled down to 5 °C. Assume a bandpass width of 1 Hz. 

---

The frill width at half maximum of the autocorrelation signal, 21 fs, corresponds to a pulse width of 13.5 fs if a sech shape for the l(t) fiinction is assumed. The corresponding output spectrum shown in fignre B2.1.3(T)) exhibits a width at half maximum of approximately 700 cm The time-bandwidth product A i A v is close to 0.3. This result implies that the pulse was compressed nearly to the Heisenberg indetenninacy (or Fourier transfonn) limit [53] by the double-passed prism pair placed in the beam path prior to the autocorrelator. 

The intensity autocorrelation measurement is comparable to all of the spectroscopic experunents discussed in the sections that follow because it exploits the use of a variably delayed, gating pulse in the measurement. In the autocorrelation experiment, the gating pulse is just a replica of the time-fixed pulse. In the spectroscopic experiments, the gating pulse is used to mterrogate the populations and coherences established by the time-fixed pulse. 

An interferometric method was first used by Porter and Topp [1, 92] to perfonn a time-resolved absorption experiment with a -switched ruby laser in the 1960s. The nonlinear crystal in the autocorrelation apparatus shown in figure B2.T2 is replaced by an absorbing sample, and then tlie transmission of the variably delayed pulse of light is measured as a fiinction of the delay This approach is known today as a pump-probe experiment the first pulse to arrive at the sample transfers (pumps) molecules to an excited energy level and the delayed pulse probes the population (and, possibly, the coherence) so prepared as a fiinction of time. 

In an ambitious study, the AIMS method was used to calculate the absorption and resonance Raman spectra of ethylene [221]. In this, sets starting with 10 functions were calculated. To cope with the huge resources required for these calculations the code was parallelized. The spectra, obtained from the autocorrelation function, compare well with the experimental ones. It was also found that the non-adiabatic processes described above do not influence the spectra, as their profiles are formed in the time before the packet reaches the intersection, that is, the observed dynamic is dominated by the torsional motion. Calculations using the Condon approximation were also compared to calculations implicitly including the transition dipole, and little difference was seen. 

The same idea was actually exploited by Neumann in several papers on dielectric properties [52, 69, 70]. Using a tin-foil reaction field the relation between the (frequency-dependent) relative dielectric constant e(tj) and the autocorrelation function of the total dipole moment M t] becomes particularly simple ... 

Another approach employing the autocorrelation coefficients as descriptors was suggested by Gasteiger et al, [22]. They used the neural networks as a working tool for solving a similarity problem. 

In order to transform the information fi om the structural diagram into a representation with a fixed number of components, an autocorrelation function can be used [8], In Eq. (19) a(d) is the component of the autocorrelation vector for the topological distance d. The number of atoms in the molecule is given by N. 

We denote the topological distance between atoms i and j (i.e., the number of bonds for the shortest path in the structure diagram) dy, and the properties for atoms i and j are referred to as pi and pj, respectively. The value of the autocorrelation function a d) for a certain topological distance d results from summation over all products of a property p of atoms i and j having the required distance d. 

Here, the component of the autocorrelation vector a for the distance interval between the boundaries dj (lower) and (upper) is the sum of the products of property p for atoms i and j, respectively, having a Euclidian distance d within this interval. 

The component of the autocorrelation vector for a certain distance interval between the boundaries 4 and du is the sum of the products of the property p x,) at a point Xi on the molecular surface with the same property p Xj) at a point Xj within a certain distance d Xj,Xj) normalized by the number of distance intervals 1. All pairs of points on the surface are considered only once. 

When the resolution of the autocorrelation vector is decreased, some signals, e.g., those for the methyl groups in m- and / -xylene, may collapse. In such a case, one cannot distinguish between these two isomers. 

These first components of the autocorrelation coefficient of the seven physicochemical properties were put together with the other 15 descriptors, providing 22 descriptors. Pairwise correlation analysis was then performed a descriptor was eliminated if the correlation coefficient was equal or higher than 0.90, and four descriptors (molecular weight, the number of carbon atoms, and the first component of the 2D autocorrelation coefficient for the atomic polarizability and n-charge) were removed. This left 18 descriptors. 

Another important characteristic aspect of systems near the glass transition is the time-temperature superposition principle [23,34,45,46]. This simply means that suitably scaled data should all fall on one common curve independent of temperature, chain length, and time. Such generahzed functions which are, for example, known as generalized spin autocorrelation functions from spin glasses can also be defined from computer simulation of polymers. Typical quantities for instance are the autocorrelation function of the end-to-end distance or radius of gyration Rq of a polymer chain in a suitably normalized manner ... 

Step 4 - Calculate the Autocorrelation Function R(t) Assuming that cr ... 

The set of central second moments is related to the autocorrelation function by means of the following simple formula,... 

The last equation, which expresses the autocorrelation function of Y(t) in terms of h(t), is often referred to as Campbell s58 theorem. It is useful to note that the autocovariance function of Y(t) is given by the simpler expression... 

The function h(t ) i(t + r)dt is often referred to as the autocorrelation function of the Amotion h(t) however, the reader should be careful to note the difference between the autocorrelation function of h(t)—an integrable function—and the autocorrelation function of Y(t)—a function that is not integrable because it does not die out in time. With this distinction in mind, Campbell s theorem can be expressed by saying that the autocovariance function of a shot noise process is n times the autocorrelation function of the function h(t). 

In this connection, it should be carefully noted that, even if X(t) is not a gaussian process, the mean and the autocorrelation function of the output of a linear, time-invariant filter are related to the mean and autocorrelation function of the input process according to Eqs. (3-293) and (3-294).64 This is an important fact of which use will be made in the next section. 

As a further application of the Wiener-Khinchine theorem, we shall now calculate the power density spectrum of the shot noise process. The autocorrelation function for such a process is given by Campbell s theorem, Eq. (3-262), repeated below... 

In order to evaluate the autocorrelation function we again exploit the ergodic hypothesis and replace the average over phase space ( ) by a time average writing,... 

Exercise 4.5. Evaluate the autocorrelation function and the autocorrelation time for the function exp(-f) cos(2f). 

An examination of the autocorrelation function (0(0) <2(0) annucleophilic attack step in the catalytic reaction of subtilisin is presented in Fig. 9.4. As seen from the figure, the relaxation times for the enzymatic reaction and the corresponding reference reaction in solution are not different in a fundamental way and the preexponential factor t 1 is between 1012 and 1013 sec-1 in both cases. As long as this is the case, it is hard to see how enzymes can use dynamical effects as a major catalytic factor. 

FIGURE 9.4. The autocorrelation function of the time-dependent energy gap Q(t) = (e3(t) — 2(0) for the nucleophilic attack step in the catalytic reaction of subtilisin (heavy line) and for the corresponding reference reaction in solution (dotted line). These autocorrelation functions contain the dynamic effects on the rate constant. The similarity of the curves indicates that dynamic effects are not responsible for the large observed change in rate constant. The autocorrelation times, tq, obtained from this figure are 0.05 ps and 0.07ps, respectively, for the reaction in subtilisin and in water. 

Therefore, from the analyses of the asymmetrical and symmetrical fluctuations in Sections HL 1 and m.4, it is concluded that the polishing-state pits discussed here, which appear beyond the critical pitting potential, have only one representative length (i.e., the autocorrelation distance of the asymmetrical fluctuations), which suggests that the morphology of the... 

---