Nonlinear susceptibilities function

FIGURE 6.3 Nonlinear susceptibility of the solution as a function of Ti02 nanoparticles concentration. (From Shcheslavskiy, V., Petrov, G., Yakovlev, V. V. Appl. Phys. Lett., 82(22) 3982-3984, 2003. Used with permission.)... 

The results of numerical calculations of the dynamic nonlinear susceptibilities for a random assembly of monodisperse grains along formulas (4.99)-(4.111) are presented in Figure. 4.2. We show them as the functions of the temperature parameter a oc 1 /T, taking coio = 10-8 as the reference value. This choice seems reasonable since the low-frequency magnetic measurements are typically carried out at co about 102 — 103 Hz, and the customary reference value of To lies in the 10 10 — 10 s range. The curves obtained by the approximate formula (4.112) are plotted with thin dashed lines that visually almost coincide with the curves corresponding to the exact solution. 

In a composite material where small particles (inclusions) are distributed in a host material, an overall third-order ncxilinear susceptibility should be a function of both host and inclusions third-order nonlinear susceptibilities. Here we consider a composite topology where the inclusions are sphere of radius a, and we define a... 

In a composite material, as described here, the effective third-order nonlinear susceptibility should depend linearly with the concentration of the inclusions in a low filling fraction regime. In that way, the nonlinear absorption coefficient of the medium, associated to the Im[ (ru)] and consequently to the two-photon absorption processes, should also be a function of the inclusions concentration. 

The derivation of formulae for the frequency-dependent nonlinear susceptibilities of nonlinear optics from the time-dependent response functions can be found in a number of sources, (Bloembergen, Ward and New, Butcher and Cotter, Flytzanis ). Here it is assumed that the susceptibilities can be expressed in terms of frequency-dependent quantities that connect individual (complex) Fourier components of the polarization with simple products of the Fourier components of the field. What then has to be shown is how the quantities measured in various experiments can be reduced to these simpler parameters. 

The electric field components in equations 1 and 2 can be associated with the same or with different frequency components and, in some situations, can resonate with electronic or vibrational oscillations in the medium. This situation has led to a shorthand notation to describe the interactions leading to the various nonlinear effects of interest. Various susceptibility functions corresponding to the x and x effects of interest and their shorthand representation are listed in Table 6.1. 

In 2002, our group extended this approach to a metal surface covered with adsorbates [24]. Time-resolved SHG was applied to a cesium-covered Pt(lll) crystal surface under ultrahigh vacuum. The intensity variation of the SHG shows oscillatory components as a function of the delay time, due to coherent nuclear wavepack-et dynamics of the Cs-Pt stretching mode. The nonlinear susceptibility of the system is considered to depend on, among other things, nuclear displacements of surface normal modes, that is,... 

The convolution defined in (4.2.1) is a linear operation applied to the input function x(t). Nonlinear systems transform the input signal into the output signal in a nonlinear fashion. A general nonlinear transformation can be described by the Volterra series. It forms the basis for the theory of weakly nonlinear and time-invariant systems [Marl, Schl] and for general analysis of time series [Kanl, Pril]. In quantum mechanics, the Volterra series corresponds to time-dependent perturbation theory, and in optics it leads to the definition of nonlinear susceptibilities [Bliil]. 

Equation (4.2.11) describes the response to three delta pulses separated by ti =oi — 02 >0, t2 = 02 — 03 > 0, and t3 = 03 > 0. Writing the multi-pulse response as a function of the pulse separations is the custom in multi-dimensional Fourier NMR [Eml ]. Figure 4.2.3 illustrates the two time conventions used for the nonlinear impulse response and in multi-dimensional NMR spectroscopy for n = 3. Fourier transformation of 3 over the pulse separations r, produces the multi-dimensional correlation spectra of pulsed Fourier NMR. Foinier transformation over the time delays <7, produces the nonlinear transfer junctions known from system theory or the nonlinear susceptibilities of optical spectroscopy. The nonlinear susceptibilities and the multi-dimensional impulse-response functions can also be measured with multi-resonance CW excitation, and with stochastic excitation piul]. 

In the previous sections, we have utilized Green s function techniques to eliminate some of the summations involved in the calculations of nonlinear susceptibilities. The general expression for R(t3,t2,t1) [Eq. (49) or (60)], involves four summations over molecular states a, b, c, d. In Eq. (80) we carried out two of these summations for harmonic molecules. It should be noted that for this particular model it is possible to carry out formally all the summations involved, resulting in a closed time-domain expression for R(t3,t2,t1). This expression, however, cannot be written in terms of simple products of functions of , r2 and t3. Therefore, calculating the frequency-domain response function / via Eq. (30) requires the performing of a triple Fourier transform (rather than three one-dimensional transforms). This formula is, therefore, useful for extremely short pulses when a time-domain expression is needed. Otherwise, it is more convenient to use the expressions of Section VI, whereby only two of the four summations were carried out, but the transformation to... 

The supennolecule approach is used to study the linear and second-order nonlinear susceptibilities of the 2-methyl-4-nitroaniline ciystal. The packing effects on these properties, evaluated at the time-dependent Hartree-Fock level with the AMI Hamiltonian, are assessed as a function of the size and shape of the clusters. A simple multiplicative scheme is demonstrated to be often reliable for estimating the properties of two- and three-dimensional clusters from the properties of their constitutive one-dimensional arrays. The electronic absorption spectra are simulated at the ZINDO level and used to rationalize the linear and nonlinear responses of the 2-methyl-4-nitroaniline clusters. Comparisons with experiment are also provided as well as a discussion about the reliability of the global approach. 

In conclusion, we have introduced a neutral type of linear response experiment for nonlinear kinetics involving multiple reaction intermediates. We have shown that the susceptibility functions from the response equations are given by the probability densities of the transit time in the system. We have shown that a transit time is a sum of different lifetimes corresponding to different reaction pathways, and that in the particular case of a time-invariant system our definition of the transit time is consistent with Easterby s definition [23]. 

In conclusion, we have shown that the neutral response approach can be extended to inhomogeneous, space-dependent reaction-diffusion systems. For labeled species (tracers) that have the same kinetic and transport properties as the unlabeled species, there is a linear response law even if the transport and kinetic equations of the process are nonlinear. The susceptibility function in the linear response law is given by the joint probability density of the transit time and of the displacement position vector. For illustration we considered the time and space spreading of neutral mutations in human populations and have shown that it can be viewed as a natural linear response experiment. We have shown that enhanced (hydrodynamic) transport due to population growth may exist and developed a method for evaluating the position of origin of a mutation from experimental data. 

Landolt-Bomstein Numerictil Data tmd Functional Relationships in Science and Technology, New Series. Gn IB Crystal tmd Solid State Physics. Vol. 11. Revised and Extended Edition of Volumes 111/1 and 111/2. — Elastic, Piezoelectric, Pyroelectric, Electrooptic Consttmts, and Nonlinear Susceptibility of CrysUils, 854 p. Springer, Berlin/Heidelberg/New York (1979)... 

Equation 3 can now be employed to calculate the linear and nonlinear susceptibilities in (1) and (2), revealing very specific tensor properties exhibited by poled polymers. With the foregoing model for the potential energy, the thermodynamic averages are sometimes also expressed in terms of the Langevin functions, which are defined as... 

The linear and the nonlinear susceptibilities are functions of the wavelength, and knowledge of their spectral dispersion is of great importance for issues such as efficiency, phase matching, and acceptance bandwidth (see Section III). The spectral dispersion of the linear polarizability can be expressed assuming several electronic resonances [18]... 

In the event the pulse duration tp is longer than the molecular relaxation time, the expression in [14] corresponds to a third-order correlation function Gq - (t). Note also from equation [13] that the transient birefringence has both an instantan us and a slower response to the picosecond pump pulse, since n as the electronic component of the nonlinear susceptibility is expected to follow the laser pulse profile but the molecular component n > a combination of vibrational and orientational parts, carries with it a time-dependence characteristic of each molecular system It is through the latter that information on orientational correlations can be deduced (39). 

---